NMR relaxation in the topological Kondo insulator ${\mathrm{SmB}}_6$

${\mathrm{SmB}}_6$ has been predicted to be a strong topological Kondo insulator, and experimentally it has been confirmed that at low temperatures the electrical conductivity only takes place at the surfaces of the crystal. We study the temperature and magnetic field dependence of the NMR Knight shift and relaxation rate arising from the topological conduction states. For the clean surface the Landau quantization of the surface states gives rise to highly degenerate discrete levels for which the Knight shift is proportional to the magnetic field $B$ and inversely proportional to the temperature $T$. The relaxation rate $1/T_1$ is not Korringa-like. For the more realistic case of a surface with a low concentration of defects (dirty limit) the scattering of the electrons leads to a broadening of the Landau levels and hence to a finite density of states. The mildly dirty surface case leads to a $T$-independent Knight shift proportional to $B$ and a Korringa-like $1/T_1$ at low $T$. The wave functions of the surface states are expected to fall off exponentially with distance from the surface giving rise to a superposition of relaxation times, i.e., a stretched exponential. It is questionable that the experimental ${}^{11}\mathrm{B}$ Knight shift and relaxation rate arise from the surface states of the TKI. An alternative explanation is that the bulk susceptibility and the ${}^{11}\mathrm{B}$ NMR properties are the consequence of the in-gap bulk states originating from magnetic exciton bound states proposed by Riseborough [Phys. Rev. B 68, 235213 (2003)].

Figure 1

Density of states in arbitrary units as a function of the Landau level positions according to Eq. (5) for the parameters discussed in the text. (a) Delta functions in the clean limit, (b) slightly broadened levels due to scattering off defects, and (c) DOS in the dirty limit. In (b) and (c) the sharp Landau levels were replaced by a Lorentzian line shape, i.e., a constant imaginary part of the self energy, although the real self energy is energy dependent. Note that the defects introduce a background DOS. The shifts of the peaks due to the real part of the self energy have been neglected here for clarity.

Figure 2

The contribution of the “in-gap” state to the susceptibility ($\Delta \chi )$ (solid circles) (Ref. [12]), the temperature dependence of the 16 meV magnetic excitation intensities measured by inelastic neutron scattering $I_N$ (open squares) (Ref. [9]), and the intensity of the Raman transition $I_R$ (open triangles) (Ref. [8]) compared to electrical conductivity data (open small circles) from Ref. [52]. While the surface states contributing to the conductivity are active below 4 K, the “in-gap” states are formed at much higher $T$. (Figure adapted from Ref. [9].)

Figure 3

Low temperature spin lattice relaxation rates as a function of $T$ for different magnetic fields (adapted from Ref. [15]). The solid lines are fits to a simple model for “in-gap” states (Ref. [15]) and can be considered as guides to the eye. The data shows peaks or shoulders as a function of $T$ and the dashed curve interpolates between these maxima.