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

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.