Nuclear magnetic resonance as a probe of electronic states of Bi${}_2$Se${}_3$

We present magnetotransport and ${}^{209}$Bi nuclear magnetic resonance (NMR) data on a series of single crystals of Bi${}_2$Se${}_3$, Bi${}_2$Te${}_2$Se, and Cu${}_x$Bi${}_2$Se${}_3$ with varying carrier concentrations. The Knight shift of the bulk nuclei is strongly correlated with the carrier concentration via a hyperfine coupling of 27 $\mu$eV, which may have important consequences for scattering of the protected surface states. Surprisingly we find that the NMR linewidths and the spin-lattice relaxation rate appear to be dominated by the presence of localized spins, which may be related to the presence of Se vacancies.

Figure 1

A representative powder x-ray diffraction pattern of Bi${}_2$Se${}_3$ (black line) compared with the theoretical prediction [(blue) vertical bars]. Inset: Crystal structure of Bi${}_2$Se${}_3$. Bi (blue) and Se (green and yellow) atoms form quintuple layers.

Figure 2

(a) Carrier concentrations of Bi${}_2$Se${}_3$ as determined by Hall measurements. The stoichiometric samples (Nos. 2 and 6, marked as in Fig. 7) show the highest concentrations, as well as samples 7 and 9 (black and purple $⧫$, respectively) from batch B. Excess Se (Nos. 4 and 5, marked as in Fig. 7) can lower the concentration by an order of magnitude. (b) Resistivity versus temperature. Sample 4 was measured in a closed-cycle refrigerator; all other samples were measured in the PPMS.

Figure 3

Carrier concentration of Bi${}_2$Se${}_3$ (filled circles) and Bi${}_2$Te${}_3$ (open circles) as a function of the atomic percentage of Se/Te in the initial mixture of Bi and Se. Data for the Bi${}_2$Te${}_3$ reproduced from Ref. 18.

Figure 4

Lattice constants $a$ and $c$ as a function of the atomic percentage of Se. Solid lines are best linear fits to the data.

Figure 5

Mobilities versus temperature as determined from Hall constant and resistivity measurements.

Figure 6

$\Delta \rho$ versus inverse field $H^{-1}$ at 1.9 K in Bi${}_2$Se${}_3$, where $\Delta \rho$ is the difference between the measured resistivity and a polynomial fit as described in the text. Curves are offset vertically for clarity. NMR was performed on samples which showed oscillations (marked as in Fig. 7). Other samples from batch B (samples 7–9) showed no oscillations.

Figure 7

${}^{209}$Bi NMR spectra of Bi${}_2$Se${}_3$ single crystals in a field of 9 T with ${\mathbf{H}}_0\parallel c$ at 10 K (except for sample 2, at 20 K). The quadrupolar splitting appears to be sample independent, but the Knight shift depends on the carrier concentration. Stoichiometric samples 1 ($•$, red), 2 ($\blacksquare$, green), and 6 ($▸$, cyan) exhibit similar spectral features. Samples 3 ($▴$, pink), 4 ($⧫$, purple) and 5 ($⧫$, yellow) have excess Se; samples 3 and 4 exhibit pronounced outer satellites consistent with previous observations,[11] whereas sample 5 exhibits a more typical profile. Disorder broadens the quadrupolar satellites and washes out the spectra in the Bi${}_2$Te${}_2$Se ($▿$, measured at 281 K) and the Cu${}_x$Bi${}_2$Se${}_3$ ($⧫$, black, measured at 10 K).[20]

Figure 8

Knight shift, $K$, versus carrier concentration, $n$. The positive correlation is consistent with a hyperfine coupling to electrons. The solid line is a fit as described in the text.

Figure 9

Magnetic contribution to the linewidth $\delta {\omega }_M$ versus carrier concentration (lower axis) and versus Se concentration (upper axis).

Figure 10

Spin-lattice relaxation rates of the ${}^{209}$Bi versus temperature for various samples with different carrier concentrations. Open symbols are reproduced from Ref. 11 and exhibit a strong temperature dependence. Solid lines are predictions based on the Korringa formula as described in the text.

Figure 11

Recovery of the Bi echo after inversion recovery at the central line in sample 2 at 10 K. The solid line is a fit to Eq. (7) and the dotted line is a fit to Eq. (9).