Phonon Collapse and Second-Order Phase Transition in Thermoelectric SnSe
Since 2014 the layered semiconductor SnSe in the high-temperature phase is known to be the most efficient thermoelectric material. Making use of first-principles calculations we show that its vibrational and thermal transport properties are determined by huge non-perturbative anharmonic effects. We show that the transition from the phase to the low-symmetry is a second-order phase transition driven by the collapse of a zone border phonon, whose frequency vanishes at the transition temperature. Our calculations show that the spectral function of the in-plane vibrational modes are strongly anomalous with shoulders and double-peak structures. We calculate the lattice thermal conductivity obtaining good agreement with experiments only when non-perturbative anharmonic scattering is included. Our results suggest that the good thermoelectric efficiency of SnSe is strongly affected by the non-perturbative anharmonicity. We expect similar effects to occur in other thermoelectric materials.
Thermoelectric materials can convert waste heat into useful electricityGoldsmid (2010); Behnia (2015). The thermoelectric efficiency of a material is measured by the dimensionless figure of merit , where is the Seebeck coefficient, the electrical conductivity, the temperature, and the thermal conductivity, constituted by electronic and lattice contributions. The thermoelectric efficiency can be thus enhanced by decreasing the thermal conductivity while keeping a high power factor . Materials have been dopedKim et al. (2013); Pei et al. (2011); Heremans et al. (2008) or nanostructuredVineis et al. (2010); Minnich et al. (2009) in order to get a high power factor combined with a low thermal conductivity, yielding, i.e., in PbTeHsu et al. (2004). In the proximity to a phase transition may also soar, as in the case of CuSeLiu et al. (2013). Recently, however, Zhao et al. reported for SnSeZhao et al. (2014) the highest thermoelectric figure of merit ever reached in a material without doping, material treatment or without being in the proximity to a phase transition: above K.
SnSe is a narrow gap semiconductor that crystallizes at room temperature in an orthorhombic phase. At KZhao et al. (2014); Adouby (1998); Chattopadhyay et al. (1986); Von Schnering and Wiedemeier (1981) it transforms into a more symmetric base-centered orthorhombic structure (see Fig. 1). The order of the transition is not clear: some worksAdouby (1998); Chattopadhyay et al. (1986); Zhao et al. (2014) claim it is a continuous second-order transition and others it has a first-order characterVon Schnering and Wiedemeier (1981). A recent workDewandre et al. (2016) argues the transition occurs in two steps, where increasing temperature induces first a change in the lattice parameters that induces after a lattice instability. There is no inelastic scattering experiment so far for the high-temperature phase, which should show a prominent phonon collapse at the transition temperature if it belonged to the displacive second-order type Holt et al. (2001); Weber et al. (2011); O’Neill et al. (2017).
The most interesting thermoelectric properties appear in the high-temperature phase, where the reduction of the electronic band gap increases the number of carriers providing a higher power factor, while the thermal conductivity remains very lowZhao et al. (2014). The value of the intrinsic of SnSe remains controversial, as the extremely low isotropic W/mK value at K reported by Zhao et al.Zhao et al. (2014) could not be reproduced in other experiments, where a clear anisotropy is shown and the in-plane thermal conductivity is considerably largerIbrahim et al. (2017); Sassy et al. (2014); Cheng et al. (2014). The disagreement is possibly due to large number of Sn vacancies in the original workZhao et al. (2014). The lattice thermal conductivity of the phase has been calculated from first principles solving the Boltzmann transport equation (BTE) using harmonic phonons and third order force-constants (TOFCs) obtained perturbatively as derivatives of the Born-Oppenheimer energy surfaceCarrete et al. (2014); Skelton et al. (2016). The phase has imaginary phonon frequencies in the harmonic approximationDewandre et al. (2016); Skelton et al. (2016); Yu et al. (2016), as expected for the high-symmetry phase in a second order displacive transitionIizumi et al. (1975); Ribeiro et al. (2018); Jian et al. (2016), and it is stabilized by anharmonicityDewandre et al. (2016); Skelton et al. (2016), hindering the calculation of Skelton et al. (2016).
In this letter, by performing ab initio calculations fully including anharmonicity at a non-perturbative level, we show that the phonon mode that drives the instability collapses at the transition temperature demonstrating that the transition is second-order. Anharmonic effects are so large that the spectral function expected for some in-plane modes deviates from the simple Lorentzian-like shape and shows broad peaks, shoulders and satellite peaks, as in other monochalcogenidesLi et al. (2014a); Ribeiro et al. (2018). We calculate the lattice thermal conductivity of the phase by combining the anharmonic phonon spectra with perturbative and non-perturbative TOFCs. We show here for the first time that non-perturbative anharmonic effects are not only crucial in the phonon spectra, but also in high-order force-constants, which here have a huge impact on the calculated thermal conductivity: agrees with experimentsIbrahim et al. (2017) only with non-perturbative TOFCs. Similar strong non-perturbative effects are thus expected for other thermoelectric compounds.
The group/subgroup index of the / transition is 2, making a displacive second-order transition possibleToledano and Toledano (1987). In this scenario, the transition temperature is defined as the temperature at which the second derivative of the free energy with respect to the order parameter that transforms the structure continuously from the phase () into the () vanishes. Symmetry Orobengoa et al. (2009); Perez-Mato et al. (2010) dictates that the amplitude of the transition is dominated by the distortion pattern associated to a non-degenerate mode () at the zone border Y point with irreducible representation (see Fig. 1 for the distortion pattern). This means that is proportional to the eigenvalue of the free energy Hessian matrix associated to this irreducible representation: .
In this work we calculate the free energy Hessian matrix using the stochastic self-consistent harmonic approximation (SSCHA)Errea et al. (2014); Bianco et al. (2017); Monacelli et al. (2018), which is applied making use of ab initio density-functional theory (DFT) calculations within the Perdew-Burke-Ernzerhof (PBE)Perdew et al. (1996) or Local Density Approximation (LDA)Perdew and Zunger (1981) parametrizations of the exchange-correlation functional (see Supplementary Material for the details of the calculationsBaroni et al. (2001); Paulatto et al. (2013); Giannozzi et al. (2009, 2017); Li et al. (2014b)). The SSCHA is based on variational minimization of the free energy making use of a trial harmonic density matrix paramatrized by centroid positions and force-constants (bold symbols represent in compact notation vectors or tensors). The centroid positions determine the most probable position of the atoms and is related to the amplitude of their fluctuations around . The free energy Hessian can be calculated asBianco et al. (2017)
where and are third- and fourth-order non-perturbative force-constants obtained as quantum averages calculated with : . The force-constants are generally different from the perturbative ones obtained as derivatives of the Born-Oppenheimer potential at the minimum: . in Eq. (1) is a function of the SSCHA frequencies and polarization vectors obtained diagonalizing , with the atomic mass ( labels both an atom and Cartesian index). The frequencies obtained instead from the free energy Hessian after diagonalizing , e.g. , can be interpreted as the static limit of the physical phononsBianco et al. (2017). The contribution of is negligible with respect to the identity matrix (see Supplementary Material) and thus it is neglected throughout.
The calculated temperature dependence of is shown in Fig. 2 for LDA and PBE for two different lattice volumes in each case. In all cases is positive at high temperatures, but it rapidly decreases with lowering the temperature, vanishing at . This phonon collapse is consistent with a second-order phase transition between the and . We indeed check that a SSCHA calculation at ( K) starting from the relaxed low-symmetry phase yields the high-symmetry atomic positions for the centroids. Thus, the is not a local minimum of the free energy above , completely ruling out the first-order transition. Our result is at odds with the conclusions drawn in Ref. Dewandre et al., 2016. First, because at the calculated in Ref. Dewandre et al., 2016, which is estimated by comparing the free energies of the two structures, the mode of the phase is stable, which implies this phase is a local minimum at , and, thus, the transition is of first-order typeDewandre et al. (2016). And second, because it is arguedDewandre et al. (2016) that the instability at is produced by a slight change in the in-plane lattice parameters induced by temperature (from to ), which makes the transition a two-step process. We do not see this sudden appearance of the instability, the mode is always unstable at the harmonic level even exchanging and (see Supplementary Material).
The obtained transition temperature strongly depends on the exchange-correlation functional and the volume. Within LDA ranges between K with theoretical lattice parameters and K with experimental lattice parametersZhao et al. (2014) (see Table 1 for the lattice parameters). Within PBE barely changes between the experimental and theoretical lattice parameters. We attribute this result to the fact that the in-plane lattice parameters and are in perfect agreement with the experimental results within PBE, while LDA clearly underestimates them. The theoretical lattice parameters are estimated neglecting vibrational contributions to the free energy. In order to estimate the role of the thermal expansion, we calculate the stress tensor including vibrational contributions at the anharmonic level following the method recently developed by Monacelli et al.Monacelli et al. (2018). The in-plane contribution of the stress tensor calculated at the temperature closest to , , shows that both theoretical LDA and PBE lattices should be stretched. In the LDA case it is clear that stretching the lattice increases . In the PBE case, when we take a stretched lattice to reduce , increases from K to K. In all cases the other in-plane component of the stress tensor, , is very similar to . The LDA transition temperature with the experimental lattice parameters yields the transition temperature in closest agreement with experiments, which is estimated to be of KLi et al. (2015); Zhao et al. (2014); Adouby (1998); Chattopadhyay et al. (1986); Von Schnering and Wiedemeier (1981). The underestimation of the transition temperature may be due to the approximated exchange-correlation or the finite supercell size taken for the SSCHA. It is interesting to note that in the experiments by Ibrahim et al.Ibrahim et al. (2017), where it is stated that the samples contain much less Sn vacancies than in Ref. Zhao et al., 2014, the in-plane thermal conductivity seems to increase at around K, which may be a fingerprint of a phase transition (see Fig. 5).
The phonon collapse predicted here should be experimentally measurable by inelastic neutron scattering (INS) experiments. INS experimentsLi et al. (2015) show a softening of a zone-center optical mode of the phase upon heating, which is consistent with the condensation of the mode after the transition. By making use of a dynamical ansatzBianco et al. (2017), we calculate the mode-projected phonon anharmonic self-energy (see Supplementary Material), from which we obtain the phonon spectral function:
Peaks in represent phonon excitations observed experimentally. Replacing in Eq. (2) the Lorentzian approximation is recovered, in which each peak is represented with a Lorentzian function centered at with a linewidth proportional to Bianco et al. (2018).
Fig. 3 compares the harmonic phonon spectrum with the anharmonic one in the Lorentzian approximation obtained at K within LDA in the experimental lattice (the results below are also obtained within the LDA in the experimental lattice). The anharmonic correction is very large for most of the modes across the Brillouin zone. Within the harmonic approximation, there are five unstable modes: two (, ) at , two (, ) at and one () at . The instabilities at would cause ferroeletric transitionsHong and Delaire (2016); Skelton et al. (2016), but they suffer from a huge anharmonic renormalization that prevents it. and are also stabilized by anharmonic effects. The mode however, even if it is strongly affected by anharmonicity, remains unstable at K (see Fig. 4a), i.e., the phase is not a minimum of the free energy and the crystal distorts adopting the low-symmetry phase.
In highly anharmonic materialsDelaire et al. (2011); Li et al. (2014a); Ribeiro et al. (2018, 2018); Li et al. (2014a); Paulatto et al. (2015); Bianco et al. (2018), the spectral functions show broad peaks, shoulders and satellite peaks, strongly deviating from the Lorentzian picture. In Fig. 4 we show the spectral function keeping the full frequency dependence on the self-energy, without assuming the Lorentzian lineshape. The spectral function clearly reproduces the collapse of the mode at the transition temperature. The calculated spectral functions show that the strong anharmonicity present on the phonon frequency renormalization is also reflected on the spectral function. The strongly anharmonic features specially affect in-plane modes in the 25-75 cm energy range. For instance, at the point the mode, who describes a vibration along the in-plane axis in opposite direction for the Sn and Se atoms (see Fig. 1) and is stabilized by anharmonicity, shows a double peak structure and a broad shoulder (see Fig. 4e). The mode that describes the same vibration () but in the other in-plane direction also shows a very complex non-lorentzian shape. The overall consequently has a broad shoulder at 25 cm as marked in Figs. 4c, which is less acute as temperature increases. At the Y point there are also two modes, , whose eigenvector is plotted in Fig. 1, and , which describes the same displacement but in the other in-plane direction, that show a strongly anharmonic non-Lorentzian shape. The modes with very complex line-shapes are those that show the largest linewidth in the Lorentzian limit (see Fig. 3), for instance, cm for the half width at half maximum of the mode. These modes have strongly anomalous spectral functions and large linewidths because they can easily scatter with an optical mode close in energy and an acoustic mode close to . We identify this by directly analyzing which phonon triplets contribute more to the linewidth. It is interesting to remark that if is calculated by substituting by , the anomalies of these modes become weaker (see Supplementary Material). This underlines that in the phase the third-order derivatives of are not sufficient to calculate the phonon linewidths and that higher order terms are important, which are effectively captured by .
In Fig. 5 we present the lattice thermal conductivity calculated with the SSCHA frequencies () and non-perturbative TOFCs (). For comparison we also calculate substituting by . The calculation is performed solving the BTE assuming the single-mode relaxation time approximation (SMA), whose validity was confirmed against a more accurate iterative method Fugallo et al. (2013) (see Supplementary Material). The thermal conductivity of SnSe is very low, mainly because the contribution of optical modes is strongly suppressed by the large anharmonicity and because the contribution of acoustic modes is also reduced due to the large scattering among themselves and with the mode. We compare these results with the values obtained by Ibrahim et al.Ibrahim et al. (2017) above the possible transition at 600 K (only the in-plane are reported at these temperatures) and with the values obtained by Zhao et al.Zhao et al. (2014) above the transition at 800 K. The lattice thermal conductivity is in better agreement with experimental results using instead of , which overestimates the lattice thermal conductivity along the in-plane and directions. This is consistent with the larger phonon linewidths obtained with the non-perturbative TOFCs. The agreement for the in-plane with the measurements by Ibrahim et al.Ibrahim et al. (2017) is good in the non-perturbative limit, contrary to previous calculations that underestimate itSkelton et al. (2016). The calculated out-of-plane is also in good agreement with the results by Zhao et al.Zhao et al. (2014), but we find that their ultralow results for the in-plane , in contradiction with the values in Ref. Ibrahim et al., 2017, are underestimated. It is not surprising that the thermal conductivity in vacancy free SnSe is lower along the out-of-plane direction due to the weaker bonding. Thus, our calculations support the interpretation that the weak anisotropy and ultralow thermal conductivity measured by Zhao et al.Zhao et al. (2014) above the transition was produced by the large amount of Sn vacancies present in their samplesIbrahim et al. (2017); Wei et al. (2016); Zhao et al. (2016).
In conclusion, our first-principles calculations show that the vibrational properties of SnSe in the phase are dominated by huge non-perturbative anharmonic effects. We show how the collapse of the mode is responsible for the second-order phase transition between the high-symmetry and the low-symmetry phase. The calculated transition temperature is volume and functional dependent. The spectral functions of in-plane modes are characterized by very anomalous features, with shoulders and double peaks, clearly deviating from the standard Lorentzian-like shape. These results will eventually be crucial to interpret future INS experiments for the high-temperature phase. The calculated in-plane thermal conductivity is in good agreement with the experiments by Ibrahim et al.Ibrahim et al. (2017), but not with those by Zhao et al.Zhao et al. (2014), supporting the interpretation that in the latter experiment the thermal conductivity was lowered by Sn vacancies. Our results show for the first time that the inclusion of non-perturbative effects is crucial not only for renormalizing phonon spectra, but also for obtaining third-order force-constants that yield a lattice thermal conductivity in agreement with experiments. Similar huge non-perturbative anharmonic effects are expected in other good thermoelectric materials.
The authors acknowledge fruitful discussions with O. Delaire. Financial support was provided by the Spanish Ministry of Economy and Competitiveness (FIS2013- 48286-C2-2-P), the Department of Education, Universities and Research of the Basque Government and the University of the Basque Country (IT756-13). U.A. is also thankful to the Material Physics Center for a predoctoral fellowship. Computer facilities were provided by the Donostia International Physics Center (DIPC), the Spanish Supercomputing Network (FI-2017-2-0007) and PRACE (2017174186).
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