Quadrupole Susceptibility and Elastic Softening due to a Vacancy in Silicon Crystal

TL;DR

This study explores how a vacancy in silicon affects electronic states and elastic properties, revealing extended vacancy states that cause low-temperature elastic softening and enhanced multipole susceptibilities.

Contribution

It provides a detailed analysis of vacancy-induced electronic states and their impact on quadrupole susceptibility and elastic softening in silicon, supported by Green's function calculations.

Findings

Extended vacancy states influence quadrupole susceptibility.

Elastic softening observed at low temperatures matches experiments.

Octupole susceptibilities are significantly enhanced for specific modes.

Abstract

We investigate the electronic states around a single vacancy in silicon crystal by using the Green's function approach. The triply degenerate vacancy states within the band gap are found to be extended over a large distance $\sim20 {\rm \AA}$ from the vacancy site and contribute to the reciprocal temperature dependence of the quadrupole susceptibility resulting in the elastic softening at low temperture. The Curie constant of the quadrupole susceptibility for the trigonal mode ($O_{yz},O_{zx},O_{xy}$) is largely enhanced as compared to that for the tetragonal mode ($O_{2}^{0},O_{2}^{2}$). The obtained results are consistent with the recent ultrasonic experiments in silicon crystal down to 20 mK. We also calculate the dipole and octupole susceptibilities and find that the octupole susceptibilities are extremely enhannced for a specific mode.