Article pubs.acs.org/JPCC
Density Functional Theory Predictions of the Nonlinear Optical Properties in α-Quartz-type Germanium Dioxide P. Hermet, G. Fraysse, A. Lignie, P. Armand,* and Ph. Papet Institut Charles Gerhardt Montpellier, UMR5253 CNRS-UM2-ENSCM-UM1, C2M, UMII, CC1504, Place E. Bataillon, 34095 Montpellier Cédex 5, France ABSTRACT: We calculated the lattice dynamics, the second-order nonlinear susceptibility, and the electro-optic response of the germanium dioxide in its αquartz-type form (α-GeO2) from first-principles calculations based on density functional theory. No theoretical or experimental investigations of these nonlinear optical properties have been previously reported in the literature. The calculation of the infrared and Raman spectra of α-GeO2 allowed us to assign its experimental phonon modes, contributing to the clarification of a long-standing debate in the literature. The second-order nonlinear susceptibility and the electro-optic coefficients of α-GeO2 are predicted to be significantly higher than those reported for α-quartz. Thus, α-GeO2 should be a promissing candidate for nonlinear optical applications when compared to α-quartz.
I. INTRODUCTION The development of materials for nonlinear optical (NLO) devices is nowadays an intensive domain of research. Its main objectives are to find materials showing large nonlinearities and satisfying at the same time all the technological requirements for applications such as wide transparency range, fast response, and high damage threshold. The new development of techniques for the fabrication and growth of artificial materials is also especially important to reach these objectives. α-Quartz (α-SiO2) has been extensively used as a reference material for second harmonic generation and sum-frequency studies for two purposes: (i) it may be used to determine the phase of the sum-frequency field by generating an additional sum-frequency signal that interferes with the one from the sample in a homodyne detection scheme,1−3 and (ii) the signal generated from α-quartz may be used to quantify the unknown nonlinear response from new samples.4 The α-quartz-type structure of germanium dioxide (α-GeO2) is similar to that of the well-studied α-quartz. Nevertheless, its NLO properties have not been presently studied because α-GeO2 is metastable with respect to its rutile phase under ambient conditions.5 To overcome this problem, a new crystal growth method has been recently successfully applied,6 highlighting the possible interest of α-GeO2 for NLO applications. Indeed, unlike hydrothermal techniques, the slow cooling method in a selected inorganic flux does not require aqueous media, allowing the growth of highquality water-free α-GeO2 single crystals.6 The absence of hydroxyl groups in the crystal is of prime importance, as they will catalyze its return in the thermodynamically stable rutile structure upon heating.5 Furthermore, in contrast with α-SiO2, which presents a well-known α−β quartz transition near 846 K, the flux-grown α-GeO2 material is stable up to its melting temperature near 1389 K.6 © 2012 American Chemical Society
Theoretical investigations of NLO phenomena were restricted for many years to semiempirical approaches such as shell models7 or bond−charge models.8,10 In the past decade, significant theoretical advances have been reported concerning density functional theory (DFT) calculations on periodic systems in an external electric field11 and opened the way to direct predictions of various optical properties. In this paper, we use DFT, taking advantage of a recent implementation based on the nonlinear response formalism and the 2n + 1 theorem,12 to investigate the second-order nonlinear susceptibility and the electro-optic (EO) coefficients of α-GeO2. Infrared and Raman spectra have been also calculated by taking into consideration the frequency position and the intensity of the modes. In particular, we found that our calculated Raman spectrum of αGeO2 is in excellent agreement with the experiment, allowing clarification of the long-standing debate reported in the literature on the assignment of its Raman lines. This paper is organized as follows. First, we describe the basic ingredients involved to the calculation of the EO response and the used computational parameters. Then, we report in section IV the zone-center phonon modes of α-GeO2. Its infrared and Raman spectra are also discussed in this section. In section V, we investigate the contributions of the TO infrared modes to its static dielectric tensor. Finally, sections VI and VII respectively report the second-order nonlinear optical susceptibility and the EO coefficients of α-GeO2. For these two NLO responses, a comparison with α-quartz is given. Received: January 26, 2012 Revised: March 6, 2012 Published: March 15, 2012 8692
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Table 1. Raman (Ra) and Infrared (IR) TO Frequencies (cm−1) of α-GeO2 with Their Corresponding Assignments observed Ra refs
36a b
and
IR and
40
ref 41b
this workb,c
ref 31a
calculated ref 32a
this work
123 166
E E
123 166
E A1
123 166
E E
123 166
E A1
123 166
E A1
124
122
E
126 169
E A1
212 212
E36 A137
213
E
212 212
E39 A140
213
E
212
E
212
209
E
215
E
249
A2
247 263 328 328
E A1 A136 E37
250 264 330
A1 E E
248 264 329
E A1 E
265 332
E E
247 248 265 332
A2 E A1 E
345
A2
345
A2
443
a
refs
ref 38a
37
39a a
A1
328
423c 442 447 494
E
263 330 330
A1 A139 E40
385
E
A1
444
A1
E
492
E
442
A1
445
A1
525
E
517
E
332
491
554 584
256 349
ref 38 E E A2 A1
A1 E
416
A2
125 169
E A1
218
E
247 251 269 337
A2 E A1 E
351
A2
449
A1
449
A1
443
A1
E A2
486 512 549 565
E E A2 E
507 528
E A2
579
E
846 862 868
E A2 A1
948
E
E
518
ref 28 131 152 182 208
540
A2
519 534
585
E
586
E
E A2 A1 E
585
E
584 759 777
E
583
E
588
E
586
E
858
E
859
E
860
E
858
E
859
E
880
A1
882
A1
881
A1
883
A1
882
A1
886
882
A2
858 877 882
959
E
951
E
960
E
960
E
961
E
958
962
E
963
823 849
A2 A1
932 969
E E
Powder. bSingle-crystal. cWater-free.
II. THEORETICAL FORMULATION
rijelγ = −
The EO coefficients (Pockels effect), rijγ, describe the change of the optical dielectric tensor, εij, in a static (or low-frequency) electric field, , , through the following expression:13 −1
Δ(ε )ij =
(2)
k =γ
where ni and nj are the principal refractive indices. The origin of the ionic contribution to the EO response is the relaxation of the atomic positions due to the applied quasistatic electric field and the variations of εij induced by these displacements. It is given by14
∑ rijγ , γ γ
8π (2) χ ni 2nj 2 ijk
(1)
In this paper, we follow the convention of using Greek (respectively Roman) indexes to label the Cartesian directions of static (respectively optical) fields. We write all vector and tensor components in the system of Cartesian coordinates defined by the principal axes of the crystal under zero field. Furthermore, we neglect any modification of the unit cell shape due to the inverse piezoelectric effect. Thus, within the Born and Oppenheimer approximation, the EO tensor is only the sum of an electronic (rel) contribution and an ionic (rion) contribution. EO coefficients can be measured for frequencies of the electric field high enough to eliminate the relaxations of the crystal lattice but low enough to avoid excitations of optical phonon modes (usually above ∼100 MHz).14 The electronic contribution is due to an interaction of the electric field with the valence electrons when considering the ions artificially clamped at their equilibrium positions. It is proportional to the second-order nonlinear optical susceptibility (χ(2)):14
rijion γ
4π =− Ω 0 ni 2nj 2
∑ m
αijmpγm ωm 2
(3)
The Raman susceptibility tensor, αmij , is defined as15 αijm =
Ω0
∑ πijκ,γum(κγ) κ, γ
(4)
where the sum runs over all atoms κ and space directions γ, Ω0 is the unit cell volume, ωm and um(κγ) are respectively the frequency and the κγ-component of the mth phonon eigendisplacement vector, and πκij,γ = ((∂χ(1) ij )/(∂τκγ))|0 is a third-rank tensor describing the changes of the linear dielectric susceptibility induced by an individual atomic displacement (τ) of the κth atom in the direction γ. The mode polarity, 8693
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mode are uniform translational modes. The E-modes are both infrared and Raman active, whereas the A2- and A1-modes are respectively infrared and Raman active. The E- and A2-modes are respectively polarized perpendicularly and along the αGeO2 trigonal axis. Tables 1 and 2 summarize the TO and LO phonon frequencies of α-GeO2 reported in the literature from Raman
∑ Z*γβ(κ) um(κβ) κ, β
(5)
where Z*, the Born effective charge tensors, are directly linked to the mode oscillator strength: Smαβ = pmα pmβ . Thus, the calculation of the EO response requires several ingredients, such as the vibrational frequencies and eigenmodes obtained from the diagonalization of the dynamical matrix, the mode polarities, the Raman susceptibilities, and the secondorder nonlinear susceptibilities. All these quantities must be in relatively good agreement with the experimental ones to have a reliable prediction of the EO response. In the next sections, all these quantities will be computed, discussed, and compared to the experimental ones when available.
Table 2. Calculated and Experimental E(LO) Frequencies (in cm−1) of α-GeO2 observeda ref 36 ref 37 123 166 212 247
III. COMPUTATIONAL DETAILS α-GeO2 (respectively α-SiO2) crystallizes in the trigonal P3121 (left handed) or P3221 (right handed) space group with the lattice constants a = 4.985 Å and c = 5.647 Å (respectively a = 4.921 Å and c = 5.416 Å) at room temperature. In α-GeO2, oxygen atoms occupy the general 6c(0.397, 0.302, 0.243) Wyckoff positions, whereas germanium atoms occupy the 3a(u, 0, 1/3) positions where u = 0.451.16 In α-SiO2, oxygen and germanium atoms occupy 6c(0.413, 0.271, 0.217) and 3a(u, 0, 1/3) Wyckoff positions, where u = 0.467, respectively.17 First-principles calculations on α-GeO2 and α-SiO2 were performed within the DFT framework, as implemented in the ABINIT package.18 The exchange-correlation energy functional was evaluated within the local density approximation (LDA) using the Perdew−Wang parametrization19 of Ceperley−Alder homogeneous electron gas data.20 The all-electron potentials were replaced by norm-conserving pseudopotentials generated according to the Troullier−Martins scheme21 thanks to a package developed at the Fritz-Haber Institute (Berlin).22 Germanium (4s2, 4p2) and oxygen (2s2, 2p4) electrons were considered as valence states in the construction of the pseudopotentials. The electronic wave functions were expanded in plane-waves up to a kinetic energy cutoff of 70 Ha. Integrals over the Brillouin zone were approximated by sums over a 8 × 8 × 8 mesh of special k-points according to the Monkhorst− Pack scheme.23 Relaxations of the lattice parameters and the atomic positions were performed using the Broyden−Fletcher− Goldfarb−Shanno algorithm24 until the maximum residual forces on the atoms and stress were less than 6 × 10−6 Ha/ Bohr and 1 × 10−4 GPa, respectively. Born effective charges, dielectric tensors, and dynamical matrix (yielding the phonon frequencies and eigenvectors) were obtained within a variational approach to density functional perturbation theory.25 The Raman susceptibilities, the second-order nonlinear optical susceptibilities, and the EO coefficients were obtained within a nonlinear response formalism taking advantage of the 2n + 1 theorem.12 The infrared absorption and Raman spectra were respectively calculated as described in refs 26 and 27. In the following, the orthogonal reference system (x,y,z) is chosen such as z is aligned along the α-GeO2 trigonal axis and x is aligned along the crystallographic a axis. Due to the crystal symmetry, the different tensors are diagonal with two independent components labeled xx = yy = ⊥ and zz = ∥.
ref 38 ref 39
124 168
123 166 212
this work
ref 38
123
124
126
125
213
215
217
221
270 340
263
263
265
367
372
377
527 599
531 600
520 593
952 973
950 976
936 957
ref 40 ref 41 122 164
249 263
a
calculated this work
364
372
516 595
516 595
517 595
516 593
372 456 514 595
950 971
950 971
960
949 973
949 970
525 600 870 975
Experimental frequencies are observed from Raman spectroscopy.
and infrared spectroscopies. Raman spectroscopies have been intensively used during the last 20 years on α-GeO2 to investigate an eventual α−β phase transition similar to α-quartz. As consequence, the assignment of the A2 infrared modes have been poorly explored (see Table 1). Nevertheless, in spite of the numerous investigations performed on the α-GeO2 Raman lines, there still remains a lot of contradictions and ambiguities about their assignments. Experimentally, these controversies are a consequence of the difficulty to obtain large high-quality αGeO2 single-crystals. Theoretically, they are due to the use of empirical force field methods.28,29 However, when firstprinciples-based methods are considered, the assignment of the α-GeO2 experimental Raman lines is usually performed from their calculated positions without any consideration of their intensities.30,38 Figure 1 compares the α-GeO2 calculated Raman spectrum and the experimental one recorded at 300 K on a single-crystal using a backscattering configuration with the incident and scattered light propagating perpendicularly to the Y naturally occurying face. We calculated not only the position of the Raman lines but also their intensities for both the TO and LO modes. The frequency positions of the calculated Raman lines are predicted with a precision better than 3% in the whole frequency range. The experimental relative intensities of the Raman lines are also very well reproduced by the calculation. Thus, our calculated Raman spectrum of α-GeO2 is sufficiently accurate to be compared to the experimental one, allowing a reliable assignment of its experimental Raman lines. Our revisited assignments, given in Table 1 (TO modes) and in Table 2 (LO modes), undoubtedly clarify the long-standing debate reported in the literature. α-GeO2 experimental infrared investigations are only limited to absorption and reflectivity spectra from polycrystalline samples.31−34 In the absence of polarization measurements, the
IV. ZONE-CENTER OPTICAL PHONON MODES The zone-center phonon modes of α-GeO2 can be classified according to the irreducible representations of the D3 point group into 4A1 ⊕ 9E ⊕ 5A2, where one A2-mode and one E8694
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experimental corresponding ones reported by Madon et al.31 Frequencies and oscillator strengths of the infrared TO modes are respectively reported in Tables 1 and 3. Frequencies of the Table 3. Calculated Phonon Contributions of the α-GeO2 Infrared TO Modes to Its Static Dielectric Constant (ε0) as Well as Mode Oscillator Strengths (S̃, ×10−5 au) and Frequencies (ωm, cm−1) E-modes ωm
S⊥
126 215 248 332 519 586 858 963
0.11 1.45 5.98 12.87 6.64 6.57 61.46 1.04
∞
ε TO1 TO2 TO3 TO4 TO5 TO6 TO7 TO8 sum (phonon) total
Figure 1. Calculated and experimental Raman spectra of a α-GeO2 single-crystal for a 473 nm excitation line. A constant line width fixed at 3 cm−1 has been used to display the calculated spectrum. The experimental spectrum was recorded at room temperature in backscattering configuration perpendicularly to the Y face and using a Horiba Jobin Yvon LabRam Aramis spectrometer equipped with an Olympus microscope and a CCD cooled by a thermoelectric Peltier device.
assignment of the infrared modes cannot be considered completely reliable. Thus, we calculated the α-GeO2 infrared spectra to have a new insight into the assignment of the A2(TO) modes and to predict the A2(LO) modes that still presently remain experimentally undetected. The calculated infrared absorption spectrum of a polycrystalline α-GeO2 is displayed in Figure 2. The calculated band positions and their relative intensities are found to be in good agreement with the
A2-modes ε0⊥ 2.950 0.058 0.250 0.770 0.926 0.196 0.152 0.664 0.009 3.027 5.977
ωm
S∥
ε0∥
247 345 534 877
11.94 1.01 30.03 62.68
3.036 1.560 0.067 0.838 0.648
3.114 6.150
E(LO) modes are reported in Table 2 with the experimental ones. Our four A2(LO) frequencies are predicted at 282, 348, 589, and 981 cm−1. The α-GeO2 LO−TO splitting strength can be estimated from its infrared reflectivity spectra. They have been computed at normal incidence according to the methodology from ref 35. Figure 2 displays the calculated αGeO2 infrared reflectivity spectra of the E and A2 modes. The reflectivity related to the A2-modes is observed when the electric field is parallel to the trigonal axis of the crystal [i.e., perpendicular to the (100) or (010) surfaces]. Similarly, the Emodes can be observed when the electric field is perpendicular to the trigonal axis [i.e., perpendicular to (001) surface]. Since our approach neglects the damping of the phonon modes, the calculated reflectivities saturate to 1. This figure shows that the α-GeO2 LO−TO splitting is significant (more than 40 cm−1) for the two E-modes around 330 and 860 cm−1 and for the two last A2-modes at the highest frequencies. Our predictions of the E(LO) and A2(LO) frequencies, along with our estimate of the LO−TO splitting, are in excellent agreement with the firstprinciples calculations recently reported by Kaindl et al.38 The eigenvectors of the polar TO modes do not necessarily correspond to those of their corresponding LO modes due to the long-range Coulomb interaction. The possible mixing between a LO mode and the TO modes can therefore be calculated from the overlap matrix: ⟨u mLO|M |uTO n ⟩ =
∑ umLO(κγ)M κunTO(κγ) (6)
γ, κ
where Mκ is the mass of the κth-atom, and and are the eigendisplacement vectors of the mth LO and nth TO modes, respectively. This overlap matrix between the A2(TO) and A2(LO) modes shows that their mixing is, however, negligible, suggesting that the eigenvector between a A2(TO) mode and its corresponding A2(LO) mode is very close. In the case of Emodes, this mixing is also negligible except for the two last E(LO) modes at high frequencies centered at 950 and 976 cm−1. These latter arise from the mixing between the E(TO7) and E(TO8) modes respectively centered at 858 and 963 cm−1. uLO m
Figure 2. Calculated infrared absorption (up) and reflectivity (down) spectra of α-GeO2. 8695
uTO n
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(TO3, TO4, TO7) contribute to the ε0⊥ ionic term. Each of these three E-modes contributes between 22 and 30%. Since the electronic and the ionic terms have similar values both perpendicularly and along the trigonal axis, ε0 is close to being isotropic: ε0⊥ = 5.977 and ε0∥ = 6.150 at 0 K. As consequence, the conductivity of α-GeO2 should be quite isotropic without a dominant electronic or ionic response.
V. ELECTRONIC AND STATIC DIELECTRIC TENSOR The electronic dielectric permittivity tensor, ε∞, describes the response of the electron gas to a homogeneous electric field if the ions are taken as fixed at their equilibrium positions. It has been computed without a scissor correction.46 Its computed ∞ values are ε∞ ⊥ = 2.950 and ε∥ = 3.036, corresponding to a refractive index of [1.72, 1.72, 1.74]. This index is in excellent agreement with the experimental value of [1.70, 1.70, 1.73] measured by Balitsky et al.42 and with the previous firstprinciples calculations.43−45 α-GeO2 is therefore a positive uniaxial trigonal crystal with a quite isotropic electronic response to a homogeneous electric field. Similarly to αGeO2, α-quartz is a positive uniaxial crystal. However, the reported values of its ε∞ components are around 15% smaller than those predicted for α-GeO2 (see Table 4).
VI. SECOND-ORDER NONLINEAR SUSCEPTIBILITY The second-order nonlinear optical susceptibility, χ(2), is a third-rank tensor related to the electronic response of the system and depends on the frequencies of the optical electric fields.47 However, in the present context of the 2n + 1 theorem applied within the LDA to (static) DFT, we neglect the dispersion of χ(2) computing the electronic response at zero frequency. Within these conditions, the χ(2)-tensor is related to a third-order derivative of a field-dependent energy functional, - = E − Ω0, 7 , where E and 7 are respectively the total energy in zero field and the macroscopic polarization.12 As a consequence, the χ(2) tensor satisfies Kleinman’s symmetry condition48 and its indices are therefore symmetric under a permutation. As usual in nonlinear optics, we report the d tensor defined as d = (1/2)χ(2). In α-quartz-type structures, the d tensor has two independent elements. The form of this tensor is given by
Table 4. Calculated Second-Order Nonlinear Optical Susceptibility Coefficient (in pm/V) and Electronic Dielectric Constants of α-GeO2 and α-SiO2a d11 ε∞ ⊥ ε∞ ∥
α-GeO2
α-SiO2
−1.456 2.950 3.036
−0.419 (0.35b) 2.487 (2.35650) 2.518 (2.38350)
a Experimental values are in parentheses when available. bAbsolute value from ref 49.
⎛ d11 −d11 · d14 · · ⎞ ⎜ ⎟ d ij = ⎜ · · · · −d14 −d11⎟ ⎜ ⎟ ⎝ · · · · · · ⎠
To include the response of the crystal lattice to the electric field, one can use a model that assimilates the solid to a system of undamped harmonic oscillator. In this context, the static dielectric constant, ε̃0 = ε̃∞ + ε̃ion, is decomposed into an electronic term and an ionic term where εion αα =
4π Ω0
∑ m
where the indices i and j denote the Cartesian components in Voigt notations. Kleinman’s symmetry rule allows one to reduce this tensor to only one independent element since in this case we have d14 = d36 = 0. The LDA computed value of d11 at 0 K is equal to −1.456 pm/V when a full relaxation of the αGeO2 unit cell is performed. This value increases at d11 = −1.563 pm/V when the experimental lattice parameters are imposed in the calculation. Unfortunately, there are no available experimental or theoretical data reported in the literature on the second-harmonic generation of α-GeO2 to estimate the reliability of this prediction. Nevertheless, the comparison of the predicted d11 values between α-GeO2 and α-SiO2 shows that α-GeO2 should have a NLO efficiency around 3 times higher than α-SiO2 (see Table 4). Thus, the predicted d11 value associated with the high thermal stability of the flux-grown α-
Sαα(m) ωm 2
(8)
(7)
The calculated phonon contributions of the α-GeO2 infrared TO modes to ε0 are reported in Table 3. From this decomposition, we observe that, except for the A2(TO2) mode, the A2(TO) modes significantly contribute to ε0∥. However, they do not equally contribute due to their different frequencies. The main contribution (50%) to the ε0∥ ionic term comes from the A2(TO1) mode whereas the A2(TO3) and A2(TO4) modes respectively represent a contribution of 27% and 21%. By contrast, only three of the eight E(TO) modes
Table 5. Calculated Electronic and Ionic Contributions of the E-Modes to the EO Tensor (in pm/V) of α-GeO2 and α-SiO2 (Frequencies, ωm, are given in cm−1) α-GeO2 electronic ionic
total
TO1 TO2 TO3 TO4 TO5 TO6 TO7 TO8 sum of ionic
α-SiO2
ωm
r11
r41
126 215 248 332 519 586 858 963 0.185
0.670 0.228 −0.013 0.275 −0.011 −0.222 0.056 −0.118 −0.010 0.101 0.855
0 −0.053 −0.154 −0.066 0.242 0.158 0.092 −0.125 0.007
8696
0.101
ωm 129 259 382 443 690 792 1062 1148
r11
r41
0.271 0.035 −0.016 0.141 −0.037 −0.024 0.019 −0.041 −0.017 0.060 0.331
0 −0.020 −0.064 −0.034 0.182 0.016 0.052 −0.091 0.009 0.050 0.050
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calculated its infrared modes and we investigated the contributions of the TO modes to its static dielectric tensor. Finally, we predicted the magnitude of the second-order nonlinear susceptibility and the EO responses of α-GeO2. We showed that these responses are expected to be significantly higher than those reported for α-quartz. By contrast to semiempirical models, the computational method used in this study is predictive and does not use experimental parameters. Thus, our calculations predict that flux-grown α-quartz-type GeO2 single crystals should be promising and interesting candidates for NLO applications when compared to α-SiO2.
quartz-type GeO2 single-crystal make this material a promising candidate for NLO applications when compared to α-SiO2.
VII. ELECTRO-OPTIC RESPONSE At this stage, we have computed all the quantities required for the calculation of the α-GeO2 EO response. These quantities have been compared to the available experimental ones, and they were found to be in excellent agreement, suggesting a reliable prediction of the α-GeO2 EO response. In α-quartztype structures, the EO tensor has two independent coefficients given by · ⎛ r11 ⎜− r · ⎜ 11 · ⎜ · rij = ⎜ r · 41 ⎜ ⎜ · −r41 ⎜ ⎝ · −r11
·⎞ ·⎟ ⎟ ·⎟ ·⎟ ⎟ ·⎟ ⎟ ·⎠
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
(9)
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where the indices i and j denote the Cartesian components in Voigt notations. These two coefficients are only linked to the E modes. Electronic and ionic contributions to the EO response are reported in Table 5. Only the r11 coefficient has an electronic contribution due to the imposition of Kleinman symmetry (d14 = d25 = 0) in the calculation of the second-order nonlinear susceptibilities. The r11 electronic contribution is mainly dominated by the E(TO1), E(TO3), E(TO5), and E(TO7) modes. The E(TO1) and E(TO3) modes have a positive ionic contribution, whereas the E(TO5) and E(TO7) modes have a negative one, yielding a positive global ionic contribution of the r11 coefficient. Taking into account that the r11 electronic contribution has a positive sign, the E(TO1) and E(TO3) modes lead to an increase of the EO response, whereas the E(TO5) and E(TO7) modes lead to a decrease. The r41 ionic contribution is dominated by the first seven E(TO) modes. Its ionic contribution is also predicted to be positive and its magnitude is around 55% smaller than that predicted for r11. In consequence, we expect a significant difference between the r11 and r41 total values, leading the αGeO2 EO response mainly governed by r11. When the calculations are performed at the experimental lattice parameters, the r11 and r41 values are not significantly altered: r11 = 0.868 pm/V and r41 = 0.067 pm/V. To the best of our knowledge, there is no theoretical or experimental value of the α-GeO2 EO coefficients in the literature. However, EO measurements are usually performed at room temperature while our predicted values are at 0 K. Thus, the influence of the temperature should be considered to compare our EO predictions to the experimental ones in the future. Both the electronic and the ionic contributions to the α-GeO2 EO response are around twice higher than those reported for αSiO2, suggesting from a theoretical point of view that α-GeO2 should be more efficient for NLO applications than α-SiO2.
VIII. CONCLUSIONS In this paper, we have first calculated the Raman spectrum of αGeO2 using density functional theory. Its excellent agreement with the experimental one was especially relevant to clarify the long-standing debate in the literature on the assignment of its Raman lines. The calculation of the intensity of the Raman lines has been performed for the first time in α-GeO2 despite the several calculations reported in the literature. Then, we 8697
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