Static Born charges and quantum capacitance in metals and doped semiconductors

TL;DR

This paper introduces a method to calculate static Born charges in metals and doped semiconductors, linking them to quantum capacitance and charge response, with implications for understanding lattice dynamics in these materials.

Contribution

The authors develop a direct calculation method for static Born charges at the zone center using the $2n+1$ theorem and relate it to quantum capacitance through an exact sum rule.

Findings

Method successfully applied to simple metals and doped semiconductors.

Static Born charges depend on electrostatic reference choices.

Quantum capacitance is crucial in the charge response analysis.

Abstract

Born dynamical charges ($\textbf{Z}^\text{dyn}$) play a key role in the lattice dynamics of most crystals, including both insulators and metals in the nonadiabatic ("clean") regime. Very recently, the so-called static Born charges, $\textbf{Z}^\text{stat}$, were introduced [G. Marchese, et al., Nat. Phys. $\mathbf{20}$, 88 (2024)] as a means to modeling the long-wavelength behavior of polar phonons in overdamped ("dirty") metals. Here we present a method to calculate $\textbf{Z}^\text{stat}$ directly at the zone center, by applying the $2n+1$ theorem to the long-wavelength expansion of the charge response to a phonon. Furthermore, we relate $\textbf{Z}^\text{stat}$ to the charge response to a uniform strain perturbation via an exact sum rule, where the quantum capacitance of the material plays a crucial role. We showcase our findings via extensive numerical tests on simple metals aluminum and copper, polar metal LiOsO$_3$, and doped semiconductor SrTiO$_3$. Based on our results, we critically discuss the physical significance of $\textbf{Z}^\text{stat}$ in light of their dependence on the choice of the electrostatic reference, and on the length scale that is assumed in the definition of the macroscopic potentials.