Static Born charges and quantum capacitance in metals and doped semiconductors

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.