Tin selenide, SnSe, has been the focus of intense interest as a thermoelectric material, with a low lattice thermal conductivity between 600 K and 800 K, and reciprocally, a large power factor 1,3-7. The structural phase transition in SnSe, from orthorhombic GeS-type with Pnma symmetry at low temperature to orthorhombic TlI-type with symmetry Cmcm at high temperature is at the heart of the low lattice thermal transport coefficient 8-13. However, the nature of the phase transition remains under investigation: is it a displacive phase transition similar to most ferroelectric materials, or an order-disorder transition with dynamic disorder in the high temperature phase? The initial identification of the phase transition as displacive, with a well-defined transition temperature following group/subgroup relationships, was based on X-ray and neutron diffraction experiments 14,15. Though sometimes still discussed as an order-disorder phenomena, the modern consensus built upon optical methods, inelastic neutron scattering, and coupled theoretical probes 13,16-20, is that the phase transition is driven by condensation of a highly anharmonic zone-boundary (Y-point) phonon within the high-temperature phase. In particular, theoretical calculations involving three-body forces have shown a double-well 0 K potential energy surface that leads to intrinsic phonon anharmonicity within the high-temperature regime12; and when third-order force constants are determined in a non-perturbative manner, the experimentally-observed phonon instability within the Cmcm phase is recovered and explains the low lattice thermal conductivity at high temperatures 16. Despite strong experimental and theoretical evidence of the soft phonon picture in the literature, the direct observation of local bonding phenomena driving this phonon instability is lacking.
The difference between a displacive phase transition and an order-disorder transition has been discussed in different texts. Examples include the ferroelectric transition in PbTiO3 which is mostly of displacive type with some local distortions, with soft phonon modes describing the transition 21-23. In contrast, the ferroelectric transition in BaTiO3 is considered order-disorder, with the titanium atom dynamically disordered along the [111] directions in the high-symmetry cubic phase 24-27. The description of the phase transition in SnSe as a displacive transition is most often represented in the literature 1,5,8,11,12,14,17,28. Thermodynamically, a displacive transition should be 2nd order with changes in the thermal behavior at the phase transition, whereas an order-disorder transition changes the configurational space, resulting in an entropy change. In the displacive transition model, atomic positions change gradually from low symmetry coordinates in the GeS-type structure (Pnma) to high symmetry coordinates in the TlI-type structure (Cmcm). The change in bonding around the Sn atom affects the band gap and could potentially create a partially filled band, where 2 Sn electrons are available for 4 orbitals. In addition, the Sn lone pair electrons are stereochemically active, producing an asymmetric bonding environment. The reduced band gap of the TlI-type structure, obtained by DFT calculations, is therefore due to charge transfer from Sn to Se. Enhanced phonon scattering decreases the lattice heat conductivity so that the power factor in SnSe reaches its highest value at temperatures near the phase transition (ZT of roughly 2.2–2.6 at 913 K).1
Several observations indicate the phase transition of SnSe may be unusual: primarily, all crystallographic studies performed using traditional Rietveld refinement methods result in larger displacement parameters for Sn than for Se 14,15. Notably, the mass difference between Sn and Se is substantial and one would therefore expect that displacement parameters for Sn be smaller than for Se. The large displacement parameters of Sn and Se (based on a harmonic approximation) indicate that the Sn atom resides in a wide potential well. The double well potential for the Sn atoms suggested recently indicates that the Sn atom can disorder into multiple positions, 12potentially resolves disparate observations of the system, as it is possible to overlay the GeS-type (Pnma) structure onto the TlI-type (Cmcm) structure, if a disordered (orientationally averaged) GeS-type (Pnma) based local model is used. In this case, the correlation length of the GeS-type atomic motif would be reduced as thermal excitations increase the disorder of the Sn atoms, which increasingly move between energetically accessible positions. A local probe is therefore needed to conclusively investigate if the local bonding in SnSe above the phase transition temperature retains the bonding observed in the GeS-type (Pnma) structure, or if there is a change in the orientation of the lone pair electrons as its average TlI type structure type might suggest. The high real-space resolution intrinsic to time-of-flight (TOF) neutron pair distribution function (PDF) studies has proven key in elucidating local bonding distortions, nanoscale correlations, and other local to long-range structural complexities in a host of functional materials 26,29-32. Herein we combine neutron TOF diffraction and PDF studies with ab initio molecular dynamics (AIMD) simulation techniques to investigate the short-range order in SnSe from room temperature to several hundred degrees above its crystallographic phase transition, thereby providing new insight into the curious nature of its phase transition.
Average Structure: A three-dimensional surface contour map with projection of neutron diffraction data measured as a function of temperature is shown in Figure 1(A). Specific diffraction peaks merge above 775 K, as highlighted in Figure 1(B), a direct observation of the structural phase transition from GeS-type (Pnma) to TlI-type (Cmcm) SnSe. Rietveld refinement results by GSAS II 33 obtained at all temperatures are consistent with previous reports 14,15. Refinements of two temperature points (400 K and 800 K) are shown in Figure 1(C). For comparison, the GeS-type (Pnma) structure model is applied up to 1000 K, and the TlI-type (Cmcm) structure model is applied from 750 K to 1000 K. The evolution of lattice parameters as well as atomic anisotropic displacement parameters (ADP) is shown in Figure S1, while the agreement factor (Rw), lattice parameters and atomic position parameters are provided in Table S1. A phase transition between 775 K and 800 K is clearly observed from the variation of lattice parameters, which correspond to the vanishing of (002) and (220) reflections in Figure 1(B) and 2(C). Furthermore, the temperature evolution of average Sn-Se bond distances derived from Rietveld fits, are shown in Figure 1(D) (corresponding labels are indicated in (G)). In this work and in previous Rietveld analysis studies, the application of a strict crystallographic model (with infinite correlation length) throughout the temperature range biases the analysis in favor of a static structure with large displacement factors. There is a consistent observation of larger displacement parameters for the Sn atoms with respect to the significantly lighter Se atoms within the low symmetry GeS-type (Pnma)structure, and persisting in the TlI-type (Cmcm) structure, as shown in Figure 1(E) and (F). Such behavior is often associated with disorder, where the heavy Sn atoms are displaced from their average position in the lattice, with static and/or dynamic disorder present.
Neutron PDF and small-box modeling: Figure 2(A) shows a contour plot of PDFs measured as a function of temperature, with the corresponding 1D datasets aligned below as Figure 2(B). A number of important features can be directly observed in real space. First, significant changes in the real-space structure begin far below and continue gradually up to the crystallographic phase transition (for example, the first and second nearest-neighbor maxima below 3.75 Å merge into one broad asymmetric maximum between RT and 800 K, and a strong minimum emerges gradually at 25 Å as another disappears gradually at ~28 Å approaching ~675 K). These trends are juxtaposed with the trends in correlations between 3.75 Å and 12 Å, and specific higher-r correlations (for example at ~ 15 Å and 24 Å) where no changes in position or intensity are observed as a function of temperature. Above the crystallographic phase transition, ~800 K, the only significant trend is a shift of all peaks to higher r, consistent with thermal expansion of the lattice.
A more in-depth interpretation can be made by modeling the local atomic structure of SnSe across the data series. Local (low-r) range fits to the 1.5-10 Å PDF region and medium (high-r) range fits to the 10-30 Å PDF region were initially fit in the small-box modeling program PDFgui 34 with the determined average structure models at each temperature (results summarized in supporting information Figure S2 and S3). The GeS type (Pnma) average structure model describes the medium range structure well leading up to the crystallographic phase transition, and the TlI-type (Cmcm) average structure model does the same above the phase transition. Model parameters extracted from full-range PDF refinements (1.5 to 30 Å, Figure S4) provide close agreement to Rietveld refinement results. The story is different when fitting solely the low-r PDFs; select results are shown in Figure 2(C-E). The partial PDFs corresponding to the Sn-Sn, Sn-Se, and Se-Se pair-pair correlations extracted from small-box modeling are displayed below the data and fits. It is evident that the GeS-type (Pnma) model gives a high-quality fit to the PDF at 400 K in Figure 2(C). By contrast, the low-r PDF for 800 K is not well fit with the higher symmetry TlI-type (Cmcm) model, particularly the peak centered at 2.9 Å, corresponding to the Sn-Se nearest-neighbor correlation (Figure 2(D)). The first and second Sn-Se correlations are changed significantly between 400 K and 800 K; notably the two distinct pair correlations merge into one asymmetric peak above 800 K. The persistent presence of this asymmetry confirms the need for a lower symmetry model above the crystallographic phase transition. Thus, the GeS-type (Pnma) model was applied to the whole temperature range, resulting in similar quality fits at all temperatures (the result for 800 K is shown in Figure 2(E)). The lower symmetry model specifically provides a substantial improvement to the peak fit centered at 2.9 Å, with the refinement agreement factor Rw dropping from 0.104 in Figure 2(D) to 0.058 in Figure 2(E). The partial PDFs indicate this comes about through retained asymmetric bonding involving the Sn-Se nearest neighbors.
The temperature-dependent partial PDFs of the Sn-Sn, Sn-Se, and Se-Se correlations extracted from the local structure models are presented in Figure 3(A-C). It is clear that Sn-Sn correlations change little over the full temperature range, while Sn-Se correlations shift/merge significantly. A natural extension is to explore the length scale for this local ordering motif and the extent to which it survives above the crystallographic phase transition. Figure 3(D) compares the refinement agreement factor from variable r range modeling using the lower and higher symmetry variants of the SnSe structure for data collected at 600 K, 800 K, and 1000 K. Results for additional temperatures are summarized in supporting information Figure S5 and S6. At 600 K (below the crystallographic phase transition), the lower symmetry GeS-type (Pnma) model is the best description over the whole real-space range, meaning that the local structure matches the average structure. Above the transition, at 800 K and 1000 K, the GeS-type (Pnma) model also performs significantly better below ~12 Å and slightly better up to ~24 Å. Beyond that range, the high-temperature average TlI-type (Cmcm) model provides a similar quality of fit to the data. The detailed nature of Sn-Se bonding is found to be particularly sensitive to the length scale of the applied real-space analysis. Figure 3(E) displays the refined Sn-Se bond distances resulting from variable r-range PDF fits for 800 K data. The distances within the layers (d1, d2 and d3) remain nearly constant and equal to distances in the refined average structure with variation in the maximum real-space range probed (rmax). However, Sn-Se bonding to adjacent layers is found to be length-scale dependent, with d4 decreasing and d5 increasing as the rmax (or probed correlation length scale) decreases. This is brought about by strong local Sn displacement, creating one shorter bond and one longer bond to Se atoms in adjacent layers. The average structure is thus realized through a superposition of the locally ordered states. These results demonstrate that SnSe maintains something of its low-temperature local symmetry across the whole temperature range, yet the coherence of these structural distortions decreases as temperature is increased. At the highest temperature probed here, 1000 K, this coherence approaches just 12 Å, or approximately 2 lattice lengths a of the SnSe structure. A technique that is sometimes applied to the analysis of Bragg diffraction data to give additional insight into phase transitions from lower-symmetry to higher-symmetry parent structures is the so-called symmetry-mode or distortion-mode Rietveld refinement.35 Symmetry-mode analysis has recently been incorporated into a number of PDF data modeling workflows,36,37 and has been applied here in order to follow the length-scale dependence of the group-subgroup relationship between the Cmcm and Pnma structures. Results are shown in Figure 2(D, E) and Figure S7. Four displacive mode amplitudes (a1, a2 for Sn and a3, a4 for Se, displayed in an inset) are applied here (within the TOPAS v6 software suite38 in conjunction with models from ISODISTORT39) to model the changes of atom positions across the structural phase transition. The amplitude of displacements a1 and a3 stay constant and low across the temperature range in both sequential Rietveld and r-dependent PDF refinements. By contrast, the a2 and a4 displacements show a sharp decline in their amplitudes at ~ 800 K, in good agreement with the ADP values from Rietveld refinement. Importantly, the amplitudes of the a2 and a4 modes retain elevated values for PDF refinements below 10 Å, in agreement with the maintained local symmetry discussed above.
A non-biased combinatorial appraisal of transition states (CATS) analysis 40,41 of changes in the data is shown in Figure S7, and it confirms the above assessment: the SnSe structure changes continuously and gradually from RT up to its crystallographic phase transition, and it does so in a non-uniform manner across real-space correlations (with pair correlations changing very little below 10 Å, and with significant higher-r changes peaking at the phase transition yet spanning the entire temperature regime). Further description of the CATS analysis is given in the Methods section. Overall, there are suggestions of both order-parameter-like transitions (gradual bonding changes as a function of T) and order-disorder type structural rearrangements (rigid/unchanging local ordering motifs that lose their coherence across the lattice with increasing T) at play in SnSe. The layer-type nature of the structure provides a mechanism for coherent Sn distortions within a layer, and reduced correlation/registry with neighboring layers. Se atoms will be affected as well, enabling easier disordering of the adjacent Sn atoms. The thermal energy allowing the Sn atom to disorder is therefore expected to manifest itself in structural changes well below the phase transition temperature. This is clearly apparent in Figure 2 and Figure S7, where changes in longer distance correlations appear 100 K to 200 K below the nominal phase transition. The result across and above the SnSe phase transition is an average structure with higher observed symmetry and anomalously high ADP (consistent with the Rietveld analyses in our and others work). These phenomena are entirely consistent with the soft phonon view of the phase transition observed with inelastic neutron scattering and supported with theoretical work.
Large-box and ab initio MD modelling: We pursued large box Reverse Monte Carlo (RMC) modeling using RMCProfile 42 and Ab Initio Molecular Dynamics (AIMD) simulations to further investigate the nature of the large ADP and the origins of local-to-long range complexity in SnSe structures. The RMC modeling was completed on 400 K and 800 K data with a supercell more than 100 Å on each side, as shown in Figure 4(A) and 4(B), with corresponding folded unit cell ‘point clouds’ on the right. Excellent fits for both temperatures were obtained for the whole PDF range, as shown in Figure S9, with fits to the normalized structure factor, F(Q), and Bragg diffraction data shown in Figure S10. We further performed AIMD simulations for theoretical support (see Figure S11). The radial distribution function (RDF) for all pair-pair correlations obtained from AIMD simulations are in good agreement with RDFs calculated from the RMC models (Figure S11). ADPs resulting from Rietveld analysis are compared with the atomic ‘point clouds’ and trajectories from RMC analysis and AIMD simulations in Figure S12, respectively, for (a) 800 K and (b) 400 K datasets. Note that the shape and size of ‘point clouds’ resulting from RMC modeling are comparable to the shape and size of ADP thermal ellipsoids from Rietveld refinement. The RMC ‘point clouds’ of Sn and Se exhibit dispersive quasi-prolate ellipsoid distributions, similar to the ADP distributions. However, the density plots from RMC and AIMD analyses indicate Se atom sites that are small ellipsoids (see Supporting Information Figure S12), and Sn atom sites that are larger ellipsoids asymmetric in nature (Figure 4(C, E, F)).
In divalent Sn, the non-zero overlap of the 5s and 5p orbitals tends to produce a bonding asymmetry, which is usually ascribed to a stereochemically active lone pair. The bonding asymmetry results in three shorter bonds that tend to be nearly orthogonal, and three or more longer bonds, that are also affected by the orientation of the lone pair electrons. The asymmetry in the Sn atom locations in SnSe is shown in more detail in Figure 4(C), where the distributions are projected along different unit cell directions. It is clear at low temperature, via ADP, RMC, and AIMD analyses, that neighboring Sn atoms distort/rotate to avoid end-to-end bonding, resulting in coherent/cooperative tilting of their locations. At higher temperatures, the Sn atoms continue to be off-center, retaining their local bonding configurations, but in such a way that the distributions are no longer coherent (they do not displace in the same directions in all unit cells). The Sn probability distribution grows in the (ab)-plane at high temperature, retaining a similar shape. However, a triangle shape Sn probability distribution is formed at high temperature in the (ac)-plane, with Sn atoms distorting/rotating in both directions. The disorder of the Sn atoms affects the coordinating Se atoms and is expected to change the form of the local minima. Thus, the existence of orientationally averaged local distortions in the SnSe system is confirmed leading up to and persisting above its global crystallographic transition. An illustrated comparison of the displacive and order-disorder phase transition models is given in Figure 4(D), with experimentally determined (from RMC analysis) probability density distributions of Sn (in grey) overlaid with average (from Rietveld analysis) first coordination polyhedron in Figure 4(E) and 4(F). The observed long-range structure at high temperature derives from locally off-centered motifs that are orientationally averaged, similar to the dynamic order-disorder phase transition observed in other Jahn-Teller and lone-pair driven phase transition phenomena.43 It should be noted that order-disorder behavior need not rule out a displacive phase transition; its signatures are observed during the displacive-type phase transitions of the perovskite PbTiO344 and even in cases where a crystallographic phase transition is absent upon warming, such as in the cubic rock-salt PbTe.45 In the latter case, an unusual correlated local structure dipole formation linked to coupled soft optical and acoustic modes were proposed.46 The order-disorder nature of SnSe is similarly important to its electrical and thermal transport properties.
It is instructive to consider several limitations of the present study. First, the neutron total scattering method used in this work is an energy integrated scattering approach, meaning that the data incorporates pair correlations from elastic (long-lifetime) correlations as well as from dynamic correlations. Future experimental work with the dynamic PDF technique (completed with an inelastic scattering instrument)47,48 could discriminate the energy-dependence of the high temperature local atomic distortions uncovered. Remarkably large and anisotropic dynamics have recently been uncovered and linked to the contradictory thermal and electronic conductivity properties in the cubic thermoelectric GeTe through energy-resolved PDF techniques.49 However, such an endeavor is challenged as present-day inelastic instruments with the requisite energy resolution and energy range lack the Q-range required for high quality real-space PDF. Second, as a powder-averaged technique, the orientational dependence of local atomic correlations is lost. Future experimental work with 3D-PDF methods46,50 on single crystal SnSe could further clarify the present work. To be clear, the present work establishes persisting local Sn dipoles, but cannot differentiate whether they are fluctuating/dynamic or static and orientationally averaged, nor in what crystallographic directions and to what extent the Sn ions propagate and fluctuate in time.
This work affirms that a model of the phase transition in SnSe must include the following characteristics:
- The Sn-Se local structure remains asymmetric at all temperatures
- The correlation length over which one Sn position affects the adjacent Sn position becomes shorter with increasing temperature
- The potential energy surface of the Sn is likely a multi-minima surface, where additional energy minima become accessible with increasing temperature
- At a high enough temperature, the Sn atoms are likely dynamically disordered over multiple local minima positions, wherein the average structure is consistent with the TlI-type motif.
We further note that observations of such a dynamic and/or orientationally averaged atomic configuration will be biased by the specific length- and time-scale sensitivities of the employed probes. The nature uncovered here in SnSe through examination of its local to long-range structural motifs thus fills in key details among previous work addressing its phase transition.
In summary, we find that the structural phase transition in SnSe features an asymmetric Sn-Se coordination environment consistent with the GeS-type structure at all temperatures probed. Above the crystallographic phase transition, thermal excitations allow the Sn atom to access alternate higher energy positions, reducing the length scale over which the locally off-centered Sn atoms are correlated. The dynamic disorder destroys the long-range coherence, allowing a satisfactory description of the average atomic arrangement in the TlI-type structure, albeit with large anisotropic ADPs for Sn. Additionally, indications of the phase transition are already apparent at temperatures well below 800 K. The correlation length is found to be strongly temperature dependent and diverges at the phase transition temperature. The dynamic disorder, described as “rattling” of the Sn atoms, creates a strong phonon scattering cross section that couples efficiently to the phonon spectrum increasing the backscattering rate, and thus decreasing the lattice thermal conductivity. The change in correlation length of the local Sn motif observed via neutron total scattering reconciles previously juxtaposed views of the phase transition in thermoelectric SnSe.