Zanuccoli Mauro tesi

Zanuccoli Mauro tesi
Alma Mater Studiorum – Università di Bologna
DOTTORATO DI RICERCA IN
TECNOLOGIE DELL'INFORMAZIONE
Ciclo XXIV
Settore Concorsuale di afferenza: 09/E
Settore Scientifico disciplinare: ING-INF/01
TITOLO TESI
ADVANCED NUMERICAL SIMULATION OF
SILICON-BASED SOLAR CELLS
Presentata da:
Mauro Zanuccoli
Coordinatore Dottorato
Relatore
Prof. Claudio Fiegna
Prof. Claudio Fiegna
Esame finale anno 2012
Acknowledgements
During my doctoral course period I have been constantly supported and advised by
several people. I want to express my gratitude to all of them. I am deeply grateful to
my supervisor, Prof. Claudio Fiegna and to Prof. Enrico Sangiorgi for the opportunity
and the support. I would like to thank all my colleagues who have supported the activity
with helpful discussions, Dr. Paolo Magnone and Raffaele De Rose. During the threeyear period I had the great opportunity to visit scientific and technological partners
like Applied Material, Inc. in Santa Clara, CA, USA (AMAT) Minatec Grenoble-Inp.,
France and the Institute of Physics and Technology of the Russian Academy of Sciences
(FTIAN-RAS), Moscow. My work has been supported by all these partners, therefore
I would like to thank Dr. Michel Frei and Dr. Mukul Agrawal (AMAT), Dr. Igor
Semenikhin, Prof. Vladimir Vyurkov (FTIAN-RAS) and Prof. Anne Kaminski, Prof.
Mirelle Mouis and Prof. Davide Bucci (IMEP-LAHC) for the support and inputs on
technological aspects.
i
ii
Publications of the author
Journals
M. Zanuccoli, R. De Rose, P. Magnone, E. Sangiorgi and C. Fiegna
Performance analysis of rear point contact solar cells by three-dimensional numerical
simulation.
IEEE Trans. Elec. Devices, vol. 59 (5), DOI: 2012, 10.1109/TED.2012.2187297
I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi and C. Fiegna
Computational efficient RCWA method for simulation of thin film solar cells.
Optical and Quantum Electronics, DOI: 10.1007/s11082-012-9560-5
International workshops and conferences
M. Zanuccoli, R. De Rose, P. Magnone, M. Frei, H.-W. Guo, M. Agrawal, E. Sangiorgi and C. Fiegna
Numerical simulation and modeling of rear point contact contact solar cells.
Conference Records of 37th IEEE Photovoltaic Specialist Conference (PVSC), Seattle, Washington, US, 2011
P. Magnone, R. De Rose, M. Zanuccoli, D. Tonini, M. Galiazzo, G. Cellere, H.-W.
Guo, M. Frei, E. Sangiorgi and C. Fiegna
Understanding the impact of Double Screen-Printing on Silicon Solar Cells by 2-D Numerical Simulations.
Conference Records of 37th IEEE Photovoltaic Specialist Conference (PVSC), Seattle, Wash-
iii
ington, US, 2011
R. De Rose, M. Zanuccoli, P. Magnone, D. Tonini, M. Galiazzo, G. Cellere, M.
Frei, H.-W. Guo, C. Fiegna and E. Sangiorgi
2-D numerical analysis of the impact of the highly-doped profile on selective emitter
solar cell performance.
Conference Records of 37th IEEE Photovoltaic Specialist Conference (PVSC), Seattle, Washington, US, 2011
M. Zanuccoli, H.-W. Guo, E. Sangiorgi, C. Fiegna
Three-dimensional numerical simulation of rear point contact crystalline silicon solar
cells.
Renewable Energy & Power Quality Journal, No.9, 12th May 2011 ISSN 2172-038X
M. Zanuccoli, P.F. Bresciani, M. Frei, H.-W. Guo, H. Fang, M. Agrawal, C. Fiegna,
E. Sangiorgi
2-D numerical simulation and modeling of monocrystalline selective emitter solar cells.
Conference Records of 35th IEEE Photovoltaic Specialist Conference (PVSC), Honolulu, Hawaaii,
US, 2010
M. Zanuccoli, P.F. Bresciani, M. Frei, H.-W. Guo, H. Fang, M. Agrawal, C. Fiegna,
E. Sangiorgi
2-D numerical simulation and modeling of monocrystalline selective emitter solar cells.
Conference Records of 35th IEEE Photovoltaic Specialist Conference (PVSC), Honolulu, Hawaaii,
US, 2010 pp. 2262 - 2265, DOI: 10.1109/PVSC.2010.5617010, ISSN : 0160-8371, Print ISBN:
978-1-4244-5890-5
I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi and C. Fiegna
Application of a computationally efficient implementation of the RCWA method to numerical simulations of thin film amorphous solar cells.
Proceedings of 11th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 2011 pp. 117 - 1185, DOI: 110.1109/NUSOD.2011.6041168, ISSN : 2158-3234,
Print ISBN: 978-1-61284-876-1
iv
R. De Rose, M. Zanuccoli, P. Magnone, E. Sangiorgi and C. Fiegna
Open issues for the numerical simulation of silicon solar cells.
Proceedings of 12th International Conference on Ultimate Integration on Silicon (ULIS), 2011
pp. 1 - 4, DOI: 1110.1109/ULIS.2011.5757961, E-ISBN : 978-1-4577-0089-7, Print ISBN: 9781-4577-0090-3
I. Semenikhin, V. Vyurkov, M. Zanuccoli, E. Sangiorgi and C. Fiegna
Efficient implementation of the Fourier modal method (RCWA) for the optical simulation of optoelectronics devices.
Proceedings of 114th International Workshop on computational Electronics (IWCE), 2010 pp.
1 - 4, DOI: 10.1109/IWCE.2010.567798, Print ISBN: 978-1-4244-9383-8
I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi and C. Fiegna
Application of RCWA method to optoelectronic numerical simulations of 2D nanostructures.
Proceedings of 14th International Workshop on computational Electronics (IWCE), 2010 pp. 1
- 2, DOI: 110.1109/INEC.2011.5991728, ISSN: 2159-3523, Print ISBN: 978-1-4577-0379-9
v
vi
List of Symbols
Symbol
Unit
Name or description
A
[cm2 ]
area of the solar cell
s−1 ]
B
[cm3
c
Cn
Cp
[m s−1 ]
[cm6 s−1 ]
[cm6 s−1 ]
Dn
Dp
[cm2
[cm2
E
E0
EC
EV
Eg
E0,A
E0,D
EA
ED
EA,M G (E)
ED,M G (E)
EF
EFi
Eph
Et
[V
s−1 ]
s−1 ]
[eV ]
m−1 ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
[eV ]
radiative recombination coefficient
speed of light in vacuum
material-specific Auger coefficient for electrons
material-specific Auger coefficient for holes
diffusion coefficient for electrons
diffusion coefficient for holes
energy
electric field strength
bottom edge of the conduction band
top edge of the valence band
energy band-gap
position of the minimum for the acceptor-like exponential-tail states
position of the minimum for the donor-like exponential-tail states
energy slope for acceptor-like exponential-tail states
energy slope for donor-like exponential-tail states
energy position of Gaussian acceptor-like mid-gap states
energy position of Gaussian donor-like mid-gap states
Fermi level
intrinsic Fermi level
photon energy
energy trap level
fA (E)
fD (E)
-
probability of occupation of acceptor-like states
probability of occupation of donor-like states
FF
-
Fill Factor
vii
Symbol
Unit
g(E)
gA (E)
gD (E)
BT (E)
gA
BT
gD (E)
M G (E)
gA
M G (E)
gD
GA,BT (E)
GD,BT (E)
GOP T
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 s−1 ]
h
H
HR
HT
[J s]
[A m−1 ]
-
Jsc
Jph
J0
Jn
Jp
[A
[A
[A
[A
[A
cm−2 ]
cm−2 ]
cm−2 ]
cm−2 ]
cm−2 ]
short-circuit current density
photo-generated current density
diode saturation current density
electron current density
hole current density
KB
k
[J
K −1 ]
-
Boltzmann constant
extinction coefficient (imaginary part of the complex refractive index)
I0
[W
m−2 ]
[A]
Input irradiance
Maximum Power-Point Current
Ln
Lp
LT
[cm]
[cm]
[cm]
diffusion length of electrons
diffusion length of holes
transport length
n
ñ
nn0
np 0
n(P )
ni
ni,ef f
nT
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
IM P P
Name or description
total density of states
total density of acceptor-like states
total density of donor-like states
density of acceptor-like exponential-tail states
density of donor-like exponential-tail states
density of acceptor-like mid-gap states
density of donor-like mid-gap states
minimum value of acceptor-like exponential-tails states
minimum value of donor-like exponential-tails states
optical generation rate
Planck constant
magnetic field strength
Haze parameter for reflected light
Haze parameter for transmitted light
real part of the complex refractive index
complex refractive index
electron conc. in n-type semiconductors in equilibrium
electron conc. in p-type semiconductors in equilibrium
concentration of free electrons
intrinsic carrier density concentration
effective intrinsic carrier concentration
total concentration of trapped electrons
viii
Symbol
Unit
Name or description
N
NA
NC
ND
NAM G (E)
M G (E)
ND
NT
Nsub
NV
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
net doping concentration
acceptor doping concentration
effective density of states in conduction band
donor doping concentration
total number of acceptor-like mid-gap states
total number of donor-like mid-gap states
total concentration of traps
substrate doping concentration
effective density of states in valence band
pn0
pp0
p(P )
pT
[cm−3 ]
[cm−3 ]
[cm−3 ]
[cm−3 ]
hole conc. in n-type semiconductors in equilibrium
hole conc. in p-type semiconductors in equilibrium
concentration of hole
total concentration of trapped hole
Rnet
RS
RSH
[cm−3 s−1 ]
Ω
Ω
Net recombination rate per time and volume unit
series resistance
shunt resistance
q
C
S
Se0
Sh0
S(E)
[W m−2 ]
[cm s−1 ]
[cm s−1 ]
[W m−2 nm−1 ]
t
T
[s]
[K]
vth
VA
VM P P
V0
VOC
[cm s−1 ]
[V ]
[V ]
[V ]
[V ]
W
WDEP
WM
WSE
WSU B
[µm]
[µm]
[µm]
[µm]
[µm]
elementary charge
Poynting vector strength
surface recombination velocity for electrons
surface recombination velocity for holes
spectral irradiance
time
absolute temperature
thermal velocity of carriers
applied voltage
Maximum Power-Point Voltage
Built-in potential
open-circuit voltage
wafer thickness
depletion region width
front finger width
lateral extension of the HDOP region
front contact pitch
ix
Symbol
Unit
α
[cm−1 ]
Name or description
absorption coefficient of the radiation
cm−1 ]
cm−1 ]
electromagnetic permittivity
permittivity of vacuum
η
ηC
-
conversion efficiency
carrier collection efficiency
θ
[◦ ]
λ
[nm]
µ
µ0
µn
µp
[H cm−1 ]
[H cm−1 ]
[cm2 V −1 s−1 ]
[cm2 V −1 s−1 ]
ρ(r)
ρBC
ρc
ρm
ρS
[C cm−3 ]
[Ω cm2 ]
[Ω cm2 ]
[Ω cm]
[Ω cm]
σA
σD
σA,n
σA,p
σD,n
σD,p
σRM S
[eV ]
[eV ]
[cm2 ]
[cm2 ]
[cm2 ]
[cm2 ]
[cm2 ]
τe
τh
[s]
[s]
ε
ε0
φ(λ)
φ(r, t)
φn
φp
ω
[F
[F
angle of incidence of the radiation
s−1
[V ]
[eV ]
[eV ]
[P hotons
[rad
s−1 ]
wavelength of the radiation
magnetic permeability
magnetic permeability in vacuum
electron mobility
hole mobility
charge density
specific back contact resistivity
specific contact resistivity
metal resistivity
substrate resistivity
standard deviation of Gaussian mid-gap acceptor-like states
standard deviation of Gaussian mid-gap donor-like states
capture cross-section of acceptor-like traps for electrons
capture cross-section of acceptor-like traps for holes
capture cross-section of donor-like traps for electrons
capture cross-section of donor-like traps for holes
root mean square value of surface heights (for rough interfaces)
carrier lifetime for electrons
carrier lifetime for holes
m−2 ]
incident photon flux
electrostatic potential
Quasi-Fermi levels for electrons
Quasi-Fermi levels for holes
angular frequency
x
xi
List of Abbreviations
ADF
AFM
ARC
ARS
a-Si
a-Si:H
BSF
BSRV
BSRVef f
BSRVpass
c-Si
CdTe
CIGS
CPV
CPW
Cz
DOS
FDTD
EMA
EQE
FZ
HE
HDOP
IBR
IBRef f
IQE
LDOP
LFC
MCz
PDE
PERC
PERL
RCWA
RPC
RT
Angular Distribution Function
Atomic Force Microscopy
Anti-Reflection Coating layer
Angular Resolved Scattering
amorphous silicon
hydrogenated amorphous silicon
Back Surface Field
Back Surface Recombination Velocity
Effective Back Surface Recombination Velocity
Back Surface Recombination Velocity at passivated interface
crystalline silicon
Cadmium Telluride
Copper Indium Gallium Diselenide
Concentrator PhotoVoltaic
cells per wavelength
Czochralski
density of states
Finite Difference Time Domain
Effective Media Approximation
External Quantum Efficiency
Floating Zone
Homogenous Emitter
heavy and deep phosphorus diffusion
Internal Bottom Reflectivity
Effective Internal Bottom Reflectivity
Internal Quantum Efficiency
lowly-doped and shallow diffusion
Laser Firing Contact
Magnetic Czochralski
Partial Differential Equation
Passivated Emitter and Rear Cell
Passivated Emitter, Rear Locally diffused cell
Rigorous Coupled-Wave Analysis
Rear Point Contact
Ray Trace
xii
SE
SEM
SiN
SR
SRH
SLT
TCAD
TCO
TE
TEM
TIS
TM
TMM
Selective Emitter
Scanning Electron Microscopy
Silicon Nitride
Spectral Response
Shockley Read Hall
Single Level Trap
Technology Computer-Aided Design
Transparent Conductive Oxide
Transverse Electric
Transverse Electro Magnetic
Total Integrating Scattering
Transverse Magnetic
Transfer Matrix Method
xiii
xiv
Contents
List of Figures
xix
1 Introduction
1
2 Solar cell device physics and technology review
7
2.1
Absorption of light in semiconductors . . . . . . . . . . . . . . . . . . .
7
2.2
Recombination mechanisms . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.1
Radiative recombination . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.2
Auger recombination . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.3
Trap-assisted recombination . . . . . . . . . . . . . . . . . . . . .
13
2.2.4
Effettive carrier lifetime . . . . . . . . . . . . . . . . . . . . . . .
14
Semiconductor Model Equations . . . . . . . . . . . . . . . . . . . . . .
15
2.3.1
Drift-Diffusion Transport Model . . . . . . . . . . . . . . . . . .
15
2.3.2
Semiconductor equations . . . . . . . . . . . . . . . . . . . . . .
16
2.3.3
1-D single-junction silicon solar cell model . . . . . . . . . . . . .
17
2.3.4
Output characteristic and figures of merit . . . . . . . . . . . . .
23
2.3.5
Other figures of merit . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3.6
Optical figures of merit: absorbance, reflectance and transmittance 26
2.3.7
Quantum efficiency and spectral response . . . . . . . . . . . . .
26
2.4
Efficiency Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5
Effect of temperature and loss mechanisms on main figures of merit of a
2.3
solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5.1
Effect of temperature . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5.2
Optical losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.5.3
Open circuit voltage and FF losses . . . . . . . . . . . . . . . . .
32
xv
CONTENTS
2.5.4
Accounting for contact resistances . . . . . . . . . . . . . . . . .
34
2.6
Light Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.7
Geometrical light trapping . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.7.1
Coherent light trapping schemes for thin film solar cells . . . . .
40
2.8
Antireflection coating layers . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.9
Standard crystalline solar cells manufacturing technology . . . . . . . .
45
2.9.1
Wafer preparation . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.9.2
Front contact and emitter design . . . . . . . . . . . . . . . . . .
47
2.9.3
Base and rear surface . . . . . . . . . . . . . . . . . . . . . . . .
49
3 Numerical Simulation of Solar Cells
3.1
.
51
3.1.1
Drift-Diffusion transport equations . . . . . . . . . . . . . . . . .
52
3.1.2
Electrical Boundary Conditions . . . . . . . . . . . . . . . . . . .
53
3.1.3
Effective Intrinsic Carrier Concentration and Band Gap structure 54
3.1.4
Carrier Mobility Models . . . . . . . . . . . . . . . . . . . . . . .
56
3.1.5
Recombination mechanisms . . . . . . . . . . . . . . . . . . . . .
57
3.1.6
Auger recombination . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.3
Propagation of electromagnetic waves . . . . . . . . . . . . . . . . . . .
62
3.3.1
From Maxwell equations to homogeneous Helmholtz equation . .
62
3.3.2
Solution of Helmholtz equations for a normal homogenous medium
3.4
Introduction to numerical solution of semiconductor device equations
51
without sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.3.3
TE and TM plane waves . . . . . . . . . . . . . . . . . . . . . . .
66
3.3.4
Geometrical optics and raytracing . . . . . . . . . . . . . . . . .
68
Finite Difference Time Domain (FDTD) Method . . . . . . . . . . . . .
73
4 Modeling of advanced crystalline silicon solar cells
4.1
77
Selective emitter solar cells . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.1.1
Introduction to selective emitter solar cells . . . . . . . . . . . .
77
4.1.2
Selective emitter solar cells: simulation setup and simulated devices 79
4.1.3
Homogeneous emitter solar cell simulation . . . . . . . . . . . . .
4.1.4
Selective emitter solar cell: impact of LDOP and HDOP profiles
on main figures of merit . . . . . . . . . . . . . . . . . . . . . . .
xvi
81
83
CONTENTS
4.2
4.1.5
Selective emitter solar cell: loss analysis . . . . . . . . . . . . . .
85
4.1.6
Selective emitter solar cell: conclusions . . . . . . . . . . . . . . .
86
Rear point contact solar cells . . . . . . . . . . . . . . . . . . . . . . . .
88
4.2.1
Introduction to rear point contact solar cells
89
4.2.2
Rear Point Contact solar cells: simulation methodology and sim-
4.2.3
. . . . . . . . . . .
ulated devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Technological and geometrical options . . . . . . . . . . . . . . .
94
4.2.3.1
Comparison between PERL and PERC solar cells . . .
96
4.2.3.2
Dependence of the main figures of merit of PERL cells
on the back-contact diameter . . . . . . . . . . . . . . . 101
4.2.3.3
PERL cells: impact of back-contact pitch on performance104
4.2.3.4
PERL cells: analysis of the dependence of the main
figures of merit on the specific back contact resistivity . 105
4.2.4
Rear Point Contact solar cells: conclusions . . . . . . . . . . . . 107
5 Optical Simulation by Fourier-Modal Methods
111
5.1
Introduction to RCWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2
RCWA implementation based on discretization of the Fourier series . . . 114
5.3
5.4
5.5
5.2.0.1
Implementation for TE polarization . . . . . . . . . . . 115
5.2.0.2
Implementation for TM polarization . . . . . . . . . . . 120
RCWA implementation based on auxiliary functions . . . . . . . . . . . 122
5.3.1
Auxiliary functions and boundary conditions . . . . . . . . . . . 123
5.3.2
Solving for the auxiliary function . . . . . . . . . . . . . . . . . . 124
5.3.3
Determining the electromagnetic field . . . . . . . . . . . . . . . 126
5.3.4
Solution in homogenous layers . . . . . . . . . . . . . . . . . . . 128
RCWA implementation based on solution of an eigenvalues problem . . 130
5.4.1
Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4.2
Expansion of the field in Fourier series . . . . . . . . . . . . . . . 131
5.4.3
Solution within homogenous layer . . . . . . . . . . . . . . . . . 138
5.4.4
Calculation of the optical generation rate . . . . . . . . . . . . . 138
Validation of RCWA method and accuracy . . . . . . . . . . . . . . . . 139
5.5.1
2-D RCWA simulator . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.2
3-D RCWA simulator . . . . . . . . . . . . . . . . . . . . . . . . 141
xvii
CONTENTS
5.6
Applications of RCWA tool . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.6.1
Application: impact of surface morphology on absorbance . . . . 143
5.6.2
Application: nano-rod based solar cell . . . . . . . . . . . . . . . 154
6 Modeling of amorphous silicon thin film solar cells
159
6.1
Thin film solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.2
Continuous distribution of defect density in energy . . . . . . . . . . . . 161
6.3
Optical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3.1
Optical confinement in thin-film solar cells: modeling by methods
based on scalar scattering theory . . . . . . . . . . . . . . . . . . 165
6.4
Application: dependence of main figures of merit of the solar cell on the
morphology of the internal interfaces . . . . . . . . . . . . . . . . . . . . 169
6.4.1
Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4.2
Dependence of figures of merit on ITO/p-type interface roughness
level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7 Conclusions
181
References
185
xviii
List of Figures
2.1
Interface between a dielectric and an absorber material (z ≥ z0 ). The
light is propagating from left to right along z-axis. . . . . . . . . . . . .
2.2
Dependence of the absorption coefficient α of crystalline silicon on the
photon energy at T = 300K. . . . . . . . . . . . . . . . . . . . . . . . .
2.3
8
9
Optical generation rate GOP T calculated at λ=500nm as function of the
distance from the front interface in case of a silicon slab featuring infinite
thickness. The amplitude E0 of the incident field is 1V m−1 .
2.4
. . . . . .
11
Optical generation rate GOP T calculated at λ=500nm as function of
the distance from the front interface in case of a silicon slab featuring
thickness comparable to the wavelength of the radiation. Minima and
maxima of GOP T can be observed. The amplitude E0 of the incident
field is 1V m−1 .
2.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
(A) Mono-dimensional (1-D) sketch of an abrupt p-n junction with the
associated charge density ρ(x) profile along the x-axis (B). The pint
x = 0 denotes the metallurgical junction interface. −XDN and XDP are
the boundaries of the depletion region at the n-type and p-type sides,
respectively. NA and ND denote the uniform doping concentrations of
the p-type and n-type regions, respectively. The electric field plot, the
energy bands diagram and the carrier concentration plots are shown in
(C), (D) and (E), respectively. In (D) V0 denotes the built-in potential.
2.6
19
Two-diode circuit model of a solar cell. Diode (A) represents the current
due to recombination within the quasi-neutral regions. Diode (B) models
the recombination occurring in the depletion region according to eq. 2.64. 24
xix
LIST OF FIGURES
2.7
I-V characteristic (red solid curve) of a solar cell. The parameters VOC
and ISC are the open-circuit voltage and the short-circuit current, respectively. The output power is represented by the black dashed curve,
which reaches the maximum value at V = VM P P and I = IM P P . . . . .
2.8
Conventional spectrum AM1.5G
(1000W m−2 )
for direct sun illumina-
tion. The plot reports the spectral irradiance. . . . . . . . . . . . . . . .
2.9
25
28
Dependence of the short-circuit current on the band-gap energy Eg for
direct illumination without external reflectance of the top interface calculated assuming the input spectrum AM1.5G (1000W m−2 ) at T=300K.
All photo-generated electron-hole pairs are assumed to be collected by
electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.10 Dependence of the open-circuit voltage VOC on the band-gap energy
Eg for direct illumination without external reflectance of the top interface and calculated assuming the spectrum AM1.5G (1000W m−2 ) at
T=300K. The saturation current is in agreement with the lower bound
limit proposed by M.A. Green in (1). All photo-generated electron-hole
pairs are assumed to be collected by electrodes. . . . . . . . . . . . . . .
30
2.11 Dependence of the conversion efficiency η on the band-gap energy Eg
for direct illumination without external reflectance of the top interface,
calculated assuming the spectrum AM1.5G (1000W m−2 ) at T=300K.
The saturation current is chosen in agreement with the lower bound
limit proposed by M.A. Green in (1). All photo-generated electron-hole
pairs are assumed to be collected by electrodes. . . . . . . . . . . . . . .
31
2.12 Contact scheme for the front surface of a silicon solar cell. Bus bars and
fingers are aimed at delivering current to the external circuit. . . . . . .
2.13 Silicon solar cell featuring the front surface textured by pyramids.
. . .
32
33
2.14 Effect of the series parasitic resistance on the I-V characteristic of a solar
cell. The current is normalized to the short-circuit current density JSC .
35
2.15 Effect of the shunt parasitic resistance on the I-V characteristic of a solar
cell. The current is normalized to the short-circuit current density JSC .
35
2.16 Sketch of a simple solar cell with the lumped-element circuit to model
the series parasitic resistances. . . . . . . . . . . . . . . . . . . . . . . .
xx
36
LIST OF FIGURES
2.17 Theoretical prediction of the dependence of the absorbance of a crystalline silicon slab featuring a finite thickness on the photon energy in
case of single-pass scheme and in case of a light trapping scheme by means
of front surface texturing (pathlength equal to 4n2 where n is the refractive index of silicon). The one-pass absorption curve of the 200µm-thick
silicon layer is comparable to that associated to the textured 20µm-thick
layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.18 Vertical grooves formed by [111] crystallographic plane orientation. . . .
40
2.19 Cross sections of (A) light trapping scheme proposed by Redfield (2),
(B) symmetric vertical grooves and (C) asymmetrical grooves. . . . . . .
41
2.20 Three dimensional upright pyramids: (A) periodic scheme and (B) random pyramids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.21 Photonic crystals in 2-D scheme (A) and 3-D scheme (B). . . . . . . . .
43
2.22 Sketch of a simple anti reflection coating layer (ARC) featuring one slab
with refractive index n1 . The refractive indices of the superstrate and
substrate are n0 and n2 , respectively.
. . . . . . . . . . . . . . . . . . .
2.23 Examples of ARCs: calculated reflectance.
. . . . . . . . . . . . . . . .
44
45
2.24 Refractive indices and extinction coefficients of the crystalline silicon (A)
and of SiO2 , TiO2 and MgF2 (B) used to calculate the reflectance of Fig.
2.23.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.25 Homogeneous emitter (HE) solar cell. Under the contacted surface and
between fingers the same emitter doping profile is adopted. LF F , HM ET
and WM ET denote the front finger length, height and width, respectively.
WF P is the distance between front fingers.
. . . . . . . . . . . . . . . .
48
2.26 Selective emitter (SE) solar cell. A heavily doped emitter doping profile
(n++ ) is present under the contacted surface and a lowly-doped profile
(n+ ) between fingers. LF F , HM ET and WM ET denote the front finger
length, height and width, respectively. WF P is the distance between
front fingers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxi
48
LIST OF FIGURES
3.1
Effective intrinsic carrier density accounting for band-gap narrowing
(BGN) as function of the total doping density for different models, including the revised BGN model adopted for numerical simulations in
Sentaurus TCAD.
3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Bulk lifetime versus boron doping concentration for the empirical fitting
suggested by Glunz et al. (3) and for the Sentaurus default Scharfetter
relation. The bulk lifetime curve provided by the modified relation is
also reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3
Single box centered at node i-th for the numerical solution of PDEs. . .
60
3.4
Thin-layer-stack boundary condition used to simulate ARCs with RayTracing in combination with the TMM solver. . . . . . . . . . . . . . . .
71
3.5
Reflection, transmission and refraction of plane waves at a planar interface. 72
3.6
Absorbance for the 2µm-thick (A) and the 1µm-thick (B) slabs of crystalline silicon calculated by raytracer (RT), FDTD and RCWA methods.
4.1
74
(A) Sketch of the cross section of the 2-D simulated selective emitter
(SE) solar cell. HDOP and LDOP denote the highly-doped and the
lowly-doped emitter profiles, respectively. W is the wafer thickness and
WM ET = 100µm and WSE = 130µm are the front-contact finger width
and the lateral extension of the HDOP diffusion under the finger, respectively. (B) Emitter doping-profiles adopted for the simulations: the
figure shows the doping profile of the homogeneous emitter (HE) solar
cell (39Ω/sq), that of the SE under the contact (HDOP) and the emitter
profile under the passivated interface between fingers (LDOP) featuring
a sheet resistance of 39Ω/sq and 109Ω/sq, respectively. . . . . . . . . . .
4.2
80
Calculated figures of merit for the SE and the HE (39Ω/sq) cells. The SE
are simulated with the profile HDOP featuring 34Ω/sq and LDOP featuring sheet resistance values within the range 47Ω/sq-215Ω/sq. Main
figures of merit calculated by Sentaurus: (A) Short-circuit current density (JSC ), (B) Open-circuit voltage (VOC ), (C) Fill Factor (FF) and (D)
conversion efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxii
82
LIST OF FIGURES
4.3
Comparison of collection efficiency of photo-generated carriers (ηC ) between selective emitter SE (WSU B = 1800µm, HDOP 46Ω/sq and LDOP
109Ω/sq) and the reference homogenous emitter solar cell HE (WSU B =
2100µm, 39Ω/sq). Larger ηC within the blue region of the spectrum
(300nm-600nm) is obtained by the SE cell. . . . . . . . . . . . . . . . .
4.4
86
2-D Auger recombination rate maps (in cm−3 s−1 ) in the passivated emitter region close to the front surface (A) for the homogeneous emitter HE
cell (WSU B = 2100µm, 39Ω/sq) and (B) for the optimized selective emitter scheme SE (WSU B = 1800µm, HDOP 46Ω/sq and LDOP 109Ω/sq).
In (C) the 1-D Auger recombination rate is reported as function of the
distance from the front interface in the region between front-contact fingers.
4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
(A) 3-D sketch of the simulated solar cell with the rear point contact
scheme. WSU B and W denote the front contact pitch and the wafer
thickness, respectively. The parameter WM is the width of the front
contact finger. The hole pitch is denoted by p. The simulation domain
is highlighted in red (Lx = p/2, Ly = WSU B /2). (B) 2-D cross section
of the device with rear surface passivation by silicon nitride (SiN). . . .
4.6
90
Example of 3-D mesh grid adopted for the device numerical simulation.
In this case the hole pitch p is 500µm and the hole diameter s is 50µm.
The simulation domain is chosen according to section 4.2.2. Up to 300000
vertices are typically required. . . . . . . . . . . . . . . . . . . . . . . . .
4.7
95
Example of 3-D output maps calculated by the device numerical simulator. In this case the hole pitch p is 500µm and the hole diameter s
is 50µm. The figure shows the 3-D conduction current density map in
short-circuit conditions under illumination. . . . . . . . . . . . . . . . .
4.8
95
Short circuit current density Jsc (left) and open circuit voltage Voc (right)
for PERC and PERL solar cells with hole pith p = 500µm and substrate
resistivity ρs = 0.5Ωcm. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
98
Calculated absorbance (A) and reflectance (R) characteristics of a RPC
cell featuring f0 = 0.2% compared to those of the uniformly contacted
cell (f = 100%).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxiii
98
LIST OF FIGURES
4.10 Fill Factor (left) and efficiency (right) for PERC and PERL solar cells
with hole pitch p = 500µm and substrate resistivity ρs = 0.5Ωcm. In
case of the PERC two different values of the specific back contact resistivity ρBC are assumed: ρBC = 3mΩcm2 and ρBC = 30mΩcm2 ; if the
former value of ρBC is adopted for both PERC and PERL, the curves of
the FF are indistinguishable. . . . . . . . . . . . . . . . . . . . . . . . .
99
4.11 Collection efficiency of photo-generated carriers (ηC ) from 600nm up to
1100nm for PERC and PERL solar cells with hole pitch p = 500µm
and substrate resistivity ρs = 0.5Ωcm; ηC is calculated at f = 100%
(uniformly contacted cell) and at the optimum metallization fraction
(PERC: f0 = 3.14%, s = 100µm; PERL: f0 = 7.07%, s = 150µm). . . . 101
4.12 Short circuit current density Jsc , open circuit voltage Voc , Fill Factor
and conversion efficiency η for PERL solar cells with constant hole pith
p = 500µm and specific back contact resistivity ρBC = 3mΩcm2 , for
different substrate resistivity values. . . . . . . . . . . . . . . . . . . . . 103
4.13 Fill Factor (up) and efficiency (bottom) for PERL solar cells with substrate resistivity ρS = 5Ωcm (left) and ρS = 1Ωcm (right). Specific back
contact resistivity ρBC = 3mΩcm2 . Hole diameter s as parameter and
variable hole pitch.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.14 Conversion efficiency versus hole pitch p for PERL solar cells with substrate resistivity ρS = 5Ωcm (left) and ρS = 1Ωcm (right). Specific back
contact resistivity ρBC = 3mΩcm2 . Hole diameter s as parameter. . . . 108
4.15 Fill Factor (left) and efficiency (right) for PERL solar cells with hole
diameter s = 50µm, substrate resistivity ρS = 5Ωcm (up) and ρS =
1Ωcm (bottom). Specific back contact resistivity ρBC as parameter. . . 109
5.1
Sketch of the 2-D simulation domain which includes the superstrate, the
substrate and the grating region (left). A typical photovoltaic device
is represented by a multi-layer structure (right) in which each layer is
homogenous and separated by smooth or rough interfaces. The rough
interfaces are treated as grating regions within which the permittivity
is not a uniform function of the position. The refractive index of the
superstrate is nI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xxiv
LIST OF FIGURES
5.2
Cross section of a homogenous layer with the coefficients of the incident,
reflected and transmitted waves. . . . . . . . . . . . . . . . . . . . . . . 129
5.3
Schematic of the incidence plane for 3-D optical simulations. The direction of wave propagation vector k1 is defined by angles θ and φ with
respect to z-axis and x-axis, respectively.
5.4
. . . . . . . . . . . . . . . . . 131
Sketch of the simulation domain (the top figure reports a 3-D pyramid)
split in layers within which the permittivity is approximated by a step
function. The bottom figure shows a schematic of a single layer with the
Fourier coefficients of the electric field at the boundaries.
5.5
. . . . . . . . 134
Sketch of the 2-D rectangular grating adopted as reference test structure
for the validation and the analysis of the accuracy of the RCWA simulator (a = 25nm, b1 = b2 = 12.5nm, h1 = 100nm, h2 = 450nm and
periodicity p = 50nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6
Comparison of the absorbance calculated by FDTD and RCWA within
the range of wavelengths 380nm − 880nm for the grating test structure
(Fig. 5.5) in case of TE (A) and TM (B) polarizations. For FDTD simulations performed by Synopsys EMW, 400 cells per wavelength(CPW)
have been adopted. RCWA simulations have been carried out by using
NF = 60 and ∆z = 1nm.
5.7
. . . . . . . . . . . . . . . . . . . . . . . . . . 140
Accuracy of the relative absorption for 2D-RCWA simulator in case of
the test structure (rectangular grating of Fig. 5.5). The accuracy has
been calculated for both TE and TM polarizations and for two values of
the wavelength of the radiation (450nm and 750nm). In (A) and (B) the
accuracy of relative absorbance is calculated as function of the number
of Fourier modes NF with ∆z = 1nm and keeping the number of points
Nx for the Fourier transform along x-axis constant (Nx = 1024). In (C)
and (D) the accuracy of relative absorbance is calculated as function of
∆z −1 adopting NF = 201. The dashed line denotes the minimum value
of NF (∆z −1 ) above which the accuracy of the relative absorbance is
smaller than 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
xxv
LIST OF FIGURES
5.8
3-D Photonic Crystal: comparison of the transmission coefficient calculated by 3D-RCWA and that from literature (Fan et al. (4)) versus the
periodicity p of the cell normalized to the wavelength of the radiation
λ. Thickness and hole diameter are set to 0.5p and 0.05p, respectively.
The real part of the permittivity of the dielectric slab is 12. RCWA simulations have been performed by adopting θ = 0, φ = 0 (Fig. 5.3) and
NF x = 43, NF y = 43 number of Fourier modes along x-axis and y-axis,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.9
Sketch of the rectangular 3-D grating adopted as reference test structure
for the analysis of the accuracy of RCWA simulator (side length L =
500nm and periodicity p = 500nm).
. . . . . . . . . . . . . . . . . . . . 144
5.10 Accuracy of the relative absorption for 3D-RCWA simulator in case of
the test structure (3-D rectangular grating of Fig. 5.9) as function of the
number of Fourier modes NF x , NF y along x-axis and y-axis, respectively.
The accuracy has been calculated for θ = 0, φ = 0 (Fig. 5.3). The dashed
line denotes the minimum value of NF above which the accuracy of the
relative absorbance is smaller than 10−3 . . . . . . . . . . . . . . . . . . . 144
5.11 Sketch of the structures considered for the analysis of the impact of the
surface morphology on the absorbance: silicon slab with (A) flat, (B)
random rough (Gaussian distribution of heights with root mean square
value σRM S =50nm and correlation length Lc =25nm), (C) triangular
(h=500nm), (D) rectangular (h=500nm) and (E) circular (r=400nm)
texturing of the front interface. The periodicity p is 1000nm for all
structures and the volume of the absorbing material is the same for all
slabs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.12 Absorbance characteristics of the silicon slab with smooth front interface
calculated at different angles of incidence θ of the light. The absorbance
decreases with increasing θ. . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.13 Absorbance characteristics of the silicon slab with Gaussian random
roughness at the front interface calculated at different angles of incidence
θ of the light. The absorbance decreases with increasing θ similarly to
the flat interface case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
xxvi
LIST OF FIGURES
5.14 Comparison between the absorbance characteristics of the silicon slabs
for different kinds of front interface morphology calculated at normal
incidence (θ=0), for TE (A) and TM (B) polarization. The absorbance of
the slab with smooth interface is remarkably lower than those calculated
by considering the texturing. In case of TE polarization, for λ < 750nm
the maximum absorbance is obtained with the random rough interface
while for TM polarization the maximum absorbance is reached by the
random rough interface for λ < 600nm and by the triangular texturing
above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.15 Monochrome optical generation rate Gopt maps per unit volume and
time calculated for the slabs featuring smooth (on the left) and random
roughness (on the right) interfaces at λ = 600nm by assuming normal
incidence. On the bottom side the profiles of Gopt versus z for the vertical
slice at x = 500nm are compared. The optical generation rate Gopt in
case of random roughness is significantly larger than that of the flat
interface throughout the volume of the absorbing layer. . . . . . . . . . 149
5.16 Total absorbance Aw weighted by conventional AM1.5G spectrum of the
silicon slabs with different kinds of roughness at the front interface, calculated as function of the angle of incidence θ of the light, for TE (A) and
TM polarization (B). For TE polarization Aw decreases with increasing
θ for all kinds of surface morphology with the exception of the rectangular case, which features a maximum value of Aw between θ=20 and
θ=30 degrees; the random roughness features the maximum value of Aw
within the whole considered range of incidence angles. In addition, the
figure reports Aw calculated for the periodic triangular surface texture
featuring the same root mean square height of the random roughness
(HRM S =50nm), at normal incidence and for two different values of the
periodicity (p=300nm and p=20nm). In (B), for TM polarization, similar trends are observed with the exception of the rectangular-textured
and of the smooth interface, for which Aw increases with increasing θ. . 150
xxvii
LIST OF FIGURES
5.17 2-D optical generation rate maps inside the c-Si slab featuring circular,
rectangular and triangular texturing (Fig. 5.11) at λ = 450nm, TE
(left) and TM (right) polarization. Simulation performed by RCWA
(NF = 40, ∆z = 1nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.18 2-D optical generation rate maps inside the c-Si slab featuring circular,
rectangular and triangular texturing (Fig. 5.11) at λ = 750nm, TE (left)
and TM (right) polarization. Simulation performed by RCWA (NF =40,
∆z = 1nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.19 2-D optical generation rate maps inside the c-Si slab featuring triangular
(left) and rectangular (right) texturing (Fig. 5.11) at λ = 750nm, for
different angle of incidence of radiation (in A,D θ = 0 degree, in B,E
θ = 30 degree and in C,F θ = 60 degree). Simulation performed by
RCWA (NF =40, ∆z = 1nm). . . . . . . . . . . . . . . . . . . . . . . . . 153
5.20 Sketch of the 3-D cylindrical nano-rod. Nano-rod diameter D, height H,
substrate thickness T0 and periodicity P are geometrical parameters. . . 155
5.21 Absorbance of the silicon cylindrical nano-rod of Fig. 5.9 calculated by
3D-RCWA from λ = 350nm to λ = 950nm with θ = 0, φ = 0. Nano-rod
diameter D, height H and periodicity P as geometrical parameters. All
structures exhibit the same volume of the absorber material. Simulations
have been performed adopting NF x = 43, NF y = 43 number of Fourier
modes along x-axis and y-axis, respectively. Substrate thickness 10µm
(A) and 100µm (B). The highest absorbance is obtained by the structure
featuring P = 2120nm, D = 1000nm and H = 4000nm. . . . . . . . . . 155
5.22 Monochromatic optical generation rate of a cylindrical nano-rod featuring P = 2120nm, D = 1000nm and H = 4000nm (Fig. 5.20) calculated
by the 3D-RCWA simulator at λ = 850nm with θ = 0, φ = 0. The 3-D
optical generation rate has been mapped and interpolated on the same
mesh grid provided as input to the device simulator Sentaurus. The
nano-rod features a radial junction with junction depth of 200nm. Since
the illumination is direct, the cylindrical symmetry of the geometry is
exploited in order to save computational resources. Therefore only one
quarter of the cylinder is considered. . . . . . . . . . . . . . . . . . . . . 156
xxviii
LIST OF FIGURES
6.1
Cross-sectional schematic of a (glass) superstrate single-junction amorphous silicon thin film solar cell, with the transparent conductive oxide
(TCO) and the internal rough interfaces. . . . . . . . . . . . . . . . . . . 162
6.2
Density of states distribution in energy (DOS) with exponential bandtail states and Gaussian shaped mid-gap states in amorphous silicon.
EACP and EDON denote the acceptor Gaussian peak energy measured
positive from the edge of the valence band EV and the donor Gaussian
peak energy measured positive from the edge of the conduction band
EC , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3
Effective media approximation (EMA): the rough interface between two
media of refractive index n1 and n2 is replaced by a stack of one or more
equivalent virtual layers featuring planar interfaces. . . . . . . . . . . . . 166
6.4
Light scattering from rough interface: specular and diffuse reflection
(transmission) occur at non-planar interfaces exhibiting roughness featuring size in wave regime. HR and HT are the refection and transmission
haze parameters, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5
Typical haze parameters versus wavelength and angular distribution
functions (ADF) for air/glass/ZN:O:Al/air system in thin film solar cells
with roughness level σRM S as paramater. . . . . . . . . . . . . . . . . . 168
6.6
Optical properties of the media of the simulated single-junction thin film
superstrate solar cell: real part (left) and imaginary part (right) of the
complex refractive indices. . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.7
Accuracy of the relative absorption for 2-D RCWA in case of the simulated superstrate thin film solar cell with one rough interface (ITO/pregion interface with σRM S = 50nm). The accuracy has been calculated
for both TE and TM polarizations and for two values of the wavelength of
the radiation (λ = 450nm and λ = 750nm). In (A) and (B) the accuracy
of relative absorbance is calculated as function of ∆−1
z for NF = 201. In
(C) and (D) the accuracy of relative absorbance is calculated as function
of the number of Fourier modes NF when ∆z = 1nm. The dashed line
denotes the upper bound limit of the accuracy for the conditions adopted
to perform the optical simulation (NF = 121, ∆z = 1nm). . . . . . . . . 172
xxix
LIST OF FIGURES
6.8
Calculated electric field (A), optical generation rate map (in cm−3 s−1 )
(B), carrier concentrations (C) and recombination rate due to band-tails
and mid-gap traps as function of the depth in short-circuit conditions
for the p-i-n cell with σRM S = 0nm . . . . . . . . . . . . . . . . . . . . . 173
6.9
Calculated absorbed power within the amorphous silicon layer under
direct AM1.5G illumination (1000W m−2 ) adopting the roughness level
σRM S as parameter. The absorbed power increases with increasing σRM S .176
6.10 Calculated 2-D full-spectrum optical generation (in cm−3 s−1 ) maps within
the a-Si layer for σRM S = 25nm and σRM S = 50nm. . . . . . . . . . . . 176
6.11 Collection efficiency ηC of the photo-generated carriers for σRM S = 0nm,
σRM S = 25nm and σRM S = 50nm. At short wavelengths the collection
efficiency decreases with increasing roughness level σRM S . . . . . . . . . 177
6.12 External quantum efficiency (EQE) of the simulated solar thin film solar
cell for σRM S = 0nm, σRM S = 25nm and σRM S = 50nm. . . . . . . . . 177
6.13 2-D maps of optical generation rates (per unit time and volume) inside
the amorphous silicon absorber layer at wavelength λ = 400nm (A)
and λ = 750nm (B) for all considered values of the roughness σRM S .
Different spatial distributions of the photo-generated electron-hole pairs,
depending on σRM S and on the wavelength of the radiation, are observed.178
xxx
1
Introduction
Photovoltaic conversion is the direct production of electrical energy from sun. This
conversion does not involve the emission of polluting substances and mechanical movements, therefore it requires minimal maintenance expenses to operate. Although the
first basic solar cell has been introduced in 1950s, only in 1960s the photovoltaic technology has been intensively used especially for space applications and starting from
mid 1970s the silicon solar technology became attractive for terrestrial use. However,
in order to satisfy the continuously increasing demand of energy at constantly growing
costs and to be competitive with other energy sources, the photovoltaic technology
have to win the challenge of the reduction of costs ensuring at the same time adequate
conversion efficiencies. In fact, for decades, many approaches and device architectures
aimed at obtaining theoretical enhancements of the conversion efficiency of the solar
cells, like those that will be reviewed in the present work, have not being considered as
competitive due to the still higher manufacturing costs.
Although that of new materials is one of the most attractive topic of the research
community in photovoltaic, silicon is still the most widely used semiconductor for mass
production devices and, according to roadmaps, its adoption in solar cell manufacturing, will be competitive for next decades. Recently, an industry-driven reawakening of
advanced silicon solar cell architectures has really motivated the interest of researchers
in investigating design concepts of 1980s and 1990s with the objective of reducing
significantly the cost for volume production. On the other hand, new designs and technologies have been investigated providing the basis for the future generation devices.
1
1. INTRODUCTION
One promising way to lower the cost of the photovoltaic device is to decrease the volume
of material required to convert the light into electric energy.
Second generation of photovoltaic devices, like thin film devices using low-cost substrates and semiconductors, are just aimed at reducing the amount of material at the
expense of a significant penalization in terms of performance with respect to the first
generation of solar cells. For this class of devices, the material quality -which introduces relatively more electrical losses in comparison with standard silicon technology
solar cells of the first generation featuring thick silicon substrates- and the typically
limited capability to collect photons from the sun, are the main causes of performance
limitation. In order to overcome this limitation by keeping the cost competitive, the
most adopted approach is the enhancement of the light trapping or photon management
strategy, which allows a more efficient absorption of energy from the external source.
This opened the way to third-generation devices which are aimed at saving costs and
at improving the conversion efficiency, reaching performance levels comparable or even
better than those obtainable by first-generation solar cells. Modeling of photovoltaic
devices has become more and more strategic and helpful not only to predict the performance of future generations devices but also to provide new and more specific guidelines
to industry in order to reduce costs and time when designing new fabrication lines and
new process setups. Prior to last decade, the most effective way to model photovoltaic
devices relied mainly on analytical approaches partially supported by the aid of calculators. As the performance of calculators and the availability of computational resources
became adequate to provide the support to cpu-intensive numerical simulations, researchers have begun to adopt the numerical device simulator as the most effective
tools to perform modeling of semiconductor devices and of solar cells. First numerical
simulations were aimed at modeling mono-dimensional (1-D) devices. Many simulators
have been specifically introduced to the 1-D numerical simulation of photovoltaic devices and, among them, at least one became a reference in the photovoltaic community;
probably the most popular of such tools is PC1-D (5) developed in late 1990s at the
University of North South Wales (USNW), Australia. PC1-D has been widely adopted
by researchers and designers to predict the efficiency of standard and even advanced
solar cells. The main advantage of such easy-to-use tool is the low requirement of
computational resources and the quite intuitive user interface. However, an accurate
analysis of losses and the prediction of performance of advanced solar cells are often
2
not possible because of the mono-dimensionality of the simulation domain which can be
handled by PC1-D, while non standard device architectures are mostly characterized
by two-dimensional (2-D) or even three-dimensional (3-D) geometries. On the other
hand, during the last decade, multi-dimensional and general-purpose numerical device
simulators (Technology Computer-Aided Design TCAD) have been introduced mostly
for the analysis of CMOS devices. However, such simulators are not often easy-to-use;
due to this reason, the photovoltaic community has not really accepted their introduction as aid for design and investigation of solar cells.
The simulation of a device under illumination requires the knowledge of the spatial distribution of the amount of photo-generated carriers per time and volume unit
within the absorber material. The optical generation rate is coupled to the transport
and continuity equations which are part of the semiconductor devices modeling framework. The calculation of the optical generation map can be performed by means of
methods based on the geometrical optics constraint like ray tracers, which are widely
used also in other fields involving electromagnetic propagation, like the design and the
simulation of optical systems including mirrors and lenses. However, the validity of the
methods based on the geometrical optics approximation is limited to spatial domains
in which the wavelength of the radiation is much larger than the size of the features of
the geometry of the structure. This is just the case of first-generation silicon solar cells
for which the substrate thickness and the features of the morphology of the surfaces
(like the textured front surface) are not comparable to the wavelength of the radiation.
In case of thin film devices or nanotextured solar cells like those based on nano-wires
(6), nano-clusters (7), nano-rods (8) and diffraction gratings (9), the methods based
on the geometrical optics approximation are no longer valid. In thin film solar cells
the modeling of the light scattering from rough interfaces (for which the size of the
features is comparable to the wavelength) has been treated in literature on the basis
of methods derived from the well-known scalar scattering theory, originally developed
by Spizzichino and Beckmann in 1960s (10). However, also such methods are limited
in the dimensionality of the problem (1-D) and in accuracy, since they are not rigorous
solvers of the Maxwell equations. Although these methods have been calibrated in order to be in agreement with experimental data by means of semi-empirical determined
coefficients, the scattering from rough interfaces is an open issue unless the optical
3
1. INTRODUCTION
simulation is not carried out by rigorous solvers of the Maxwell equations. Among
these solvers, the most widespread is the Finite Difference Time Domain (FDTD) for
simulation domains of any geometrical dimension. However, the FDTD, especially for
2-D and 3-D problems, is really cpu and memory-intensive. The requirement in terms
of processing resources is particularly critical for problems in which the size of the simulation domain is significantly larger than the wavelength of the radiation. Recently,
in optical simulation of solar cells, the Fourier Modal Method also known as Rigorous
Coupled-Wave Analysis (RCWA), has gained relevance. RCWA may be used for the
simulation of a wide range of structures and devices, including image sensors, light
emitter diodes (LED) and solar cells.
The objective of this thesis is to provide a contribution to the numerical simulation
and modeling of advanced silicon solar cells -including third-generation devices- by
exploiting a state-of-the-art TCAD simulator featuring the capability to handle 2-D
and 3-D geometries. The thesis describes the work carried out during the three-year
period in which a simulation procedure -including the optical part- has been defined.
In particular, the focus of the first part of the work is on the device simulation setup.
The physical models of interest for the device simulation (like that for the band-gap
narrowing occurring when high doping concentration is assumed and that for the carrier lifetime) have been tuned ad-hoc in order to fit data from experiments and from
literature. The simulation framework has been successfully applied to the simulation
of advanced crystalline silicon cells, like the selective emitter and the rear point contact
one, for which the multi-dimensionality of the transport model is required in order to
properly account for all physical competing mechanisms occurring in such solar cells.
In order to complete the simulation flow, in the second part of the thesis, the optical problem is discussed, as contribution to reach the main objective of the research
activity. Two novel and computationally efficient implementations of an electromagnetic simulator for 2-D simulation domains based on Fourier modal expansion of the
electromagnetic field have been presented, as result of an intensive collaboration with
the Institute of Physics and Technology (FTIAN) of the Russian Academy of Sciences
(RAS), Moscow. A third method -still based on Fourier modal approach- has been implemented with the purpose of modeling the light propagation in 3-D structures. The
4
proposed implementations have been validated in terms of accuracy, numerical convergence, computation time and correctness of results, by means of a comparison with
literature data and with those provided by a commercial Finite Difference simulator in
case of similar structures. Promising results in terms of computational efficiency have
been obtained. The first application of 3-D optical simulation have been presented as
basis of future research activities which will involve nanostructured photovoltaic devices.
In the following, the detail of the content of each chapter of the thesis is summarized. After a review of the basic semiconductor transport model in chapter 2, the
basis of the simulation of 2-D and 3-D silicon-based solar cells is introduced in chapter
3. In particular the approach adopted to calibrate the physical models of a well-known
general purpose device simulator (Sentaurus Device by Synopsys) is shown. The simulation flow and the main standard methods commonly adopted to solve the optical
problem in solar cell modeling are described.
Chapter 4 describes two examples of numerical simulation of advanced silicon solar
cells: the selective emitter which requires the solution of the transport equations in
a 2-D simulation domain and the rear point contact solar cells, which requires true
3-D simulations. The selective emitter and the rear point contact approaches are not
mutually exclusive, on the contrary these two designs -which involve different portions
of the solar cell- can be adopted simultaneously, enhancing the conversion efficiency
towards the record efficiency that has been reached in 1990s by small-area solar cells
fabricated in laboratory. The modeling activity of advanced silicon solar cells has been
performed in the frame of the collaboration between University of Bologna and Applied
Materials, Inc. US, since 2008.
Chapter 5 presents a review of the state-of-the-art Fourier-based algorithms for optical simulation, which are rigorous solvers of the Maxwell equations and lead to an
acceptable trade-off between accuracy and computational resources requirement. In
addition, chapter 5 describes the details of two novel implementations of RCWA aimed
at solving the Maxwell equations in 2-D spatial domains that are not based on the
solution of an eigenvalue problem. These methods feature better performance in terms
5
1. INTRODUCTION
of cpu simulation time with respect to standard implementations. They have been
validated by means of a comparison with a FDTD commercial tool. For 3-D spatial
domains an implementation of RCWA based on standard approaches -but more general
in terms of validity- has been described. Some applications of the RCWA methods, like
the investigations of the impact of different surface morphologies of thin-film devices as
well as of the main geometrical parameters of nano-rod based solar cells on the photon
absorbance are presented.
Finally chapter 6 outlines the basics of the device physics of thin film amorphous
silicon solar cells. Moreover, an application of a 2-D numerical simulation of a singlejunction device exhibiting nanorough interfaces, for which the RCWA tool has been
successfully adopted to perform the optical simulation, is presented.
6
2
Solar cell device physics and
technology review
In this chapter a review of the semiconductor modeling theory is presented. The
main loss mechanisms as well as the absorption process in semiconductor materials
are described emphasizing the impact of such phenomena on the conversion of energy
from the sun. The main figures of merit of a solar cell are introduced highlighting the
theoretical upper bound limit of the conversion efficiency under some simple assumptions. Finally, the main technological options adopted in the design of silicon solar
cells, including the most used techniques aimed at enhancing the photons absorption,
are presented.
2.1
Absorption of light in semiconductors
If a monochromatic electromagnetic wave characterized by a given intensity and by
a wavelength impinges on an interface between a dielectric and an absorber material
(Fig. 2.1), part of the incident power is reflected by the interface and the remainder
is transmitted into the absorber material. The light passing through the absorber
material is attenuated. The entity of the attenuation of the light propagating through
an absorbing material depends on the wavelength λ of the radiation and on the material
absorption characteristics. It can be demonstrated that, for a simple homogeneous
infinite layer of absorber material, the attenuation of the light intensity follows an
7
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
exponential decay according to Beer-Lambert Law:
I(x, λ) = I0 (0, λ)e−α(λ)(x−x0 )
(2.1)
where I0 is the intensity of the light at the interface (x = x0 ) and α is the absorption
coefficient which is related to the extinction coefficient k of the material according to:
α(λ) =
4πk
λ
(2.2)
Figure 2.1: Interface between a dielectric and an absorber material (z ≥ z0 ). The light
is propagating from left to right along z-axis.
The extinction coefficient k is the imaginary part of the complex refractive index
ñ of the material. An attenuation of the intensity of the light after passing through
a distance equal to α−1 corresponding to e−1 occurs when an electromagnetic wave
propagates along the z-direction (electromagnetic propagation basics are described in
chapter 3). If n denotes the real part of the refractive index, n and k can be expressed as
function of the real and the imaginary part of the permittivity (ε1 and ε2 respectively):
sp
ε21 + ε22 + ε1
n=
2
sp
k=
ε21 + ε22 − ε1
2
(2.3)
(2.4)
The dependence of the absorption coefficient α of the crystalline silicon at T=300K
on the photon energy is shown in Fig. 2.2.
8
2.1 Absorption of light in semiconductors
Figure 2.2: Dependence of the absorption coefficient α of crystalline silicon on the photon
energy at T = 300K.
When a photon is absorbed, an electron of the valence band is excited and leaves
a hole by means of a transition to the conduction band. In direct band-gap semiconductors, the crystal momentum is conserved when a transitions to the conduction band
occurs because the photon has a smaller momentum compared to that of the crystal.
However, the transition is characterized by the following energy balance:
Ef − Ei = hf
(2.5)
in which the difference between the energy that is associated to the final and to
the initial states before the transition (Ef and Ei , respectively) must be equal to the
photon energy Eph = hf , where h is the Planck constant and f is the photon frequency.
An approximated theoretical model for the absorption relates the absorption coefficient
for direct band-gap material to the photon energy according to:
A(hf ) ∝
p
Eph − Eg
(2.6)
where Eg is the energy band-gap of the semiconductor. A first consequence of
this simple model is that, if Eph is much larger than Eg , the probability of photon
9
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
absorption is significantly higher in the first few wavelengths distance passing through
the semiconductor.
In indirect bad-gap semiconductors like silicon, a photon energy much larger than
the forbidden bad-gap is required in order to allow a direct transition of electrons from
the valence band to the conduction band. The crystal momentum is not conserved
during a transition. The absorption process in an indirect band-gap semiconductor
involves a third particle, the phonon, which is characterized by low energy and high
momentum. Therefore the probability of light absorption in these semiconductors is
significantly lower than that of direct band-gap materials. In order to allow a transition
of an electron, the following energy balance must be satisfied:
hf = Eg − Ep
(2.7)
where Ep is the energy of the phonon with a given momentum.
Other absorption processes are possible, like that occurring when the photon energy is
so high to allow the excitation across the forbidden bad-gap or the two-step absorption
in direct band-gap semiconductors, which involves the emission or the absorption of
phonons.
In order to simulate an optoelectronic device such as a solar cell, the calculation
of the optical generation rate which gives the amount of photo-generated electronhole pairs per volume and time unit GOP T inside the absorber material is required.
The Beer-Lambert law (eq. 2.1) is valid only for monodimensional structures having
linear dimension significantly larger than the absorption length at a given wavelength.
When such assumption is not satisfied, effects due to coherent propagation of the light
(like interfence) occur and the Beer-Lambert law is not valid. In Fig. 2.3 the optical
generation rate per time and volume unit calculated at λ = 500nm is reported in
case of a silicon slab of infinite thickness. In such case the eq. 2.1 is an acceptable
approximation. However, if a thinner slab of silicon is considered (e.g. 100nm like that
of Fig. 2.4), due to the multiple internal reflections of the light occurring at the front
and at back interfaces of the slab, maxima and minima of the light intensity and hence
of the optical generation rate can be observed and the eq. 2.1 is not suitable.
10
2.1 Absorption of light in semiconductors
Figure 2.3: Optical generation rate GOP T calculated at λ=500nm as function of the
distance from the front interface in case of a silicon slab featuring infinite thickness. The
amplitude E0 of the incident field is 1V m−1 .
Figure 2.4: Optical generation rate GOP T calculated at λ=500nm as function of the
distance from the front interface in case of a silicon slab featuring thickness comparable
to the wavelength of the radiation. Minima and maxima of GOP T can be observed. The
amplitude E0 of the incident field is 1V m−1 .
11
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
2.2
Recombination mechanisms
While under illumination conditions the carrier concentration is in excess with respect
to their value in equilibrium, in dark the concentration is reduced down to the equilibrium value through the recombination mechanisms which annihilate the electron-hole
pairs. In the following, the most common recombination mechanisms -which contribute
to the electrical losses in a solar cells- are outlined.
2.2.1
Radiative recombination
The radiative recombination occurs under thermal equilibrium involving a transition
from the conduction band down to the valence band; this process leads to the emission
of a photon having energy equal to the difference between the energy level before the
transition and that after the process. The radiative recombination rate per unit time
and volume can be expressed as:
Rrad = Bnp
(2.8)
where B is a semiconductor property, n and p are the concentrations of electrons
and holes, respectively. The radiative recombination is negligible in indirect band-gap
semiconductors since it involves, as the light absorption mechanism, a two-step process
with a phonon. Under thermal equilibrium, for which np = ni 2 , where ni is the intrinsic
carrier density concentration, when the device is in dark conditions and in the absence
of an external stimulus, the radiative recombination is balanced by a generation process,
therefore the net recombination rate is given by:
N ET
Urad
= B(np − n2i )
2.2.2
(2.9)
Auger recombination
The Auger recombination is the reverse process of the impact ionization. When an
electron recombines with a hole, its energy is given to a second electron which leads to
a phonon emission in order to acquire again its original energy level. The probability
of Auger recombination is particularly significant in case of highly doped materials
(in particular when doping concentration is above 1017 cm−3 ) because of the electron
excitations in the majority carrier band. Auger recombination is one among the main
12
2.2 Recombination mechanisms
loss mechanisms which should be accounted for in designing highly doped regions (like
emitters) of solar cells, as it will be shown in chapter 4.
The Auger recombination rate can be expressed as:
RAug = (Cn n − Cp p)(np − n2i )
(2.10)
where Cn and Cp are material-specific parameters.
2.2.3
Trap-assisted recombination
Impurities and defects in semiconductor materials lead to allowed energy levels within
the forbidden energy bang-gap. The trap-assisted recombination mechanism occurs
when an electron relaxes down to the defect level from the conduction band and then
from the defect level to the valence band by exploiting a two-step process. In case of
a single level trap (SLT), under the assumption of stationary trapped charge density
into the recombination centers, the net recombination can be expressed, according to
the Schokley-Read-Hall (SRH) recombination model, as:
net
USRH
=
(np − n2i )
τhSLT,0 (n + n1 ) + τeSLT,0 (p + p1 )
(2.11)
where
n1 = NC exp
ET RAP − EC
kT
p1 = NV exp
EV − ET RAP
kT
and ET RAP is the energy level associated to the defect, τeSLT,0 and τhSLT,0 are the
SRH carrier lifetimes for electrons and holes, respectively. NC and NV are the effective
density of states in conduction band and in the valence band, respectively. The edges
of the conduction and valence bands are denoted by EC and EV , respectively. It is
net occurs when E
worth noting that the peak value of USRH
T RAP is exactly in the mid
of the forbidden energy gap, for which the probability of transition is maximum. The
single level trap approximation is suitable for crystalline silicon but not for materials
like amorphous silicon, for which, an extension of the SRH model accounting for a
continuous energy level distribution of defects in energy is typically adopted.
13
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Similarly, it is possible to define a model to take into account the recombination
occurring at surfaces where defects act like recombination centers, for which the net
recombination rate can be expressed by:
net
USRH,SU
RF =
Se0 Sh0 (np − n2i )
Se0 (n + n1 ) + Sh0 (p + p1 )
(2.12)
where Se0 , Sh0 are the surface recombination velocities (measured in cm/s). In this
net
case USRH,SU
RF is a net recombination rate for unit time and area.
2.2.4
Effettive carrier lifetime
Independently on the specific recombination mechanism, under the assumption of operating conditions close to equilibrium, for a n-type semiconductor, by using the Taylor
approximation, it is possible to reformulate the net recombination function as:
Unet (n, p) u U (pn0 , nn0 ) +
∂U
∂p
(p − pn0 ) +
0
∂U
∂n
(n − nn0 ) ,
(2.13)
0
where pn0 and nn0 denote the equilibrium hole and electron concentrations, respectively. In eq. 2.13 the derivatives are calculated at p = pn0 , n = nn0 . Carrier lifetimes
are then introduced and defined as:
τe =
1
,
τh = ∂U
∂n 0
1
∂U
∂p
.
(2.14)
0
The terms p − pn0 and n − nn0 represent the excess of carrier concentrations with
respect to their equilibrium concentrations. The eq. 2.13 reads:
Unet (n, p) =
p − pn0
n − nn0
+
.
τp
τn
(2.15)
Therefore, an expression of lifetimes for each recombination mechanism can be
derived:
τAU G =
1
Cn n2n0
(2.16)
τRAD =
1
Bnn0
(2.17)
14
2.3 Semiconductor Model Equations
The single-trap level SRH carrier lifetimes of eq. 2.11 can be written as:
τpSRH,BU LK =
1
(2.18)
σp vT H NT
where NT is the concentration of traps, vT H the thermal velocity of carriers and σp
the capture cross section (which is proportional to the carrier capture probability) of
the recombination center. Similary for a p-type doped semiconductor.
The effective lifetime can be defined as:
1
τef f
2.3
2.3.1
=
1
τSRH,BU LK
+
1
τRAD
+
1
τAU G
.
(2.19)
Semiconductor Model Equations
Drift-Diffusion Transport Model
Due to diffusion mechanism, in the absence of any external forces, electrons and holes
in semiconductors move from regions of high concentration to regions of low concentration leading to a flux of particles in the direction normal to the surfaces with same
concentration. If φp and φn denote the flux density of holes and electrons, respectively,
the diffusion current densities are given by
Jp = −qDp ∇p
Jn = qDn ∇n
(2.20)
where n = n(r), p = p(r) are the position-dependent electron and holes concentrations and Dp , Dn are the hole and electron diffusion coefficients, respectively.
By applying an electric field E across a uniformly doped semiconductor, an upward
bending of energy bands occurs. As consequence of such electric field, electrons in
the conduction band, being negative charges, move in the opposite direction of the
applied field. On the contrary, the holes in the valence band. Carriers are therefore
accelerated by the electric field. At macroscopic level, carriers appear to move at a
constant velocity, the drift velocity vd , which is directly proportional to the electric
field. The drift current densities for holes and electrons can be written as
Jp = qµp pE,
Jn = qµn nE.
15
(2.21)
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
where µn and µp are the electron mobility and the hole mobility, respectively. In
thermal equilibrium the drift and diffusion currents must exactly balance. In nondegenerate materials, the Einstein relations link the diffusion coefficients to the carrier
mobilities:
D p = µp
kT
,
q
Dn = µn
kT
.
q
(2.22)
The total current densities, for each carrier, are the superposition of the drift and
diffusion contributions:
Jp = qµp pE − qDp ∇p,
Jn = qµn nE + qDn ∇n.
(2.23)
The total current is then
J = Jp + Jn + JD
(2.24)
where the displacement current density is
JD = ε
∂E
∂t
(2.25)
ε denotes the electric permittivity of the semiconductor. The displacement current
is typically neglected in solar cells modeling since, in such case, the main figures of
merit are static parameters.
2.3.2
Semiconductor equations
Transport equations link the current densities to the carrier concentrations. Since the
unknowns of the problem are n = n(r, t), p = p(r, t) and E = E(r), more equations
are required to define the device modeling framework. Two equations, among those
required, are the hole and the electron continuity equations ((11)):
(
1
q∇
1
q∇
· Jp = GOP T − Rp − ∂p
∂t ,
· Jn = Rn − GOP T + ∂n
∂t .
(2.26)
where GOP T is the optical generation rate of electron-hole pairs and Rp and Rn
include also the thermal generation.
16
2.3 Semiconductor Model Equations
Under quasi-stationary conditions the electrostatic potential φ(r, t) is related to the
spatial charge density ρ(r) by the Poisson’s equation:
∇2 φ(r, t) = −
ρ(r, t)
(2.27)
and if N = ND − NA denotes the total doping concentration (donor and acceptor),
ρ = (ND − NA + p − n) = (N + p − n).
(2.28)
Under steady state conditions, in case of a mono-dimensional (1-D) device,
∂E
q
= (p − n + N )
∂x
(2.29)
In case of uniform spatial doping concentration, the mobility and the diffusion
coefficients are constant, therefore:
(
2.3.3
2
d p
d
qµp dx
(pE) − qDp dx
2 = q(G − R),
2
d
qµn dx
(nE) + qDn ddxn2 = q(R − G).
(2.30)
1-D single-junction silicon solar cell model
In this subsection, following the approach described in (12), the current-voltage characteristic at electrodes of a 1-D single junction silicon solar cell (Fig. 2.5A) is derived
by adopting the semiconductor equations discussed in the previous subsection.
A simple solar cell is formed by a n+ -p junction in which the heavy doped n-type
diffusion is the emitter (−TN < x < 0) and the p-type substrate is the base (0 < x <
TP ) of the solar cell. In thermal equilibrium no net current flows and the Fermi energy
does not depend on the position. Due to the gradient of concentration between the two
types of semiconductors, holes diffuse from the p-type region into the n-type region.
Similarly, electrons from the n-type material diffuse into the p-type region. An electric
field is therefore produced limiting the diffusion of carriers. The space-charge region
(in Fig. 2.5B for −XDN < x < XDP ) is depleted of both holes and electrons. Under
the assumption that the p-type and the n-type regions are thick enough, the regions
on either sides of the depletion region (that for x < −XDN and that for x > XDP ) are
quasi-neutral. The electrostatic potential difference across the junction is the built-in
voltage V0 . The electrostatics of the single-junction system is defined by the Poissons
17
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
equation, that, within the depletion region, under the assumption that p0 , n0 << |N |
(where p0 and n0 are the equilibrium concentration of holes and electrons, respectively)
can be written as:
(
d2 φ
dx2
d2 φ
dx2
= qε NA
if
q
= − ε ND if
− XDN < x < 0,
0 < x < X DP .
(2.31)
while in the quasi-neutral regions charge neutrality is assumed (depletion approximation):
d2 φ
= 0.
dx2
(2.32)
As a direct consequence of the definition of V0 ,
Z
XDP
Z
Edx = −
−XDN
XDP
−XDN
dφ
dx = φ(−XDN ) − φ(XDP ) = V0 .
dx
(2.33)
If conventionally φ(XDP ) = 0, the solution of the previous equations reads:

V0



2
D
V0 − qN
2 (x + XDN )
qNA
2

(x − XDP )

 2
0
x ≤ −XDN ,
− XDN < x ≤ 0,
0 ≤ x < XDP ,
x ≥ XDP .
if
if
if
if
(2.34)
Due to the continuity of the electrostatic potential φ and of the electric field at
x = 0,
V0 −
qND 2
qNA 2
xDN =
X ,
2
2 DP
xDN ND = xDP NA .
(2.35)
Solving the previous equations under equilibrium conditions, the depletion region
width can be derived as:
s
WDEP = xDN + xDP =
2
q
ND + NA
ND NA
V0 .
(2.36)
If an external bias voltage VA is applied, the previous equation reads:
s
WDEP = xDN + xDP =
2
q
18
ND + NA
ND NA
(V0 − VA ).
(2.37)
2.3 Semiconductor Model Equations
Figure 2.5: (A) Mono-dimensional (1-D) sketch of an abrupt p-n junction with the
associated charge density ρ(x) profile along the x-axis (B). The pint x = 0 denotes the
metallurgical junction interface. −XDN and XDP are the boundaries of the depletion
region at the n-type and p-type sides, respectively. NA and ND denote the uniform doping
concentrations of the p-type and n-type regions, respectively. The electric field plot, the
energy bands diagram and the carrier concentration plots are shown in (C), (D) and (E),
respectively. In (D) V0 denotes the built-in potential.
19
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Since under thermal equilibrium the net current density of holes and electrons is
zero, the expression of the built-in potential can be easily derived:
xDP
Z
V0 =
Z
Edx =
−xDN
xD P
−xDN
kT 1 dp0
kT
dx =
ln
q p0 dx
q
ND NA
n2i
.
(2.38)
The charge density, the electric field and the energy bands for the p-n junction are
shown in Fig. 2.5B, Fig. 2.5C and Fig. 2.5D, respectively, while the plots of the carrier
concentrations are shown in Fig. 2.5E. It is worth noting that the maximum electric
field intensity EM AX is reached at x = 0, therefore
EM AX
dφ
= |E(x = 0)| = | |x=0 =
dx
s
2q
1
1 −1
(V0 − VA )
+
.
ND
NA
(2.39)
In order to solve the equations of the single-junction solar cell, proper boundary
conditions have to be set at the emitter and at the base contacts; if the emitter contact
is assumed to be an ideal ohmic contact:
∆p(−TN ) = 0.
(2.40)
Modeling the emitter contact with an effective (finite) surface recombination velocity SF makes the boundary condition more realistic:
d∆p
SF
|x=−TN =
∆p(−TN ).
dx
Dp
(2.41)
It is worth noting that, if SF → ∞ the previous boundary conditions is reduced to
eq. 2.40.
Similarly for the back contact:
d∆n
SB
|x=TP =
∆n(TP ).
dx
Dn
(2.42)
By assuming the semiconductor nondegenerate, under nonequilibrium and low-level
injection conditions, it is possible to assume the quasi-Fermi energies constant within
the depletion region:
qVA = FN (x) − FP (x),
20
−xDN ≤ x ≤ xDP .
(2.43)
2.3 Semiconductor Model Equations
The boundary conditions at the edges of the depletion region are known as the law
of the junction (12):
pN (−xDN ) =
n2i qVA
e kT ,
ND
nP (xDP ) =
n2i qVA
e kT .
NA
(2.44)
If the solar cell is illuminated, it is possible to express the optical generation rate
inside the device by using the Beer-Lambert law (eq. 2.1):
Z
α(λ)e−λ(x+TN ) dλ.
G(X) =
(2.45)
λ
The minority carrier diffusion equations (eq. 2.20) are solved in the quasi-neutral
regions; for the p-type side, in case of a monochromatic wave:
Dn
d2 ∆nP
∆nP
−
= −Gλ (x),
dx2
τn
XDP ≤ x ≤ TP .
(2.46)
where
Gλ (x) = I0 αe−λ(x+TN ) .
(2.47)
∆nP = nP − np0 .
(2.48)
and
The eq. 2.46 is a second order differential non-homogeneous equation; its solution
is:
∆pN (x) = C1,P
sinh
x − XDP
Ln
+ C2,P
cosh
x − XDP
Ln
+ Gλ,P (x).
(2.49)
where
Gλ,P (x) =
τn
2
Ln α(λ)2
−1
α(λ)e−λ(x+TN ) .
(2.50)
and Ln is the diffusion length of electrons defined as Ln =
√
Dn τn . Similarly for
n-type side:
∆pP (x) = C1,N sinh
x + XDN
Lp
+ C2,N cosh
21
x + XDN
Lp
+ Gλ,N (x).
(2.51)
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
where
Gλ,N (x) =
τp
2
Lp α(λ)2
−1
α(λ)e−λ(x+TN ) .
and Lp is the diffusion length of holes defined as Lp =
(2.52)
p
Dp τp .
The arbitrary constants Cj,N , Cj,P (with j = 1, 2) are determined by using the
boundary conditions expressed by eq. 2.41, eq. 2.42 and eq. 2.44.
The minority carrier current densities in the quasi-neutral regions are those associated to the diffusion components:
N
Jp,N (x) = −qDp d∆p
dx ,
P
Jn,P (x) = qDn d∆n
dx .
(2.53)
The total current is J(x) = Jp (x) + Jn (x), but the equations 2.53 give the current
density in different positions. By integrating the electron continuity equation over
the depletion region, it is immediate to link the electron current at the edges of the
depletion region:
Z
Jn (−XDN ) = Jn (XDP ) + q
XDP
Z
G(X)dx − q
−XDN
XDP
RDEP dx.
(2.54)
−XDN
where
qVA
ni e 2kT − 1
RDEP =
τDEP
.
(2.55)
is the trap-assisted SLT SRH recombination rate within the depletion region; such
recombination rate is assumed constant (12) and calculated at the point x = xC for
which p(xC ) = n(xC ) (Fig. 2.5E). The related (constant) carrier lifetime is τDEP .
Therefore, expanding the previous equation,
Jn (−XDN ) =
h
i
WDEP ni qVA
e 2kT − 1 .(2.56)
Jn (XDP ) + q e−λ(TN −XDN ) − e−λ(TN +XDP ) − q
τDEP
In order to calculate the current density at contacts, the solution to the minority
carrier diffusion equations 2.53 is used.
22
2.3 Semiconductor Model Equations
2.3.4
Output characteristic and figures of merit
A common expression for the current voltage characteristic of a single junction solar
cell is:
qVA
qVA
J = JSC − J0A e kT − 1 − J0B e kT − 1 .
(2.57)
where JSC denotes the short circuit current density, which is the sum of the contributions from the n-type quasi neutral region, the depletion region and the p-type quasi
neutral region:
JSC = JSCN + JSCDEP + JSCP .
(2.58)
The detailed expressions of these contributions are reported in (12).
The terms J0A and J0B are the dark saturation current densities in the quasi-neutral
regions and within the depletion region, respectively.
J0A = J0A,P + J0A,N .
(2.59)
where
J0A,P = q
n2i
D
Dp 
ND Lp
p
Lp sinh
N −XDN
T
Dp
Lp cosh
Lp
N −XDN
Lp
T
+ SF cosh
N −XDN
T
Lp

T −X  .
+ SF sinh N Lp DN
(2.60)
Similarly for J0A,N . In case of a long base diode (Ln TP ) the expression of J0A,N
can be simplified as:
J0A,N = q
n2i Dn
.
NA Ln
(2.61)
It is worth noting that there is no influence of the boundary condition (in particular
of the back surface recombination velocity SB ) at the back contact on J0A,N under long
base approximation. On the contrary, when Ln TP (short base):
J0A,N = q
n2i
Dn
SB
.
Dn
NA TP − XDP SB + T −X
p
D
P
23
(2.62)
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
In addition, if SB 1 (i.e. no BSF), the following short-base approximation is
obtained:
n2i
Dn
.
(2.63)
NA TP − XDP
The saturation current density due to recombination in the depletion region is:
J0A,N = q
WDEP ni
(2.64)
τDEP
the dependence on the applied bias is through WDEP , which is bias-dependent.
J0B = q
In J0B
Eq. 2.57 suggests the two-diode circuit equivalent model of Fig. 2.6, in which the
ideal current generator represents the light polarization and the two diodes in parallel
the recombination current terms. It is worth noting that the ideality factor of the
two diodes are different: for the recombination in the quasi-neutral regions it is equal
to 1 while for the recombination in the depletion regions it is equal to 2. The latter
contribution is commonly neglected especially at large bias voltage.
Figure 2.6: Two-diode circuit model of a solar cell. Diode (A) represents the current due
to recombination within the quasi-neutral regions. Diode (B) models the recombination
occurring in the depletion region according to eq. 2.64.
In Fig. 2.7 the current-voltage I(V ) and the output power POU T (V ) characteristics
of the cell are shown, where I = AJ, ISC = AJSC and A is the area of the solar cell.
In Fig. 2.7 VM P P and IM P P denote the maximum output power voltage and current,
respectively. At V = VM P P , I = IM P P the output power reaches the maximum value
M AX = V
POU
M P P IM P P . The open circuit voltage VOC at which I(VOC ) = 0 is:
T
VOC
kT
≈
ln
q
ISC
I0A
24
,
ISC I0A ,
(2.65)
2.3 Semiconductor Model Equations
Figure 2.7: I-V characteristic (red solid curve) of a solar cell. The parameters VOC and
ISC are the open-circuit voltage and the short-circuit current, respectively. The output
power is represented by the black dashed curve, which reaches the maximum value at
V = VM P P and I = IM P P .
where I0A = AJ0A . The ratio of the maximum output power to the theoretical
maximum output power is the Fill Factor:
FF =
M AX
POU
VM P P IM P P
T
=
.
VOC ISC
VOC JSC
(2.66)
Ideally the Fill Factor (FF) is 100. For a silicon solar cell the FF typically ranges
between 77.0 and 82.0.
The conversion efficiency η is the most important figure of merit and it is defined
as the ratio of the maximum output power to the incident power:
EF F =
2.3.5
M AX
POU
F F VOC ISC
T
=
.
PIN
PIN
(2.67)
Other figures of merit
Figures of merit like open circuit voltage, Fill Factor and short circuit current are not
helpful to understand in detail which are the main loss mechanisms that contribute to
25
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
degradation of the efficiency. The short circuit current of a solar cell is limited by recombination mechanisms or by poor light trapping strategy effectiveness. In addition,
recombination mechanisms may be dominant within a particular portion of the irradiance spectrum and the photon absorption may be limited by the reflectance properties
of the external top interface rather than the absorption properties of the semiconductor.
When an analysis of loss mechanisms is performed, some figures of merit are helpful to
address the design of the solar cell.
2.3.6
Optical figures of merit: absorbance, reflectance and transmittance
Reflectance R(λ) and transmittance T (λ) are intuitively defined as
R(λ) =
PREF L
,
PIN C
T (λ) =
PT RAN S
PIN C
(2.68)
where PIN C is the incident power and PREF L , PT RAN S are the portion of the
incident power which are reflected and transmitted by the solar cell, respectively. If an
ideal back reflector is adopted (internal bottom reflectivity equal to 1.0) then T (λ) = 0.
In order to satisfy the optical conservation law,
R(λ) + T (λ) + A(λ) = 1.
(2.69)
A(λ) is the absorbance of the solar cell at a given wavelength λ.
2.3.7
Quantum efficiency and spectral response
The external quantum efficiency EQE(λ) is defined as the ratio of the number of
carriers which contribute to the current at electrodes under short-circuit conditions to
the number of incident photons. If φ(λ) denotes the incident power flux (number of
photons per time and area unit) and I0 is the input irradiance (in W/m2 ),
φ(λ) =
I0 (λ)
I0 (λ)
= hc ,
Eph (λ)
λ
(2.70)
where Eph (λ) is the photon energy, h and c are the Planck Constant and the speed
of light (in vacuum), respectively.
26
2.4 Efficiency Limits
The number of incident photons NIN C per unit time is therefore given by φA where
A is the area of the solar cell. Therefore,
EQE(λ) =
Jsc (λ)
Jsc (λ)
Jsc (λ)
=
=
.
qNIN C (λ)
qφ(λ)
JIN (λ)
(2.71)
The internal quantum efficiency IQE(λ) is defined as the number of carriers which
contribute to the output current in short-circuit conditions to the total number of
photo-generated electron-hole pairs within the volume of the absorber material. Hence
IQE(λ) can be expressed as (13):
IQE(λ) =
EQE(λ)
.
1 − R(λ)
(2.72)
The collection efficiency ηC of the photo-generated carriers is defined as:
ηC (λ) =
Jsc (λ)
.
Jph (λ)
(2.73)
ηC reports about the property of electrodes to collect the carriers once generated
in the absorber material. Carriers recombination leads to reduction of ηC .
Finally, the spectral response SR(λ) is defined as the ratio of the current generated
by the solar cell to the incident power; if QE(λ) denotes the quantum efficiency (either
external or internal),
SR(λ) =
2.4
qQE(λ)
.
Eph (λ)
(2.74)
Efficiency Limits
In this section the theoretical upper bound limits of the main figures of merit of a solar
cell are estimated under some simply assumptions described in the following. Under
ideal conditions, all photo-generated carriers are collected by electrodes contribute to
the short-circuit current density, therefore, if S(E) and A(E) denote the spectral irradiance and the absorbance at a given photon energy E, respectively, the short-circuit
current density may be calculated as:
Z
JSC = q
∞
S(E)A(E)dE.
0
27
(2.75)
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Ideally all photons having energy higher than the band-gap Eg are absorbed by the
semiconductor, hence:
Z
∞
S(E)dE.
JSC = q
(2.76)
Eg
Figure 2.8: Conventional spectrum AM1.5G (1000W m−2 ) for direct sun illumination.
The plot reports the spectral irradiance.
In Fig. 2.9 the dependence of the short-circuit current on the band-gap energy under
the assumptions described above is reported. The curve is calculated by adopting direct
illumination, neglecting the external reflectance of the top interface and assuming the
conventional spectrum AM1.5G (1000W m−2 ) (14) (Fig. 2.8). It is worth noting that
for c-Si (Eg = 1.12eV ) the upper bound limit of the JSC is 43.8mAcm−2 while for the
hydrogenated amorphous silicon (a-Si:H, Eg = 1.72eV ) is 21.9mAcm−2 . Comparing
the c-Si limit with the typical short-circuit current density exhibited by the record
efficiency cells (laboratory samples), which is about 42mAcm−2 (15), the conclusion
is that for thick substrates, light-trapping strategies for silicon technology are enough
effective to ensure an absorbance close to the maximum achievable. Concerning the
a-Si:H, typical JSC values are within the range 12-16mAcm−2 (16) while the upper
bound limit is above 20mAcm−2 . These discrepancies are due to light trapping scheme
limitations, to the non-zero reflectance of the external front interface, to the finite
28
2.4 Efficiency Limits
Figure 2.9: Dependence of the short-circuit current on the band-gap energy Eg for direct
illumination without external reflectance of the top interface calculated assuming the input
spectrum AM1.5G (1000W m−2 ) at T=300K. All photo-generated electron-hole pairs are
assumed to be collected by electrodes.
thickness of the absorbing layer as well as to the recombination losses which reduce the
short-circuit current.
M. A. Green in (1) proposed the following estimation of the lower bound limit of
the saturation current density:
EG
J0 = 1.5 × 10 exp −
kT
5
Acm−2 .
(2.77)
This lower bound bang-gap dependent value of the saturation current density is
helpful to calculate the open circuit voltage VOC as well as the conversion efficiency η
by using the I-V characteristic of eq. 2.57. Results are presented in Fig. 2.10 and in
Fig. 2.11. It is possible to observe that for c-Si the maximum expected VOC is about
700mV; depending upon the technology, process conditions and the particular design
of the cell, practical devices exhibit VOC within the range 600-670mV.
In case of c-Si, the upper bound limit of the efficiency is 25.9% while for a-Si:H
is 25.4%. In order to obtain a value of conversion efficiency close to the peak value,
the bang-gap energy should be within the range 1.4-1.6eV. It is worth noting that for
29
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.10: Dependence of the open-circuit voltage VOC on the band-gap energy Eg
for direct illumination without external reflectance of the top interface and calculated
assuming the spectrum AM1.5G (1000W m−2 ) at T=300K. The saturation current is in
agreement with the lower bound limit proposed by M.A. Green in (1). All photo-generated
electron-hole pairs are assumed to be collected by electrodes.
GaAs the band-gap is close the optimum value (Eg = 1.43eV, max η = 28%). In
addition, small volume laboratory c-Si devices are reported to reach an efficiency closer
to the upper bound limit (15); for single junction a-Si:H thin film solar cells, the gap
between practical efficiency values (about 9.5%, (16)) and the theoretical upper bound
limit (above 25%) is larger, due to the poor material quality and to the limited light
trapping scheme effectiveness.
2.5
Effect of temperature and loss mechanisms on main
figures of merit of a solar cell
2.5.1
Effect of temperature
The temperature may significantly vary during the operation of a photovoltaic device.
Despite the short-circuit current is not significantly affected by change in temperature,
it is worth noting that the absorption coefficient is temperature-dependent (17). In
addition, the absorption coefficient is doping dependent (18), especially within the
30
2.5 Effect of temperature and loss mechanisms on main figures of merit of
a solar cell
Figure 2.11: Dependence of the conversion efficiency η on the band-gap energy Eg for
direct illumination without external reflectance of the top interface, calculated assuming the
spectrum AM1.5G (1000W m−2 ) at T=300K. The saturation current is chosen in agreement
with the lower bound limit proposed by M.A. Green in (1). All photo-generated electronhole pairs are assumed to be collected by electrodes.
infrared region of the spectrum and, in case of a-Si:H, it is strictly dependent on the
dilution of silane by hydrogen in plasma deposition (19).
For this reason the short circuit current is expected to increase slightly with increasing temperature. The dependence of the figures of merit on temperature is particularly
interesting in concentrator solar cells (CPV)(20), for which, in addition to the change
of temperature due to the environmental conditions, large current densities lead to
self-heating effects.
However, the impact of temperature on VOC is not negligible. M.A. Green in (1)
reports the following approximated expression for the change in open circuit voltage as
consequence of a change in temperature:
dVOC
=−
dT
VG0 − VOC + γ
T
kT
q
,
(2.78)
which predicts a decrease (approximately linear) in VOC with increasing temperature (a typical value of
dVOC
dT
for c-Si solar cells is about −2.3mV /C). In eq. 2.78 γ is
a temperature-dependent parameter.
31
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
2.5.2
Optical losses
Main optical losses include the external front reflectance (although the front reflectivity
of bare silicon is significantly reduced by texturing and by anti-reflection coating layers
as shown in section 2.8), the front contact shadowing (the impact of the solely fingers
on the photo-generated current density is typically around 5 percent if realistic front
contact pitches -for instance 2mm- and 100µm-wide fingers are assumed) and the light
trapping mechanism limitations. The presence of a bus bar (Fig. 2.12) additionally
lowers the photo-generated current -and hence the short circuit current-: state-of-the
art commercial c-Si solar cells featuring a side of 156mm exhibit three bus bars 2mmwide, therefore the ratio of the bus bars total area to the cell area is about 3.8%.
Figure 2.12: Contact scheme for the front surface of a silicon solar cell. Bus bars and
fingers are aimed at delivering current to the external circuit.
2.5.3
Open circuit voltage and FF losses
VOC is strongly affected by recombination losses, especially those at interfaces and at
bulk level. The back surface field (BSF), by introducing a built-in electric field at the
p-p+ transition, helps to separate carriers photo-generated by large-wavelength photons
close to the contacted back-surface and hence leads to an increase of VOC . However,
32
2.5 Effect of temperature and loss mechanisms on main figures of merit of
a solar cell
Figure 2.13: Silicon solar cell featuring the front surface textured by pyramids.
due to the large doping concentration of the BSF diffusion, the Auger recombination
plays a crucial role in determining the trade-off between the reduced recombination
losses at the back surface and the increase of losses at bulk level. The macroscopic
effect of the introduction of a BSF is the lower effective back surface recombination
velocity with respect to that of the cell without any BSF. But the VOC is also limited
by recombination losses in the depletion region (the contribution to the dark saturation
current by the second diode in the equivalent circuit of Fig. 2.6). Several techniques,
like the passivation of front surfaces and the rear point contact scheme (which will be
discussed in chapter 4), are helpful to reduce the recombination losses and hence to
enhance the VOC .
Starting form the two-diode circuit model of a solar cell, it is possible to reframe
the expression of the I-V characteristic by adopting an effective ideality factor of the
diode:
J = JSC − J0EF F
qV
A
nEF F kT
−1 .
e
(2.79)
It is worth noting that the effective ideality factor nEF F varies within the range 1−2
33
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
depending upon the value of the current. The value of nEF F affects the FF; following
VOC
the approach of M.A. Green (1), for normalized voltages vOC = q nEF
> 10,
F kT
FF =
vOC − ln(vOC + 0.72)
.
vOC + 1
(2.80)
However the main contribution to FF is given by parasitic resistive losses like the
series resistance RS and the shunt resistance RSH . The former contribution is due to
the front contact resistance (fingers, bus bars) as well as to the back contact one (in
uniformly back contacted solar cells the latter contribution to the series resistance is
typically negligible, but it may be significant if the metallization fraction -defined as the
ratio of the contacted area to the total area- is significantly smaller than unity, like in
case of rear point contact solar cells (21). The shunt resistance (which is ideally infinite)
is caused by the current leakage across the p-n junction at the edge of the device, by
crystal defects and by precipitates of impurities. In order to quantify the impact of the
parasitic resistance on FF, M.A. Green (1) proposed the following models:
RS
F F = F F0 1 −
.
RCH
(2.81)
(vOC + 0.7) F F0
.
F F = F F0 1 −
vOC
rSH
(2.82)
and
where F F0 denotes the ideal FF (when RS = 0 and RSH = 0), RCH =
a characteristic resistance of the solar cell and rSH =
RSH
RCH
VOC
ISC
is
is the normalized shunt
resistance. The previous expressions of the FF are valid under the assumption that
vOC > 10, rS < 0.4 and rSH > 2.5.
The effects of the series resistance and of the shunt resistance on FF are shown in
Fig. 2.14 and Fig. 2.15, respectively.
2.5.4
Accounting for contact resistances
The total parasitic resistance of a solar cell is given by the sum of several contributions (Fig. 2.16). The contact resistance of the metal-semiconductor interfaces at rear
(Rbsm ) and at the front surface (Rf sm ) should be accounted for. That of ohmic contact
is an acceptable approximation in case of a metal directly deposited onto a highly-doped
34
2.5 Effect of temperature and loss mechanisms on main figures of merit of
a solar cell
Figure 2.14: Effect of the series parasitic resistance on the I-V characteristic of a solar
cell. The current is normalized to the short-circuit current density JSC .
Figure 2.15: Effect of the shunt parasitic resistance on the I-V characteristic of a solar
cell. The current is normalized to the short-circuit current density JSC .
35
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.16: Sketch of a simple solar cell with the lumped-element circuit to model the
series parasitic resistances.
semiconductor; a narrow space charge region is created and electrons can move through
such region by tunnelling. If the metal is deposited onto a lowly-doped semiconductor, the charge transfer occurs by thermionic effect (22). A theoretical relationship
between the specific contact resistivity ρc and the doping level of the semiconductor
at the interface has been proposed in (22). When tunnelling occurs, the contact resistance decreases with increasing doping, while, in case of thermionic effect, the contact
resistance is dependent only on the barrier height. In case of a front contact finger,
if LT denotes the transport length and RSHT the sheet resistance of the contacted
semiconductor, a general expression for the contact resistance is:
√
Rf sm =
RSHT ρc
coth W
L
s
RSHT
ρc
!
,
(2.83)
where L and W are the finger length and width, respectively. If LT ≥ 2W , then
Rf sm
ρc
=
=
LLT
36
√
ρc RSHT
.
L
(2.84)
2.6 Light Trapping
In case LT ≤ 2W , the equation can be approximated by
Rf sm =
ρc
.
LW
(2.85)
It is worth noting that, if LT ≤ 0.5W the resistance does not depend on W .
The second contribution to the total series resistance is the contact finger resistance,
which can be expressed as (22):
1
L
Rf = ρm
,
3 HW
(2.86)
where H is the height of the finger and ρm is the resistivity of the metal.
The bus bar resistance, similarly to that of the finger, is expressed by (22):
Rf b =
1 LB
ρm ,
6 HB WB
(2.87)
where LB , HB and WB are the busbar length, height and width, respectively.
2.6
Light Trapping
In crystalline Si (c-Si) and in compound semiconductor photovoltaic technologies, the
cost of materials used to fabricate the device is one of the dominant contributions
to the total cost of the solar cell. The main challenge in such technologies is the
reduction of the cost of the device preserving the performance. The major consequence
of the reduction of volume of the semiconductor is the decrease of the amount of
absorbed photons from the external source. In case of normal incidence of the light
on a simple slab of silicon with finite thickness T , the absence of internal bottom
reflection leads to only one passage of the light within the absorber layer, therefore
the pathlength is equal to the thickness T . The single-pass absorption scheme is the
reference case adopted for the following discussion. In case of a two-pass light trapping
scheme, the pathlength is simply twice than that obtainable with only one passage of
the radiation. In order to enhance the absorption of photons within the volume of
the semiconductor, the probability of photon absorption has to be larger than that
resulting when a single passage of the incident photons occurs. A simple light trapping
scheme is implemented also in thick-wafer silicon solar cells: the back contact (the
aluminum plate) acts like a back reflector (even though its reflectivity is significantly
37
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
smaller than that of the ideal reflector (23)) and standard industrial solar cells feature
a textured front interface, which, in addition to the anti reflective coating (ARC) layer,
contributes to the reduction of the external front reflectivity as well as to the increase of
the internal reflectivity (at bottom and at top interfaces). Therefore multiple bounces
of light are expected within the absorbing layer, hence enhancing the probability of
photon absorption. Two kinds of light trapping schemes are reported in literature:
the randomizing or Lambertian light trapping and the geometrical light trapping. The
former exploits the features of its surface to randomize the direction of the reflected
light; the latter is an attempt to deviate the reflection angle from the normal direction
by means of regular geometric patterns on the surface.
In case of a Lambertian reflector, the reflected radiant intensity varies following
the rule of cos(θ), where θ is the angle of the scattered light (24). It is possible to
demonstrate that the pathlength is about 4n2 times that of the reference case (single
pass) (25), where n is the refractive index of the absorber material. This geometrical
limit of the light trapping is referred as the well-known Yablonovitch limit. In case
of crystalline silicon, for which n is approximately 3.5, 4n2 is 50. The pathlength is a
typical figure of merit of a light trapping scheme. The Yablonovitch limit represents
the optimum performance that can be obtained by a light trapping scheme in terms of
absorbance enhancement in the geometric optics regime for a structure exhibiting an
acceptance cone spanning a full hemisphere.
In case of crystalline silicon solar cells, in order to obtain a reasonable value of absorbance, the active layer thickness should be higher than 100µm. A simple calculation
shows that the optimal thickness for a crystalline Silicon cell without any light trapping
scheme is about 200µm. If a light trapping strategy is adopted, the optimal thickness
required can be consequently reduced. In Fig.2.17 it is possible to observe that the onepass absorption curve of the 200µm-thick silicon layer is comparable to that associated
to the textured 20µm-thick one, confirming that texturing enhances the absorption of
the silicon layer for a given thickness. However, this approach works for wafers featuring a thickness that is not comparable to the wavelengths of the radiation, because the
limit 4n2 has been derived under the geometrical optics approximation. Therefore the
Yablonovitch limit is not helpful to predict the enhancement of absorption for a thin
film solar cell and it is not possible to state which is the best light trapping scheme for
such kind of solar cells.
38
2.7 Geometrical light trapping
Figure 2.17: Theoretical prediction of the dependence of the absorbance of a crystalline
silicon slab featuring a finite thickness on the photon energy in case of single-pass scheme
and in case of a light trapping scheme by means of front surface texturing (pathlength
equal to 4n2 where n is the refractive index of silicon). The one-pass absorption curve of
the 200µm-thick silicon layer is comparable to that associated to the textured 20µm-thick
layer.
2.7
Geometrical light trapping
In case of geometrical light trapping the direction of the incident sunlight is predictable
by analytic formulae. The simpler geometrical light trapping scheme was proposed by
Redfield (2); it consists of an oblique rear surface (Fig. 2.19A) with limited slope to
take into account for the narrow escape cone from silicon. In addition, the slope of
the rear surface must be inverted because the thickness of the cell is supposed to be
limited. In the worst case, when the illumination is direct, only four passages of the
light within the absorber are expected. Depending upon the position of the reflection
on the rear surface, the number of passages may vary.
A more effective scheme has been introduced by grooves formed by (111) crystallographic plane orientation (Fig. 2.18). If the grooves are symmetric (Fig. 2.19B),
it can be easily demonstrated that the typical number of passages is two. In case of
asymmetry, due to a slight tilt (Fig. 2.19C), the number of passages may increase
significantly.
Three dimensional upright pyramids are formed by anisotropically etching (100);
39
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
this light trapping scheme is more effective than the two-dimensional groove. In (13) a
regular upright 3-D pyramid scheme (Fig. 2.20A) is compared to randomized geometry
of upright pyramids (Fig. 2.20B) and to the Lambertian surface. Random pyramids
lead to better absorption performance than the case of the regular pyramids. In addition, it is not possible to manufacture regular pyramids in real processes due to the
difficulties in controlling the exact geometry. Smith and Roghati (26) demonstrated
that an even better performance is obtained by using 3-D inverted pyramids. Other
schemes have been proposed providing higher performance for direct illumination and
less sensitivity to the angle of incidence (27).
Figure 2.18: Vertical grooves formed by [111] crystallographic plane orientation.
2.7.1
Coherent light trapping schemes for thin film solar cells
Mostly adopted coherent light trapping schemes for thin film solar cells exploit periodic
photonic crystals. Structures featuring size in wavelength regime typically cause strong
angular and wavelength selective scattering. On the contrary, only when the geometric
features of texturing structures are not comparable to the wavelength, a complete
randomization of light can be obtained. In thin film solar cells, like hydrogenated
amorphous Si (a-Si:H) and micro-crystalline Si (mc-Si) cells -which typically feature
textured SnO2 : F or ZnO2 : Al coated on glass substrate-, the geometrical approach
is less effective than that of thick devices. In addition, in thin film solar cells, the
front contact is typically made by transparent conductive oxide (TCO) to ensure an
electric contact and to allow the light to pass it through. However, TCO absorbs
part of the light. In order to enhance the absorption in solar cells featuring thin
substrates, Sheng et al. (28) proposed one (1-D) or two (2-D) dimensional metal
40
2.7 Geometrical light trapping
Figure 2.19: Cross sections of (A) light trapping scheme proposed by Redfield (2), (B)
symmetric vertical grooves and (C) asymmetrical grooves.
41
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.20: Three dimensional upright pyramids: (A) periodic scheme and (B) random
pyramids.
gratings as back reflectors for a-Si:H cells. Stiebig et al. (29) have experimentally
characterized 1-D rectangular grating structures in 450nm thick a-Si:H and 1µm thick
mc-Si:H cells, providing a comparison with the randomly textured substrates; however,
they reported an enhancement of absorption that is approximately half of that achieved
by random texturing. Feng et al. (30) and Bermel et al. (31) have investigated the
adoption of photonic crystals as back reflectors and discussed their optimization. In
particular, Bermel and co-authors considered 2µm-thick c-Si cells exhibiting 1-D and
2-D metal gratings, 1-D and 2-D dielectric gratings in combination with distributed
Bragg reflectors (32) and photonic crystals; they reported that, cells with 2-D dielectric
gratings with Bragg reflectors provide better performance with reference to the photonic
crystals (33). In Fig. 2.21, typically adopted photonic crystals in 2-D and 3-D schemes
are shown.
Plasmonics is another phenomenon potentially interesting for photovoltaic. When
light is incident and scatters from a metal structure featuring nanometric size, a cooperative electron quantized oscillation occurs leading to a coupling of the scattered
radiation into the waveguide modes exhibited by such nanometric structures. The absorption can be strongly enhanced by plasmonic effect resulting in a set of wavelengths
which, for a given metal nanoparticle, allows light to couple into scattering effect. For
instance, for Pd and Au the plasmonic wavelengths are 250nm and 525nm, respectively,
but their values is also dependent on substrate, waveguide and medium effects which
can shift the value of the plasmonic wavelength. Randomly arranged metal nanoparticles and plasmon crystals (periodic structures) are used to model the electric field
42
2.8 Antireflection coating layers
Figure 2.21: Photonic crystals in 2-D scheme (A) and 3-D scheme (B).
intensity inside structures by exploiting the plasmonic effect.
Agrawal (34) theoretically demonstrated that coherent optical schemes are not expected to provide better enhancement of the absorbance with respect to the geometrical
optics.
2.8
Antireflection coating layers
As mentioned in the previous section, the reduction of the external reflectance can be
achieved by means of a stack of one or more thin layers of different media (Fig. 2.22)
(AntiReflection Coating, ARC) or by surface texturing, which, by exploiting multiple
reflections and different scattering angles of the light, allows most of the light to be
absorbed in the cell. In this section the ARC option is discussed.
An electromagnetic wave exhibits a phase shift of λ/2 upon entry into an optically
denser medium. If the optical path (i.e. the product of refractive index n and layer
thickness L) is equal to a quarter of the wavelength (nL =
λ
4 ),
then the destructive
interference is achieved, which means that light of such wavelength featuring normal
incidence is completely extinguished.
According to Fresnel’s formula, the reflection factor is:
R=
r12 + r22 + 2r1 r2 cos(2θ)
.
1 + r12 r22 + 2r1 r2 cos(2θ)
(2.88)
where θ is the incidence angle; if n0 denotes the refractive index of the uppermost
layer (air or glass), n1 denotes the refractive index of the antireflection layer and n2
43
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.22: Sketch of a simple anti reflection coating layer (ARC) featuring one slab
with refractive index n1 . The refractive indices of the superstrate and substrate are n0 and
n2 , respectively.
the refractive index of the silicon,
r1 =
n0 − n1
.
n0 + n1
(2.89)
r2 =
n1 − n2
.
n1 + n2
(2.90)
θ=
2πn1 L
.
λ
(2.91)
Finally,
In order to minimize the reflection (at θ = 0),
RM IN =
n21 − n0 n2
n21 + n0 n2
2
.
(2.92)
The reflection is zero when
n1 =
√
n0 n2 .
(2.93)
In Fig. 2.23 the reflectance of bare crystalline silicon is compared to those of a
silicon substrate coated by a single silicon dioxide (SiO2 ) 70nm-thick layer, by a single
70nm-thick titanium dioxide (TiO2 ) layer and by a stack of MgF2 (110nm-thick) and
44
2.9 Standard crystalline solar cells manufacturing technology
TiO2 (70nm-thick) layers. The latter ARC is that minimizes the reflectance within the
considered range of wavelength (300-1200nm). The real and the imaginary parts of
the refractive index of the silicon and of the other coating materials are shown in Fig.
2.24A and Fig. 2.24B, respectively.
Figure 2.23: Examples of ARCs: calculated reflectance.
In conclusion, by properly designing the ARC, it is possible to enable the destructive interference between reflected and incident light, eliminating reflection at both
interfaces; however this technique works only for a single wavelength with direct incidence, so it is not really effective over most of the spectrum. ARCs exhibiting more
layers (typically two or three) may be more effective within extended portions of the
spectrum, but the ideal coating would have a continuously graded refractive index.
2.9
2.9.1
Standard crystalline solar cells manufacturing technology
Wafer preparation
Despite mono-crystalline float zone (FZ-Si) material allows high crystalline perfection
and low contamination level of impurities leading to longest post-processing SRH life-
45
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.24: Refractive indices and extinction coefficients of the crystalline silicon (A)
and of SiO2 , TiO2 and MgF2 (B) used to calculate the reflectance of Fig. 2.23.
times (in the millisecond range) and Magnetic Czochralski (MCz) material exhibits a
good trade-off between cost and performance (35), industrial solar cells commonly use
Czochralski (Cz-Si) wafers because of their availability. The main drawback of Cz-Si
wafers is the presence of high concentration of oxygen that limits the carrier lifetime
(36). Commercial devices widely use multi-crystalline (mc-Si) substrates which, in
addition to crystal defects, such as grain boundaries and dislocations, exhibit higher
defect levels of metallic impurities because of lower segregation to the melt during the
faster solidification process leading to lower lifetimes. Lifetimes typically change during the cell fabrication process, in fact contamination typically occurs especially during
high-temperature steps like furnace cleaning. The integration of gettering steps (37)
in the fabrication flow is helpful in order to reduce the contamination leading to final
substrate lifetimes of industrial mc-Si cells within the range 1-10µs.
In (38) the gettering obtained with phosphorus and aluminum diffusions for emitter
and BSF layers is investigated. Diffusion techniques for p-type doping with P OCl3 and
P H3 leave a dead layer of electrically inactive phosphorus near the surface leading to
a reduced ultra-violet (UV) response of the cells in case such layer is not etched away
(39).
Techniques like sputtering, vacuum evaporation or screen printing allow the aluminum deposition on silicon; the annealing over the eutectic temperature forms a liquid
Al-Si layer which is helpful to segregate impurities because of their enhanced solubility
46
2.9 Standard crystalline solar cells manufacturing technology
(40) leading to bulk lifetime improvement. Bulk passivation by means of hydrogen is
another techniques to improve material quality (41). Industrial cells use boron-doped
substrates mainly for practical motivations (42). Intrinsic substrates show effective
lifetimes close to those expected in presence of the solely Auger recombination mechanism (the well-known Auger limit), but also large resistivity (as shown in chapter 4,
the parasitic series resistance affects the Fill Factor, therefore one way to improve the
performance is just to decrease the base resistivity); on the other hand, higher doping
concentration leads to larger SRH recombination rates.
Industrial cells are fabricated with substrate doping concentration values within
the range 1015 cm−3 -1016 cm−3 . The optimum substrate thickness is chosen depending
upon optical considerations ( light trapping scheme, light absorption properties of the
semiconductor) as well as electric parameters, like the carrier lifetimes. As the wafer
becomes thinner, the light trapping strategy becomes fundamental in order to maintain
acceptable levels of the photo-generated current density. If the diffusion length is longer
than the thickness, the most important issue is the surface recombination at the back
contact (as shown in chapter 4) which plays a crucial role on the open circuit voltage.
The thickness of current standard industrial silicon solar cells is within the range 180280µm. The diffusion length of Cz-Si, MCz and Fz-Si industrial cells is typically larger
than the standard thickness.
2.9.2
Front contact and emitter design
Metal grids are used at the front face to collect the photo-generated carriers. Standard
schemes of metal grids for mono-crystalline and mc-Si solar cells include bus bars and
fingers as shown in Fig. 2.25. In order to reduce the parasitic series resistance and
in particular that associated to the front contact, one way is to increase the contacted
emitter surface. The front contact resistance depends also on the emitter doping concentration at interface (43). However, a trade-off between parasitic series resistance and
the shadowing effect caused by fingers and busbars (which limits the photo-generated
current) is carefully considered in the design of the cell. Laboratory fabricated cells
exploit photolithographic techniques and evaporation to form narrow metal fingers (1020µm-wide) with Ti/Pd/Ag structures to lower the contact resistance. Mass production
devices are mostly based on screen printed Ag pastes (44) resulting in 100-200µm-wide
fingers.
47
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
Figure 2.25: Homogeneous emitter (HE) solar cell. Under the contacted surface and
between fingers the same emitter doping profile is adopted. LF F , HM ET and WM ET denote
the front finger length, height and width, respectively. WF P is the distance between front
fingers.
Figure 2.26: Selective emitter (SE) solar cell. A heavily doped emitter doping profile
(n++ ) is present under the contacted surface and a lowly-doped profile (n+ ) between fingers. LF F , HM ET and WM ET denote the front finger length, height and width, respectively.
WF P is the distance between front fingers.
48
2.9 Standard crystalline solar cells manufacturing technology
Under the metal lines, the substrate must be heavily doped to make the contact selective. However, high doping concentration leads to large recombination losses (Auger
and trap-assisted) as well as to significant surface recombination losses. On the other
hand, lowly doped emitters lead to large emitter resistance and hence to performance
degradation. Lowly doped emitter exhibiting a sheet resistance up to 100Ω/square can
be contacted (44). Standard industrial emitters feature sheet resistance values within
the range 50-70Ω/square. The homogenous emitter (Fig. 2.25) scheme is characterized by the same emitter doping profile between front fingers and under the contacted
portion of the front side of the cell. The emitter should be deep enough (0.4-1µm) to reduce the recombination of the photo-generated carriers at front interface and to reduce
the emitter resistance. Advanced emitter schemes exhibit a double-diffused emitter (or
selective emitter) as described in chapter 4 and as it shown in Fig. 2.26; according
to this scheme, a deep and heavily-doped diffusion is adopted under the metal finger
to ensure a good ohmic contact. However, very highly-doped regions exhibit a dead
layer due to the presence of precipitates resulting in a degraded collection efficiency
of short-wavelength light (which leads to photo-generation close to the front surface).
Heavy phosphorus diffusions produce very effective gettering. Selective emitters can be
fabricated by adopting screen-printing techniques (44).
In literature other advanced solutions concerning metallization schemes featuring
combination of pastes aimed at reducing the specific contact resistivity are reported
(45); these techniques, in addition to reduce the manufacturing costs, enhance the
aspect ratio of the fingers leading to reduced contact shadowing effect and to low
metallization fraction with the purpose of reducing the contacted area (the trade-off
is between the reduction of recombination losses at the metal-silicon interface and the
increase of the contact parasitic resistance).
2.9.3
Base and rear surface
The base contact of industrial solar cells is usually made by printing and subsequent
firing of aluminum with Ag-conducting paste. The p+ diffused layer (BSF, back surface
field, Fig. 2.25) is helpful in order to reduce the recombination in the base of the device,
as previously explained. The BSF is more effective for relatively thin wafer cells for
which the carrier diffusion length is comparable to the base thickness. Rear point
contact schemes like that of Passivated Emitter and Rear Cell (PERC) (13) with rear
49
2. SOLAR CELL DEVICE PHYSICS AND TECHNOLOGY REVIEW
surface passivation, which will be extensively described in chapter 4, are aimed at
reducing the effective surface recombination velocity at the back of the solar cell. In
addition, the presence of a local p+ BSF diffusion may be restricted to point contacts
as in Passivated Emitter Rear Locally diffused (PERL) cells (13). However, the main
drawback in reducing the metallization fraction of the back contact is the increase of
the parasitic series resistance, due to the spreading base resistance and to the back
contact resistance. PERC and PERL solar cells allow to reach the record efficiency for
silicon solar cells (up to 25%). As general remark, the optimization of the rear section
is dependent on the emitter configuration; once the emitter is optimized, a subsequent
optimization of the rear is performed.
50
3
Numerical Simulation of Solar
Cells
In the first part of this chapter a review of the simulation methodology by means of
numerical Technology Computer-Aided Design (TCAD) is presented. In particular, the
approach used to calibrate the physical models adopted in the numerical modeling is
described. In the second part, a basic theory of electromagnetic wave propagation is
proposed in order to introduce the main approaches to optical simulation of solar cells.
In particular, the main limitations and the applicability of the Finite Difference Time
Domain (FDTD) and of the raytracing methods are discussed, motivating the need to
adopt more computationally efficient tools to solve the optical problem in modeling of
advanced solar cells.
3.1
Introduction to numerical solution of semiconductor
device equations
In this work numerical simulations have been performed by using the Sentaurs TCAD
by Synopsys (46), a general purpose finite element method simulator for semiconductor
devices. Solar cells are simulated by adopting the drift-diffusion transport model described in chapter 2. In the following, only isothermal analyses are discussed. The three
governing equations for charge transport in semiconductor devices are the Poisson equation (eq. 2.27), the electron and hole continuity equations (eq. 2.26) and the transport
equations (eq. 2.23); they provide the general framework for device simulation. The
51
3. NUMERICAL SIMULATION OF SOLAR CELLS
drift-diffusion model is widely used for the simulation of carrier transport in semiconductors featuring low-power density; it is one of the simplest charge transport models
since it does not introduce any independent variable in addition to the electrostatic
potential and to the carrier concentrations. The drift-diffusion (D-D) model has been
considered adequate for most technologically feasible devices in past decade. However,
the D-D approximation becomes less accurate for smaller feature sizes like nano-metric
field effect transistors (FET), but in case of solar cells, it can be successfully applied.
3.1.1
Drift-Diffusion transport equations
The current densities can be expressed by means of the quasi-Fermi levels φn and φp
as:
Jn = −nqµn ∇φn ,
Jp = −pqµp ∇φp .
(3.1)
Under the Boltzmann transport theory approximation:
n = ni,ef f e
n = ni,ef f e
q(ψ−φn )
kTL
(3.2)
q (ψ−φp )
− kT
L
(3.3)
where TL is the lattice temperature and ni,ef f is the effective intrinsic carrier concentration:
ni,ef f =
E
p
− g
NC NV e 2kTL
(3.4)
and NC , NV are the conduction band and the valence band effective densities of
states, respectively.
The quasi-Fermi potential can be therefore rewritten as:
kTL
φn = φ −
ln
q
kTL
ln
q
φn = φ +
52
n
(3.5)
ni,ef f
p
ni,ef f
(3.6)
3.1 Introduction to numerical solution of semiconductor device equations
where φ is the electrostatic potential. A conventional expression for current densities
is:
Jn = qnµn En + qDn ∇n,
Jn = qpµp Ep − qDp ∇p.
(3.7)
where the effective electric fields En,ef f and Ep,ef f are
En,ef f
kTL
ln(ni,ef f )
= −∇ φ +
q
Ep,ef f
3.1.2
kTL
= −∇ φ −
ln(ni,ef f )
q
(3.8)
(3.9)
Electrical Boundary Conditions
By default, if no electrodes are specified, on all boundaries of the device the following
reflective ideal Neumann boundary conditions are assumed:
Jn · n̂ = 0,
Jp · n̂ = 0
(3.10)
where n̂ denotes the normal to the surface.
At electrodes for which an Ohmic contact is set, charge neutrality and equilibrium
are assumed; if n0 and p0 denote the equilibrium concentration of carriers,
n = n0 ,
p = p0
n0 p0 = n2i,ef f
(3.11)
(3.12)
When a finite recombination velocity ve (vh ) for electrons (holes) is specified, the
boundary condition at electrodes becomes:
Jn · n̂ = qve (n − n0 )
(3.13)
Jp · n̂ = −qvh (p − p0 )
(3.14)
53
3. NUMERICAL SIMULATION OF SOLAR CELLS
3.1.3
Effective Intrinsic Carrier Concentration and Band Gap structure
The intrinsic carrier density ni is a fundamental quantity in semiconductor physics
which plays a significant role in the simulation of silicon solar cells due to the presence of optical excitation. In fact, ni has a strong impact on the recombination losses
which limit the ultimate conversion efficiency of solar cells. Therefore, in order to
perform realistic simulations of solar cells, ni must be accurately determined. Prior
to 1990, the most commonly adopted value for crystalline silicon at T = 300K was
ni = 1.45 × 1010 cm−3 which was affected by significant deviations from experiments.
In 1991 Sproul and Green (47) measured the intrinsic concentration providing the value
ni = 1 × 1010 cm−3 , which is still the most widely accepted value within the PV community. However, recent theoretical investigations (48) reinterpreted Sproul and Green
experiment, demonstrating that their measurements were influenced by the band-gap
narrowing (BGN), although at relatively low doping densities. A theoretical study
based on the finite-temperature full random-phase approximation model of BGN proposed in (49) revealed that ni = 9.65 × 109 cm−3 for undoped silicon, consistently with
the experiments by Misiakos and Tsamakis (50) performed by capacitance measurements.
The temperature dependent mobility bang gap can be modeled as (51):
Eg (T ) = Eg (0) −
αT 2
T +β
(3.15)
where α and β are material dependent parameters and Eg (0) denotes the band-gap
energy at T = 0K, which can be expressed as:
Eg (0) = Eg0 + ∆Eg,0
(3.16)
where ∆Eg,0 and Eg0 are determined by the particular band-gap narrowing model
used.
At a given temperature T , the effective band-gap is
Eg,ef f (T ) = Eg (T ) − EBGN
(3.17)
EBGN = ∆Eg0 + ∆EgF ERM I
(3.18)
54
3.1 Introduction to numerical solution of semiconductor device equations
The parameter ∆EgF ERM I is a correction factor to account for the adopted carrier
statistics (46).
A widely used model to account for band-gap narrowing in n-type silicon is that of
del Alamo (52)(53)(54)(55)(56), according which:
(
∆Eg0
=
EREF
ln
NT OT
NREF
0
if NT OT ≥ NREF ,
(3.19)
if NT OT < NREF .
where NREF = 7×1017 cm−3 and EREF = 18.7×10−3 eV . In eq. 3.19 NT OT denotes
the total doping concentration.
Models like Bennett-Wilson (57), del Alamo, Slotboom (58)(59)(60)(61) and JainRoulston (62) are all doping-induced band-gap narrowing models. They do not depend
on free carrier concentration. However, high carrier concentration due to optical excitation or high electric field injection may lead to band-gap narrowing effect (plasmainduced band-gap narrowing).
In order to correctly reproduce the increase of ni occurring at large doping density
and to account for large carrier injection due to optical excitation, the value of the
parameters of Schenk band-gap narrowing model with Fermi-Dirac statistics can be
properly set to obtain the value ni = 9.65 × 109 cm−3 at low doping density.
In Fig. 3.1 the effective intrinsic carrier density accounting for band-gap narrowing is reported as function of the total doping density for different models (BennettWilson, Slotboom, del Alamo, Schenk). The default implementation of Bennett and
Schenk models provides at very low doping levels ni = 1.075 × 1010 cm−3 , whereas the
Slotboom and del Alamo models employ a correction factor, leading to higher values,
namely 1.180 × 1010 cm−3 and 1.412 × 1010 cm−3 , respectively. Fig. 3.1 reports also the
values of the effective intrinsic concentration calculated by the revised Schenk banggap narrowing model that provides very good agreement with the experimental data
reviewed by Altermatt et al. in (48). In Sentaurus it is possible to specify the total
doping-dependent band-gap narrowing by using a table, in which the band gap narrowing for acceptors and donors is specified separately; the band-gap narrowing is then
calculated as the sum of both contributions. A drawback of this approach to model the
band-gap narrowing, is the lack of the dependence on carrier concentration.
55
3. NUMERICAL SIMULATION OF SOLAR CELLS
Figure 3.1: Effective intrinsic carrier density accounting for band-gap narrowing (BGN)
as function of the total doping density for different models, including the revised BGN
model adopted for numerical simulations in Sentaurus TCAD.
3.1.4
Carrier Mobility Models
Electrons and holes are accelerated by electric fields, but their momentum is reduced
because of various scattering processes like from phonons, impurity ions, other carriers
and surfaces. However, in equations 2.21 all these microscopic effects are accounted
by introducing the carrier mobilitiy in the transport equations. The carrier mobility
is strongly linked to local electric field, lattice temperature and doping concentration.
In presence of low electric fields, carriers are almost in equilibrium with the lattice
and the mobility depends on phonon and impurity scattering which leads to mobility
degradation. In case of high electric field intensity the carrier mobility is reduced
because the carriers -that gain energy- have more probability to take part of more
scattering processes. In fact, the mean drift velocity saturates at a constant velocity
vsat with increasing field intensity. The saturation velocity is strongly dependent on
lattice temperature. Modeling mobility in bulk material involves the knowledge of the
low field mobility for a given doping and lattice temperature as well as the dependence
of the saturation velocity on the lattice temperature. In addition, surface scattering,
carrier-carrier scattering as well as quantum mechanical size quantization effects occur
56
3.1 Introduction to numerical solution of semiconductor device equations
in inversion layers.
The Philips unified mobility model (63) describes the temperature dependence of
the mobility accounting for electron-hole scattering, screening of ionized impurities by
charge carriers and clustering of impurities. According to the Philips unified mobility
model, a first contribution to the carrier mobility is represented by phonon scattering
µP HON ON and by a second contribution µSCAT T (46), which accounts for all other bulk
scattering mechanisms like those of free carriers and ionized dopants. Such contributions, for electrons (e) and holes (h), can be accounted for by the following expression:
1
1
1
+
,
=
µc,ef f
µc,P HON ON
µc,SCAT T
c = e, h
(3.20)
According to Matthiessens rule, for a given temperature T :
µc,P HON ON (T ) =
3.1.5
AX
µM
c
T
300K
θc
,
c = e, h
(3.21)
Recombination mechanisms
As discussed in chapter 2, carrier generation-recombination is the process through
which the semiconductor material attempts to return to equilibrium conditions after being disturbed from it (applied bias, device under illumination). The modeling
of generation-recombination processes should account for phonon transitions, Auger
transitions, surface recombination and impact ionization. Recombination through deep
defect levels within the gap is known as Shockley-Read-Hall (SRH) recombination.
When Maxwell-Boltzmann statistics is adopted,
SRH
RN
ET =
τp
np − n2i,ef f
ET
E
− kTT
kT
n + ni,ef f e
+ τn p + ni,ef f e
(3.22)
where ET is the difference between the defect level and intrinsic one (ET = 0 for
crystalline silicon).
The doping dependence of the SRH lifetimes is accounted through the Scharfetter
relation:
τc (N ) = τc,min +
τc,max − τc,min
γ ,
N
1 + NREF
c
57
c = e, h
(3.23)
3. NUMERICAL SIMULATION OF SOLAR CELLS
The temperature dependence is expressed by
τc (T ) = τc,0 +
T
300K
αc
,
c = e, h
(3.24)
The minority carrier lifetime in standard boron-doped Czochralski silicon (B-Cz-Si)
is strongly degraded under illumination or carrier injection due to the metastable Czspecific defects. Nevertheless, the defect concentration and consequently the lifetime
degradation can be significantly reduced by optimized high-temperature annealing steps
using conventional or rapid thermal processing as described by Glunz et al. in (3) in
which, the measured doping dependence of the bulk lifetime in standard B-Cz-Si is
described by a simple empirical fitting. By default, Sentaurus simulator assumes the
following values for p-type silicon: τmin = 0µs, τmax = 10µs, γ = 1 and NREF =
1016 cm−3 . In order to realistically model the carrier lifetimes in the boron-doped base
of silicon solar cell, the parameters in eq. 3.23 have been modified as: τmin = 0µs,
τmax = 134ms, γ = 1.407 and NREF = 1014 cm−3 ; the empirical fitting of experimental
data for standard B-Cz-Si reported in (3) is therefore reproduced. Fig. 3.2 reports the
bulk lifetime versus boron doping concentration for the empirical fitting suggested by
Glunz et al. (3) and for the Sentaurus default Scharfetter relation. The bulk lifetime
curve provided by the modified relation is also reported in Fig. 3.2, showing a good
agreement with the fit to experimental data proposed by Glunz.
At interfaces the SRH model is extended by
SRH,surf
RN
ET
np − n2i,ef f
=
ET
E
− kTT
−1
−1
kT
Sh n + ni,ef f e
+ Se p + ni,ef f e
(3.25)
The recombination velocities Se and Sh depend on the concentration of dopants at
the surface (64) as well as on process parameters that determine the interface quality.
The doping dependence of surface recombination velocities is expressed by:
γc Ns
,
Sc = S0,c 1 + SREF,c
NREF
c = e, h
(3.26)
where Ns denotes the doping concentration at interface, and γc , NREF and SREF
are tuneable parameters of the model.
58
3.1 Introduction to numerical solution of semiconductor device equations
Figure 3.2: Bulk lifetime versus boron doping concentration for the empirical fitting
suggested by Glunz et al. (3) and for the Sentaurus default Scharfetter relation. The bulk
lifetime curve provided by the modified relation is also reported.
3.1.6
Auger recombination
In Sentaurus, the band-to-band Auger recombination rate is expressed by
AU G
2
RN
ET = np − ni,ef f [Ce (T, n)n + Cp (T, p)p]
(3.27)
where the temperature-dependent Auger coefficients Ce (T ) and Ce (T ) are given by
Ce (T, n) =
C1,e + C2,e
Ch (T, p) =
C1,p + C2,p
T
T0
T
T0
+ C3,e
+ C3,p
T
T0
2 !
T
T0
2 !
HeIN J (n)
(3.28)
HpIN J (p)
(3.29)
Details about the coefficients HeIN J (n) and HpIN J (p), which account for effects due
to large carrier injection, are reported in (46).
59
3. NUMERICAL SIMULATION OF SOLAR CELLS
3.2
Numerical methods
The partial differential equations (PDEs) eq. 2.23, eq. 2.27 and eq. 2.26 are discretized
by using a numerical grid. A review of the most widely adopted methods of discretization of PDEs for the device simulation is presented in (65). Burgler et al. in (66)
discuss the box discretization scheme to solve numerically the Poisson and the continuity equations. A symmetry element of the device is discretized by means of division of
the spatial simulation domain in boxes. Rectangular or more effective triangular boxes
are typically used for 2-D devices.
Figure 3.3: Single box centered at node i-th for the numerical solution of PDEs.
For an arbitrary i-th vertex (Fig. 3.3) a single box Ωi is constructed by choosing
box boundaries that are perpendicular bisectors of the lines binding adjacent nodes. In
this way it is possible to cover the whole symmetry element. If a PDE is given in the
form
∇Γ(r) = W (r),
(3.30)
where Γ(P ) is a vector field and W (P ) is a scalar field, such PDE is integrated over
the single box volume. In case of the Poisson equation, the electric field is conservative,
60
3.2 Numerical methods
therefore it is possible to apply the Gaussian Theorem:
Z
Z
Z
[∇Γ(r) − W (r)] dτ =
Ωi
∇Γ(r)dn̂(r) −
∂Ωi
W (r)dτ = 0
(3.31)
Ωi
where n̂ is the normal vector to the box boundary. Therefore the discretized PDE
reads:
X
Γij dij − Wi τi = 0
(3.32)
j6=i
Γij is the projection of the vector field Γ onto the edge lij (3.3) and dij is the length
of the normal bisector between edges j and i. Wi and τi denote the value of the scalar
field W in node i and the volume of the element Ωi , respectively. Applying this scheme
to the Poisson equation,
X
Eij dij − ρi τi = 0
(3.33)
j6=i
or, by expressing the electric field E as a difference of electrostatic potential between
nodes i and j along lij :
X dij
j6=i
lij
(φi − φj ) − Vi (pi − ni + Ni ) = 0
(3.34)
Heiser (67) proposed an approach to discretize the continuity equations:
X dij
j6=i
lij
µnij [nj B(φj − φi ) − ni B(φi − φj )] − Vi (ri − Gi ) = 0,
where B denotes the Bernoulli function B(x) =
x
ex −1
(3.35)
and the mobility µij is assumed
constant on the edge lij . Similarly for the continuity equation for holes.
The Poisson’s, the continuity and the transport equations are discretized in each
node of the grid mesh. The unknowns are the carrier concentrations n, p and the
electrostatic potential φ. The system of PDEs is then solved by the iterative Newton
method, which, at each computation step, calculates a new solution until a breaking
criterion is reached.
61
3. NUMERICAL SIMULATION OF SOLAR CELLS
3.3
3.3.1
Propagation of electromagnetic waves
From Maxwell equations to homogeneous Helmholtz equation
A portion of space with a given distribution of current density Ji (r, t) is considered.
The vector r defines the position and t is the time. The electromagnetic field produced
by Ji (r, t) is given by the time and position dependent intensity of electric field E(r, t)
and by the intensity of magnetic field H(r, t) which satisfy the following system of
differential equations:
(
∇ × H(r, t) = ∂D(r,t)
+ JC (r, t) + Ji (r, t),
∂t
∂B(r,t)
∇ × E(r, t) = − ∂t .
(3.36)
which are known as Maxwell equations. D(r, t) denotes the electric displacement
while its partial derivative with respect to the time is the displacement current density.
Jc (r, t) and B(r, t) are the conduction current density and the magnetic induction,
respectively. Jc (r, t), B(r, t) and D(r, t) describe the reaction of the material to the
electromagnetic field in terms of free charge transport, magnetic polarization and electric polarization of atoms and molecules, respectively. For a generic material we can
write:
D(r, t) = D [E(r, t), H(r, t), t]
(3.37)
B(r, t) = B [E(r, t), H(r, t), t]
(3.38)
JC (r, t) = J [E(r, t), H(r, t), t]
(3.39)
where D, B and J are integral-differential operators which can be non linear, in the
most general case. Eq. 3.37, eq. 3.38 and 3.39 are the constitutive relations of the
material describing the electrical response of various media to the electromagnetic field.
The trivial problem of the solution to the Maxwell equation can be simplified under
some assumptions. The first one is the condition of time-invariant medium, which
requires that the material does not change its electromagnetic parameters during an
arbitrary period of time and that its position never changes with respect to an observer.
These assumptions lead to the existence of operators which depends only on the position
62
3.3 Propagation of electromagnetic waves
r. If the material is nondispersive, time-invariant, linear and anisotropic, equations
3.37, 3.38 and 3.39 read:
D(r, t) = ε̄(r) · E(r, t)
(3.40)
B(r, t) = µ̄(r) · H(r, t)
(3.41)
JC (r, t) = c̄(r) · E(r, t)
(3.42)
where the third equation is the Ohm’s Law, ε̄(r), µ̄(r) and c̄(r) are called permittivity, magnetic permeability and conducibility, respectively; for a linear and anisotropic
medium they are second-order tensors, therefore, for instance, in case of the permeability,

 

µ11 (r) µ12 (r) µ13 (r)
H1 (r, t)
B(r, t) = µ̄(r) · H(r, t) = µ21 (r) µ22 (r) µ23 (r) = H2 (r, t) .
µ31 (r) µ32 (r) µ33 (r)
H3 (r, t)
(3.43)
where H1 , H2 and H3 are the scalar components of the vector field H and µ(r) is a
3 × 3 matrix of elements µij , i = 1..3,
j = 1..3. An anisotropic material is a lossless
or non-dissipative material if, in every point of the space, µ(r) and ε(r) are symmetric
tensors and c(r) is antisymmetric, or
ε̄T (r) = ε̄(r),
µ̄T (r) = µ̄(r),
c̄T (r) = c̄(r).
(3.44)
Finally, the medium is homogenous if the tensors in eq. 3.39, 3.38 and 3.37 are not
dependent on the position.
If H(r, t), D(r, t), Jc (r, t), Ji (r, t), B(r, t) and E(r, t) are sinusoidal functions of
time with the same angular frequency ω, the following complex vector field
T̃ (r, t) = Re T(r, t)eiωt
63
(3.45)
3. NUMERICAL SIMULATION OF SOLAR CELLS
can be introduced, in which T denotes one of the vector fields H(r, t), E(r, t),
D(r, t), JC (r, t), Ji (r, t) or B(r, t). By introducing the complex permittivity εc (r, ω)
defined by
εc (r, ω) = ε(r, ω) +
c(r, ω)
,
jω
(3.46)
it is possible to reformulate the Maxwell equations as (68):
∇ × H(r, ω) = iωεc (r, ω)E(r, ω) + Ji (r, ω)
∇ × E(r, ω) = −iωµ(r, ω)H(r, ω)
(3.47)
In case of normal and homogenous materials without sources, the Maxwell equations
read:
∇ × H(r, ω) = iωεc (ω)E(r, ω)
∇ × E(r, ω) = −iωµ(ω)H(r, ω)
(3.48)
Since it is possible to derive that for homogeneous materials (68)
∇ · H(r, ω) = 0,
(3.49)
σ 2 (ω) = −ω 2 µ(ω)εc (ω),
(3.50)
and by defining
the following equations
∇2 E(r, ω) − σ 2 (ω)E(r, ω) = 0
∇2 H(r, ω) − σ 2 (ω)H(r, ω) = 0
(3.51)
can be derived. Such equations are known as homogenous Helmholtz equations.
They are called homogenous since no charges are present in the space. On the contrary, in case of presence of electromagnetic sources, if the medium is homogeneous,
similar equations can be derived. The main advantage of the Helmholtz equations with
respect to the Maxwell equations is their easier integration in order to calculate the
electromagnetic field. However, eq. 3.51 is a 4th-order system of differential equations
while eq. 3.48 is a system of 2nd-order differential equations, therefore it is possible
to find solutions of eq. 3.51 that are not solutions of eq. 3.48. The eq. 3.51 can be
considered a reformulation of the Maxwell equations of the sinusoidal electromagnetic
field to describe the propagation of waves in homogeneous media.
64
3.3 Propagation of electromagnetic waves
3.3.2
Solution of Helmholtz equations for a normal homogenous medium
without sources
In the following, a portion of the space which does not include sources and that is
occupied by a passive, normal and homogenous medium (for which µ, ε and c are not
dependent on the position) is considered. In addition, the sinusoidal electromagnetic
field with angular frequency ω is supposed to be dependent only on the y-coordinate.
Therefore, from eq. 3.51, for each Cartesian component g(r) of the complex electromagnetic field:
d2 g(r)
− σ 2 g(r) = 0,
dy 2
(3.52)
σ 2 = −ω 2 µε + iωµc.
(3.53)
where
For a homogenous medium, µ > 0, > 0 and c ≥ 0. For a passive medium, if
σ = α + iβ denotes the intrinsic propagation constant of the material, then the general
solution of eq. 3.52 is
g(r) = G1 e−σy + G2 eσy
(3.54)
where G1 and G2 are arbitrary integration constants. Therefore the time-dependent
solution is:
g(r, t) = G1 e−αy cos(ωt − βy) = G1 e−αy cos [β (y − vp t)]
(3.55)
that represents a progressive sine wave propagating towards positive y with speed
vp =
ω
.
β
(3.56)
Such wave propagates with an attenuation factor exp(−αy) where α is the absorption coefficient (already defined in chapter 2) or the intrinsic attenuation coefficient of
the medium.
The phase of the sinusoidal wave is defined by φ = −βy, where β is the phase
constant (intrinsic) of the medium. If the surface on which the phase is constant is a
plane, eq. 3.55 represents a uniform plane wave.
65
3. NUMERICAL SIMULATION OF SOLAR CELLS
3.3.3
TE and TM plane waves
A more general solution of the Helmholtz equation
∇2 g − σ 2 g =
∂2g
∂2g ∂2g
+
+
− σ2g = 0
∂x2 ∂y 2 ∂z 2
(3.57)
is a linear combination of solutions of the form (68):
g(x, y, z) = g0 e−Ux x e−Uy y e−Uz z ,
where Uj ,
Ux2
+
Uy2
+
Uz2
(3.58)
j = x, y, z are arbitrary constants which must satisfy the condition
= σ 2 and u0 is an arbitrary integration constant.
If the position vector is expressed as r = xîx +y îy +z îz , the wave propagation vector
is defined as K = Kx îx + Ky îy + Kz îz . The most general solution of the Helmholtz
equations for the field intensity are therefore given by superposition of the following
vector fields
E(r) = E0 e(−K·r) ,
H(r) = H0 e(−K·r) ,
(3.59)
where H0 and E0 are constant complex vectors. In addition it is possible to write
K = K1 + iK2 and
K1 = kK1 kŝ1 ,
K2 = kK2 kŝ2
(3.60)
If ŝ0 = ŝ1 = ŝ2 then K = (α + iβ)ŝ0 = σŝ0 , therefore
E(r) = E0 e(−σŝ0 ·r) ,
H(r) = H0 e(−σŝ0 ·r) ,
(3.61)
ŝ0 · H0 = 0,
(3.62)
and
ŝ0 · E0 = 0,
E0 = ηH0 × ŝ0
(3.63)
1
ŝ0 × E0 ,
η
(3.64)
H0 =
66
3.3 Propagation of electromagnetic waves
where
η=
σ
=
jωεc
r
µ
εc
(3.65)
is the intrinsic impedance of the homogenous medium.
In particular sˆ0 · r (the surfaces with same value of the phase -or wavefronts- are
planes) is constant and the eq. 3.61 represent plane waves with propagation direction
sˆ0 . In addition, on the wavefronts the intensity is constant; these waves are called
transverse electromagnetic waves or TEM waves. It is possible to demonstrate that
these solutions are not physically possible since they carry an infinite power along the
propagation direction. Their importance is only theoretical.
If the fields can be expressed as
E = E0 exp(−Kn1 ŝ1 · r)exp(−iKn2 ŝ2 · r)
(3.66)
H = H0 exp(−Kn1 ŝ1 · r)exp(−iKn2 ŝ2 · r)
(3.67)
where Knq , q = 1, 2 are arbitrary constants, the solution is still a plane wave, being
the locus of the points with the same phase (wavefront surface) still planes perpendicular to ŝ2 . In this case, the intensity of the fields is not constant on the wavefront;
however, on surfaces for which ŝ1 · r is constant, the vector field norms are constant
and such planes are normal to the direction of attenuation of the wave. Therefore
these waves are characterized by different direction of propagation and attenuation. In
particular if
E0 = E0 ŝ1 × ŝ2
(3.68)
then ŝ2 · E0 = 0 and H is elliptically polarized on the plane defined by ŝ1 and ŝ2 .
In addition, H exhibits always a component along ŝ2 . This waves is called transverse
electric (TE). Similarly for a transverse magnetic (TM) wave:
H0 = H0 ŝ1 × ŝ2
(3.69)
The most general plane wave is obtained by superposition of TE and of TM waves.
67
3. NUMERICAL SIMULATION OF SOLAR CELLS
3.3.4
Geometrical optics and raytracing
In the following, a portion of space without sources and featuring a normal medium
without losses is considered. Since a non-homogeneous medium is assumed, µ = µ(r)
and ε = ε(r).
In addition, the conditions that allows to obtain solutions of Maxwell equations for
which, for every point of the spatial domain, it is possible to define an arbitrary small
region with finite volume within which the field intensities satisfies equations similar to
eq. 3.61, eq. 3.62, eq. 3.63 and eq. 3.64, are discussed.
The goal is to determine solutions in the form
E(r) = E0 (r)exp [−β0 V (r)]
(3.70)
H(r) = H0 (r)exp [−β0 V (r)]
(3.71)
√
where V (r) is called iconal function and β0 = ω µ0 ε0 is the intrinsic phase constant
in vacuum. If the refractive index of the medium is defined as
s
n(r) =
µ(r)ε(r)
,
µ 0 ε0
(3.72)
the Maxwell equations can be reframed as:
(
n(r)
η(r) E0 (r)
− iβ10 ∇ × H0 (r) = H0 (r) × ∇V (r)
n(r)η(r)H0 (r) + iβ10 ∇ × E0 (r) = ∇V (r) × E0 (r).
(3.73)
If
kλ0 η(r)∇ × H0 (r)k kE0 (r)k
(3.74)
kλ0 ∇ × E0 (r)k kη(r)H0 (r)k
(3.75)
therefore
(
E0 (r) = η(r)H0 (r) ×
η(r)H0 (r) =
where λ0 =
∇V (r)
n(r)
2π
β0 .
68
∇V (r)
n(r)
× E0 (r)
(3.76)
3.3 Propagation of electromagnetic waves
It is possible to demonstrate that the eq. 3.76 represents a local plane wave of TEM
type with respect to the direction given by ∇V (r) (68). In order to make the eq. 3.76
valid, it is necessary to ensure that:
|λ0 ∇ · E0 (r)| kE0 (r)k
(3.77)
|λ0 ∇ · H0 (r)| kH0 (r)k
(3.78)
|λ0 ∇ε(r)| ε(r)
(3.79)
|λ0 ∇µ(r)| µ(r).
(3.80)
In conclusion, the electromagnetic field can be described in terms of local TEM
plane waves when its components and the intrinsic parameters of the medium are
not significantly dependent on the position within any spatial variation of coordinates
comparable to λ0 : it means that, the approximation introduced by eq. 3.70 and by
eq. 3.71, is valid when the size of the structures -that contributes to the definition of
the electromagnetic field distribution within the portion of interest of the domain- is
significantly larger than the wavelength of the radiation.
By solving for E0 (r) and E0 (r):
∇V (r) · ∇V (r) − n2 (r) E0 (r) = 0,
(3.81)
∇V (r) · ∇V (r) − n2 (r) H0 (r) = 0.
(3.82)
These equations are not coupled and do not allow non-banal solutions if
∇V (r) · ∇V (r) = n2 (r).
(3.83)
The eq. 3.83 is the fundamental equation of geometrical optics; it is a non-linear
partial differential equation which admits an infinite number of possible solutions. Each
69
3. NUMERICAL SIMULATION OF SOLAR CELLS
particular integral of such equation defines a wave according to the given distribution
of the refractive index. The related wavefront is described by the equation
V (r) = C0
(3.84)
where C0 is an arbitrary real constant.
The normal trajectories to the surfaces defined by eq. 3.84, for any value of the
real constant C0 , are called electromagnetic rays. For a given distribution of the refractive index n(r), the optical problem is solved by determining the rays by means of
a technique called raytracing.
The geometrical optics approximation is not valid when discontinuities of the medium
are assumed. This problem is solved by assuming that each ray, impinging on a surface
of discontinuity, acts like a local plane wave which behaves exactly as a uniform wave
plane exhibiting the same conditions of incidence in the whole surface. Therefore, rays
follow the reflection and transmittance rules defined by the Snell’s Law for the plane
waves (69).
Since intensity, direction and polarization are the sole information associated to
rays, raytracing is not suitable to describe phenomena such as interference and diffraction. The raytracer algorithm uses a recursive method which starts with a source ray
and builds a binary tree that tracks the transmission and reflection of the ray at interfaces between media featuring different refractive indices. At interfaces, incident rays
split into reflected and transmitted rays: the TE component of the polarization vector
maintains the same direction, whereas the TM component changes direction.
Raytrace extensions proposed in commercial simulators like Sentaurus (46) and Silvaco Atlas (70) allow to define special boundary conditions with in order to model
interfaces with constant reflectivity/transmittivity, multilayer ARC layers and diffusive
reflections. In (70), by default, the diffusive reflection features a Lambertian distribution (exhibiting constant luminance with respect to the angle). Gaussian or Lorenz
distributions can be chosen. The raytracer has been successfully adopted to calculate
the optical behaviour of a textured silicon wafer in (71).
The Monte Carlo ray trace algorithm (72) is helpful to simulate random textured
interfaces like those of thin film solar cells. The light scattering from rough interfaces
is determined by using probabilistic estimation through random number generation.
Each ray is traced in the direction of propagation until it scatters from an interface;
70
3.3 Propagation of electromagnetic waves
consequently, reflection and transmission probabilities are estimated by means of standard equations used to calculate transmittance and reflectance. The random number
generated by assuming a uniform distribution is compared against the transmission or
reflection probability in order to discriminate the sole ray to trace at each scattering
event. Interfaces may be described by angular distribution functions (ADFs), which
associate a value of intensity of the light to each scattering angle. Arbitrary ADFs
can be defined. The modified angle is used to calculate the transmission and reflection
probabilities as well as the transmission and reflection angles.
Figure 3.4: Thin-layer-stack boundary condition used to simulate ARCs with RayTracing
in combination with the TMM solver.
A thin-layer-stack boundary condition is helpful to simulate ARCs (Fig. 3.4). The
angle of incidence of the ray -with respect to the normal to the coating layer surface- is
passed as input to the Transfer Matrix Method (TMM) solver (46) which can calculate
the reflectance, the transmittance, and the absorbance for thin layers featuring parallel
interfaces. The angle of refraction is calculated by the raytracer according to Snell’s
law in order to ensure the phase matching.
According to Fig. 3.5, which shows the sketch of an infinitely spatially extended
planar interface between two lossless media of refractive indices n1 (for the incident
and reflected rays) and n2 (for transmitted rays), the incident, the reflected and the
transmitted angles are denoted by θi , θr and θt , respectively. The scattering of rays
71
3. NUMERICAL SIMULATION OF SOLAR CELLS
Figure 3.5: Reflection, transmission and refraction of plane waves at a planar interface.
from smooth interfaces is governed by the the Snell’s law of the specular reflection.
Therefore the transmitted and the reflected angles are linked to that of incidence by
the following equations:
θr = θi .
(3.85)
n1 sin(θi ) = n2 sin(θt ).
(3.86)
The amplitudes of the electric field for the TE and the TM polarizations are described by
TE
ER
=
n2 cos(θi ) − n1 cos(θt ) T E
E .
n2 cos(θi ) + n1 cos(θt ) i
(3.87)
TE
ER
=
2n1 cos(θi )
ET E .
n2 cos(θi ) + n1 cos(θt ) i
(3.88)
TM
ER
=
n1 cos(θi ) − n2 cos(θt ) T M
E .
n1 cos(θi ) + n2 cos(θt ) i
(3.89)
TM
ER
=
2n1 cos(θi )
ET M .
n1 cos(θi ) + n2 cos(θt ) i
(3.90)
72
3.4 Finite Difference Time Domain (FDTD) Method
Thus, the reflectance for the TM case is
2
ErT M
.
(3.91)
R
=
EiT M
Since no absorption occurs at interface, by applying the conservation of energy, the
TM
transmission is then expressed by
R
3.4
TM
=1−R
TM
=
EtT M
EiT M
2
n2 cos(θt )
.
n1 cos(θi )
(3.92)
Finite Difference Time Domain (FDTD) Method
FDTD (73) is one of the most popular numeric methods for computational electromagnetics. It is widely applied in many different areas like electromagnetic wave propagation, antennas design and guided wave propagation. In FDTD the electromagnetic
quantities are represented in the time domain and the time step is limited by a stability criterion depending upon the size of the smallest cell in the simulation grid, hence
limiting the effective applicability of the method for a given frequency range. In fact,
small geometric features lead to smaller time steps and hence, a larger amount of time
steps are required to simulate a certain number of periods of the wave at a given frequency. Consequently, FDTD results effective with structures featuring size comparable
to the wavelength of the radiation. FDTD simulates the continuous electromagnetic
waves in spatial region of finite dimension by sampled-data. If the wavelength is much
greater than the size, both the spatial sampling factor (or the number of grid-points
per wavelength) and the temporal sampling factor (number of time steps per period)
are required to be large.
FDTD is based on Maxwell curl equations in derivative form in the time domain
which are expressed in a linearized form by means of central finite differences.
The Yee algorithm (74) for Maxwell’s curl equations assumes a basic uniform Cartesian grid without using any potential functions. It is based on spatial sampling of the
unknown near-field distribution over a period of time. The sampling in space is performed in the Nyquist sense. The sampling in time is selected to ensure numerical
stability of the algorithm.
FDTD is based on second-order accurate grid-based algorithms exhibiting concurrent schemes for time-marching the six-vector components of the electromagnetic near
73
3. NUMERICAL SIMULATION OF SOLAR CELLS
field. The computational burden for the FDTD involves the number of volumetric grid
cells NC -in which the six vector electromagnetic field components must be updated at
every time step-, the number of time steps NS , and the cumulative propagation errors.
4
Overall, a computational burden of order of NC NS of about NC3 is estimated for FDTD
(75).
Figure 3.6: Absorbance for the 2µm-thick (A) and the 1µm-thick (B) slabs of crystalline
silicon calculated by raytracer (RT), FDTD and RCWA methods.
As examples to put in evidence the limitations of the conventional RT method,
a 2µm-thick slab of crystalline silicon featuring ideal smooth interfaces is considered.
The absorbance of the silicon slab has been calculated within the range of wavelengths
350nm-850nm. Since the layer thickness is comparable to the wavelength, light diffraction and interference play a crucial role, therefore the optical generation rate inside
the silicon layer calculated by raytrace (RT) is affected by significant error. Fig. 3.6A
reports the absorbance as calculated by means of RT, RCWA (which will be described
in detail in chapter 5) and FDTD simulations. The absorbance calculated by RT differs
from those calculated by RCWA and FDTD, as it features an absolute maximum at
around 500nm and a monotonic trend up to 850nm. The absorbance curves, calculated by means of methods based on the solution of Maxwell’s equations like RCWA and
FDTD, are almost coincident. They feature a set of maxima and minima at relatively
large wavelengths, due to the finite thickness of the silicon layer. For this particular
example, by assuming a standard AM1.5G spectrum, the total absorbed energy integrated over the whole spectrum is essentially the same for all the considered methods.
If the silicon thickness of the layer is further reduced (Fig. 3.6B) the absorbed energy
74
3.4 Finite Difference Time Domain (FDTD) Method
weighted by the AM1.5G spectrum provided by raytrace is significantly overestimated
in comparison with that calculated by RCWA (or by other rigorous solvers of the
Maxwell’s equations).
75
3. NUMERICAL SIMULATION OF SOLAR CELLS
76
4
Modeling of advanced crystalline
silicon solar cells
This chapter is a contribution towards the 2-D and 3-D electro-optical numerical simulation of advanced crystalline silicon solar cells. Two applications of numerical TCAD
simulations are presented. The first application concerns the modeling of a selective
emitter solar cell which, due to the adopted scheme of doping profiles in the emitter section of the cell, requires a 2-D simulation. The second application is about a
rear point contact solar cell. Rear point contact schemes are used in high-efficiency
devices like PERC and PERL solar cells (introduced in chapter 2) and require true
3-D cpu-intensive numerical simulations. In this chapter the simulated devices and the
related simulation setups are described in detail. Finally, the simulation results and
the conclusions are discussed.
4.1
4.1.1
Selective emitter solar cells
Introduction to selective emitter solar cells
The improvement of the emitter and of the front contact scheme of crystalline solar cells
is a challenging goal which requires to account for several trade-offs in the optimization
of the device. A paramount requirement of the emitter of high-performance solar cells
is the capability to collect photo-generated carriers with high efficiency; this goal may
be achieved by designing emitters featuring good transport properties and low recombination losses. An ideal emitter electrode requires high doping levels to be an ohmic
77
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
contact; on the other hand, large doping concentration leads to high recombination
losses (Auger, trap-assisted as well as surface recombination). Several approaches have
been proposed to improve the emitter of solar cells: optimized silver pastes (44)(76)
and seed-and-plate approach (77)(78) are promising techniques to allow the contact of
high-sheet resistance emitters (above 65Ω/sq).
Another approach to boost the solar cell efficiency for low-cost and high-volume
productions is the selective emitter design. Selective emitter (SE) (or double-diffused)
solar cells, in contrast with standard screen-printed industrial homogeneously diffused
emitter (HE) cells are characterized by different doping profiles in the emitter region
(Fig. 4.1). A first lowly-doped and shallow diffusion (LDOP), under the passivated
uncontacted front-side interface between fingers is followed by a heavy and deep phosphorus diffusion (HDOP) in correspondence of the contacted interface under the metal
front fingers (79).
The advantages of the SE cell over the conventional HE cells, are provided by
reduced Auger and surface recombination effects in the passivated LDOP surface region
and by enhanced spectral response in the blue region (80)(81) because of the reduced
doping concentration. However, the adoption of a lowly-doped diffusion between frontcontact fingers in the illuminated area, leads to relatively large emitter resistance which
contributes to the reduction of the FF, as shown in chapter 2. In SE solar cells a good
ohmic contact is ensured by the HDOP diffusion in the non-illuminated region.
Another crucial trade-off in the front-side design of the solar cell is that between the
front-contact pitch (or the distance between fingers) and the emitter resistance. Large
front contact pitch leads to relatively higher photo-generated current density because
of the reduced front contact shadowing. However, the distance between fingers has a
strong impact on emitter resistance and hence on FF. Since the emitter resistance, for
a given emitter geometry is dependent on the emitter sheet resistance, SE and HE cells
designs lead to different optimum values of the front contact pitch.
In previous works the impact of the emitter doping profile has been analyzed by
means of analytical models or by 1-D numerical simulations (82)(83).
In this section, 2-D numerical simulations have been successfully used to optimize
the emitter design and to investigate the main loss mechanisms which lead to the above
described trade-offs. In the following a homogeneous emitter solar cell (optimized in
78
4.1 Selective emitter solar cells
terms of front contact pitch) is used as reference cell for the SE design, allowing to
quantify the predicted enhancement of performance obtainable by the SE. In particular, the dependence of the main figures of merit of the cell on the front metal contact
pitch (WSU B ) is investigated by using as simulation parameters the doping and geometrical parameters of the HDOP and LDOP diffusions. 2-D simulations are helpful to
appreciate the effects of recombination mechanisms which are strongly dependent on
the doping concentration as well as to take advantage of the more realistic modeling
of the multi-dimensional transport. Indeed, in addition to discriminate regions with
different doping concentration (like that between front-contact fingers and that of the
non-illuminated regions under the front-contact fingers), a 2-D analysis allows to take
into account for the contact shadowing, differently from 1-D simulations. Therefore,
for this kind of structures, the lack of 2-D capability of mono-dimensional simulators,
like PC-1D (5), which is a well-known and widely used tool in photovoltaic community,
may result in inaccurate predictions of expected performance.
4.1.2
Selective emitter solar cells: simulation setup and simulated
devices
All the considered cells feature a p-type base region, wafer thickness DSU B = 180µm,
front-contact finger width WM ET = 100µm and a lateral width of the HDOP diffusion
under the finger WSE = 130µm (Fig. 4.1A). Doping profiles are chosen as described in
the following: the boron-doped base doping concentration is equal to NSU B = 1016 cm−3
(which corresponds to a substrate resistivity of 1.33Ωcm assuming a constant mobility
for holes equal to 470.5cm2 V −1 s−1 ), the boron p+ BSF diffusion (common for SE and
HE cells) is described by an error function of the depth (starting from the back conpk
tact interface) featuring a peak doping concentration CBSF
= 1020 cm−3 and junction
depth equal to 0.6µm. The HDOP profile is described by an analytical function with
parametric junction depth and peak doping in order to obtain different values of the
sheet resistance. The LDOP profiles are described by analytical error functions with
constant junction depth (0.27µm) and variable peak doping. Finally, the emitter profile
in the HE cell features a sheet resistance of 39Ω/sq resulting from an error function
pk
profile characterized by peak doping concentration equal to CHE
= 3.7 × 1020 cm−3 and
junction depth of 0.39µm. All sheet resistance values are calculated by using the Arora
79
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
model (chapter 2) with Sentaurus’ default parameters. The emitter profiles for the SE
and the HE solar cells are shown in Fig. 4.1B.
Surface recombination velocity is set to 105 cm/s (constant) for the metal front and
back contacts. The surface recombination velocity at the front passivated interface is
assumed to be dependent on the surface doping concentration of the emitter according
to the surface SRH model of Sentaurus described in chapter 3, eq. 3.25, with S0 =
20cm/s, NREF = 1016 cm−3 and SREF = 10−3 . For instance, if the surface doping
concentration is 1020 cm−3 , S = 220cm/s.
Electrical simulations take into account Auger recombination, doping dependent
Shockley-Read-Hall (SRH) bulk and surface recombination, radiative recombination,
band-gap narrowing (del Alamo model), doping dependence of carrier mobility (Philips
Unified mobility model) and mobility degradation at high fields (Canali Model). The
standard Sentaurus’ model for intrinsic carrier concentration is adopted (leading to
ni,ef f = 1.41164 × 1010 cm−3 at T = 300K in base). Fermi statistics is adopted in
this analysis in order to correct deal with heavy doping concentrations. All the models
accounted in the device simulation are discussed in chapter 3.
Figure 4.1: (A) Sketch of the cross section of the 2-D simulated selective emitter (SE)
solar cell. HDOP and LDOP denote the highly-doped and the lowly-doped emitter profiles,
respectively. W is the wafer thickness and WM ET = 100µm and WSE = 130µm are the
front-contact finger width and the lateral extension of the HDOP diffusion under the finger,
respectively. (B) Emitter doping-profiles adopted for the simulations: the figure shows the
doping profile of the homogeneous emitter (HE) solar cell (39Ω/sq), that of the SE under
the contact (HDOP) and the emitter profile under the passivated interface between fingers
(LDOP) featuring a sheet resistance of 39Ω/sq and 109Ω/sq, respectively.
The optical generation resulting from a spectral illumination source (standard AM1.5G
80
4.1 Selective emitter solar cells
spectrum 1000W m−2 , direct illumination), is modeled by superimposing the spectrally
resolved generation rates. The light at the front surface is assumed to be Lambertian
distributed accounting for textured interface coated by 70nm silicon nitride. The external medium is air. The multiple bounces of light inside the device are described
analytically in terms of a geometric progression. External reflectivity, internal top and
bottom reflectivity coefficients, which are wavelength dependent, are calculated by using
the Transfer Matrix Method (TMM) (46). The shadowing under front-contact finger
is assumed ideal. The calculated main figures of merit include the short circuit current
density (Jsc ), the open circuit voltage (Voc ), the fill factor (FF) and the efficiency (η).
The parasitic series resistances are accounted using eq. 2.85 and eq. 2.86, hence the
predicted electrical output power of the cell and the FF are affected by resistive losses.
In particular the maximum output power is given by:
2
PM = PM0 − IM
P P R,
(4.1)
where PM0 is the effective maximum output power of the cell without resistive losses
and IM P P is the maximum power point current. The total series resistance is denoted
by R. The metal-semiconductor-contact resistance and the contact finger resistance
are calculated as discussed in chapter 2, assuming as length and thickness of the grid
finger 3cm and 12µm, respectively. The length of the finger is chosen according to the
assumption of considering a cell 12.5x12.5cm2 for simulations. For the calculations, the
sheet resistivity of the metal and the contact resistivity are set to 6 × 10−6 Ωcm and
10−3 Ωcm2 , (independently on the emitter doping peak concentration), respectively.
The simulations have been performed by adopting as simulation domain the 2-D
symmetry element with lateral size defined by half front contact pitch and height equal
to the wafer thickness. Typical numerical grids include 50000 vertices.
4.1.3
Homogeneous emitter solar cell simulation
The HE solar cell has been simulated by varying the front contact pitch WSU B within
the range 1400µm-2600µm in order to optimize the front contact pattern geometry. As
discussed before, for a given front-contact metal finger width, increasing WSU B results
in larger JSC . However, large WSU B leads to larger emitter resistance that degrades
the FF (Fig. 4.2C). The resulting optimum front contact pitch for the HE cell is
approximately 2100µm, for which the calculated efficiency is 16.81% (Fig. 4.2D).
81
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Figure 4.2: Calculated figures of merit for the SE and the HE (39Ω/sq) cells. The SE are
simulated with the profile HDOP featuring 34Ω/sq and LDOP featuring sheet resistance
values within the range 47Ω/sq-215Ω/sq. Main figures of merit calculated by Sentaurus:
(A) Short-circuit current density (JSC ), (B) Open-circuit voltage (VOC ), (C) Fill Factor
(FF) and (D) conversion efficiency.
82
4.1 Selective emitter solar cells
4.1.4
Selective emitter solar cell: impact of LDOP and HDOP profiles
on main figures of merit
In order to optimize the LDOP profile for a given HDOP diffusion (34Ω/sq, peak
doping concentration 3.7 × 1020 cm−3 , junction depth 0.82µm) the parameters affecting
the analytical function which describe the doping profile LDOP are varied, resulting
in sheet resistance values within the range 47Ω/sq-215Ω/sq (Tab. 4.1). In particular,
since a constant junction depth of 0.27µm is assumed, only the peak doping of LDOP
is changed from 5.0 × 1019 cm−3 to 3.0 × 1020 cm−3 . The calculated dependence of the
figures of merit on the front contact pitch is reported in Fig. 4.2 for the most significant
LDOP profiles.
The trend of JSC and VOC are shown in Fig.4.2A and Fig. 4.2B, respectively.
Table 4.1: Peak doping and sheet resistance values of the considered lowly-doped region
profiles (LDOP) for the SE solar cell. Simulated LDOP profiles are described by analytical
error functions. For all LDOP profiles the junction depth is set to 0.27µm.
Peak Doping [cm−3 ]
Sheet Resistance [Ω/sq]
1020
3.00 ×
2.00 × 1020
1.50 × 1020
1.15 × 1020
1.00 × 1020
9.00 × 1019
5.00 × 1019
47
68
87
109
123
136
215
For a given front contact pitch value, both Jsc and Voc of the SE solar cells are
larger with respect to those of the HE cell, because the lower doping concentration of
LDOP profiles, compared to that of HE cell, results in reduced Auger, trap-assisted
and surface recombination losses in the illuminated passivated region between fingers.
Hence, the highest values of Jsc and Voc are obtained by the SE solar cell featuring the
LDOP profile with the lowest doping concentration (sheet resistance 215Ω/sq) within
the considered range. Moreover, as already observed for the HE solar cell, increasing
front contact pitch leads to larger Jsc (and consequently higher Voc ) due to reduced
contact shadowing.
83
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
However, increasing sheet resistance of LDOP results in degraded FF due to the
strong impact of doping concentration on emitter resistance (Fig. 4.2C). In addition,
the increase of the front contact pitch leads to larger emitter resistance and the strongest
dependence of FF on WSU B is observed for the largest sheet resistance of the LDOP
profile. For a given value of front-contact pitch, the lowest FF is obtained for the LDOP
profile with the largest sheet resistance (215Ω/sq). The trade-off between the emitter
resistance, the front contact shadowing and the recombination losses in the emitter
region are highlighted in Fig. 4.2D. For lower values of WSU B , the efficiency benefits
from the increase of both Jsc and Voc for increasing WSU B , but for higher WSU B ,
the emitter resistance losses dominate and hence the efficiency degrades. The above
discussed trade-off results in an optimum value of the front contact pitch which strongly
depends on the sheet resistance of the LDOP profile. In particular, it is worth noting
that the optimum WSU B moves towards lower values of the pitch for higher emitter
sheet resistance of LDOP. The maximum predicted efficiency is 17.59%, obtained by
the SE cell characterized by LDOP with sheet resistance 109Ω/sq. It is worth noting
that, as expected, since the resulting emitter resistance is higher than that of the HE
cell, a smaller optimum value of WSU B results, in comparison with the homogenous
emitter case, as shown in Tab. 4.2, in which an enhancement in terms of conversion
efficiency with respect to the HE cell equal to 0.78%Abs is reported. In addition, it can
be observed that, although the optimum WSU B is lower in case of the SE with respect
to the HE cell, the resulting short-circuit current (and hence the open-circuit voltage)
is higher than that of the reference cell.
Table 4.2: Main figures of merit of the simulated solar cells calculated by Sentaurus: shortcircuit current density (Jsc ), open-circuit voltage (Voc ), Fill Factor (FF) and efficiency for
homogeneous emitter cell (WSU B = 2100µm, 39Ω/sq) and the selective emitter SE cell
(WSU B = 1800µm, HDOP 34Ω/sq and LDOP 109Ω/sq).
Cell Type
WSU B [µm]
JSC [mAcm−2 ]
VOC [mVolt]
FF
Efficiency [%]
HE
SE
2100
1800
34.32
35.13
610
623
80.37
80.35
16.81
17.59
The sheet resistance of the HDOP profiles has been varied from 34Ω/sq to 63Ω/sq
(Tab. 4.3), keeping the LDOP profile constant (109Ω/sq) and WSU B = 1800µm. The
simulation results show that there is only a slight dependence of efficiency on HDOP
84
4.1 Selective emitter solar cells
parameters (peak doping, junction depth). The maximum value of conversion efficiency
(17.61%) is obtained by the 46Ω/sq HDOP profile. The emitter resistance is dominated
by the diffusion between front-contact fingers (LDOP) and the HDOP profile has only
a negligible impact on FF. The conclusion of the analysis of the dependence of the main
figures of merit on the features of HDOP is that, under the adopted assumptions, in
particular for low values of WSE , the impact of the HDOP profile on the efficiency is
weak. On the contrary, when the lateral extension of the HDOP diffusion is significantly
larger than that of the front-contact finger, larger recombination losses are expected
(in particular due to the Auger mechanism) because of the presence of a heavily-doped
region exposed to the illumination. Ideally WSE = WM ET .
Table 4.3: Peak doping, junction depth and sheet resistance values of the considered
HDOP doping profiles.
4.1.5
Peak Doping [cm−3 ]
Junction Depth [µm]
Sheet Resistance [Ω/sq]
3.5 × 1020
2.0 × 1020
1.0 × 1020
2.0 × 1020
2.0 × 1020
0.98
0.98
0.98
1.31
0.60
34
46
63
41
54
Selective emitter solar cell: loss analysis
The SE (featuring the doping profiles HDOP 46Ω/sq and LDOP 109Ω/sq) and the HE
(39Ω/sq) solar cells are compared in terms of carrier collection efficiency (ηC ) defined
in chapter 2, by eq. 2.73. The results are reported in Fig. 4.3.
The SE cell features a better collection efficiency in the blue region of the spectrum
(300nm-600nm) resulting in higher short-circuit current (JSC = 35.17mA/cm2 for SE,
JSC = 34.32mA/cm2 for the HE cell). As remarked previously, a larger short-circuit
current density is reached by the optimized SE cell although its optimum front contact
pitch value is smaller than that of the HE cell. The enhancement in terms of collection efficiency is due to the lower doping concentrations in the emitter, which leads
to reduced Auger recombination and to a shallow junction in the passivated emitter
region, which improves separation for electron-hole pairs generated by photons at lower
85
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Figure 4.3: Comparison of collection efficiency of photo-generated carriers (ηC ) between
selective emitter SE (WSU B = 1800µm, HDOP 46Ω/sq and LDOP 109Ω/sq) and the
reference homogenous emitter solar cell HE (WSU B = 2100µm, 39Ω/sq). Larger ηC within
the blue region of the spectrum (300nm-600nm) is obtained by the SE cell.
wavelengths that are absorbed close to the front surface. Furthermore, lower doping
concentrations lead to reduced surface recombination rates at the front passivated interfaces. Simulations highlight the influence of Auger recombination on efficiency as
the major loss mechanism of a homogeneous solar cell with heavy and deep emitter
diffusions. By selectively disabling the Auger recombination effect, the SE and HE
solar cells increase their efficiencies by an absolute 0.60% and 1.24%, respectively; this
means that reduced Auger recombination is the main reason for higher efficiency of the
SE cell with respect to the reference HE cell. The 2-D maps of the Auger recombination rates in the region close to the front surface for SE and HE cells are compared in
Fig. 4.4A and Fig. 4.4B, respectively. In addition Fig. 4.4C reports the 1-D profile of
the Auger recombination rates of the HE and the SE cells between the front-contact
fingers, confirming the relatively larger Auger recombination rate for the HE.
4.1.6
Selective emitter solar cell: conclusions
The main figures of merit of a selective emitter solar cell have been investigated as
function of the front contact pitch and the emitter doping profiles, providing some
guidelines aimed at optimizing the front side of the device. The analysis has been
performed by means of 2-D numerical TCAD simulations by using the state-of-the-art
86
4.1 Selective emitter solar cells
Figure 4.4: 2-D Auger recombination rate maps (in cm−3 s−1 ) in the passivated emitter
region close to the front surface (A) for the homogeneous emitter HE cell (WSU B = 2100µm,
39Ω/sq) and (B) for the optimized selective emitter scheme SE (WSU B = 1800µm, HDOP
46Ω/sq and LDOP 109Ω/sq). In (C) the 1-D Auger recombination rate is reported as
function of the distance from the front interface in the region between front-contact fingers.
87
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
models described in chapter 3. According to performed simulations, a selective emitter
solar cell may provide an efficiency boost of approximately 0.8% with respect to a
standard homogeneous emitter device; such enhancement depends also on the emitter
sheet resistance of the considered HE baseline cell (for instance, if a 65Ω/sq emitter
cell is assumed as reference, a reduced efficiency boost is expected, approximately 0.5%
absolute). The advantages of double-diffused emitters result from the enhancement
of the collection efficiency, especially in the blue region of the spectrum, and from
reduced Auger recombination due to a lighter emitter doping concentration in the
region between fingers. The doping profile under the contacted region should satisfy
the requirement of ensuring a good ohmic emitter contact. However, a careful design
of the heavy-doped region profile should be considered depending upon the lateral
extension of the HDOP diffusion; in presence of a relatively small lateral extension of
the HDOP diffusion, a negligible impact of the features of the HDOP profile on the
main figures of merit of the cell is observed.
4.2
Rear point contact solar cells
The adoption of Rear Point Contacts (RPCs) schemes at the back surface of high
efficiency mono-crystalline silicon solar cells is one of the most promising approaches
aimed at reducing the recombination losses at the rear side of the device. However, a
drawback of the reduction of the rear contact surface is the increase of series resistance
losses which, as described in chapter 2, leads to degradation of the Fill Factor. This
section presents an extensive analysis based on 3-D electro-optical numerical device
simulations of RPCs in order to highlight the dependence of the figures of merit of
the solar cell on the main geometrical and technological parameters of the device,
like the hole pitch, the size of the rear point contacts and the substrate resistivity.
The TCAD simulator has been successfully adopted in order to accurately solve the
transport equations in the semiconductor by taking into account all the loss mechanisms
that are crucial in order to address the design of the cell. In subsection 4.2.2 the
main issues associated with the numerical simulation of 3-D structures as well as the
simulation set-up and the adopted physical models are described. Subsection 4.2.3
reports the results of an extensive analysis of technological options such as local BSF,
88
4.2 Rear point contact solar cells
substrate resistivity, specific contact resistivity and geometrical features of the backcontact scheme like the contact size and the hole pitch (Fig. 4.5).
4.2.1
Introduction to rear point contact solar cells
An effective approach in c-Si solar cells to improve the conversion efficiency is the
adoption of rear point contacts schemes which are adopted in high-efficiency PERC
(Passivated Emitter and Rear Cell) and PERL (Passivated Emitter, Rear Locally Diffused) solar cells (13),(1) (reported efficiency values are above 23% (84)).
In standard solar cells exhibiting a uniformly contacted rear surface, significant recombination losses at the back metal-contacted interfaces occur. The main advantage
of the RPC scheme derives from the reduction of the effective Back Surface Recombination Velocities (BSRVef f ) with decreasing metallization fraction (ratio of the contacted
to the total back surface area). In addition the rear passivation allows to enhance
the effective Internal Bottom Reflectivity (IBRef f ) due to the larger reflectivity of cSi/dielectric/metal stack interface (typically above 0.90, depending upon the dielectric)
with respect to the metal/c-Si one (approximately 0.65) (23)(85)(86)(87)(88). Disregarding the increased manufacturing costs, the main drawback of RPCs is the increase
of the parasitic series resistances (due to two main contributions: the spreading base
resistance and the back contact resistance). In this section only Cz (Czochralski) silicon wafers are considered. Low-cost and high-volume manufacturing processes for RPC
solar cells use the Laser Firing Technique (LFC) to form local back-surface-field (BSF)
diffusions (89)(90)(91). When LFC techniques are adopted, holes featuring diameters
within the range 25-50µm and featuring relatively small specific contact resistivity
(0.05-3mΩcm2 ) can be fabricated (91)(92) for a wide substrate resistivity range (0.01100Ωcm). In rear point contact solar cells (Fig. 4.5), the non-contacted back surface
is typically passivated by dielectrics like silicon nitride (SiN) (93) or by silicon dioxide
(SiO2 ) leading to low surface recombination velocities, within the 1-100cm/s-range,
depending on the process parameters and on the substrate resistivity.
Passivation techniques are not limited to those involving the formation of SiN or
SiO2 layers, indeed the future trend toward ultra-thin c-Si solar cells has increased
the demand for low-cost rear surface passivation techniques like that by means of
amorphous hydrogenated silicon nitride (SiNx) deposited by low-temperature PlasmaEnhanced Chemical Vapour Deposition (PECVD). Very low surface recombination ve-
89
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
WSUB
WM
A
W
Lx
p
z
p
SIMULATION
DOMAIN
Ly
B
y
x
Front Contact Finger
n+
p-type c-Si
SiN
p+ diffusion
W
SiN
Back Contact
p
Figure 4.5: (A) 3-D sketch of the simulated solar cell with the rear point contact scheme.
WSU B and W denote the front contact pitch and the wafer thickness, respectively. The
parameter WM is the width of the front contact finger. The hole pitch is denoted by p.
The simulation domain is highlighted in red (Lx = p/2, Ly = WSU B /2). (B) 2-D cross
section of the device with rear surface passivation by silicon nitride (SiN).
90
4.2 Rear point contact solar cells
locities (SRVs) have been obtained with SiNx films on low-resistivity p-type silicon
wafer (94), (95).
The RPC pattern design requires the optimization of the parameters which define
its geometry. Due to the complexity of the structure, modeling by means of TCAD
numerical simulation is helpful to analyze the trade-offs between several competing
physical mechanisms which have to be accounted for in the optimization of rear point
contact solar cells. The rear point contact geometry leads to 3-D conduction paths
that are particularly important especially for large substrate resistivity, requiring a
rigorous 3-D modelling as already adopted in (96) for the optimization of rear contact
geometries and in (97) for the analysis of back contact solar cells (BCSC).
In other works, 1-D numerical simulations have been adopted to approximate the
effects of 3-D geometries on electrical trasnport and on recombination losses by means
of semi-empirical models to calculate the BSRVef f and the spreading resistance (85)
(98) (99). In (100) (101) (102) the analysis has been simplified by considering twodimensional (2-D) spatial domains leading to potentially misleading and inaccurate
predictions, mainly due to incomplete information to calculate correctly the resistive
losses and to properly model the 3-D transport. In (103) the optimization of a PERL
solar cell has been performed by a simplified 3-D Finite Difference simulation under
the following assumptions: low injection conditions and ideal emitter. In (104) a simulation method based on the numerical solution by means of Fast Fourier Transform of
the minority and majority carrier transport equations in three dimensions in the base
region of the PERC cell has been proposed. In this section, the state-of-the-art TCAD
Sentaurus is adopted to perform true 3-D finite element device simulations of RPCs
solar cells.
4.2.2
Rear Point Contact solar cells: simulation methodology and
simulated devices
The 3-D sketch of the simulated solar cell is shown in Fig. 4.5A. The wafer thickness is
W = 180µm and the front contact pitch WSU B = 2mm. The front contact finger width
(WM ) and the height (HM ) are 100µm and 20µm, respectively. The impact of the rear
point contact geometrical parameters on JSC , VOC , Fill-Factor and conversion efficiency
(η) is investigated by varying the hole pitch p and the hole diameter s (only circularshaped back-contact holes are assumed). In the following, f denotes metallization
91
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
fraction. In Fig. 4.5A, Lx and Ly denote the width and the length of the simulation
domain which are equal to half hole pitch and to half front contact pitch, respectively.
The height of the simulation domain is equal to the wafer thickness W . The assumption
of choosing the ratio of the front contact pitch to the hole pitch as an integer number is
necessary to limit the lateral extension Ly of the simulation domain to WSU B /2. The
boron-doped wafer resistivity ρS is varied within the range 0.5-10Ωcm. The emitter
is homogeneously doped and its doping profile is described by a Gaussian function
featuring a junction depth equal to 0.4µm and a sheet resistance equal to 75Ω/sq (the
emitter peak doping concentration is therefore adjusted depending on the considered
substrate resistivity in order to keep the junction depth and emitter sheet resistance
constant for each value of wafer doping concentration).
In PERL cells, the p+ BSF regions, introduce a built-in electric field at the p-p+
transition allowing the separation of carriers photo-generated by large-wavelength photons close to the contacted back-surface; the doping profile of the p+ BSF is described
by a Gaussian function with a peak doping concentration of 2.5 × 1019 cm−3 and junction depth in the range 6.7-7.1µm, depending on wafer resistivity. As shown in Fig.
4.5B, the front surface and the uncontacted rear surface are assumed to be passivated
by a SiN layer 70nm thick. In order to achieve realistic predictions, the physical models
implemented in the TCAD simulator have been calibrated as described in chapter 3.
The Schenk band-gap narrowing (BGN) model with Fermi-Dirac statistics is adopted
to obtain the value ni = 9.65 × 109 cm−3 at low doping density. The revised BGN
model provides good agreement with data reviewed by Altermatt (48) at temperature
T=300K as described in section 3.1.3. The considered physical models include the
Philips-Unified Mobility Model with carrier drift velocity saturation at high electric
fields.
The main recombination losses (trap-assisted SRH, Auger and surface SRH mechanisms) have been accounted for, including their dependence on doping concentration
as shown in section 3.1.5. The surface recombination velocities (SRVs) at front and
rear passivated interfaces are calibrated according to (64) in which experimental values of SRV between 10cm/s and 100cm/s for bulk doping concentrations within the
range 1014 cm−3 to 1017 cm−3 have been reported. The recombination velocity at front
contact and at back holes is set to 106 cm/s. Tab. 4.4 summarizes the values of the
main physical parameters involved in calculations; as shown in the table, significantly
92
4.2 Rear point contact solar cells
different situations in terms of bulk carrier lifetimes τn and hence of minority carrier
diffusion length Ln are considered (for instance for ρS = 0.5Ωcm, τn = 39.75µsec and
Ln = 303µm; when ρS ≥ 5Ωcm, τn = 1136µsec and Ln = 1875µm). On the other
hand, the back surface recombination velocity modeled according to (64), only slightly
depends on the base doping concentrations, ranging from 11cm/s to 14cm/s for the
considered base resistivity range. As in the case of the selective emitter solar cell (4.1),
the figures of merit of the solar cell (FF and η), are corrected in order to account for
the losses due to the parasitic series resistance associated to contacts and metal fingers. The resistance associated to front contacts and to metal fingers is 0.182Ωcm2 .
Relatively large specific back contact resistivity ρBC values within the range 1mΩcm2 10mΩcm2 depending on several LFC processing conditions (like annealing temperature,
laser power and passivation material) have been reported in recent experimental works
(90), (85). Hence, in the following ρBC = 3mΩcm2 is adopted. The impact of different
values of the specific contact resistivity of the back contact is investigated and discussed
in subsection 4.2.3.4.
Table 4.4: Physical and electrical parameters adopted for the performed simulations for
different type of considered cells and for different substrate resistivity values ρS . Nsub
denotes the substrate doping concentration. τn , µn and Ln are the bulk minority carrier lifetime, the low-field minority carrier mobility at low doping concentration and the
corresponding calculated diffusion length in the base, respectively. The back surface recombination velocity at the passivated rear surface is denoted by BRSVpass .
Cell Type
PERC
PERL
PERL
PERL
PERL
PERL
ρS
Nsub
τn
BRSVpass
µn
Ln
[Ωcm]
[cm−3 ]
[µsec]
[cm/s]
[cm2 s−1 V −1 ]
[µm]
0.5
0.5
1.0
2.0
5.0
10.0
3.255 × 1016
3.255 × 1016
1.513 × 1016
7.216 × 1015
2.781 × 1015
1.368 × 1015
39.75
39.75
115.0
230.0
1136
1136
14.58
14.58
12.66
11.79
11.31
11.15
896
896
1027
1113
1195
1195
303
303
553
814
1875
1875
The optical simulation has been performed by a 1-D optical simulator; the calculated
optical generation map is then interpolated on the 3-D grid (Fig. 4.6) provided as input
for the device simulator. Optical generation rate profiles are calculated on the basis of
93
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
the propagation of monochromatic plane waves through layered media. The multiple
bounces of light inside the device are described analytically in terms of a geometric
progression of plane waves. The internal reflectance (at top and at bottom interfaces)
is assumed to be the same for subsequent light bounces. The optical generation rate
profiles are calculated assuming direct illumination with a standard AM1.5G spectrum
(input irradiance 1000W/m2 ).
The textured top surfaces coated by a 70nm thick anti-reflection coating layer of
SiN is modeled by a measured external reflectivity. In addition, ideal shadowing by
front fingers is assumed. The internal bottom reflection coefficient Rbi,p of the silicondielectric-metal stack interface is set to 0.90 and the internal bottom reflection coefficient of the Al/p-Si interface Rbi,m is assumed equal to 0.65 (23). To simplify the
optical modeling, the optical simulation is performed in one dimension, therefore the
internal bottom reflection coefficient is assumed to be uniform at the rear interface and
is set as an average value weighted by the metallization fraction f .
Fig. 4.6 and Fig.4.7 report examples of 3-D mesh grid adopted for the numerical
simulation and of a 3-D output map calculated by the device simulator; in particular
a 3-D conduction current density map in short-circuit conditions under illumination is
shown. The figure allows to observe the 3-D conduction path of the current exhibiting
the typical branching in the base and the current crowding occuring close the back
contact holes.
4.2.3
Technological and geometrical options
In the following the results of an extensive analysis of the impact of different technological and geometrical parameters of the RPC scheme are discussed. In subsection
4.2.3.1 the PERL and PERC solar cells are compared and the focus of the analysis is
the impact of the presence the p+ local BSF diffusion on the optimum value of the
metallization fraction and on the figures of merit.
In subsection 4.2.3.2 a PERL cell featuring a constant hole pitch p is analyzed and
the hole diameter s is adjusted in order to vary the metallization fraction, assuming
as simulation parameter the substrate resistivity ρS in order to investigate how the
parasitic spreading base resistance as well as the doping-dependent bulk recombination
losses affect the optimization of the cell.
94
4.2 Rear point contact solar cells
Figure 4.6: Example of 3-D mesh grid adopted for the device numerical simulation. In
this case the hole pitch p is 500µm and the hole diameter s is 50µm. The simulation
domain is chosen according to section 4.2.2. Up to 300000 vertices are typically required.
Figure 4.7: Example of 3-D output maps calculated by the device numerical simulator.
In this case the hole pitch p is 500µm and the hole diameter s is 50µm. The figure shows
the 3-D conduction current density map in short-circuit conditions under illumination.
95
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
In subsection 4.2.3.3 the analysis is performed by varying the hole pitch and by
assuming as simulation parameter the hole size for two different values of ρS . Finally,
in subsection 4.2.3.4 the impact of the back contact specific resistivity on the main
figures of merit is investigated.
4.2.3.1
Comparison between PERL and PERC solar cells
PERL and PERC solar cells have been compared in order to investigate the impact of
the local p+ BSF diffusion on the figures of merit of the solar cell. The characteristics
of the doping profile of the BSF diffusion of the PERL cell are described in section
4.2.2. For both PERL and PERC devices ρs = 0.5Ωcm is chosen, since for larger ρS
values the Al/p-Si interface is rectifying (100). The hole diameter s is varied within
the range 25µm-400µm (by keeping the hole pitch p constant to p = 500µm) in order
to investigate the dependence of the figures of merit on metallization fraction f . The
chosen value of hole pitch is comparable to that reported by other works (105), (106) in
which experimental data related to PERC cells are presented. In addition, the choice
of p = 500µm is motivated by the requirement, previously discussed, to limit the simulation domain to WSU B /2. By decreasing the metallization fraction f , a monotonic
increase of both Voc and Jsc due to the reduced BSRVef f and to the increase of the
IBRe f f is observed (Fig. 4.8). In Fig. 4.9 the absorbance and the reflectance characteristics of a RPC cell featuring a significantly low metallization fraction (f0 = 0.2%)
are compared to those of the uniformly contacted cell (f = 100%); in the former
case, due to the higher BSRVef f (approximately 0.90) with reference to the baseline
(IBR = 0.65), the cell exhibits a lower reflectance, in particular at longer wavelengths
(λ > 1000nm), for which the influence of the internal reflection at the bottom interface on the total reflectance is remarkable. Hence, the absorbance of the RPC cell is
larger than that of the baseline. Therefore the RPC solar cell exhibits a larger photogenerated current density (Jph = 39.542mA/cm2 ) than that of the uniformly contacted
cell (Jph = 38.611mA/cm2 ). In addition, due to the smaller losses in the rear side of
the device, both Voc and Jsc are larger for the PERL cell with reference to the PERC
device, indeed, the local BSF in the PERL cell, enhances carrier separation leading
to reduced recombination at the highly-defective back metal-semiconductor interfaces
thanks to the electric field in the p-p+ transition region. However, by reducing the metallization fraction, the impact of the metal-semiconductor interfaces is less dominant
96
4.2 Rear point contact solar cells
and thus, the difference in terms of Voc and Jsc between PERL and PERC cells become
smaller (Fig. 4.8). In addition, in case of the PERC, the absence of the local BSF leads
to steeper dependence of both Voc and Jsc on f , compared to the PERL cell, because
of the stronger impact of the BSRVef f . The calculated Fill Factor (FF) for the PERL
and the PERC cells is reported as a function of f in Fig. 4.10. The base parasitic
spreading and the contact series resistances increase with decreasing f , leading to a
degradation of the FF. Indeed, the spreading series resistance is associated to the 3-D
current branching occurring when the extension of the metallized interface at the back
interface is significantly smaller than the total cell area; moreover, the contact series
resistance, for a given specific contact resistivity, is inversely proportional to the area
of the contacted surface. Since the hole pitch p is assumed to be constant, lower f
leads to larger distance between the metallized surfaces due to reduced diameter of the
holes. Moreover, lower f means larger back contact resistance due to a lower contacted
area. Since, due to a lower peak doping concentration at the metal-silicon interface, a
larger ρBC in case of the PERC is expected, simulations of the PERC cell have been
performed by assuming as specific back contact resistivity ρBC = 3mΩcm2 (value common to both PERL and PERC to allow a direct comparison without introducing the
influence of the specific back contact resistivity) and ρBC = 30mΩcm2 (the specific
back contact resistivity of the PERC cell is expected to be significantly larger than
that of the PERL device due to the absence of the BSF diffusion). It is worth noting
that, when PERC and PERL are compared by assuming the same value of specific
back contact resistivity, the FF curves are indistinguishable; however, in case of PERC
cell, if ρBC = 30mΩcm2 is assumed, the FF is strongly degraded with reference to the
PERL cell because of the significant parasitic series resistance introduced by the rear
point contact pattern; in addition its dependence on f is steeper.
Concerning the curves of conversion efficiency (Fig. 4.10), a typical bell-shape is
obtained due to a trade-off between two competing mechanisms: when f decreases, Voc
and Jsc enhance, on the contrary the FF degrades significantly leading to an optimum
value of metallization fraction f0 for which the efficiency reaches its maximum value.
Moreover f0 is different for the PERC and PERL: the maximum efficiency for the
PERL cell is 19.68% (at f0 = 7.07%), while in case of the PERC, the calculated
values of maximum efficiency are 19.31% (f0 = 3.14%) and 18.74% (f0 = 9.62%) for
ρBC = 3mΩcm2 and ρBC = 30mΩcm2 , respectively. The optimum RPC configuration,
97
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Figure 4.8: Short circuit current density Jsc (left) and open circuit voltage Voc (right)
for PERC and PERL solar cells with hole pith p = 500µm and substrate resistivity ρs =
0.5Ωcm.
Figure 4.9: Calculated absorbance (A) and reflectance (R) characteristics of a RPC cell
featuring f0 = 0.2% compared to those of the uniformly contacted cell (f = 100%).
98
4.2 Rear point contact solar cells
Figure 4.10: Fill Factor (left) and efficiency (right) for PERC and PERL solar cells with
hole pitch p = 500µm and substrate resistivity ρs = 0.5Ωcm. In case of the PERC two
different values of the specific back contact resistivity ρBC are assumed: ρBC = 3mΩcm2
and ρBC = 30mΩcm2 ; if the former value of ρBC is adopted for both PERC and PERL,
the curves of the FF are indistinguishable.
with reference to the case of the uniformly back contacted solar cell (f = 100%), allows
an enhancement in terms of efficiency of 0.88%abs and 2.14%abs (ρBC = 3mΩcm2 ) for
the PERL and the PERC cells, respectively. The larger enhancement in case of the
PERC cell is due to the significant reduction of the losses in the base region thanks to
the low metallization fraction. However, the reduction of the losses in the base region
is not so remarkable for the PERL cell, indeed the BSF mitigates the dependence of
the figures of merit on the boundary conditions at the back interface almost for all
values of f , even for the case of the uniformly contacted solar cell. In particular, as
shown in Fig. 4.10, a weak dependence of the efficiency on f within the range 0.8-12%
is observed in the case of the PERL cell, consistently with the small dependence of Voc
and Jsc on f . Hence, the performance of PERL cells is expected to be less sensitive to
the uncertainties affecting geometrical parameters.
In the considered solar cells the values of the obtainable short-circuit current density
and hence of the calculated values of conversion efficiency are lower than those reported
in (1) and (84) (Jsc significantly above 40mA/cm2 and efficiencies above 23%). The
main reason of these discrepancies relies on the absence in the device simulated in this
work of photo-lithographic techniques for texturing (for instance inverted pyramids) and
for contact definition, that allow the reduction of the front reflectance and of the front
99
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
contact shadowing effect, respectively. On the contrary, photo-lithographic techniques
are adopted in the cited works. In addition, in the mentioned high-performance solar
cells, FZ wafers are adopted; as observed in chapter 2.9.1, such high-quality substrates
allow to reach longer minority carrier lifetimes, even in millisecond range.
The calculated main figures of merit for the simulated PERC and PERL cells are
summarized in Tab. 4.5.
Table 4.5: Results of the simulations for PERL and PERC solar cells with p = 500µm
and substrate resistivity ρs = 0.5Ωcm. The PERC solar cell has been simulated with
back contact specific resistivities ρBC = 3mΩcm2 and ρBC = 30mΩcm2 . The parameter s
denotes the hole diameter. The table reports only the main figures of merit of the uniformly
contacted cell and those of the RPC solar cell calculated at the optimum metallization
fraction f0 . The related efficiency boost ∆η (with respect to the baseline) is reported in
the last column.
Cell Type
ρBC
f0
Jsc
Voc
[µm]
[%]
[mAcm−2 ]
[mV olt]
3
3
150
-
7.07
100
37.64
36.04
643
637
PERC
PERC
3
3
100
-
3.14
100
37.34
33.93
PERC
PERC
30
30
175
-
9.62
100
36.93
33.93
[mΩcm2 ]
PERL
PERL
f
FF
η
∆η
[%]
[%abs ]
81.26
81.54
19.68
18.70
0.88
640
622
80.84
81.31
19.31
17.17
2.14
636
622
79.76
81.18
18.74
17.14
1.60
In order to perform a more detailed analysis about the different impact of loss
mechanisms on PERC and PERL solar cells, the collection efficiency ηC of the photogenerated electron-hole pairs (defined in section 2.3.7) has been calculated within the
range of wavelength 600nm-1100nm (below 600nm the collection efficiency is not affected by the RPC scheme since high-energy photons are absorbed in the emitter region). In Fig. 4.11 the collection efficiency has been calculated in correspondence of
the optimum metallization fraction f0 value and for f = 100% (uniformly contacted
solar cell). For long wavelengths, for which photons are mostly absorbed in the base
region close to the rear surface, ηC is significantly larger with reference to the uniformly
contacted counterpart in case of low metallization fraction because of the remarkable
reduction of the BSRVef f obtained by the RPC scheme. In addition, the collection
100
4.2 Rear point contact solar cells
efficiency of the optimized RPC with respect to the baseline cell is larger for the PERC,
because, in PERL solar cells, the presence of the BSF, as already discussed, leads to
reduced losses in the base region also in case of uniformly contacted devices.
Figure 4.11: Collection efficiency of photo-generated carriers (ηC ) from 600nm up to
1100nm for PERC and PERL solar cells with hole pitch p = 500µm and substrate resistivity
ρs = 0.5Ωcm; ηC is calculated at f = 100% (uniformly contacted cell) and at the optimum
metallization fraction (PERC: f0 = 3.14%, s = 100µm; PERL: f0 = 7.07%, s = 150µm).
4.2.3.2
Dependence of the main figures of merit of PERL cells on the backcontact diameter
In this section the dependence of the figures of merit of the PERL cell on the metallization fraction -by assuming as simulation parameter the substrate resistivity ρS are investigated. The substrate resistivity has a strong impact on both the spreading
resistance and the recombination losses. In this analysis, the peak doping concentration
of the local BSF diffusion at the metallized interface is assumed to be independent on
ρS . The following values of ρS are assumed: 0.5, 1, 2, 5 and 10Ωcm. The metallization
factor varies as consequence of a change in hole diameter s within the range 25µm400µm. On the contrary, the hole pitch is assumed to be constant (p = 500µm). The
curves showing the dependence of the figures of merit on the metallization fraction f
of the analyzed cells are shown in Fig. 4.12. Both Voc and Jsc are strongly affected
101
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
by the substrate resistivity; indeed, smaller substrate resistivity values corresponds to
larger bulk recombination losses as shown in Tab. 4.4 where, for each considered substrate resistivity value, the doping dependent carrier lifetime is reported. However, the
bulk doping concentration weakly affects also the surface recombination velocity at the
passivated interfaces. In addition, in case of higher substrate doping concentration, the
local p+ diffusion depth decreases, leading to larger BSRVef f . Therefore, increasing
ρS leads to larger Voc and Jsc . For ρS > 5Ωcm such dependence is weaker because the
carrier lifetime saturates for a Cz wafer (Tab. 4.4). In addition a slight dependence of
Voc on f is observed for low ρs values. Indeed, the lower is ρs , the lower is the carrier
lifetime, therefore the minority carrier diffusion length degrades and the impact of the
boundary conditions at rear interface is less dominant.
A remarkable degradation of the FF is observed with decreasing f , in particular for
large ρs values. Indeed, larger substrate resistivity values lead to enhanced spreading
base resistance, hence to a more significant impact of the metallization fraction on FF.
An approximate calculation of the spreading resistance as function of the metallization
fraction and of the base resistivity by using the model proposed by Cox and Strack
(107) shows that, at lower metallization fraction (around 1%), for relatively low values of ρs , the effect of the back contact resistance on the FF is almost of the same
magnitude of that of the spreading resistance; however, the relative contribution of the
base parasitic series resistance is significantly higher, (even above 80%) for larger ρs
(for instance 10Ωcm); this represents one of the main drawbacks occurring when the
substrate doping concentration in RPC designs is decreased. As a consequence of two
competing mechanisms -reduced losses at large ρs and increased base resistance at low
ρs - a strong dependence of the optimum metallization fraction f0 and of the maximum
efficiency value on ρs is observed. The calculated maximum efficiency is obtained for
ρS = 5Ωcm (f0 = 12.32%, s = 200µm). For ρS = 10Ωcm the maximum achievable efficiency is 20.00% at f0 = 7.07%. It is worth noting that for ρS = 10Ωcm the efficiency
gain between the optimum rear point contact geometry and the uniformly contacted
counterpart is 0.59%abs ; however, for ρS = 1Ωcm the enhancement in terms of efficiency
is even higher: 1.03%abs . The calculated main figures of merit of the simulated PERL
cells for different values of ρS are reported in Tab. 4.6.
102
4.2 Rear point contact solar cells
Figure 4.12: Short circuit current density Jsc , open circuit voltage Voc , Fill Factor and
conversion efficiency η for PERL solar cells with constant hole pith p = 500µm and specific
back contact resistivity ρBC = 3mΩcm2 , for different substrate resistivity values.
103
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Table 4.6: Results of the simulations for PERL solar cells with p = 500µm, specific back
contact resistivity ρBC = 3mΩcm2 and substrate resistivity as parameter. The parameter s
denotes the hole diameter. The table reports only the main figures of merit of the uniformly
contacted cell and those of the RPC solar cell calculated at the optimum metallization
fraction f0 . The related efficiency boost ∆η (with respect to the baseline) is reported in
the last column.
ρS
4.2.3.3
s
f0
Jsc
Voc
[mV olt]
FF
η
∆η
[%]
[%abs ]
[mΩcm]
[µm]
[%]
[mAcm−2 ]
10
10
250
-
19.63
100
38.43
37.39
652
640
79.44
80.71
19.91
19.32
0.59
5
5
200
-
12.32
100
38.49
37.35
655
641
80.18
81.15
20.21
19.42
0.79
2
2
175
-
9.62
100
38.38
27.09
649
637
80.79
81.30
20.13
19.21
0.92
1
1
150
-
7.07
100
38.15
36.65
646
636
81.10
81.43
20.00
18.97
1.03
0.5
0.5
150
-
7.07
100
37.64
36.04
643
636
81.26
81.53
19.68
18.70
0.98
PERL cells: impact of back-contact pitch on performance
In this section the dependence of the efficiency of a PERL solar cell on the contact
pitch p is analysed for two values of the substrate resistivity: ρS = 1Ωcm and ρS =
5Ωcm. The adopted simulation paramater is the hole diameter s; three values of s
are considered: 50, 100 and 175µm. The hole pitch p, differently from the previous
analyses, is varied in order to control the metallization fraction f . Voc and Jsc are
only weakly dependent on the hole diameter s. Indeed, the dependence of Voc and Jsc
on f is mainly due to the back-surface recombination losses, hence to the area of the
contacted back-surface. The calculated trend of the FF is shown in Fig. 4.13; it is worth
noting that a steep degradation of the FF occurs at large s and for low metallization
fractions (f ≤ 10%). The reason of such dependence is that, for a given f , larger
hole diameter means larger distance between the adjacent rear contacts, consequently
leading to larger spreading resistance. A first conclusion is that, in order to achieve
high values of efficiency, small hole diameter values are required, as shown in Tab. 4.7
104
4.2 Rear point contact solar cells
and Fig. 4.13.
In case of ρS = 5Ωcm, the maximum calculated efficiency (20.34%) is achieved for
s = 50µm. The efficiency boost (with respect to the uniformly contacted counterpart) which derives by the RPC featuring substrate resistivity ρS = 5Ωcm is 0.92%abs
(s = 50µm). However, the lower bound limit of the hole size is determined by technological limitations in opening small laser fired holes and by the increase of back contact
resistance.
Qualitatively similar results have been obtained in the case ρS = 1Ωcm. The
maximum efficiency for the optimized RPC geometry (p = 181.12µm, f = 5.94%, Fig.
4.14) is lower than that obtained assuming ρS = 5Ωcm. The efficiency gain of the
cell featuring substrate resistivity ρS = 1Ωcm with respect to the conventional cell is
1.05%abs for s = 50µm.
Finally, it is worth noting that, on the basis of the predicted trend of the efficiency on
the metallization fraction, if realistic uncertainties in terms of hole pitch and hole size for
a standard laser opening process are accounted for (85), in case of an optimized PERL
cell design with ρS = 1Ωcm, the resulting metallization fraction may vary from 4% to
9% (the calculated metallization fraction for the optimized RPC pattern is 5.94%), thus
the efficiency may be affected by uncertainty up to 0.02%abs . In case of ρS = 5Ωcm
the expected uncertainty affecting the efficiency is slightly higher (0.04%abs ).
4.2.3.4
PERL cells: analysis of the dependence of the main figures of merit
on the specific back contact resistivity
In previous analyses the adopted value of specific back contact resistivity ρBC is
3mΩcm2 . However, as mentioned in subsection 4.2.2, experimental works report a
relatively extended range of values of specific back contact resistivity. In this section a
PERL cell featuring a hole diameter s = 50µm and substrate resistivity ρS = 1Ωcm and
ρS = 5Ωcm has been simulated adopting as simulation parameter ρBC . The simulated
values of ρBC are: ρBC = 0 (ideal case), 0.6, 3, 6, 9 and 15mΩcm2 . The simulation
results are reported in Fig. 4.15 . As expected, for a given metallization fraction,
the degradation of the FF is steeper with increasing ρBC , while both Voc and Jsc are
almost independent on ρBC . Therefore the optimum metallization fraction f0 moves
towards higher values with increasing ρBC . For ρS = 1Ωcm and ρBC = 0mΩcm2 the
maximum calculated efficiency 20.11% is achieved at f0 = 2.41% (F F = 81.32) while
105
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Figure 4.13: Fill Factor (up) and efficiency (bottom) for PERL solar cells with substrate
resistivity ρS = 5Ωcm (left) and ρS = 1Ωcm (right). Specific back contact resistivity
ρBC = 3mΩcm2 . Hole diameter s as parameter and variable hole pitch.
106
4.2 Rear point contact solar cells
Table 4.7: Simulations results for PERL solar cells with hole diameter s as parameter,
specific back contact resistivity ρBC = 3mΩcm2 , substrate resistivity ρS = 1Ωcm and
ρS = 5Ωcm. The parameter p denotes the hole pitch of the rear point contact pattern.
The table reports only the main figures of merit of the uniformly contacted cell and those
of the RPC solar cell calculated at the optimum metallization fraction f0 .
ρS
s
p
f0
Jsc
Voc
FF
η
[mΩcm]
[µm]
[µm]
[%]
[mAcm−2 ]
[mV olt]
5
50
100
175
154
250
400
8.30
12.6
15.0
100
38.54
38.49
38.47
37.35
656
655
654
641
80.46
80.52
80.44
81.15
20.34
20.30
20.25
19.42
1
50
100
175
181
286
500
5.94
9.62
9.62
100
38.13
38.08
38.10
36.65
646
646
646
636
81.21
81.30
81.20
81.43
20.02
20.00
19.99
18.97
[%]
for ρBC = 15mΩcm2 the maximum efficiency (19.83%) is obtained at f0 = 11.04%
(F F = 80.79). For ρS = 5Ωcm, the impact of back contact resistance is lower, with
respect to the case of ρS = 1Ωcm, due to the larger contribution of the base spreading
resistance. The calculated boost of efficiency with reference to the uniformly backcontacted solar cell is remarkably significant also for the largest specific back contact
resistivity within the considered range.
4.2.4
Rear Point Contact solar cells: conclusions
In this section the dependence of the main figures of merit of rear point contact solar
cells on the geometrical parameters of the rear contact pattern configuration, on the
back specific contact resistance and on the substrate resistivity ρS , has been investigated. The performed analyses confirmed the effectiveness of the local BSF diffusion
of the PERL cell. In addition, the dependence of the efficiency on the metallization
fraction at constant hole pitch, variable contact-hole diameter and substrate resistivity
as simulation parameter has been analyzed. Simulations highlighted that the metallization fraction is a crucial parameter in the design of a rear point contact cell. For
relatively small values of metallization fraction, the contribution of the series back con-
107
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
Figure 4.14: Conversion efficiency versus hole pitch p for PERL solar cells with substrate
resistivity ρS = 5Ωcm (left) and ρS = 1Ωcm (right). Specific back contact resistivity
ρBC = 3mΩcm2 . Hole diameter s as parameter.
tact resistance and of the base parasitic spreading resistance to the significant reduction
of FF is remarkable. On the other hand, large values of metallization fraction lead to
dominant recombination losses as well as lower effective internal bottom reflectivity.
Therefore, the value of the metallization fraction which optimizes the RPC design is
strongly dependent on the presence of the BSF local diffusion, on the specific back
contact resistivity, on the base doping concentration as well as on the contact size.
In a rear point contact design the goal of enhancing the efficiency can be achieved
by reducing both the specific back contact resistivity and the hole size. However, it
is worth noting that the specific back contact resistivity is the most sensitive to the
optimization of the RPC cell. Once the specific back contact resistivity and of the hole
size are minimized, the substrate resistivity still plays a remarkable role in determining
the maximum obtainable efficiency. Moreover, the performed simulations reported that
the optimum design of a RPC cell is only weakly sensitive to the uncertainties of the
main geometrical parameters which lead to the optimized configuration, such as the
hole size and the hole pitch.
For the best considered configuration (PERL cell, ρS = 5Ωcm), the maximum
108
4.2 Rear point contact solar cells
Figure 4.15: Fill Factor (left) and efficiency (right) for PERL solar cells with hole diameter s = 50µm, substrate resistivity ρS = 5Ωcm (up) and ρS = 1Ωcm (bottom). Specific
back contact resistivity ρBC as parameter.
109
4. MODELING OF ADVANCED CRYSTALLINE SILICON SOLAR
CELLS
achievable calculated efficiency is η = 20.34% and the associated optimum metallization
fraction is f0 = 8.30%; the improvements in terms of efficiency is 0.92%abs with respect
to the uniformly contacted counterpart.
110
5
Optical Simulation by
Fourier-Modal Methods
Modeling of optoelectronic devices and in particular of solar cells, requires an accurate
prediction of the optical generation rate per volume and time unit inside the semiconductor. An effective way to improve the performance of photovoltaic devices is the
photon management strategy, which is aimed at increasing the absorbance of the device within the largest possible portion of the solar spectrum. Nano-metric structures
and nano-rough interfaces are commonly adopted in advanced photovoltaic devices. As
discussed in chapter 3, the ray tracer tool, which is commonly adopted in modeling of
crystalline silicon solar cell exhibiting thick wafers, is not suitable for modeling of nanostructured devices and thin film solar cells. In addition, as it will be discussed in chapter
6, the solution to the optical problem in multi-layer devices featuring nano-rough interfaces (like amorphous silicon thin film solar cells) can be supported by methods based
on the scalar scattering theory (10), which exhibit several theoretical limitations, due
to the approximations which these methods rely on. Practically, methods based on
scalar scattering theory can be applied only when the wavelength of the radiation is
significantly larger than the size of the roughness. However, such methods are commonly adopted in commercial 1-D tools specifically aimed at simulating thin film solar
cells in which the scattering parameters are calibrated by empirical data in order to fit
experiments.
Rigorous numerical solvers of the Maxwell equations are required to treat the near
field optical problem. The most common approaches to the solution of the Maxwell
111
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
equations are the Finite Difference Time Domain method (FDTD) (73), the Finite
Element Method (108), the Beam Propagation Method (BPM) (109) and the Fourier
Modal Method (FMM), also known as Rigorous Coupled-Wave Analysis (RCWA) (110).
RCWA is a general and accurate method based on a Fourier expansion in terms of
the spatial harmonics of the electromagnetic field and the permittivity, that can be
applied to a wide variety of optical problems. RCWA leads to a good trade-off between
computational resources requirement and accuracy, which depends on the number of
Fourier modes included in the calculations.
5.1
Introduction to RCWA
Many implementations of the RCWA method have been presented in the literature; the
first formulation for planar dielectric gratings was proposed by Moharam and Gaylor
(111); subsequently the original implementation was extended to metallic and dielectric
surface relief gratings (112)(113). The electromagnetic field in the external media (substrate and superstrate), expressed in terms of a Rayleigh expansion (34), was required
to match the in-plane space harmonics of the field within the grating region by means
of phase-matching conditions. The problem of determining out-of-plane variations in
terms of amplitudes of the space harmonics within the diffraction grating and the subsequent matching in order to find the amplitude of the Rayleigh terms of the field in
the external media, has been solved by using the space-state method (111) in combination to a transmittance matrix formulation (T-Matrix). When T-Matrix formulation
is adopted, however, the numerical stability is critical, as discussed in section 5.2, in
which a new implementation of RCWA based on T-Matrix is presented.
Recently, RCWA has been used to study near field optical problems in corrugated
waveguides (114), nanostructured light emitting diodes (115), arbitrary aperiodic geometries (116) and to model the light propagation in solar cells (117). A detailed
review of the state-of-the-art implementations of RCWA is reported in (34), where the
author proposes a computationally efficient formulation of RCWA based on the solution
of an eigenvalue problem associated to a second order coupled-wave equation system
which reframes the Maxwells equations. In (118), a more general solution still based
on eigenvalues calculation is proposed, employing Jordan factorization. However, as
112
5.1 Introduction to RCWA
discussed in (118), the computational effort is a serious open issue for methods based
on eigenvalue calculation.
In this work three implementations of the RCWA method are presented. Section
5.2 and section 5.3 describe two approaches adopted for the solution of the electromagnetic problem in 2-D simulation domains previously proposed for the solution of the
Schrödinger equation. The proposed implementations are more computationally efficient with respect to standard solvers based on eigenvalue problems, since they do not
require the calculation of matrix exponentials and square roots. The first implementation, in particular, is computationally efficient for structures exhibiting thin layers
(with respect to the wavelength of the radiation) and rough surfaces. The second
method is more efficient for structures featuring thick grating regions, within which
the permittivity is not a uniform function of the position, although it requires more
memory than the first method. Both proposed implementations are not restricted to
periodic structures, but they can be applied to arbitrary geometries.
In section 5.4, an implementation of the RCWA aimed at solving the Maxwell
equations in 3-D simulation domains based on a more standard eigenvalue problem, is
presented. In such implementation the problem of finding the Fourier coefficients associated to the electromagnetic field is formulated as a second order differential equations
system. The proposed implementation is more general than that presented in (34), in
which the authors assume that the eigenvectors associated to the second order differential equations problem are not linearly independent; this assumption -which is however
reasonable for most practical applications- limits the applicability of the tool to materials (or to range of the wavelength of the radiation) for which the imaginary part of
the dielectric constant is zero (or significantly smaller with respect to the real part).
In the last section of this chapter, a validation by means of a comparison with
a commercial FDTD simulator as well as a discussion on accuracy and convergence
properties of the tools are presented. In addition, some applications of the proposed
implementations are described.
113
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
5.2
RCWA implementation based on discretization of the
Fourier series
In this section, a new and numerically stable RCWA implementation based on finite
difference approximation of the wave equations, in which the electromagnetic field and
the permittivity are expanded by means of truncated Fourier series, is presented. The
proposed T-Matrix formulation of the problem, which has been previously used to
numerically solve the Schrödinger equation in (119), has been successfully applied to
accurately model the light propagation in devices featuring particular kinds of front
surface texturing like grooves, rectangular texturing and non-periodic geometries like
random rough interfaces. The proposed method, unlike standard implementations proposed in literature, that are based on the solution of an eigenvalue problem, does not
involve the calculation of matrix exponentials or square roots; it simply requires fast
matrix multiplications; therefore, it is computationally efficient especially in case of
near field optical problems involving thin film layers with rough interfaces.
However, when the T-Matrix approach is adopted, an instability problem occurs
because of the presence of both growing and evanescent modes in the general solution
of the problem, that leads to a loss of significant digits when a large quantity is to
be summed to a significantly smaller one. This situation is particularly critical for
deep grating structures (34). To remove the instability problem, Moharam and other
authors proposed an improved T-Matrix method (110) that, however, features convergence problems when transverse magnetic polarization is assumed. In this work the
problem of the numerical instability is solved by using an arbitrary number of digits
(multi-precision arithmetic) with ad-hoc libraries for the implementations of critical
calculations.
The method is described in detail for the solution of the Maxwell equations in 2-D
simulation domains. However it can be straightforwardly extended to 3-D problems. In
order to describe the general solution of the electromagnetic problem, both the Transverse Electric (TE) and the Transverse Magnetic (TM) polarizations are considered.
The simulation domain is represented by a superstrate and a substrate regions where
the permittivity ε is constant and by a grating region where ε depends on position (Fig.
5.1).
114
5.2 RCWA implementation based on discretization of the Fourier series
5.2.0.1
Implementation for TE polarization
In the TE case, the electric field E = E(0, Ey , 0) is parallel to the y-axis and H lies in
the x-z plane. For the TM polarization, H is parallel to the y-axis and E lies in the
x-z plane. We start the description of the method from the Maxwell’s equations for a
linear isotropic and magnetically homogenous material without free charges.
In case of TE polarization, under the assumptions previously mentioned, the Maxwell’s
equations for the single nonzero components Ey of the electric field can be reformulated
by the following wave equation:
∆Ey = −k02 ε(x, z)Ey ,
Ex = Ez = 0,
(5.1)
where ε(x, y) is the relative permittivity, k02 = ε0 µ0 ω 2 = ω 2 c−2 = (2π)2 λ−2 , c is the
speed of light in vacuum, ω is the angular frequency, ε0 and µ0 are the permittivity
and the permeability of free space, respectively.
We consider a plane wave propagating in the superstrate region (Fig. 5.1). The
incident wave exhibiting a propagation direction forming an angle θ with the normal
to the device surface is described by
Eyinc = eik0 nI (x sin(θ)+z cos(θ)) ,
(5.2)
where nI is the refractive index of the superstrate. We assume that the grating
region features a periodicity p; therefore, lateral translation from the coordinate x to
x + p leads to multiplication of the incident wave by an exponential phase factor, as
expressed by the Floquet boundary condition (118):
Ey (x + p, z) = eik0 nI p sin(θ) Ey (x, z),
(5.3)
Therefore, if C(x, z) is a periodic function of the spatial coordinate x with periodicity p, by using the Fourier expansion of the electric field we obtain:
Ey (x, z) =
+∞
X
= eik0 nI x sin(θ) C(x, z) = eik0 nI x sin(θ)
s=−∞
115
cs (z)e
is(2π x
)
p
=
+∞
X
s=−∞
cs (z)eikxs x ,(5.4)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
λ
where kxs = k0 nI sin(θ) + s 2π
=
k
n
sin(θ)
+
s
and cs (z) are the Fourier
0
I
p
p
coefficients independent of x.
Therefore:
d2
2
cs (z) = k12 cs (z).
− 2 + kxs
dz
(5.5)
In addition, according to the spatial Fourier expansion of the permittivity along x:
ε(x, z) =
+∞
X
ih
εh (z)e
2πx
p
.
(5.6)
h=−∞
By applying the finite version of Laurent’s rule (120), adopting the simultaneous
truncation approximation in order to handle the product of the two Fourier expansions:
d2
d2
− 2 − 2 eik0 nI x sin(θ) C(x, z) = k02 ε(x, z)eik0 nI x sin(θ) C(x, z),
dz
dx
(5.7)
X
Mx
Mx
X
d2
d2
cs (z)eikxs x ,
− 2− 2
cs (z)eikxs x = k02 ε(x, z)
dz
dx
(5.8)
Mx
X
d2
2
− 2 + kxs cs (z) = k02
εs,h (z)ch (z).
dz
(5.9)
s=−Mx
s=−Mx
s=−Mx
Equating the corresponding terms on the left and right-hand side for s = −Mx . . . Mx :
+M
Xx
d2 cs (z)
=
As,h (z)ch (z).
dz 2
(5.10)
h=−Mx
where As,h (z) are linear combinations of the Fourier coefficients of the permittivity.
Differently from previous implementations of the RCWA, this system of ordinary differential equations is solved for the unknown coefficients ch (z) through discretization
on a uniform mesh with nodes in zk , k = 1 . . . N , and step ∆z. The finite difference
approximation leads to:
cs (zk−1 ) = −cs (zk+1 ) + 2cs (zk ) + ∆z 2
X
j
116
As,j (zk )cj (zk )
(5.11)
5.2 RCWA implementation based on discretization of the Fourier series
Figure 5.1: Sketch of the 2-D simulation domain which includes the superstrate, the
substrate and the grating region (left). A typical photovoltaic device is represented by a
multi-layer structure (right) in which each layer is homogenous and separated by smooth
or rough interfaces. The rough interfaces are treated as grating regions within which
the permittivity is not a uniform function of the position. The refractive index of the
superstrate is nI .
By adopting the vector notation:
c(zk−1 ) = −c(zk+1 ) + 2c(zk ) + ∆z 2 A(zk )c(zk ) = −c(zk+1 ) + Ã(zk )c(zk )
(5.12)
where c denotes the vector of the 2Mx + 1 coefficients in eq. 5.5. Therefore
c(zk−1 )
c(zk )
=
Ã(zk ) −I
I
0
c(zk )
c(zk+1 )
= T(zk )
c(zk )
c(zk+1 )
(5.13)
where I is a unity matrix. By applying the previous equation to all nodes zk we
obtain the following expression for the set of coefficients c at the top boundary of the
grating region:
c(z−1 )
c(z0 )
=
N
Y
k=0
T (zk )
c(zN )
c(zN +1 )
=T
c(zN )
c(zN +1 )
(5.14)
This equation relates the coefficients c at the top boundary with those at the bottom
one through the total transfer-matrix T which is obtained by multiplying the transfermatrices T(zk ) calculated at each node zk . This leads to a particularly computationally
117
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
efficient algorithm that does not involve to solve an eigenvalue problem, therefore,
differently from (34) and (118), it does not require the calculation of matrix exponentials
and roots, but only the multiplications of matrices. In fact, it is worth noting that the
matrices T(zk ) are simply calculated as linear combination of the Fourier coefficients
of the permittivity expansion.
In order to properly set the boundary conditions we express the coefficients cs in
terms of amplitudes of ingoing and outgoing waves:
−
cs (z) = c+
s (z) + cs (z)
(5.15)
In the substrate and superstrate, where the relative permittivity ε = εl with l=1,2
is constant:
+i
cs,l (z + ∆z) = c+
s,l (z)e
√
2 ∆z
k02 εl −kxs
−i
+ c−
s,l (z)e
√
2 ∆z
k02 εl −kxs
(5.16)
where l = 1 denotes the superstrate and l = 2 the substrate. In addition,
s,l
s,l
c+
s,l (z) = b11 cs,l (z) + b12 cs,l (z + ∆z)
(5.17)
s,l
s,l
c−
s,l (z) = b21 cs,l (z) + b22 cs,l (z + ∆z)
(5.18)
where
bs,l
11 =
bs,l
11 =
1 − e2i
−2i
1−e
1
√ 2
√
1
2 ∆z
k02 εl −kxs
2 ∆z
k0 εl −kxs
,
,
bs,l
12 =
bs,l
22 =
e
−i
ei
√
√
1
2 ∆z
k02 εl −kxs
− e−i
1
2 ∆z
k02 εl −kxs
+i
−e
√
√
2 ∆z
k02 εl −kxs
2 ∆z
k02 εl −kxs
,
= −bs,l
12 ,
Moreover
s,l −
+
cs,l (z) = b̃s,l
11 cs,l (z) + b̃12 cs,l (z)
(5.19)
s,l −
+
cs,l (z + ∆z) = b̃s,l
21 cs,l (z) + b̃22 cs,l (z)
(5.20)
where
118
5.2 RCWA implementation based on discretization of the Fourier series
b̃s,l
11 = 1,
+i
b̃s,l
21 = e
√
2 ∆z
k02 εl −kxs
b̃s,l
12 = 1,
−i
b̃s,l
22 = e
,
√
2 ∆z
k02 εl −kxs
,
The eq. 5.13 can be reformulated as:
c(z−3 )
c(z−2 )
= T(z−2 )
= T(z−2 ) T(z−1 ) T(z0 )
=
N
+2
Y
c(z−2 )
c(z−1 )
c(z0 )
c(z1 )
T(zk )
k=−2
= T(z−2 ) T(z−1 )
c(z−1 )
c(z−0 )
= T(z−2 )T(z−1 )T(z0 )
c(zN +2 )
c(zN +3 )
c(zN +2 )
c(zN +3 )
=T
c(z0 )
c(z1 )
=
= ... =
(5.21)
By applying the eq. 5.19, the eq. 5.20 and using the eq. 5.21,
c+ (z−3 )
c− (z−3 )
=B
= B · T · B̃
c(z−3 )
c(z−2 )
c+ (zN +2 )
c− (zN +2 )
=B·T
= T̃
c(zN +2 )
c(zN +3 )
c+ (zN +2 )
c− (zN +2 )
=
(5.22)
where B̃ is a (4Mx + 2)-order square matrix:





B̃ = 



x
b−M
11
0
0
−Mx
b12
0
0
0
0
... 0
x
0 bM
11
0
0
... 0
x
0 bM
21
−Mx
b12
0
0
x
b−M
22
0
0
0
0
... 0
x
0 bM
12
0
0
... 0
x
0 bM
22





.



(5.23)
At the top and at the bottom of the grating region, for z = z−3 and z = zN +2 ,
respectively, the following boundary conditions are adopted:


0
...
0




+
ik0 nI z−3 cos(θ)
c (z−3 ) = 
 e

0


...
0
119




,




c− (zN +2 ) = 0,
(5.24)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
which allow to obtain:
c+ (z−3 )
0
=
T11 0
T21 −I
=B·T
c+ (zN +2 )
c− (z−3 )
(5.25)
where c− (z−3 ), c+ (zN +2 ) are reflection and transmission coefficients, respectively.
The coefficients c+ (z−3 ), which are defined by the eq. 5.24, represent the incident
wave. To solve the well known numerical stability issues of the T-matrix method,
multi-precision arithmetic is adopted by dynamically adjusting the amount of digits
in numbers with multi-precision representation. However, the use of the arbitrary
multi-precision is limited to some critical calculations (for instance to accumulate the
resulting transfer matrix in eq. 5.14) and double precision is adopted otherwise to save
computational time. The proposed method may work with complicated and arbitrary
geometrical configurations of the grating regions.
5.2.0.2
Implementation for TM polarization
In case of TM polarization, the Maxwell’s equations for the single nonzero components
Hy of the electric field leads to the following equation:
1
∆H = −εµω 2 H − ∇(ε) × ∇ × H,
ε
(5.26)
that can be expressed as
∂
∂z
1 ∂Hy
ε(x, z) ∂z
∂
=−
∂x
1 ∂Hy
ε(x, z) ∂x
− k02 Hy ,
Hx = Hz = 0,
(5.27)
The incident wave, similarly to the TE polarization is
Hyinc = eik0 nI (x sin(θ)+z cos(θ)) .
(5.28)
The component Hy (x, z) can be expressed as a truncated Fourier expansion by
means of coefficients cs (z):
Hy (x, z) = eik0 nI x sin(θ)
+M
Xx
is(2π x
)
p
cs (z)e
s=−Mx
=
+M
Xx
s=−Mx
120
cs (z)eikxs x ,
(5.29)
5.2 RCWA implementation based on discretization of the Fourier series
∂Hy
1
ε(x,z) and (x, z) → ∂x
1 ∂Hy
ε(x,z) ∂x is continuous
In 5.27 the functions (x, z) →
however the function (x, z) →
are only piecewise continuous,
(118), therefore the following
Fourier expansion can be defined:
e−ik0 nI x sin(θ)
+M
Xx
1 ∂Hy
is(2π x
)
p ,
=
c1,s (z)e
ε(x, z) ∂x
(5.30)
s=−Mx
where the coefficients c1 are given by
c1 (z) = i [ε(z)]−1 Kc(z),
(5.31)
and K is the diagonal matrix with kxs elements.
Following the approach proposed in (120),
+M
Xx
i ∂Hy
c1s (z)eikxs x ,
=
ε(x, z) ∂x
(5.32)
s=−Mx
∂
∂x
1 ∂Hy
ε(x, z) ∂x
+
k02 Hy
=
+M
Xx
ic1s (z)kxs + k02 cs (z) eikxs x .
(5.33)
s=−Mx
From eq. 5.27 and eq. 5.33,
∂
∂z
+M
Xx dcs (z)
1
eikxs x
ε(x, z)
dz
!
=−
s=−Mx
+M
Xx
ic1s (z)kxs + k02 cs (z) eikxs x .
(5.34)
s=−Mx
The finite difference approximation of eq. 5.34, for each k-th node is:
+M
Xx
!
cs (zk+1 ) − cs (zk ) cs (zk ) − cs (zk−1 )
−
eikxs x =
ε(x, zk+ 1 )
ε(x, zk− 1 )
s=−Mx
2
2
ik x
P+Mx
2
2
xs ,
= −∆z
s=−Mx ic1s (z)kxs + k0 cs (z) e
(5.35)
that can be expressed in matrix notation as:
"
#
1
ε(zk+ 1 )
2
"
(c(zk+1 − c(zk )) −
#
1
ε(zk− 1 )
2
121
c(zk − c(zk−1 ) = A1 c(zk ),
(5.36)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
where
A1 = ∆z 2 K [ε(zk )]−1 K − k02 I .
(5.37)
By rearranging all terms, similarly to the TE case,
c(zk−1 )
c(zk )
=
T1,1 T1,2
I
0
"
T1,1 =
#−1
1
c(zk )
c(zk+1 )
"
ε(zk− 1 )
T1,2 = −
= T(zk )
c(zk )
c(zk+1 )
(5.38)
!
#
1
+ I,
+ A1
2
#−1 "
1
ε(zk− 1 )
2
5.3
ε(zk+ 1 )
2
"
#
1
ε(zk+ 1 )
.
(5.39)
2
RCWA implementation based on auxiliary functions
In this section the electromagnetic field within the grating region is expressed by means
of an expansion through auxiliary functions which are determined by using a numerical
discretization technique. The amplitude of the harmonics within the grating region are
then matched with those of the field in the external region by using Dirichlet boundary
conditions on the auxiliary functions as well as on the amplitude and on the first spatial
derivative of the field amplitude with respect to the propagation direction. The adoption of the auxiliary functions is reported also in (121) for the solution of the Schrödinger
equations; in the mentioned work, however, the auxiliary functions are determined analytically since they are applied in regions featuring simple and known geometries, like
rectangular-shaped domains. This approach allows to limit the extension of the discretization grid only within the grating region, differently from the method presented in
section 5.2, for which, in order to numerically solve the approximated wave functions,
the discretization is adopted throughout the simulation domain. An advantage of this
approach, in comparison with the first implementation, is the enhanced computational
efficiency for structures exhibiting grating regions in combination to relatively thick
(with reference to the wavelength of the radiation) homogenous layers, within which,
in order to save simulation time, the solution can be calculated by using an analytical
approach.
122
5.3 RCWA implementation based on auxiliary functions
5.3.1
Auxiliary functions and boundary conditions
In the following a common approach to the solution of the electromagnetic problem
for TE and TM polarization is described. The function φ(x, z) denotes either the component Ey (x, z) or the component Hy (x, z) of the filed depending upon the particular
polarization, TE or TM, respectively. The function φ(x, z) can be therefore expressed
as:
 P
√
√
2 (z−z )
2 (z−z )
N
− −i k12 −kxs
+ i k12 −kxs
1
1

+
c
e
eikxs x , z ≤ z1
c
e

1,s
 Ps=−N 1,s
N
φ(x, z) =
z 1 ≤ z ≤ z2
s=−N (d
1,s χ1,s
√(x, z)2 + d2,s χ2,s (x, z))√, 2 2


+ i k22 −kxs
− −i k2 −kxs (z−z1 )
(z−z2 )
ik
x
 PN
xs
+ c2,s e
e
, z ≥ z2
s=−N c2,s e
(5.40)
√
where kl = k0 εl , l = 1, 2 (ε1 and ε2 are the values of permittivity of the superstrate and of the substrate, respectively. According to eq. 5.40, the function φ(x, z)
is expanded into 2N + 1 terms within the grating region (z1 ≤ z ≤ z2 ) by means of
the auxiliary functions χj,s , j = 1, 2, s = −N.. + N . The following Dirichlet boundary
conditions are applied at the boundaries of the grating regions (z = z1 , z = z2 ) for
s = −N.. + N :
χ1,s |z=z1 = eikxs x ,
χ1,s |z=z2 = 0,
χ2,s |z=z1 = 0 χ2,s |z=z2 = eikxs x .
(5.41)
In addition, the following periodic Floquet boundary condition is applied along the
x-axis:
χj,s (x + p, z) = eik0 NI psin(θ) χj,s (x, z),
j = 1, 2,
s = −N.. + N.
(5.42)
In case of TE polarization, at each boundary of the grating region, the continuity
of the electric field and of its first derivative with respect to z have to be satisfied:
∂Ey
∂Ey
|z=zi−0 =
|z=zi+0 ,
(5.43)
∂z
∂z
where i = 1 for the top boundary (z = z1 ) and i = 2 for the bottom boundary
Ey |z=zi−0 = Ey |z=zi+0 ,
(z = z2 ) of the grating region. Similarly for the TM polarization:
Hy |z=zi−0 = Hy |z=zi+0 ,
1 ∂Hy
1 ∂Hy
|z=zi−0 =
|z=zi+0 ,
ε(x, z) ∂z
ε(x, z) ∂z
123
(5.44)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
5.3.2
Solving for the auxiliary function
In the following the algorithm used to solve the eq. 5.40 for the auxiliary function
is described. The auxiliary functions χs,j with s = −N..N,
j = 1, 2 are required
to be solution of the Helmholtz equation 5.1 (for TE polarization) or of the eq. 5.26
(for TM polarization). The proposed method to solve the Helmholtz equations relies
on the Fourier expansion of the auxiliary functions along x-axis, by using, in order to
simplify the algorithm, the same truncation order N adopted in eq. 5.40. Therefore, if
a uniform discretization grid of Nz points is adopted along the propagation direction
z, for each node zk with k = 0, 1, ..., Nz − 1:
χs,j (x, zk ) =
N
X
= χ̂(zk )eikxh x ,
(5.45)
h=−N
where χ̂(zk ) denote the Fourier coefficient of the auxiliary function expansion. The
Helmholtz equations are then discretized on the uniform mesh along z; for each node
zk :
N
N
X
X
χ̂h (zk−1 ) − 2χ̂h (zk ) + χ̂h (zk+1 ) ikxh x
2
χ̂h (zk )eikxh x =
−Kxh
e
+
(∆z)2
h=−N
h=−N
=
k02 ε(x, zk )
N
X
χ̂h (zk )eikxh x .
(5.46)
h=−N
Both terms of eq. 5.46 are multiplied by e−ikxj x ,
i, j = 1..N and integrated
over the x-axis from x = 0 to x = p, where p is the periodicity of the grating region.
Therefore:
Z
p
e
i(kxh −kxj )x
dx = δ(h, j) =
0
1 if
0 if
h = j,
h 6= j.
(5.47)
Eq. 5.46 can be then reformulated as:
N
X
χ̂j (zk−1 ) − 2χ̂j (zk ) + χ̂j (zk+1 )
2
2
−
K
χ̂
(z
)
=
k
εh,j (zk )χ̂h (zk ),
xj j k
0
(∆z)2
h=−N
124
(5.48)
5.3 RCWA implementation based on auxiliary functions
where
Z
p
εh,j (zk ) =
ε(x, zk )ei(kxh −kxj )x dx.
(5.49)
0
If k = 0 (top boundary), by applying the boundary conditions expressed by eq.
5.41,
χ1,s (x, z0 ) = eikxs x =
N
X
χ̂h (zk )eikxh x
(5.50)
h=−N
which requires that χh=s (z0 ) = 1,
χh6=s (z0 ) = 0. Similarly for the bottom
boundary of the grating region (z = zNz −1 ).
For k = 1, 2, ..., Nz − 2, multiplying by ∆z 2 and rearranging eq. 5.48,
−χ̂j (zk−1 ) +
+N
X
2
2 − ∆z 2 kxj
δhj − ∆2z k02 εh,j (zk ) χ̂j (zk ) − χ̂j (zk+1 ) = 0
(5.51)
i=−N
Similarly, for k = 0, using eq. 5.50, eq. 5.48 reads
+N
X
2
2 − ∆z 2 kxj
δhj − ∆2z k02 εh,j (z0 ) χ̂j (z0 ) − χ̂j (z1 ) = δs,h
(5.52)
h=−N
Finally, if k = Nz − 1,
+N
X
2
2 − ∆z 2 kxj
δhj − ∆2z k02 εh,j (zNz −1 ) χ̂j (zNz −1 ) − χ̂j (zNz −3 ) = 0.
(5.53)
h=−N
In vector notation, equations from 5.51 to 5.53 can be written as:
à · χ̂ = B
(5.54)
where à is the tridiagonal block matrix of elements [ai,j ] ,




A=



[a1,1 ] [a1,2 ]
0
···
0
[a2,1 ] [a2,2 ] [a2,3 ]
0
···
0
[a3,2 ] [a3,3 ] [a3,4 ] 0
0
0
···
··· ···
···
···
···
··· ···
0
0
0
0
0
0
0
···
0
···
0
0
0
···
···
i, j = 1 · · · Nz−1 :
0
0
0
0
···
[aNz −4,Nz −4 ] [aNz −3,Nz −3 ] [aNz −2,Nz −2 ]
125




.



5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
2
[an,n−1 ] = − Kjx
2
[an,n ] = 2 − ∆z 2 kxj
δij − ∆2z k02 εi,j (zNz −1 )
2
[an,n+1 ] = − Kjx
 h
2
K−N,x



B=


i
0
h
0
2
K−N
+1,x
i
0
0
0
0
0
0
0
···
···
···
···
0
0
0
···
0
[χ̂s=−N (z1 )]
[χ̂s=−N +1 (z1 )]
 [χ̂s=−N (z2 )]
[χ̂s=−N +1 (z2 )]
χ̂ = 

···
···
[χ̂s=−N (zNz −2 )] [χ̂s=−N +1 (zNz −2 )]
···
···
···
···



.

h ··· i 
2
KN,x
0
···



[χ̂s=+N (z1 )]
[χ̂s=+N (z2 )] 
.

···
[χ̂s=+N (zNz −2 )]
The system of linear equations 5.54 is solved by means of standard algorithms in
order to determine the Fourier coefficients χ̂i and subsequently the auxiliary functions
χs,j .
5.3.3
Determining the electromagnetic field
Once determined the auxiliary function as described in the previous subsection, the
problem of finding the coefficients cj,s and dj,s (with j = 1, 2 and s = −N · · · + N ) of
the system of equations 5.40 can be solved by applying the matching conditions 5.43
or 5.44, depending upon the polarization; for TE polarization, by applying the first of
the eq.5.43 :
+N X
c+
1,s
+
c−
1,s
ikxs x
e
s=−N
+N X
s=−N
N
X
=
(d1,s χ1,s (x, z1 ) + d2,s χ2,s (x, z1 )) ,
(5.55)
s=−N
c+
2,s
+
c−
2,s
ikxs x
e
=
N
X
(d1,s χ1,s (x, z2 ) + d2,s χ2,s (x, z2 )) .
−N
By applying the second of the eq. 5.43,
126
(5.56)
5.3 RCWA implementation based on auxiliary functions
+N q
X
−
ikxs x
2
i k12 − kxs
c+
−
c
=
1,s
1,s e
s=−N
N X
∂χ2,s (x, z)
∂χ1,s (x, z)
|z=z1 + d2,s
|z=z1 ,
=
d1,s
∂z
∂z
(5.57)
−N
+N q
X
−
ikxs x
2
i k22 − kxs
c+
−
c
=
2,s
2,s e
s=−N
=
N
X
s=−N
∂χ2,s (x, z)
∂χ1,s (x, z)
|z=z2 + d2,s
|z=z2 .
d1,s
∂z
∂z
(5.58)
By multiplying by eikxs1 x and integrating the eq. from 5.55 to 5.58 over x-axis from
x = 0 to x = p,
−
c+
1,s1 + c1,s1 = d1,s1 ,
(5.59)
−
c+
2,s1 + c2,s1 = d2,s1 ,
(5.60)
q
+
−
2
i k12 − kxs
c
−
c
1,s1
1,s1 =
1
=
N
X
s=−N
1
d1,s
p
Z
p
e−ikxs1 x
0
d2,s
1
p
p
Z
∂χ1,s (x, z)
|z=z1 dx +
∂z
e−ikxs1 x
0
∂χ2,s (x, z)
|z=z1 dx,
∂z
(5.61)
q
+
−
2
i k22 − kxs
c
−
c
2,s1
2,s1 =
1
=
N
X
s=−N
1
d1,s
p
Z
p
e−ikxs1 x
0
1
d2,s
p
Z
p
∂χ1,s (x, z)
|z=z2 dx +
∂z
e−ikxs1 x
0
127
∂χ2,s (x, z)
|z=z2 dx,
∂z
(5.62)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
which can be expressed in vector-matrix notation as:
 +
c + c−

1 = d1

 1+
c2 + c−
2 = d2
+
 F1 c1 − c−
1 = A1,1 d1 + A1,2 d2


−
F2 c+
−
c
1
1 = A2,1 d1 + A2,2 d2
(5.63)
or
−
+
−
+
−
F1 c+
1 − c1 = A1,1 c1 + c1 + A1,2 c2 + c2 −
+
−
+
−
F2 c+
1 − c1 = A2,1 c1 + c1 + A2,2 c2 + c2
(5.64)
In S-matrix formulation:
c+
2
c−
1
=S
c+
1
c−
2
.
(5.65)
where
S=
A1,2
A1,1 + F1
A2,2 − F2
A2,1
−1 F1 − A1,1
−A1,2
−A2,1
−A2,2 − F2
.
(5.66)
In T-matrix formulation:
c+
2
c−
2
c+
1
c−
1
T1,1 T1,2
T2,1 T2,2
=T
.
(5.67)
.
(5.68)
where T, in block matrix representation, is:
T=
and
S=
5.3.4
−1
T1,1 − T1,2 T−1
2,2 T2,1 T1,2 T2,2
−1
−1
−T2,2 T2,1
T2,2
.
(5.69)
Solution in homogenous layers
Within homogenous layers, in which the permittivity ε is uniform, the matrix S can be
calculated by using an analytical formula, leading to reduced computation time. For a
homogenous layer of thickness ∆z (Fig. 5.2),
128
5.3 RCWA implementation based on auxiliary functions
"
=
e+i
√
c+ (z + ∆z)
c− (z)
2 ∆z
kl2 −kxs
0
c+ (z)
=S
=
c− (z + ∆z)
#
0
c+ (z)
√ 2 2
.
c− (z + ∆z)
e+i kl −kxs ∆z
(5.70)
Figure 5.2: Cross section of a homogenous layer with the coefficients of the incident,
reflected and transmitted waves.
If the scattering matrices of two layers are known (S(1) and S(2) , respectively), the
total scattering matrix S(12) is
"
S(12) = S(1)
O
S(2) =
(12)
S1,2
(12)
S2,2 .
S1,1
S2,1
(12)
(12)
#
.
where
(12)
(2)
(1) (2) −1 (1)
S1,1 = S1,1 1 − S1,2 S2,1
S1,1 ,
(12)
(2)
(2)
(1) (2) −1 (1) (2)
S1,2 = S1,2 + S1,1 1 − S1,2 S2,1
S1,2 S2,2 ,
(12)
(1)
(1) (2)
(1) (2) −1 (1)
S2,1 = S2,1 + S2,2 S2,1 1 − S1,2 S2,1
S1,1 ,
129
(5.71)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
(12)
S2,2
=
(1)
S2,2
(2)
(1) (2) −1 (1)
(2)
S2,1 1 − S1,2 S2,1
S1,2 + 1 S2,2 .
Finally, if S denotes the total scattering matrix of the entire structure, it is possible
to relate the coefficients of the incident (cIN ), reflected (cRL ) waves to those of the
transmitted wave (cT R ):
cT R
cRL
=S
cIN
0
.
(5.72)
where the amplitude of the incident wave from the substrate is supposed to be zero.
5.4
RCWA implementation based on solution of an eigenvalues problem
In this section, a RCWA implementation based on a more standard eigenvalues calculation problem for 3-D structures is described. The method follows the same approach
proposed in (34), by removing some assumptions that potentially limit its applicability
to a restricted range of structures.
5.4.1
Problem definition
The following incident plane wave is considered according to Fig. 5.3:
EIN C (r) = E0 eik1 r = E0 eik0 nI (xsin(θ)cos(φ)+ysin(θ)cos(φ)+zcos(θ)) ,
(5.73)
where k02 = ε0 µ0 ω 2 = (2πλ−1 )2 and nI is the refractive index of superstrate. The
grating region is supposed to be p-periodic, where p = px îx + py îy is a vector of the
xy-plane. According to the Floquet’s theorem for diffraction gratings,
E(r) = eik0 nI (xîx +yîy ) S(r) = eik1 (xîx +yîy ) S(r) = eik1 p S(r),
(5.74)
where S(r) is a p-periodic function of the position. Similarly for the magnetic field:
H(r) = −i
ε0
µ0
1
130
2
eik1 p U(r).
(5.75)
5.4 RCWA implementation based on solution of an eigenvalues problem
Finally, the Maxwell equations, in case of time-harmonic fields, assuming µ = 1 and
σ = 0, read (eq. 3.48 ):
∇ × E(r) = jωµ0 H(r)
∇ × H(r) = −jωε(r)ε0 E(r)
(5.76)
Figure 5.3: Schematic of the incidence plane for 3-D optical simulations. The direction of
wave propagation vector k1 is defined by angles θ and φ with respect to z-axis and x-axis,
respectively.
5.4.2
Expansion of the field in Fourier series
The fields S(r) and U(r) defined in the previous section are expanded according to
Fourier sums, therefore the expressions of the electromagnetic field read:
E(r) = eik1 p S(r) =
∞
X
∞
X
Six ,iy (z)ei(kx,ix x+ky,iy y) ,
(5.77)
ix =−∞ iy =−∞
H(r) = −i
= −i
ε0
µ0
1 X
∞
2
∞
X
ε0
µ0
1
2
eik1 p U(r) =
Uix ,iy (z)ei(kx,ix x+ky,iy y) ,
ix =−∞ iy =−∞
131
(5.78)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
where Six ,iy and Uix ,iy denote the Fourier coefficients, kx,ix = k0 nI sin(θ)cos(φ) + ix pλx
and ky,iy = k0 nI sin(θ)cos(φ) + iy pλy . The expansions of eq. 5.77 and eq. 5.78
are truncated to a finite order, Mx and My for the x-axis and the y-axis, respectively.
Therefore, by replacing the eq. 5.77 and 5.78 into eq. 5.76, the following scalar
equations are obtained:
∂Sy,ix ,iy (z)
= k0 Ux,ix ,iy
∂z
∂Sx,ix ,iy (z)
−ikx,ix Sz,ix ,iy (z) +
= k0 Uy,ix ,iy
∂z
ikx,iy Sy,ix ,iy (z) − iky,iy Sx,ix ,iy (z) = k0 Uz,ix ,iy
iky,iy Sz,ix ,iy (z) −
ky,iy Uz,ix ,iy (z) + i
(5.79)
(5.80)
(5.81)
X
∂Uy,ix ,iy (z)
= −ik0
Sx,jx ,jy A1,ix ,iy ,jx ,jy
∂z
(5.82)
jx ,jy
X
∂Ux,ix ,iy (z)
= −ik0
Sy,jx ,jy A2,ix ,iy ,jx ,jy
∂z
jx ,jy
X
kx,ix Uy,ix ,iy (z) − iky,iy Ux,ix ,iy (z) = −ik0
Sz,jx ,jy A3,ix ,iy ,jx ,jy
−kx,ix Uz,ix ,iy (z) − i
(5.83)
(5.84)
jx ,jy
The matrix A is a Lifeng Li’s permittivity matrix (122).
The tangential components of the electric field are continuous because of the absence
of surface current density, therefore the component Ex of the electric field is a continuous
function of y and z; similarly, Ez is continuous along x and y. In addition, the normal
components of the electric displacement are continuous due to the absence of surface
charge density and consequently the product εEx is continuous along x-axis, εEy is
continuous along y-axis and εEz is continuous along z-axis. Since the permittivity
ε(r) may exhibit a discontinuity, the Fourier transform of
1
ε(r)
is calculated along the
x-axis, where also the electric field is discontinuous.
Then the permittivity matrix
h
i−1 1
coefficients have the form An,ix ,jx ,iy ,jy = ε(r)
for n = 1. In order to obtain
ix ,jx i ,j
y y
the coefficients An,ix ,jx ,iy ,jy ,
n = 2, the Fourier series associated to
1
ε(r)
along y-
axis is calculated. Subsequently,
for each value of x, the inverse matrix is calculated,
h
i−1 1
therefore A2,ix ,jx ,iy ,jy = ε(r)
. Since Ez is continuous along x and y, simply
iy ,jy i ,j
x
x
h
i
A3,ix ,jx ,iy ,jy = [ε(r)]iy ,jy
.
ix ,jx
132
5.4 RCWA implementation based on solution of an eigenvalues problem
Eq. 5.79 and 5.82 in matrix notation are:
∂sy (z)
= iky sz (z) − ux ,
k0 ∂z
∂sx (z)
= ikx sz (z) + uy ,
k0 ∂z
uz = ikx sy (z) − iky sx (z),
∂uy (z)
= −A1 sx (z) + iky uz (z),
k0 ∂z
∂ux (z)
= A2 sy (z) + ikx uz (z),
k0 ∂z
sz (z) = iA−1
3 (kx uy (z) − ky ux (z)) ,
where kx =
1
k0
[kx,ix ] and ky =
1
k0
(5.85)
(5.86)
(5.87)
(5.88)
(5.89)
(5.90)
ky,iy are diagonal matrices. By removing the
z-component from eq. 5.85 and 5.88,
∂sx (z)
−1
= kx A−1
3 ky ux (z) − kx A3 kx − I uy (z)
k0 ∂z
∂sy (z)
−1
= ky A−1
3 ky − I ux (z) − ky A3 kx uy (z)
k0 ∂z
∂ux (z)
= kx ky sx (z) + A2 − k2x sy (z)
k0 ∂z
∂uy (z)
= − A1 − k2y sx (z) − ky kx sy (z)
k0 ∂z
(5.91)
or, in more compact notation,
∂s(z)
= Qu(z)
k0 ∂z
∂u(z)
= Fs(z).
k0 ∂z
(5.92)
Each system of equations in 5.92 is a set of 2Mx My equations.
As shown in Fig. 5.4, the simulation domain is split in N layers with interfaces
parallel to the xy-plane. Within each layer, the permittivity ε(r) is replaced by a step
function for which the permittivity is constant along z-axis. This approximation allows
133
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
to treat an arbitrary surface profile as a set of 2-D rectangular grating regions. The
thickness ∆zq ,
q = 1..N of each layer must be carefully chosen, with regarding to
the spatial variation of the surface and the wavelength of the radiation. In order to
reduce by a factor 2 the number of equations, the first two systems of equations in 5.91
are differentiated with respect of the variable z, then the derivatives of the magnetic
field coefficients are replaced into the third and the fourth equation. In addition, due to
the approximation, within each layer, the spatial derivative of the permittivity matrix
is zero. Therefore:
"
∂ 2 sx (z)
∂ z̃ 2
∂ 2 sy (z)
∂ z̃ 2
#
=
2
kx A−1
3 kx − I A1 +ky
ky A−1
3 kx A1 − kx
kx A−1
sx (z)
3 ky A2 − ky
,
2
sy (z)
ky A−1
3 ky − I A 2 + kx
or
∂ 2 s(z)
= −Ks(z)
∂ z̃ 2
(5.93)
where z̃ = k0 z.
Figure 5.4: Sketch of the simulation domain (the top figure reports a 3-D pyramid) split
in layers within which the permittivity is approximated by a step function. The bottom
figure shows a schematic of a single layer with the Fourier coefficients of the electric field
at the boundaries.
134
5.4 RCWA implementation based on solution of an eigenvalues problem
The solution in terms of spatial harmonics of the electric field of the problem of
eq. 5.93 within each homogenous layer (zq ≤ z ≤ zq+1 = zq + ∆zq ), calculated at the
generic point (zq + ∆z) is (Fig. 5.4):
s(zq + ∆z) = eik0
√
s (z) + e−ik0
K∆z +
√
K∆z −
s (z)
(5.94)
As in the standard implementation of the RCWA method, the coupled-wave equations of eq. 5.93 are solved by finding the eigenvalues and eigenvectors of K defined in
eq. 5.93, which is a matrix of rank 2Mx My . It is worth noting that, the matrix K, in
the most general case, is not an Hermitian matrix, as assumed in (34), therefore it is
not possible to write K as a diagonal matrix, in which, the diagonal elements are the
positive square roots of the eigenvalues of the matrix. By removing this assumption,
which requires that the eigenvectors associated to the matrix K are not linearly independent, the implementation proposed in this work, is the most general possible; the
assumption of considering K a Hermitian matrix -which is however reasonable for most
practical applications- limits the applicability of the tool to materials (or to range of
the wavelength of the radiation) for which the imaginary part of the dielectric constant
is zero (or significantly smaller with respect to the real part).
At the boundaries of each layer the continuity of the tangential components of the
electric and magnetic fields must be ensured, as expressed by:
Ex |z−0 = Ex |z+0 ,
Ey |z−0 = Ey |z+0 ,
(5.95)
Hx |z−0 = Hx |z+0 ,
Hy |z−0 = Hy |z+0 ,
(5.96)
which lead to the following condition on the Fourier coefficients of the tangential
components of the field:
Sx,ix ,iy |z−0 = Sx,ix ,iy |z+0 ,
Sy,ix ,iy |z−0 = Sy,ix ,iy |z+0 ,
Ux,ix ,iy |z−0 = Ux,ix ,iy |z+0 ,
Uy,ix ,iy |z−0 = Uy,ix ,iy |z+0 ,
(5.97)
u|z−0 = u|z+0 ,
(5.98)
or in vector form:
s|z−0 = s|z+0 ,
135
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
In order to calculate the coefficients s and u at the bottom boundary of the q-th
layer (zq ≤ z ≤ zq + ∆q , Fig. 5.4), from eq. 5.92 and 5.94, by applying the boundary
conditions of eq. 5.97:
s(zq + ∆zq − 0) = eik0
√
Kq ∆zq +
sq (z)
+ e−ik0
√
Kq ∆zq −
sq (z)
=
−
= s+
q+1 + sq+1 = s(zq + ∆zq + 0),
(5.99)
∂s
u(zq + ∆zq − 0) = Q−1
|z +∆zq −0 =
q
k0 ∂z q
√
p ik √K ∆z +
q
q
= iQ−1
Kq e 0
sq (z) − e−ik0 Kq ∆zq s−
q (z) =
q
p
−
+
= iQ−1
q+1 Kq+1 sq+1 (z) − sq+1 (z) =
= Q−1
q+1
∂s
|z +∆zq +0 = u|zq +∆zq +0
k0 ∂z q
(5.100)
or
−
+
−1 −
Pq s+
q + Pq sq = sq+1 + sq+1
−
+
−1 −
−
s
Pq s +
−
P
s
=
W
s
q+1
q
q
q
q+1 ,
q+1
Wq
(5.101)
where
Pq = eik0
√
Kq ∆z
,
Wq = Q−1
q
Wq+1 = Q−1
q+1
p
Kq ,
p
Kq+1 .
(5.102)
The Fourier coefficients of the electric field at the boundaries of the q-th layer can
be linked by the scattering matrix S(q→q+1) :
s+
q+1
s−
q
(q→q+1)
=S
s+
q
s−
q+1
.
(5.103)
where
"
S(q→q+1) =
(q→q+1)
S1,2
(q→q+1)
S2,2
S1,1
S2,1
136
(q→q+1)
(q→q+1)
#
.
(5.104)
5.4 RCWA implementation based on solution of an eigenvalues problem
(q→q+1)
S1,1
(q→q+1)
S1,2
(q→q+1)
S2,1
= 2 (Wq + Wq+1 )−1 Wq Pq ,
= − (Wq + Wq+1 )−1 (Wq − Wq+1 ) ,
= Pq (Wq + Wq+1 )−1 (Wq − Wq+1 ) Pq ,
(q→q+1)
S2,2
= 2Pq (Wq + Wq+1 )−1 Wq+1 .
When all layers are accounted (n = 1 · · · N ),
s+
N
s−
1
=S
(1→N )
s+
1
s−
N
.
(5.105)
where the scattering matrix S(1→N ) is calculated similarly to that of eq. 5.71.
Finally, the coefficients of the incident field (sI ) are linked to those of the reflected and
transmitted fields (sR and sT , respectively) by the following equation:
sT
sR
=S
B,z=0
O
S
(1→N )
sI
0
.
(5.106)
where the matrix SB,z=0 is given by:
"
SB,z=0 =
SB,z=0
SB,z=0
1,1
1,2
B,z=0
S
SB,z=0
2,2
2,1
#
.
= 1 + (Wq + Wq+1 )−1 (Wq − Wq+1 ) ,
SB,z=0
1,1
SB,z=0
= 2 (Wq + Wq+1 )−1 Wq+1 − 1,
1,2
SB,z=0
= (Wq + Wq+1 )−1 (Wq − Wq+1 ) ,
2,1
SB,z=0
2 (Wq + Wq+1 )−1 Wq+1 ,
2,2
and the symbol
N
denotes the matrix operator already defined in eq. 5.71.
137
(5.107)
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
5.4.3
Solution within homogenous layer
Within homogenous layers with parallel front and bottom interfaces, the solution can
be significantly simplified. Indeed, the coefficient of the magnetic and electric fields
are not coupled, therefore eq. 5.91 represents in this case a system of independent
equations. The permittivity matrix can be replaced by the (constant) permittivity εl
of the layer, hence:
"
∂ 2 sx,ix ,iy (z)
∂ z̃ 2
∂ 2 sy,ix ,iy (z)
∂ z̃ 2
#
1
= 2
k0
"
2
2
kx,i
+ ky,i
− kl2
0
x
y
2
2
0
kx,ix + ky,i
− kl2
y
#
sx,ix ,iy (z)
sy,ix ,iy (z)
, (5.108)
where z̃ = k0 z and kl2 = k02 εl . Matrices Q, K and P are in the form:
Ql,ix ,iy =
1
kl2
Kl,ix ,iy =
Pl,ix ,iy = e
2
+ kl2
kx,ix ky,iy −kx,i
x
2
− kl2 −kx,ix ky,iy
ky,i
y
2
2
+
k
kl2 − kx,i
y,iy
x
k02
i
,
(5.109)
I,
r
2
2
+ky,i
∆z
k02 εl − kx,i
y
x
(5.110)
I.
(5.111)
where I denotes the unity matrix of size 2. Therefore matrix W can be written as:
Wl,ix ,iy =
Q−1
l
k02
5.4.4
1
p
Kl =
r
2
2
kl2 − kx,i
+
k
y,iy
x
−kx,ix ky,iy
2
kl2 − ky,i
y
2
−kl2 + kx,i
x
kx,ix ky,iy
. (5.112)
Calculation of the optical generation rate
The flux of energy through a plane perpendicular to z-axis (Fig. 5.3) is:
1
Sz = Re Ex Hy∗ − Hx∗ Ey
2
(5.113)
where Sz is the z-component of the Poynting vector, defined as:
1
S = Re [E × H∗ ] .
2
Integrating with respect of x, y:
138
(5.114)
5.5 Validation of RCWA method and accuracy
px Z py
Z
Sz (x, y, z)dxdy =
0
1
=
2
r

ε0
px py Im 
µ0
∞
X
h
0

i
sy,ix ,iy (z)u∗x,ix ,iy (z) − sx,ix ,iy (z)u∗y,ix ,iy (z)  .
(5.115)
ix ,iy =−∞
In order to calculate the energy absorption per unit time and volume, the divergence
of the Poynting vector must be calculated:
1
1
S = Re (∇, [E, H∗ ]) =
Im (E, ∇ × ∇ × E∗ )
2
2µ0 ω
(5.116)
Since, from Maxwell equations (Eq. 3.36) ∇x∇xE∗ (r) = ω 2 µ0 ε0 ε(r)E(r), and being
the optical generation rate equal to the absorbed energy to the photon energy,
Gopt (ω) = −
2πdivS
21piε0 Im(ε)|E|2
=
,
hω
2h
(5.117)
where h is the Planck constant.
5.5
5.5.1
Validation of RCWA method and accuracy
2-D RCWA simulator
The proposed RCWA simulators have been checked by means of a comparison with
a commercial FDTD tool in terms of calculated absorbance (Sentaurus Device Electromagnetic Wave Solver EMW by Synopsys, (123)) with a similar 2-D test structure
(crystalline silicon grating surrounded by air shown in Fig. 5.5, with a = 25nm,
b1 = b2 = 12.5nm, h1 = 100nm, h2 = 450nm and period p = 50nm) within the range
of wavelength 380nm-880nm. A good agreement has been observed for both TE and
TM polarizations (Fig. 5.6). The maximum accuracy of the relative absorbance is
comparable to that obtained by the commercial simulator on the basis of the expertise
on FDTD simulations with EMW. For the FDTD calculations, 400 cells per wavelength
(CPW) have been adopted. The size of the tensor grid cell used by the FDTD simulator
is determined by the ratio of the wavelength of the radiation to CPW.
The accuracy of the RCWA simulator versus the number of Fourier modes (NF )
and versus the discretization step ∆z along the propagation direction (Fig. 5.7) within
139
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.5: Sketch of the 2-D rectangular grating adopted as reference test structure
for the validation and the analysis of the accuracy of the RCWA simulator (a = 25nm,
b1 = b2 = 12.5nm, h1 = 100nm, h2 = 450nm and periodicity p = 50nm).
Figure 5.6: Comparison of the absorbance calculated by FDTD and RCWA within the
range of wavelengths 380nm − 880nm for the grating test structure (Fig. 5.5) in case of
TE (A) and TM (B) polarizations. For FDTD simulations performed by Synopsys EMW,
400 cells per wavelength(CPW) have been adopted. RCWA simulations have been carried
out by using NF = 60 and ∆z = 1nm.
140
5.5 Validation of RCWA method and accuracy
the grating region has been investigated for two different values of the wavelengths of
the radiation (λ = 450nm and λ = 750nm) in case of the grating test structure (Fig.
5.5).
The accuracy of relative absorbance for both TE and TM polarizations (the absorbance calculated adopting NF = 301 is used as reference ) has been plotted as
function of the number of modes in Fig. 5.7A and 5.7B at wavelengths of the radiation
equal to 450nm and 750nm, respectively, by adopting ∆z = 1nm. For the considered
structure the accuracy of the relative absorbance is below 0.001 for NF > 10.
The accuracy of relative absorbance for both TE and TM polarizations (the absorbance calculated adopting ∆z = 0.1nm is used as reference) has been plotted as
function of ∆−1
in Fig. 5.7C and 5.7D at wavelengths of the radiation 450nm and
z
750nm, respectively, by adopting NF = 201. The relative absorbance is below 0.001
for ∆z ≤ 1nm for the considered rectangular grating.
The simulation time of the RCWA implementation presented in section 5.3 has been
compared to that of FDTD for the 2-D test grating (Fig. 5.5). The calculation of the
absorbance by FDTD requires about 30 times more time then RCWA algorithm to
achieve the same accuracy (for RCWA NF = 121 and ∆z = 1nm have been adopted
while for FDTD, CP W = 200). For the multi-layer structures with rough interfaces
the method is almost 10 times faster then the classical RCWA (eigenvalue calculations)
with the same number of modes and the same accuracy.
5.5.2
3-D RCWA simulator
The 3D RCWA simulator results have been compared to those provided by Fan et al.
(4). The test structure is a 3-D photonic crystal exhibiting a single slab (permittivity
ε = 12) with circular air holes arranged in a square lattice with periodicity p = 200nm
(Fig. 5.8). The thickness of the slab and the radius of holes are equal to 0.5p and
0.05p, respectively. The slab is lossless (extinction coefficient of the material equal to
zero). The transmission coefficient calculated by RCWA shows a good agreement to
data reported by Fan et al. (4), as shown in Fig. 5.8.
Fig. 5.10 reports the accuracy of relative absorbance for the test structure of Fig. 5.9
(cuboid of silicon with lateral side equal to 500nm surrounded by air) versus the number
of Fourier modes along x-axis and y-axis, denoted by NF x and NF y , respectively. The
reference absorbance is calculated by adopting NF x = 89, NF y = 161 and θ = 0, φ = 0
141
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.7: Accuracy of the relative absorption for 2D-RCWA simulator in case of the
test structure (rectangular grating of Fig. 5.5). The accuracy has been calculated for both
TE and TM polarizations and for two values of the wavelength of the radiation (450nm
and 750nm). In (A) and (B) the accuracy of relative absorbance is calculated as function
of the number of Fourier modes NF with ∆z = 1nm and keeping the number of points Nx
for the Fourier transform along x-axis constant (Nx = 1024). In (C) and (D) the accuracy
of relative absorbance is calculated as function of ∆z −1 adopting NF = 201. The dashed
line denotes the minimum value of NF (∆z −1 ) above which the accuracy of the relative
absorbance is smaller than 10−3 .
142
5.6 Applications of RCWA tool
(Fig. 5.3) at λ = 350nm. It is worth noting that, for the reference structure and for the
given wavelength of the radiation, in order to obtain a relative accuracy below 10−3 ,
more than 90 Fourier modes are required.
Figure 5.8: 3-D Photonic Crystal: comparison of the transmission coefficient calculated
by 3D-RCWA and that from literature (Fan et al. (4)) versus the periodicity p of the
cell normalized to the wavelength of the radiation λ. Thickness and hole diameter are set
to 0.5p and 0.05p, respectively. The real part of the permittivity of the dielectric slab is
12. RCWA simulations have been performed by adopting θ = 0, φ = 0 (Fig. 5.3) and
NF x = 43, NF y = 43 number of Fourier modes along x-axis and y-axis, respectively.
5.6
5.6.1
Applications of RCWA tool
Application: impact of surface morphology on absorbance
The 2-D RCWA simulator has been applied to model the light propagation in crystalline silicon layers with different kinds of front texturing exhibiting micrometric and
nanometric features, such as the triangular, rectangular, circular texturing and the
Gaussian random roughness (Fig. 5.11). For each considered kind of texturing the
absorbance of the structure -featuring a nominal thickness of the silicon slab equal to
10.25µm- has been calculated for different angles of incidence θ of the radiation (Fig.
5.1). Simulations have been performed for both TE and TM polarizations by adopting
as numerical parameters NF = 121 and ∆z = 1nm. Absorbance characteristics have
143
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.9: Sketch of the rectangular 3-D grating adopted as reference test structure for
the analysis of the accuracy of RCWA simulator (side length L = 500nm and periodicity
p = 500nm).
Figure 5.10: Accuracy of the relative absorption for 3D-RCWA simulator in case of the
test structure (3-D rectangular grating of Fig. 5.9) as function of the number of Fourier
modes NF x , NF y along x-axis and y-axis, respectively. The accuracy has been calculated
for θ = 0, φ = 0 (Fig. 5.3). The dashed line denotes the minimum value of NF above
which the accuracy of the relative absorbance is smaller than 10−3 .
144
5.6 Applications of RCWA tool
been calculated within the spectral range 350nm-950nm. The absorbance characteristics calculated for the flat surface and the Gaussian-random one are reported in Fig.
5.12 and Fig. 5.13, in case of TE polarization; as expected, the absorbance decreases
with increasing θ; similar results have been obtained for all other considered kinds of
surface morphology. In Fig. 5.14 the absorbance characteristics of the silicon slabs
for different types of surface texturing are reported, assuming normal incidence (θ = 0
degree) and TE polarization of the light. It is worth noting that the surface texturing
increases the absorbance of the silicon slab and that the best curve of absorbance is
obtained with the Gaussian random rough roughness.
Fig. 5.15 reports the optical generation rate per volume and time unit calculated at
λ = 600nm for the slabs featuring flat and random rough interfaces under normal light
incidence, TE polarization. It is possible to observe that, in case of random roughness,
the optical generation rate Gopt is significantly larger than that of the flat interface
throughout the volume of the absorbing layer. The enhancement in terms of absorption due to the random roughness with nanometric features is exploited in thin film
solar cells, for which, typical interfaces between amorphous silicon and transparent conductive oxide exhibit nano-roughness with root mean square height within the range
10-100nm (124).
In order to compare the different structures, in order to introduce a more realistic
figure of merit from the point of view of view of practical applications, the following
performance parameter is defined as the total absorption weighted by the Air Mass 1.5
Global spectrum (AM1.5G) and normalized by the total input irradiance:
P
Aw =
S(λi )A(λi )∆λi
P
λi S(λi )∆λi
λi
(5.118)
where A(λi ) and S(λi ) are the calculated absorbance and the spectral irradiance
for a given wavelength λi , respectively. Aw is equals unity when the absorbance is
maximum within the entire spectrum. In Fig. 5.16 and Tab. 5.1 the calculated Aw for
different kids of front surface morphology and different angles of incidence θ of the light
are reported, for both TE and TM polarizations. It is worth noting that, in case of TE
polarization (Fig. 5.16A), Aw decreases with increasing θ for all the considered surface
geometries, with the exception of the rectangular case, which features a maximum value
145
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
of Aw between θ = 20 and θ = 30 degrees; the random roughness reaches the maximum
value of Aw within the whole considered range for θ. In addition, if a periodic triangular
surface texturing featuring the same root mean square height of the random roughness
(HRM S = 50nm) is considered, by reducing the periodicity p down to a value close to
that of the random roughness (average spatial periodicity approximately 20nm), the
value of the parameter Aw reaches values closer to that of the random roughness as
shown in Fig. 5.16A, where, in addition, the values of Aw calculated for periodicity
p = 300nm and p = 20nm at normal incidence are reported (black symbols). For
TM polarization (Fig. 5.16B), similar trends are observed with the exception of the
rectangular textured and the smooth interfaces, for which Aw increases with increasing
θ.
Examples of optical generation maps calculated by RCWA are shown in Fig. 5.17
and Fig. 5.18, in which triangular, rectangular and circular surface texturing of Fig.
5.11 are considered. Monochromatic maps of Gopt are reported for both TE and TM
polarizations. The effect of the angle of incidence of the radiation θ is considered in
Fig. 5.19, in which the Gopt maps are reported for θ = 0 (direct illumination), θ = 30
degrees and θ = 69 degrees.
Table 5.1: Total normalized absorbance Aw (defined in eq. 5.118) weighted by conventional AM1.5G spectrum for the considered kinds of morphology of the front interface
calculated at different values of θ.
Polarization
θ
Flat
Random
Triangular
Rectangular
Circular
TE
0
0.5354
0.7972
0.6763
0.6179
0.6664
TE
15
0.5255
0.7927
0.6731
0.6472
0.6648
TE
30
0.4956
0.7748
0.6481
0.6442
0.6336
TE
45
0.4395
0.7335
0.6161
0.5954
0.5775
TE
60
0.3490
0.6462
0.5468
0.4892
0.4638
TM
0
0.5354
0.8156
0.8448
0.6321
0.7082
TM
30
0.5767
0.8022
0.8104
0.7290
0.7567
TM
60
0.7258
0.6986
0.6883
0.7589
0.7702
146
5.6 Applications of RCWA tool
Figure 5.11: Sketch of the structures considered for the analysis of the impact of the
surface morphology on the absorbance: silicon slab with (A) flat, (B) random rough
(Gaussian distribution of heights with root mean square value σRM S =50nm and correlation length Lc =25nm), (C) triangular (h=500nm), (D) rectangular (h=500nm) and (E)
circular (r=400nm) texturing of the front interface. The periodicity p is 1000nm for all
structures and the volume of the absorbing material is the same for all slabs.
Figure 5.12: Absorbance characteristics of the silicon slab with smooth front interface
calculated at different angles of incidence θ of the light. The absorbance decreases with
increasing θ.
147
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.13: Absorbance characteristics of the silicon slab with Gaussian random roughness at the front interface calculated at different angles of incidence θ of the light. The
absorbance decreases with increasing θ similarly to the flat interface case.
Figure 5.14: Comparison between the absorbance characteristics of the silicon slabs for
different kinds of front interface morphology calculated at normal incidence (θ=0), for
TE (A) and TM (B) polarization. The absorbance of the slab with smooth interface
is remarkably lower than those calculated by considering the texturing. In case of TE
polarization, for λ < 750nm the maximum absorbance is obtained with the random rough
interface while for TM polarization the maximum absorbance is reached by the random
rough interface for λ < 600nm and by the triangular texturing above.
148
5.6 Applications of RCWA tool
Figure 5.15: Monochrome optical generation rate Gopt maps per unit volume and time
calculated for the slabs featuring smooth (on the left) and random roughness (on the right)
interfaces at λ = 600nm by assuming normal incidence. On the bottom side the profiles
of Gopt versus z for the vertical slice at x = 500nm are compared. The optical generation
rate Gopt in case of random roughness is significantly larger than that of the flat interface
throughout the volume of the absorbing layer.
149
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.16: Total absorbance Aw weighted by conventional AM1.5G spectrum of the
silicon slabs with different kinds of roughness at the front interface, calculated as function
of the angle of incidence θ of the light, for TE (A) and TM polarization (B). For TE
polarization Aw decreases with increasing θ for all kinds of surface morphology with the
exception of the rectangular case, which features a maximum value of Aw between θ=20
and θ=30 degrees; the random roughness features the maximum value of Aw within the
whole considered range of incidence angles. In addition, the figure reports Aw calculated
for the periodic triangular surface texture featuring the same root mean square height of
the random roughness (HRM S =50nm), at normal incidence and for two different values of
the periodicity (p=300nm and p=20nm). In (B), for TM polarization, similar trends are
observed with the exception of the rectangular-textured and of the smooth interface, for
which Aw increases with increasing θ.
150
5.6 Applications of RCWA tool
Figure 5.17: 2-D optical generation rate maps inside the c-Si slab featuring circular,
rectangular and triangular texturing (Fig. 5.11) at λ = 450nm, TE (left) and TM (right)
polarization. Simulation performed by RCWA (NF = 40, ∆z = 1nm).
151
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.18: 2-D optical generation rate maps inside the c-Si slab featuring circular,
rectangular and triangular texturing (Fig. 5.11) at λ = 750nm, TE (left) and TM (right)
polarization. Simulation performed by RCWA (NF =40, ∆z = 1nm).
152
5.6 Applications of RCWA tool
Figure 5.19: 2-D optical generation rate maps inside the c-Si slab featuring triangular
(left) and rectangular (right) texturing (Fig. 5.11) at λ = 750nm, for different angle of
incidence of radiation (in A,D θ = 0 degree, in B,E θ = 30 degree and in C,F θ = 60
degree). Simulation performed by RCWA (NF =40, ∆z = 1nm).
153
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
5.6.2
Application: nano-rod based solar cell
3D-RCWA simulator has been adopted to calculate the absorbance characteristic of
cylindrical silicon nano-rods (Fig. 5.20) in order to determine the best optimum configuration (by varying geometrical parameters like periodicity, diameter and height) in
terms of optical performance. In this work, due to the relatively high computation
cost of the simulation of the absorbance characteristics, only few geometries have been
investigated: in particular only two values of the diameter (1000nm and 2000nm) and
two values of height (4µm and 8µm) are accounted for. The periodicity P has been
chosen in order to maintain the volume of the absorber material constant independently on the value of the geometrical parameters. Two different thicknesses of the
silicon substrate have been accounted for: 10µm and 100µm; the related absorbance
characteristics have been reported in Fig. 5.21A and Fig. 5.21B, respectively. The
absorbance has been compared to that of the smooth substrate featuring the same volume of silicon. Simulations have been performed adopting θ = 0 degree, ψ = 0 degree
(Fig. 5.3) and NF x = 53, NF y = 53 number of Fourier modes along x-axis and y-axis,
respectively. For both considered values of substrate thickness, the highest absorbance
is obtained by the structure featuring P = 2120nm, D = 1000nm and H = 4000nm,
within the range of wavelength 350nm-950nm.
Fig. 5.22 report an example of calculated monochromatic optical generation map
for a cylindrical nano-rod featuring P = 2120nm, D = 1000nm and H = 4000nm
calculated by the 3D-RCWA simulator at λ = 850nm (θ = 0, φ = 0). The 3-D optical
generation rate has been mapped and interpolated on the same mesh grid provided
as input to Sentaurus TCAD. The nano-rod features a radial junction with depth of
200nm and peak doping concentration at the external surface of the cylinder equal to
1019 cm−3 ; the doping concentration of the core of the cylinder is 1018 cm−3 . Since the
illumination is direct, the cylindrical symmetry of the geometry is exploited in order to
save computational resources. Therefore only one quarter of the cylinder is considered.
The numerical mesh grid exhibits about a million vertices.
154
5.6 Applications of RCWA tool
Figure 5.20: Sketch of the 3-D cylindrical nano-rod. Nano-rod diameter D, height H,
substrate thickness T0 and periodicity P are geometrical parameters.
Figure 5.21: Absorbance of the silicon cylindrical nano-rod of Fig. 5.9 calculated by 3DRCWA from λ = 350nm to λ = 950nm with θ = 0, φ = 0. Nano-rod diameter D, height
H and periodicity P as geometrical parameters. All structures exhibit the same volume
of the absorber material. Simulations have been performed adopting NF x = 43, NF y = 43
number of Fourier modes along x-axis and y-axis, respectively. Substrate thickness 10µm
(A) and 100µm (B). The highest absorbance is obtained by the structure featuring P =
2120nm, D = 1000nm and H = 4000nm.
155
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
Figure 5.22: Monochromatic optical generation rate of a cylindrical nano-rod featuring
P = 2120nm, D = 1000nm and H = 4000nm (Fig. 5.20) calculated by the 3D-RCWA
simulator at λ = 850nm with θ = 0, φ = 0. The 3-D optical generation rate has been
mapped and interpolated on the same mesh grid provided as input to the device simulator
Sentaurus. The nano-rod features a radial junction with junction depth of 200nm. Since
the illumination is direct, the cylindrical symmetry of the geometry is exploited in order
to save computational resources. Therefore only one quarter of the cylinder is considered.
156
5.6 Applications of RCWA tool
157
5. OPTICAL SIMULATION BY FOURIER-MODAL METHODS
158
6
Modeling of amorphous silicon
thin film solar cells
As discussed in chapter 2, an effective method to reduce the cost of photovoltaic devices
is the adoption of thin semiconductor films instead of high-purity materials. Several
materials have been considered for thin film solar cells, like copper indium gallium diselenide (CIGS) (125), dye-sensitized nanocrystalline titanium dioxide nc − T iO2 (126),
amorphous silicon (a-Si) (127) and cadmium telluride (CdTe) (128). Amorphous silicon is the most mature and widely used thin film for commercial panels. However
several trade-offs have to be accounted for in designing thin film devices. Aside from
considerations of material stability and relatively large recombination losses with reference to crystalline materials, light confinement is one of the most challenging issues
in thin film solar cells. Although the absorption length of the radiation is remarkably
higher for a-Si, the thickness of thin film devices cannot be increased arbitrarily because
of the large defect density; indeed, the relative thicker absorber layer leads to reduced
collection efficiency. In order to take into account the several competing physical mechanisms occurring in thin film solar cells, TCAD simulators can be helpfully adopted to
realistically predict trends and performance of such devices. State-of-the-art simulators
-specifically developed for the numerical simulation of thin film devices- are mostly 1-D
tools (ASA (129), SUNSHINE (130), AMPS-1D (131)); this means that, not only the
optical problem is solved in one dimension, but also the transport is mono-dimensional.
In addition, simulators like ASA, which can be considered as the most mature simulator
in thin film solar cells field, adopt an optical simulation method based on the scalar
159
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
scattering approach which, being a non rigorous solver of the Maxwell equations, is
not theoretically suitable to solve the near field optical problem in presence of nanostructured features like those exhibited by typical interfaces between layers of a thin
film solar cell (132); however, for most practical devices an acceptable agreement with
experiments is obtained by means of calibration of the light scattering coefficients from
rough interfaces.
This chapter outlines the basics of thin film devices with specific reference to amorphous silicon solar cells. In addition, the physical models commonly adopted for device
simulation are described and a review of the approaches to optical modeling adopted
by state-of-the-art 1-D simulators for thin film devices is presented. The chapter focus
is on the application and the calibration of Sentaurus TCAD simulator for 2-D a-Si
solar cells modeling as well as on the optical simulation performed by the implemented
RCWA simulator described in detail in chapter 5. The presented application concerns
the investigation of the impact of internal surface morphology on performance of a
single junction a-Si thin film solar cell. Transport and light propagation are modeled
in two dimensions as in (133), where only smooth interfaces are accounted for. Finally,
an investigation of the loss mechanisms by device simulation is presented.
6.1
Thin film solar cells
Commonly, in superstrate solar cells (Fig.6.1) a p-doped/intrinsic/n-doped (p-i-n) junction is deposited on a thick glass substrate coated with transparent conductive oxide
(TCO). The front electrode material is typically Sn-doped In2 O3 (ITO), boron or
aluminum doped ZnO or F-doped SnO2 (127)(134). Amorphous silicon is deposited
by plasma enhanced chemical vapour deposition (PEVCD) in the temperature range
150−250◦ C. The typical thickness of the intrinsic region is within the range 200−750nm
(135)(136) and that of the doped layers is in the range 10 − 30nm in order to reduce the impact of high defective semiconductor on device performance; indeed, large
doping concentration leads to remarkable low electric mobility in a-Si. The intrinsic
layer absorbs photons from the sun and allows the transport of the photo-generated
carriers due to the relatively higher mobility with respect to the doped layers. The
spectral response of a-Si is determined by the optical band-gap EGo which is typically
in the range 1.55eV -1.75eV (137) for optimized devices, like those containing bonded
160
6.2 Continuous distribution of defect density in energy
hydrogen concentration. High-quality amorphous silicon solar cells use hydrogenated
materials (a-Si:H) (138). Since the diffusion length in a-Si:H is typically within the
range 100nm-200nm, the dominant transport mechanism is drift. Differently from
crystalline materials, amorphous silicon exhibits a continuous distribution of states in
energy throughout the band-gap, which is defined, for this materials, as the difference
between the energy levels which bounds the localized and the non-localized transport
(conventionally called conduction and valence band edges) (139). The concentration of
defect states increases towards the boundaries of the intrinsic layer leading to relatively
high space charge regions at the p/i and i/n interfaces. Therefore, the highest field
intensity is reached in the portions of the device close to the electrodes. Hence, if the
intrinsic layer is too thick, in the middle part of such region the field intensity may
be too low to sustain the drift, therefore in such case only diffusion ensure transport;
however, due to the upper bound value of the carrier mobility in a-Si, thick i-layers
lead to degraded collection efficiency. Due to well-known Staebler-Wronski effect (140),
the presence of strained bonds (weak bonds between silicon atoms) leads to high probability of bond breaking with consequent creation of metastable defects because of the
release of energy due to electron-hole recombination events boosted by illumination.
The increase of dangling bonds concentration reduces the electric field intensity and
enhances the recombination losses leading to conversion efficiency degradation. However, this effect is self-limiting, therefore the performance of the cell typically reach
stabilized values (approximately 20-30% lower than the initial value) within the first
year of life of the module (141).
6.2
Continuous distribution of defect density in energy
In amorphous materials only the short-range crystallographic order can be assumed to
be the same of crystalline counterparts. Since lattice periodicity cannot be defined in
such materials, the wave vector is not helpful to describe the allowed electronic states.
The singularities in the density of states distribution at the boundaries of the bands
are replaced by gradually decreasing distributions called band-tails, which extension is
a measure of the entity of the disorder in the atomic structure. Because the lack of
periodicity, these states are localized. In amorphous silicon the model of traps assumes
the presence of donor-like states (charged for electrons) and of acceptor-like states
161
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Figure 6.1: Cross-sectional schematic of a (glass) superstrate single-junction amorphous
silicon thin film solar cell, with the transparent conductive oxide (TCO) and the internal
rough interfaces.
(charged for holes). The density of states in energy (DOS) consists of two exponential
BT (E)) in superposition with Gaussian-distributed middistributions of tail states (gA,D
M G (E)) accounting for dangling bonds (Fig. 6.2). Therefore the total
gap states (gA,D
DOS can be defined as superposition of four distributions of states (142):
BT
BT
MG
MG
g(E) = gD
(E) + gA
(E) + gD
(E) + gA
(E),
(6.1)
where
BT
gD
(E)
GD,BT
E − E0,D
exp −
,
=
2
ED
BT
gA
(E)
GA,BT
E − E0,A
=
exp
,
2
EA
(6.2)
and
MG
gD
(E)
"
#
ND,M G
(E − ED,M G )2
exp −
,
=
σD
2σD
(6.3)
BT
gA
(E)
"
#
NA,M G
(E − EA,M G )2
=
exp −
,
σA
2σA
(6.4)
where EA,M G , ED,M G are the energy positions for the Gaussian peaks, NA,M G ,
ND,M G are the total number of states of the Gaussian distributions and σD,A their
162
6.2 Continuous distribution of defect density in energy
standard deviations. For acceptor-like states the probability of occupation is defined
by the following function:
h
i
n + CA NV exp EVkT−E
h
i
h
i,
fA (E) =
EV −E
C
n + CA p + NC exp E−E
+
C
N
exp
A
V
kT
kT
(6.5)
where EC and EV are the edges of the conduction and valence bands, NC and NV
the density of states in conduction and valence bands, CA =
σA,p
σA,n
and σA,p , σA,n the
sections of charged traps (holes for acceptor-like states) and neutral ones (electron for
acceptor-like states); similarly for the donor-like states:
i
h
nCD + NV exp EVkT−E
h
i
h
i,
fD (E) =
EV −E
C
CD n + p + CD NC exp E−E
+
N
exp
V
kT
kT
(6.6)
In case of amorphous materials, in order to extend the SRH model for single-level
traps (described in section 2.11) to a continuous distribution of energy levels with
arbitrary shapes, the Taylor-Simmons statistics is applied (143). Therefore the concentrations of trapped electrons and holes are given by:
Z
EC
Z
fA (E)gA (E)dE,
nT (n, p) =
EC
[1 − fD (E)] gD (E)dE.
pT (n, p) =
(6.7)
EV
EV
Hence, the extended SRH model, under steady state conditions, can be formulated
as:
Rp (n, p) = Rn (n, p) =
= Cvth σn np −
n2i
Z
EC
EV
CD n + p + NV exp
h
gD (E)
i
EV −E
kT
+ CD NC exp
h
E−EC
kT
i+
gA (E)
i
h
i dE, (6.8)
h
C
n + CA p + CA NV exp EVkT−E + NC exp E−E
kT
where C is the ratio of the section of charged traps to neutral ones and vth is the
thermal velocity of carriers.
The expression of the net recombination rate due to traps in Sentaurus is different
from that of eq. 6.8, which is adopted by AMPS-1D. Indeed, in Sentaurus device,
163
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
instead of NC and NV , the effective carrier concentration ni,ef f is used; this difference
has to be considered in order to calibrate the model for trap definition in Sentaurus
with that implemented in AMPS-1D (144).
Figure 6.2: Density of states distribution in energy (DOS) with exponential band-tail
states and Gaussian shaped mid-gap states in amorphous silicon. EACP and EDON denote
the acceptor Gaussian peak energy measured positive from the edge of the valence band
EV and the donor Gaussian peak energy measured positive from the edge of the conduction
band EC , respectively.
6.3
Optical modeling
In this section the main issues related to the optical confinement in thin film solar cells
are outlined. The focus of the section is on state-of-the-art optical modeling techniques
adopted in commercial tools for thin film devices. Among these 1-D approximated techniques, the Bruggeman’s effective media approximation (EMA) (145) and the Monte
Carlo ray-tracing (described in chapter 3) are commonly used. According to EMA
approach, the region of the simulation domain interested by the roughness (or discontinuity of the permittivity) is replaced by virtual homogenous layers with smooth
interfaces and refractive indices calculated as average of the refractive indices of the
materials separated by the interface and weighted by the spatial dependence of the surface heights (Fig. 6.3). However, the most adopted approach to optical simulation of
164
6.3 Optical modeling
thin film solar cells is that based on the scalar scattering theory or on its modifications
with respect to the original formulation (10). This method, adopted by well-known
simulators like ASA and SUNSHINE, is described in the next section.
6.3.1
Optical confinement in thin-film solar cells: modeling by methods based on scalar scattering theory
One effective way to enhance light confinement in thin film solar cell is the introduction of rough and nanotextured interfaces in the multi-layer structure; this results in
an enhanced average optical path in the layers due to light scattering from rough interfaces which introduces higher angular dependent reflectance than those occurring in
case of normal incidence or scattering from smooth interfaces. Rough interfaces are
introduced in the multi-layer structure by using substrates with textured surface. For
supestrate solar cells, a stack of relatively thick glass (with thickness in the millimeter
range) and textured TCO film (for instance magnetron sputtered ZnO:Al film (146))
is used to form the front contact; different surface roughness levels and morphologies
can be obtained by wet chemical etching in diluted hydrochloric acid (HCl). The deposition of thin a-Si:H films helps to transfer the roughness to all internal surfaces. As
result of the stack of different materials and rough interfaces, a multiple light reflection
mechanism exhibiting interference effects occurs inside the structure. Practical surface morphologies exhibit random roughness featuring height root mean square values
(σRM S ) within the range 20nm-100nm (147) and which can be adequately described by
Gaussian spatial distributions of heights. The height root mean square value is defined
by:
σRM S
v
u
N
u1 X
=t
(zi − z̃)2 ,
N
(6.9)
i=1
where N is the number of samples of the surface heights, zi is the data point and z̃
is the average surface level.
A critical issue in optical modeling of these structures is the description of the scattering parameters of light from random rough interfaces depending on the morphology.
In (148)(147) two types of descriptive scattering parameters are introduced: the haze
parameter (HT and HR for transmission and reflection, respectively) and the angular
distribution function (ADF). The haze parameter describes which is the portion of the
165
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Figure 6.3: Effective media approximation (EMA): the rough interface between two media
of refractive index n1 and n2 is replaced by a stack of one or more equivalent virtual layers
featuring planar interfaces.
Figure 6.4: Light scattering from rough interface: specular and diffuse reflection (transmission) occur at non-planar interfaces exhibiting roughness featuring size in wave regime.
HR and HT are the refection and transmission haze parameters, respectively.
166
6.3 Optical modeling
light intensity that is diffused to the total incident intensity at the interface (Fig. 6.4).
The ADF relates the intensity distribution of the scattered light to the scattering angle. Typically, these parameters are experimentally determined. Approaches aimed at
modifying and at measuring the ADFs and the haze parameters of internal interfaces
are also described in (147). Applying different etching times it is possible to control
the roughness level of ZnO:Al films. Characterization and analysis of surfaces are performed by atomic force microscopy (AFM) and scanning electron microscopy (SEM).
The latter technique allows to determine statistical parameters of the random rough
interfaces like height distribution and correlation length. Total integrating scattering
(TIS) technique allows the measurement of the total and diffused transmittance (reflectance) of an interface, Ttot (Rtot ) and Tdif f (Rdif f ), respectively, therefore the haze
parameters can be calculated as HT =
Tdif f
Ttot
and HR =
Rdif f
Rtot .
In order to measure
ADFs, the angular resolved scattering (ARS) technique is commonly used. Typical
examples of normalized haze parameters and ADFs are shown in Fig. 6.5A and Fig.
6.5B, respectively. In (147), the trend of normalized haze parameters for glass/ZnO:Al
interfaces are reported to increase with increasing σRM S and to decrease with increasing wavelength λ of the radiation, indicating that the scattering from such interfaces
is more effective for short wavelength and large roughness level. In addition, ADFs
are reported to decrease with increasing σRM S , therefore for relatively large roughness
levels, the scattering into smaller angles is more remarkable; on the contrary, when
σRM S is lower, the light scattering is more effective into shallow angles. However, direct measurement of scattering parameters is difficult. The scalar scattering theory
(10) allows to define a link between optical properties of bulk materials, haze parameters and morphology of the interface. The approach described in (147) assumes that
the total intensity of transmitted and reflected light does not depend on the roughness
level. If φ1 and φ2 denote the incident and the outgoing angle of the specular beam,
respectively, and n1 and n2 are the real part of the complex refractive indices of the
materials separated by the interface, the haze parameters are given by:
" #
4πσRM S CR (λ, σRM S ) n1 cos(φ1 ) 2
HR = 1 − exp −
,
λ
4πσRM S CT (λ, σRM S ) |n1 cos(φ1 ) − n2 cos(φ2 )| a
HT = 1 − exp −
,
λ
167
(6.10)
(6.11)
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
where CT (λ, σRM S ) and CR (λ, σRM S ) are correction wavelength and morphologydependent parameters which are introduced into the original scalar scattering theory in
order to fit experimental data and a is a power factor for HT (typically within the range
2-3). Haze parameters and ADFs can be used to reformulate the Fresnel coefficients in
presence of rough interfaces extending the well-know theory valid for planar interfaces
between homogeneous media (149).
Figure 6.5: Typical haze parameters versus wavelength and angular distribution functions
(ADF) for air/glass/ZN:O:Al/air system in thin film solar cells with roughness level σRM S
as paramater.
Original scalar scattering theory is assumed to be an acceptable approximation of
the haze parameters only for a restricted range of wavelength of the radiation and not
for any arbitrary surface morphology (147). Indeed, the scalar scattering theory is a
good approximation if the distribution of the surface heights is completely random and
the correlation length of the interface morphology is much larger than the wavelength
(in order to neglect shadowing and multiple scattering effects at the rough interface).
Typically, these assumptions are not satisfied by practical interfaces. Therefore rigorous
solvers of the Maxwell equations like the RCWA method described in chapter 5 can be
successfully adopted to perform the optical simulation of thin film solar cells.
168
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
6.4
Application: dependence of main figures of merit of
the solar cell on the morphology of the internal interfaces
In the following the dependence of the main figures of merit of a thin film a-Si superstrate solar cell (Fig. 6.1) on the morphology of the rough TCO/a-Si interface is
investigated by means of 2-D electro-optical simulations. In subsection 6.4.1 the simulation methodology adopted by using Sentaurus Device as TCAD numerical tool is
described; a simple solar cell with 1-D symmetry is simulated in order to validate the
simulation setup by means of a comparison with the simulation results obtained by
AMPS-1D (144), when similar devices and physical parameters are adopted. The remainder of the section is the discussion of the results from Sentaurus 2-D simulations
of the cell exhibiting the rough internal interface as well the analysis of the loss mechanisms occurring with increasing roughness level. The optical simulations of the devices
featuring rough interfaces have been carried out by the 2-D RCWA tool.
6.4.1
Simulation setup
The simulated device is a superstrate thin film solar cell featuring a 500nm-thick a-Si
layer deposited on a stack of 1mm-thick SiO2 substrate and 200nm-thick transparent
conductive oxide (TCO). The back contact is made of silver (Ag). All internal interfaces
are smooth (layers with parallel interfaces) with the exception of the TCO/a-Si interface
for which heights are described by a spatial Gaussian distribution with correlation
length equal to 25nm. Three root mean square (RMS) values of roughness height
have been considered in this work: σRM S = 0 (smooth interface), σRM S = 25 and
σRM S = 50nm. The a-Si layer forms a p-doped/intrinsic/n-doped p-i-n junction with
uniformly doped regions (NA = 3 × 1019 cm−3 , ND = 1 × 1019 cm−3 ); the thickness of
the p-doped and n-doped regions are 8nm and 15nm, respectively. Doping parameters
are comparable to those presented in (133)(139)(150). The stack of layers ITO/ptype/intrinsic region is a conformal structure, while the interface between the intrinsic
and the p-type regions is flat.
The mobility band-gap is 1.72eV (independent on doping concentration). Optical
properties of the materials are shown in Fig. 6.6. For the 2-D optical simulation carried
out by RCWA, 121 Fourier modes (NF ) and a discretization step ∆z = 1nm have been
169
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
adopted (section 5.3); with this set of numerical parameters, the accuracy of the relative
absorption of the a-Si layer, in case of the multi-layer structure with one rough interface
(σRM S = 50nm) for both TE and TM polarizations at λ = 450nm and at λ = 750nm,
is below 10−3 , as shown in Fig. 6.7A and Fig. 6.7B -in which the accuracy is plotted
as function of ∆−1
z (adopting NF = 201)- and in Fig. 6.7C and Fig. 6.7D, where the
relative accuracy is reported as function of the number of Fourier modes (assuming
∆z = 1nm). It is worth noting that a relatively higher number of Fourier modes are
required -with respect to the rectangular grating considered in chapter 5- in order to
attain the same relative accuracy, due to the more complicated geometry of the surface
profile.
The optical generation maps are calculated by assuming normal incidence, TE polarization and the AM1.5G spectrum (1000W m−2 ). The results from the optical simulator
have been imported into a 2-D drift-diffusion device simulator by means of interpolation
over the numerical grid of Sentaurus.
The density of states as function of energy in the amorphous silicon is described
by continuous exponential and Gaussian distributions of trap levels (section 6.2). The
parameters of the defect energy levels distribution, which are comparable to those proposed in (142)(139), are reported in Tab. 6.1. Recombination at electrodes (bottom and
ITO/p-region interfaces) is accounted by setting a recombination velocity for minority
carriers equal to 107 cms−1 . The carrier mobility is constant and equal to 20cmV −1 s−1
and 2cmV −1 s−1 for electrons and holes, respectively. Since the work function difference
between TCO and a-Si p-layer causes band bending and builds a potential barrier in
the p-layer resulting in an imperfect ohmic contact (151), front electrode is modeled
by a Schottky contact with barrier set to 1.54eV. Non ideality of the back contact
is accounted for by assuming a barrier of 0.2eV. The figures of merit calculated by
means of the electro-optical TCAD simulations of the cell with smooth ITO/p-region
interface are reported in Tab. 6.2 (σRM S = 0nm). The simulation of such simple device featuring 1-D symmetry has been carried out in order to validate the simulation
setup and the physical models adopted in Sentaurus by means of a comparison of the
main figures of merit as well as of the internal maps of electrostatic potential, electric
field, recombination rates, generations rates and free carrier concentrations with those
calculated by AMPS-1D. In order to calibrate the physical models in Sentaurus, since
AMPS-1D adopts a different model to calculate the net trap recombination rate of eq.
170
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
6.8 as discussed in section 6.2, the peak concentration of the mid-gap Gaussian defects
(for donors and acceptors) has been set to 5 × 1016 cm−3 in AMPS-1D, differently to the
value adopted in Sentaurus shown in Tab. 6.1. Under these conditions, a good agreement between the two simulators has been obtained. Fig. 6.8 shows the electric field,
the optical generation rate, the carrier concentrations and the trap recombination rates
in short-circuit conditions calculated by Sentaurus for the cell featuring σRM S = 0nm.
Figure 6.6: Optical properties of the media of the simulated single-junction thin film
superstrate solar cell: real part (left) and imaginary part (right) of the complex refractive
indices.
6.4.2
Dependence of figures of merit on ITO/p-type interface roughness level
Tab. 6.2 reports the main figures of merit of the simulated cell for different values
of roughness level of the ITO/p-region interface. The photo-generated current density
Jph increases with the roughness level (Tab. 6.2) which leads to reduced reflectance;
this trend is consequence of that of the absorbed power within the a-Si layer, which is
reported in Fig. 6.9, where the roughness level is a simulation parameter. Fig. 6.10
reports the 2-D calculated full-spectrum optical generation rate maps inside the a-Si
layer for σRM S = 25nm and σRM S = 50nm.
In Tab. 6.2 it is worth noting that the maximum short-circuit current density
Jsc and the maximum conversion efficiency η, are obtained for σRM S = 25nm while,
for σRM S = 50nm, the Jsc and the conversion efficiency are smaller than those for
σRM S = 25nm. In order to explain this trend Tab. 6.3, Fig. 6.11 and Fig. 6.12, which
171
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Figure 6.7: Accuracy of the relative absorption for 2-D RCWA in case of the simulated superstrate thin film solar cell with one rough interface (ITO/p-region interface with
σRM S = 50nm). The accuracy has been calculated for both TE and TM polarizations and
for two values of the wavelength of the radiation (λ = 450nm and λ = 750nm). In (A) and
(B) the accuracy of relative absorbance is calculated as function of ∆−1
z for NF = 201. In
(C) and (D) the accuracy of relative absorbance is calculated as function of the number
of Fourier modes NF when ∆z = 1nm. The dashed line denotes the upper bound limit
of the accuracy for the conditions adopted to perform the optical simulation (NF = 121,
∆z = 1nm).
172
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
Figure 6.8: Calculated electric field (A), optical generation rate map (in cm−3 s−1 ) (B),
carrier concentrations (C) and recombination rate due to band-tails and mid-gap traps as
function of the depth in short-circuit conditions for the p-i-n cell with σRM S = 0nm .
173
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Table 6.1: Density of states in energy for Gaussian shaped and exponential states for the
simulated thin film solar cell.
Symbol
Unit
p-region
i-region
n-region
GD,BT
E0,D
ED
cm−3
1021
1021
eV
eV
0
0.05
0
0.05
1021
0
0.05
GA,BT
E0,A
EA
cm−3
eV
eV
1021
0
0.03
1021
0
0.03
1021
0
0.03
ND,M G
ED,M G
σD
cm−3
eV
eV
5 × 1019
1.24
0.15
7 × 1016
1.12
0.15
1019
1.12
0.15
ND,M G
ED,M G
σD
cm−3
eV
eV
5 × 1018
1.14
0.15
7 × 1016
1.02
0.15
1019
1.02
0.15
BT
σn,A
BT
σp,A
cm2
cm2
10−17
10−15
10−17
10−15
10−17
10−15
MG
σn,A
MG
σp,A
cm2
cm2
10−15
10−14
10−15
10−14
10−15
10−14
BT
σn,D
BT
σp,D
cm2
cm2
10−15
10−17
10−15
10−17
10−15
10−17
MG
σn,D
MG
σp,D
cm2
cm2
10−14
10−15
10−14
10−15
10−14
10−15
show the collection efficiency and the external quantum efficiency (EQE) characteristics
for different roughness levels, respectively, can be successfully considered. It is possible
to observe that the contribution of the lowest portion (300nm-600nm) and of that
from 600nm up to 1000nm of the AM15G spectrum to the photo-generated current
density Jph is of the same magnitude for all simulated roughness levels. In addition,
the collection efficiency calculated within the range 300nm-600nm is clearly degraded
for σRM S = 50nm with respect to that for σRM S = 25nm and to that for the smooth
interface (Fig. 6.11).
At short wavelengths (Fig. 6.13A, λ = 450nm) the optical generated electron-hole
pairs are clearly spatially concentrated close to the front side of the absorber layer
174
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
Table 6.2: Calculated figures of merit (photo-generated current density Jph , short-circuit
current density Jsc , open-circuit voltage Voc , Fill Factor and efficiency η ) of the simulated thin film solar cell for different roughness levels σRM S of the internal ITO/p-region
interface.
σRM S
Jph
Jsc
Voc
[nm]
[mAcm−2 ]
[mAcm−2 ]
[V olt]
0
25
50
15.34
17.40
17.84
12.16
13.58
12.82
0.897
0.902
0.897
FF
η
[%]
56.50
56.78
57.84
6.16
6.95
6.62
due to the wavelength-dependent absorption profile of a-Si, while at longer wavelength
(Fig. 6.13B, λ = 750nm), the optical generation is spatially distributed over the entire absorber layer; the morphology of the interface featuring σRM S = 50nm further
enhances the amount of photo-generated carriers in the p-layer. As consequence of
the different spatial distribution of the photo-generated electron-hole pairs, at lower
wavelengths the recombination at the front electrode is significantly larger than that
occurring at longer wavelengths and its magnitude increases with increasing roughness
leading to a decreasing trend of the collection efficiency ηC versus the roughness level.
At longer wavelengths, the contribution of the recombination at the front electrode to
the total losses is negligible, while the main contribution to electrical losses is provided
by defects at bulk level, which increases with increasing σRM S similarly to the case
of shorter wavelengths. However, at longer wavelengths ηC is enhanced by the roughness level since the increase of photo-generated current is far larger than that of the
recombination current because of defects.
Moreover, simulations reveal that, for each value of σRM S , both loss mechanisms
(bulk defects and recombination at top electrode) play a paramount role in determining
the collection of carriers at electrodes and that increasing σRM S leads to different
relative contribution of recombination mechanisms to the total recombination current.
175
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Figure 6.9: Calculated absorbed power within the amorphous silicon layer under direct
AM1.5G illumination (1000W m−2 ) adopting the roughness level σRM S as parameter. The
absorbed power increases with increasing σRM S .
Figure 6.10: Calculated 2-D full-spectrum optical generation (in cm−3 s−1 ) maps within
the a-Si layer for σRM S = 25nm and σRM S = 50nm.
176
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
Figure 6.11: Collection efficiency ηC of the photo-generated carriers for σRM S = 0nm,
σRM S = 25nm and σRM S = 50nm. At short wavelengths the collection efficiency decreases
with increasing roughness level σRM S .
Figure 6.12: External quantum efficiency (EQE) of the simulated solar thin film solar
cell for σRM S = 0nm, σRM S = 25nm and σRM S = 50nm.
177
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
Figure 6.13: 2-D maps of optical generation rates (per unit time and volume) inside the
amorphous silicon absorber layer at wavelength λ = 400nm (A) and λ = 750nm (B) for
all considered values of the roughness σRM S . Different spatial distributions of the photogenerated electron-hole pairs, depending on σRM S and on the wavelength of the radiation,
are observed.
178
6.4 Application: dependence of main figures of merit of the solar cell on
the morphology of the internal interfaces
Table 6.3: Contributions of the lower part (300nm-600nm) and of the highest portion
T OT
(600nm-1000nm) of the spectrum to the total photo-generated current density (Jph
)
calculated within the range of wavelengths from 300nm up to 1000nm for different values
of σRM S .
σRM S
300−600
T OT
Jph
/Jph
600−1000
T OT
Jph
/Jph
[nm]
[%]
[%]
0
25
50
53.4
49.5
48.0
46.6
50.5
52.0
179
6. MODELING OF AMORPHOUS SILICON THIN FILM SOLAR
CELLS
180
7
Conclusions
One of the most challenging goal in photovoltaic industry and research is the manufacturing of low-cost, high-efficiency and reliable solar cells. The PV market is, so far,
largely dominated by crystalline silicon devices, and recently a reawakening of advanced
solar cell architectures has driven the research towards process and design optimizations in order to enhance the conversion efficiency. On the other hand, thin film solar
cells are the most effective way towards low-cost photovoltaic, but the main drawbacks
associated to these second-generation devices are the poor material quality -which leads
to higher losses- and the significantly limited optical absorption. In this thesis, after an
outline of the main physical mechanisms which limit the ultimate conversion efficiency,
the basis of the numerical modeling of solar cells -including both device and optical
simulation- has been discussed with the emphasis on the main issues and limitations of
the state-of-the-art tools - mostly 1-D or 2-D- aimed at simulating photovoltaic devices.
In the first part of the thesis, as a main contribute towards the goals of this work,
the physical models adopted as well as the methodology used to perform the tuning
of the models from experimental data in order to set up a simulation framework by
means of the state-of-the-art TCAD Sentaurus by Synopsys have been discussed. In
particular, a simulation procedure for 2-D and 3-D spatial domains has been introduced
and two solar cell architectures of interest have been investigated. The first considered
scheme is a double-diffused emitter solar cell which has been investigated and optimized by means of 2-D electro-optical simulations. The second application concerns
the optimization of the rear point contact scheme which has been carried out by true
181
7. CONCLUSIONS
3-D numerical simulations. In this work, the numerical simulation has not been limited
to the optimization of the device in terms of doping and geometrical parameters, which
results from trade-offs between competing physical mechanisms, but it has been successfully exploited to investigate the main loss mechanisms which limit the conversion
efficiency of solar cells. With respect to 1-D simulators, the adoption of 2-D and 3-D
tools in combination to ad-hoc tuned physical models, allows to predict realistic results
and trends highlighting the effectiveness of each parameter involved in the device design
as well as the sensitiveness of the main figures of merit to such parameters. The numerical modeling activity of advanced crystalline solar cells has been carried out in the
framework of the collaboration between University of Bologna and Applied Materials,
Inc.
The second part of the thesis is on the optical simulation. After a review of the main
approaches to optical modeling of optoelectronic devices (FDTD, Raytracing, effective
media approximation and scalar scattering approach) with particular reference to solar
cells, for which the theoretical limitations of non-rigorous methods to model the light
propagation in practical devices featuring nanostructured geometries have been highlighted, two novel implementations of Fourier modal (or RCWA) solvers of the Maxwell
equations aimed at simulating 2-D structures have been presented. Besides featuring
new algorithms, the proposed approaches are computationally efficient with respect
to the state-of-the-art implementations of RCWAs, especially when processing devices
involving multi-layers structures with nanorough interfaces. Finally a third method
aimed at solving the near field optical problem in 3-D simulation domains, based on a
standard -but more general in terms of validity- approach has been implemented. All
the proposed methods, developed with the strategic collaboration with the Institute of
Physics and Technology of RAS, Moscow, which provided support to the implementation of the code as well as to the solution of trivial numerical problems, have been
validated by means of comparison with either commercial finite difference tools and
literature data. In addition, the RCWA method has been applied to the optical simulation of thin film amorphous silicon solar cells featuring nanorough interfaces as well
as nano-rods.
The adoption of proposed simulation techniques as well as of the implemented tools
for the optical problem will be the basis for several future activities which will involve
182
the analysis and the optimization of advanced photovoltaic devices of second and third
generations.
183
7. CONCLUSIONS
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