Journal of ELECTRONIC MATERIALS 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 MEHRDAD DADSETANI1 and REIHAN NEJATIPOUR1,2,3 1.—Department of Physics, Lorestan University, email@example.com. 3.—e-mail: firstname.lastname@example.org Khorramabad, Iran. 2.—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 INTRODUCTION 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. COMPUTATIONAL PROCEDURES 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 Approaches 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 X j exc j j Hvck;v ð1Þ 0 c0 k0 A 0 0 0 ¼ E Avck : vck 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 110. 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 contributions: RESULTS AND DISCUSSION diag exc dir x Hvck;v þ Hvck;v 0 c 0 k0 ¼ H 0 c0 k0 þ cx Hvck;v0 c0 k0 ; vck;v0 c0 k0 ð2Þ 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: 2 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 Approaches 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) u 2.948 2.948 18.529 0.1139 h-AgGaO2 (P63/mmc) Others 2.989 2.989 18.534 0.1135 (Exp.)a (Exp.)a (Exp.)a (Exp.)a This work 3.019 3.019 12.359 0.0812 Others 2.990 2.990 12.357 0.0810 (Comp.)b (Comp.)b (Comp.)b (Comp.)b o-AgGaO2 (Pna21) This work Others 5.535 7.102 5.443 5.568 (Exp.)c 7.147 (Exp.)c 5.468 (Exp.)c a 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 orbitals. 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 Phase r-AgGaO2 h-AgGaO2 o-AgGaO2 LDA-PW G 0W 0 Lowest Direct Gap: G0W0 (LDA) Exp. 0.62 0.61 0.28 2.35 2.23 2.07 3.75 (1.56) 3.58 (1.56) 2.07 (0.30) 2.4a – 2.1a a 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 dominant. 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 Approaches 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 energies. Calculation of Electronic and Optical Properties of AgGaO2 Polymorphs Using Many-Body Approaches 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 agreement. 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 80). 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 Approaches 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. CONCLUSIONS 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. 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