one-micron thick silicon photonic crystal to achieve a power conversion efficiency of 17.5%,

Coupled optical and electrical modeling of solar cell based on conical pore
silicon photonic crystals
Alexei Deinega, Sergey Eyderman, and Sajeev John
Citation: J. Appl. Phys. 113, 224501 (2013); doi: 10.1063/1.4809982
View online: http://dx.doi.org/10.1063/1.4809982
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i22
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JOURNAL OF APPLIED PHYSICS 113, 224501 (2013)
Coupled optical and electrical modeling of solar cell based on conical
pore silicon photonic crystals
Alexei Deinega,a) Sergey Eyderman, and Sajeev John
Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 1A7, Canada
(Received 25 March 2013; accepted 27 May 2013; published online 10 June 2013)
We compare the efficiency of thin film photonic crystal solar cells consisting of conical pores and
nanowires. Solving both Maxwell’s equations and the semiconductor drift-diffusion in each geometry,
we identify optimal junction and contact positions and study the influence of bulk and surface
recombination losses on solar cell efficiency. We find that using only 1 lm of silicon, sculpted in the
form of an inverted slanted conical pore photonic crystal film, and using standard contact recombination
velocities, solar power conversion efficiency of 17.5% is obtained when the carrier diffusion length
exceeds 10 lm. Reducing the contact recombination velocity to 100 cm s1 yields efficiency up to
22.5%. Further efficiency improvements are possible (with 1 lm of silicon) in a tandem cell with
C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4809982]
amorphous silicon at the top. V
I. INTRODUCTION
Silicon is the most widely used material for solar cell
production due to its abundance, nontoxity, reliability, and
mature fabrication process. At the present time, solar cells
based on silicon occupy most of the photovoltaics market.
Currently, the best efficiency of commercial silicon solar
modules is around 17.5%.1–3 Silicon is an indirect-bandgap
semiconductor which results in moderate absorption length.
To absorb enough of the solar spectrum, the thickness of the
silicon planar cell is typically more than 100 lm. In order to
collect all generated carriers before they recombine, the carrier diffusion length in silicon should be comparable to the
cell thickness. This requirement for a large volume of high
quality silicon constitutes a significant part of silicon solar
cell cost. Thin film technology is a promising way to avoid
these cost issues. However, best thin film samples of thickness less than 10 lm yield an efficiency of around 10%,
which is low compared to wafer-based cells.4,5 This is due to
poor light absorption in thin silicon films.
Some enhancement in solar absorption is achieved by
texturing the surface of silicon thin films with nanocones or
nanoholes. This provides an antireflective mechanism at the
top surface of the cell for incident light. In the long wavelength regime, the texture acts like a graded index film with
low reflection above the active region of the solar cell.6–11 In
the short wavelength regime, suppression of the reflection
occurs because rays must be reflected many times before
being backscattered.12,13 In the visible range, textured surfaces
with the periodicity of the optical wavelength are highly
efficient.14–18 Recently, antireflective properties of textured
surfaces in a whole range of size-to-wavelength ratios, including effective medium and geometric optics regimes, were
studied.18 For each of these cases, texturing is performed outside of the active light-absorbing region of the solar cell.
Another effect of surface texturing is the trapping of certain light rays by total internal reflection. For a random
a)
Electronic mail: [email protected]
0021-8979/2013/113(22)/224501/9/$30.00
surface texture, leading to Lambertian probability distribution of deflected light rays, the effective path length of light
within the absorbing medium is enhanced by a factor of 4n2 ,
where n is refractive index of the medium.19,20 This leads to
the so called Lambertian light trapping limit, an important
benchmark for solar light trapping that has proved difficult
to surpass in most solar cell designs. However, by texturing
the interior, active regions of the solar cell, this Lambertian
limit can be surpassed. Recently, it was shown that photonic
crystal (PC) light trapping in a wave optics regime enables
exceptional solar absorption in thin films.21–28 Most strikingly, it was shown that for slanted conical pore photonic
crystals, it is possible to surpass the longstanding 4n2
Lambertian limit even after the absorption spectrum is integrated over the range of 400–1100 nm.28 With a single
micron of silicon (equivalent bulk thickness), this solar
absorption corresponds to a maximum achievable photocurrent density (MAPD) of nearly 35 mA/cm2. In more traditional solar cell architectures, such a large MAPD would
require more than 100 lm of silicon.
Silicon nanowire architectures offer an alternative type
of thin film solar cell. There are many reports of design principles to minimize reflection and efficiently absorb sunlight
in nanowires.29–34 Optimized solar absorption (for a fixed
volume of silicon) can be obtained by arrangement of the
nanowires into a photonic crystal with appropriate lattice
constant and wire diameter. Additional modulation of the
nanowire diameter along the vertical axis provides gradedindex antireflection effects at the top of the array, light trapping in the middle section, and Bragg reflection near the
bottom of the wires.35 As in the case of conical nanopore
photonic crystal, part of the light trapping occurs as a result
of parallel-to-interface refraction (PIR) effects into slow
group velocity modes.36 While the optimized MAPD for
nanowires is less than that of a conical pore photonic crystal,
the nanowire provides opportunities for more efficient charge
carrier collection.
Nanostructuring of the solar cell gives opportunities for
nontrivial choice of pn-junction architecture and contact
113, 224501-1
C 2013 AIP Publishing LLC
V
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Deinega, Eyderman, and John
positioning. For example, a radial junction geometry in
nanowires can be exploited to increase power conversion efficiency with poor quality silicon due to decoupling of light
absorption and carrier collection directions.37–44 To calculate
efficiency for nanostructured cells, optical and electrical
modeling must be combined. The solution of Maxwell’s
equations provides the absorption profile inside the structure
that defines the charge carrier generation profile. This profile
is used as the input for the semiconductor drift-diffusion
equations to calculate solar cell efficiency, including bulk
and surface recombination losses.45–52
In this paper, we present results for coupled optical and
electrical modeling for nanopore photonic crystal solar cells.
As a reference case, we also discuss a simple nanowire geometry. We consider photonic crystals with straight and
slanted conical nanopores. We study the influence of recombination losses on the efficiency of the solar cell and identify
optimal junction and contact geometries. We show that
power conversion efficiency of 17.5% can be achieved in a
slanted conical pore photonic crystal solar cells using only
1 lm of equivalent bulk thickness of silicon and significant
contact recombination losses.
The rest of the paper is organized as follows. In Sec. II,
we compare results for solar absorption distribution and carrier generation profile inside different nanostructured cells.
In Sec. III, we calculate how the current and voltage of the
solar cell vary with position of the p-n interface and contacts
for various choice of the carrier diffusion lengths.
Conclusions are given in Sec. IV.
II. SOLAR ABSORPTION MODELING
We first compare absorption properties for nanowires
and nanopore PC films (Fig. 1). The equivalent bulk thickness for all considered structures is 1 lm. The geometrical
parameters for two specific architectures are chosen in order
to achieve maximal solar absorption using given equivalent
bulk thickness. These parameters were found as a result
J. Appl. Phys. 113, 224501 (2013)
of optimization procedure performed in our previous
works.28,35
1. We consider a nanopore PC film of height 1.6 lm containing a square array of conical holes with radius r ¼ 0:5 lm,
filled with silica (refractive index n ¼ 1.5). The lattice constant of this PC is a ¼ 0:85 lm. The tip of each cone
touches the center of the square unit cell (straight cones)
or center of its side (slanted cones). The bottom metal contact is modeled as a perfect electric conductor that reflects
all sunlight that reaches it. The top contact is an indium tin
oxide (ITO) net of the height 50 nm and width 150 nm (refractive index n ¼ 1.8). The top of the solar cell is covered
with partial silica hemispheres with height 0:2 lm above
the silicon film and radius 0:5 lm.
2. We consider nanowires with the radius r ¼ 0:082lm and
height h ¼ 6:3lm, arranged in a square lattice photonic
crystal with period a ¼ 0:35lm. The silicon nanowire PC
is fully embedded in a silica slab (refractive index
n ¼ 1.5) of height equal to nanowires. The bottom metal
contact is modelled as a perfect electric conductor. The
top contact is an ITO film of thickness 50 nm, touching
each nanowire.
We note that the nanowires surface shape can be separately optimized to provide considerable improvement in
MAPD. In previous work,35 we showed that by appropriate
modulation of the nanowire surface, light trapping enhancement occurs by parallel-to-interface refraction into slow light
modes. For our present comparative purposes, we consider
only straight cylindrical nanowires.
We calculate solar absorption using the finite-difference
time-domain (FDTD) method53 with the help of the
Electromagnetic Template Library.54 We assume that the
entire solar spectrum is collimated into a normal angle of
incidence. (Oblique incidence case can be simulated with the
help of our FDTD iterative technique.55,56) We use the standard FDTD scheme, where a plane wave impulse is directed
onto the structure, fields are recorded, transformed to the
FIG. 1. From left to right, side view of (i)
slanted conical pore PC film, (ii) straight
conical pore PC film, (iii) nanowire with radial p-n junction, and (iv) nanowire with
axial p-n junction. Lower left figure depicts
top view of ITO top contact in conical pore
PC geometry. Different scale is used for vertical and horizontal dimensions.
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Deinega, Eyderman, and John
J. Appl. Phys. 113, 224501 (2013)
frequency domain, and normalized to the incident spectrum,
to calculate the frequency dependent absorption at each point
r inside the structure
aðx; rÞ ¼
x ImðeÞjEðx; rÞj2
:
c Re½Einc ðx; rÞ; Hinc ðx; rÞ
(1)
Here, c is the speed of light in vacuum, x is the frequency of
light and e is silicon dielectric function. For nanowire and
straight conical pore PC, we model linear light polarization
along one periodic direction. For the slanted conical pore
PC, we perform two numerical experiments for linear polarization along the two distinguishable periodic directions, and
then average the results.
We use a subpixel smoothing technique57 to eliminate
any staircase effect caused by our rectangular FDTD mesh.
This technique has been shown to significantly improve the
accuracy of the FDTD calculations for arbitrary shaped scatterers. For reducing numerical reflection from the artificial
absorbing perfectly matched layer (PML),53 we use additional back absorbing layers technique.58
Experimental data on the silicon dielectric permittivity
eðxÞ are taken from Ref. 59. The frequency dependence of
eðxÞ is assigned in FDTD by considering a modified Lorentz
approximation where the dielectric polarization depends
both on the electric field and its first time derivative60
2
X
Deðx2p ic0p xÞ
eðxÞ ¼ e1 þ
;
x2 2ixcp x2
p¼1 p
(2)
1
c0p
with (xp , cp , and in units of (lm) , and the speed of light
is unity): e1 ¼ 1; De1 ¼ 8:93; De2 ¼ 1:855; x1 ¼ 3:42 (corresponding to wavelength k1 0:292lm), x2 ¼ 2:72
(k2 0:368lm), c1 ¼ 0:425; c2 ¼ 0:123; c01 ¼ 0:087; c02 ¼
2:678. This model provides an accurate fit to the response of
bulk crystalline silicon to sunlight over the wavelength range
from 300 to 1000 nm, while conventional Debye, Drude, and
Lorentz approximations fail. Fitting of the silicon dielectric
function is found with the help of an open MATLAB script.61
The modified Lorentz approximation is implemented in FDTD
using the auxiliary differential equation (ADE) technique.60
We assume that each absorbed photon of energy larger
than the silicon electronic bandgap leads to generation of an
electron-hole pair. The electrons and holes are assumed to
rapidly lose energy by scattering from phonons and occupy
energy levels near the conduction and valence band edges,
respectively. Subsequently the electron and hole dynamics is
described by the drift-diffusion model.62,63 The total charge
carrier generation rate per unit volume is obtained by integration of the calculated absorption aðk; rÞ with incident solar Air Mass 1.5 Global Spectrum64 intensity IðkÞ, in units of
energy per unit area per unit time, per unit wavelength, over
the wavelength range of 350–1000 nm
ð kmin
k
IðkÞaðk; rÞdk:
(3)
GðrÞ ¼
hc
kmin
Here, the photon energy is
Planck’s constant.
hc
k,
wavelength k ¼ 2pc
x , and h is
The MAPD, in which all generated carriers are assumed
to be collected, is calculated by integrating the generation
rate (3) over the photonic crystal unit cell volume Vs and
dividing by the surface area a2 of the unit cell
ð
1
(4)
MAPD ¼ 2 eGðrÞdr:
a Vs
Here, a is the PC lattice constant and e is electron charge.
The case of 100% solar absorption over the 350–1000 nm
range in crystalline silicon corresponds to MAPD
¼ 42:3 mA=cm2 .
In Fig. 2, we present the calculated absorption profile
aðk; r; zÞ for the slanted conical pore PC film. Calculated
MAPD for our slanted conical pore PC film with silica packaging is 33:3 mA=cm2 . For the straight conical pore PC film,
this drops to 30:4 mA=cm2 , and for our cylindrical nanowires, the MAPD is 20:7 mA=cm2 . As shown in previously,35 by appropriate modulation of the nanowire surface
the latter MAPD can improved up to 27:7 mA=cm2 .
III. JUNCTION GEOMETRY OPTIMIZATION
To model electrical transport, we simultaneously solve
the Poisson equation and continuity equations for electrons
and holes
q
r2 w ¼ ðp n þ ND NA Þ;
e
(5)
rJn ¼ rJp ¼ qðR GÞ;
(6)
Jn ¼ qDn rn qln nrw;
(7)
Jp ¼ qDp rp qlp prw;
(8)
where w is the electrostatic potential, q is the elementary
electronic charge, e is the dielectric function, n and p are
electron and hole densities, ND and NA are concentrations of
ionized donors and acceptors, R and G are recombination
FIG. 2. Total charge carrier generation rate profile (see Eq. (3)) inside (a)
straight and (b) slanted conical pore PC film. Different scale for vertical and
horizontal dimensions is used.
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Deinega, Eyderman, and John
J. Appl. Phys. 113, 224501 (2013)
R¼
FIG. 3. Photocurrent density distribution inside (a) central slice and (b) side
slice in the straight conical pore PC film. Different scale for vertical and horizontal dimensions is used.
and generation rates, Ji, Di, and li are current densities, diffusion coefficients, and mobilities, i ¼ n, p.
The generation rate profile, calculated using Eqs. (1)
and (3), is used as an input to the continuity equation (6). All
transport calculations are performed using our 3D semiconductor device modeling library Microvolt.65 Nanowire modeling is simplified using an angular average of the generation
profile. This is a good approximation which reduces dimensionality of the problem from 3D (x, y, z) to 2D (r, z). This is
solved using a simple radial finite difference scheme.45
In our calculations, we use silicon parameters found in
Ref. 37:
(i)
(ii)
doping concentration Nd ¼ Na ¼ 1018 cm3,
electron and hole mobilities ln ¼ 270 cm2 V1 s1
and lp ¼ 95 cm2 V1 s1, respectively. The diffusion
coefficients Dn and Dp are then calculated using the
Einstein relation D ¼ kTe l (temperature T is assumed
to be 300 K throughout the solar cell).
As in earlier literature,37,46 we consider only ShockleyReed-Hall (SRH) recombination from a single-trap level that
lies near the middle of the bandgap
np n2i
:
sp ðn þ ni Þ þ sn ðp þ ni Þ
(9)
Here, ni is intrinsic charge carrier concentration. We choose
the lifetime of the minority electrons in the p-region equal to
spffiffiffiffiffiffiffiffiffi
. In ffithe
the lifetime of minority holes in the n-region sn ¼ p
following, we consider the diffusion length Ln ¼ sn Dn as
an independent variable. Including Auger and radiative
recombination1 in our model for R results primarily in a
change of the diffusion length Ln. Rather than adding these
other recombination channels into the microscopic expression (9), for simplicity we subsume these processes as done
previously46 into the overall diffusion length Ln.
We impose a surface recombination velocity of
100 cm s1 at each Si-SiO2 interface, consistent with available experimental data.2,40 As shown previously,46 increasing
the surface recombination velocity at the Si-SiO2 interface
from 100 cm s1 to 1000 cm s1 leads to only slight (1%–2%)
reduction of short-circuit current density and open-circuit
voltage. Contacts are assumed to be ohmic with a larger surface recombination velocity s which we vary in the range
103 cm s1 s 1. We model the p-n junction and contact
geometries as in Fig. 1. For our conical pore PC, the junction
interface is parallel to the bottom surface. For nanowires we
consider both radial and axial junction geometries.
For illustration, we present typical photocurrent distributions inside the straight conical pore PC film in Fig. 3. The
short circuit current density Jsc is calculated as the current
flux through the contact (bottom or top) area inside the
square lattice unit cell normalized to this unit cell area a2
(a ¼ 750 nm for conical pore PC film and a ¼ 350 nm for
nanowire PC).
We calculate the short circuit current density Jsc and
open circuit voltage Voc for different values of Ln and s,
varying the distance d between the p-n junction and top surface of the PC film (Fig. 4). If the diffusion length is smaller
than film width (Ln ¼ 0:1lm in Fig. 4), the results are almost
independent of surface recombination. In this case, only carriers that are close to the p-n junction contribute to the photocurrent. These carriers are quickly separated across the
junction and have minimal subsequent opportunity to recombine. Other carriers recombine mostly in the bulk, not at the
surface. Therefore, the optimal junction position is located in
FIG. 4. Short circuit current density Jsc and open circuit voltage Voc as a function of distance between junction and top surface of the slanted conical pore PC
film for various diffusion lengths Ln and surface recombination at the contacts s values. All geometrical parameters are listed at the beginning of Sec. II.
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Deinega, Eyderman, and John
J. Appl. Phys. 113, 224501 (2013)
FIG. 5. Short circuit current density Jsc and open circuit voltage Voc for slanted conical pore PC film, radial, and axial nanowires as a function of diffusion
length Ln for various contact surface recombination velocities s. All geometrical parameters are listed at the beginning of Sec. II.
the region where most of the light is absorbed and most carriers are generated. The highest absorption rate is at the top
surface of PC film, however, the highest filling fraction of
silicon is at the bottom surface. The competition between
these two factors determines the optimal junction position at
d ¼ 1:2 lm below the top surface (see maximum of blue and
red curves in Fig. 4, left). If the carrier diffusion length is
longer than the film depth (Ln ¼ 100lm in Fig. 4) and surface recombination is high (s ¼ 1 in Fig. 4), generated carriers are most likely to recombine at the contacts. Since the
surface area of the bottom contact is larger than the surface
area of the top one, photocurrent improves when the junction
position shifts to the bottom contact. This enables carriers
generated near the bottom of the photonic crystal to “escape”
from the bottom contact and be separated by the junction
(green curve in Fig. 4, left). However, as we increase the top
contact area, the improvement is less pronounced. To illustrate this, we consider a situation where the top contact
extends to the surface of the conical hole z 0:9 lm, where
the coordinate z increases from 0 (bottom of the film) to
1:6 lm (top of the film), see Fig. 6(i). For this geometry, we
use the charge carrier generation profile obtained for the geometry in Fig. 1(i), in order to isolate the recombination
influence of the extended contact from possible absorption
profile changes. Results are represented by the magenta
curve in Fig. 4, left. If the recombination losses are negligible (Ln ¼ 100 lm and s ¼ 103 cm=s), then Jsc and Voc are
almost independent of junction position. In this case, Jsc is
approximately equal to the MAPD (black curve in Fig. 4),
since nearly all generated carriers contribute to the
photocurrent.
The open circuit voltage Voc depends only weakly on
junction position d, but in the opposite way: Voc slightly
improves if the junction shifts closer to the top surface (Fig. 4,
right). In this case, the junction area becomes smaller (Fig. 1)
resulting in reduced recombination of excess carriers in the
depletion region. As discussed below, this improves Voc .
In Fig. 5, we compare Jsc and Voc for slanted conical
pore PC films and nanowires with radial and axial junction
geometries. Results are given as a function of diffusion
lengths Ln for small (s ¼ 103 cm/s) and high (s ¼ 1) surface
recombination rates at the contacts. The distance between
the p-n junction and top surface of the conical pore PC film
is chosen to be d ¼ 1:2 lm. This is close to the optimal value
for small diffusion lengths (see maximum of blue and red
curves in Fig. 4, left). The radial junction of the nanowire
consists of two parts (Fig. 1): horizontal surfaces 0:1lm
from the top and bottom contacts and vertical surface
0:05lm from the nanowire surface. Our axial junction model
separates the nanowire into two layers of equal height.
For small Ln, Jsc , and Voc are independent of surface
recombination, since most of the recombination occurs in the
bulk. In this case, only carriers generated close enough to the
junction contribute to the photocurrent. Carriers generated
further than Ln from the junction are highly susceptible to
bulk recombination. In other words, the effective volume of
silicon producing photocurrent is the volume within the distance Ln to the junction. The radial junction provides the
highest value of Jsc for small carrier diffusion lengths, since
it has the most developed junction area with largest catchment volume for separating newly generated carriers.
Nanowires with axial junction provide the smallest Jsc since
the effective volume of silicon providing photocurrent is a
small fraction of the whole nanowire volume.
FIG. 6. (i) Slanted conical pore PC film with extended top contact and planar bottom contact, (ii) slanted conical pore PC film with reduced top contact (see Fig. 1) and bottom contact consisting of square lattice of protruding
dots and intervening SiO2 layer. Different scale for vertical and horizontal
dimensions is used.
224501-6
Deinega, Eyderman, and John
If the diffusion length Ln and the surface recombination
velocity s are large, most of the recombination occurs at the
metal contacts. The radial junction yields the largest Jsc since
carriers are more likely to “escape” contact recombination
and reach the junction where they become separated. For
long diffusion lengths Ln and small surface recombination
velocity s (no recombination losses), Jsc approaches the
MAPD for each of the structures. The conical pore PC produces the highest Jsc , since it has highest MAPD.
The radial junction nanowire produces the smallest open
circuit voltage Voc of all architectures considered. This is due
to the negative influence of excess carrier recombination in
the depletion region.1 When these carriers recombine, additional carriers diffuse to the depletion region to make up for
this loss. This diffusive flow of carriers constitutes a recombination current that is opposite to the required photocurrent
drift. The backflow recombination current is greater if
the depletion (and junction) region is more developed.
Therefore, a more developed junction area produces a
smaller open circuit voltage Voc . This explains the smaller
values of Voc for the radial junction nanowire geometry.
Using silicon of high quality in the depletion region around
the junction can improve Voc , even if the silicon quality elsewhere is small.46
For the axial nanowire geometry, the volume of silicon
within a diffusion length of the junction is much smaller than
the total silicon volume. In the case of small diffusion length,
axial nanowires produce smaller Voc than the conical pore
PC film. This is explained by the smaller photocurrent Jsc for
axial nanowires. For each value of applied voltage, the total
current is a sum of the photocurrent and the forward bias current,1 which is opposite to the photocurrent. Forward bias
current is caused by diffusion of carriers to the region with
opposite doping, which increases with applied voltage.1 In
the case of smaller photocurrent Jsc , a correspondingly
smaller voltage Voc is required to produce forward bias current with magnitude equal to the photocurrent, so the total
open circuit current is zero.
For long diffusion lengths Ln and small surface recombination velocity s (no recombination losses), all structures exhibit a similar Voc value 0.73 V.
We now compare the power conversion efficiency of
straight and slanted conical pore PC films, nanowires with
axial and radial junctions, and a planar silicon solar cell of
thickness 1 lm. The geometrical parameters in all cases are
chosen to use the same amount of silicon as in the planar
cell. The surface recombination velocity at the contacts is
chosen to be 105 cm s1, as in Refs. 37 and 46.
For a conical pore PC film geometry with diffusion length
exceeding the film thickness, carriers recombine mostly at the
contact surface (compare red and blue curves in Fig. 5). The
influence of contact recombination can be suppressed by
reducing the semiconductor-metal interface area and passivating the remaining rear surface with silica. This has been demonstrated for wafer-based silicon solar cells using a square
array of dot shaped rear contacts of the radius r ¼ 10 lm,
spaced by a ¼ 200 lm from each other.1 The back contact
area fraction for this geometry pr 2 =a2 is then less than 1% of
the total back-surface area. In our model we simulate reduced
J. Appl. Phys. 113, 224501 (2013)
rear contact surface area using a periodic array of cylindrical
dots of radius 50 nm and height 50 nm, corresponding to a contact area 1% of the total back-surface area, see Fig. 6(ii).
We also use a highly doped region (Na ¼ 5 1018 cm–3) with
thickness 50 nm near rear contact (back surface field). The
interface between the higher and lower doped regions behaves
like a p-n junction where the resulting electric field makes a
potential barrier to minority carrier flow to the surface.1 As a
result, these carriers are more likely to cross the p-n junction
into the region with opposite doping and contribute to the photocurrent. On the other hand, the interface between higher and
lower doped regions enhances the flow of majority carriers in
the direction of the photocurrent.
We obtain the generation rates for conical pore PC and
nanowires from our FDTD simulations of Maxwell’s equations. The generation rate for the planar cell is estimated
using Beer’s law for absorption and assuming no reflection
from the front (perfect antireflective coating) and back sides
of the cell, as in Ref. 37. The calculated efficiencies as a
function of diffusion length are plotted in Fig. 7. Results for
the planar cell are in excellent agreement with analytical
results.37 The photonic crystal solar cells exhibit higher efficiency than the planar cell due to light-trapping and
enhanced absorption. The radial junction shows the best efficiency for very small diffusion lengths due to the more
developed junction area. For medium to long diffusion
lengths, the best efficiency of 17.5% is demonstrated by the
slanted conical pore PC film due to its higher solar absorption. This is comparable with the best efficiency of commercial planar silicon solar modules with the cell thickness more
than 100 lm.1–3
As in the case of conventional solar cells, a smaller doping concentration N increases the saturation current J0 (leakage of thermally generated minority carriers across the
junction) that, in turn, leads to smaller open-circuit voltage
FIG. 7. Power conversion efficiency as a function of diffusion length Ln for
different cells with the same volume of silicon, equivalent to the volume of
a planar cell of 1 lm thickness. All geometrical parameters are listed at the
beginning of Sec. II. For the planar cell, a perfect antireflection coating is
assumed. Surface recombination velocity at the contact s is chosen to be
105 cm s1.
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Deinega, Eyderman, and John
FIG. 8. Power conversion efficiency as a function of back contact filling
fraction for the slanted conical pore PC film. Diffusion length Ln ¼ 10 lm
and surface recombination velocity at the contact s is 105 cm s1.
and lower efficiency.1 By lowering the doping concentration
in our slanted conical pore PC from 1018 cm3 to 1017 cm3
and 1016 cm3, the efficiency drops from 17.5% to 15.6%
and 13.7% correspondingly. In the limit of zero doping, there
is no electric field in the junction area to provide photocurrent and power conversion efficiency goes to zero.
As discussed above, reducing the back contact area
leads to improved efficiency. The rate of improvement is
highest for a back contact fill fraction of 100%, corresponding to a flat contact (Fig. 8). For our choice of back contact
(cylindrical dots of the radius 50 nm) the rate of further
improvement is very small. In other words, further reduction
of the contact area does not lead to significant efficiency
improvement.
IV. DISCUSSION
Our calculated power conversion efficiency of 17.5%
for a 1 lm (equivalent bulk thickness) slanted conical pore
silicon PC solar cell provides only a baseline result. The efficiency can be improved in various ways. A simple improvement is achieved using amorphous and microcrystalline
silicon in the top and bottom regions, respectively, to form a
tandem solar cell. Both forms of silicon can be manufactured
using the same technology.1 Such tandem cells have been
studied previously for planar66 and nanowire67,68 geometries.
Amorphous silicon and crystalline silicon have bandgaps
well suited for tandem cells (1.7 eV and 1.12 eV, respectively). The absorption coefficient of amorphous silicon is
about an order of magnitude greater than in crystalline silicon at visible wavelengths. However, amorphous silicon has
poor transport properties with a short carrier diffusion length
and carriers photogenerated in p or n layers are unlikely to
contribute to the photocurrent. To overcome this problem, a
p-i-n junction with undoped or intrinsic region is used.1 The
built-in bias is dropped across the undoped region, creating
an electric field there. Carriers photogenerated in the
undoped region are driven by this electric field to produce a
photocurrent.
Our initial crystalline silicon slanted conical pore PC
film already absorbs 95% of sunlight with energy higher
than the electronic bandgap of amorphous silicon.
Therefore, using amorphous silicon at the top does not
increase the overall amount of absorbed sunlight. It simply
redistributes the carrier generation profile along the depth of
conical pore PC film. The thickness of the amorphous silicon
J. Appl. Phys. 113, 224501 (2013)
segment is ideally chosen such that the same amount of photocurrent is produced in the amorphous and crystalline silicon layers. Employing separate p-i-n and p-n junctions for
each segment yields two solar cells connected in series.
Adjacent n-doped and p-doped regions of these subcells can
be heavily doped to form a tunnel junction between the subcells. Current passes through the tunnel junction between the
subcells by quantum tunneling.1 The photocurrent in the
resulting tandem cell is equal to the photocurrent in each
subcell, while the total voltage is a sum of subcell voltages.
For a crystalline silicon subcell, we expect Voc ¼ 0:6V,
whereas for the amorphous silicon subcell, we expect
Voc ¼ 0:9V according to recent experimental69,70 and numerical71 results. Therefore, the total open circuit voltage is
1.5 V. Photocurrent in the tandem cell is one half that of the
initial silicon conical PC film cell, since the same amount of
sunlight is absorbed by the two subcells. Therefore, we
obtain around 25% improvement in efficiency relative to the
initial silicon conical pore PC film. This suggests a tandem
cell with 21.9% power conversion efficiency.
Improvement in the contact regions can also provide
substantial gain in power conversion efficiency. For diffusion lengths longer than the thickness of the conical pore PC
film, photogenerated carriers recombine mostly at the contacts. As we found numerically, if the surface recombination
velocity at the contacts is lowered until 100 cm s–1, the efficiency improves from 17.5% to 22.5% when diffusion length
Ln is equal to 10 lm. Similarly, for the tandem solar cell we
expect efficiency improvement from 21.9% to 25.5%. The
possibility of achieving small surface recombination velocity
at the contacts by passivation with an intrinsic amorphous
buffer layer has recently been presented.72
We note finally that the buffer region of SiO2 between
the structured metallic back contact (Fig. 6(ii)) and the lightabsorbing silicon photonic crystal may be replaced with
material that up-converts photons in the 1100–1500 nm
wavelength range to photons of energy higher than the silicon band gap. Nearly 19% of incoming solar power is lost
due to photons below the silicon band gap that cannot otherwise be absorbed. Dye-sensitized, erbium nano-particles
may provide a key to such broadband up-conversion.73 A
significant feature of the conical pore PC film is that it provides not only light-trapping but also strong light concentration in specific regions with intensity enhancement of about
100 times the average intensity.28 A suitably structured metallic back-contact can provide further intensity enhancement
through plasmonic resonances. The resulting strong solar
concentration effects can facilitate nonlinear up-conversion
of sub-bandgap light. If in this way solar power conversion
efficiency could reach 30% or more with only 1 lm of silicon, it would be a major advance in solar cell technology.
ACKNOWLEDGMENTS
This work was supported in part by the United States
Department of Energy Contract No. DE-FG02-10ER46754,
the Natural Sciences and Engineering Research Council of
Canada, and the Canadian Institute for Advanced Research.
224501-8
1
Deinega, Eyderman, and John
J. Nelson, The Physics of Solar Cells (Imperial College Press, London,
2003).
2
M. Green, Silicon Solar Cells: Advanced Principles and Practice (Centre
for Photovoltaic Devices and Systems, Sydney, 1995).
3
M. Green, Third Generation Photovoltaics (Springer, Berlin, 2006).
4
Nanotechnology for Photovoltaics, edited by L. Tsakalakos (CRC Press,
2010).
5
G. Beaucarne, “Silicon thin-film solar cells,” Adv. OptoElectron. 2007,
36970 (2007).
6
C. G. Bernhard, “Structural and functional adaptation in a visual system,”
Endeavour 26, 79–84 (1967).
7
P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the
‘Moth Eye’ principle,” Nature 244, 281–282 (1973).
8
W. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8,
584–586 (1983).
9
C.-H. Sun, P. Jiang, and B. Jiang, “Broadband moth-eye antireflection
coatings on silicon,” Appl. Phys. Lett. 92, 061112 (2008).
10
C.-H. Chang, J. Dominguez-Caballero, H. Choi, and G. Barbastathis,
“Nanostructured gradient-index antireflection diffractive optics,” Opt.
Lett. 36, 2354–2356 (2011).
11
Q. Yang, X. A. Zhang, A. Bagal, W. Guo, and C.-H. Chang,
“Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24, 235202 (2013).
12
O. Bucci and G. Franceschetti, “Scattering from wedge-tapered
absorbers,” IEEE Trans. Antennas Propag. 19, 96–104 (1971).
13
B. L. Sopori and R. A. Pryor, “Design of antireflection coatings for textured silicon solar cells,” Sol. Cells 8, 249–261 (1983).
14
Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings
fabricated upon silicon substrates,” Opt. Lett. 24, 1422–1424 (1999).
15
S. A. Boden and D. M. Bagnall, “Tunable reflection minima of nanostructured antireflective surfaces,” Appl. Phys. Lett. 93, 133108 (2008).
16
D. Poxson, M. Schubert, F. Mont, E. Schubert, and J. K. Kim, “Broadband
omnidirectional antireflection coatings optimized by genetic algorithm,”
Opt. Lett. 34, 728–730 (2009).
17
B. Paivanranta, T. Saastamoinen, and M. Kuittinen, “A wide-angle antireflection surface for the visible spectrum,” Nanotechnology 20, 375301
(2009).
18
A. Deinega, I. Valuev, B. Potapkin, and Yu. Lozovik, “Minimizing light
reflection from dielectric textured surfaces,” J. Opt. Soc. Am A 28(5),
770–777 (2011).
19
E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907
(1982).
20
P. Campbell and M. A. Green, “Light trapping properties of pyramidally
textured surfaces,” J. Appl. Phys. 62, 243–249 (1987).
21
P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos,
“Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express 15, 16986–17000 (2007).
22
D. Zhou and R. Biswas, “Photonic crystal enhanced light-trapping in thin
film solar cells,” J. Appl. Phys. 103, 093102 (2008).
23
R. Dewan and D. Knipp, “Light trapping in thin-film silicon solar cells
with integrated diffraction grating,” J. Appl. Phys. 106, 074901 (2009).
24
Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18, 366–380 (2010).
25
S. E. Han and G. Chen, “Toward the Lambertian limit of light trapping in
thin nanostructured silicon solar cells,” Nano Lett. 10, 4692–4696 (2010).
26
S. B. Mallick, M. Agrawal, and P. Peumans, “Optimal light trapping in
ultra-thin photonic crystal crystalline silicon solar cells,” Opt. Express 18,
5691–5706 (2010).
27
S. Sandhu, Z. Yu, and S. Fan, “Detailed balance analysis of nanophotonic
solar cells,” Opt. Express 21, 1209–1217 (2013).
28
S. Eyderman, S. John, and A. Deinega, “Solar light trapping in slanted
conical-pore photonic crystals: Beyond statistical ray trapping,” J. Appl.
Phys. 113, 154315 (2013).
29
L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire
arrays for photovoltaic applications,” Nano Lett. 7(11), 3249–3252 (2007).
30
J. S. Li, H. Y. Yu, S. M. Wong, X. C. Li, G. Zhang, P. G. Q. Lo, and D. L.
Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell
application,” Appl. Phys. Lett. 95, 243113 (2009).
31
C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon
nanowire arrays with a large lattice constant for photovoltaic
applications,” Opt. Express 17, 19371–19381 (2009).
32
V. Sivakov, G. Andra, A. Gawlik, A. Berger, J. Plentz, F. Falk, and S. H.
Christiansen, “Silicon nanowire-based solar cells on glass: Synthesis, optical properties, and cell parameters,” Nano Lett. 9(4), 1549–1554 (2009).
J. Appl. Phys. 113, 224501 (2013)
33
E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,”
Nano Lett. 10, 1082–1087 (2010).
34
S. Han and G. Chen, “Optical absorption enhancement in silicon nanohole
arrays for solar photovoltaics,” Nano Lett. 10, 1012–1015 (2010).
35
G. Demesy and S. John, “Solar energy trapping with modulated silicon
nanowire photonic crystals,” J. Appl. Phys. 112, 074326 (2012).
36
A. Chutinan and S. John, “Light trapping and absorption optimization in
certain thin-film photonic crystal architectures,” Phys. Rev. A 78, 023825
(2008).
37
B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device
physics principles of planar and radial p-n junction nanorod solar cells,”
J. Appl. Phys. 97, 114302 (2005).
38
B. Z. Tian, X. L. Zheng, T. J. Kempa, Y. Fang, N. F. Yu, G. H. Yu, J. L.
Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and
nanoelectronic power sources,” Nature 449, 885 (2007).
39
L. Tsakalakos, J. Balch, J. Fronheiser, B. A. Korevaar, O. Sulima, and J.
Rand, “Silicon nanowire solar cells,” Appl. Phys. Lett. 91, 233117 (2007).
40
M. D. Kelzenberg, D. B. Turner-Evans, B. M. Kayes, M. A. Filler, M. C.
Putnam, N. S. Lewis, and H. A. Atwater, “Photovoltaic measurements in
single-nanowire silicon solar cells,” Nano Lett. 8, 710 (2008).
41
Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu,
J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W.
Ager, and A. Javey, “Three dimensional nanopillar array photovoltaics on
low cost and flexible substrates,” Nature Mater. 8, 648 (2009).
42
H. Yoon, Y. Yuwen, C. Kendrick, G. Barber, N. Podraza, J. Redwing, T.
Mallouk, C. Wronski, and T. Mayer, “Enhanced conversion efficiencies
for pillar array solar cells fabricated from crystalline silicon with short minority carrier diffusion lengths,” Appl. Phys. Lett. 96, 213503 (2010).
43
K.-Q. Peng and S.-T. Lee, “Silicon nanowires for photovoltaic solar
energy conversion,” Adv. Mater. 23, 198–215 (2011).
44
J. Christesen, X. Zhang, C. Pinion, T. Celano, C. Flynn, and J. Cahoon,
“Design principles for photovoltaic devices based on Si nanowires with
axial or radial p-n junctions,” Nano Lett. 12, 6024 (2012).
45
A. Deinega and S. John, “Finite difference discretization of semiconductor
drift-diffusion equations for nanowire solar cells,” Comput. Phys.
Commun. 183, 2128 (2012).
46
A. Deinega and S. John, “Solar power conversion efficiency in modulated
silicon nanowire photonic crystals,” J. Appl. Phys. 112, 074327 (2012).
47
M. D. Kelzenberg, M. C. Putnam, D. B. Turner-Evans, N. S. Lewis, and
H. A. Atwater, “Predicted efficiency of Si wire array solar cells,” in
Proceedings of the 34th IEEE PVSC (2009).
48
R. Kapadia, Z. Fan, and A. Javey, “Design constraints and guidelines for
CdS/CdTe nanopillar based photovoltaics,” Appl. Phys. Lett. 96, 103116
(2010).
49
R. R. LaPierre, “Numerical model of current-voltage characteristics and efficiency of GaAs nanowire solar cells,” J. Appl. Phys. 109, 034311 (2011).
50
J. Foley, M. Price, J. Feldblyum, and S. Maldonado, “Analysis of the operation of thin nanowire photoelectrodes for solar energy conversion,”
Energy Environ. Sci. 5, 5203–5220 (2012).
51
M. Deceglie, V. Ferry, A. Alivisatos, and H. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano
Lett. 12, 2894 (2012).
52
N. Huang, C. Lin, and M. L. Povinelli, “Limiting efficiencies of tandem
solar cells consisting of III-V nanowire arrays on silicon,” J. Appl. Phys.
112, 064321 (2012).
53
A. Taflove and S. H. Hagness, Computational Electrodynamics: The
Finite Difference Time-Domain Method (Artech House, Boston, 2005).
54
See http://fdtd.kintechlab.com for Electromagnetic Template Library
(EMTL), Kintech Lab Ltd.
55
I. Valuev, A. Deinega, and S. Belousov, “Iterative technique for analysis
of periodic structures at oblique incidence in the finite-difference time-domain method,” Opt. Lett. 33, 1491–1493 (2008).
56
I. Valuev, A. Deinega, and S. Belousov, “Implementation of the iterative
finite-difference time-domain technique for simulation of periodic structures at oblique incidence,” Comput. Phys. Commun. (unpublished).
57
A. Deinega and I. Valuev, “Subpixel smoothing for conductive and dispersive media in the FDTD method,” Opt. Lett. 32, 3429–3431 (2007).
58
A. Deinega and I. Valuev, “Long-time behavior of PML absorbing boundaries for layered periodic structures,” Comput. Phys. Commun. 182,
149–151 (2011).
59
M. A. Green and M. Keevers, “Optical properties of intrinsic silicon at
300 K,” Prog. Photovoltaics 3, 189–192 (1995).
60
A. Deinega and S. John, “Effective optical response of silicon to sunlight in
the finite-difference time-domain method,” Opt. Lett. 37, 112–114 (2012).
224501-9
61
Deinega, Eyderman, and John
See http://fdtd.kintechlab.com/en/fitting for Program for fitting of dielectric function.
62
S. Selberherr, Analysis and Simulation of Semiconductor Devices
(Springer, Wien, 1984).
63
R. Kircher and W. Bergner, Three-Dimensional Simulation of
Semiconductor Devices (Birkhauser Verlag, 1991).
64
See http://rredc.nrel.gov/solar/spectra/am1.5 for Air Mass 1.5 Global
Spectrum.
65
See https://sites.physics.utoronto.ca/sajeevjohn/software/microvolt for
Microvolt software for semiconductor devices modeling.
66
J. Meier, S. Dubail, S. Golay, U. Kroll, S. Fay, E. Vallat-Sauvain, L.
Feitknecht, J. Dubail, and A. Shah, “Microcrystalline silicon and the
impact on micromorph tandem solar cells,” Sol. Energy Mater. Sol. Cells
74, 457–467 (2002).
67
J. Zhu, Z. Yu, G. F. Burkard, C. M. Hsu, S. T. Connor, Y. Xu, Q. Wang,
M. McGehee, S. Fan, and Y. Cui, “Optical absorption enhancement in
amorphous silicon nanowire and nanocone arrays,” Nano Lett. 9, 279
(2009).
J. Appl. Phys. 113, 224501 (2013)
68
M. M. Adachi, M. P. Anantram, and K. S. Karim, “Optical properties of
crystalline-amorphous core-shell silicon nanowires,” Nano Lett. 10,
4093–4098 (2010).
69
J. Kim, A. J. Hong, J.-W. Nah, B. Shin, F. M. Ross, and D. K. Sadana,
“Three-dimensional a-Si:H solar cells on glass nanocone arrays patterned
by self-assembled Sn nanospheres,” ACS Nano 6, 265 (2012).
70
C.-M. Hsu, C. Battaglia, C. Pahud, Z. Ruan, F.-J. Haug, S. Fan, C. Ballif,
and Y. Cui, “High-efficiency amorphous silicon solar cell on a periodic
nanocone back reflector,” Adv. Energy Mater. 2, 628 (2012).
71
Z. Pei, S.-T. Chang, C.-W. Liu, and Y.-C. Chen, “Numerical simulation on
the photovoltaic behavior of an amorphous-silicon nanowire-array solar
cell,” IEEE Electron Device Lett. 30, 1305–1307 (2009).
72
P. J. Rostan, U. Rau, V. X. Nguyen, T. Kirchartz, M. B. Schubert, and J. H.
Werner, “Low-temperature a-Si:H/ZnO/Al back contacts for high-efficiency
silicon solar cell,” Sol. Energy Mater. Sol. Cells 90, 1345 (2006).
73
W. Zou, C. Visser, J. A. Maduro, M. S. Pshenichnikov, and J. C.
Hummelen, “Broadband dye-sensitized upconversion of near-infrared
light,” Nat. Photonics 6, 560–564 (2012).

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