Thermal and vibrational properties of thermoelectric ZnSb: Exploring the origin of low thermal conductivity

The intermetallic compound ZnSb is an interesting thermoelectric material largely due to its low lattice thermal conductivity. The origin of the low thermal conductivity has so far been speculative. Using multitemperature single crystal x-ray diffraction (9–400 K) and powder x-ray diffraction (300–725 K) measurements, we characterized the volume expansion and the evolution of structural properties with temperature and identified an increasingly anharmonic behavior of the Zn atoms. From a combination of Raman spectroscopy and first principles calculations of phonons, we consolidate the presence of low-energy optic modes with wave numbers below $60{\mathrm{cm}}^{-1}$. Heat capacity measurements between 2 and 400 K can be well described by a Debye-Einstein model containing one Debye and two Einstein contributions with temperatures ${\mathrm{\Theta }}_{\mathrm{D}}=195\mathrm{K}, {\mathrm{\Theta }}_{\mathrm{E}1}=78\mathrm{K}$, and ${\mathrm{\Theta }}_{\mathrm{E}2}=277\mathrm{K}$ as well as a significant contribution due to anharmonicity above 150 K. The presence of a multitude of weakly dispersed low-energy optical modes (which couple with the acoustic, heat carrying phonons) combined with anharmonic thermal behavior provides an effective mechanism for low lattice thermal conductivity. The peculiar vibrational properties of ZnSb are attributed to its chemical bonding properties, which are characterized by multicenter bonded structural entities. We argue that the proposed mechanism to explain the low lattice thermal conductivity of ZnSb might also control the thermoelectric properties of other electron poor semiconductors, such as ${\mathrm{Zn}}_4{\mathrm{Sb}}_3$, CdSb, ${\mathrm{Cd}}_4{\mathrm{Sb}}_3, {\mathrm{Cd}}_{13-x}{\mathrm{In}}_y{\mathrm{Zn}}_{10}$, and ${\mathrm{Zn}}_5{\mathrm{Sb}}_4{\mathrm{In}}_{2-\delta }$.

Figure 1

(a) The orthorhombic crystal structure of ZnSb built from layers (dark and pale colors) of rhomboid rings ${\mathrm{Zn}}_2{\mathrm{Sb}}_2$. Cyan and red balls denote Zn and Sb atoms, respectively. (b) Local five-coordination of Zn and Sb atoms. The inserted numbers indicate interatomic distances in Angstroms at room temperature. (c) Rhomboid ring ${\mathrm{Zn}}_2{\mathrm{Sb}}_2$ and its connectivity to neighboring rings. Distances are distinguished as c (connecting) and r (ring) types. The center of a ring corresponds to a center of inversion. The thermal ellipsoids correspond to 90% probability density at room temperature obtained from SCXRD measurements. (d) Idealized ${\mathrm{Zn}}_6{\mathrm{Sb}}_5$ framework of rhombohedral β-${\mathrm{Zn}}_4{\mathrm{Sb}}_3$ along the [001] direction. The framework is built from chains of condensed rhomboid rings ${\mathrm{Zn}}_2{\mathrm{Sb}1}_{2/2}$ [cf. panel (e)] and consists of channels that are stuffed by Sb2 atoms (depicted as blue balls). Sb2-Zn bonds are omitted for clarity. (e) Structural fragments and local coordination of Zn, Sb1, and Sb2 atoms in β-${\mathrm{Zn}}_4{\mathrm{Sb}}_3$.

Figure 2

(a) Lattice parameters and (b) unit cell volume of ZnSb as a function of temperature. Squares and circles denote parameters obtained from SCXRD and PXRD refinements, respectively. The solid line represents a linear fit to the PXRD data. (c) Variation of interatomic distances in ZnSb as a function of temperature. Data were obtained from refinements of SCXRD data including anharmonic atomic displacement parameters (using a Gram-Charlier expansion). For the assignment of the individual bond lengths, see Fig. 1. The size of the symbols used in panels (a)–(c) exceeds by far the standard deviation of the corresponding data point.

Figure 3

PXRD patterns (Cu Kα radiation) of the ZnSb bulk sample at temperatures between 300 and 723 K in a 2Θ range of 25–42.5°. At 573 K, reflections from elemental antimony occur (marked with boxes). At 723 K, the sample decomposition is already severe, and a mixture of ZnSb and elemental Sb ($R\overline{3}m$) with a molar ratio of approximately 1:6 is observed. Bragg reflections of the Sb impurity phase are marked by their corresponding Miller indices.

Figure 4

Temperature dependent heat capacity $C_p$($T$) (per formula unit) of crystalline and SPS consolidated ZnSb samples (pink diamonds and red triangles, respectively) and their corresponding fits (model 2) as pink and red solid lines, respectively. The Dulong-Petit limit of $3R$ is marked as a black broken horizontal line. Earlier measurements of Danilenko et al. [44] and Mamedova et al. [43] have been added for comparison (black circles and green squares, respectively; interconnection lines are a guide for the eye). DFT calculated heat capacities $C_v$ and $C_p$ of ZnSb are shown as dashed black and dashed-dotted blue lines, respectively. The theoretical $C_p$ values have been estimated via the thermal expansion coefficient from the PXRD measurements (see below). The inset shows the ratio of experimental and theoretical $C_p$ values.

Figure 5

(a) Heat capacity $C_p/T$ (per formula unit) of ZnSb (black diamonds) including Debye-Einstein fit (model 2, solid black line) and ${\mathrm{Zn}}_4{\mathrm{Sb}}_3$ (red circles) including the original fit of Schweika et al. [11] (solid red line) as well as the improved fit using model 1 (solid green line). The DFT calculated data of ZnSb is shown as a dotted blue line. (b) $C_p/T^3$ (per formula unit) representation of panel (a) with additional fit for ZnSb (model 1, dashed black line) for comparison. The $y$ axis range is scaled according to the number of atoms per formula unit in ZnSb and ${\mathrm{Zn}}_4{\mathrm{Sb}}_3$ by the ratio 2:7 for both $C_p/T$ and $C_p/T^3$.

Figure 6

(a) Calculated phonon dispersion curves (left) and PDOS of ZnSb (right). The partial Sb atom contribution to the PDOS is indicated by the cyan area. (b) PDOS model based on the Debye and Einstein temperatures as extracted from the measured heat capacity. (c) PDOS constructed from fitting model 2 (cf. Table 1).

Figure 7

Raman spectrum of ZnSb and elemental antimony recorded with an excitation wavelength of 532 nm and 785 nm. The listed values on the right-hand side are given in wave numbers (${\mathrm{cm}}^{-1}$) for the labeled bands 1–11 for ZnSb and A-B for elemental antimony. The wave numbers of the individual optical modes were obtained from the DFT calculation at the Γ point and are shown as bars at the bottom of the spectra.

Figure 8

(a), (b) Theoretical harmonic ADPs obtained from DFT (solid lines) in comparison with experimental values; (c) comparison of $U_{\mathrm{eq}}$ values, which are calculated as the mean of the diagonal elements of the harmonic ADP tensor; and (d) comparison of the ADP anisotropy factor, which is defined as the ratio between maximum $U_{ii}$ and minimum $U_{ii}$ values. A ratio of 1 would correspond to an isotropic displacement.

Figure 9

Difference electron density before (left) and after (right) anharmonic refinement for temperatures of 400 K, 300 K, and 150 K. The positive (red) and negative (blue) contour values are shown in $0.2\mathrm{e}/{\AA }^3$ steps. The probability density distribution (middle) is given with contour values ($2,4,8\times {10}^n{\AA }^{-3},n=-2,-1,0,1,2,3$).

Figure 10

Third order Gram-Charlier ADP expansion coefficients for Zn (a) vs $T$ ($T$ axis scaled as $T^2$). Estimated standard uncertainties of the refined parameters are shown as error bars and (b) isosurface (surface value 0.1 ${\AA }^{-3}$) representation of the nuclear probability density of the ${\mathrm{Zn}}_2{\mathrm{Sb}}_2$ entity at 400 K.