Hyperfine coupling and spin polarization in the bulk of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

Nuclear magnetic resonance (NMR) and transport measurements have been performed at high magnetic fields and low temperatures in a series of $n$-type ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ crystals. In low-density samples, a complete spin polarization of the electronic system is achieved, as observed from the saturation of the isotropic component of the ${}^{209}\mathrm{Bi}$ NMR shift above a certain magnetic field. The corresponding spin splitting, defined in the phenomenological approach of a 3D electron gas with a large (spin-orbit-induced) effective $g$ factor, scales as expected with the Fermi energy independently determined by simultaneous transport measurements. Both the effective electronic $g$ factor and the “contact” hyperfine coupling constant are precisely determined. The magnitude of this latter reveals a nonnegligible $s$ character of the electronic wave function at the bottom of the conduction band. Our results show that the bulk electronic spin polarization can be directly probed via NMR and pave the way for future NMR investigations of the electronic states in Bi-based topological insulators.

Figure 1

Raw frequency-swept NMR spectra of sample P3-$6(n_e=7.5\times {10}^{17}{\mathrm{cm}}^{-3})$ as a function of the magnetic field $B\parallel c$ axis at $T=5$ K. The vertical offset between the spectra is linear in $B$. The open circles denote the fitted NMR line positions, while the connecting dashed lines are guides to the eye. The frequency axis origin is defined by $f_0$ = ${}^{209}\gamma B$, the resonance for the bare ${}^{209}\mathrm{Bi}$ nuclei ${(}^{209}\gamma =6.84184$ MHz/T [19]).

Figure 2

NMR frequency shift $(f-f_0)$ of the Bi central line as a function of the magnetic field $B$ applied along the $c$ axis, for three samples having different carrier concentrations. $T=5$ K.

Figure 3

Decomposition of the total frequency shift into isotropic $\left(\Delta f_{\mathrm{iso}}\right)$ and axial parts $\left(\Delta f_{\mathrm{axial}}\right)$ (see text) for sample P3-6. To compare the two components on the same graph, $-\Delta f_{\mathrm{axial}}$ is plotted. The black dotted line is a simulation in a 3D electron gas model $(B_s$ indicates the field corresponding to full spin polarization). The red solid line further includes effects of the Landau quantization and of disorder.