Ionic and electronic properties of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ investigated via $\beta$-detected nuclear magnetic relaxation and resonance of ${}^8\mathrm{Li}$

We report measurements on the high-temperature ionic and low-temperature electronic properties of the three-dimensional topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ using ion-implanted ${}^8\mathrm{Li}\beta$-detected nuclear magnetic relaxation and resonance. With implantation energies in the range $5–28\mathrm{keV}$, the probes penetrate beyond the expected range of the topological surface state, but are still within $250\mathrm{nm}$ of the surface. At temperatures above $\sim 150\mathrm{K}$, spin-lattice relaxation measurements reveal isolated ${}^8\mathrm{Li}^{+}$ diffusion with an activation energy $E_A=0.185\left(8\right)\mathrm{eV}$ and attempt frequency ${\tau }_0^{-1}=8\pm 3\times {10}^{11}{\mathrm{s}}^{-1}$ for atomic site-to-site hopping. At lower temperature, we find a linear Korringa-type relaxation mechanism with a field-dependent slope and intercept, which is accompanied by an anomalous field dependence to the resonance shift. We suggest that these may be related to a strong contribution from orbital currents or the magnetic freeze-out of charge carriers in this heavily compensated semiconductor, but that conventional theories are unable to account for the extent of the field dependence. Conventional NMR of the stable host nuclei may help elucidate their origin.

Figure 1

Crystal structure of ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ [8], consisting of $\mathrm{Te}\text{−}\mathrm{Bi}\text{−}\mathrm{Se}\text{−}\mathrm{Bi}\text{−}\mathrm{Te}$ layers. These quintuple layers are weakly coupled through van der Waals interactions, giving rise to a characteristic gap between adjacent $\mathrm{Te}$ planes. This atomic arrangement is analogous to transition metal dichalcogenides, where it is possible to insert foreign atoms and small molecules within the van der Waals gap. The structure was drawn using vesta [20].

Figure 2

${}^8\mathrm{Li}$ spin-lattice relaxation data in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ at high and low magnetic fields. The characteristic kink at $t=4\mathrm{s}$ corresponds to the trailing edge of the ${}^8\mathrm{Li}^{+}$ beam pulse. The ${}^8\mathrm{Li}$ relaxation is strongly field dependent, increasing with decreasing magnetic field, and increases nonmonotonically with increasing temperature. The solid black lines are the result of a global fit to all spectra at a given field to a stretched exponential relaxation function, Eq. (1), convoluted with the square beam pulse [49, 50].

Figure 3

The spin-lattice relaxation rate for ${}^8\mathrm{Li}$ in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ at high magnetic fields with $B_0\parallel \left(001\right)$ using a stretched exponential analysis. $1/T_1$ is strongly field dependent, increasing with decreasing magnetic field, but also increases nonmonotonically with temperature. At each field, a clear $1/T_1$ maximum can be identified where the average fluctuation rate of the dynamics inducing relaxation matches the ${}^8\mathrm{Li}$ Larmor frequency. The solid black lines are fits to the model in Eqs. (2, 3, 4), consisting of a linear $T$ dependence with a nonzero intercept and a term due to ionic diffusion. The highlighted gray region denotes the lower limit of measurable $1/T_1$ due to the ${}^8\mathrm{Li}$ probe lifetime.

Figure 4

Temperature dependence of the ${}^8\mathrm{Li}^{+}1/T_1$ in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ at low magnetic field $[7.5\mathrm{mT}\perp (001\left)\right]$. The relaxation rate is orders of magnitude larger than at high fields, with both a larger apparent $T$-linear slope and intercept. There is some evidence of a small $1/T_1$ peak (see inset) near $\sim 150\mathrm{K}$ superposed on the dominant linear dependence. Above $200\mathrm{K},1/T_1$ increases substantially, suggestive of another relaxation mechanism absent at higher fields.

Figure 5

Field dependence of the intercept $a$ and slope $b$ describing the low-temperature electronic relaxation in Eq. (2). (Top panel) The $T\to 0$ intercept of $1/T_1$ falls with increasing field as $1/{\omega }_0^2$, behavior consistent with a low-field relaxation due to fluctuating host lattice nuclear spins [55]. The solid red line indicates a fit to a simple BPP model [37], yielding a coupling constant ${\overline{C}}_0^2=1.5\left(3\right)\times {10}^6{\mathrm{s}}^{-2}$ and correlation time ${\tau }_c=3.2\left(7\right)\times {10}^{-5}{\mathrm{s}}^{-1}$, the latter being remarkably close to the ${}^{209}\mathrm{Bi}$ NMR $T_2$ time in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ [26]. The inset shows a zoom of the small high-field intercepts, at the lower limit measurable due to the ${}^8\mathrm{Li}$ probe lifetime highlighted in gray. (Bottom panel) Field dependence of the low-temperature slope of $1/T_1$, exhibiting a strong orientation dependence, but a common field dependence $\propto {\omega }_0^{-1.5}$, indicated by the solid colored lines.

Figure 6

${}^8\mathrm{Li}$ NMR spectra in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ with $6.55\mathrm{T}\parallel \left(001\right)$ shown as solid black lines. The vertical scale is the same for all the spectra, but the baselines are offset to match the absolute temperature indicated on the left. The spectra reveal a small, temperature-independent quadrupole splitting, centered about an unsplit Lorentzian line, whose amplitude grows above $100\mathrm{K}$, becoming the dominant feature at higher temperatures. The solid colored lines are global fits to a sum of this Lorentzian plus $2I=4$ quadrupolar satellites of the quadrupole split resonance (see text for further details). The reference frequency, determined by the ${}^8\mathrm{Li}$ resonance position in $\mathrm{MgO}$ (100) at room temperature, is indicated by the vertical dashed gray line. Note the small negative shift of the ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ resonances as the temperature is lowered.

Figure 7

Fit results for ${}^8\mathrm{Li}$ resonance in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ at high field with $B_0\parallel \left(001\right)$, following the fitting procedure described in the text. The upper panel shows the resonance shift $\delta$ relative to $\mathrm{MgO}$. Above $\sim 150\mathrm{K}$, the shift is nearly field independent and scattered about zero; however, at lower temperatures, the trends at each field diverge, revealing a field-dependent shift whose magnitude is maximized at the lowest temperatures. The lower panel shows the change in amplitude of the central Lorentzian and quadrupole satellites (averaged over all four lines), normalized by the off-resonance baseline [52]. The amplitude of the unsplit line grows above $\sim 100\mathrm{K}$, contrasting the temperature independence of the quadrupole satellite amplitudes. The inflection point $T^{*}$ of this trend is indicated by the dashed vertical black line. The solid black lines are drawn to guide to the eye.

Figure 8

Arrhenius plot of the ${}^8\mathrm{Li}$ fluctuation rate extracted from the NMR frequency-dependent positions of $1/T_1\left(T_{\max }\right)$ in Figs. 3 and 4. The solid red line is a fit to Eq. (4) with the activation energy $E_A$ and prefactor ${\tau }_0^{-1}$ indicated in the inset and the fit's $\pm 1\sigma$ uncertainty band is highlighted in gray. The kinetic parameters extracted from this minimally model-dependent analysis are in good agreement with those obtained from fits of the $1/T_1$ data to Eqs. (2, 3, 4), given in Table 1.

Figure 9

Arrhenius plot of the diffusion coefficient $D$ for ${}^8\mathrm{Li}^{+}$ in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$. Values for $D$, shown in black, were estimated using the Einstein-Smoluchowski expression [Eq. (9)], taking $f=1,d=2$, and $l=4.307\AA$, with the values of ${\tau }^{-1}$ taken from Fig. 8. For comparison, diffusion coefficients for ${\mathrm{Li}}^{+}$ [65, 72, 74], ${\mathrm{Ag}}^{+}$ [71], and ${\mathrm{Cu}}^{+}$ [70, 73] in structurally related materials are also included. It is clear that the diffusivity of isolated ${}^8\mathrm{Li}^{+}$, free from repulsive ${\mathrm{Li}}^{+}\text{−}{\mathrm{Li}}^{+}$ interactions, is exceptional, significantly exceeding that in the well-known fast ion conductor $h\text{−}{\mathrm{Li}}_x{\mathrm{TiS}}_2$ [74, 75].

Figure 10

Stopping distribution and range for ${}^8\mathrm{Li}^{+}$ implanted in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ calculated using the srim Monte Carlo code [106]. The histogram profiles, shown on the left, are generated from simulations of ${10}^5$ ions. The ion range and straggle at each implantation energy are shown on the right. All measurements in this study correspond to average stopping depths $\ge 30\mathrm{nm}$ below the crystal surface, well below where the topological surface state is expected to be important.

Figure 11

Typical helicity-resolved ${}^8\mathrm{Li}$ resonance spectra in ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ with $6.55\mathrm{T}\parallel \left(001\right)$, revealing the fine structure of the line. Four quadrupole satellites, split asymmetrically in each helicity about a “central” Lorentzian line, are evident. Note that, in contrast to conventional NMR, the satellite amplitudes are determined primarily by the high degree of initial polarization [48]. The solid colored lines are global fits to a sum of five Lorentzians with positions given by Eqs. (5, 6, 7) (see Sec. 3c for further details). Upon combining helicities, a nearly symmetric line about the resonance center of mass, shown in the bottom panel, is obtained.

Figure 12

High-symmetry sites within the van der Waals gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$, shown as colored circles, in a plane perpendicular to the trigonal $c$ axis. The Wyckoff position is indicated for each site and the boundary of the unit cell is marked by solid black lines. Neighboring $3b$ (blue) sites, enclosed by $\mathrm{Te}$ quasioctahedra, are connected indirectly through $6c$ (green) sites with quasitetrahedral $\mathrm{Te}$ coordination, and directly through $9d$ (pink) sites of lower symmetry. The structure was drawn using vesta [20].