Phase stability, electronic structure, and optical properties of indium oxide... S. Zh. Karazhanov, P. Ravindran, P. Vajeeston,

Phase stability, electronic structure, and optical properties of indium oxide... S. Zh. Karazhanov, P. Ravindran, P. Vajeeston,
PHYSICAL REVIEW B 76, 075129 共2007兲
Phase stability, electronic structure, and optical properties of indium oxide polytypes
S. Zh. Karazhanov,1,2 P. Ravindran,1 P. Vajeeston,1 A. Ulyashin,3 T. G. Finstad,4 and H. Fjellvåg1
1Department
of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway
Institute, 2B Mavlyanov Street, Tashkent 700084, Uzbekistan
3
Institute for Energy Technology, P.O. Box 40, NO-2027 Kjeller, Norway
4Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
共Received 17 January 2007; revised manuscript received 13 July 2007; published 29 August 2007兲
2Physical-Technical
Structural phase stability, electronic structure, optical properties, and high-pressure behavior of polytypes of
In2O3 in three space group symmetries I213, Ia3̄, and R3̄ are studied by first-principles density-functional
calculations. From structural optimization based on total energy calculations, lattice and positional parameters
have been established, which are found to be in good agreement with the corresponding experimental data
except for I213, where the symmetry analysis for optimized structure indicates that it arrived at the Ia3̄ phase.
In2O3 of space group symmetry Ia3̄ is found to undergo a pressure-induced phase transition to the R3̄ phase at
pressures around 3.8 GPa. From the analysis of band structure coming out from the calculations within the
local density and generalized gradient approximations, it is found that In2O3 of space group symmetry I213 and
R3̄ are indirect band gap semiconductors, while the other phase of space group Ia3̄ is having direct band gap.
The calculated carrier effective masses for all these three phases are compared with available experimental and
theoretical values. From charge-density and electron localization function analysis, it is found that these phases
have dominant ionic bonding with noticeable covalent interaction between indium and oxygen. The magnitudes of the absorption and reflection coefficients for In2O3 with space groups Ia3̄ and R3̄ are small in the
energy range 0 – 5 eV, indicating that these phases can be regarded and classified as transparent.
DOI: 10.1103/PhysRevB.76.075129
PACS number共s兲: 71.15.⫺m, 71.22.⫹i
I. INTRODUCTION
Transparent conducting oxides 共TCOs兲 have increased
scientific interest because of their many useful features such
as transparency in the visible spectrum of incident light and
high electrical conductivity. The main interest for the investigation of TCO is their extensive applications such as window layers in solar cells, sensors, front electrodes in flat
panel displays, low emissive windows, electrochromic materials in rare-view mirrors of automobiles, smart windows,
etc. Investigations on TCO have lead to the formation of a
new field in optoelectronic device technology called transparent electronics or invisible electronics.1–7 Here, a combination of n- and p-type homo-p-n-junction based on TCO
could lead to a functional window, which transmits visible
portion of solar radiation and, at the same time, generates
electricity by the absorption of the ultraviolet part of
sunlight.6
One of the materials widely used as TCO is Sn doped
In2O3 共commonly referred to as ITO兲. In2O3 can exist in
three different phases characterized by space group symmetries I213, Ia3̄, and R3̄, which we refer to throughout the
paper as phases In2O3-I, -II, and -III, respectively. Among
them, the In2O3-I is rarely studied. Only a few papers8 are
available on crystal structure studies of this phase. In2O3-II
with band gap of Eg = 3.7 eV 共Ref. 9兲 is widely studied
both theoretically and experimentally. Until recently,
In2O3-III was also rarely studied because it is a high-pressure
phase and therefore not easily available. However, recent
works have changed this situation. Nanoparticles of In2O3 in
rhombohedral structure have been synthesized by hydrothermal method, followed by postannealing at moderate temperatures and pressures.10 Phase selective growth of this
phase was also demonstrated11 by means of metal organic
1098-0121/2007/76共7兲/075129共13兲
chemical vapor deposition method at high substrate temperatures 共⬎550 ° C兲 and low 共⬎4 ␮mol/ min兲 trimethylindium
flow rates. Furthermore, it is found12 that this phase can be
formed upon annealing in air at ⬎550 ° C from the amorphous indium zinc oxide.
The electronic structures of ideal and defective In2O3 with
oxygen vacancies have been studied by the discrete variational X␣ method, and it was found9 that vacancy levels
appear in the band gap. From comparative analysis of electronic structure of In2O3, ZnO, and SnO2, it was shown that
they possess a unique feature that the holes are heavier than
conduction band 共CB兲 electrons.13 Ab initio studies of In2X3
共X = O , S , Se, Te兲 using the tight-binding linear muffin-tin orbital 共LMTO兲 method show that compression of the lattice
increases the band gap.14 Further, this method has been applied to analyze the x-ray photoelectron, bremsstrahlung isochromat, and optical spectra of In2O3 and ITO. Based on
band-structure calculations for In2O3 and ITO15,16 using the
density-functional full-potential 共FP兲 LMTO and fullpotential linearized augmented plane wave method, criteria
for transparent conducting behavior have been formulated,
and it is found that position, dispersion, and character of the
lowest conduction band are responsible for electro-optical
properties. By a systematic study17 using the densityfunctional theory 共DFT兲 within the local density 共LDA兲 and
generalized gradient approximations 共GGA兲 with the multiorbital mean-field Hubbard potential 共LDA+ U and GGA+ U兲,
it is found that the band gap of In2O3-II calculated within
pure LDA and GGA is direct, while that calculated within
LDA+ U and GGA+ U with and without the spin-orbit coupling is indirect. The latter approximation is found to correct
also energy location of the In 4d electrons. From firstprinciples molecular orbital calculations, native donors such
as the oxygen vacancy 共VO兲, indium interstitial 共Ini兲 and the
075129-1
©2007 The American Physical Society
PHYSICAL REVIEW B 76, 075129 共2007兲
KARAZHANOV et al.
(a)
VO-Ini complex were studied18 in undoped In2O3, and it is
found that Ini generates a shallow donor level, VO-Ini creates
even shallower level, while VO forms a very deep donor level
and facilitates emergence of Ini.
Schematic energy band model for ITO was constructed19
based on x-ray photoelectron spectroscopy 共XPS兲 studies. A
systematic study of the electronic structure as well as optical
and transport properties for cubic ITO was performed20–22
using the ab initio software CRYSTAL. Dependence of carrier
mobility and film resistance with carrier concentration in the
range 1018 – 1022 cm−3 共Ref. 21兲, optical spectra in the energy
range 0.0– 3.1 eV 共Ref. 22兲, and carrier concentrations of
1018 – 1022 cm−3 共Ref. 20兲 were analyzed, and good agreement with experimental data was achieved. The electronic
band structure and the cohesive energies of In4Sn3O12 and
In5SnSbO12 were calculated by DFT within GGA, and it is
found23 that incorporation of Sb into the In4Sn3O12 matrix
broadens the conduction band.
There still exist some points which should be clarified: 共i兲
Among the In2O3-I, -II, and -III polymorphs, only In2O3-II is
widely studied; 共ii兲 charge density, Bader and Voronoi
charges, as well as electron localization function analysis
were not performed; 共iii兲 structural properties of In2O3-I, -II,
and -III and possibility of phase transition between these
phases were not studied; 共iv兲 optical spectra are available
only for films, and In2O3-II and -III in a narrow energy range
0 – 6 eV; and 共v兲 there are no experimentally measured effective masses for In2O3-I and -III. The aim of this paper is to
study the structural, electronic, and optical properties of
In2O3-I, -II, and -III by ab initio calculations.
(b)
(c)
In1
In2
In3
O
II. STRUCTURAL ASPECTS AND COMPUTATIONAL
DETAILS
A. Crystal structures
Crystal structures 共Fig. 1兲 and lattice parameters 共Table I兲
for In2O3-I, -II, and -III taken from Ref. 24 are used as input
for the structural optimization. In all the three polymorphs,
the coordination is sixfold for In atoms and fourfold for O
atoms.
In2O3-I crystallizes in a cubic bixbyite-type structure
共space group No. 199兲 with 8 f.u. per unit cell, with those
containing three types of In and two types of O atoms occupying 8a, 12b, 12b, 24c, and 24c Wyckoff positions, respectively. In Fig. 1, the three types of In atoms are indicated by
different colors and all are surrounded by oxygen in trigonal
biprism coordination. The In2O3-II phase also crystallizes in
a cubic bixbyite-type structure with space group No. 206 and
8 f.u. per unit cell. Distinct from In2O3-I, it has different
atomic arrangements and bond lengths, and consists of two
types of In 共they are surrounded by oxygen in the octahedral
and trigonal prismatic coordinations alternatively, as shown
in Fig. 1兲 and one type of O atoms located at Wyckoff positions 8b, 24d, and 48e, respectively. It may be noted that the
polyhedra surrounding the two types of In atoms differ from
those of In2O3-I. In2O3-III is of corundum structure with
2 f.u. per unit cell 共space group No. 167兲. It consists of one
type of In 共surrounded by oxygen in trigonal biprism coordination兲 and one type of O atoms occupying 12c and 18e
FIG. 1. 共Color online兲 Crystal structures of 共a兲 In2O3-I 共experimental structural parameters兲, 共b兲 In2O3-II, and 共c兲 In2O3-III. Different types of In atoms are marked by different colors.
Wyckoff positions, respectively. This is a high-pressure
phase10 and, consequently, has been rarely produced, but it
has been claimed10,11 that the stability and conductivity of
this phase should be favorable to that of cubic phase.
B. Computational details
The band-structure calculations have been performed using the VASP projector-augmented wave 共PAW兲 package,27
which calculates the Kohn-Sham eigenvalues in the framework of DFT within LDA. For the exchange-correlation potential and energy, we have used the Perdew-Zunger
075129-2
PHYSICAL REVIEW B 76, 075129 共2007兲
PHASE STABILITY, ELECTRONIC STRUCTURE, AND…
TABLE I. Equilibrium lattice parameters 共a, b, and c兲 and volumes 共V兲, positional parameters, bulk modulus 共B0兲, and its pressure
derivative 共B⬘0兲 for different phases of In2O3 derived from the total energies obtained from present DFT calculations. Values given in
parentheses refer to experimental data 共Ref. 24兲.
Compound
space group
Unit cell
共Å兲
Atom
Site
x
y
B0⬘
共0.2500兲
共0.2500兲
共0.2500兲
共0.3950兲
共0.3730兲
172.44
4.79
172.87
4.75
183.61
4.62
In2O3-I
I213
a = 10.080 共10.120兲a
b = 10.080 共10.120兲a
c = 10.080 共10.120兲a
V = 1024.26 共1036.43兲a Å3
In1
In2
In3
O1
O2
8a
12b
12b
24c
24c
0.2500
0.0335
0.5338
0.1182
0.1543
In2O3-II
a = 10.077 共10.117兲b
b = 10.077 共10.117兲b
In1
In2
8b
24d
0.2500 共0.2500兲
0.4665 共0.4668兲
0.2500 共0.2500兲
0.0000 共0.0000兲
0.2500 共0.2500兲
0.2500 共0.2500兲
c = 10.077 共10.117兲b
V = 1023.28 共1035.51兲b Å3
O
48e
0.3900 共0.3905兲
0.1544 共0.1529兲
0.3820 共0.3832兲
a = 5.4928 共5.4870兲c
b = 5.4928 共5.4870兲c
In
O
12c
18e
0.0000 共0.0000兲
0.2961 共0.2980兲
0.0000 共0.0000兲
0.0000 共0.0000兲
0.3576 共0.3573兲
0.2500 共0.25000兲
Ia3̄
In2O3-III
R3̄c
共0.2500兲
共0.0210兲
共0.5420兲
共0.1250兲
共0.1000兲
B0
共GPa兲
z
0.2500
0.0000
0.0000
0.1099
0.3821
共0.2500兲
共0.0000兲
共0.0000兲
共0.1350兲
共0.3580兲
0.2500
0.2500
0.2500
0.3456
0.3901
c = 14.4242 共14.5100兲c
V = 376.89 共378.33兲c Å3
a
Experimental values from Ref. 8.
values from Ref. 25.
c
Experimental values from Refs. 10 and 26.
bExperimental
interpolation28 of the many-body calculations of Ceperley
and Alder.29 To clarify whether In2O3-II possesses direct or
indirect band gap, we have performed additional calculations
using generalized gradient approximations 共GGA兲 where
Perdew-Burke-Ernzerhof exchange-correlation functional30
has been used. The interaction between electrons and atomic
cores was described by means of non-norm-conserving
pseudopotentials implemented in the VASP package. The
pseudopotentials are generated in accordance with the PAW
method.31,32 The use of the PAW pseudopotentials addresses
the problem of the inadequate description of the wave functions in the core region—a problem common to all pseudopotential approaches.33 The application of the pseudopotentials allow us to construct orthonormalized all-electron-like
wave functions for the In 4d and 5s, and O 2s and 2p valence
electrons. Except spin-orbit coupling, all the other relativistic
effects are included in the present calculations.
The diagonal elements of the effective mass tensor for the
conduction band 共CB兲 electrons are calculated by
1 ⳵2E共k兲
1
= 2
mc共k兲 ប ⳵k2
共1兲
in different directions in the k space from the ⌫ point toward
the other high-symmetry points in the Brillouin zone of
In2O3-I, -II, and -III. We use this to give an indication of the
conduction, since the CB minimum of the band dispersions
of all the compounds considered 关see Sec. III B兴 are located
at the ⌫ point of the Brillouin zone. The band edge energies
E共k兲 have been extracted from DFT calculations, and ninth
order polynomial fitting has been performed. From the fitted
polynomial, second order derivative was calculated, which
was used in the effective mass calculations in Eq. 共1兲. We
have studied the effective masses along the crystallographic
directions 关001兴, 关011兴, and 关111兴 corresponding to the directions ⌫ → H, ⌫ → N, and ⌫ → P, respectively, for In2O3-I
and -II and ⌫ → L, ⌫ → F, and ⌫ → Z for In2O3-III.
The equilibrium lattice parameters and bulk modulus for
In2O3-I, -II, and -III are determined from the total energies
obtained as a function of volume calculated including cell
shape as well as atom position relaxations into the computation using the universal equation of state fitting.34 The calculations have been performed on a ⌫ centered 10⫻ 10⫻ 10 k
grid in the entire Brillouin zone.
The imaginary part of the optical dielectric function ⑀2共␻兲
has been derived from DFT results by summing interband
transitions from occupied to unoccupied states for energies
much higher than those of phonons. The real part of the
dielectric function ⑀1共␻兲 is calculated using the KramersKronig transformation. The knowledge of both the real and
imaginary parts of the dielectric tensor allows one to calculate other important optical parameters. In this paper, we
present and analyze the reflectivity R共␻兲, the absorption coefficient ␣共␻兲, the refractive index n共␻兲, and the extinction
coefficient k共␻兲. More details about the optical calculations
were discussed in Refs. 35 and 36.
For simplification of analysis of the optical spectra, the
labels E0, E1, E2, d1, and d2 have been used. Here, the subscript 0 in E refers to transitions occurring at ⌫, 1 for transitions at points in the 关111兴 direction, and 2 for transitions in
the 关100兴 direction of the k space.37 These notations are used
based on those of Ref. 37 to describe the reflectivity spectra
of semiconductors of wurtzite and zinc blende structures.
075129-3
PHYSICAL REVIEW B 76, 075129 共2007兲
Total energy (eV/f.u.)
KARAZHANOV et al.
In2O3-I
In2O3-II
In2O3-III
-30.8
-31.2
-31.6
55
60
65
3
Volume (Å /f.u.)
FIG. 2. Dependence of total energy Etot with cell volume V per
formula unit cell for In2O3-I, In2O3-II, and In2O3-III.
III. RESULTS AND DISCUSSION
A. Ground-state properties
Using the experimentally determined crystal structure information as input, structural optimization has been performed for In2O3-I, -II, and -III. Positional and lattice parameters derived from the DFT calculations for the equilibrium
lattices are given in Table I together with the experimentally
determined values. Analysis of the Table I shows that deviation of the calculated equilibrium volumes are ⬍1.2% of the
experimentally determined values, which indicates that the
theoretical calculations are reliable in predicting structural
parameters for such complex systems. Furthermore, symmetry of the thus optimized lattices is checked for In2O3-I, -II,
and -III, and it is found that it is the same as that of the
corresponding experimentally determined one. Although for
the In2O3 phases studied in this paper the calculated and
experimentally determined lattice parameters do not deviate
much from each other, such a structural study is still needed
because the optimized atomic positions deviate from experiment for In2O3-I.
min
for In2O3-III is larger
Analysis of Fig. 2 shows that Etot
than that of In2O3-I and -II. Total energies, volumes, and
crystal structures for In2O3-I and -II are nearly the same and
the difference is within the accuracy of the calculations. So,
In2O3-I can be regarded as distorted In2O3-II, and small
variation of temperature or pressure can cause structural
transformation of In2O3-I into In2O3-II. The transition is well
demonstrated in Table II about atomic distances of the relaxed and unrelaxed lattices of In2O3-I, -II, and -III. It is seen
that after optimization, positional parameters for In2 and In3
atoms of In2O3 become almost the same when we use LDA
and exactly the same for GGA. Furthermore, the positional
parameters obtained from calculations within GGA are closer
to experimental data than the LDA derived values. This result is in agreement with the general opinion that structural
optimization using GGA is more preferable than that using
LDA. However, as we noted above, underestimation of the
equilibrium volume even within LDA is ⬍1.2%, which indicates that the relaxation within LDA, in this particular case,
shall not lead to large deviations of electronic structure and
optical spectra from experimental values except for In2O3-I.
Due to the difference in the optimized and experimental positional parameters for In2O3-I, there is large difference in
the electronic structure and optical properties between experimental and theoretically optimized structures for this
phase.
From Fig. 2, it is seen that equilibrium volume for
In2O3-III is smaller, but the total energy is higher than that
for In2O3-II, which indicates that In2O3-III is less stable than
In2O3-II. Furthermore, at lower volumes, and, respectively,
higher pressures, the dependence Etot共V兲 for In2O3-III
crosses that for In2O3-II. It indicates that, upon compression,
In2O3-II can be transformed into In2O3-III. In order to estimate the pressure value for the transitions from In2O3-II to
-III, we have plotted the dependence of Gibbs free energy
共defined as G = U + PV, where U is the total energy for the
particular pressure P at volume V兲 difference 共⌬G兲 as a function of the applied pressure 共Fig. 3兲. The pressure value at
which ⌬G = 0 is the transition pressure at the structural phase
transition point. According to our findings in Fig. 3, the
pressure-induced phase transitions from In2O3-II to -III occurs at P = 3.83 GPa.
TABLE II. Interatomic distances 共multiplicity in parentheses兲 for relaxed and unrelaxed lattices of In2O3-I, -II, and -III calculated by
DFT within the LDA and GGA. The interatomic distances of the experimentally determined crystal structures correspond to unrelaxed
lattices.
Phase
In2O3-I 共expt.兲
In1-O1
In1-O1
In2-O1
In2-O2
In2-O2
In3-O1
In3-O1
In3-O2
2.247
2.260
2.232
2.268
2.273
2.264
2.285
2.261
共3 ⫻ 兲
共3 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
In2O3-I 共LDA兲
In2O3-I 共GGA兲
2.166 共3 ⫻ 兲
2.167 共3 ⫻ 兲
2.121 共2 ⫻ 兲
2.186 共2 ⫻ 兲
2.212 共2 ⫻ 兲
2.120 共2 ⫻ 兲
2.186 共2 ⫻ 兲
2.211 共2 ⫻ 兲
2.213
2.213
2.165
2.235
2.255
2.165
2.235
2.255
共3 ⫻ 兲
共3 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
共2 ⫻ 兲
In2O3-II 共expt.兲
In2O3-II 共LDA兲
In2O3-III 共expt.兲
In2O3-III 共LDA兲
2.191 共6 ⫻ 兲
2.166 共6 ⫻ 兲
2.116 共3 ⫻ 兲
2.258 共3 ⫻ 兲
2.120 共3 ⫻ 兲
2.248 共3 ⫻ 兲
2.120 共2 ⫻ 兲
2.192 共2 ⫻ 兲
2.210 共2 ⫻ 兲
2.118 共2 ⫻ 兲
2.187 共2 ⫻ 兲
2.211 共2 ⫻ 兲
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PHYSICAL REVIEW B 76, 075129 共2007兲
PHASE STABILITY, ELECTRONIC STRUCTURE, AND…
In2O3-II to -III
∆G (kJ/mol)
0.10
3.83
0.05
0.00
-0.05
3.0
4.0
3.5
4.5
Pressure (GPa)
5.0
FIG. 3. Gibbs free-energy difference for In2O3-III relative to
In2O3-II 共open circles兲 as a function of the applied pressure. Structural phase transition point is marked with an arrow and the corresponding pressure value 共in GPa兲 is also stated.
The pressure-induced structural transition is well demonstrated in the total energy variation with volume curves 共see
Fig. 2兲, and from that, the dependence of the cell volume on
applied pressure is derived 共Fig. 4兲. It is seen that the volume
shrinkage at the transition pressure is 1.72 Å3. The phase
transition In2O3-II-to-III is accompanied by breaking and
transformations of chemical bonds between atoms and reconstruction of anion and cation sublattices.
3
cell volume (Å /f.u.)
65
In2O3-II
In2O3-III
64
63
62
3.83
61
60
59
0
1
2
3 4 5
Pressure (GPa)
6
7
FIG. 4. Calculated cell volume as a function of pressure for
In2O3-II and -III. The pressure-induced phase transition point is
marked with an arrow and the corresponding pressure value at the
transition point 共in GPa兲 is also stated.
Phase transition from In2O3-II to -III was studied experimentally in the literature before. However, the reported transition pressure value from different experimental studies differs noticeably. In Ref. 38, such a transition has been
demonstrated experimentally at 6.5 GPa and 1250 ° C. However, at pressures up to 6 GPa and temperatures 550, 1000,
and 1450 ° C, In2O3 has been retained39 in phase -II. According to experiments of Ref. 26, the mixture of In2O3 and Co
possesses the same structure as In2O3-III. The experimental
results from Ref. 26 for pure In2O3 show that the formation
of In2O3-III was possible at 800 ° C provided that the sample
is initially preheated at 1250 ° C. Without the preheating, formation of In2O3-III occurred at 1000 ° C only. In contrast to
the experimental report from Refs. 26, 38, and 39, the
pressure-induced phase transition obtained in the present
work is small and, however, it corresponds to 0 K. It may be
noted that the phase selective growth of In2O3-III in the experiments of Ref. 11 always occurred at very low pressures
共20 kPa兲.
Bulk modulus is the parameter characterizing the compressibility of a solid. The bulk modulus of In2O3-III is
larger than those of In2O3-I and -II. Consequently, the
phases -I and -II are more easily compressible than In2O3-III.
No experimental data on bulk modulus values for pure
In2O3-I, -II, and -III have been reported so far. So, the calculated bulk modulus values are compared to 99± 32 GPa of
ITO films.40 It is seen that our calculated bulk moduli for
In2O3 phases 共Table I兲 are much larger than that for the ITO
films and, consequently, the pure In2O3 phases are hardly
compressible than the Sn doped In2O3 films.
B. Band structure
For a better understanding of the electronic and optical
properties and chemical bonding of the polytypes of indium
oxide, the analysis of band structures can be quite helpful.
Band structures for In2O3-I, -II, and -III calculated in this
work are presented in Fig. 5. Although we noted above that
In2O3-I is distorted In2O3-II, we still plotted band structure
for both In2O3-I and -II to demonstrate how lattice distortion
can drastically change the band structure. For example, a
well distinct band region is seen in the bottom of the CB of
In2O3-I 关Fig. 5共a兲兴 compared with other two polymorphs.
This distinct band region is split from the rest of the CB, and
it is located in the energy range 0.009– 2.391 eV above the
topmost valence band 共VB兲. This band region can be called
as intermediate band 共IB兲. Since IB is well dispersive, recombination of electrons and holes through this band is not
expected to be very high. As this band region is completely
empty, it can be useful for photoemission of electrons from
VB to the IB. The small difference in total energy of In2O3-I
and -II and large difference in the band features at the CB
minimum in these two modifications indicate that one can
drastically change the optical properties by different preparatory conditions by stabilization of different proportions of
these two phases.
Despite the large difference in crystal structures, In2O3-I,
-II, and -III have some similar features. Particularly, the bottommost CB of In2O3-I, -II, and -III is dispersive and is
075129-5
PHYSICAL REVIEW B 76, 075129 共2007兲
KARAZHANOV et al.
4
3
2
1 (a)
0
Energy (eV)
Γ
4 (b)
3
2
1
0
Γ
H
P
Γ
N
2.86 eV
1.10 eV
H
4
3
2
(c)
1
0
F
N
N
P
Γ
N
L
Γ
3.12 eV
1.25 eV
Γ
Z
F
FIG. 5. Band structure for 共a兲 unrelaxed In2O3-I, 共b兲 In2O3-II,
and 共c兲 In2O3-III near the VB maximum and CB minimum. The
Fermi level is set to zero.
located at the ⌫ point, while the topmost VB is flat, which
are the important properties inherent to TCO materials. The
VB consists of three regions in all the three phases. The
lowest one is located below −15 eV from the topmost VB.
The bands at the intermediate energy region is located between −14 and −10 eV. Both these band energy regions are
very narrow, while the topmost VB region is quite broad.
The VB maximum is located at ⌫ point for In2O3-I and -II
and at L point for In2O3-III. Consequently, one can conclude
that the In2O3-III is an indirect band gap material, while
In2O3-I and -II are direct band gap materials.
The direct band gaps 共Edir
g 兲 coming out from the LDA
calculations are presented in Table III along with the experimentally determined values. Analysis of Table III shows that
the calculated Edir
g for In2O3-I, -II, and -III are underestimated due to the well-known problem of the DFT. Experimental data for the band gap of In2O3-I are not yet available.
Our calculated Edir
g for In2O3-II is in good agreement with
0.9 eV,13 1.0 eV,15 and ⬍1.0 eV,14 reported from DFT calculations. The second band gap 共Esg兲, which is the energy
difference between the two lowest CBs, has been calculated.
We found that Esg of In2O3-III is larger than that of In2O3-I
and -II. So, one can expect transparency in a wider energy
range of the solar spectra from In2O3-III than from In2O3-I
and -II.
The character 共direct or indirect兲 and value of the band
gap are important factors for the characterization of TCO
materials. Until recently, band-structure results obtained
from different theoretical studies of In2O3-II have been controversial. From the band-structure calculations using the
LMTO method within the atomic sphere approximation
共ASA兲 based on the LDA, it is found14 that the band gap of
In2O3-II is indirect with CB minimum at the ⌫ point and VB
maximum at H point. However, DFT calculations of Refs.
13, 15, and 42 show that the band gap of In2O3-II is direct
with CB minimum and VB maximum located at the ⌫ point.
The band gap value of 1.0 eV was found for experimental
lattice constant and 1.5 eV for theoretically determined lattice constants.15 Recent systematic study of Ref. 17 by DFT
calculations within LDA+ U and GGA+ U has shed light on
the problem. It is found that the band gap of In2O3-II calculated within pure LDA and GGA is direct, while that calculated within LDA+ U and GGA+ U for U − J = 7 eV is indirect. Our results given in Table III are in agreement with
these findings. We have calculated the band structure of
In2O3-II using the theoretically and experimentally determined lattice constants within LDA and GGA. It shows that
the CB minimum and VB maximum are located at the ⌫
point. Hence, we conclude that the band gap of In2O3-II
calculated within pure LDA and GGA is direct. Our experience shows that the indirect band gap predicted by LMTO
method is associated with the use of ASA in the calculation.
From room temperature fundamental absorption edge
studies, it was found43 that direct and indirect band gaps of
single crystal In2O3-II are 3.750 and 2.619 eV, respectively.
Similar band gap values have been found44 from absorption
measurements for In2O3 films, which decreased with increasing film thickness from 2.65 eV for 100 nm to 2.40 eV for
400 nm. The band gap also decreased with increasing deposition temperature from 2.50 eV at 20 K to 2.30 eV at
270 K. Band gap reduction from 2.46 to 2.30 eV was reported to be caused also by increase of the annealing temperature from 200 to 500 K. From these reports, it appears
that structural defects play an important role in defining the
band gap of In2O3 and hence optical properties.
C. Density of states
Insight into the bonding interaction between constituents
can be obtained from the site and orbital projected partial
density of states 共DOS兲 analysis. Moreover, analysis of the
projected density of states 共PDOS兲 gives detailed information about the origin of CB and VB electronic energy levels.
Such analysis has been performed for all the phases of In2O3
considered in the present work 共Fig. 6兲. The main feature in
Fig. 6 is that the DOS distribution of In and O are energeti-
075129-6
PHYSICAL REVIEW B 76, 075129 共2007兲
PHASE STABILITY, ELECTRONIC STRUCTURE, AND…
TABLE III. Second band gap 共in parentheses兲 and fundamental band gap 共in eV兲 for In2O3-I, -II, and -III
calculated from DFT within LDA 共ELDA
兲 and GGA 共EGGA
兲 using the lattice parameters determined theoretig
g
cally 共a0兲 and experimentally 共ae兲 along with experimentally measured values 共Eexpt.
g 兲.
ELDA
g
Phase
In2O3-I
In2O3-I
In2O3-II
EGGA
g
a0
ae
1.16
共2.93兲
1.17
共2.86兲
0.009
1.100
共2.860兲
In2O3-III
1.250
共3.120兲
a0
ae
Eexpt
g
1.33
共2.80兲
1.29
共2.84兲
3.70a
3.60b
3.70c
3.80d
3.00c
a
Reference 9.
Reference 41.
c
Reference 11.
dReference 26.
b
-1
-1
PDOS (States eV atom )
cally not separated, indicating that the nature of chemical
bonding between In and O is not pure ionic and it contains
some covalency, consistent with our charge-density analysis
given below. Since the basic feature of In2O3-I has already
been discussed in the previous section, below we concentrate
our attention on the analysis of In2O3-II and -III.
The VB of In2O3-II and -III can be divided into three
regions. It is seen in Fig. 6 that the bands at the lowest
0.4
0.2
In1
0.4
0.2
In2
0.4
0.2
O
EF
s
p
d
(a) In2O3-II
0.4
0.2
In
0.4
0.2
O
EF
0
-15 -10 -5
(b) In2O3-III
Energy (eV)
5
FIG. 6. 共Color online兲 The orbital and site-projected DOS for 共a兲
In2O3-II and 共b兲 -III. The Fermi level is set to zero.
energy region 共below −15 eV from the topmost VB, we call
it VB1兲 are mainly contributed by O 2s electrons with noticeable In 4d electrons. The bands at the intermediate energy
region 共located between −14 and −10 eV, we call it VB2兲 are
basically from In 4d electrons with small contribution from
O 2s and 2p states due to hybridization interaction. Both
these band energy regions are very narrow, while the topmost
VB region is quite broad.
The states around −5 eV in the VB 共we call VB3兲 are
equally contributed by both In 5s and O 2p electrons. The
main contribution to the topmost valence band of In2O3-III is
from O 2p electrons, which has noticeable hybridization interaction with In 4d / 5p electrons. The results from the
PDOS analysis are in good agreement with those of previous
ab initio calculations9,14,15 and recent soft x-ray emission
共SXE兲 and soft x-ray absorption 共SXA兲 spectra
measurements45 for In2O3-II. The character of the topmost
VB is similar to that of ZnO, which is also a transparent
conducting oxide 共see, e.g., Ref. 46兲.
The position, dispersion, and character of the lowest CB
carry the key features responsible for electro-optic properties
of TCO materials.15 The bottommost CBs of In2O3-II and
-III are mainly originating from In 5s electrons with finite
O 2p contribution. Orbital and site decomposed DOS analysis is used for the detailed characterization of the contributions of electrons from s, px, py, pz, dxy, dxz, dyz, dx2−y2, and
dz2 orbitals into the bands of particular interest. From such
analysis, it is found that the topmost VB of In2O3-II and -III
is contributed by electrons from O 2px, 2py, 2pz orbitals,
which slightly hybridized with electrons from In 4dxy, 4dxz,
4dyz orbitals.
The origin for the CB minimum of In2O3-II and -III is
more or less the same, and it basically originates from electrons of In 5s orbitals with small O 2s, 2px, 2py, and 2pz
character. Based on this analysis and the basic formulations
suggested in Ref. 15 regarding highly dispersed and single
character s-type band at the bottommost CB, the features that
favor TCO behavior, one can conclude that In2O3 possesses
the feature of s-electron based TCO.
075129-7
PHYSICAL REVIEW B 76, 075129 共2007兲
Intensity (arb. units)
-1
-1
TDOS (States eV f.u. )
KARAZHANOV et al.
8 (a)
6
4
2
In2O3-II
EF
8 (b)
VB1
6
4
2
(c)
VB3
VB2
BIS (×10)
XPS (×10)
XPS (×20)
-15
In2O3-III
-10
0
-5
Energy (eV)
- - SXE
SXA
5
FIG. 7. Calculated total DOS for 共a兲 In2O3-II and 共b兲 -III as
compared with 共c兲 experimental data obtained from different techniques: BIS 共open triangles兲 共Ref. 47兲, XPS 共closed triangles, Ref.
47兲, XPS 共thick solid lines, Ref. 48兲, SXE 共Ref. 45兲, and SXA 共Ref.
45兲. The Fermi level is set to zero.
The total DOSs for In2O3-II and -III are displayed in Fig.
7 along with XPS, bremsstrahlung isochromat spectroscopy
共BIS兲, SXE, and SXA measurements. The DOSs exhibit
sharper peaks than the experimental spectra, since we have
not included the lifetime broadening and instrumental resolution. As noted in the Introduction, there are no experimental XPS studies available on In2O3-I. So, comparative analysis with experimental data has been performed only for
In2O3-II and -III.
As noted in the previous section, there are three distinct
regions in the VB of the In2O3 phases considered in the
present work. According to the band dispersion and total
DOS analysis, the widths of the band regions corresponding
to VB1, VB2, and VB3 are 2.22, 1.52, and 6.38 eV for
In2O3-I, 1.5, 1.77, and 5.46 for In2O3-II, and 0.81, 1.74, and
5.5 eV for In2O3-III, respectively. These results are in fair
agreement with that of Ref. 13, where 2.33 eV for the VB2
and 5.70 eV for the topmost valence band are reported. This
small discrepancy may be related to different approximations
and methods used in the calculations. It should also be noted
that the locations of the In 4d bands in In2O3-II and -III
coming out from the DFT calculations are shifted toward
higher energies and their intensity is also higher compared to
experimental data 关Fig. 7共d兲兴.
Analysis of Fig. 7 shows that the widths of VB3 of
In2O3-II and -III more or less agree with SXE measurements,
which show a well-defined intensive peak closer to the topmost VB. Such a peak is smeared out in other experimental
spectra shown in Fig. 7. However, the experimental spectroscopic measurements show only one peak centered around
−15 eV, whereas the DFT calculation show two 共VB1 and
VB2兲 around this energy range. As mentioned in the PDOS
analysis, the VB1 is mainly originating from O 2s electrons
the photoionization cross section of which is much smaller
than that of In 4d electrons; hence, VB1 band feature is not
observed experimentally. So, the peak centered around
−15 eV in the spectroscopic measurements can be concluded
as the band features arising from In 4d electrons. However,
the calculated 4d band 共VB2兲 position is shifted around
2.5 eV toward the valence band maximum.
It may be noted that there is discrepancy in the XPS results of different groups. For example, the binding energy of
the In 4d band is found to be at 14 eV in Ref. 47 and 18 eV
in Ref. 48. This discrepancy may be associated with the use
of different reference points in estimating the peak
positions.45 The width of the calculated VB1 band agrees
with experimental data of Refs. 45, 47, and 48. Furthermore,
in our calculations, these two peaks are much more intense
than the topmost valence band, which disagrees with experimental data but agrees with theoretical calculations of Refs.
9 and 14. The reason for the discrepancy can be that the
photoionization cross section is important in accounting for
the intensity. So, only locations of the calculated total DOS
have been compared with the experimental measurements. It
is important to note that the pseudocore d bands are not
placed correctly in systems such as ZnO 共Ref. 49兲 and CdTe
共Ref. 50兲 by the present type of calculations. So, one should
go beyond the LDA/GGA calculations by taking into account
the Coulomb correlation effects from the In 4d electrons
through methods such as LDA+ U, LDA+ SIC, etc. Recent
study17 of band structure of In2O3 showed that within LDA
+ U and GGA+ U calculations for U − J = 7 eV, the In 4d levels have been shifted toward lower energies and split the
O 2s band into two parts.
D. Conduction band effective masses
Effective masses characterize band dispersion and are one
of the important parameters linking electronic structure with
transport properties of solids. As noted above, the lowest
CBs of In2O3-II and -III are more dispersive than the topmost VB, which means that the CB electrons are lighter than
holes. Almost flat topmost VB indicates that valence electrons are tightly bound to the atoms. Consequently, the dominant intrinsic charge carriers in In2O3 are expected to be
electrons and not holes, consistent with experimental observations. Hence, we focus on the CB electron effective
masses, and the calculated results are presented in Table IV.
Analysis shows that the calculated masses for In2O3-II and
-III are almost isotropic. The observed small anisotropy can
be related to the effect of p and d orbitals hybridizing with s
orbitals. The deviation in the calculated effective masses
from the experimentally measured values are in the range
23%–33%. This deviation can be related to the deficiency of
DFT in the calculation of the eigenvalue of electrons. So, not
only the band gap but also the band dispersion comes out
incorrectly, and this effect is generally well pronounced in
075129-8
PHYSICAL REVIEW B 76, 075129 共2007兲
PHASE STABILITY, ELECTRONIC STRUCTURE, AND…
TABLE IV. Calculated effective masses of CB electrons 共in
units of the free-electron mass m0 and in the directions indicated兲
for In2O3-II and -III are compared with available experimental and
calculated values.
In2O3-II
0.23
0.20
0.23
0.42
0.30
0.36
0.30
0.36
0.41
0.30
共⌫-H兲
共⌫-N兲
共⌫-P兲
共⌫-H兲c
共⌫-N兲c
共⌫-P兲c
共⌫-H兲f
共⌫-N兲f
共⌫-P兲f
共⌫-P兲g
In2O3-III
ZnO
0.16 共⌫-F兲
0.14 共⌫-L兲
0.15 共⌫-Z兲
0.23 共E 储 c兲a
0.21 共E ⬜ c兲a
0.24b
0.28 共E 储 c兲d
0.32 共E ⬜ c兲d
0.14 共E 储 c兲e
0.13 共E ⬜ c兲e
FIG. 8. 共Color online兲 共a兲 Charge density 共in e / Å3兲 and 共b兲 ELF
for In2O3-II in the 共011兲 plane.
aFP
LMTO, Ref. 51.
Ref. 56
c
LMTO ASA, Ref. 14.
dLCAO, Ref. 57.
eVASP-PAW, Ref. 49.
f
DMOL3, Ref. 58
gExperiment, Refs. 41 and 59.
bExperiment,
transition metal oxides 共see, e.g., Refs. 51–55兲. Experimental
data for CB electron effective masses are available for
In2O3-II only. The calculated effective masses from the
present study are compared with available experimental and
other theoretical values available for In2O3-II and ZnO 共other
well-known TCO兲 in Table IV. Overall, the presently calculated CB effective masses for In2O3-II are smaller than those
from experimental and other theoretical values. In particular,
the masses calculated by the LMTO ASA 共Ref. 14兲 method
agree fairly with experimental data and are more anisotropic
than our results.
Analysis shows that the calculated effective masses for
In2O3-II are considerably larger than those for In2O3-III.
Hence, carrier mobility in In2O3-III is expected to be larger
than that in In2O3-II. The calculated effective masses for
In2O3-II are much closer to experimental data than that for
ZnO. This difference in accuracy of the calculated masses
comes from Coulomb correlation effects, which are strong in
ZnO.49 In DFT calculations within LDA, the Zn 3d bands are
located inappropriately close to the topmost VB, falsifying
the band dispersion and, consequently, effective masses. As
noted in the analysis of the band dispersion 共Fig. 5兲, In 4d
band is located much below the topmost VB than the 3d
band in ZnO. So, Coulomb correlation effects in In2O3-II
and -III are not as strong as that in ZnO to falsify the band
masses significantly.
distribution shows that the highest charge density is residing
in the immediate vicinity of the atomic sites, reflecting the
dominant ionic-type bonding. If the bonding interaction between In and O is purely ionic, one could expect negligible
charge-density distribution between these ions. From Fig. 8,
it is clear that there are finite charges distributed in between
In and O in all the polytypes. Also, the charge around In and
O atoms is not distributed spherically. These features indicate that apart from dominant ionic bonding, finite covalentlike bonding interaction is present between In and O. The
observed hybridization interaction is responsible for the
mixed nature of the electronic structure of this material discussed in the previous section. This suggestion is also confirmed by the Born-effective-, Bader-, and Voronoi-charge
analyses in the following section.
The ELF analysis is one of the ways to measure the probability of distribution of paired electrons in solids.60–62 It is
seen from Fig. 8 that the maximum value of ELF is around O
and the minimum around In atoms, which confirms the
dominant ionic bonding. Between the In and O atoms, the
ELF does not show any maximum value, and this indicates
that there is no dominant covalent-type bonding interaction
present between these atoms. However, nonspherical distri-
E. Charge-density and electron localization function analysis
For qualitative characterization of the bonding interaction
between constituents, charge-density and electron localization function 共ELF兲 analysis have been performed for
In2O3-II 共Fig. 8兲 and -III 共Fig. 9兲. Analysis of the charge
FIG. 9. 共Color online兲 共a兲 Charge density 共in e / Å3兲 and 共b兲 ELF
for In2O3-III in the 共100兲 plane.
075129-9
PHYSICAL REVIEW B 76, 075129 共2007兲
KARAZHANOV et al.
TABLE V. Atomic charges 共in units of the electron charge e−兲
around In and O atoms calculated according to Voronoi and Bader
topological analyses for In2O3-II and -III.
Compound
Atom
Voronoi
Bader
In2O3-II
In1
In2
O
2.00
2.04
−1.36
1.81
1.84
−1.23
In2O3-III
In
O
2.09
−1.39
1.81
−1.20
bution of ELF at the O site and small but finite value of ELF
in between In and O reflect the presence of non-negligible
hybridization interaction between these atoms. Unlike intermetallic phases,60 no isolated spot in the interstitial regions
with high ELF is found in any of the In2O3 phases considered here.
F. Bader- and Voronoi-charge analyses
For quantitative characterization of chemical bonding in
solids, Bader- and Voronoi-charge analyses63–65 are widely
used. In the Bader-charge analysis, each atom of a compound
is surrounded by a surface 共called Bader regions兲 that runs
through the minima of the charge density and the total charge
of the atom is determined by integration within the Bader
region. In the Voronoi-charge analysis, each grid point of
charge density is assigned to the nearest atom, but each of
these distances is not scaled by the radius of each atom. So,
the Voronoi charges in the present paper are used just for a
sanity check. Table V presents the charges at each atomic site
for In2O3-II and -III calculated according to the Bader and
Voronoi schemes. As expected, In atoms donate electrons
while O atoms accept them. Based on the amount of the
transferred charges, one can conclude that chemical bonding
is of dominant ionic type.
G. Born effective charges
Born effective charges 共BECs兲 are the fundamental parameters characterizing the change in electron polarization
upon displacement of atoms. They are related to microscopic
electron currents produced in a system by a change of the
atomic position. These currents may contribute to the BEC,
which is unrelated and additional to the static electron charge
carried out by the atoms during the displacement, so that
BECs can be much larger than their static counterpart, i.e.,
higher than formal valence charge. In this case, BECs are
referred to as anomalous. BECs are helpful in understanding
the ferroelectric and piezoelectric properties of materials. For
the present study, BECs were obtained from finite differences of macroscopic polarization induced by small displacements of the atomic sublattices using the Berry phase
approach.66,67 The calculated BEC for In2O3-III is given in
Table VI. The formal valences for In and O are +3 and −2,
respectively. The calculated BECs reflect this formal valence
picture that the average diagonal components of the BEC at
the In and O sites are 3.27 and −2.17, respectively. These
values are slightly larger than the formal valence, indicating
that there is small anomalous charge 共also called dynamical
charge兲 added to the static charge in the atoms. The diagonal
components of the BEC are almost the same, indicating that
the bonding interaction is more isotropic reflecting ionic-type
bonding. It is seen that for In atoms, the off-diagonal elements of the BEC tensor are negligible, reflecting the ionictype interaction. However, there are non-negligible offdiagonal elements of BEC at the O site, indicating that there
is finite covalentlike bonding interaction present between O
atoms. This result is in agreement with charge density and
ELF analysis 共Fig. 9兲, which demonstrates that there is finite
directional bonding between nearest neighbor O atoms.
H. Optical properties
Optical properties of In2O3-I, -II and -III are studied. The
real and imaginary parts of the optical dielectric response
function, absorption coefficient, reflectivity, refractive index,
and the extinction coefficients have been analyzed. Since the
crystal structure of In2O3-I and -II is cubic, the optical spectra are isotropic along the crystallographic a, b, and c axes.
So, only one of the components is sufficient for the analysis.
We also found that the optical spectra of In2O3-I with optimized lattice and In2O3-II are almost the same.
Figure 10 presents the calculated optical spectra for
In2O3-II and -III. All the calculated optical spectra of
In2O3-II and -III are shifted toward lower energies compared
with experimental results owing to the underestimation of
the band gap in the DFT calculations. One of the ways to
rectify it is to use the scissors operator technique, i.e., rigid
shift of all the CB states, so that the optical spectra shall also
be shifted accordingly.55,68 The search of literature shows
that experimental data are available for In2O3-II in Ref. 41,
where reflectivity and transmittance spectra were measured
by spectrophotometry at room temperature. The other optical
spectra such as absorption coefficient, refractive index, and
the extinction coefficient are calculated using the KramersKronig relation. These five optical spectra have been compared in Fig. 10共a兲 with those calculated from the present
band-structure results. The optical spectra of In2O3-II were
measured41 for a narrow energy range 0 – 6 eV, which contains only the E0 peak induced by electronic transitions from
TABLE VI. Calculated Born-effective-charge-tensor elements 共Z*兲 for the constituents of In2O3-III.
Atom
In1
O1
xx
yy
zz
xy
yz
zx
xz
zy
yx
3.300
−2.260
3.306
−2.136
3.206
−2.130
0.009
0.098
0.002
0.432
0.000
0.186
0.001
0.257
0.000
0.308
0.002
0.432
075129-10
PHYSICAL REVIEW B 76, 075129 共2007兲
PHASE STABILITY, ELECTRONIC STRUCTURE, AND…
ε2(ω)
ε1(ω)
6
In2O3-II
4
4
2
2
6
6
4
4
2
2
R(ω)
In2O3-III
E || c
E⊥c
d1 d
2
E0
0.2
d2
0.2
0.1
E1
0.1
E0 E1 E2 d1
E2
2
3
α(ω)
6
2
Expt.
1
k(ω)
n(ω)
1
2
2
Expt.
1
1
1.2
1.2
0.8
0.8
0.4
0.4
5
10
15
5
Energy (eV)
10
15
FIG. 10. Optical spectra of In2O3-II and In2O3-III for directions
of the electric field 共E兲 parallel 共E 储 c兲 and perpendicular 共E ⬜ c兲 to
the crystallographic c axes. Calculated imaginary parts of the optical dielectric function 关␧2共␻兲兴, absorption coefficient 关␣共␻兲 in cm−1
multiplied by 10−5兴, reflectivity 关R共␻兲兴, refractive index 关n共␻兲兴, and
extinction coefficients 关k共␻兲兴 for In2O3-II have been compared with
experimental data of Ref. 41. Calculated absorption coefficient for
In2O3-III has been compared with experimental results of Ref. 11.
the VB maximum to CB minimum occurring at the ⌫ point.
So, in the present work, these experimental data were used
for rigid shift of the calculated optical spectra toward higher
energies by 1.65 eV to fit to location of the E0 peak. Since
the location of the E0 peak is clearly seen in the spectra of
n共␻兲, fitting was performed for n共␻兲. The other spectra have
been shifted accordingly.
From the comparison of the calculated spectra with experimentally measured spectra, one can see that the calculated n共␻兲 and k共␻兲 agree well with experimental data. Calculated reflectivity spectra agree with experimental data only
at higher energies well above the band gap. At lower energies, near the band gap, experimental reflectivity is higher
than that calculated by DFT, and the experimentally observed low energy peak may be associated with excitonic
effect, which is not considered in the present calculations.
The calculated absorption spectra agree with the measured
spectra at lower energies. However, at energies exceeding the
location of the E0 peak, the calculated absorption coefficient
is found to be much smaller than the experimentally determined values. The origin for the deviation between experimental and theoretical absorption coefficient spectra is not
known.
Because of the rhombohedral symmetry, there is optical
anisotropy present in In2O3-III. The two components of the
dielectric functions are calculated for In2O3-III corresponding to the electric field parallel E 储 c and perpendicular E ⬜ c
to the crystallographic c axes. The first experimental data for
In2O3-III are already available.11 The absorption coefficient
was measured for the energy range 0 – 6 eV, which contains
the E0 peak. The calculated absorption spectra were shifted
to 1.35 eV toward higher energies to fit the E0 peak in order
to match with experimental spectra. The other calculated optical spectra have also been shifted accordingly. The anisotropy is well pronounced at energies from ⬃7 to 9 eV,
whereas the optical spectra are more or less isotropic at other
energies considered. The locations of all the major peaks of
the optical spectra corresponding to both the directions are
almost the same except the E2 peak. The magnitude of the E2
peak at around 7.5 eV is higher for E 储 c and its shoulder at
8.3 eV is larger than that for E ⬜ c. Absorption and reflectivity spectra for this compound are low in the energy range
0 – 5 eV, indicating that it will be optically transparent.
The intensity of the calculated optical spectra differs from
the experimentally measured one for both In2O3-II and -III.
The reason of this deviation may be related to the factors
such as overestimation of the momentum matrix elements,
neglect of the Coulomb interaction between free electrons
and holes 共excitons兲, and local-field and finite lifetime effects. Furthermore, in the calculations of the imaginary part
of the dielectric response function 关␧2共␻兲兴, only the direct
optical transitions from occupied to unoccupied states are
considered. Also, the lifetime effects and the experimental
resolution smear out many fine features. Moreover, the experimental measurements11,41 are made on thin films at room
temperatures where one could expect considerable amount of
defects which will cause changes in optical properties. However, the theoretical optical spectra are valid only for defect
free single crystals at low temperatures.
IV. CONCLUSION
Structural properties, electronic structure, high-pressure
behavior, and optical properties of In2O3 of space group
symmetries I213, Ia3̄, and R3̄ are studied by DFT calculations. From the structural optimization using total energy cal-
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PHYSICAL REVIEW B 76, 075129 共2007兲
KARAZHANOV et al.
culations, the equilibrium lattice and positional parameters of
all the three polytypes of In2O3 have been found, which are
in good agreement with experimental data except for the
I213 phase where the positional parameters deviate from experiment. In2O3 in space group of I213 is found to be a
distorted one of space group Ia3̄. Upon optimization of the
lattice, the former is transformed into the latter structure. The
present calculations predicted pressure-induced phase transition from In2O3-II to -III. The calculated transition pressure
is compared with available experimental values. Based on
the analysis of band structure calculated within LDA and
GGA, it is found that In2O3-III has indirect band gap,
whereas In2O3-II has direct band gap. Intermediate band is
found in In2O3-I located right above the valence band, and it
is isolated from the valence and conduction bands. From the
orbital decomposition analysis and orbital as well as siteprojected density of states, the characters of the valence and
conduction bands have been analyzed. It is shown that the
topmost valence band consists of O 2p states hybridized with
In 4d states. Further, we found that the conduction band
minimum of In2O3-II consists of In 5s orbitals hybridized
with O 2s orbitals, whereas that for In2O3-III is mainly In 5s
states. From orbital and site decomposed band analysis, it is
shown that the topmost valence bands of In2O3-II and -III are
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ACKNOWLEDGMENTS
This work has been supported by FUNMAT under Project
No. 101043. Supercomputing support was provided through
the Research Council of Norway. The authors gratefully acknowledge Ch. Y. Wang 共Institute of Micro- and Nanotechnologies, Technical University Ilmenau, Ilmenau, Germany兲
for fruitful discussions and sending his experimental optical
data used for comparison with the present theoretical results.
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