Critical mode and band-gap-controlled bipolar thermoelectric properties of SnSe

The reliable calculation of electronic structures and understanding of electrical properties depends on an accurate model of the crystal structure. Here we have reinvestigated the crystal structure of the high-$zT$ thermoelectric material tin selenide, SnSe, between 4 and 1000 K using high-resolution neutron powder diffraction. Symmetry analysis reveals the presence of four active structural distortion modes, one of which is found to be active over a relatively wide range of more than $\pm 200$ K around the symmetry-breaking $Pnma\text{−}Cmcm$ transition at 800 K. Density functional theory calculations on the basis of the experimental structure parameters show that the unusual, steplike temperature dependencies of the electrical transport properties of SnSe are caused by the onset of intrinsic bipolar conductivity, amplified and shifted to lower temperatures by a rapid reduction of the band gap between 700 and 800 K. The calculated band gap is highly sensitive to small out-of-plane Sn displacements observed in the diffraction experiments. SnSe with a sufficiently controlled acceptor concentration is predicted to produce simultaneously a large positive and a large negative Seebeck effect along different crystal directions.

Figure 1

(a) Crystal structure of SnSe below 800 K in the $Pnma$ setting. (b) Temperature dependence of the lattice parameters of SnSe. (c) Neutron powder diffraction pattern of SnSe at 900 K $(Bbmm$ phase) and Rietveld fit. Measured data are shown by black circles, the Rietveld fit by red lines, and the difference curve by the green line at the bottom. Tick marks show the fitted peak positions for SnSe (black, upper row) and a $\sim 1$ wt.% ${\mathrm{SnO}}_2$ impurity that was also observed (gray, lower row). Three small regions in $d$ spacing with artifacts from the TOF detection were excluded from the fits.

Figure 2

(a) Atomic displacement patterns of the active symmetry-adapted distortion modes of SnSe. The arrows are not to scale. (b)–(d) Amplitudes of the distortion modes as a function of temperature, see text for details. Solid black lines in (b) and (c) are fits to the data as explained in the text, the dashed line in (c) shows a quadratic extrapolation of the 4–400 K data, and the dotted lines in (b)–(d) are guides to the eye.

Figure 3

Calculated band structure of SnSe at 300 K (solid lines, blue) and 900 K (dotted lines, red). The special points refer to the $Pnma$ Brillouin zone, which was used for both calculations to allow for a direct comparison.

Figure 4

(a) Evolution of the calculated indirect band gap of SnSe with temperature. Solid symbols show results based on the crystal structure as measured, and open symbols where the peak of the Sn-$\mathrm{\Gamma }1$ distortion mode around 800 K was suppressed [dashed line in Fig. 2]. The band gap is that of the static lattice. Small symbols represent additional computational results where interpolated structural parameters were used. (b) Difference in total energy between that of the hypothetical structure with the Sn-$\mathrm{\Gamma }1$ anomaly suppressed and that of the crystal structure as measured. (c) Static band gap and gap adjusted for thermal motion after [35], and chemical potential $E_F$ relative to the valence band maximum $E_V$ for an extrinsic hole concentration of $n_p=5\times {10}^{17}{\mathrm{cm}}^{-3}$.

Figure 5

Correlation of the calculated band gap of SnSe with the amplitude of the Sn-$Y2$ critical mode. Open symbols mark data points below 400 K and above 800 K; small symbols indicate results where interpolated structural parameters were used in the calculations.

Figure 6

(a) Calculated electrical conductivity $\sigma$ divided by the scattering time $\tau$ of $p$-SnSe and (b) calculated Seebeck coefficient as a function of temperature. A hole-doping concentration of $n_p=5\times {10}^{17}{\mathrm{cm}}^{-3}$ at ambient temperature and the adjusted band gaps from Fig. 4 were used in both cases. (c) and (d) Corresponding experimental results by Zhao et al. [1] (solid symbols) and Ibrahim et al. [36] (open symbols).

Figure 7

Inverse Hall coefficients $1/\left(e{R}_H\right)$ and Sn vacancy concentrations $C_{{\text{V}}_{\text{Sn}}}$ in SnSe as a function of temperature. Red solid circles show the average inverse Hall coefficient obtained here from Boltzmann transport calculations assuming hole-doping concentration of $n_p=5\times {10}^{17}{\mathrm{cm}}^{-3}$ at ambient temperature and the adjusted band gaps from Fig. 4. Blue open squares show the corresponding experimental data by Zhao et al. [1]. Cyan diamonds and stars show the carrier concentration obtained from the ${\mathrm{Sn}}^{2-}$ vacancy model by Dewandre et al. [13] for the $Pnma$ and $Cmcm/Bbmm$ phases, respectively.