Large thermopower in topological surface state of Sn-BSTS topological insulators: Thermoelectrics and energy-dependent relaxation times

Topological surface Dirac states (TSDSs) generated in three-dimensional topological insulators (3D-TIs) are currently of significant interest for new science and advanced technologies. In contrast to many other thermoelectric materials, 3D-TIs exhibit a significant potential to achieve a large enhancement in thermoelectric power factor ($PF=\sigma S^2$) due to their special topological symmetry. However, only limited experiments and discussions have been made so far for elucidating the thermoelectric properties of TSDS. Herein, we report a large $S$ and $\mathit{PF}$ observed for high-quality single-crystal flakes of $\mathrm{Sn}\text{−}\mathrm{B}{\mathrm{i}}_{1.1}{\mathrm{Sb}}_{0.9}\mathrm{Te}{\mathrm{S}}_2$ (Sn-BSTS). Accurate interpretations that the energy-dependent relaxation times $\tau \left(E\right)$ play an important role in thermoelectrical transport of 3D-TIs are provided. Among 3D-TIs, Sn-BSTS has the highest bulk insulation and shows intrinsic TSDS transport without bulk contributions, along with its hallmark of quantum integer Hall effect at high temperatures. Based on the longitudinal/transverse electrical transport and the thermoelectric coefficient, $\tau \left(E\right)\propto E^{0.21}$ is accurately deduced. As a consequence of the energy-dependent $\tau \left(E\right)$, a large enhancement in both $S$ and $\mathit{PF}$ is obtained ($S=58\mu \mathrm{V}{\mathrm{K}}^{-1}$ and $PF=5.0\mathrm{mW}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$ at 77 K), leading to a large increase of 160% for $S$ and 280% for $\mathit{PF}$ when compared to those of graphene at 77 K. The potential thermoelectric performance of the pure TSDS is discussed based on the Boltzmann transport equations.

Figure 1

Crystal structure and electronic resistivity of Sn-BSTS flakes. (a) Crystal structure of Sn-BSTS. One quintuple layer (QL) has two sets of Te (green), Bi/Sb (pink) layers, and one S (yellow) layer at the center. The Sn atom is doped into the Bi/Sb layer (not shown in the figure). (b) Evolution of electronic sheet resistance as a function of $T$ of four Sn-BSTS flakes with different thicknesses of 113, 26, 7, and 3 μm. (c) Colored contour map of surface dominant conductance ($G_{\mathrm{s}}/G_{\mathrm{tot}}$: the ratio of sheet conductance of the surface to the total sheet conductance) as a function of $T$ and sample thickness. $G_{\mathrm{tot}}$ was calculated from the experimental sheet resistance $R_{\mathrm{sh}}$, and $G_{\mathrm{s}}$ was estimated from the fitting analyses of sheet resistance, as described in the main text.

Figure 2

Thermopower $\left(S\right)$ of Sn-BSTS flakes. Evolution of Thermopower (Seebeck coefficient) as a function of $T$ of four Sn-BSTS flakes with different thicknesses of 113, 26, 7, and 3 μm.

Figure 3

Hall resistance of Sn-BSTS flakes. Hall resistance of Sn-BSTS flakes of 113, 26, 7, and 3 μm thicknesses at 2 K. The $R_{yx}$ of the 7-μm flake saturates at $R_{yx}=h/e^2$, and that of the 3-μm flake shows plateaus at $R_{yx}=h/2e^2$ and $R_{yx}=h/e^2$ of QHE in 3D-TIs. The inset is the enlarged view of the $R_{\mathrm{yx}}$ of the 113-μm flake.

Figure 4

The Shubnikov–de-Haas (SdH) oscillations of 7-μm Sn-BSTS flake. (a) The SdH oscillations of 7-μm flake. The inset is the FFT of the SdH oscillations. Two oscillation frequencies of 2.2 T (red bars) and 5.5 T (black bars) are detected. (b) Fan-diagram plots of the two SdH oscillations.

Figure 5

Intrinsic TE properties of TSDSs in Sn-BSTS. (a) A schematic band picture of the 3-μm and the 7-μm Sn-BSTS flakes calculated from analysis of the SdH oscillations. The position of $E_{\mathrm{F}}$ from the NDP of the top surface ($E_{\mathrm{t}}$) is 44 meV for the 3-μm and 40 meV for the 7-μm flake, respectively. The $E_{\mathrm{F}}$ of the bottom surface ($E_{\mathrm{b}}$) is 26 meV (red lines) for both flakes. The chemical potential of the bulk (blue line) is estimated as the average of the two TSDSs. (b) A schematic model of TE transport in 3D-TIs with a parallel circuit channel model. (c), (d) Analytical fitting of thermopower $\left(S\right)$ for the 3-μm and 7-μm flake, respectively.

Figure 6

Deconvolution of GS between TSDSs and insulating bulk (iB). Each term of TSDS and iB was calculated based on the Boltzmann equations using the accurate band picture derived from the experimental results. The total GS values are almost identical to the experimental values.

Figure 7

$\mathit{PF}$ among 2D and 3D thermoelectric materials. $\mathit{PF}$ values of 3D bulk crystals (green closed circles) at 300 K [24, 25, 26, 27, 28]; $\mathit{PF}$ values of the 2D material thin films (blue ball) [3, 4, 5, 29, 30, 31, 32, 33]. $\mathit{PF}$ of Sn-BSTS is plotted on the right side as a function of $T$ (red ball), calculated from the sheet resistance and Seebeck coefficient in Fig. 1. They increase with an increase in $T$ below 80 K. The saturation above 80 K is caused by the contribution of the bulk carriers thermally excited at high $T\mathrm{s}$. The red dashed line in the right panel represents the intrinsic $\mathit{PF}$ values of pure TSDS of Sn-BSTS at high $T\mathrm{s}$ evaluated from the deconvolution analysis of GS shown in Fig. 6 (details are described in the main text).

Figure 8

Comparison of TE properties on TSDSs: Sn-BSTS vs graphene. (a) Seebeck coefficients and (b) $\mathit{PF}\mathrm{s}$ of TSDSs on 3-μm Sn-BSTS and graphene/hBN. The solid line is the connection of the deconvoluted values for TSDSs [discussed in the text and shown in Fig. 3]. The experimental values of Gr/hBN (blue) and Gr/${\mathrm{SiO}}_2$ (black) are taken from Ref. [5]. (c) A schematic image showing structural features of the two Dirac systems of 3D-TI and Gr. Charged impurities of a substrate (red closed circles) perturbed fluctuations in the intrinsic Coulomb scattering potential, which decreases γ to suppress the enhancement in $S$ and $\mathit{PF}$ of Dirac-electron states.