arXiv2017
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 $μ$. When $μ$ is inside the band gap, $(1/T_1)_{\rm orb} \sim T ^3 \log (2 T/ω_0)$ for temperatures above the band gap, where $ω_0$ is the nuclear Larmor frequency. When $μ$ lies in the conduction or valence bands, $(1/T_1)_{\rm orb} \propto T k_{\rm F}^2 \log (2 |v_{\rm F}| k_{\rm F}/ω_0)$ 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 $μ$. It is shown that $K_{\rm orb}$ is negative and its magnitude significantly increases with decreasing temperature when $μ$ 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.