Collapse of electrons to a donor cluster in SrTiO$_3$

TL;DR

This paper investigates how electrons collapse onto large charged donor clusters in SrTiO$_3$, driven by its nonlinear dielectric response, leading to charge renormalization and potential experimental tests with surface charge patterns.

Contribution

It reveals a novel electron collapse mechanism in SrTiO$_3$ due to nonlinear dielectric effects, distinct from relativistic vacuum pair production, and provides quantitative thresholds for charge renormalization.

Findings

Electron collapse occurs when cluster charge exceeds Z_c ≈ R/a.

Net charge Z_n grows with Z until Z > Z^*, then saturates.

Predictions can be tested with surface charge patterns.

Abstract

It is known that a nucleus with charge $Ze$ where $Z>170$ creates electron-positron pairs from the vacuum. These electrons collapse onto the nucleus resulting in a net charge $Z_n<Z$ while the positrons are emitted. This effect is due to the relativistic dispersion law. The same reason leads to the collapse of electrons to the charged impurity with a large charge number $Z$ in narrow-band gap semiconductors and Weyl semimetals as well as graphene. In this paper, a similar effect of electron collapse and charge renormalization is found for donor clusters in SrTiO$_3$ (STO), but with a very different origin. At low temperatures, STO has an enormously large dielectric constant. Because of this, the nonlinear dielectric response becomes dominant when the electric field is not too small. We show that this leads to the collapse of surrounding electrons into a charged spherical donor cluster with radius $R$ when its total charge number $Z$ exceeds a critical value $Z_c\simeq R/a$ where $a$ is the lattice constant. Using the Thomas-Fermi approach, we find that the net charge $Z_ne$ grows with $Z$ until $Z$ exceeds another value $Z^*\simeq(R/a)^{9/7}$. After this point, $Z_n$ remains $\sim Z^*$. We extend our results to the case of long cylindrical clusters. Our predictions can be tested by creating discs and stripes of charge on the STO surface.