Nonadiabatic Born Effective Charges in Metals and the Drude Weight

In insulators, Born effective charges describe the electrical polarization induced by the displacement of individual atomic sublattices. Such a physical property is at first sight irrelevant for metals and doped semiconductors, where the macroscopic polarization is ill defined. Here we show that, in clean conductors, going beyond the adiabatic approximation results in nonadiabatic Born effective charges that are well defined in the low-frequency limit. In addition, we find that the sublattice sum of the nonadiabatic Born effective charges does not vanish as it does in the insulating case, but instead is proportional to the Drude weight. We demonstrate these formal results with density functional perturbation theory calculations of Al and electron-doped ${\mathrm{SnS}}_2$ and ${\mathrm{SrTiO}}_3$.

Figure 1

Demonstration of nonadiabatic Born effective charge (NABEC) sum rule for Al versus $k$ points in the Brillouin zone of the conventional 4-atom cell. Blue dot-dashed curve is the NABEC, the black solid (pink dotted) curve is minus the cell volume multiplied by the modified (normal) Drude weight, and red dashed (green dotted) curve is the sum of the two. Inset is a magnification around 0 $e$ for the largest $k$ mesh.

Figure 2

(a) Nonadiabatic Born effective charges versus doping above the conduction-band minimum (CBM) in the in-plane $x$ direction and out-of-plane $z$ direction. Solid (dashed) lines are for the excess ionic charge associated with the Sn (S) sublattice. (b) Demonstration of the NABEC sum rule, where the sublattice sum of the BECs equals the Drude weight. Blue shaded region is the DOS of the conduction band calculated from the Wannier interpolation.

Figure 3

Nonadiabatic Born effective charges of the sublattices in ${\mathrm{SrTiO}}_3$ versus doping above the conduction-band minimum on the (a) Sr, (b) Ti, or (c) O sublattice. Panel (d) demonstrates the NABEC sum rule, where the sublattice sum of the BECs cancels with the Drude weight. Blue shaded region is the density of states of the Wannierized Ti $t_{2g}$ manifold.