Quasi-two-dimensional thermoelectricity in SnSe

Stannous selenide is a layered semiconductor that is a polar analog of black phosphorus and of great interest as a thermoelectric material. Unusually, hole doped SnSe supports a large Seebeck coefficient at high conductivity, which has not been explained to date. Angle-resolved photoemission spectroscopy, optical reflection spectroscopy, and magnetotransport measurements reveal a multiple-valley valence-band structure and a quasi-two-dimensional dispersion, realizing a Hicks-Dresselhaus thermoelectric contributing to the high Seebeck coefficient at high carrier density. We further demonstrate that the hole accumulation layer in exfoliated SnSe transistors exhibits a field effect mobility of up to $250{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ at $T=1.3\mathrm{K}$. SnSe is thus found to be a high-quality quasi-two-dimensional semiconductor ideal for thermoelectric applications.

Figure 1

SnSe crystal and band structure. (a) Crystal structure and Brillouin zone (BZ). (b) DFT band structure along high-symmetry lines, with hole pockets 1–4 identified. (c) DFT calculated position and shape of the hole pockets in the $\mathrm{\Gamma }YZ$ plane of the BZ, with valleys identified. An isoenergy line 0.03 eV below the local valence-band maximum is shown for each valley. The energy of each hole valley band maxima is identified. (d) ARPES isoenergy cuts at $E=0.3$, 0.5, and $0.7\mathrm{eV}$ showing the location of the valleys 1, 2, and 4 in the BZ $\left(h\nu =74\text{eV}\right)$. (e) Momentum cuts along $\mathrm{\Gamma }Z$ of valleys 1 and 2. The ARPES intensity maxima and the effective mass are also shown. (f) Momentum cuts along $\mathrm{\Gamma }Y$ of the pockets 1 and 2 along with the ARPES intensity maxima and the effective masses. (g) Momentum cuts parallel to $\mathrm{\Gamma }Z$ (upper panel) and along $\mathrm{\Gamma }Y$ (lower panel) of pocket 4. (h) Out-of-plane $\left(K_a\right)$ dependence of the ARPES intensities of valleys 1 and 2. The crosses indicate the VBM and allow for extraction of the effective masses along $K_a$ (see text). This momentum cut also shows that valley 3 is the extension of valley 2 along $XU$. (i) SnSe optical conductivity spectrum measured at room temperature with unpolarized light at normal incidence. The model fit (blue line) is that for a 2D indirect gap semiconductor.

Figure 2

Zero-field resistivity. (a) Schematic of exfoliated SnSe FET with Si back-gate. (b) Optical micrograph and AFM image of a representative exfoliated SnSe FET with a thickness of $180\pm 16\mathrm{nm}$. (c) A comparison of current-voltage characteristics of exfoliated SnSe with Ti, Pd, and Pt contact electrodes, where Pt gives Ohmic conduction at $T=1.5\mathrm{K}$ and above.

Figure 3

Field effect. (a) Optical micrograph of an exfoliated SnSe flake with a thickness of $320\pm 30\mathrm{nm}$. (b) The resistivity $R_{xx}$ vs temperature $T$ during cooldown with and without illumination from a red $\left(\lambda =630\mathrm{nm}\right)$ light emitting diode. (c) The sheet conductance $G_{xx}$ vs gate voltage $V_{GS}$ at $T=1.3\mathrm{K}$. A hole accumulation layer threshold is observed at $V_{GS}=-37\mathrm{V}$. (d) The FET mobility ${\mu }_{\text{FET}}$ and the Hall mobility ${\mu }_H$ vs gate voltage $V_{GS}$ at $T=1.3\mathrm{K}$.

Figure 4

Magnetoresistance. (a) The resistance $R_{xx}$ vs temperature $T$ of a bulk SnSe crystal of thickness $170\pm 12\mu \mathrm{m}$. (b) The resistance $R_{xx}$ vs magnetic field $B$ at different temperatures $T$, showing a clear positive magnetoresistance that increases at low temperature. (c) Excellent agreement with a two-carrier model is found for $R_{xx}$ at $T=50$ and $100\mathrm{K}$. (d) Kohler plot of $\mathrm{\Delta }{R}_{xx}/R_0=[R_{xx}\left(B\right)-R_0]/R_0$ vs $B/R_0$. The measured data do not fall on a single universal curve, explicitly showing a violation of Kohler's rule.

Figure 5

Dimensionality, density of states, and Seebeck coefficient. (a) The isoenergy surfaces of valley 1, with anisotropic 3D dispersion, and valley 2, with a quasi-2D dispersion that is flat out of plane. (b) The density of states $\partial p/\partial E$ vs energy $E$ for the anisotropic 3D dispersion of valley 1 with $m_x=0.80m_0,m_y=0.25m_0$, and $m_z=0.22m_0$ and the quasi-2D dispersion of valley 2 with $m_y=0.17$ and $m_z=0.17$. The well-known enhancement of density of states with reduced dimensionality is apparent. (c) The Seebeck coefficient $S$ vs total hole density $p$ at room temperature calculated with an energy independent scattering approximation and a scattering time $\tau \propto E^{-1.3}$. Comparison is made against experimental measurements of the Seebeck coefficient [9, 11, 38, 39, 40, 41, 42].