Static Born charges and quantum capacitance in metals and doped semiconductors

Born dynamical charges (Zdyn) 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, Zstat, were introduced [G. Marchese, Nat. Phys. 20, 88 (2024)1745-247310.1038/s41567-023-02203-3] as a means to modeling the long-wavelength behavior of polar phonons in overdamped ("dirty") metals. Here we present a method to calculate Zstat 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 Zstat 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 LiOsO3, and doped semiconductor SrTiO3. Based on our results, we critically discuss the physical significance of Zstat 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.