Tight-binding theory of NMR shifts in topological insulatorsBi2Se3andBi2Te3

Motivated by recent nuclear magnetic resonance (NMR) experiments, we present a microscopic $s{p}³$ tight-binding model calculation of the NMR shifts in bulk ${\mathrm{Bi}}₂{\mathrm{Se}}₃$ and ${\mathrm{Bi}}₂{\mathrm{Te}}₃$. We compute the contact, dipolar, orbital and core polarization contributions to the carrier-density-dependent part of the NMR shifts in $²⁰⁹\mathrm{Bi},¹²⁵\mathrm{Te}$, and $⁷⁷\mathrm{Se}$. The spin-orbit coupling and the layered crystal structure result in a contact Knight shift with strong uniaxial anisotropy. Likewise, because of spin-orbit coupling, dipolar interactions make a significant contribution to the isotropic part of the NMR shift. The contact interaction dominates the isotropic Knight shift in $²⁰⁹\mathrm{Bi}$ NMR, even though the electronic states at the Fermi level have a rather weak $s$-orbital character. In contrast, the contribution from the contact hyperfine interaction to the NMR shift of $⁷⁷\mathrm{Se}$ and $¹²⁵\mathrm{Te}$ is weak compared to the dipolar and orbital shifts therein. In all cases, the orbital shift is at least comparable to the contact and dipolar shifts, while the shift due to core polarization is subdominant (except for Te nuclei located at the inversion centers). By artificially varying the strength of spin-orbit coupling, we evaluate the evolution of the NMR shift across a band inversion but find no clear signature of the topological transition.