Averaging of the electron effective mass in multicomponent transparent conducting
oxides
Julia E. Medvedeva∗
Department of Physics, University of Missouri–Rolla, Rolla, MO 65409 (USA)
We find that layered materials composed of various oxides of cations with s2 electronic config-
uration, XY2O4, X=In or Sc, Y=Ga, Zn, Al, Cd and/or Mg, exhibit isotropic electron effective
mass which can be obtained via averaging over those of the corresponding single-cation oxide con-
stituents. This effect is due to a hybrid nature of the conduction band formed from the s-states of all
cations and the oxygen p-states. Moreover, the observed insensitivity of the electron effective mass
to the oxygen coordination and to the distortions in the cation-oxygen chains suggests that similar
behavior can be expected in technologically important amorphous state. These findings significantly
broaden the range of materials as efficient transparent conductor hosts.
Transparent conducting oxides (TCO) – the vital part
of optoelectronic devices – have been known for a century
and employed technologically for decades [1, 2, 3, 4, 5].
Yet, the current TCO market is dominated by only three
materials, In2O3, SnO2 and ZnO, and the research efforts
are primarily focused on the oxides of post-transition
metals with (n − 1)d10ns2 electronic configuration. De-
spite excellent optical and thermal properties as well as
low cost, oxides of the main group metals, such as Al2O3,
SiO2, MgO and CaO, have never been considered as can-
didates to achieve useful electrical conductivity due to
the challenges of efficient carrier generation in these wide-
bandgap materials [6, 7, 8].
Multicomponent TCO with layered structure, e.g.,
InGaZnO4 [9, 10, 11, 12, 13, 14], drew attention due to a
possibility to separate carrier donors (traditionally, oxy-
gen vacancies or aliovalent substitutional dopants) and
the conducting layers where carriers are transfered with-
out charge scattering on the impurities. In InGaZnO4,
octahedrally coordinated In layers alternate with double
layers of oxygen tetrahedrons around Ga and Zn, Fig. 1.
Because octahedral oxygen coordination of cations was
long believed to be essential for a good transparent con-
ductor [9, 15, 16, 17, 18], it has been suggested that in
InGaZnO4 the charge is transfered within the InO1.5 lay-
ers while the atoms in GaZnO2.5 layers were proposed as
candidates for efficient doping [9, 10, 11]. Conversely, it
has been argued that InGaO3(ZnO)m is a Zn 4s conduc-
tor [12].
To understand the role of local symmetry in the intrin-
sic transport properties of TCO’s and to determine the
functionality of structurally and chemically distinct lay-
ers in InGaZnO4, we employ ab-initio density functional
approach to study the electronic properties of various sin-
gle and multi-cation oxides. Further, using InGaZnO4 as
a test model which examplifies not only the structural
but also combinatorial peculiarities of complex TCO’s,
we survey other ns2 cations – beyond the traditional In,
Zn and Ga – for a possibility of being effectively incor-
porated into novel multicomponent TCO hosts.
Isotropy of the electronic properties in InGaZnO4.
FIG. 1: The unit cell of InGaZnO4 where three similar blocks
consisting of one InO1.5 and two GaZnO2.5 alternate along the
[0001] direction. Ga and Zn atoms are distributed randomly.
Three-dimensional interatomic (background) charge density
distribution is evident from the contour plot calculated in the
(011) plane. The plotted charge density corresponds to the
carrier concentration of ∼1×1018cm−3.
The electronic band structure calculations for InGaZnO4
show that the atoms from both InO1.5 and GaZnO2.5
layers give comparable contributions to the conduction
band, fig. 2, leading to a three-dimensional distribution
of the charge density, fig. 1. Moreover, the isotropy of
the electronic properties in this layered material mani-
fests itself in the electron effective masses being nearly
the same in all crystallographic directions (table I).
The conduction band in InGaZnO4 consists of a set
of highly dispersed parabolic bands, fig. 3(a). Since the
band gap values in the corresponding single metal oxides
are different, one may expect that each band is attributed
to a certain cation in this multicomponent compound.
However, we find that each band cannot be assigned to
a state of a particular atom since all atoms in the cell,
including the oxygen atoms, give non-negligible contri-
butions to the conduction band wavefunction, fig. 2 and
http://arxiv.org/abs/0704.1499v2
1.2 1.6 2 2.4 2.8
Energy, eV
0.005
0.01  
     
    0
0.005
     
    0
0.005
InGaZnO4
In s−states
Ga s−states
Zn s−states
O1 s, p−states
O2 s, p−states
0.4 0.8 1.2 1.6 2
Energy, eV
     
    0
0.005
0.01 
     
0.005
0.01 
     
InAlMgO4
In s−states
Al s−states
Mg s−states
O1 s, p−states
O2 s, p−states
FIG. 2: Partial density of states at the bottom of the conduc-
tion band for InGaZnO4 and InAlMgO4. Atoms from both
In-O1 and Ga-Zn-O2 (or Al-Mg-O2) layers give non-negligible
contributions.
table I.
The conduction band dispersion calculated along the
[0001] crystallographic direction for a single unit cell,
fig. 3(b), reveals that the multiple bands can be at-
tributed to a “folding” of one parent band. Triple unfold-
ing of the conduction band corresponds to the three-time
reduction (expansion) of the conventional unit cell (Bril-
louin zone) in the z direction. Since the block of three
layers (one InO1.5 and two GaZnO2.5 layers, fig. 1) is
repeated in the unit cell via translation, the splitting be-
tween the resulting bands [for the k-vector equal to π
, ..., fig. 3(c)] is negligible. Although the subsequent
unfolding into three individual layers is not justified be-
cause the three layers are structurally and chemically dis-
similar, fig. 1, we find that the band can be unfolded
again, fig. 3(c). The resulting highly dispersed band is in
accord with the stepless increase of the density of states,
fig. 3(d). Thus, the conduction band can be “unfolded”
nine times that corresponds to the total number of layers
in the unit cell. Therefore, the electronic properties of
the individual layers are similar.
Unconventional s2-cations at work. A two-dimensional
electronic structure could be expected in InAlMgO4 and
ScGaZnO4 since the band gap values in Sc2O3, Al2O3
and MgO are at least twice larger than those in In2O3,
Ga2O3, CdO and ZnO and, hence, the unoccupied s-
states of Sc, Al and Mg should be located deeper in the
conduction band. From the analysis of the partial den-
sity of states for InGaMgO4, InAlCdO4, ScGaZnO4 and
InAlMgO4, we find that although the contributions to
the bottom of the conduction band from Sc, Al and Mg
atoms are notably reduced, these states are available for
electron transport, fig. 2. Thus, similar to InGaZnO4
where the cations s-states are energetically compatible,
in all multicomponent oxides considered, the conduction
band wavefunction is a combination of the s-states of all
cations and the p-states of the oxygen atoms. The con-
tributions from the chemically distinct layers are compa-
rable, table I, and, consequently, these complex oxides
exhibit three-dimensional network for the electron trans-
port and isotropic electron effective mass, table I.
Comparison to single-cation TCO’s. The unfolded
conduction band in the layered multicomponent mate-
rials resembles those of single-cation TCO’s, e.g., In2O3,
cf. Figs. 3 (c) and (e). Such a highly-dispersed single
conduction band is the key attribute of any conventional
[19] n-type TCO host [9, 17, 20, 21, 22, 23]. Upon proper
doping, it provides both high mobility of extra carriers
(electrons) due to their small effective mass, and low
optical absorption due to high-energy inter-band tran-
sitions from the valence band, Ev, and from the par-
tially filled conduction band, Ec, fig. 4. Even in rela-
tively small bandgap oxides, e.g., CdO where the optical
band gap, Eg, is 2.3 eV, the high energy dispersion en-
sures a pronounced Fermi energy displacement with dop-
ing (so called Burstein-Moss shift) which helps to keep
the intense transitions from the valence band out of the
visible range. However, large carrier concentrations re-
quired for good electrical conductivity may result in an
increase of the optical absorption due to low-energy tran-
sitions from the Fermi level up into the conduction band
as well as plasma frequency. Application-specific optical
properties and desired band offsets (work functions) can
be attained in a multicomponent transparent conductor
with a proper composition.
In both single and multi-cation oxides, the conduc-
tion band is formed from the empty s-states of the metal
atoms and the oxygen antibonding p-states, e.g., fig. 2.
For multicomponent oxides we find that even at the bot-
tom of the conduction band, i.e., at Γ point, the contri-
butions from the oxygen p-states are significant, table I.
Thus, the key feature of the conduction band in a conven-
tional TCO – its high energy dispersion – originates in
a strong interaction between the cation s-states and the
anion antibonding p-states [17]. The direct s-s overlap
is insignificant, fig. 1, and therefore, the s-s interactions
which were commonly assumed to play to key role in the
electronic properties of TCO [12, 14, 16], do not gov-
ern the transport properties in these oxides. Indeed, the
small electron effective mass in s2-cation oxides is deter-
mined by the strong s-p interactions [26].
From orbital symmetry considerations, one can see
that the oxygen coordination of cations does not affect
the s-p overlap. Instead, the largest overlap should be at-
tained in materials where the oxygen atom is coordinated
octahedrally by cations with the extended s-orbitals [17].
Our systematic comparison of the calculated electron ef-
fective mass in the oxides of metals with s2 electronic con-
figuration, fig. 4, shows that the mass slightly decreases
as the ionic radii of the cations from the same group or
the symmetry of the same-cation oxide increases. How-
 [010]   Γ  [100] 
 (a) 
 Γ  [001] 
 (b) 
 9k        6k        3k        0 
 (c) 
0 5 10
 H  Γ  P 
 (e)  
−2k −k 0 k 2k
β1 β ββ 1 2 22
ε ε ε01 2 n
FIG. 3: Electronic band structure of single and multi-cation oxides. (a) Conduction band dispersion calculated in the ab-plane
and (b) along the [0001] direction in InGaZnO4. (c) The conduction band unfolded nine times that corresponds to the number
of [0001] layers in the unit cell, k=π
, c is the lattice parameter. The resulting single free-electron-like band is in accord with
the stepless total density of states (d), in states/eV. (e) Conduction band of In2O3 is given for comparison. (f) Tight-binding
conduction band (solid line) calculated for one-dimensional atomic chain depicted above the plot. Two types of metal atoms
(red and purple spheres) alternate with oxygen atoms (blue spheres) and only the nearest-neighbor hopping β is assumed. To
illustrate the effective mass averaging, cf. eq. (1), the conduction bands for the corresponding single-metal oxide chains (dashed
lines) are aligned with (ε1 + ε2)/2.
FIG. 4: The key electronic features of n-type TCO hosts. High energy dispersion of the conduction band (left) is essential for
both high mobility of extra carriers due to their small effective mass and low optical absorption. For complete transparency in
the visible range, the inter-band transitions Ev and Ec should be >3.1 eV, while the intra-band transitions as well as plasma
frequency should be <1.8 eV. Such a dispersed conduction band along with the above-visible optical transitions at the Γ point
are found in a variety of oxides with s2-cation(s). The electron effective mass calculated for different oxide phases shows little
dependence on the oxygen coordination and is isotropic (δ=(ma +mb)/2mc − 1). The highlighted areas are to guide the eye;
red circles represent currently known TCO hosts.
TABLE I: Net contributions from the states of the atoms that belong to the X-O1 or Y2-O2 layers (X=In or Sc, Y=Ga, Zn,
Al, Cd and/or Mg) to the conduction band wavefunction at Γ point, in per cent; and the electron effective masses m, in me,
calculated from the band structure of the layered oxides and the components of the electron effective-mass tensor, ma,b and
mz, calculated from eqs. (1) and (2) using the effective masses of the corresponding single-cation oxides.
XY2O4 NX NO1 NY2 NO2 m[100] m[010] m[001] mab mz
InGaZnO4 23 25 29 23 0.23 0.22 0.20 0.23 0.23
InAlCdO4 27 27 18 28 0.26 0.25 0.20 0.27 0.27
InGaMgO4 27 31 21 21 0.27 0.27 0.24 0.28 0.29
InAlMgO4 33 40 12 15 0.32 0.31 0.35 0.31 0.34
ScGaZnO4 8 19 40 33 0.33 0.33 0.34 0.33 0.53
ever, variations in the oxygen coordination as well as
strong distortions in the metal-oxygen chains in different
oxide phases lead to insignificant changes in the effec-
tive mass. For example, for cubic (octahedral coordina-
tion) and hexagonal (tetrahedral) ZnO or for corundum
(distorted tetrahedral) and monoclinic β-phase (both dis-
torted tetrahedral and trigonal) Ga2O3 the correspond-
ing electron effective masses vary by 15% or 9%, respec-
tively. The largest deviation in the effective mass values
for various SiO2 phases is 26%. Furthermore, the effec-
tive mass remains isotropic for all phases of the s2-cation
oxides – including silica ITQ-4 zeolite with large pore
channels [24]. These observations explain the success of
amorphous transparent conducting oxides – in marked
contrast to the amorphous Si where the directional inter-
actions between the conduction p-orbitals lead to strong
anisotropy of the transport properties which are sensitive
to the orbital overlap and hence to the distortions in the
atomic chains [14]. Finally, we note that the fact that the
calculated as well as the observed [25] isotropic effective
mass in rutile SnO2 where the direct overlap between Sn
s-orbitals is possible (only) along the [001] direction, cor-
roborate our conclusion that the s-s interactions do not
govern the transport properties as discussed above.
Effective mass averaging. Because the local symme-
try (nearest neighbors), the cation-oxygen bond lengths
and, hence, the s-p overlap are similar in the single and
multi-cation oxides, the intrinsic transport properties in
the layered materials should be related to those in the
single-cation oxides. Moreover, due to the hybrid nature
of the conduction states in the multicomponent oxides,
the states of all cations should give the same order of
magnitude contributions to the effective mass. Thus, we
expect the later to be an “effective” average over the ef-
fective masses of the corresponding single-cation oxides.
Indeed, we find that in case of one-dimensional atomic
chain where two types of metal atoms alternate with oxy-
gen atoms, the effective mass averaging can be derived
analytically, fig. 3(f).
We formulate a simple approach which allows one to
estimate the effective mass of the multicomponent ox-
ides as follows. With proper doping, the Fermi energy
is shifted up into the conduction band (Burstein-Moss
shift). When the extra electrons propagate along the z
direction, i.e., across the layers, the resulting resistivity
is a sum of the resistivities of each layer. Therefore, the
z component of the average effective-mass tensor can be
found as:
mz = (m1 +m2 +m3)/3, (1)
wherem1,2,3 are the effective masses of the corresponding
single metal oxides – In2O3, Ga2O3 and ZnO in the case
of InGaZnO4.
For the in-plane charge transport in the layered mate-
rials, the effective-mass tensor components can be found
in a parallel fashion. Note, that one needs to average the
effective mass for the mixed GaZnO2.5 layers:
(m2 +m3)
. (2)
The resulting ma,b and mz are presented in table I. We
find that the increase of the electron effective masses
in the order InGaZnO4 < InAlCdO4 < InGaMgO4 <
InAlMgO4 < ScGaZnO4 is well reproduced by the above
averaging. Moreover, thema,b andmz values nearly coin-
cide with the corresponding effective masses of the multi-
cation oxides with the exception of the Sc case. In fact,
for the Sc-containing compounds the effective mass aver-
aging is not legitimate due to the presence of the empty
d-states of Sc near the conduction band edge. In Sc2O3,
the Sc d-states are located at 0.5 eV above the conduction
band edge, and therefore give significant contributions to
the effective mass which is about the mass of an electron.
In the multi-cation oxide, the Sc d-states are found to be
at ∼2 eV above the conduction band edge, so that the
resulting small effective mass, table I, is determined pri-
marily by the s − p interactions – similar to the rest of
the s2-cation oxides.
The effective mass averaging procedure, eqs. (1) and
(2), can be generalized for materials consisting of any
number of layers, e.g., for InGaO3(ZnO)m, m=integer.
Furthermore, since the intrinsic transport properties are
determined entirely by the local symmetry, i.e., the near-
est neighbors, the effective mass averaging should apply
to TCO’s in amorphous state. In this case, one needs
to average the components of the effective-mass tensor,
mamorph=(ma+mb+mz)/3.
Importance of carrier generation. Upon doping of a
TCO host material, the resulting conductivity depends
not only on the effective mass but also on the carrier
generation mechanism, carrier concentration and carrier
relaxation time. Doping of a structurally anisotropic
material may lead to non-uniform distribution of the
carrier donors and, therefore, the isotropic behavior
of the host may not be maintained as, for example,
in oxygen-deficient β-Ga2O3 [27]. In multicomponent
InGaO3(ZnO)m, different valence states (In
3+ and Ga3+
vs Zn2+) and oxygen coordination (octahedral for In vs
tetrahedral for Ga and Zn) are likely to result in pref-
erential arrangement of aliovalent substitutional dopants
or oxygen vacancies. Consequently, an anisotropic mo-
bility should be expected in the layered materials due
to the spatial separation of the carrier donors and the
layers where the extra carriers are transfered efficiently,
i.e., without charge scattering on the impurities. While
targeted doping can help make either or both struc-
turally distinct layers conducting, the amorphous com-
plex oxides readily offer a way to maintain isotropic
transport properties. Indeed, experimental observa-
tions that the mobility and conductivity are indepen-
dent of the large variations in the composition in amor-
phous InGaO3(ZnO)n with n≤4 [12] and that the effec-
tive masses of amorphous and crystalline InGaZnO4 are
nearly the same [28] support our conclusions.
For efficient doping of wide-bandgap oxides such as
MgO, CaO, SiO2 and Al2O3, novel carrier generation
mechanisms should be sought. A non-traditional ap-
proach has already yielded promising results in calcium
aluminates [29, 30, 31] – a conceptually new class of
transparent conductors [19]. Multicomponent oxides –
such as those considered in this work, ordered ternary
oxides [32] as well as their solid solutions and amorphous
counterparts – represent an alternative way to utilize the
abundant main-group elements such as Ca, Mg, Si and
Al towards novel TCO hosts with a predictable effective
mass and optical and transport properties controllable
via the composition.
Acknowledgement. The work is supported by Uni-
versity of Missouri Research Board.
∗ Electronic address: juliaem@umr.edu
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APPENDIX
Theoretical methods First-principles full-potential linearized
augmented plane wave method [33, 34] with the local density ap-
proximation is employed for electronic band structure investiga-
tions of the XY2O4 compounds, X = In or Sc and Y = Ga, Zn,
Al, Cd and/or Mg, and single-cation oxides. Cut-offs of the plane-
wave basis (16.0 Ry) and potential representation (81.0 Ry), and
expansion in terms of spherical harmonics with ℓ ≤ 8 inside the
muffin-tin spheres were used. Summations over the Brillouin zone
were carried out using 14 special k points in the irreducible wedge.
Structure optimization. XY2O4 compounds have rhombohedral
R3̄m crystal structure of YbFe2O4 type [35, 36]. Indium (or scan-
dium) atoms substitute Yb in 3(a) position, while both Y 3+ and
Y 2+ atoms replace Fe in 6(c) position and are distributed ran-
domly [37]. Our total energy calculations for several structures in
the (a,2a,c) supercell with various arrangements of the Y 2+ and
Y 3+ atoms suggest that their arrangement is not ordered but ran-
dom – in agreement with the experiment. We note here that the
electronic band structure features are similar for different spatial
distributions of the Y atoms for the reasons discussed in the paper.
Since the valence state and ionic radii of Y 2+ and Y 3+ are dif-
ferent, the site positions of these atoms as well as their oxygen sur-
rounding should be different. Because the exact internal positions
of atoms are unknown, we used those of the YbFe2O4 [35] as the
starting values and then optimized each structure via the total en-
ergy and atomic forces minimization. During the optimization, the
lattice parameters were fixed at the experimental values [36, 38].
We find that the optimized cation-anion distances correlate with
ionic radii of the cations.
Single-cation oxides. For the single-cation oxides, the following
phases have been calculated: Fm3̄m for MgO, CaO, SrO, BaO
and YbO; Ia3̄ for Sc2O3 and Y2O3; Fm3̄m and P63mc for ZnO;
Fm3̄m for CdO; R3̄c for Al2O3; R3̄c and C2/m for Ga2O3; Ia3̄,
R3̄c and I213 for In2O3; P3221, P21/c, P63/mmc, P41212 and
I2/m for SiO2; P41212 and P42/mnm for GeO2; P42/mnm and
Pbcn for SnO2. For each structure, the internal positions of all
atoms have been optimized via the total energy and atomic forces
minimization, while the lattice parameters were fixed at the exper-
imental values.
One-dimensional model of complex oxides. The effective mass
averaging, cf. eq. (1), can be shown analytically using a one-
dimensional model in the tight-binding approximation. To capture
the key features of complex oxides, we consider a chain consisting
of two types of metal atoms which alternate with oxygen atoms,
fig. 3(f), and assume only the nearest-neighbor interactions given
by the hopping integrals β1 and β2. The Hamiltonian of this model
system is:
|n, l〉εl〈n, l|+
n,n′,l,l′
|n′, l′〉βl〈n, l|. (3)
Here, l is the atom index in the unit cell, n enumerates the cells
and n′, l′ in the second sum run over the nearest neighbors. For
the bottom of the conduction band, the dispersion relation can be
simplified to
ε(k) =
ε1 + ε2
if |ε1 − ε2| < 2
. Here ε0, ε1 and ε2 are the atomic level
energies of the oxygen and two types of metal atoms, respectively,
and it is assumed that ε0 < ε1,2 and ε1 ∼ ε2; ∆ =
(ε1 + ε2)− ε0
and a is a half of the lattice parameter. Similar considerations
for the chain consisting of only one type of metal atoms alternat-
ing with oxygen atoms show that the quantity ∆
represents the
effective mass of the system. Therefore, eq. (4) represents the ef-
fective mass averaging over those of the corresponding single-metal
“oxide” chains, cf., fig. 3(f) – in agreement with the results of our
first-principles calculations. The following parameters were used to
plot fig. 3(f): ε0 = 1.00, ε1 = 2.00, ε2 = 2.05, β1 = 0.4 and β2 =

  We find that layered materials composed of various oxides of cations with
$s^2$ electronic configuration, $XY_2$O$_4$, $X$=In or Sc, $Y$=Ga, Zn, Al, Cd
and/or Mg, exhibit isotropic electron effective mass which can be obtained via
averaging over those of the corresponding single-cation oxide constituents.
This effect is due to a hybrid nature of the conduction band formed from the
s-states of {\it all} cations and the oxygen p-states. Moreover, the observed
insensitivity of the electron effective mass to the oxygen coordination and to
the distortions in the cation-oxygen chains suggests that similar behavior can
be expected in technologically important amorphous state. These findings
significantly broaden the range of materials as efficient transparent conductor
hosts.

Introduction to Solid State Physics, p. 253
(John Wiley and Sons, Inc., 2005).
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APPENDIX
Theoretical methods First-principles full-potential linearized
augmented plane wave method [33, 34] with the local density ap-
proximation is employed for electronic band structure investiga-
tions of the XY2O4 compounds, X = In or Sc and Y = Ga, Zn,
Al, Cd and/or Mg, and single-cation oxides. Cut-offs of the plane-
wave basis (16.0 Ry) and potential representation (81.0 Ry), and
expansion in terms of spherical harmonics with ℓ ≤ 8 inside the
muffin-tin spheres were used. Summations over the Brillouin zone
were carried out using 14 special k points in the irreducible wedge.
Structure optimization. XY2O4 compounds have rhombohedral
R3̄m crystal structure of YbFe2O4 type [35, 36]. Indium (or scan-
dium) atoms substitute Yb in 3(a) position, while both Y 3+ and
Y 2+ atoms replace Fe in 6(c) position and are distributed ran-
domly [37]. Our total energy calculations for several structures in
the (a,2a,c) supercell with various arrangements of the Y 2+ and
Y 3+ atoms suggest that their arrangement is not ordered but ran-
dom – in agreement with the experiment. We note here that the
electronic band structure features are similar for different spatial
distributions of the Y atoms for the reasons discussed in the paper.
Since the valence state and ionic radii of Y 2+ and Y 3+ are dif-
ferent, the site positions of these atoms as well as their oxygen sur-
rounding should be different. Because the exact internal positions
of atoms are unknown, we used those of the YbFe2O4 [35] as the
starting values and then optimized each structure via the total en-
ergy and atomic forces minimization. During the optimization, the
lattice parameters were fixed at the experimental values [36, 38].
We find that the optimized cation-anion distances correlate with
ionic radii of the cations.
Single-cation oxides. For the single-cation oxides, the following
phases have been calculated: Fm3̄m for MgO, CaO, SrO, BaO
and YbO; Ia3̄ for Sc2O3 and Y2O3; Fm3̄m and P63mc for ZnO;
Fm3̄m for CdO; R3̄c for Al2O3; R3̄c and C2/m for Ga2O3; Ia3̄,
R3̄c and I213 for In2O3; P3221, P21/c, P63/mmc, P41212 and
I2/m for SiO2; P41212 and P42/mnm for GeO2; P42/mnm and
Pbcn for SnO2. For each structure, the internal positions of all
atoms have been optimized via the total energy and atomic forces
minimization, while the lattice parameters were fixed at the exper-
imental values.
One-dimensional model of complex oxides. The effective mass
averaging, cf. eq. (1), can be shown analytically using a one-
dimensional model in the tight-binding approximation. To capture
the key features of complex oxides, we consider a chain consisting
of two types of metal atoms which alternate with oxygen atoms,
fig. 3(f), and assume only the nearest-neighbor interactions given
by the hopping integrals β1 and β2. The Hamiltonian of this model
system is:
|n, l〉εl〈n, l|+
n,n′,l,l′
|n′, l′〉βl〈n, l|. (3)
Here, l is the atom index in the unit cell, n enumerates the cells
and n′, l′ in the second sum run over the nearest neighbors. For
the bottom of the conduction band, the dispersion relation can be
simplified to
ε(k) =
ε1 + ε2
if |ε1 − ε2| < 2
. Here ε0, ε1 and ε2 are the atomic level
energies of the oxygen and two types of metal atoms, respectively,
and it is assumed that ε0 < ε1,2 and ε1 ∼ ε2; ∆ =
(ε1 + ε2)− ε0
and a is a half of the lattice parameter. Similar considerations
for the chain consisting of only one type of metal atoms alternat-
ing with oxygen atoms show that the quantity ∆
represents the
effective mass of the system. Therefore, eq. (4) represents the ef-
fective mass averaging over those of the corresponding single-metal
“oxide” chains, cf., fig. 3(f) – in agreement with the results of our
first-principles calculations. The following parameters were used to
plot fig. 3(f): ε0 = 1.00, ε1 = 2.00, ε2 = 2.05, β1 = 0.4 and β2 =
