Electron and hole contributions to normal-state transport in the superconducting system ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$

Indium-doped SnTe has been of interest because the system can exhibit both topological surface states and bulk superconductivity. While the enhancement of the superconducting transition temperature is established, the character of the electronic states induced by indium doping remains poorly understood. We report a study of magnetotransport in a series of ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$ single crystals with $0.1\le x\le 0.45$. From measurements of the Hall effect, we find that the dominant carrier type changes from holelike to electronlike at $x\sim 0.25$; one would expect electronlike carriers if the In ions have a valence of $+3$. For single crystals with $x=0.45$, corresponding to the highest superconducting transition temperature, pronounced Shubnikov–de Haas oscillations are observed in the normal state. In measurements of magnetoresistance, we find evidence for weak antilocalization (WAL). We attribute both the quantum oscillations and the WAL to bulk Dirac-like hole pockets, previously observed in photoemission studies, which coexist with the dominant electronlike carriers.

Figure 1

First-principles band structure of (a) SnTe and (b) ${\mathrm{Sn}}_{0.5}{\mathrm{In}}_{0.5}\mathrm{Te}$. Applying the VCA method to the occupancy of the Sn $5s$ orbital, the impact of the In substitution is to push the Fermi level down into the valence band, reaching a level of 0.8 eV below the top of the valence band. The overall band structure remains intact, but with an enhancement of the band inversion.

Figure 2

Transport measurements on ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$ samples. (a) Sketch of the contact locations on the sample. Inset shows a typical single crystal sample prepared for transport measurements (length $\sim 6$  mm). (b) Resistivity vs temperature for various In concentrations; note that the superconducting transition increases with $x$. (c) Temperature dependent Hall carrier concentration $N_{\mathrm{H}}$ calculated using single band model $N_{\mathrm{H}}=1/e{R}_{\mathrm{H}}$. A change of dominant carrier type occurs between $x=0.2$ and 0.3. (d) Transverse resistance $R_{xy}$ as a function of magnetic field $B$, at $T=50$  K. (e) Hall coefficient at 5 K vs In doping (circles); line is a fit with the two-band model described in the text. (f) Plot of carrier concentrations $N_h$ (green line) and $N_e$ (blue line) assumed in the model calculation; dashed line represents the $N_e$ multiplied by the squared ratio of mobilities, as discussed in the text. Circles indicate $1/e{R}_{\mathrm{H}}$ data; magenta line is the model calculation. (g) Resistivity at 5 K (squares), compared with the model calculation (line). (Data point at $x=0$ from [13].) (h) Hole and electron mobilities used in the model calculations.

Figure 3

Cartoon of the hole (blue) and electron (red) densities of states vs energy for ${\mathrm{Sn}}_{1-x}{\mathrm{In}}_x\mathrm{Te}$, as discussed in the text: (a) $x=0$; (b) $x\sim 0.1$; (c) $x\sim 0.4$.

Figure 4

(a) SdH oscillations in ${\mathrm{Sn}}_{0.55}{\mathrm{In}}_{0.45}\mathrm{Te}$ transverse resistance measured at temperatures of 1.5 to 10 K plotted vs inverse magnetic field, after subtraction of conventional Hall response. The 10 K data are multiplied by 3, and curves have been offset vertically. The assigned Landau level indices are indicated by the numbered vertical dashed lines. Inset shows fast Fourier transform (FFT) spectrum of the 5 K data. (b) Plot of inverse field for oscillation extrema vs Landau level index; linear fit yields an intercept of $0.32\pm 0.07$. (c) The temperature dependence of the oscillation amplitude at $n=5$ fitted by Lifshitz-Kosevich theory (dashed line), yielding a cyclotron mass $m_{\mathrm{cycl}}$ of $0.185m_e$.

Figure 5

Normalized longitudinal resistivity for the ${\mathrm{Sn}}_{0.55}{\mathrm{In}}_{0.45}\mathrm{Te}$ single crystal. (a) Magnetoresistance curves exhibit a sharp cusp at $B=0$ with an amplitude that diminishes with increasing temperature. (b) Phase diagram of ${\mathrm{Sn}}_{0.55}{\mathrm{In}}_{0.45}\mathrm{Te}$ showing the upper critical field vs temperature [15]. (c) Zero-field resistivity over an extended temperature range; inset shows that there is a plateau below 20 K, where the MR develops.

Figure 6

Angle-dependent magnetoresistance measurements for ${\mathrm{Sn}}_{0.55}{\mathrm{In}}_{0.45}\mathrm{Te}$ at 5 K. The cusp appears to be independent of orientation of applied magnetic field. Inset defines the orientation angles of the applied magnetic field relative to the sample surface and direction of applied current.

Figure 7

Conductance change with magnetic field for ${\mathrm{Sn}}_{0.55}{\mathrm{In}}_{0.45}\mathrm{Te}$ at 5 K. Circles denote experimental data; line is a fit to the WAL formula (see text) with $\alpha =0.82$ and $l_{\varphi }=80$  nm. Inset shows the temperature dependence of $\alpha$; dashed line shows an extrapolation to low temperature.