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/T1)orb and (1/T1)Q, which are due to the orbital and quadrupole interactions, respectively, and show that (1/T1)Q gives a negligible contribution compared with (1/T1)orb. It is found that (1/T1)orb exhibits anomalous dependences on temperature T and chemical potential μ. When μ is inside the band gap, (1/T1)orb∼T3log(2T/ω0) for temperatures above the band gap, where ω0 is the nuclear Larmor frequency. When μ lies in the conduction or valence bands, (1/T1)orb∝TkF2log(2vFkF/ω0) for low temperatures, where kF and vF are the Fermi momentum and Fermi velocity, respectively. The Knight shift Korb due to the orbital interaction also shows anomalous dependences on T and μ. It is shown that Korb is negative and its magnitude significantly increases with decreasing temperature when μ is located in the band gap. Because the anomalous dependences in Korb is caused by the interband particle-hole excitations across the small band gap while (1/T1)orb is governed by the intraband excitations, the Korringa relation does not hold in the Dirac electron systems.