Vacancies in SnSe single crystals in a near-equilibrium state

The development of intrinsic vacancies in SnSe single crystals was investigated as a function of annealing temperature by means of positron annihilation spectroscopy accompanied by transport measurements. It has been demonstrated that two types of vacancies are present in single-crystalline SnSe. While Sn vacancies dominate in the low-temperature region, Se vacancies and vacancy clusters govern the high-temperature region. These findings are supported by theoretical calculations enabling direct detection and quantification of the most favorable type of vacancies. The experiments show that Sn vacancies couple with one or more Se vacancies with increasing temperature to form vacancy clusters. Interestingly, the clusters survive the α→β transition at ≈800 K and even grow in size with temperature. The concentration of both Se vacancies and vacancy clusters increases with temperature, similar to thermoelectric performance. This indicates that the extraordinary thermoelectric properties of SnSe are related to point defects. We suggest that either these defects vary the band structure in favor of high thermoelectric performance or introduce an energy-dependent scattering of free carriers realizing, in fact, energy filtering of the free carriers. Cluster defects account for the glasslike thermal conductivity of SnSe at elevated temperatures.

Figure 1

(a) Structure of the SnSe Pnma phase; Sn and Se ions are indicated by red and green spheres, respectively. (b) Coordination of Sn ion. Interatomic distances between Sn and Se ions are denoted by labels.

Figure 2

Configurations of vacancy clusters considered in ab initio calculations (a) ${\mathrm{V}}_{\mathrm{Sn}}+{\mathrm{V}}_{\mathrm{Se}}$, (b) ${\mathrm{V}}_{\mathrm{Sn}}+2{\mathrm{V}}_{\mathrm{Se}}$, and (c) ${\mathrm{V}}_{\mathrm{Sn}}+3{\mathrm{V}}_{\mathrm{Se}}$. Sn and Se ions are indicated by red and green spheres, respectively. Vacancies are denoted by open squares.

Figure 3

The development of the mean positron lifetime ${\tau }_{\mathrm{mean}}$ as a function of AT. The dashed lines show the calculated positron lifetimes for a perfect SnSe lattice (bulk) and for various types of defects.

Figure 4

The development of the lifetimes ${\tau }_i$ of the exponential components resolved in the LT spectra as a function of AT. The quantity ${\tau }_{\mathrm{f}}$ calculated using Eq. (5) is plotted in the figure as well. The dashed lines show calculated lifetimes for various positron states.

Figure 5

The development of the relative intensities $I_i$ of the exponential components resolved in the LT spectra as a function of AT. Dashed lines show extrapolation to zero intensity in case where the corresponding component could not be resolved in the spectrum.

Figure 6

The development of the concentrations of point defects detected by LT spectroscopy in SnSe SCs plotted as a function of AT.

Figure 7

The calculated CDB ratio curves (related to pure Al) for (a) the perfect SnSe crystal, (b) ${\mathrm{V}}_{\mathrm{Sn}}$, and (c) ${\mathrm{V}}_{\mathrm{Se}}$. Partial contributions of the positrons annihilated by Sn and Se electrons are plotted with dashed lines.

Figure 8

Experimental CDB ratio curves (related to well-annealed pure Al) for SnSe SCs. The ratio curves for reference samples of pure Sn and Se are plotted in the figure as well.

Figure 9

Calculated CDB ratio curves (related to Al) for a perfect SnSe crystal and various point defects in SnSe.

Figure 10

Hall concentration $h$ of holes of SnSe as a function of AT for three temperatures. Note the significant increase in the hole concentration between 300 and 400 K for 673 K AT, which is indicative of thermal excitation. The dotted line represents the α→β transition temperature (≈810 K) of SnSe. The inset documents the eventual impact of heavy holes (HH) if combined with light holes (LH) in terms of carrier mobility. The curves are calculated using $R_H=\frac{1}{\left|e\right|}\frac{\left(h_{HH}{\mu }_{HH}^2+h_{LH}{\mu }_{LH}^2\right)}{{\left(h_{HH}{\mu }_{HH}+h_{LH}{\mu }_{LH}\right)}^2}$, where $h$ and µ are the respective hole concentration and mobility.

Figure 11

Hall mobility ${\mu }_{\mathrm{H}}$ of SnSe as a function of AT for three temperatures. The dotted line represents the α→β transition temperature (≈810 K) of SnSe.

Figure 12

PF as a function of AT. The dotted line represents the α→β transition temperature (≈810 K) of SnSe.

Figure 13

The Seebeck coefficient S as a function of the Hall concentration of holes (Pisarenko plot) for various ATs (not shown are the samples annealed above the α→β transition temperature ≈800 K). The solid lines are fits using a single parabolic model assuming scattering of FCs by acoustic phonons. The sample annealed at 793 K (green triangle) is on the verge of the α→β transition. The red square represents the standard (nonannealed) sample. The values were measured at $T=200\mathrm{K}$.

Figure 14

A typical PXRD pattern of a powdered SnSe single crystal. No traces of foreign phases are detectable. The red arrow shows the $K_{\beta }$ reflection. The inset documents the variation of UCV with AT. The maximum deviation of the UCV is $\pm 0.1{\mathrm{A}}^3$.