Probing the bulk electronic states of Bi${}_2$Se${}_3$ using nuclear magnetic resonance

We report a nuclear magnetic resonance (NMR) study of Bi${}_2$Se${}_3$ single crystals grown by three different methods. All the crystals show nine well-resolved peaks in their ${}^{209}$Bi NMR spectra of the nuclear quadrupolar splitting, albeit with an intensity anomaly. Spectra at different crystal orientations confirm that all the peaks are purely from the nuclear quadrupolar effect, with no other hidden peaks. We identify the short nuclear transverse relaxation time ($T_2$) effect as the main cause of the intensity anomaly. We also show that the ${}^{209}$Bi signal originates exclusively from bulk, while the contribution from the topological surface states is too weak to be detected by NMR. However, the bulk electronic structure in these single crystals is not the same, as identified by the NMR frequency shift and nuclear spin-lattice relaxation rate ($1/T_1$). The difference is caused by the different structural defect levels. We find that the frequency shift and $1/T_1$ are smaller in samples with fewer defects and a lower carrier concentration. Also, the low-temperature power law of the temperature-dependent $1/T_1$ ($\propto T^{\alpha }$) changes from the Korringa behavior $\alpha =1$ in a highly degenerate semiconductor (where the electrons obey Fermi statistics) to $\alpha <1$ in a less degenerate semiconductor (where the electrons obey Boltzmann statistics).

Figure 1

${}^{209}$Bi NMR frequency-swept spectra in three Bi${}_2$Se${}_3$ single crystals Nos. 1–3 at 4.2 K for a magnetic field of 65.77 kG applied along the $c$ axis. The dashed line denotes the position of the unshifted Larmor frequency $f_0$(MHz)$=(\gamma /2\pi )65.77$, where $\gamma /2\pi =0.684$ MHz/kG is the ${}^{209}$Bi nuclear gyromagnetic ratio. (Inset) ${}^{209}$Bi field-swept spectra of our early Bi${}_2$Se${}_3$ crystals Nos. 4 and 5 (by the Bridgeman[12] and CVT methods, respectively), where the broad linewidth suggests their low quality.

Figure 2

${}^{209}$Bi NMR frequency-swept spectra of Bi${}_2$Se${}_3$ for (a) sample No. 2 and (b) sample No. 3, with magnetic fields applied at different angles with respect to the $c$ axis. The dashed line denotes the unshifted Larmor frequency position.

Figure 3

The curve fittings for the ${}^{209}$Bi NMR frequency-swept spectra of Bi${}_2$Se${}_3$, sample No. 3, for the magnetic field directions B$\parallel$c, B$\angle$57${}^{\circ }$, and B$\perp$c. (Circles: spectrum points; red curve: fitting curve; dotted curve: individual quadrupolar peaks.)

Figure 4

(a) The decay of ${}^{209}$Bi NMR intensity as a function of $T_2$ delay time (see text) for the Bi${}_2$Se${}_3$ sample No. 3 at 65.77 kG and 4.2 K. The left panel is for peaks near the center. Right panel is for the satellite peaks on the high-frequency side. (b) The $T_2$ relaxation curves at frequencies 45.04, 45.56, and 45.71 MHz. The red lines are the curve fittings for the points at the peaks of each curve. The relaxation rates $T_2^{-1}$, obtained from the fits, are labeled next to the curves.

Figure 5

Temperature-dependent resistivity for Bi${}_2$Se${}_3$ samples No. 2 (open circles) and No. 3 (solid circles). The corresponding ARPES spectra (insets) show a significant shift in the chemical potential between these two samples, indicating higher carrier concentration in sample No. 2. The spectra were recorded along the $\Gamma$K line in the surface Brillouin zone.

Figure 6

Temperature dependence of nuclear spin-lattice relaxation rate ($1/T_1$) for Bi${}_2$Se${}_3$ crystals.

Figure 7

Temperature-dependent magnetization curves for Bi${}_2$Se${}_3$ No. 3. (Inset) ${}^{209}$Bi NMR spectra at temperatures of 4.2, 40, and 120 K. The dashed line denotes the reference field for ${}^{209}$Bi at 45.7 MHz.