Normal-to-topological insulator martensitic phase transition in group-IV monochalcogenides driven by light

A material potentially exhibiting multiple crystalline phases with distinct optoelectronic properties can serve as a phase-change memory material. The sensitivity and kinetics can be enhanced when the two competing phases have large electronic structure contrast and the phase change process is diffusionless and martensitic. In this work, we theoretically and computationally illustrate that such a phase transition could occur in the group-IV monochalcogenide SnSe compound, which can exist in the quantum topologically trivial Pnma-SnSe and nontrivial Fm3¯m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Fm\bar 3m$$\end{document}-SnSe phases. Furthermore, owing to the electronic band structure differences of these phases, a large contrast in the optical responses in the THz region is revealed. According to the thermodynamic theory for a driven dielectric medium, optomechanical control to trigger a topological phase transition using a linearly polarized laser with selected frequency, power and pulse duration is proposed. We further estimate the critical optical electric field to drive a barrierless transition that can occur on the picosecond timescale. This light actuation strategy does not require fabrication of mechanical contacts or electrical leads and only requires transparency. We predict that an optically driven phase transition accompanied by a large entropy difference can be used in an “optocaloric” cooling device. Researchers now theorize that memory devices which normally use temperature effects to store or retrieve information can be improved using laser-sensitive materials. Often found in rewritable optical disks, phase-change memory materials use heat variations to flip between glassy and crystal states. However, such changes can damage sensitive electronics. Jian Zhou from Xi’an Jiaotong University in China, Ju Li at the Massachusetts Institute of Technology, Cambridge, United States, and colleagues propose that phase transformations in metal compounds such as tin–selenium (SnSe) may be initiated with lasers. Using quantum theory calculations, the team identified a SnSe phase transition that can be performed in just picoseconds. This topological phase transition between two quantum phases could lead to high-performance electronics. Thermodynamic analysis revealed that the rapid changes between the phases might also make SnSe materials effective refrigerants. Topologically electronic state receives great attention during the past decades, owing to their topologically protected metallic edge states. Topological phase transition between nontrivial and trivial states is thus intriguing for not only information storage but also for dissipationless information transport, which is usually driven by pressure or gate voltage. Triggering topological phase transition without physical contact fascinates the scientific community. Due to contrast band topology, the trivial and nontrivial topological materials possess different optical responses. Based on this, a novel non-contacting optical illumination scheme is predicted by first-principles calculations to drive topological phase transition in IV–VI semiconductors.


Introduction
Phase change memory (PCM) materials are one of the key components of next-generation information storage 1 . Currently, the most widely used commercialized PCM materials are Ge-Sb-Te alloys 2 . These materials can exist in both crystalline and amorphous phases with distinct optical or electronic transport features, which are separated by a significant energy barrier. The phase transformation between these phases is reconstructive owing to the diffusional displacements of atoms. The drawback of a temperature-driven transformation is that due to the equipartition theorem, all phonon modes are involved, and thus, the total driving energy and the heat dissipation are significant. Furthermore, the phase change occurs on a timescale on the order of a few to a few hundred nanoseconds. Hence, one would like to find a PCM material that can simultaneously satisfy the following requirements: (i) it can exist in at least two structural phases, (ii) these phases have distinct physical properties that can be easily measured, preferably in a contact-free manner, (iii) the phase transformation is displacive/martensitic, where the atomic motions are diffusionless, and (iv) the directed phase transition preferably occurs within a single phonon oscillation period (picosecond) in the barrier-free limit. The main idea is to strongly bias only the reaction coordinates of the structural transition, instead of exciting all the phonon modes, such as in a temperature-driven transition. With these requirements fulfilled, the phase transition would have the benefits of low energy consumption, low heat load and fast switching speed.
The discovery of topological materials during the past decade has offered ample interesting physical effects 3 . Transitions between topologically different states have attracted tremendous attention both theoretically and experimentally. One example is group-VI transition metal dichalcogenide (TMD) monolayers, such as MoTe 2 and its analogues, which exist in the 2H (normal insulator, NI) 4 and 1T′ (topological insulator, TI) phases 5 . Theoretical predictions have shown that the 2H → 1T′ phase transformation can occur under lithiation 6 , electron doping 7 , or tensile strain 8 . Later experimental efforts showed that such a phase transformation can occur under Ar plasma bombardment 9 or gate voltage 10 . During this process,~1/6 of the Mo-Te chemical bonds need to be broken, and then, the same number of new Mo-Te bonds will be formed. Hence, even though the energy difference between 2H-MoTe 2 and 1T′-MoTe 2 is small, the energy barrier separating the two phases is high (~2000 J/cm 3 when we take an energy barrier of 1.05 eV/f.u. and a MoTe 2 monolayer thickness of 7.74 Å 5,8 ). Most other topological phase transitions in a single material occur via modulation of their electronic band dispersion under strain engineering 11,12 , electrostatic gating [13][14][15] , or temperature change 16,17 . Some of these phases have a short lifetime (volatile) once the external field is removed. In addition, these phase transitions usually require electrochemical, electrical, or mechanical contacts and patterning. This stimulates us to explore topological phase change materials whose phase transition can be triggered through a noncontacting scheme with intermediate energy barriers 18 .
In this work, based on thermodynamic analysis and first-principles density functional theory (DFT) calculations, we propose that a group-IV monochalcogenide compound (such as SnSe or other analogues) is a good platform to realize nonvolatile topological phase transition by optomechanical stimulation. This group-IV monochalcogenide compound family possesses a few interesting properties 19,20 . The SnSe compound can exist in multiple phases, such as in the space groups of Pnma, Fm3m, and Cmcm. While many studies focus on the hightemperature ( > 800 K) phase transition from Pnma to Cmcm 21,22 , here, we study the two energetically lower polymorphs, namely, Pnma (ground state) and Fm3m (metastable). Geometrically, Pnma-SnSe is a layered structure. This phase has been attracting great interest owing to its good thermoelectric property [22][23][24] . Fm3m-SnSe shows a rocksalt geometry and has been proven theoretically 25,26 and experimentally 27,28 to be a novel type of quantum topological materiala topological crystalline insulator (TCI) with a large change in near band-edge (NBE) optical susceptibilities. Hence, the topological phase transition between Pnma and Fm3m would be interesting and promising for potential applications. Nevertheless, except for foreign atom doping 29,30 , theoretical studies on such phase transitions are very rare. Here, we apply the random phase approximation (RPA) to calculate the optical response features. Owing to the topological band inversion feature of Fm3m-SnSe, its dielectric screening effect is much stronger than that of semiconducting Pnma-SnSe. Therefore, according to the thermodynamic theory of a driven dielectric medium, when the sample is exposed to a linearly polarized laser (LPL) with THz frequency, the free energy of Fm3m-SnSe reduces faster with greater alternating electric field strength (the optical tweezers effect) than that of Pmna-SnSe. Hence, we predict a phase transformation between Pnma and Fm3m under an intermediate intensity LPL. Such a transformation can even be barrier-free, with significantly enhanced kinetics. Owing to the entropy difference between the two phases, we can even design a thermodynamic phase-change cycle to utilize the "optocaloric" effect of SnSe as a refrigeration medium.

Density functional theory
We employ DFT calculations 31,32 with generalized gradient approximation (GGA) 33 treatment of the exchange-correlation energy term using the Vienna ab initio simulation package (VASP) 34,35 . The projector augmented wave (PAW) method 36 and a planewave basis set with a cutoff energy of 300 eV are applied to treat the core and valence electrons in the system, respectively. The first Brillouin zone is represented by a Γ-point-centered Monkhorst-Pack k-mesh 37 with a grid density of 2π × 0.02 Å −1 along each dimension. The convergence of these parameters is well tested. To correct the underestimation of long-range dispersion in the GGA treatment, we include the semiempirical quantum chemical method DFT-D3 term in the total energy expression 38 . Other treatments give similar results. The convergence criteria of the total energy and force component are set to 1 × 10 −7 eV and 0.001 eV/Å, respectively. Spin-orbit coupling (SOC) is self-consistently included due to the high Z of Sn and Se atoms. We perform ab initio molecular dynamics (AIMD) simulations and confirm that both phases are thermally stable at room temperature ( Fig. S1 in the Supplementary Information). To evaluate the finite temperature behavior, we also calculate phonon dispersions and the frequency density of states using the finite displacement method, as implemented in the Phonopy code 39 .

Results and discussion
When an LPL containing an alternating electric field (Ẽ ω 0 ; t ð Þ¼E 0 e Àiω 0 tê i , where eî is a unit vector along the i-th direction) is applied to the system, its contribution to the free energy can be evaluated via the thermodynamic theory of an external reservoir bath and Legendre transformation. By taking the electric field magnitude as the thermodynamic variable, one adds an additional term into the free energy density (in units of energy/volume) 40 , Here, ε (1) (ω 0 ) is the real part of the dielectric function tensor at optical frequency ω 0 . Note that here, the optical frequency ω 0 is much larger than the vibrational frequency ω vib (typically < 6 THz); thus, the ion motion contribution to the dielectric function is neglected. Microscopically, the incident photon carries an optical electric field and excites electron subsystems. Then, phonons are stimulated by electron-phonon coupling.
When the phonon displacement is large enough, the system can go beyond elastic deformation and undergo a phase transition. We start our discussions with the geometric and energetic features of the SnSe compound in both the Fm3m and Pnma phases (Fig. 1). Pnma-SnSe shows a layered structure. According to previous works, each SnSe layer possesses an in-plane (along the y-direction) ferroelectricity (P 0 = 1.5 × 10 −10 C/m) 41,42 . In the bulk Pnma-SnSe compound, the layers are antiferroelectrically stacked, exhibiting a centrosymmetric structure with no net spontaneous polarization. This structure has been demonstrated to be a semiconductor with an optical band gap of 0.9 eV, showing high-performance thermoelectric behavior (ZT > 2.2) 22,23 . In contrast, in the rocksalt Fm3m-SnSe, all the Sn and Se atoms are ionically bonded. Its band dispersion calculation reveals an inverted band order near the Fermi level, which is topologically protected by spatial mirror symmetry 25 . Thus, this structure belongs to the TCI family, with a calculated band gap of0 .2 eV 43 . Our DFT-D3 scheme yields Pnma-SnSe lattice parameters of a = 4.169 Å, b = 4.515 Å, and c = 11.558 Å (each supercell contains two SnSe layers) and Fm3m-SnSe lattice parameters of a = b = c = 4.247 Å. These results agree well with experimental data (Pnma-SnSe: a = 4.15 Å, b = 4.45 Å, and c = 11.50 Å; Fm3m-SnSe: a = b = c = 4.24 Å) 44,45 . From the atomic structures, one observes that when the even and odd numbered layers in Pnma-SnSe alternatively translate along the ± y-direction by 0.18 × b (green arrows in Fig. 1a), this phase can change into Fm3m-SnSe. During this internal coordinate shuffling, the supercell transformation strain is actually small (compared with, e.g., ref. 46 ); thus, little mesoscopic elastic constraint strain energy is required when a certain embedded region undergoes transformation. The volume difference between the two phases is within 0.5%. These facts demonstrate that the phase transition is martensitic and displacive, without requiring random atomic diffusion. Hence, the time scale of the phase transition can be greatly reduced, and the energy input to drive the phase transition is also small.
Our DFT calculations show that Pnma-SnSe has a lower ground state energy than Fm3m-SnSe by 17 meV per formula unit (meV/f.u.). This agrees well with previous DFT works 43 . To estimate the energy barrier between the two phases and phase transition kinetics, we apply the cell-variable nudged elastic band (NEB) method 47,48 to calculate the reaction path versus total energy. As shown in Fig. 2a, the energy barrier from Pnma-SnSe to Fm3m-SnSe is calculated to be 42 meV/f.u., corresponding to 123 J/cm 3 . This value is much smaller than the energy barrier from monolayer 2H-MoTe 2 to 1T′-MoTe 2 (~2000 J/cm 3 ). The phonon spectrum of the saddle point crystal structure is shown in Fig. 2b, where only one large imaginary optical branch can be observed (at Γ). This imaginary mode corresponds to reaction coordinate shuffling to the Pnma phase, while the imaginary modes at Z (0, 0, 1/2) correspond to the transition to Fm3m. When the system transits back from Fm3m-SnSe to Pnma-SnSe, the energy barrier will be 25 meV/f.u. (73 J/cm 3 ). By applying the Arrhenius law to estimate the (where E b is the energy barrier, N is the total number of formula units, and ν 0 = 10 −12 s is the trial frequency), we note that Fm3m-SnSe can exist for over 20 years at room temperature (T = 300 K) when N = 60 (~3 nm 3 ). On the other hand, the moderate energy barrier value per unit also indicate that the phase change between the two phases may be triggered by intermediate external field energy input with fast kinetics. One fascinating feature during this structural phase transformation is the accompanying topological electronic phase transition. As previously discussed, Pnma-SnSe is a topologically trivial semiconductor, with its highest valence band and lowest conduction band located far from the Fermi level [ Fig. 3a, left panel]. In contrast, the topologically nontrivial band gap insulator Fm3m-SnSe has a valence and conduction band inversion at the L-point in the Brillouin zone [ Fig. 3b, left panel]. We plot the calculated band dispersions in the Supplementary Information, Fig. S2. Such a band topology change leads to quite a few important contrasting properties between the two phases, such as robust metallic surface states without backscattering. During a topological phase transition, a band gap closing and reopening process usually occurs. Thus, we also calculated the band gap value variation along the reaction path [ Fig. 2a]. The band gap closing structure lies in the Fm3m-SnSe energy basin. Here, we show that the optical responses to an LPL of the two phases are very different. One can examine the electron-contributed dielectric function based on the random phase approximation (RPA) 49,50 Here, ju n;k i and E n,k are the periodic part of the Bloch wavefunction and its corresponding eigenvalue of band n at k, with indices v and c representing valence and conduction bands, respectively. ξ is a phenomenological damping parameter representing the temperature and disorder effects of the sample. From the above equation, one sees that the real part of the dielectric function is determined by two factors: (i) the direct energy band gap (E c,k -E v,k ) and (ii) the Berry connectioñ A nm k ð Þ ¼ ihu n;k j∇ k ju m;k i. Hence, compared with the NI Pnma-SnSe, the TCI Fm3m-SnSe has not only a relatively small band gap but also a larger Berry connection magnitude owing to its inverted band order with an enhanced band correlation. These facts indicate that the magnitude of ε (1) of Fm3m-SnSe should be much larger than that of Pnma-SnSe, giving a large contrast of their NBE optical responses.
We thus calculate the dielectric functions of the two phases by choosing ξ = 0.025 eV, as shown in the right panels of Fig. 3a (Pnma-SnSe) and Fig. 3b (Fm3m-SnSe). Due to the geometric anisotropy, the x-, y-, and z-components of ε(ω) of Pnma-SnSe are slightly different. In our current setup, when the incident photon energy is less than the direct band gap (ℏω < 0.70 eV, where ε (2) (ω) is zero), the real parts of the dielectric functions of Pnma-SnSe almost remain constant, with ε xx (1) (ω) = 19.9, ε yy (1) (ω) = 17.5, and ε zz (1) (ω) = 17.9. On the other hand, the real parts of the dielectric functions of isotropic Fm3m-SnSe below the direct band gap are much larger. For example, at an incident energy ℏω 0 = 0.17 eV, the calculated values are ε xx (1) (ω 0 ) = ε yy (1) (ω 0 ) = ε zz (1) (ω 0 ) = 109.9, which are nearly six times those of Pnma-SnSe at the same incident photon energy. We also plot the optical response of the DFT-calculated saddle point in the Supplementary Information (Fig. S3). We use a simplified k·p model to illustrate the topological contribution to the optical responses of Fm3m-SnSe (see the Supplementary Information for details). We find a relationship between the band gap at L-point (1/2, 1/2, 1/2) (measured as 2|m| in the model) and the dielectric constant, ε 1 ð Þ 0 ð Þ ¼ 1 þ α m j j Àγ . For the topologically nontrivial band structure, the fitting coefficient α is much larger than that of the trivial band structure, indicating a large Berry connection contribution, as previously discussed.
Owing to the centrosymmetric geometric feature of both phases, one can hardly apply a static electric field to drive the phase transition, as in ferroelectrics. Alternatively, here, we propose a noncontacting optomechanical approach. According to our previous discussions, the free energies G E (Eq. 1) under LPL pulse (ℏω 0 = 0.17 eV) exposure of the two phases are different. Taking a y-directional LPL as an example (the other directions are similar), we calculate the free energy difference between the two phases under different electric field magnitudes (Fig. 4), Here, Ω is the volume of a formula unit, and ε 0 is the vacuum permittivity. As the electric field magnitude ε 0 increases, the Gibbs free energy difference ΔG quadratically reduces to zero. We see that the critical electric field is 0.35 V/nm. Above this value, LPL exposure makes  Fm3m-SnSe more stable, enabling a martensitic topological phase transition. To drive the opposite phase transition (from Fm3m-SnSe to Pnma-SnSe), even though the Pnma phase has a lower energy, we can apply a laser frequency at ℏω 0 ′ = 0.25 eV to accelerate the process. At this incident energy, the calculated ε xx (1) (ω 0 ′) values are 20.1, 41.2, and 6.5 for Pnma-SnSe, the transition saddle point, and Fm3m-SnSe, respectively. We estimate the electric field required to drive an ultrafast barrier-free phase transformation. For the Pnma to Fm3m transition, the incident LPL pulse can be ℏω 0 ′′ = 0.12 eV along the ydirection, and the calculated dielectric functions are ε yy (1) (ω 0 ′′) = 17.2, 107.5 and 78.7 for Pnma, the saddle point and Fm3m, respectively. Hence, the electric field strength of 0.56 V/nm would drive a barrier-free phase transformation. For the Fm3m to Pnma transition, an electric field along the x-direction of 0.69 V/nm (incident energy ℏω 0 ′ = 0.25 eV) can suppress the energy barrier. When the bulk energy barrier (in units of meV/f.u.) vanishes, the transition kinetics is similar to that of spinodal decomposition, but in terms of the phonon displacement 48 instead of the chemical composition order parameter. With such a driving force that causes a vanishing bulk energy barrier, the typical nucleationgrowth kinetics can be avoided, and the transition process could occur within picoseconds. We also illustrate this process (see the Supplementary Information for details), which reveals that the barrier-free phase transition occurs with a discontinuity of the electric displacement. Note that the phase transition is not temperature driven. Therefore, it does not require a large amount of latent heat to trigger the phase transition, as in the conventional temperature-driven process.
We then propose that this optomechanically driven phase transformation of SnSe will lead to a promising cooling mechanism 57 , which we refer to as the optocaloric effect. We calculate the entropy (S = S ph + S el , where S ph and S el are the phononic and electronic entropies, respectively) difference between the two phases (Fig. 5a). The calculation details can be found in the Supplementary Information. The largest entropy change appears at T = 40 K, with ΔS = -0.23 k B /f.u. At room temperature (T = 300 K), Fm3m-SnSe has a higher entropy than Pnma-SnSe by 0.01 k B /f.u. (k B is the Boltzmann constant, corresponding to 0.5 J K −1 kg −1 ). Such latent heat (TΔS) between two phases (one stable, one metastable, at T when optical field is turned off) can be used to drive cooling, similar to producing ice from liquid water in a summer day using a specially made mechanical shocking device, and then letting ice → liquid water for cooling purposes on that summer day. This is not a Carnot cycle (Fig. 5b), which is based on a single-phase medium (ideal gas) and consisting of four quasi-static thermodynamic legs (two isothermal which contributes sensible heat, and two adiabatic legs, and all four legs require gradual pressure/volume changes). Here, instead, we propose to use latent heat rather than sensible heat during a twolegged cycle (Fig. 5c): in one leg of the loop we use optomechanical shock to non-quasistatically drive the phase transition from the stable → metastable phase (stable and metastable phases defined when the optical field is turned off, for that T), and in the other leg we allow the metastable phase to relax back to the stable phase thermally. This will be a two-step loop. During the first step, the Fm3m-SnSe phase absorbs latent heat (Q in ) and quasi-statically (and isothermally) transforms to Pnma-SnSe, like ice melting in a cold drink. In the second step, we apply a laser pulse to drive Pnma-SnSe back to the Fm3m-SnSe phase. This leg is not a quasi-static process (out of local thermal equilibrium, since the reaction coordinate gains a lot more kinetic energy than the other vibrational degrees of freedom), which cannot be represented by a thermodynamic state curve in an equilibrium thermodynamic macro-state plot like Fig. 5b.
We also perform calculations for other analogous group-IV monochalcogenide compounds (Fig. S7). During geometric optimization, Pnma-PbSe and Pnma-PbTe relax to the Fm3m structure owing to the large size of the Pb ion. Fm3m-PbS is also energetically more stable than Pnma-PbS. Hence, alloyed materials, such as Pb 1−x Sn x X (X = S, Se, Te) 58 compounds, are possible candidates for tuning Pnma ↔ Fm3m transitions. For example, when x = 0.30, experimental evidence shows that the topological transition temperature is~250 K in Pb 1−x Sn x Se 59 . We also know that in the Fm3m-GeTe and Fm3m-SnTe compounds, there is a spontaneous cubic-torhombohedral structural phase transition in which the sublattice Ge/Sn and Te atoms displace along the [111] direction 60,61 . This forms a ferroelectric state, which is topologically trivial with a wide band gap. Under hydrostatic pressure, the rocksalt Fm3m phase can be dynamically stable. Hence, unless under pressure, the phase transition between the two phases is not accompanied by topological changes in the electronic band structure, but nonetheless has certain dielectric response contrast. The GeSe, GeS, SnS, and PbS compounds all have similar results to SnSe, whose Pnma and Fm3m phases are locally stable when not optically pumped. The optical responses of these compounds are also similar to those of SnSe (Fig.  S8).

Conclusion
In conclusion, we systematically calculate the energetics and optical responses of the SnSe compound and chemical analogues in their low temperature phases, namely, normal insulator Pnma and topological crystalline insulator Fm3m. We propose that the phase transition can also be driven when one applies a pulsed laser with a selected frequency. Such a topological phase transition, similar to the transformation of TMD monolayers from 2H to 1T′ but with lower energy barrier and energy input, belongs to martensitic displacive phase transitions, but with very little transformation strain during and after the transformation, and thus has low elastic constraint penalty energy, ensuring fast kinetics and reversibility. We predict a promising optocaloric effect in SnSe based on the phase-change cycle, taking advantage of the low dissipation and stress of such rapid optically driven transitions, which could be useful as a refrigerant. Schematic of the optocaloric process for SnSe. a Entropy difference between two phases as a function of temperature. b Entropytemperature (S-T) in a Carnot thermodynamic cycle. c Phase-change cooling cycle, that is significantly different from the Carnot cycle. Here solid lines denote quasi-static processes, whereas dashed line denotes out-of-thermal-equilibrium process.