Static Born charges and quantum capacitance in metals and doped semiconductors

Born dynamical charges (${\mathbf{Z}}^{\mathrm{dyn}}$) 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, ${\mathbf{Z}}^{\text{stat}}$, were introduced [G. Marchese et al., Nat. Phys. 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 ${\mathbf{Z}}^{\text{stat}}$ 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 ${\mathbf{Z}}^{\text{stat}}$ 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 ${\mathrm{LiOsO}}_{3}$, and doped semiconductor ${\mathrm{SrTiO}}_{3}$. Based on our results, we critically discuss the physical significance of ${\mathbf{Z}}^{\text{stat}}$ 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.