Piezoelectricity

The piezoelectric stress tensor is the change of the polarization \(P = d/\Omega\) (dipole moment \(d\) of the cell per volume) with strain,

\[ e_{iJ} = \frac{\partial P_i}{\partial \varepsilon_J} \]

in C/m\(^2\), for Cartesian direction \(i\) and Voigt strain component \(J\). The strain coefficients follow with the relaxed-ion compliance \(S = C^{-1}\),

\[ d = e\, S \]

in pC/N, and the Born effective charges are the dipole change per atomic displacement,

\[ Z^*_{n, \alpha\beta} = \frac{\partial d_\alpha}{\partial u_{n\beta}} \]

in units of the elementary charge. Charge neutrality requires \(\sum_n Z^*_n = 0\) (acoustic sum rule); minelastic reports the sum as a convergence check.

Proper response. A Berry-phase polarization is defined modulo a quantum. minelastic writes the dipole of each cell in lattice coordinates, \(d = \sum_a \phi_a a_a\), brings the \(\phi_a\) of all cells of a pattern to one branch (\(\phi \to \phi - q\, \mathrm{round}((\phi - \phi_{\mathrm{prev}})/q)\), in order of increasing strain), and differentiates

\[ e_{iJ} = \frac{1}{\Omega_0} \sum_a a^0_{a,i}\, \frac{\partial \phi_a}{\partial \varepsilon_J}, \]

with the lattice vectors \(a^0_a\) and the volume \(\Omega_0\) of the reference cell (Vanderbilt, J. Phys. Chem. Solids 61, 147 (2000)). Unlike the derivative of \(P\) itself (the improper tensor), this proper tensor does not depend on the branch of the polarization.

Clamped ions. The dipoles of the strained cells give the clamped-ion tensor \(e^0\) with the same finite-difference formula as the stresses. The polarization is a polar vector, \(P(R\varepsilon R^T) = R\, P(\varepsilon)\) for every point-group rotation \(R\), so the responses of the computed patterns determine the whole tensor by least squares over their symmetry orbits, as for the internal-strain tensor.

Relaxed ions. The displaced atoms give \(Z^*\) from their dipoles. With the relaxation per unit strain \(u = \Phi^+ \Lambda\) (Relaxed ions), the polarization \(P = e^0 \varepsilon + Z^* u/\Omega\) gives the total (relaxed-ion) tensor

\[ e = e^0 + \frac{1}{\Omega}\, Z^* \Phi^+ \Lambda . \]

With \(Z^*\) in units of \(e\), \(\Phi^+\Lambda\) in Å and \(\Omega\) in Å\(^3\), the second term is in \(e\)/Å\(^2\) (1 \(e\)/Å\(^2\) = 16.02 C/m\(^2\)).

The dipoles can come from a Berry-phase calculation (read from VASP with LCALCPOL), from any function of the atoms (calculate(..., dipole=f)), or be passed to ElasticSetup.compute(..., polarization=...). Instead, \(Z^*\) and \(e^0\) can be taken from a DFPT run of the reference structure.