Piezoelectric Response in α-Quartz-Type GeO2 - American Chemical

Dec 16, 2015 - contrast to α-quartz, α-GeO2 has also a sizable e14 value that reaches 0.169 C/m2. ... Among the α-quartz-type TIVO2 (T = Si, Ge), g...
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Piezoelectric Response in α‑Quartz-Type GeO2 P. Hermet* Institut Charles Gerhardt Montpellier, UMR-5253, CNRS, Université de Montpellier, ENSCM, Place E. Bataillon, 34095 Montpellier Cédex 5, France ABSTRACT: Density functional perturbation theory is used to study the mechanism of the piezoelectric response in α-GeO2 and to estimate its magnitude. To date, its piezoelectric-stress elements have never been calculated using this level of theory nor measured on a high-quality water-free single crystal. Our calculations predict that the value of e11 piezoelectric-stress is about 50% higher than in α-quartz. This enhancement is due to larger tetrahedra distortions and/or a lower frequency of the dominant polar E-mode. In contrast to α-quartz, α-GeO2 has also a sizable e14 value that reaches 0.169 C/m2.

I. INTRODUCTION Nowadays, piezoelectric materials are integral to numerous applications and devices and form the basis for a multibillion dollar worldwide market.1 Examples are found in high voltage and power applications, actuators, sensors, motors, atomic force microscopes, energy harvesting devices, and medical applications. These technologies all rely on the conversion of voltage to mechanical deformation or vice versa. α-Quartz (α-SiO2) is the second most abundant mineral and a commonly employed material in piezoelectric devices. Nevertheless, the existence of an α-quartz to β-quartz phase transition2 near 846 K restricts its temperature range for technological applications since the β-phase is weakly piezoelectric. Several investigations have been devoted to find other α-quartz homeotypes as alternative.3−7 The study of the αquartz-type TIVO2 (T = Si, Ge) and MIIIXVO4 (M = Al, Ga, Fe; X = P, As) families showed that the piezoelectric properties are linked to the structural distortion with respect to the β-quartz phase.7−10 In particular, the piezoelectric efficiency and the thermal stability increase as a function of structural distortion.7 Among the α-quartz-type TIVO2 (T = Si, Ge), germanium dioxide (α-GeO2) has the most distorted structure.11,12 Its piezoelectric efficiency is therefore estimated to be significantly higher than in α-SiO2 although it has never been measured on a high-quality water-free single crystal. In addition, α-GeO2 does not present an α−β phase transition until its melting near 1380 K.13 In this article, we use the density functional perturbation theory to understand the mechanism of the piezoelectric effect in α-GeO2. We also report its piezoelectric tensors and the figure-of-merit of its electromechanical couplings. For these purposes, the piezoelectric-stress tensor is separated into a clamped-ion contribution and an internal-strain contribution. The first one represents the effect of the strain on the electronic structure whereas the second contribution quantifies the © 2015 American Chemical Society

additional relaxation of the relative atomic coordinates that would be induced by the strain. In order to obtain further insight into the major piezoelectric contribution, we also decomposed the piezoelectric internal-strain into contributions from individual atoms, as well as contributions from phonon normal modes. Throughout this article, the contributions above cited will be compared to that of α-quartz to answer to these two questions: (i) is the same mechanism at the origin of the piezoelectric effect between these two homeotypes and (ii) why is the piezoelectric response higher in α-GeO2?

II. COMPUTATIONAL DETAILS The piezoelectric tensors are calculated within a variational approach to density functional perturbation theory as implemented in the ABINIT package.14 The exchangecorrelation energy functional is evaluated using the generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof.15 The all-electron potentials are replaced by norm-conserving pseudopotentials generated according to the Troullier-Martins scheme. Si(3s2, 3p2), Ge(4s2, 4p2), and O(2s2, 2p4)-electrons are considered as valence states. The electronic wave functions are expanded in plane-waves up to a kinetic energy cutoff of 70 Ha and integrals over the Brillouin zone are approximated by sums over a 10 × 10 × 10 mesh of special k-points according to the Monkhorst−Pack scheme.16 These k-point sampling and kinetic energy cutoff give converged results. Lattice parameters and atomic positions were fully relaxed using a Broyden-Fletcher-Goldfarb-Shanno algorithm until the maximum stresses and residual forces were less than 7 × 10−4 GPa and 6 × 10−6 Ha/Bohr, respectively. In Received: November 5, 2015 Revised: December 15, 2015 Published: December 16, 2015 126

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intertetrahedral angle, T-O-T, is in excellent agreement with the experiment (0.2%) in α-GeO2 and deviates by 2.8% in αSiO2. Such a deviation seems to be related to the use of the GGA (see Table 1 of Demuth et al.18). This intertetrahedral angle is also linked to an empirical thumb of rule to classify the α-quartz-type TIVO2 (T = Si, Ge) and MIIIXVO4 (M = Al, Ga, Fe; X = P, As) families according to their piezoelectric efficiency:7 the smaller this angle, the higher the piezoelectric efficiency. B. Piezoelectric-stress tensor. For the first-principles study of the piezoelectricity, the quantity that is generally computed is the zero-field derivative of the polarization with ⎛ ∂P ⎞ respect to strain, eα , μν = ⎜ ∂η α ⎟ , usually reduced to the Voigt ⎝ μν ⎠, = 0 form, eαj with j = (μ,ν) = 1, 2, ..., 6. In the case of α-quartz-type homeotypes, the piezoelectric-stress has only two independent elements, namely e11 and e14. Their magnitudes are given in Table 2 for α-GeO2 and α-SiO2. Although the piezoelectric-

the following, the crystals are oriented according to the IEEE 1978 standard on piezoelectricity.

III. RESULTS AND DISCUSSION A. Structure. TIVO2 (T = Si, Ge) 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 for α-GeO2 at 294 K,11 and a = 4.916 Å and c = 5.408 Å for α-SiO2 at 298 K.17 This structure consists of a continuous random network of rigid TO4 tetrahedral units connected through flexible bridging T-OT bonds (Figure 1). The local short-range order is usually characterized7,11 by the intratetrahedral O-T-O angle and the intertetrahedral T-O-T bridging angle.

Table 2. Calculated Independent Piezoelectric-Stress Components (in C/m2) in TIVO2 (T = Si, Ge)a α-GeO2 Clamped-ion Internal-strain Total (present) Total (DFT10,24) Total (expt.19) Total (expt.20) Total (expt.21)

Figure 1. Structure of α-GeO2 and α-SiO2 highligthing the tetrahedra with the atom labels. T1 is the tetrahedron centered on the Ge1 (or Si1) atom formed by the O1, O6, O2, and O5 oxygens. T2 is the tetrahedron centered on the Ge2 (or Si2) atom formed by the O2, O3, O4, and O5 oxygens. T3 is the tetrahedron centered on the Ge3 (or Si3) atom formed by the O1, O3, O4, and O6 oxygens.

a (Å) c (Å) c/a T2-O3 (Å) T2-O2 (Å) O3-T2-O5 (deg) O4-T2-O2 (deg) T-O-T (deg)

calc. (0 K)

expt.17 (298 K)

5.077 5.752 1.133 1.7621 1.7673 106.77 111.17 130.53

4.985 5.647 1.133 1.7341 1.7405 107.88 110.48 130.22

5.022 5.508 1.097 1.6210 1.6236 108.24 108.95 147.70

4.916 5.408 1.100 1.6074 1.6132 108.75 109.33 143.62

e11

e14

−0.101 0.249 0.148 0.179 0.171 0.172 0.149

−0.068 0.077 0.009 −0.060 −0.041 −0.039 −0.057

stress elements in α-SiO2 were intensively studied,19−23 those in α-GeO2 have never been measured on a high-quality waterfree single crystal, only estimated. Our calculated values for αSiO2 are in acceptable agreement with the experimental ones19−21 and previous DFT calculations using hybrid functionals.10,24 For α-GeO2, our calculations give: e11 = 0.220 C/m2 and e14 = 0.169 C/m2. These values are consistent with the calculation of El-Kelany et al.10 using the modern theory of polarization and the finite difference method although they found a negative sign for e14. This sign problem could be related to a different crystal orientation. The calculated values of e11 are in excellent agreement with the estimated value, eest 11 = 0.24 C/m2, obtained from Brillouin spectroscopy.25 Thus, we expect that e11 is about 50% higher in α-GeO2. Furthermore, e14 is not weak in α-GeO2 and its value is even similar to that predicted for e11. Piezoelectric constants can be written in different ways which are more suitable for applications and we can transform from one set to the other using thermodynamic transformations.26 The two most useful expressions are the piezoelectric-strain tensor, dαj = ∑keαkSkj where S̃ is the compliance tensor, and the

α-SiO2

expt.11 (294 K)

e14 −0.133 0.302 0.169 −0.145

Experimental measurements are also reported when available. The DFT calculations of refs 24 and 10 use hybrid functionals.

Table 1. Calculated and Experimental Selected Structural Parameters in α-Quartz-Type TIVO2 (T = Si, Ge) α-GeO2

e11 −0.186 0.407 0.220 0.241

a

Table 1 lists selected structural parameters as the lattice constants including the c/a ratio, the T-O bond lengths, and

calc. (0 K)

α-SiO2

the intra- and intertetrahedral angles. Our relaxed lattice parameters at 0 K slightly overestimate the experimental ones measured around room temperature by about 2% as usually observed with GGA exchange−correlation functionals. The calculated c/a ratios are however in excellent agreement with the experiments. There are two distinct Ge−O (respectively Si−O) bond lengths in GeO4 (respectively SiO4) tetrahedra, as two distinct intratetrahedral angles. The discrepancy experiment-calculation for these parameters is also within 2%. The deviations of the O−Ge−O bond angles from the ideal tetrahedral angle (109.5°) are larger than those for O−Si−O, resulting in distorted GeO4 tetrahedra. The calculated

figure-of-merit of the electromechanical coupling, kαj =

|eαj|

(σ ) Cjj εαα

where C̃ is the elastic tensor and ε̃(σ) is the free-stress dielectric constant. No scissors correction was used to compute ε̃(σ). Using the values reported in Table 3, we get for α-GeO2: d11 = 5.627 pC/N, d14 = 4.646 pC/N, k11 = 12%, and k14 = 12%. In the literature, the piezoelectric-strain elements have been only 127

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The Journal of Physical Chemistry C Table 3. Calculated Elastic, Compliance, and Dielectric Constants in TIVO2 (T = Si, Ge)a α-GeO2 present

calc.10

C11 C12 C13 C14 C33 C44

58.45 20.02 23.38 0.91 97.23 34.20

54.73 18.08 20.75 3.15 94.68 33.69

S11 S12 S13 S14 S33 S44

20.46 −5.59 −3.58 −0.69 12.00 29.28

21.71 −6.02 −3.44 −2.60 12.07 30.20

ε∞ 11 ε∞ 33 ε(σ) 11 ε(σ) 33

3.04 3.14 6.64 6.46

2.48 2.60

α-SiO2 exp.29

exp.25

Elastic Constants (GPa) 64 68−69 22 25.1 32 2 ∼0 118 118.8 38 38.6 Compliance Constants (TPa−1) 19.25 −4.68 −3.95 −1.29 10.62 27.17 Dielectric Constants 2.89b 2.99b

present

calc.10

exp.28

82.01 −3.95 7.40 18.38 89.80 52.12

93.20 14.22 20.38 14.97 120.79 61.34

86.79 6.79 12.01 18.12 105.79 58.21

13.36 −0.33 −1.07 −4.83 11.32 22.56

11.95 −2.00 −1.68 −3.40 8.85 17.99

12.78 −1.77 −1.25 −4.53 9.74 20.00

2.40 2.42 4.60 4.76

2.13 2.16

2.36c 2.38c 4.514d

a Experimental values are given when available. The DFT calculation of El-Kelany et al.10 uses the PBE0 hybrid functional. bReference 31. cReference 30. dReference 7.

reported on a hydrothermally grown α-GeO2 containing OHgroups.27 Thus, the associated experimental values, dexp 11 = 4.04 pC/N and dexp 14 = 3.82 pC/N, are not very reliable as OHgroups significantly alter the piezoelectric response. In addition, since the electronic dielectric constant, and therefore ε̃(σ), is usually overestimated in DFT-based calculations using GGA functionals, our calculated values of k could be underestimated with respect to experiments. For α-SiO2, our calculated values, d11 = 2.063 pC/N and k11 = 8%, are in good agreement with the exp reported experimental values,7 dexp 11 = 2.31 pC/N and k11 = 8%. The excellent agreement obtained for k11 is a consequence of the small error obtained in the computation of ε(σ) 11 . Note that the PBE0 hybrid functional underestimates ε∞ with respect to the GGA, but the experimental elastic properties28−31 are well described by the two classes of functionals (see Table 3). At this stage, we decompose the piezoelectric-stress tensor into a clamped-ion and an internal-strain contributions such as26 eαj = eclαj + estr αj . The clamped-ion (or homogeneous-strain) contribution is evaluated at vanishing internal strain (i.e., without allowing the additional relaxation of the relative atomic coordinates that would be induced by the strain) and represents the influence of electron delocalization on the piezoelectric properties. The internal-strain contribution measures the additional relaxation of the relative atomic coordinates induced by the applied strain and it is simply calculated as the difference between the total piezoelectric-stress and the clamped-ion contribution. Table 2 reports the results of this decomposition. As expected, the absolute values of the clamped-ion contributions are the highest in α-GeO2 because of a better polarizability of Ge-atoms against Si-atoms. For both compounds, e11 is strongly dominated by the internal-strain contribution while the clamped-ion contribution decreases the magnitude of e11 by its negative sign. For e14, the same trend is observed in α-GeO2. However, in the case of α-SiO2, we observe that the internal-strain and the clamped-ion contributions are compensated, yielding to a weak e14 (∼0.009 C/m2).

In the following part of this article, only the sizable values of the piezoelectric-stress tensor will be analyzed. C. Atom Contributions to the Piezoelectric InternalStrain. The piezoelectric internal-strain contribution is related to the effective charges, Z*, and to the displacement-response internal-strain, du/dη, that describes the first-order displacements resulting from a first-order strain:26 eαstrj =

1 Ω0

* (κ ) ∑ Zαβ β ,κ

duβ(κ ) dηj

(1)

where the sum runs over all the direction β and atoms κ in the primitive unit cell and Ω0 is the unit cell volume. It follows that a sizable piezoelectric internal-strain contribution arises either from (i) a dielectric effect, namely a large value of the effective charges Z*, or (ii) an elastic effect, namely a large response of the internal coordinates to a macroscopic strain. The substitution in eq 1 of the α-GeO2 effective charges by those str of α-SiO2 changes estr 11 (or e14) by only 2%. This means that the piezoelectric response in α-GeO2 results mainly by a change of atomic positions within the structure. Table 4 reports the atom contributions to the piezoelectric internal-strain in TIVO2 (T = Si, Ge). The absolute values of estr 11 are quite different between both compounds but they share the same sign. estr 11 is dominated by O atoms: ∼80% for both compounds. In contrast, the Ge-atoms dominate estr 14 by 89% in α-GeO2. The contribution of the tetrahedra to piezoelectric internal-strain can be also estimated as str eαstrj (T) i = eαj (Gei /Sii) +

1 2

∑ O ∈ Ti

eαstrj (O) (2)

where Ti labels the tetrahedron centered on the Gei or Sii atoms and the one-half factor takes into account that one O atom is linked to two tetrahedra (see Figure 1 for the labels of the tetrahedra). Results of this decomposition are shown in Table 5. In α-GeO2, the T2 and T3 tetrahedra dominate estr 11 by 91% 128

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displacement-response internal-strain vectors, du/dη1, when a strain is applied along the x-direction. This configuration is linked to estr 11. For the T1 tetrahedra in α-GeO2, the atoms move in the same direction and sense (Figure 2a). In α-SiO2, these atomic displacements are slightly oblique to the strain direction but the sense of these displacements is the same for all atoms (Figure 2b). The piezoelectric contribution of the oxygens is therefore counterbalanced in both compounds by the contribution of germanium/silicon atoms as they carry opposite charges, and as a consequence T1 tetrahedra have a weak contribution to estr 11. For T2 and T3 in α-GeO2, we observe a rotation of the tetrahedra against each other without a significative displacement of the germanium or silicon atoms. Thus, the contribution of the oxygens to estr 11 is no longer counterbanlanced by the cations, and T2 and T3 dominate estr 11 with equal contributions. However, since this rotation is more frustrated in α-GeO2 (O1-atoms do not move), the charges, which are initially balanced in both compounds, are now much more unbalanced in the case of α-GeO2, and a higher polarization appears. The contribution of the T2 and T3 tetrahedra to piezoelectric internal-strain is therefore higher in α-GeO2 than in α-SiO2, and the estr 11 value is the highest in αGeO2. The situation is somewhat different for estr 14 in α-GeO2. Indeed, the T2 and T3 tetrahedra bring a positive contribution (0.489 C/m2) to estr 14, whereas the T1 tetrahedra tend at present to decrease the magnitude of estr 14 because of their negative piezoelectric contributions (−0.187 C/m2). Nevertheless, the T2 and T3 tetrahedra are those contributing the most to the str internal-strain as estr 14 has a positive sign. The magnitude of e14 is also sizable because all the tetrahedra have a frustrated rotation when a shear strain is applied in the yz plane (Figure 2c). These explanations to understand the reason why the piezoelectric efficiency is the highest in α-GeO2 are rather qualitative and a quantitative description requires the phonon contributions to the piezoelectric internal-strain to be calculated. D. Phonon Contributions to the Piezoelectric Internal-Strain. The optical zone-center phonon modes of TIVO2 (T = Si, Ge) can be classified according to the irreducible representations of the D3 point group into: 4A1⊕8E⊕4A2. The A2 and A1 representations are infrared and Raman active, respectively. The doubly degenerate E representation is both infrared and Raman active. Phonon modes belonging to the A2 representation are polarized along the trigonal axis whereas those belonging to the E representation are polarized perpendicularly to this axis. Thus, only the modes belonging str to the E irreducible representation contribute to estr 11 and e14. The phonon contributions to the piezoelectric internal-strain are estimated as

Table 4. Calculated Atom Contributions to the Piezoelectric Internal-Strain (in C/m2) in TIVO2 (T = Si, Ge)a α-GeO2 Ge1/Si1 Ge2/Si2 Ge3/Si3 O1 O2 O3 O4 O5 O6 total Ge/Si total O a

α-SiO2

estr 11

estr 14

estr 11

0.416 −0.166 −0.166 −0.025 −0.354 0.541 0.541 −0.025 −0.354 0.084 0.324

−0.304 0.286 0.286 0.029 0.088 −0.100 −0.100 0.029 0.088 0.268 0.034

0.637 −0.300 −0.300 −0.205 −0.415 0.725 0.725 −0.205 −0.415 0.037 0.210

See Figure 1 for the atom labels given in this table.

Table 5. Calculated Contribution of Each Tetrahedron to the Piezoelectric Internal-Strain (in C/m2) in TIVO2 (T = Si, Ge)a α-GeO2

α-SiO2

tetrahedron

estr 11

estr 14

estr 11

T1 T2 T3

0.037 (−0.379) 0.1855 (0.3515) 0.1855 (0.3515)

−0.187 (0.117) 0.2445 (−0.0415) 0.2445 (−0.0415)

0.017 (−0.620) 0.115 (0.415) 0.115 (0.415)

a

See Figure 1 for the definition of the tetrahedra. Contribution of the oxygen atoms are given between parentheses.

while the T1 tetrahedra contribute by 9% only. In α-SiO2, the same trend occurs but the magnitude of piezoelectric internalstrain associated with each tetrahedron is smaller, yielding to a smaller estr 11. To understand the reason why the T2 and T3 tetrahedra contribute the most to estr 11, we displayed in Figure 2 the

eαstrj

the yz plane. We use the convention

∑ m

pαm g jm ωm2

(3)

where the sum runs over the transverse optical modes and the phonon-strain coupling

Figure 2. Calculated displacement-response internal-strain vectors, du/dη, when a strain is applied along the x-direction in (a) α-GeO2 and (b) α-SiO2, and (c) when a shear strain is applied in α-GeO2 on du (κ ) that ∑ακ dαη =0. −1

1 = Ω0

g jm =

Eigendisplace-

∑ Λβj (κ )Uβ(κ , m) βκ

ment vectors of the mode centered (d) at 774 cm in α-GeO2 and (e) at 1032 cm−1 in α-SiO2. Eigendisplacements are normalized according to ⟨U|M|U⟩ = 1, where M is mass matrix. For all pictures, the arrowheads are proportional to the magnitude of the response.

(4)

is linked to the force-response internal-strain vector (Λ), and to the frequency (ωm) and the βκth eigendisplacement vector (Uβ(κ,m)) of the mth normal mode obtained from the 129

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The Journal of Physical Chemistry C diagonalization of the dynamical matrix. The mode polarity, pmα , is given by pαm =

shown in their study. Considering these elements, the agreement experiment-theory is however satisfactory. A deeper analysis of the infrared modes of α-GeO2 can be found in ref 35. The different terms of eq 3 are listed in Tables 6 and 7 for αGeO2 and α-SiO2, respectively. In both compounds, estr 11 is

* (κ )Uβ(κ , m) ∑ Zαβ (5)

βκ

estr αj

Eq 3 shows that is sizable if we have a low frequency polar mode close to zero associated with a large mode effective charge and a strong coupling with strain. The reliability of the mode polarities can be estimated by a comparison between the calculated infrared spectra and the experimental ones (see eq 6 of ref 32). These calculated spectra are reported in Figure 3 and

Table 7. Calculated Contributions of the E-Modes to the estr 11 Piezoelectric Internal-Strain in α-SiO2a

a

ωm

pm1

gm1

estr 11

estr 11

cm−1

×103 e Bohr

×103

×103 e/Bohr2

C/m2

113 247 377 417 658 756 1032 1134

0.24 −1.84 −9.80 20.34 −3.20 9.49 38.98 3.40

0.43 −0.04 0.88 0.52 −2.47 0.33 1.99 6.00

0.28 0.05 −2.40 1.86 0.71 0.30 2.92 0.54

0.01 0.00 −0.14 0.10 0.04 0.02 0.17 0.04

See eqs 4 and 5 for the definitions of gmj and pmα , respectively.

dominated by three modes. They are centered at 237, 468, and 774 cm−1 in α-GeO2, and at 377, 417, and 1032 cm−1 in αSiO2. The first mode decreases the value of estr 11 by its negative sign which is related to a negative polarity whereas the two other modes have a positive contribution. The mode centered at 774 cm−1 in α-GeO2 is the one who gives the most important contribution (0.33 C/m2) to estr 11. In αSiO2, the corresponding mode is at 1032 cm−1 and it contributes by 0.17 C/m2. Their large frequency difference (Δω = 258 cm−1) is at the origin of the higher estr 11 value in αGeO2. Indeed, if the frequency at 774 cm−1 in α-GeO2 is replaced in eq 3 by 1032 cm−1, then we get a value of 0.26 C/ str m2 for estr 11 which is very close to that obtained in α-SiO2 (e11 = 0.25 C/m2, Table 2). The eigendisplacement vectors of these two modes, displayed in the bottom of Figure 2, show that the atomic motions of oxygens are quite similar and the main difference lies in silicon atoms which move more than the germanium atoms. This is consistent with the tetrahedra description discussed in the previous section where the piezoelectric contribution of the oxygen atoms is counterbalanced by silicon atoms in α-SiO2 while this is not the case in α-GeO2. For estr 14 in α-GeO2, five of the eight optical E-modes are mainly involved. Their frequencies are at 194, 237, 302, 535, and 774 cm−1.

Figure 3. Calculated infrared absorbance spectra of α-SiO2 and αGeO2 using a Lorentzian line shape and a constant line width fixed at 2 cm−1. Corresponding experimental spectra can be found in refs 33 and 34, respectively.

they are dominated by a doublet in the 1200−900 cm−1 (respectively 900−700 cm−1) range and a multiband structure in the 800−600 and 500−300 cm−1 (respectively 600−450 and 350−150 cm−1) for α-SiO2 (respectively α-GeO2). The doublet has the most intense bands in the both quartz isotypes. For αSiO2, the agreement between the experiment33 and the calculation is excellent both for the position of the bands and their relative intensities. Indeed, this compound has been intensively studied and the experimental infrared spectra reported in literature are very reliable. In contrast, the infrared lattice dynamics of α-GeO2 have been little studied and the only experimental spectrum has been reported by Madon et al.34 Their experimental spectrum is less resolved than in αSiO2 and we cannot exclude OH-contaminations in their sample as the high-frequency range (4000−2600 cm−1) is not

Table 6. Calculated Contributions of the E-Modes to the Piezoelectric Internal-Strain in α-GeO2a ωm cm

−1

113 194 237 302 468 535 774 878 a

pm1

gm1

×10 e Bohr

×10

−0.95 −4.90 −8.05 13.22 −8.14 8.47 32.54 −2.86

0.33 −0.21 0.44 0.22 −1.75 0.13 2.99 −3.13

3

gm4 3

estr 11

×10

×10 e/Bohr

−0.28 −0.51 −0.65 −0.79 0.60 −2.21 2.61 1.01

−1.35 1.22 −3.46 1.17 3.09 0.20 5.76 0.48

3

3

estr 11 2

C/m

estr 14 2

−0.08 0.07 −0.20 0.06 0.18 0.01 0.33 0.03

×10 e/Bohr 3

1.15 2.92 5.10 −4.26 −1.07 −3.47 5.03 −0.15

estr 14 2

C/m2 0.07 0.16 0.29 −0.25 −0.06 −0.20 0.29 −0.01

See eqs 4 and 5 for the definitions of gmj and pmα , respectively. 130

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The Journal of Physical Chemistry C

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IV. CONCLUSIONS The density functional perturbation theory is used to study the mechanism of the piezoelectric effect in α-GeO2 and to identify if this mechanism is fundamentally different from that of αSiO2. We predict that the independent elements of the piezoelectric-stress in α-GeO2 are e11 = 0.220 C/m2 and e14 = 0.169 C/m2. The magnitude of e11 is therefore expected to be about 50% higher in α-GeO2. In contrast to α-quartz, e14 is not weak in α-GeO2 and its magnitude is even similar to that predicted for e11. In both compounds, the piezoelectric internalstrain is the dominant contribution to the piezoelectric-stress response. The atom contributions to the piezoelectric internal-strain show that estr 11 is mainly associated with the tetrahedra centered on the Ge 2 /Si2 and Ge 3/Si 3 atoms. Considering this tetrahedron model, the piezoelectric efficiency is higher in αGeO2 due to a better polarization induced by a frustrated rotation of the GeO4 tetrahedra with strain. The phonon contributions to the piezoelectric internal-strain indicate that estr 11 is dominated for both compounds by three phonons modes. They are centered at 237, 468, and 774 cm−1 in α-GeO2 and at 377, 417, and 1032 cm−1 in α-SiO2. The frequency difference between the mode centered at 774 cm−1 in α-GeO2 and the mode centered at 1032 cm−1 in α-SiO2 explains why the value str of estr 11 in α-GeO2 is the highest. For e14 in α-GeO2, five of the eight optical E-modes are mainly involved. Their frequencies are at 194, 237, 302, 535, and 774 cm−1. Basically, the mechanism of the piezoelectric effect in αGeO2 is the same than in α-SiO2. However, the value of the piezoelectric e11 constant is higher in α-GeO2 due to larger tetrahedra distortions and/or to a lower frequency of the dominant polar E-mode at 774 cm−1.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jpcc.5b10843 J. Phys. Chem. C 2016, 120, 126−132

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DOI: 10.1021/acs.jpcc.5b10843 J. Phys. Chem. C 2016, 120, 126−132