Comparative Analysis of the Electronic Structure and Nonlinear

Article pubs.acs.org/JPCC

Comparative Analysis of the Electronic Structure and Nonlinear Optical Susceptibility of α‑TeO2 and β‑TeO3 Crystals E. M. Roginskii,*,†,‡ V. G. Kuznetsov,‡ M. B. Smirnov,‡ O. Noguera,§ J.-R. Duclère,§ M. Colas,§ O. Masson,§ and P. Thomas§ †

Ioffe Institute, Polytekhnicheskaya 26, 194021 St. Petersburg, Russia Faculty of Physics, St Petersburg State University, Petrodvoretz, 194508 St. Petersburg, Russia § Laboratoire de Science des Procédés Céramiques et de Traitements de Surface UMR 7315 CNRS, Centre Européen de la Céramique, Université de Limoges, 12 rue Atlantis, Limoges 87068 Cedex, France ‡

ABSTRACT: The hypothesis of the huge optical nonlinearity of the crystalline TeO3, recently advanced on the basis of the quantummechanical simulations, is tested. Electronic band gaps of α-TeO2 and β-TeO3 crystals are determined by diffuse reflectance measurements. The DFT+U method is applied to calculate electronic band gap Eg and the third-order nonlinear dielectric susceptibility χ(3). The χ(3)(TeO3) is about two times lower than the χ(3)(TeO2) in spite of the fact that the Eg(TeO3) is narrower than Eg(TeO2). It is shown that this peculiarity is related to the 5s(Te) electronic states which are occupied in TeO2 and are vacant in TeO3. This distinction is due to the specific electronic state related to the electron lone pairs localized on the Te(IV) atoms.

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INTRODUCTION

predicted for this crystal third-order susceptibility 10 times higher than that of α-TeO2. One could consider this result as an outstanding achievement which opens wide perspectives for elaborating new highly nonlinear materials. However, no experimental confirmation was reported up to now. The matter is that the crystalline βTeO3 is available only in powder form. Besides, this compound is unstable and under moderate heating decomposes into TeO2 and O2. Thus, the hypothesis suggested in ref 12 was based on purely theoretical argumentation drawn from the quantummechanical simulations. As a rule quantum-mechanical calculations of optical characteristics (linear and nonlinear) are based on the perturbation theory, in which the energy gap between valence and conduction bands plays a central role.13,14 Traditional DFT methods (such as LDA or GGA) severely underestimated the band gap value for semiconductors and insulators.15−17 It should be noted that the calculations presented in ref 12 gave for β-TeO3 an electronic band gap three times less than the experimental estimate. This puts in doubt the predicted optical properties and necessitates a re-examination. To overcome the band gap problem various improvements were proposed. The quasi-particle GW Approximation (GWA) yields very accurate band gap values (see, e.g., refs 18−23) but leads to rather time-consuming calculations, especially if done

Since the beginning of the 1990s the extraordinarily high optical nonlinearity of tellurium dioxide TeO2 in both crystalline and glassy states has attracted a lot of attention of specialists searching for advanced materials for new optoelectronic devices.1 It was found that the nonlinear refractive index of glassy TeO2 is maximal among other oxide glasses and is 50 times larger than that of the silica glass.2 A large body of studies were devoted to searching a chemical composition of tellurite glasses with optimal nonlinear optical (NLO) properties.3 Much of the attention was paid to the pure tellurium dioxide and, first of all, to its ground-state polymorph paratellurite, αTeO2, due to its outstanding optoacoustic, piezoelectric, and electro-optic properties.4,5 The electronic structure and dielectric properties of this material were analyzed in many theoretical studies based on quantum-mechanical calculations.6−12 The first theoretical estimations of the third-order NLO susceptibility χ(3) of α-TeO2 and of α-SiO2 have confirmed that the former compound has χ(3) of about 40 times larger than that of the latter.9 Later on, similar results were obtained for other TeO2 polymorphs.11 Thereupon, the dielectric properties of other tellurium oxides were investigated. The mixed Te(IV)/Te(VI) oxide system (i.e., the TeO2−TeO3 binary systems including the α-TeO2, Te4O9, Te2O5, and β-TeO3 compounds) was examined in ref 12. The last member of this family, the tellurium trioxide βTeO3, was suggested to possess a rather high dielectric nonlinearity. Numerical estimations based on DFT simulations © XXXX American Chemical Society

Received: February 24, 2017 Revised: April 12, 2017 Published: May 9, 2017 A

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The Journal of Physical Chemistry C self-consistently. The DFT+U approach24 is now considered as a rather effective tool to treat the strong on-site Coulomb interaction of localized electrons, which is not correctly described by LDA or GGA. It is computationally as cheap as the LDA and GGA functional and allows one to reproduce reasonably well the excited electron states. The main problem which hampers use of this method is that the U value is usually unknown and should be fitted for any particular compound. This causes an unresolvable problem in the lack of sufficient experimental data. In this study we have chosen an approach which combines adequacy of the GWA and feasibility of the DFT+U method. The electronic band structure was simulated by the GWA and the DFT+U method. These results are compared with each other, and the optimal U value was estimated providing the best agreement. For an additional justification of the chosen U value, the DFT+U results were compared with the whole body of available experimental data: structural parameters, dielectric constants, and phonon frequencies. Finally, the DFT+U approach was used to simulate the nonlinear optical susceptibilities. The present study is aimed to obtained theoretical estimations of the nonlinear dielectric susceptibility of the tellurium (Te(IV) and Te(VI)) oxide crystals and to justify their credibility. The main objective was a verification of the hypothesis of the high optical nonlinearity of the β-TeO3 crystal. In order to compare credibility of different computational approaches we began with another well-known tellurium oxide α-TeO2. Structure and electronic and optical properties of paratellurite α-TeO2 have been thoroughly studied experimentally4 and theoretically.6−12 This provides us with good opportunity to test the chosen computational schemes.

constructed with explicit treatment of the 4d 5s 5p states for the tellurium atoms and the 2s 2p states for the oxygen atoms. In all calculations, the atomic positions and lattice parameters have been optimized via independent relaxation until the atomic forces and stresses were reduced below 10−5 Ha/Bohr and 0.2 Kbar, respectively. The results were checked for the convergence with respect to the size of k-point sampling integration grid and to the plane-wave kinetic energy cutoff. It was found that the convergence of total energy within 0.1 mHa was achieved with the energy cutoff of 35 Ha. The k-point grids were chosen according to the Monkhorst−Pack scheme37 as 6 × 6 × 4 for α-TeO2 and 6 × 6 × 6 for β-TeO3 for in the LDA and LDA+U calculations. The 4 × 4 × 4 Monkhorst−Pack grid was found to be acceptable for the GWA calculations. The fully nonlocal two-projector norm-conserving pseudopotentials28 with PBEsol functional38 were used for GWA calculations. These calculations were performed self-consistently with the use of numerical integration by the contour deformation method.39 Within GWA the band structure was determined as a linear correction with respect to the Kohn− Sham energy for the k-points where it has not been calculated explicitly. To compute the effective Born charges and dielectric constants, both the linear-response theory40,41 and the finite electric field calculations were used. The latter approach uses the Berry phase approach42−44 implemented in the ABINIT package. The GWA method is accurate for the electronic structure calculations, but it is very time-consuming. Besides, it is not yet adapted to the phonon state and the finite-field calculations. Consequently, the phonon states in the center of the Brillouin zone and the dielectric constants were calculated by using both LDA and LDA+U approximations within the linear-response theory, while the third-order susceptibilities were obtained by fitting the polarization vs field dependences obtained in the finite-field calculations. For LDA+U methods, we select two sets of the Coulomb (U) and exchange (J) parameters in the model Hamiltonian: U = 10 eV, J = 1 eV for α-TeO2 and U = 6 eV, J = 0.6 eV for βTeO3. These parameters resulted in the best agreement with the experimental phonon spectra and the band gap values obtained within GWA.

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EXPERIMENTAL AND COMPUTATIONAL DETAILS Diffuse reflectance measurements were performed with powder samples of β-TeO3, for which the population of grains with sizes larger than 5 μm were carefully selected, this condition being mandatory in order to apply the Kubelka−Munk formula.25 The optical band gap value was extracted from the diffuse reflectance measurements carried out in the 250−2500 nm (0.5−5 eV) range, using an integration sphere. The reflectance data were treated by applying the Kubelka−Munk formula.25 This formula relates the ratio of the absorption and scattering coefficients with the reflection coefficient and allows one to represent the absorption−scattering ratio as a function of the photon energy. The optical band gap is extracted by extrapolating this function to the zero absorption coefficient. Details of the method can be found elsewhere.26 All calculations reported in this paper were based on the DFT pseudopotential method with plane-wave basis set as realized in the ABINIT code.27 Exchange and correlation were treated within both the GGA and LDA approximation with the Perdew−Wang28 and Perdew−Burke−Ernzerhof (PBE) exchange-correlation functionals.29 The projector-augmented wave (PAW) method30,31 was used to describe the interactions between the core and valence electrons. Electronic structures of both α-TeO2 and β-TeO3 have been simulated within the LDA, LDA+U, and GWA. The LDA+U approach32 allows us to take into account the localization of tellurium 4d and oxygen 2p states and cures the band gap problem to a large extent. The quasiparticle self-consistent GWA33,34 enables us to overcome the deficiencies of DFT to treat excited states. All calculations, except GWA, were performed using PAW data sets generated using the GPAW software package.35,36 All data sets were

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RESULTS Crystal Structure. Both the α-TeO2 and β-TeO3 structures have been thoroughly studied experimentally. 45,46 The structures consist of a 3D framework of the corner-sharing polyhedra: the TeO4 polyhedron units specified as disphenoids in α-TeO2 and the TeO6 octahedra in β-TeO3 (see Figure 1) First we turn to the crystal structure of paratellurite α-TeO2. Structural parameters of paratellurite optimized within different approximations are compared with experimental data in Table 1 The results presented in Table 1 show that the LDA method underestimates the specific volume, whereas the PBEsol approximation (which is used later in GWA) overestimates it. In any case the deviation is about 7%. This is a common situation earlier noted for other oxide crystals.47,48 Another important characteristic of the α-TeO2 structure is the ratio of the two nonequivalent Te−O bond lengths. The basic structural units of this lattice are the TeO4 polyhedra with the shape of distorted trigonal bipyramids (see Figure 2a) in which the equatorial plane is occupied by a lone pair of electrons (shown by the arrow) and by two oxygen atoms (Oeq) with the Te−Oeq bond length of 1.88 Å and the Oeq− B

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allowed us to suggest a quasi-molecular nature of the bounding in paratellurite.49 To the best of our knowledge, none of the previous DFT studies could correctly reproduce this structural peculiarity. The calculated bond length ratio was always found to be markedly larger. This disagreement evidently distorted the electron density distribution. It is remarkable that our PBEsol calculations perfectly reproduce this value. The crystal structure of β-TeO3 also is a three-dimensional framework built of the corner-sharing polyhedra. Unlike the αTeO2 structure, the polyhedra in β-TeO3 are almost ideal TeO6 octahedra (see Figure 2). It was shown49 that this structure can be considered as a cubic ReO3-like lattice distorted by a concerted rotation of the octahedra around the (111) axis. Such distortion results in the formation of trigonal lattice with D3d symmetry.46 Due to high symmetry of this structure, all Te−O bonds are equivalent. Structural parameters computed for this structure within different approximations are compared with experimental data in Table 2. Table 2. Optimized Structural Parameters of β-TeO3 in Comparison with Experimental Dataa Figure 1. Crystal structures of α-TeO2 (a) and β-TeO3 (b). a (Å) c (Å) Vs (Å3) Te−O (Å)

Table 1. Optimized Structural Parameters of α-TeO2 in Comparison with Experimental Dataa a (Å) c (Å) Vs (Å3) Te−Oeq (Å) Te−Oax (Å) (Te−Oeq)/(Te−Oax)

exptl45

LDA

LDA+U

PBEsol

4.808 7.612 43.99 1.8762 2.1228 0.8838

4.732 7.324 41.00 1.9153 2.1046 0.9101

4.645 7.396 39.90 1.9105 2.0679 0.9239

4.972 7.617 47.08 1.9115 2.1544 0.8873

exptl46

LDA

LDA+U

PBEsol

4.901 13.03 45.17 1.9109

4.870 12.972 44.41 1.9088

4.834 12.819 43.24 1.8798

5.036 13.183 48.25 1.9466

a

Space symmetry group is R3̅2/c (No 167). Specific volume Vs is the volume per one TeO3 unit.

In this case, the LDA and LDA+U calculations also underestimate the specific volume and the bond length, whereas the PBEsol overestimates them providing the average deviation below 7%. Lattice Dynamics. In order to test the credibility of the used computational approaches we have calculated the zonecenter phonon frequencies and compared them with experimental data. The phonon frequency distributions calculated by LDA and LDA+U methods are shown in Figure 3. The GWA method was not tested because it is not adapted to the phonon state calculations. Figure 3 shows that the phonon frequencies calculated by the LDA+U method agree with the experimental data better than the LDA results: the mean deviation is 5.8% in LDA+U (vs 6.1% in LDA) for αTeO2 and 4.5% (vs 7.4%) for β-TeO3. The most marked disagreement between the experiment and the LDA takes place in the high-frequency region, where the calculated frequencies are systematically smaller. The maximum deviation reaches 100 cm−1 for the A2u mode in β-TeO3. The LDA+U results show a better agreement with the mean deviation within 20 cm−1 (with the only exception of the highest A2g mode in β-TeO3 whose calculated frequency is 48 cm−1 higher than in the experiment). Electronic Structure. Figure 4 plots the results of diffuse reflectance measurements. The band gap value was determined as 3.25 ± 0.05 eV for β-TeO3 using the Kubelka−Munk function. To the best of our knowledge, this value is the first experimental determination of this basic characteristic. For αTeO2, the Eg value was found to be close to 3.75 eV in ref 51. Computed electron band structures for both α-TeO2 and βTeO3 crystals are shown in Figure 5.

a

Space symmetry group is P41212 (No. 92). Specific volume Vs is the volume per one TeO2 unit.

Figure 2. Basic structural units of the α-TeO2 (a) and β-TeO3 (b) lattices: TeO4 dtp (a) and TeO6 octahedron (b). Big solid brown atoms are Te; small red atoms are O; and the arrow visualizes the lone pair.

Te−Oeq angle of 103°. The axial positions are occupied by two axial oxygens (Oax) with Te−Oax bond length of 2.12 Å and the Oax−Te−Oax angle of 168°. According to experimental data, the length ratio (Te−Oeq)/(Te−Oax) is equal to 0.884. Such a large difference in lengths of the Te−Oeq and Te−Oax bonds C

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Figure 3. Calculated zone-center phonon frequencies in comparison with experimental data: (a) α-TeO2, experimental data from ref 49; (b) β-TeO3, experimental data from ref 50. Symmetry representations are discriminated by using different colors.

and indirect band gaps in both crystals is insignificant. The main divergence between band structures calculated by different approximations is the value of the energy gap between valence and conduction bands (these values are listed in Table 3). The calculated band gap value for α-TeO2 within GWA is Eg = 3.68 eV. This is close to the experimental value (3.75 eV). The LDA approach gives markedly lower value of 2.82 eV. This approach evidently fails to reproduce the band gap value. We have tried to overcome this shortcoming by using the LDA+U method. This allowed us to reach the value of band gap Eg = 3.26 eV which is much closer to the experimental data. Corresponding U value was equal to 10 eV. Further increase of the U parameter perverted the phonon spectrum. The band gap values for β-TeO3 calculated within different computational schemes also differ significantly. The GWA calculation reveals the value of band gap Eg = 2.74 eV which is not far from the experimental value (3.25 eV). The LDA approach gives a two times lower value of 1.5 eV. Similar disagreement is inherent to the GGA calculations. For example, calculations in ref 12 gave Eg = 0.99 eV which is three times lower than the experimental value. Such a big discrepancy is obviously due to the significant contribution of quasiparticles to the electronic structure of β-TeO3. Thus, the absorption spectra are expected to demonstrate the feature related to exciton. In order to correct the failure of the semilocal functionals, we have applied the LDA+U method in all further calculations. This method gives Eg = 2.27 eV which is much closer to experimental data than the LDA and GGA results. The corresponding U value was fixed at 6 eV. Larger values of the U parameters also perverted the phonon spectrum. Dielectric Properties. For both crystals the linear dielectric responses have been calculated by using the DFPT method.40,41 Thus, calculated orientation averaged dielectric susceptibilities

Figure 4. Diffuse reflectance data for the powder β-TeO3 sample: Kubelka−Munk function25 versus the photon energy. The intersection of the tangent line with the abscissa axis defines the optical band gap value (Eg).

The electronic structures obtained with the LDA and LDA +U methods are quite similar. The only marked distinctions between them are the band gap values. Therefore, Figure 5 presents only the LDA results. In Figure 5a and 5b one can also see that the band structures of α-TeO2 calculated by LDA+U and GWA are alike. Both are similar to that reported in refs 6 and 7. As concerns the top of the valence band, there are several BZ points with almost equal band energy. However, the density of states near the top of the valence band is markedly higher in the GWA results. Contrariwise, the bottom part of the conduction band is similar in both the calculations. The energy minimum is well localized in the vicinity of the (1/4, 1/4, 0) point located at the center of the Γ → M path. The direct gap also is situated in the vicinity of this point in both calculations. In β-TeO3 (see Figure 5c and 5d), the top of valence band also is not clearly determined: there are several BZ points with almost equal electronic energy. At the same time, the bottom of the conduction band is well localized in Γ point. The direct gap also is situated in the Γ point. The difference between direct

χ (1) =

1 (1) (χ + χyy(1) + χzz(1) ) 3 xx

are presented in Table 3. One can see that the χ(1) values calculated by different methods are markedly scattered: from 3.34 up to 5.32 for αTeO2 and from 2.46 up to 4.11 for β-TeO3. It is, however, D

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Figure 5. Band structures of α-TeO2 (a, b) and β-TeO3 (c, d) calculated within various approximations: LDA+U (a, c) and GWA (b, d).

and estimate the components of χ(n) tensors. It should be recalled that 7 in eq 1 implies only the electronic polarization; i.e., the atoms remain fixed at their equilibrium positions. The Berry phase approach imposes some limitations on the possible field magnitude. When the electric field is chosen to be too large, the electric enthalpy functional will lose its minima.53 A safe calculation condition demands that the magnitude of |,| must be several times smaller than

Table 3. Linear Dielectric Susceptibility χ(1) and Band Gap Eg (in eV) Computed by Different ab Initio Methods in Comparison with Available Experimental Dataa α-TeO2 χ

Eg exptl LDA LDA+U GWA

3.75 2.82 3.26 3.68

β-TeO3 (1)

51

52

4.05 5.32 4.42 (4.25) 3.34

Eg

χ(1)

(1) χ(1) TeO2/χTeO3

3.25 1.50 2.27 2.74

4.11 3.42 (3.34) 2.46

0.77 0.77 0.74

,max =

The χ values given in parentheses were obtained within the finitefield method (see next section). a

(1)

eaN

(2)

where Eg is the band gap; e is the electron charge; and a and N are the unit cell dimension and the number of k-points in the direction of the , vector, respectively. In our case the average values are Eg ∼ 3.3 eV, a = 5 Å, and N = 4. This leads to the estimation ,max ≈ 0.003 a.u. Accordingly, the variation of |,| was limited to the value of 0.0012 au. In order to obtain maximum accuracy with minimal computational cost, the calculation of 7(,) was performed by sequentially increasing |,| values. The wave function calculated at the previous step was used as the initial approximation for the subsequent SCF calculations. The strength of the electric field was increased stepwise by 0.0001 au, and the calculations were stoppped when the difference between χ(3) values computed in two subsequent steps was less than 0.01 au. The procedure being applied to α-TeO2 with the field directed in x-direction allowed estimating χ(3) xxxx = 298.08 ± 0.18 au. Similar calculations with the field directed along the z-

noticeable that variation of the calculated χ(1) value is in line with variation of the calculated Eg value. An alternative approach to estimate the dielectric response is related with the direct calculation of the polarization induced by an external electric field. This approach is based on the Berry phase method42 and allows studying dependence of the polarization vector 7 on the strength of external field , . Such calculations are more time-consuming in comparison with the DFPT method. However, this is the only reliable way to calculate the third-order dielectric susceptibility. Steadily rising the field strength , one progressively accumulates the 7i(,i) points which characterize the 7(,) dependency. Using the cumulated points one can apply the interpolation formula 7 = χ (1) , + χ (2) , ·, + χ (3) , ·, ·, + ...,

Eg

(1) E

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Figure 6. Partial density of states (PDOS) for α-TeO2 (a) and β-TeO3 (b) projected on the atomic orbitals according to the LDA+U calculations. Positive values are Te atom PDOS, and negative ones are inverted O atom PDOS. Black, red, and green lines are projection on s, p, and d orbitals accordingly. Filled areas denote correlated states of these compounds. Fermi energy is shifted to zero for both structures.

being transformed in SI units are 119.59 and 57.65 (in 10−22 m2/V2) for α-TeO2 and β-TeO3, respectively. The χ(1) values obtained within the finite-field calculations are reported in Table 3 in parentheses. It is noticeable that they well agree with the values calculated by the DFTP scheme. No direct measurements of χ(3) tensor components were published up to now. The only reference value found in the literature concerns the third-order susceptibility of a pure αTeO2 glass, reported by Kim et al.2 In that work, the authors extracted from third harmonic generation measurements the n2 value of a TeO2 glass and compared it with that of fused silica used as a reference sample. The ratio χ(3)(α-TeO2):χ(3)(SiO2) was found equal to ∼50. Employing the ratio and considering a reference n2 value55 of 2.73 × 10−16 cm2/W for the fused silica (i.e., χ(3) value of ∼2.04 × 10−22 m2/V2), one then derives a χ(3) value for α-TeO2 roughly equal to 102 × 10−22 m2/V2. Therefore, one can see that our LDA+U calculations provide the χ(3) value in a good agreement with the experimental value.

axis gave χ(3) zzzz = 159.30 ± 0.13 au. Analogous calculations for β(3) TeO3 gave χ(3) xxxx = 84.98 ± 0.18 au and χzzzz = 194.1 ± 0.53 au. (3) Transformation of the χ values to the SI units can be done by using the relation54 χ (3) (SI) = (4π )3 ε02

ao4 e2

χ (3) (au)

= 0.475 × 10−22 m 2/V2χ (3) (au)

(3)

where a0 − Bohr radius, e - elementary charge and ε0 - vacuum permittivity constant. Both the crystal structures are uniaxial, and thus, the third-order nonlinear dielectric susceptibility (3) obeys the relation χ(3) xxxx = χyyyy. The calculated orientation averaged values χ (3) =

1 (3) (3) (3) (χ + χyyyy + χzzzz ) 3 xxxx F

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Moreover, it looks like the conduction band of β-TeO3 above 8 eV resembles the conduction band of α-TeO2 above 4 eV, and the two lowest branches of β-TeO3 have no counterparts in the α-TeO2 phase. What is the origin of these “additional” branches with quasi-parabolic dispersion? Let us look at the partial DOS projected on the atomic orbitals (see Figure 6). One can see that the bottom of the conduction band in α-TeO2 primarily consists of the 5p(Te) states. The same states contribute to the conduction band of β-TeO3, however not at its bottom but at much higher energies (>6 eV) (see black arrow in Figure 6). At the same time, a large spectral feature which is well detected in the DOS of β-TeO3 (see Figure 6b) has no analogue in the DOS of α-TeO2. The contribution to the peak marked by an asterisk originates from the two lowest branches in the conduction band of β-TeO3. Analysis of the partial DOS evidences that these branches consist primarily of the 5s(Te) contributions. It is noticeable that the 5s(Te) electrons contribute markedly to the states near the top of the valence band in α-TeO2 and are almost absent in the valence band of β-TeO3. It looks as if these electrons do not participate in the bonding states of the β-TeO3 phase. The finding correlates with a structural distinction between the two compounds. As Figure 2 shows, the main difference between the two structures is the presence in α-TeO2 (and absence in β-TeO3) of the specific electron stateslone pairs localized on the Te atoms. These are two electrons which do not leave the “parent” atom for the Te−O valence bond formation. Accordingly, the valence state of Te atoms in αTeO2 is referred to as Te(IV). It is remarkable that these lone pairs consist primarily of the 5s(Te) orbitals.56 Hence, the 5s(Te) orbitals are occupied and belong to the valence band states in the α-TeO2 crystal. In the β-TeO3 crystal, these electrons leave the “parent” Te atom for forming two additional Te−O bonds. Accordingly, the valence state of Te atoms in β-TeO3 is referred to as Te(VI). The electron states corresponding to the Te−O bonds in TeO6 octahedra do not include the 5s(Te) contributions (owing to directional character of those bonds). Hence, the 5s(Te) orbitals remain unoccupied and migrate to the conduction band. However, the orbital energy of the 5s(Te) electrons is close to that of the 2p(O) states. Therefore, the unoccupied 5s(Te) orbitals in the β-TeO3 are situated not far from the top of the valence band and give rise to the additional (as compared with the α-TeO2) unoccupied electron states with energy close to the top of the valence band. This consideration explains the relation Eg(α-TeO2) > Eg(β-TeO3). At the same time the 5s(Te) electron states cannot give any significant contributions to the decomposition (eq 1). The top of the valence band primarily consists of the 2p(O) states which correspond to the electron lone pairs localized on the oxygen atoms. Thus, the probability of transitions (eq 6) between the top of the valence band and the bottom of the conduction band (mainly the 5s(Te) electron states) is very low because of the nonoverlapping character of these states. Hence, the main contributions to polarizability of β-TeO3 give the conduction band states with much higher energy (filled areas in Figure 6b). This reasoning can explain the relation χ(1)(α-TeO2) > χ(1)(βTeO3). The electronic dielectric function ε(ω) describes interaction of a crystal with the electromagnetic radiation of frequency ω in the visible and UV regions. The imaginary part of the ε(ω)

The previous GGA12 and B3LYP9 calculations gave the values more than two times larger. Thus, we can consider the obtained result as a quite satisfactory one. Recall that the main goal of the study is not to reproduce the known experimental data for α-TeO2 but to predict the unknown dielectric characteristics of β-TeO3. Our calculations show that the χ(3) value of β-TeO3 is about two times lower than that of α-TeO2. This result strongly disagrees with the results of ref 12 which predicted a 10-fold increase of χ(3) for βTeO3 as compared with paratellurite.

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DISCUSSION The computational results represented in Table 3 obey the two relations Eg (α ‐TeO2 ) > Eg (β ‐TeO3) and χ (1) (α ‐TeO2 ) > χ (1) (β ‐TeO3)

(4)

This is a quite challenging result because the general trend reported in numerous publications is just the opposite, i.e., the narrower the band gap the larger is the dielectric susceptibility. The trend is also confirmed by analysis of a large body of experimental data (see, e.g., ref 51), and it is in line with the first-order perturbation theory42 which provides us with the following expression for the dielectric susceptibility χ(1) χ (1) =

e2 Vc

M

∞

∑∑ ∑ k

m=1 n=M+1

|Pnm(k)|2 εnm(k)

(5)

where k is wave vector; n and m numerate the electron states in the conduction and valence bands, respectively; M is the number of occupied bands; εnm(k) = εn(k) − εm(k) is the energy difference between the states, and Pnm(k) = ⟨Ψ kn|r |Ψ mk ⟩

(6)

is the matrix elements of the coordinate operator. Terms with the minimum value of the denominator in eq 5 give the main contribution to the sum, and since min εnm(k) = Eg, it follows 1 that χ (1) ∼ E . Therefore, the first-order perturbation theory g

predicts that Eg (A) > Eg (B) ⇔ χ (1) (A) < χ (1) (B)

(7)

It is worth discussing the reason why the case under study is an exception of the general rule (eq 7). To understand the origin of the exception a thorough analysis of the electronic band structures of the two crystals is needed. One can see that the dispersion branches in the upper part of the valence band are almost flat in both compounds, whereas the bottom parts of the conduction bands differ drastically. In the conduction band of α-TeO2 the dispersion is weak. and the branches are flat. There are 12 branches with energy below 8 eV (i.e., three electronic states per one TeO2 unit). In the conduction band of β-TeO3 the dispersion is strong, and the branches have parabolic shape. There are only 2 branches with the energy below 8 eV (i.e., one electronic state per one TeO3 unit). In view of these differences, one can conclude that the sum (eq 5) for the α-TeO2 crystal contains three times more terms with small denominators (