\[ c _ { i j k l } = \frac { 1 } { 2 v } \sum _ { p , p ^ { \prime } } A _ { i p , k p ^ { \prime } } ^ { ( 2 ) j l } . \eqno ( 1 2 ) \]

The technical details of the calculations are as follows. The calculations were done for cubic STO exploiting the Quantum ESPRESSO (QE) [14] ab initio package within the GGA PBE exchange-correlation functional with ultrasoft pseudopotentials [15]. We have used an automatically generated uniform 16x16x16 grid of k-points, the kinetic energy cutoff for wavefunctions was 80 Ry. The calculations of phonon spectrum are done using Density Functional Perturbation Theory (DFPT) [16] as implemented in the PHonon code. The energy threshold for self-consistency was chosen to be equal to \( 1 0 ^ { - 2 0 } \) Ry with the help of convergence tests for gamma point phonons.

To find the independent components of tensors \( \chi _ { i j } = \chi \delta _ { i j } \) , \( M _ { i j } = M \delta _ { i j } \) , \( f _ { i j k l } \), and \( c _ { i j k l } \) (\( \chi \), M, \( f _ { 1 1 } \), \( f _ { 1 2 } \), \( f _ { 4 4 } \), \( c _ { 1 1 } \), \( c _ { 1 2 } \), \( c _ { 4 4 } \)), we evaluated the wave-vector dependence of the dynamical matrix \( A _ { i p , i ^ { \prime } p ^ { \prime } } ( \vec { q } ) \) for the [100] and [110] directions of the reciprocal space. This provided us with the components of the \( A _ { i p , i ^ { \prime } p ^ { \prime } } ^ { ( 0 ) } \) and \( A _ { i p , i ^ { \prime } p ^ { \prime } } ^ { ( 2 ) j l } \) matrices needed to finalize the calculations using Eqs. (8)-(12).

The results of the calculations are presented in Tables I and II. A remarkable observation, which forms the main message of this Letter, is a very strong renormalization of the flex-ocoupling tensor by dynamic flexoelectricity: it is seen that the corresponding components of the tensors \( f _ { i j k l } \) and \( f _ { i j k l } ^ { \mathrm { t o t } } \) can drastically differ. This implies that the traditional neglect of dynamic flexoelectricity in the dynamic electromechanical simulations incorporating flex-oelectricity is inadmissable. Notably, the dynamic flexoelectricity can play an essential role in the frequency dependence of the electromechanical response close to the frequencies of mechanical resonances of the sample. The point is that such response is expected to be sensitive to the dynamic flexoelectricity only above the relevant resonance frequency [6].

The results of our calculations of the dynamical matrix provide us with an alternative way of evaluating some components of the material tensors of STO directly from the phonon dispersion curves. This will provide a cross-check of the above results. In addition, the method presented below will offer a way of evaluating these tensors from the experimental phonon spectra of the material. We suggest fitting the long-wavelength part of the low-energy spectrum of the crystal to that obtained from the continuum theory, incorporating dynamic flexoelectricity.

The suggested approach uses the spectra of transverse acoustic (TA) and soft-mode trans-

<table><tr><td>2
VM (×10−8 s2 )m</td><td>6 \( \pm \) 0.5</td></tr><tr><td>f11 (V)</td><td>1.11 \( ^ + \) 0.05</td></tr><tr><td>f12 (V)</td><td>−1.30 ± 0.05</td></tr><tr><td>f44 (V)</td><td>−0.28 ± 0.05</td></tr><tr><td>\( f _ { 1 1 } ^ { \mathrm { t o t } } \)(V)</td><td>−2.1 ± 0.1</td></tr><tr><td>\( f _ { 1 2 } ^ { \mathrm { t o t } } \)(V)</td><td>−2.5 ± 0.1</td></tr><tr><td>tot \( I A \)
(V) 
4</td><td>−1.5 ± 0.1</td></tr><tr><td>χ/0</td><td>2400</td></tr></table>

TABLE I. Calculated static (f), dynamic (M), and total (\( f ^ { \mathrm { t o t } } \)) flexocoupling coefficients for cubic \( \mathrm { S r T i O 3 } \) (\( a \) = 3.886 ˚A) under a pressure of 78 kBar using dynamical matrix. \( \chi / \epsilon _ { 0 } \) is the relative dielectric susceptibility.

<table><tr><td>(11N
10 2 )m</td><td>(a) Dyn. mat.</td><td>(b) Ph. dis.p</td><td>(c) Ex.p</td></tr><tr><td>c11</td><td>2.77 ± 0.05</td><td>3.1 \( \pm \)  0.1</td><td>3.16</td></tr><tr><td>c12</td><td>1.06 ± 0.05</td><td>0.9 \( \pm \) 0.1</td><td>1.03</td></tr><tr><td>c44</td><td>0.92 \( + \) 0.05</td><td>1.0 \( \pm \)  0.1</td><td>1.22</td></tr><tr><td>|M|c44
(V)ρ</td><td>1.2 \( \pm \) 0.1</td><td>1.1 \( \pm \) 0.4</td><td></td></tr><tr><td>|ftotV44 | ()</td><td>1.5 \( + \) 0.1</td><td>1.5 \( \pm \)  0.2</td><td>1.2-2.2</td></tr><tr><td>|ftottt− V11 fo12 | ()</td><td>0.2 \( \pm \) 0.1</td><td>\( < \) 0.5</td><td>1.2-1.4</td></tr></table>

TABLE II. Material parameters for cubic \( \mathrm { { S r T i O } _ { 3 } } \) (\( a \) = 3.886 ˚A) under a pressure of 78 kBar obtained using the dynamical matrix and from analysis of phonon dispersion curves. c is the stiffness tensor. \( f ^ { \mathrm { t o t } } \) is defined in Eq. (20). \( \rho \) is the density \( ( \rho = 5 1 7 4 \frac { \mathrm { k g } } { \mathrm { m } ^ { 3 } } ) \)) calculated by the mass of atoms in the unit cell divided by the cell volume. (a) Calculated from dynamical matrix. (b) Obtained from the simulated phonon dispersion curves. (c) Obtained from the experimental phonon dispersion curves [17–19].

verse optic (TO) branches for the high symmetry [100], [110], and [111] directions of wavevec-tor. In this case, the phonons are not accompanied by any wave of the electric field and the continuum theory equations, Eqs. (1) and (2) with \( E _ { i } = 0 \), yield the following dispersion equation for the frequency \( \omega \) of the transverse modes [6]: