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DOI: 10.1007/s11664-017-5845-6
2017 The Minerals, Metals & Materials Society
Calculation of Electronic and Optical Properties of AgGaO2
Polymorphs Using Many-Body Approaches
1.—Department of Physics, Lorestan University, 3.—e-mail:
Ab initio calculations based on many-body perturbation theory have been used
to study the electronic and optical properties of AgGaO2 in rhombohedral,
hexagonal, and orthorhombic phases. GW calculations showed that AgGaO2 is
an indirect-bandgap semiconductor in all three phases with energy bandgap of
2.35 eV, 2.23 eV, and 2.07 eV, in good agreement with available experimental
values. By solving the Bethe–Salpeter equation (BSE) using the full potential
linearized augmented plane wave basis, optical properties of the AgGaO2
polymorphs were calculated and compared with those obtained using the GWcorrected random phase approximation (RPA) and with existing experimental
data. Strong anisotropy in the optical absorption spectra was observed, and
the excitonic structures which were absent in the RPA calculations were
reproduced in GWBSE calculations, in good agreement with the optical
absorption spectrum of the rhombohedral phase. While modifying peak positions and intensities of the absorption spectra, the GWBSE gave rise to the
redistribution of oscillator strengths. In comparison with the z-polarized response, excitonic effects in the x-polarized response were dominant. In the
x- (and y-) polarized responses of r- and h-AgGaO2, spectral features and
excitonic effects occur at the lower energies, but in the case of o-AgGaO2, the
spectral structures of the z-polarized response occur at lower energies. In
addition, the low-energy loss functions of AgGaO2 were calculated and compared using the GWBSE approach. Spectral features in the energy loss
function components near the bandgap region were attributed to corresponding excitonic structures in the imaginary part of the dielectric function.
Key words: AgGaO2 polymorphs, electronic band structure, optical
properties, Bethe–Salpeter equation (BSE), excitonic effects
Chalcopyrites are semiconducting ternary I–III–
VI2 compounds with a variety of applications,
including in optoelectronic and photovoltaic
devices,1–3 and nonlinear optics.4,5 Second-harmonic generation and infrared (IR) radiation have
been observed in these materials.6,7 They have
potential for use in energy storage,8 photocatalysis,9
and electron–hole pair transfer.10 This wide range
of electronic and optoelectronic applications has
(Received November 29, 2016; accepted September 27, 2017)
made them the subject of many experimental measurements and ab initio calculations. The optical
absorption spectrum and band structure of CuGaS2,
AgGaS2, and CuInS2 have been studied.11,12 Mn
doping has been investigated for several members of
this group.13 Electronic and optical properties of
CuXS2 (X = Al, Ga, In) and AgGaS2 have been
calculated by density functional theory (DFT) using
the generalized gradient approximation (GGA).14
Whereas copper-based delafossites have been
widely studied, silver-based ternary compounds
AgXM2 (X = Al, Ga, In; M = S, Se, Te) have been
the subject of few electronic and optical investigations; For example, the mechanism of linear and
Dadsetani and Nejatipour
nonlinear optical effects of AgGaM2 (M = S, Se, Te)
was studied by Bai et al.15 Unlike In2O3 and Ga2O3,
which are established n-type transparent conducting oxides (TCOs), chalcopyrites are p-type TCOs.
Moreover, a subclass of these materials, viz.
ternary I–III–O2 compounds (I = Cu, Ag, Au, Zn,
Cd; III = Al, Ga, and In), have many applications in
optoelectronic devices.16–18 Like ZnO, SnO2, and
In2O3, they belong to the family of TCOs, as they are
used in photocatalytic applications, touchscreens,
solar cells, and displays.19–21 They can exhibit both
n-type and p-type conductivity, promising a wide
range of optoelectrical applications. Based on the
first-principles DFT calculations, electronic and
optical properties of silver delafossite oxides, Ag(Al,
Ga, or In)O2 have been studied. It has been shown
that, in Ag(Al, Ga, or In)O2 delafossite oxides,
substitution of III-site atoms changes the bandgap
and electronic structure, altering their photocatalytic properties.22–24 In some theoretical25,26 and
experimental17,27,28 studies, the crystal structures
and electronic and optical properties of some phases
of AgGaO2 have also been investigated.
Theoretical studies on optical properties of
AgGaO2 are based on DFT calculations, whereas
experimental studies are limited to optical absorption spectroscopy. However, to use the full photovoltaic potential of Ag-based chalcopyrite systems,
knowledge on the optical response obtained using
advanced methods such as many-body perturbation
theory (MBPT) is required. MBPT is known to be a
powerful approach for studying the electronic structure and optical properties of a wide range of
compounds.29 Several DFT-based electronic structure calculations on chalcopyrite compounds,
including their band structure, density of states
(DOS), and optical properties, have been
reported,17,25–28,30,31 but to the best of the authors’
knowledge, there have been no studies on the
optical properties of the three polymorphs of
AgGaO2, focusing on the coupling of electrons and
holes resulting from electron transitions, i.e., excitonic effects. To this end, one can solve the equation
of motion of the two-particle Green’s function, the
so-called Bethe–Salpeter equation (BSE),32 as
applied in this study. AgGaO2 crystallizes in three
phases with rhombohedral (r-AgGaO2), hexagonal
(h-AgGaO2), and orthorhombic (o-AgGaO2) lattice,
whose structural and electronic properties have
been reported in some papers.30,31 The present
study compares and discusses the crystal structure,
lattice parameters, and GW-based electronic band
structure of AgGaO2 in these three phases. For
comparative study of the optical response of these
polymorphs near the bandgap, the present work
deals with excitonic effects. The focus is on the real
and imaginary parts of the dielectric as well as
energy loss functions. Although there are some
experimental studies on the absorption spectrum of
AgGaO2 with rhombohedral and orthorhombic
structure, they only deal with the absorption edge,
i.e., optical gaps in the energy range from 0 eV to
3 eV.27
The rest of this article is organized as follows:
First, the computational procedures and detailed
calculations are described. Then, the trends in the
electronic structure and optical properties from
r-AgGaO2 to h-AgGaO2 then o-AgGaO2 are investigated, focusing on excitonic effects in the optical
response near the bandgap of the three polymorphs.
Calculation Parameters
The exciting code33 was used to calculate the
electronic band structure and optical properties in
the random phase approximation (RPA) and using
the BSE.32 The full potential linearized augmented
plane wave (FPLAPW) method was employed for
electronic and optical calculations, in which no
shape approximation is made regarding the potential or electronic charge density and space is divided
into nonoverlapping atomic spheres centered on
atoms and an interstitial region. For ground-state
calculations, the local density approximation Perdew–Wang (LDA-PW) exchange–correlation functional34 was employed. The muffin-tin radius for Ag,
Ga, and O atoms is 2.25 a.u., 2.0 a.u., and 1.4 a.u.,
respectively. RMTKMax = 7, and lMax = 10 were used
for the structural and electronic properties of all
three phases. The GMax parameter was taken to be
14 Bohr1. To calculate the optical dipole matrix
elements in the dielectric function, Brillouin zone
(BZ) integrations were performed within self-consistency cycles via a tetrahedron method. Convergence was tested with respect to the number of
k-points. The converging k meshes in the irreducible
BZ were set to 15 9 15 9 8 (15 9 12 9 15) for the
ground-state calculations of r- and h- (o-) AgGaO2,
and to 6 9 6 9 6 (5 9 4 9 5) for optical calculations.
To neglect the symmetry conditions in the applied
k-grid, and include more k-points in the BSE calculations, the origin of the k-mesh was shifted to (0.1,
0.075, 0.15). The present study adopted the experimental structural parameters of AgGaO2 as a starting point. Lattice constants and positional
parameters for the three phases were optimized up
to force convergence of 0.2 mHa/a.u. For optical
spectra, lifetime broadening of 0.1 eV was applied.
Electron–Hole Interactions
It is well known that DFT is insufficient for the
subtle description of the exact bandgap and excited
states. By neglecting the electron–hole (e–h) coupling and formation of bound e–h pairs, the RPA
normally does not reproduce the correct peak positions and oscillator strengths and therefore cannot
accurately explain electronic transitions. In addition to creation of excitonic structures within the
energy gap region, e–h pairs sometimes redistribute
the oscillator strength throughout the optical
Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body
spectra. Clearly, MBPT is an accurate theory for the
linear response regime of infinite systems, being
very useful to compare the linear response with
experimental data. By solving the BSE for the e–h
two-particle Green’s function in the framework of
advanced methods, one can calculate the optical
absorption spectrum of various ternary compounds
such as copper-based chalcopyrite CuGaS2,35 perovskite BaTiO3,36 and noncubic LiNbO3,37 achieving
good agreement with corresponding experimental
data. Our previous theoretical studies of the optical
absorption spectrum of various organic and inorganic crystals also confirmed the good accuracy of
such many-body approaches.38–41 The exciting code,
a full-potential all-electron package, employs the
linearized augmented plane wave (LAPW) families,
in addition to these methods. In this code, the
equation of motion for the two-particle Green function is solved on the basis of the FPLAPW method.
The effective eigenvalue equation, including information about two-particle excitations in the manybody system, is
j j
0 c0 k0 A 0 0 0 ¼ E Avck :
v 0 c0 k 0
volume, and Evk and Eck represent the corrected
Kohn–Sham (KS) v and c state energies, respectively. Consideration of quasiparticle energies in
Eq. 3 and the two-particle wavefunction leads to a
higher level of calculation known as GWBSE. In
addition to correcting the peak positions, use of the
GWBSE level often modifies the oscillator strengths
in the optical response. The excitonic effects in the
dielectric function of all the structures were converged by including the 20 valence bands and 25
conduction states. The number of bands used to
calculate the screened interaction was converged at
Furthermore, due to the well-known underestimation of energy bandgap values when using DFT
and its exchange–correlation approximation, the
band structure and bandgap were calculated in the
many-body G0W0 approximation, as well.42 This
approximation was also selected as the starting
point for the BSE calculations. The k-mesh for the
GW calculations was converged at 4 9 4 9 3
(4 9 3 9 4) for the r- and h (o)-AgGaO2 structures,
and the number of empty states as the input to the
GW calculations was set at 60.
The eigenenergies Ej and eigenvectors Ajvck represent the excitation energy and the coupling coefficient of the jth e–h pair, respectively. The subscripts
denote valence states by v, conduction states by c,
and a vector in the irreducible BZ by k. The effective
e–h kernel, H exc , describing all interactions in
optical processes, contains three interacting
þ Hvck;v
0 c 0 k0 ¼ H
0 c0 k0 þ cx Hvck;v0 c0 k0 ;
vck;v0 c0 k0
where the first term on the right-hand side of Eq. 2,
H diag , is the difference of quasiparticle energies.
Consideration of only this term corresponds to the
independent particle approximation. The second
term is the Coulomb attraction between the electron
and hole, whereas the third one, the exchange term,
indicates the splitting between spin singlet (cx = 2)
and spin triplet (cx = 0) excitons. The direct interaction term contains the screened Coulomb interaction, whereas the exchange term contains the bare
Coulomb interaction without the long-range part.
By using the relation between the two-particle
correlation function derived from the BSE and the
polarization, the macroscopic dielectric function is
obtained, including local field effects (LFE) as well
as e–h correlation effects:
8p2 XX j vkjpi jck A
Im eii ðxÞ ¼
d Ej x ;
X j vck vck Eck Evk ð3Þ
where hvkjpi jcki is the matrix element of the
momentum operator component pi between the
valence and conduction bands, X is the crystal
Structure and Energy Bands
The crystal structure of the studied polymorphs is
displayed in Fig. 1. Table I presents a comparison of
the experimental and theoretical results, the optimized lattice parameters, and internal atomic positions. AgGaO2 has layered structure in its three
polymorphs. Depending on the stacking of the GaO6
octahedral layers, AgGaO2 exhibits 3R (rhombohedral) or 2H (hexagonal) symmetry (AaBbCcAa in
the former and AaBbAa in the latter43), belonging to
(166) and P63/mmc (194), respecspace group R3m
tively. These are two different polytypes with
delafossite crystal structure, which is similar to
the zincblende structure. In the 3R and 2H phases,
the monovalent cation (Ag) exhibits twofold linear
coordination with oxygen atoms while the trivalent
cation (Ga) is octahedrally surrounded by six oxygen
atoms. In other words, each oxygen atom is tetrahedrally coordinated with one Ag atom and three Ga
atoms. Recently, it was shown that AgGaO2 can
appear in another crystal structure with orthorhombic symmetry (o-AgGaO2) with b-NaFeO2 crystal
structure in space group Pna21 (33), belonging to
the wurtzite lattice, with the order of cations being
on the tetrahedral site,28 in which the zinc ions of
the wurtzite zinc oxide crystal are replaced by Ag
and Ga ions. This order leads to a decrease in
symmetry from the hexagonal to orthorhombic
structure. In o-AgGaO2, monovalent and trivalent
cations exhibit fourfold tetrahedral coordination
with oxygen atoms. In the z-direction, its unit cell
ideally contains two layers of oxygen atoms at
heights 0 and 1/2, and two layers of cations at
heights 1/6 and 2/3. This crystal contains two
Dadsetani and Nejatipour
Fig. 1. Conventional unit cell (top) and extended view (bottom) of AgGaO2 with rhombohedral (left), hexagonal (middle), and orthorhombic (right)
structure. Large, medium, and small spheres are Ag, Ga, and O atoms, respectively.
symmetrically nonequivalent oxygen atoms at positions 4a (0.621, 0.827, 0.426) and (0.542, 0.417,
0.332). Cations are tetrahedrally coordinated with
four oxygen atoms, i.e., three oxygen atoms of one
type and one oxygen atom of another.
Our calculations show that, energetically, the
most stable phase of AgGaO2 is rhombohedral. The
energy difference between the r- and h-AgGaO2
structures is low, indicating a possibility that both
structures can form. Although water is the key
factor for o- to r-AgGaO2 phase transformation in
experiment, the phase transformation accelerates at
higher temperature.44 Study of the effect of pressure on acoustic phonons in r-AgGaO2 indicates
dynamical instability and phase transition in this
structure (around 17 GPa).31 In an experimental
study, Nagatani et al. showed that o-AgGaO2 is
stable up to 690C under O2 atmosphere. The phase
became a mixture of metallic silver and Ga2O3 when
heated to 800C. Research on their thermal stability
indicated that no direct transformation occurs from
o-AgGaO2 to r-AgGaO2 phase.28 Elastic calculations
for o-AgGaO2 showed that all mechanical stability
conditions are satisfied.25
The band structure calculations showed that,
similar to the delafossite-structured oxides of
Cu,22–24 the three phases of AgGaO2 are indirectbandgap semiconductors (Fig. 2). Our calculations
show that, in the LDA, r-, h-, and o-AgGaO2 have
indirect energy bandgap values of 0.62 eV
(F fi C), 0.61 eV (M fi C), and 0.28 eV (Z fi C),
in agreement with other DFT studies. The band
dispersion of the three polymorphs clearly differ.
Compared with o-AgGaO2, the valence bands of rand h-AgGaO2 are highly dispersed. Due to the
well-known underestimation when using the LDA,
these values are smaller than those obtained experimentally (Table II). In the experimental studies,
however, optical bandgaps were measured. The
electronic bandgaps obtained using computational
band structure calculations can be compared with
experimental bandgaps obtained from photoemission measurements. Thus, comparison between
indirect computational and direct optical bandgaps
Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body
Table I. Lattice parameters and atomic positions of AgGaO2 in its three polymorphs (rhombohedral,
hexagonal, and orthorhombic) compared with other values reported in computational (Comp.) and
experimental (Exp.) studies
r-AgGaO2 (R3m)
This work
a (Å)
b (Å)
c (Å)
h-AgGaO2 (P63/mmc)
This work
o-AgGaO2 (Pna21)
This work
5.568 (Exp.)c
7.147 (Exp.)c
5.468 (Exp.)c
Ref. 17.bRef. 26.cRef. 28.
is not reasonable. The optically measured bandgaps
of r- and o-AgGaO2 in polycrystalline powder form
have been reported as 2.4 eV and 2.1 eV, respectively27 (2.38 eV and 2.18 eV in Ref. 10), with these
values being related to the not dipole-allowed direct
transition at C point.10,27,45 Both fundamental
direct and indirect bandgaps of AgGaO2 have the
same parity, an anomaly similar to that observed for
Cu-based delafossites.24 In addition, the samples
used for bandgap measurements were not pure or
homogeneous. In the experimental study conducting
by Vanaja et al.,46 the 4.12 eV (optical) bandgap of
AgGaO2 refers to a thin-film sample, which normally leads to a higher bandgap value than for a
crystal. This value is close to that of 3.75 eV, the
lowest direct G0W0 bandgap at C point obtained in
the present study. The indirect nature of the
bandgap of the AgGaO2 polymorphs may be related
to the considerable deviation of the Ag–O tetrahedra from the ideal form. The other silver-based
compounds, AgGaX2 (X = S, Se, Te), which crystallize with chalcopyrite (CuFeS2) structure in space
group I 42d,
are direct-bandgap semiconductors,
with both valence-band maximum (VBM) and conduction-band minimum (CBM) at C point. Their
experimental bandgaps are 2.63 eV, 1.74 eV, and
1.32 eV for X = S, Se, and Te, respectively.15
Due to the general underestimation of DFTderived band energies, the band structure of the
three polymorphs was also calculated in the manybody G0W0 approximation (Fig. 2). Table II lists
these values, which are in good agreement with
experimental reports. The G0W0 approximation
shifts the conduction bands to higher energy and
also modifies the deeper valence states to lower
energy. In this approximation, the indirect energy
bandgap of r-, h-, and o-AgGaO2 is 2.35 eV, 2.23 eV,
and 2.07 eV, respectively, and the lowest direct
bandgaps at C point are 3.75 eV, 3.58 eV, and
2.07 eV. The difference between the indirect and
lowest direct bandgaps is much smaller for oAgGaO2 compared with r- or h-AgGaO2. It has been
demonstrated that, due to the larger energy bandgap, r-AgGaO2 shows higher photocatalytic activity
than o-AgGaO2.27 For all three polymorphs, the
VBM mostly consists of silver 4d and oxygen 2p
Optical Response
Figures 3, 4, 5, and 6 show the components of the
imaginary part of the dielectric function, Im e(x), for
the different polymorphs of AgGaO2. One can see
some differences among the optical absorption spectra of the hexagonal, rhombohedral, and orthorhombic phases, both in their general form and in the
energy positions of the main structural features.
This can be related to the differences in their crystal
symmetry, bandgap, and electronic structure. Based
on their crystal symmetry, the dielectric tensor for
the rhombohedral and hexagonal structure has two
independent elements (xx and zz), while the dielectric tensor of the orthorhombic structure has three
independent elements (xx, yy, and zz). Figure 3
compares the x- and z-polarized response of
r-AgGaO2 calculated using the BSE versus the
GWBSE and GWRPA approaches. Figure 3 includes
the optical absorption spectra calculated using the
BSE with KS eigenstates and eigenvalues in Eq. 3,
in which a rigid scissor shift equal to the difference
between the KS and G0W0 bandgaps has been
applied. One can see some differences between the
BSE and GWBSE spectra, both in their spectral
shape and energy position. This shows that, to
achieve reasonable results for the optical absorption
spectrum of AgGaO2, it is not enough to apply a
rigid scissor shift and inclusion of the quasiparticle
correction is obligatory. In addition to correcting the
peak positions, the quasiparticle correction modifies
the oscillator strengths in the optical response. As
Fig. 3 indicates, the overall shapes of the dielectric
function obtained using the full BSE (GWBSE) and
GWRPA approaches are very different. When using
the GWRPA, although some of the main spectral
features are reproduced at higher energies, both the
intensities and the overall shape of the spectra are
not in accordance with the GWBSE results. This
indicates that full solution of the BSE results in
Dadsetani and Nejatipour
Fig. 2. Band structure of AgGaO2 in (a) rhombohedral, (b) hexagonal, and (c) orthorhombic phase. Blue and red lines show the one-particle
Kohn–Sham (LDA) and many-body G0W0 band structure, respectively (Color figure online).
Table II. Calculated electronic bandgap of three polymorphs of AgGaO2 in one-particle LDA-PW and
many-body G0W0 approximations, in comparison with optical bandgaps obtained by experiment
G 0W 0
Lowest Direct Gap: G0W0 (LDA)
3.75 (1.56)
3.58 (1.56)
2.07 (0.30)
Ref. 27.
considerable redistribution of oscillator strengths in
the optical spectra and reproduces the excitonic
features. These excitonic structures are mostly
continuum rather than bound excitons, because
instead of producing bound excitons in the bandgap
region, the overall shape of the spectrum beyond the
bandgap region is changed.
Comparison of the x- and z-component of the
imaginary part of the dielectric function (Figs. 3
and 6) clearly reveals high anisotropy across the
entire range of photon energies, not only in the
r-AgGaO2 crystal, but also in h- and especially
o-AgGaO2 crystals. Compared with the z-polarized
response, the spectral features of the x-polarized
response of r- and h-AgGaO2 occur at lower energies. Figure 6 also indicates that the spectral features of the z-polarized response of o-AgGaO2 occur
at lower energies. For all three polymorphs, excitonic effects in the x-polarized response are
By means of the diffuse reflectance spectrum,
Sheets et al.47 measured the optical absorption
spectrum and thereby the optical bandgap of
r-AgGaO2. This technique is especially useful for
determination of the optical gap. Moreover, due to
the natural limitations of this technique for powder
samples, for instance, the occurrence of additional
light absorption, exact determination of the optical
bandgap is a problem. As Fig. 7 indicates, the
experimental absorption spectrum of r-AgGaO2
shows an onset at energy of 2.4 eV, which corresponds to the optical bandgap. In addition, it
exhibits a spectral peak near 3.8 eV and a dip near
4.6 eV. From a different perspective, the dip at
energy near 4.6 eV in the absorption spectrum could
be a sign of a strong peak with position near 4.6 eV
in the reflection spectrum, and therefore in the
dielectric function, according to the Kramers–Kronig relations. At the end of this paper, there is a
discussion on the reflection spectra of the materials
under study. Here, it is worth mentioning that the
expected peak in the reflection spectrum of rAgGaO2 at energy of 4.6 eV has been reproduced
well (Fig. 12a). Figure 7 shows the theoretical
absorption spectrum of r-AgGaO2, as well. This
figure shows an onset at energy of 2.4 eV, which
corresponds to the optical bandgap obtained by
experiment. In addition, the dominant experimental
Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body
Fig. 3. Imaginary part of (a) x- and (b) z-polarized dielectric function
of r-AgGaO2, calculated using BSE (dash–dotted lines),
GW-corrected BSE (GWBSE, solid lines), and GW-corrected RPA
(GWRPA, dashed lines).
Fig. 5. Imaginary part of (a) x-, (b) y-, and (c) z-polarized dielectric
function of o-AgGaO2, calculated using GWBSE (solid lines) and
GWRPA (dashed lines).
Fig. 4. Imaginary part of (a) x- and (b) z-polarized dielectric function
of h-AgGaO2, calculated using GWBSE (solid lines) and GWRPA
(dashed line).
peak in the energy range from 1.5 eV to 5.0 eV,
which is located at 3.8 eV, was reproduced by our
calculations at energy of 3.78 eV. Besides, as Fig. 3b
indicates, the imaginary part of the z-polarized
dielectric function of r-AgGaO2, obtained in the
GWBSE approximation (blue line) shows a dominant spectral feature at energy of 4.6 eV. Since
there is no analogous feature at 4.6 eV in the
spectrum obtained using the GWRPA approach,
one can attribute this feature to e–h coupling
originating from the electron transition.
Compared with r-AgGaO2, h-AgGaO2 shows
nearly similar behavior, in which the features in
the x-polarized optical response occur at lower
energies (Fig. 4). Comparison between the GWBSE
and GWRPA calculations demonstrates that use of
the GWBSE approach results in considerable redistribution of oscillator strength in the optical spectrum. In this case, the x-polarized optical absorption
is dominated by a bound excitonic state with weak
oscillator strength, occurring at 2.05 eV. Figures 3
and 4 indicate that, in contrast to the other crystal
directions, excitonic effects are dominant in the
direction of the a parameter.
In the case of o-AgGaO2, the situation is more
interesting. The anisotropy in this case, at least in
the gap region, is at the lowest level. The optical
Dadsetani and Nejatipour
Fig. 7. Experimental47 [reprinted with permission from W.C. Sheets,
E.S. Stampler, M.I. Bertoni, M. Sasaki, T.J. Marks, T.O. Mason, and
K.R. Poeppelmeier, Inorg. Chem. 47, 2696 (2008). Copyright 2008
American Chemical Society] and theoretical z-polarized absorption
spectra of r-AgGaO2. Experimental spectrum obtained by diffuse
reflectance spectrum; theoretical spectrum obtained by postprocessing with LayerOptics package,49 a tool for determination of the
dependence of optical coefficients such as absorbance and reflection on the angle h of incoming light. Inset shows dominant theoretical feature in smaller energy range.
Fig. 6. Comparison among imaginary parts of dielectric function of
(a) r-AgGaO2, (b) h-AgGaO2, and (c) o-AgGaO2. The difference in
the spectral dispersion of each polymorph indicates high anisotropy
of the optical properties of the AgGaO2 polymorphs.
absorption spectra of o-AgGaO2 obtained using the
GWRPA and GWBSE approaches differ mostly in a
slight shift rather than their oscillator strength
distribution. All the spectral features in the
GWRPA and GWBSE results are nearly the same,
with lower intensities for the former. Solving the
BSE clearly leads to a shift in the spectral features
to lower energy positions, producing strong excitonic effects in the bandgap region. Figure 5 shows
that the lowest excitonic feature belongs to the
z-polarized response, i.e., for light polarized parallel
to the crystallographic c axis. The peak of the strong
excitonic structure in the z-polarized response is
found at 1.74 eV. The first optical peak positions for
the x- and y-polarized response are 2.064 eV and
2.23 eV, respectively, beyond the bandgap.
To complete this discussion, we consider the real
part of the dielectric function of the AgGaO2 polymorphs. Figures 8, 9, and 10 show the real part of
the dielectric function, Re e(x), of r-, h-, and
o-AgGaO2, respectively, as calculated using the
GWBSE and GWRPA approaches. The changes
from GWRPA to GWBSE, in both the spectral
dispersion and physical parameters such as dielectric constants, are obvious. Although Re e(x) of rand h-AgGaO2 are nearly the same in their overall
dispersion, the dielectric constants are very different. This is true for both the x- and z-polarized
response. The dielectric constant of the x-polarized
response of r-AgGaO2 obtained using the GWBSE
(GWRPA) approach is 3.54 (2.86), whereas that of
the z-polarized response is 2.90 (2.27). In GWBSE,
the z-polarized (x-polarized) component of Re e(x) of
r-AgGaO2 has negative values for the energy range
of 5.70 eV to 10.45 eV (4.69 eV to 8.74 eV),
corresponding to complete reflectivity. This energy
range is slightly broader than that of the x-polarized
response. Instead, this energy range in the GWRPA
results is 7.2 eV to 11.17 eV (6.36 eV to 9.63 eV).
This means that, in addition to changing the
spectral dispersion, use of the GWBSE approach
applies a red-shift to Re e(x).
As shown in Fig. 9, the dielectric constant of hAgGaO2 calculated using GWBSE (GWRPA) is 3.50
(3.10) and 3.35 (2.67) for the x- and z-polarized
response, respectively. In the energy range from
4.83 eV to 8.50 eV, the reflection of the x-polarized
response is complete, whereas the corresponding
energy range for the z-polarized response is much
broader (5.46 eV to 12.68 eV) and occurs at higher
Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body
Fig. 8. Real part of (a) x- and (b) z-polarized dielectric function of
r-AgGaO2, calculated using GWBSE (solid lines) and GWRPA
(dashed lines).
Fig. 10. Real part of (a) x-, (b) y-, and (c) z-polarized dielectric
function of o-AgGaO2, calculated using GWBSE (solid lines) and
GWRPA (dashed lines).
Fig. 9. Real part of (a) x- and (b) z-polarized dielectric function of hAgGaO2, calculated using GWBSE (solid lines) and GWRPA (dashed lines).
Figure 10 shows that the dielectric constant of
o-AgGaO2 is smaller than for the other polymorphs.
According to the GWBSE results, the dielectric
constant is 2.11, 2.06, and 2.11 for the x-, y-, and
z-polarized response, respectively, whereas according
to GWRPA, they are 1.97, 1.93, and 1.97. The
difference between the dielectric constant values
obtained using the GWBSE and GWRPA approach
for all three phases corresponds to the inclusion of
e–h coupling in GWBSE and the modification of
both the oscillator strengths and peak positions.
From r-AgGaO2 to o-AgGaO2, as the energy bandgap decreases, metallic treatment and screening
will increase, and therefore e–h bonding will become
weaker, decreasing the differences between the
dielectric constants obtained by the GWBSE and
GWRPA approach. In the case of o-AgGaO2, the real
part of the dielectric function exhibits no negative
values, so the reflectivity coefficient will not reach
unity. In the first 2 eV of the spectra, the overall
dispersions are nearly the same, while those at
higher energies differ. Comparison between the
GWBSE and GWRPA results shows that, in this
case, the spectral shapes are much more in
The energy loss components of the AgGaO2
polymorphs are displayed in Fig. 11. Being directly
related to the response function of materials, the
Dadsetani and Nejatipour
Fig. 11. Components of energy loss functions of (a) r-AgGaO2, (b)
h-AgGaO2, and (c) o-AgGaO2.
electron energy loss (EEL) is a key function, being
defined as the energy loss of a fast electron which
moves through a medium and experiences inelastic
scattering. Low-loss function
is defined as
LðxÞ ¼ Imð1=eÞ ¼ e2 = e21 þ e22 .48 Based on this
function, features in the low-loss function can be
categorized into two main types, viz. plasmon peaks
and interband transition peaks. Plasmon peaks
result from zeros in the real part of the dielectric
function of the scattering material (Re eii), whereas
interband transitions originate from the most
prominent peaks of the imaginary part of the
dielectric function (Im eii). Due to the excitation of
the volume plasmon hxp, the energy loss function
will have a strong maximum if Re eii is zero and Im
eii is small. The strong anisotropy observed in
Fig. 11 is similar to that of the optical response. In
the energy range near to 3.5 eV to 4.6 eV, the loss
functions show some intense features that can be
attributed to the spectral features in the energy
range of 3.2 eV to 4.1 eV of Im e(x) for the r- and hAgGaO2 phases. The x-, y-, and z-components of
L(x) of o-AgGaO2 show some main peaks at energies
of 2.248 eV, 2.43 eV, and 1.88 eV, respectively,
Fig. 12. Parallel component of reflection coefficients of (a) r-AgGaO2, (b) h-AgGaO2, and (c) o-AgGaO2 for different orientations
between the system and incoming light (h = 0, 20, 40, 60, and
which can be assigned to energy losses due to
electron transitions as captured in the corresponding components of Im e(x) at 2.06 eV, 2.23 eV, and
1.74 eV, respectively. In addition, the spectral features in the energy range of 6.0 eV to 10.0 eV
originate from the main profiles of Im e(x) in the
energy range of 4.0 eV to 8.0 eV.
To further investigate the anisotropy effects in
AgGaO2 phases, the dependence of the optical
coefficients on the angle h of incoming light was
calculated, using the LayerOptics package49 as a
postprocessing tool to determine optical coefficients
such as absorbance and reflection. This package has
been used to study anisotropy effects in core-level
excitations of Ga2O3 with monoclinic crystal structure,50 a compound similar to those studied herein.
An effective rotation of the Euler angle a = 45, the
angle of rotation around the Cartesian z axis, was
applied to the two independent components of the
dielectric tensor, while the other two Euler angles
were set to zero. More details about the rotation
matrix can be found in Ref. 49. The results in
Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body
Fig. 12 correspond to the dependence of the parallel
polarized (p-polarized) reflection coefficient on the
angle h, the relative orientation of the incident beam
and sample. The angle h was varied from 0 up to
80, while the polarization angle was kept fixed. The
polarization of the beam is the angle d relative to the
positive y-axis. Angle d = 0 corresponds to full
parallel polarization (p-component), while d = 90
corresponds to fully perpendicular polarization (scomponent). In r-AgGaO2, for h = 0, 20, and 80,
the reflection coefficient is dominant in the energy
range of 3 eV to 5.20 eV. In this range, the contribution of h = 60 is lower than the others, while it is
dominated by an intense feature at 5.20 eV and
reaches higher values beyond 5.20 eV. Up to
6.12 eV, the contribution of h = 40 coincides with
the contributions of h = 0, 20, and 80, and beyond
6.12 eV, it slightly overtakes. For the h-AgGaO2
structure, h = 40 makes the greatest contribution
to the reflection coefficient. Up to 5.53 eV, h = 80
makes the lowest contribution, whereas beyond
5.53 eV, its contribution overtakes those of h = 0,
20, and 60. In the case of o-AgGaO2, h = 0, 20,
and 40 make the same and largest contributions to
the reflection coefficient. The contribution of h = 60
is slightly less significant, while the contribution of
h = 80 is the least.
MBPT was used to calculate and compare the
structural, electronic, and optical properties of
AgGaO2 in its three polymorphs, viz. rhombohedral,
hexagonal, and orthorhombic. In both the oneparticle Kohn–Sham and many-body G0W0 approximations, the AgGaO2 polymorphs are indirectbandgap semiconductors. The three polymorphs
show high anisotropy in their optical spectra,
including the real and imaginary parts of the
dielectric function and energy loss spectra. Depending on the relative orientation of the incident light
beam and sample, the reflection coefficients of all
three polymorphs exhibit high anisotropy. The
reproduced absorbance coefficient is in good agreement with existing experimental data. Compared
with the GWRPA approach, GWBSE results in
significant redistribution of oscillator strength in
the optical spectra. Spectral features and excitonic
effects (electron–hole coupling) were reproduced
well near the bandgap region of all three phases;
in addition to modifying the peak positions, this
coupling changed the overall shape of the spectra.
In the case of the x-polarized response, the e–h
couplings are stronger. Furthermore, the real part
of the dielectric function and energy loss spectra
were calculated and compared. Spectral analysis
showed that the main spectral features of the loss
functions near the bandgap region resulted from the
main excitonic structures in the optical absorption
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