Letter | Published:

Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals

Nature volume 508, pages 373377 (17 April 2014) | Download Citation

Abstract

The thermoelectric effect enables direct and reversible conversion between thermal and electrical energy, and provides a viable route for power generation from waste heat. The efficiency of thermoelectric materials is dictated by the dimensionless figure of merit, ZT (where Z is the figure of merit and T is absolute temperature), which governs the Carnot efficiency for heat conversion. Enhancements above the generally high threshold value of 2.5 have important implications for commercial deployment1,2, especially for compounds free of Pb and Te. Here we report an unprecedented ZT of 2.6 ± 0.3 at 923 K, realized in SnSe single crystals measured along the b axis of the room-temperature orthorhombic unit cell. This material also shows a high ZT of 2.3 ± 0.3 along the c axis but a significantly reduced ZT of 0.8 ± 0.2 along the a axis. We attribute the remarkably high ZT along the b axis to the intrinsically ultralow lattice thermal conductivity in SnSe. The layered structure of SnSe derives from a distorted rock-salt structure, and features anomalously high Grüneisen parameters, which reflect the anharmonic and anisotropic bonding. We attribute the exceptionally low lattice thermal conductivity (0.23 ± 0.03 W m−1 K−1 at 973 K) in SnSe to the anharmonicity. These findings highlight alternative strategies to nanostructuring for achieving high thermoelectric performance.

Main

The efficiency of thermoelectric materials and devices is determined by the dimensionless figure of merit (ZT), defined as ZT = (S2σ/κ)T, where S, σ and κ are the Seebeck coefficient, the electrical conductivity and the thermal conductivity, respectively1,2. The well-known interdependence of S, σ and κ complicates efforts to develop strategies for improving a material’s average ZT well above 2.5, especially using less expensive, more Earth-abundant materials2,3,4, a feat that could revolutionize the field of thermal energy conversion. Several approaches to enhance ZT have emerged in the past decade, including enhancement of Seebeck coefficients (by modifying the band structure5, heavy valence (conduction) band convergence6,7, quantum confinement effects8 and electron energy barrier filtering9), and reducing lattice thermal conductivity (by nanostructuring10 and all-scale hierarchical architecturing11) while maintaining hole mobility (band energy alignment between nano-precipitate and matrix12,13,14). Most of these approaches aim to maintain a high power factor (electrical transport properties) and/or reduce the lattice thermal conductivity. Alternatively, high performance could be sought in pristine thermoelectric compounds with intrinsically low thermal conductivity, which may arise from properties such as a large molecular weight15, a complex crystal structure16 or charge density wave distortions17.

We find that SnSe, which is a very stable and simple compound consisting of Earth-abundant elements, exhibits an intrinsically ultralow thermal conductivity. Historically, SnSe was ignored by the thermoelectric community18,19; however, its layered and anisotropic crystal structure motivated us to explore its electrical transport properties along all axial directions (unless otherwise noted, all crystallographic directions and planes mentioned in this Letter are defined with respect to its room-temperature form, with Pnma space group, #62). Surprisingly, we find that the electrical resistivity is low enough to result in a moderate power factor (along the b axis), but, even more surprisingly, we observe that the thermal conductivity of SnSe is intrinsically ultralow (<0.25 W m−1 K−1 at >800 K), resulting in ZT = 2.62 at 923 K along the b axis and 2.3 along the c axis; these represent the highest ZT values reported so far for any thermoelectric system. Along the a direction, however, ZT is significantly lower, 0.8. Here, it should be noted that SnSe along the b axis shows a room-temperature ZT = 0.12, which is comparable to the room-temperature value of 0.15 reported earlier19. SnSe, however, reveals high ZT values near and above the transition temperature of 750 K at which the structure converts from Pnma to Cmcm20,21,22. Such ultrahigh ZT along two principal directions and the observed crystallographic and ZT anisotropy prompted us to investigate the scientific underpinning of these intriguing results.

SnSe adopts a layered orthorhombic crystal structure at room temperature, which can be derived from a three-dimensional distortion of the NaCl structure. The perspective views of the room-temperature SnSe crystal structure along the a, b and c axial directions are shown in Fig. 1a–d. There are two-atom-thick SnSe slabs (along the bc plane) with strong Sn–Se bonding within the plane of the slabs, which are then linked with weaker Sn–Se bonding along the a direction20. The structure contains highly distorted SnSe7 coordination polyhedra, which have three short and four very long Sn–Se bonds, and a lone pair of the Sn2+ sterically accommodated between the four long Sn–Se bonds (Fig. 1b). The two-atom-thick SnSe slabs are corrugated, creating a zig-zag accordion-like projection along the b axis. The easy cleavage in this system is along the (l00) planes. While cooling from its high-temperature, higher symmetry phase (space group Cmcm, #63), SnSe undergoes a displacive (shear) phase transition at 750–800 K, resulting in a lower symmetry Pnma (#62) space group and concomitant matrix transformation of its axes21,22.

Figure 1: SnSe crystal structure Pnma and ZT values.
Figure 1

a, Crystal structure along the a axis: grey, Sn atoms; red, Se atoms. b, Highly distorted SnSe7 coordination polyhedron with three short and four long Sn–Se bonds. c, Structure along the b axis. d, Structure along the c axis. e, Main panel, ZT values along different axial directions; the ZT measurement uncertainty is about 15% (error bars). Inset images: left, a typical crystal; right, a crystal cleaved along the (l00) plane, and specimens cut along the three axes and corresponding measurement directions. Inset diagram, how crystals were cut for directional measurements; ZT values are shown on the blue, red and grey arrows; colours represent specimens oriented in different directions.

The ZT values along the three different crystallographic axes are shown in Fig. 1e. The inset of Fig. 1e shows the SnSe crystals and the cutting directions for typical samples in this study. It should be noted that the a, b and c axial directions were determined through X-ray diffraction (XRD; Extended Data Fig. 1) as well as electron backscatter diffraction (EBSD) analysis (Extended Data Fig. 2).

The electrical conductivities for SnSe crystals along different crystallographic directions show the same temperature-dependent trend (Fig. 2a). We observe three regions: first, metallic transport behaviour from 300 to 525 K; then a change to thermally activated semiconducting behaviour up to 800 K; and above that, a nearly temperature-independent trend up to 973 K. The first upturn above 525 K is attributed to the thermal excitation of carriers, while the second is related to the phase transition from Pnma (a = 11.49 Å, b = 4.44 Å, c = 4.135 Å) to the Cmcm (a = 4.31 Å, b = 11.70 Å, c = 4.31 Å) space group21,22. It can be readily seen that the electrical conductivities along the b and c directions are similar, whereas it is lower along the a direction. This anisotropy is due to the higher ratio of Hall coefficient to resistivity, RH/ρ (related to mobility), within the plane of the SnSe slabs than perpendicular to them (that is, along the a direction) (Extended Data Fig. 3). The Seebeck coefficients show almost isotropic behaviour, and are independent of crystallographic directions (Fig. 2b). The gradual decrease of the Seebeck coefficients above 525 K is consistent with the increasing trend in the electrical conductivity, and suggests bipolar conduction and an increasing inverse Hall coefficient, 1/RH (Extended Data Fig. 3a). This behaviour is consistent with our electronic band structure calculations (Extended Data Fig. 4), which show that the bandgap (Eg) decreases considerably from Pnma (0.61 eV) to Cmcm (0.39 eV), and thus a bipolar conduction process is expected with rising temperature.

Figure 2: Thermoelectric properties as a function of temperature for SnSe crystals.
Figure 2

a, Electrical conductivity. b, Seebeck coefficient. c, Power factor, PF. d, Total thermal conductivity, κtot. Inset, lattice thermal conductivity, κlat (same units as κtot), versus temperature (same units as main panel).

The power factor along the b axis shows the highest value (10.1 μW cm−1 K−2) compared to the other two axial directions around 850 K (Fig. 2c); the maximum power factors at 850 K along the c and a directions are 7.7 μW cm−1 K−2 and 2.1 μW cm−1 K−2, respectively. Compared to other state-of-the-art thermoelectrics2,3,4, the power factors obtained in SnSe crystals are moderate, but are much higher than those found in other thermoelectrics with intrinsically low overall thermal conductivity (for example, Yb14MnSb11, Ag6TlTe5, AgSbTe2)15,16,23. The highest power factor along b is in agreement with the highest RH/ρ of 250 cm2 V−1 s−1 obtained along this axis, which is twofold higher than that along the c axis, and tenfold higher than that along the a axis (Extended Data Fig. 3b).

The temperature dependence of total thermal conductivity (κtot) is shown in Fig. 2d. At room temperature (300 K), the values of κtot are (in W m−1 K−1) 0.46, 0.70 and 0.68 along the a, b and c axis directions, respectively. Compared to state-of-the-art thermoelectrics2,3,4, these thermal conductivity values are exceedingly low. Surprisingly, these low values continue to decrease with rising temperature, and at 973 K all fall in the range 0.23–0.34 W m−1 K−1. The ratio of lattice thermal conductivity (κlat) to κtot indicates that κtot is dominated by phonon transport (Extended Data Fig. 6d). The inset in Fig. 2d indicates that κlat falls as low as 0.20 W m−1 K−1 at 973 K along the a direction. This is a remarkably low value, which is lower than those obtained even by nanostructuring and all-scale hierarchical architecturing of PbTe-based thermoelectric materials11.

The dynamic structural behaviour of SnSe above 750 K, involving a reversible phase transition from low-temperature Pnma to a high-temperature Cmcm space group, helps to preserve the high power factor22. This is because the Cmcm phase, which is structurally closely related to the Pnma phase, exhibits a substantially reduced energy gap and enhanced carrier mobilities while maintaining the ultralow thermal conductivity. We have confirmed this transition using in situ sample heating in a transmission electron microscope (TEM). A defect-free lattice image of a SnSe specimen is shown in Fig. 3a, with the inset showing the corresponding selected area diffraction (SAD) pattern taken along the [011] zone axis. It should be noted that kinematically forbidden reflections for the Pnma space group, of the type {100}, occur owing to ubiquitous double diffraction in electron diffraction along these directions, and reflect the presence of translational symmetry elements (screw axes/glide planes) in the space group24. Other detailed crystal information, including high-resolution TEM images along multiple crystal orientations ([100], [201], [211], [021]) and SAD along [001] and [010], are shown in Extended Data Fig. 7, which confirms the single-crystal SnSe phase at room temperature.

Figure 3: High-temperature in situ TEM observations.
Figure 3

a, Main panel, high-resolution TEM image of single-crystal SnSe (scale bar, 2 nm). Bottom inset, corresponding diffraction pattern along the [011] zone axis; top inset, the line profile (distance is plotted in Å, y axis) along the dotted line AB in the main panel showing the d spacing of (100). b, Simulated crystal structures of the phase at room temperature (RT; Pnma) and at high temperature (HT; Cmcm), viewing along the [211] and [121] directions; planes (1−1−1), (−101) and (0−11) are marked by blue lines. c, Diffraction patterns obtained at different temperatures. B, zone axis. There is a difference in measured angle between (1−1−1) and (0−11) of about 2.6° between room and elevated temperatures.

We used the SAD mode (obtained after zone axis alignment with convergent beam electron diffraction) because it is sensitive and can detect subtle changes in crystal symmetry, especially angular rotations. The [211] zone axis at room temperature, which becomes the [121] zone after the transition at about 750–800 K, was chosen to resolve the evolution of the planes (1−1−1) and (0−11). As illustrated by the crystal models in Fig. 3b, viewing along [211] and [121] at room temperature and high temperature, respectively, the plane normal angle between (1−1−1) and (0−11) (marked as blue lines) is expected to change from 86.18° to 89.89° on transition. Figure 3c shows this variation of the angle on the experimentally determined SADs, and we see a good match with the simulated SADs in Extended Data Fig. 8. The diffraction patterns in Fig. 3c were obtained from the same sample area. The sample temperature was increased rapidly from room temperature to 800 K, and then held at this temperature for 30 min after which a diffraction pattern was obtained; the temperature was then increased to 820 K, and the diffraction pattern determined again after holding at this temperature for 60 min; the sample was then brought back to room temperature and the pattern obtained again. The angle between (1−1−1) and (0−11) increased by 2.6° when 800 K was reached. The high-temperature phase is stable up to 820 K for the extended duration used here. There was no noticeable change in terms of d spacing and angles between the two SADs at elevated temperatures. In addition, the phase transition was observed to be reversible by returning to room temperature and monitoring the SAD of the Pnma phase that reappears. These observations are consistent with a displacive reversible phase transition between Pnma and Cmcm space groups at about 800 K. Given the subtle angular changes in diffraction observations, it is likely that the phase transition may involve mere shuffling of the SnSe layers, as expected for a displacive second-order phase transition and consistent with the very close relationship of the two structures.

It is known that strong anharmonicity in bonding can give rise to low lattice thermal conductivity in ordered crystal structures23,25,26. The strength of the lattice anharmonicity can be estimated from the Grüneisen parameters, which characterize the relationship between phonon frequency and crystal volume change. Therefore, to clarify the origin of the intrinsically low thermal conductivity of SnSe, the phonon and Grüneisen dispersions were calculated using first-principles density functional theory (DFT) phonon calculations within the quasi-harmonic approximation. Figure 4a shows that the acoustic modes along the Γ–X Brillouin zone direction (a axis) are significantly softer (with lower Debye temperatures and smaller phonon velocities; see Extended Data Table 1) than those along both the Γ–Y (b axis) and the Γ–Z Brillouin zone direction (c axis). These soft modes along the a axis suggest weak interatomic bonding and possible strong anharmonicity. To quantitatively assess the anharmonicity along the three directions, we plot the dispersion of the Grüneisen parameters of SnSe (Fig. 4b), which shows that the Grüneisen parameters are all very large, with that along the a axis being larger than those along both the b and c axes. The average Grüneisen parameters along the a, b and c axes are 4.1, 2.1 and 2.3, respectively, as shown in the inset of Fig. 4b. Along the a axis, the maximum longitudinal acoustic (LA) Grüneisen parameter around the Γ point is extraordinarily high, 7.2. In contrast, the Grüneisen parameters are 2.05 for AgSbTe2 (ref. 23), 3.5 for AgSbSe2 (ref. 25) and 1.45 for PbTe (ref. 27), corresponding to measured lattice thermal conductivities at room temperature (in W m−1 K−1) of 0.68, 0.48 and 2.4, respectively. The anomalously high Grüneisen parameter of SnSe is a reflection of its crystal structure, which contains very distorted SnSe7 polyhedra (due to the lone pair of Sn2+) and a zig-zag accordion-like geometry of slabs in the b–c plane. This implies a soft lattice—and if this lattice were mechanically stressed along the b and c directions, the Sn–Se bond length would not change directly, but instead the zig-zag geometry would be deformed like a retractable spring or an accordion. In addition, along the a direction, the weaker bonding between SnSe slabs provides a good stress buffer or ‘cushion’, thus dissipating phonon transport laterally28. The anomalously high Grüneisen parameter is therefore a consequence of the ‘soft’ bonding in SnSe, which leads to the very low lattice thermal conductivity.

Figure 4: Theoretically calculated phonon and Grüneisen dispersions, and measured lattice thermal conductivity.
Figure 4

a, Phonon dispersion. TA, TA′, transverse acoustic phonon scattering branches; LA, longitudinal acoustic phonon scattering branch. b, Grüneisen dispersion; inset, the average Grüneisen parameters along a, b and c axes. TA, red colour; TA′, green colour; LA, blue colour. The high symmetry points in the first Brillouin zone can be found in Extended Data Fig. 4. c, The lattice thermal conductivity comparison of SnSe along the b axis (ZTmax = 2.62) and hierarchical architectured PbTe-4SrTe-2Na (ZTmax = 2.2)11.

Using our DFT-calculated quantities (Debye temperatures and phonon velocities; see Extended Data Table 1) in the amorphous limit equation29, we can calculate the minimum lattice thermal conductivity of the three directions (a, b and c axis) at 770 K: κmina = 0.256 W m−1 K−1, κminb = 0.360 W m−1 K−1 and κminc = 0.326 W m−1 K−1. The trend of the theoretically predicted minimum thermal conductivity κmin is in good agreement with the experimental measurements (κlata < κlatc < κlatb). The theoretically calculated minimal thermal conductivities are slightly larger than those from experimental measurements, which could possibly result from (1) the value of the Lorenz number, 1.5 × 10−8 V2 K2, which can vary in the range (1–2.4) × 10−8 V2 K2; (2) the values of thermal diffusivity, which depend on the details of a fit to time-dependent reflectivity curves (in which instrumental error is about 5%); (3) errors associated with the determined sample thickness, the homogeneity of that thickness, and the determined sample density; and (4) the difficulty of measuring precisely along one crystallographic axis. Indeed, the EBSD analysis shows an 11° angular deviation of the crystallographic axes from the surface normal (Extended Data Fig. 2).

It is interesting to compare the lattice thermal conductivity of SnSe with that of state-of-the-art thermoelectric systems. The previously reported nanostructured all-scale hierarchical PbTe-4SrTe-2Na (with ZT of 2.2) exhibits a lattice thermal conductivity of 0.5 W m−1 K−1 (see Fig. 4c)11. The unprecedented ZT ≈ 2.62 of SnSe comes principally from an even lower lattice thermal conductivity of 0.23 W m−1 K−1 despite the lack of nanostructuring. We find good experimental repeatability for this high ZT, as evidenced by measurements on seven separate crystals prepared independently (Extended Data Fig. 10).

The value of ZT (2.62) at 923 K in SnSe crystals suggests that bulk materials with layered structures, anharmonic bonding and intrinsically ultralow thermal conductivity are promising candidates for providing high thermoelectric performance. It is remarkable that this ultralow thermal conductivity can be realized in a simple compound such as SnSe, as it does not have high molecular weight, a complex crystal structure or a large unit cell. These attributes are generally associated with low thermal conductivity. Compared to other high-performance thermoelectrics, our results for SnSe demonstrate that a high ZT also can be realized in simple layered, anisotropic and anharmonic systems, without nanostructuring.

Methods Summary

The SnSe samples were synthesized first in the form of ingots by heating to 1,223 K over 9.5 h, soaking at that temperature for 6 h, then furnace-cooling to room temperature, followed by vertical Bridgman crystal growth by heating to 1,223 K over 9.5 h, and then cooling from 1,223 K to room temperature with the sample moving at a rate of 2 mm h−1. The electrical resistivity and Seebeck coefficient were measured simultaneously in a helium atmosphere at 300–973 K using a ULVAC-RIKO ZEM-3 instrument system. We determined carrier concentrations using a custom-made apparatus at 300–823 K. The thermal diffusivity, D, was directly measured at 300–973 K by using the laser flash diffusivity method in a commercial Netzsch LFA-457 instrument. The thermal diffusivity was measured along the same direction as the electrical transport. The heat capacity, Cp, was indirectly derived using a representative sample (Pyroceram 9606) in the range 300–973 K. The total thermal conductivity was calculated using the formula κ = DCpρ, where ρ is the sample density, which was determined using the dimensions and mass of the sample and then reconfirmed by measurements using a gas pycnometer (Micromeritics AccuPyc 1340). Transmission electron microscope (TEM) investigation was conducted with a JEOL 2100F at 200 kV. The phonon and Grüneisen dispersions were obtained by using first-principles DFT phonon calculations within the quasi-harmonic approximation.

Online Methods

Starting materials

Sn chunk (99.999%, American Elements, USA) and Se shot (99.999%, 5N Plus, Canada).

Bridgman crystal growth

Ingots (20 g) of nominal composition SnSe were synthesized by mixing appropriate ratios of high-purity starting materials (Sn and Se) in quartz tubes. The tubes were evacuated to a pressure of 10−4 torr, flame-sealed, slowly heated to 1,223 K in 10 h, soaked at this temperature for 6 h and subsequently furnace cooled to room temperature. The temperature of 1,223 K (950 °C) is determined by differential thermal analysis (DTA) of SnSe (Extended Data Fig. 9). The obtained ingots were crushed into powder and placed in a quartz tube, evacuated and flame-sealed. This charged quartz tube was placed into another, bigger, quartz tube, evacuated and flame-sealed. The outer tube is used to prevent the crystal from oxidation by air because the inner tube breaks owing to the considerable difference of thermal expansion between the crystal and quartz, near fully dense samples are achieved. SnSe crystals with dimensions of 13 mm (diameter) × 20 mm (length) were obtained. Good experimental repeatability was evidenced by seven crystals with excellent quality, in terms of showing shining flat surfaces, no cracks or obvious defects, and no porosity or other macroscopic features.

Electrical transport properties

The obtained SnSe single crystals were cut into bars along different directions with dimensions 10 mm × 2.5 mm × 2.5 mm, which were used for simultaneous measurement of the Seebeck coefficient and the electrical conductivity using an UlvacRiko ZEM-3 instrument under a helium atmosphere from room temperature to 973 K. The uncertainty of the Seebeck coefficient and electrical conductivity measurements is 5%, and is about 10% for the power factor.

Hall measurements

High-temperature Hall coefficients were measured with a custom-made high-temperature apparatus, which provides a working range from 300 to 823 K. A sample with dimensions 8 mm × 3 mm × 1 mm was mounted and protected with an argon gas atmosphere to avoid possible oxidation at high temperature. The Hall resistance was monitored with a Linear Research AC Resistance Bridge (LR-700), with constant magnetic fields of ±1 T applied by using an Oxford Superconducting Magnet.

Thermal conductivity

High-density SnSe crystals were cut and polished into coins of diameter 8 mm (or rectangular samples with side length of 6 mm) and 2 mm thickness for thermal diffusivity measurements along different directions. The samples were coated with a thin layer of graphite to minimize errors from the emissivity of the material. The thermal conductivity was calculated from κ = DCpρ, where the thermal diffusivity coefficient (D) was measured along the same direction as the electrical transport using the laser flash diffusivity method in a Netzsch LFA457, the specific heat capacity (Cp) was indirectly derived using a representative sample (Pyroceram 9606) in the range from room temperature to 973 K, and the density (ρ) was determined using the dimensions and mass of the sample and then reconfirmed using gas pycnometer (Micromeritics AccuPyc1340) measurements. The thermal diffusivity data were analysed using a Cowan model with pulse correction, and heating and cooling cycles give reproducible values for each sample. The uncertainty of the thermal conductivity is estimated to be within 5%, considering the uncertainties for D, Cp and ρ. The combined uncertainty for all measurements involved in the calculation of ZT is around 15%.

Bandgap measurements

Room-temperature optical diffuse reflectance measurements were performed on finely ground powders to probe the optical energy gap. The measurements were performed using a Shimadzu Model UV-3101PC double-beam, double-monochromator spectrophotometer (ultraviolet–visible absorption spectroscopy). BaSO4 was used as a 100% reflectance standard. The generated reflectance versus wavelength data were used to estimate the bandgap by converting reflectance to absorption data according to the Kubelka-Munk equation: α/S = (1 − R)2/(2R), where R is the reflectance, and α and S are the absorption and scattering coefficients, respectively.

Differential thermal analysis (DTA)

Differential thermal analysis was performed with a computer-controlled Shimadzu DTA-50 thermal analyser. A ground SnSe crystal (30 mg total mass) was sealed in a fused quartz ampoule under vacuum. An ampoule containing α-Al2O3 of equal mass was sealed and placed on the reference side of the detector. The SnSe sample and reference were heated to 1,173 K at a rate of 3 °C min−1 and cooled at a rate of 3 °C min−1 to 50 °C. DTA measurements were run in two successive cycles of heating and cooling.

Transmission electron microscopy (TEM)

Transmission electron microscope investigations were conducted with a JEOL 2100F at 200 kV. A Gatan double-tilt heating stage was used to carry out the in situ heating experiments. The in situ experiment was conducted under a fast temperature ramping rate (20 K min−1) and more than 30 min sample incubation time. TEM samples were prepared by a standard conventional method with polishing, dimpling, and Ar-ion milling. Crystal Maker and Single Crystal software (CrystalMaker Software)were used to simulate the crystal structure and diffraction patterns.

Electron backscattering scatter diffraction (EBSD)

The EBSD investigation was conducted using an FEI Quanta ESEM equipped with Aztec and Oxford Channel 5 software for data collection and analysis, respectively. The sample was polished until mirror-like, and data were collected from a 1 mm2 area.

X-ray diffraction (XRD)

Samples with a cleavage plane of (100) were used for XRD characterization. The XRD pattern was obtained with Cu Kα (λ = 1.5418 Å) radiation in a reflection geometry on an Inel diffractometer operating at 40 kV and 20 mA and equipped with a position-sensitive detector.

Density functional theory (DFT) calculations

We perform DFT calculations using the Vienna Ab initio Simulation Package (VASP)30 with the projector augmented wave (PAW) scheme, and the generalized gradient approximation of Perdew, Burke and Ernzerhof (GGA-PBE)31 for the electronic exchange-correlation functional. The energy cut-off for the plane wave expansion is 500 eV. The Brillouin zones of SnSe are sampled by Monkhorst-Pack32 k-point meshes of (4 × 12 × 12). Atomic positions and unit cell vectors are relaxed until all the forces and components of the stress tensor are below 0.01 eV Å−1 and 0.2 kbar, respectively. Vibrational properties are calculated using the supercell (112 atoms in the SnSe supercell) force constant method by the alloy theoretic automated toolkit (ATAT)33. In the quasiharmonic DFT phonon calculations, the system volume is isotropically expanded by +2% from the DFT relaxed volume. The Grüneisen parameter (γ) is defined aswhich characterizes the relationship between phonon frequency and volume change. The Grüneisen parameters provide an estimate of the strength of the anharmonicity in a compound. The minimum lattice thermal conductivity can be calculated using the approach developed by Cahill34:where v, Θ and n are the phonon velocity, Debye temperature and the number density of atoms, respectively.

The crystal structure of SnSe at low temperature has a Pnma space group, with a = 11.58 Å, b = 4.22 Å and c = 4.40 Å. Increasing the temperature to 800 K, SnSe undergoes a phase transition from the low-temperature Pnma phase to the high-temperature Cmcm phase. In the Cmcm phase, the experimentally measured lattice constants are a = 4.31 Å, b = 11.71 Å and c = 4.31 Å (refs 35, 36). Our theoretically relaxed low-temperature and high-temperature SnSe lattice constants are respectively a = 11.794 Å, b = 4.215 Å and c = 4.550 Å, and a = 4.30 Å, b = 12.08 Å and c = 4.30 Å, which are in good agreement with the experimental measurements. The theoretically calculated electronic band structures of low-temperature and high-temperature SnSe phases are shown in Extended Data Fig. 4a, b. The theoretically calculated bandgap of the low-temperature SnSe phase is 0.61 eV, which is smaller than the experimentally measured 0.86 eV (Extended Data Fig. 5). Underestimation of bandgaps in the DFT framework is a well-known problem in the community. Even though DFT does not give quantitatively accurate predictions of bandgaps, the trends of bandgap between different phases are more reliable. The band structures show some features that suggest potential for thermoelectric materials: such as flattened valence bands and degenerate band extrema at off-Γ points. For the high-temperature SnSe phase (Extended Data Fig. 4b), the theoretically calculated bandgap is 0.39 eV (it is experimentally difficult to measure), which is much lower than that of the low-temperature phase. The smaller bandgap at high-temperature is in good agreement with the experimentally observed reduction of the Seebeck coefficient and the rise in electrical conductivity at high temperatures.

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Acknowledgements

This work was supported in part by Revolutionary Materials for Solid State Energy Conversion, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, and Office of Basic Energy Sciences under award number DE-SC0001054 (L.-D.Z., S.-H.L., Y.Z., H.S., G.T., C.U., C.W., V.P.D. and M.G.K.). TEM work was performed in the (EPIC, NIFTI, Keck-II) facility of the NUANCE Center at Northwestern University. The NUANCE Center is supported by NSF-NSEC, NSF-MRSEC, the Keck Foundation, the State of Illinois and Northwestern University.

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Affiliations

  1. Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA

    • Li-Dong Zhao
    • , Gangjian Tan
    •  & Mercouri G. Kanatzidis
  2. Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA

    • Shih-Han Lo
    • , Yongsheng Zhang
    • , C. Wolverton
    •  & Vinayak P. Dravid
  3. Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

    • Hui Sun
    •  & Ctirad Uher

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Contributions

L.-D.Z. synthesized the samples, designed and carried out thermoelectric experiments, and wrote the paper. S.-H.L. and V.P.D. performed the TEM characterizations. Y.Z. carried out the calculations. H.S. and C.U. carried out the Hall measurements. G.T. helped with sample synthesis. L.-D.Z., S.-H.L., Y.Z., H.S., G.T., C.U., C.W., V.P.D. and M.G.K. conceived the experiments, analysed the results and co-edited the manuscript. S.-H.L. and Y.Z. contributed equally.

Competing interests

The authors declare no competing financial interests.

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Correspondence to Mercouri G. Kanatzidis.

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https://doi.org/10.1038/nature13184

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