Nuclear Magnetic Relaxation and Knight Shift Due to Orbital Interaction in Dirac Electron Systems

TL;DR

This paper investigates the nuclear magnetic relaxation rate and Knight shift in three-dimensional Dirac electron systems, revealing anomalous temperature and chemical potential dependences due to orbital interactions, and showing deviations from the Korringa relation.

Contribution

It provides rigorous calculations of relaxation rates and Knight shift in Dirac systems, highlighting the dominant orbital contribution and its unique anomalous behaviors.

Findings

Orbital relaxation rate $(1/T_1)_{orb}$ dominates over quadrupole contribution.

$(1/T_1)_{orb}$ exhibits $T^3 ext{log}$ dependence in the band gap.

Knight shift $K_{orb}$ becomes large and negative at low temperatures in the band gap.

Abstract

We study the nuclear magnetic relaxation rate and Knight shift in the presence of the orbital and quadrupole interactions for three-dimensional Dirac electron systems (e.g., bismuth-antimony alloys). By using recent results of the dynamic magnetic susceptibility and permittivity, we obtain rigorous results of the relaxation rates $(1/T_1)_{\rm orb}$ and $(1/T_1)_{\rm Q}$, which are due to the orbital and quadrupole interactions, respectively, and show that $(1/T_1)_{\rm Q}$ gives a negligible contribution compared with $(1/T_1)_{\rm orb}$. It is found that $(1/T_1)_{\rm orb}$ exhibits anomalous dependences on temperature $T$ and chemical potential $\mu$. When $\mu$ is inside the band gap, $(1/T_1)_{\rm orb} \sim T ^3 \log (2 T/\omega_0)$ for temperatures above the band gap, where $\omega_0$ is the nuclear Larmor frequency. When $\mu$ lies in the conduction or valence bands, $(1/T_1)_{\rm orb} \propto T k_{\rm F}^2 \log (2 |v_{\rm F}| k_{\rm F}/\omega_0)$ for low temperatures, where $k_{\rm F}$ and $v_{\rm F}$ are the Fermi momentum and Fermi velocity, respectively. The Knight shift $K_{\rm orb}$ due to the orbital interaction also shows anomalous dependences on $T$ and $\mu$. It is shown that $K_{\rm orb}$ is negative and its magnitude significantly increases with decreasing temperature when $\mu$ is located in the band gap. Because the anomalous dependences in $K_{\rm orb}$ is caused by the interband particle-hole excitations across the small band gap while $\left( 1/T_1 \right)_{\rm orb}$ is governed by the intraband excitations, the Korringa relation does not hold in the Dirac electron systems.