Unraveling a novel ferroelectric GeSe phase and its transformation into a topological crystalline insulator under high pressure

Germanium selenide is a promising material for electronic, photovoltaic, and thermoelectric applications; however, structural phase transitions of GeSe under pressure are controversial. Combining evolutionary algorithms, density functional theory, tight-binding method, and laser-heated diamond anvil cell experiments, pressure-induced phase transitions of GeSe are thoroughly investigated. Two novel intermediate phases are predicted to exist in between the well-known α-GeSe and the recently discovered β-GeSe under high pressure. α-GeSe is found to transform into a rhombohedral crystal structure with a space group of R3m at a low hydrostatic pressure. The R3m phase of GeSe exhibits robust ferroelectricity analogous to GeTe. By further increasing the pressure to approximately 6 GPa, the R3m phase is predicted to transform into a rock-salt structure, becoming a 3D topological crystalline insulator with an inverted band structure. The newly discovered GeSe high-pressure phases greatly enrich our knowledge of IV–VI compounds. The atoms in germanium selenide can adopt more configurations than previously thought according to a study by researchers in China. A phase change, the rearrangement of atoms in a solid from one crystal structure to another, can radically alter a material’s properties. Such a change in phase can be induced by exposing matter to high pressure. Kuo Li from the Center for High Pressure Science and Technology Advanced Research in Beijing, Yue Chen from The University of Hong Kong, and their co-workers have combined multiple computational and experimental techniques to identify the phases of germanium selenide, a material useful for energy conversion. Recently a high-pressure phase called β-GeSe was discovered in addition to the common low-pressure α-GeSe phase. The researchers predict two further phases at intermediate pressure, each with its own distinctive electrical properties. Pressure-induced phase transitions in GeSe are investigated from theory and experiment. Two new phases are predicted to exist in-between the well-known α-GeSe and the recently discovered β-GeSe under pressures. It is predicted that α-GeSe transforms into a rhombohedral structure with a space group of R3m and exhibits robust ferroelectricity at a low hydrostatic pressure. Laser-heated DAC experiments have been conducted to provide further evidence on the existence of the R3m phase. By increasing the hydrostatic pressure to approximately 6 GPa, the R3m phase is predicted to transform into a rock-salt crystal structure (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}) and become a topological crystalline insulator.


Introduction
The IV-VI binary compounds have attracted much research interest due to their promising photovoltaic and thermoelectric applications. For example, a SnSe single crystal was recently found to exhibit a record high thermoelectric figure of merit [1][2][3] . PbTe is an important material for photodetectors 4,5 and solar cells 6 that is also a well-known high-performance thermoelectric material because of its low thermal conductivity 7 . GeTe has a ferroelectric phase transition at approximately 670 K and is important for electronic memory applications 8 . Moreover, rock-salt SnTe 9 and SnSe 10,11 were found to be topological crystalline insulators (TCIs), in which the topological states are protected by crystal symmetry 12,13 .
As a promising IV-VI compound, GeSe is a narrow band gap semiconductor and possesses an orthorhombic Pnma crystal structure (GeS-type) at ambient conditions 14 . Most existing studies focus on this laminar Pnma structure, whereas the properties and structural phase transitions of GeSe under pressure are largely unexplored. Until recently, it was unknown that a new β-GeSe phase exists under a hydrostatic pressure as low as 6 GPa 15 . The β phase still exists after recovering to ambient conditions and its electrical conductivity was found to be essentially temperature independent. Bhatia et al. 16 performed high-pressure experiments on single crystal GeSe, in which an abrupt drop in the electrical resistance at 6 GPa was observed because of the transformation to a face-centered-cubic phase. Nonetheless, Hsueh et al. 17 attributed the reduction of electrical resistance to the metallization of the α phase. It is obvious that a systematic study is imperative for a more complete understanding of the high-pressure behavior of GeSe.
New phases with interesting properties may exist under high pressures 18 ; e.g., a novel intermediate phase of SnTe with the Pnma space group was recently discovered using angle-dispersive synchrotron X-ray diffraction 19 . Because high-pressure experimental characterizations are expensive and time-consuming, a theory-assisted ground state structural search is ideal for exploring potential new highpressure phases 20,21 . Ab initio evolutionary algorithms were recently shown to be extremely powerful in new crystal structure predictions. This algorithm has been proven to be effective in finding the global minimum of the potential energy by mimicking Darwin's theory of evolution. Employing the natural selection of the fittest (e.g., the phases with the lowest potential energies in this work), a new generation can be generated using operators, such as heredity, soft mutation, and transmutation. The ground-state phases may be identified after a sufficient amount of generations. Details of the operators for generating new generations can be found in ref. 20 . By using this method, a transparent sodium high-pressure phase was predicted and later confirmed in experiments 22 ; a metallic two-dimensional boron sheet was predicted and synthesized in the experiments 21 . In this work, we systematically investigate the GeSe compound under pressure from extensive evolutionary algorithm searches and laser-heated diamond anvil cell (DAC) experiments. Two new phases were found to exist in between the α and the β phases with intriguing properties.

Computational details
Structure searches were performed by combining the evolutionary algorithms and density functional theory (DFT) as implemented in USPEX 23 and Vienna ab initio simulation package (VASP) 24 . The maximum number of atoms allowed in a unit cell was 18. All existing crystal structures of GeSe were taken into account in the evolutionary searches 25 . New generations were obtained using the following schemes: heredity (40%), random (20%), softmutation/coormutation (20%), and transmutation (20%). The projector augmented wave (PAW) method 26 combined with the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) functional 27 was applied. Forces and full stress tensors were calculated and used to relax crystal structures into their ground states. The Brillouin zones were meshed using the gamma-centered Monkhorst-Pack method 28 with a density of approximately 0.03 × 2π Å −1 . A planewave energy cut-off of 350 eV was used and the convergence criterion of electronic self-consistent calculations was set to 10 −6 eV. A large 4 × 4 × 4 supercell of 128 atoms was adopted to study the phonon dispersions using the small displacement method 29 . The effect of longitudinal optical-transverse optical (LO-TO) splitting was considered in the phonon calculations. To ensure accurate phonon calculations, we applied a higher total energy convergence criterion of 10 −8 eV.

Experimental details
A piece of α-GeSe sample, sandwiched between MgO for thermal insulation, was loaded in a DAC (diamond culet size d = 400 μm). The sample was compressed to~2 GPa as calibrated by Ruby fluorescence under room temperature, and then double-sided heated to 1600 K for approximately 45 min using an ytterbium fiber laser with a wavelength of 1070 nm. The cell was then cooled down and the pressure was released. The sample was picked up from the gasket, and the X-ray powder diffraction (XRD) was measured using the powder mode of Rigaku XtaLab Pro X-ray single crystal diffractometer, under the voltage of 50 kV and the current of 24 mA, with Mo Kα as the Xray source. Rietveld refinement was conducted using the Topas 3 program with the simulated annealing method.

Results
Two energetically preferable new phases were found in our ab initio evolutionary searches under pressure. As shown in Fig. 1a, GeSe has a sophisticated phase β δ Fig. 1 Enthalpies of competing GeSe high-pressure phases and the crystal structures. a Enthalpy differences of the competing crystal structures of GeSe as functions of hydrostatic pressures. The dashed line represents an extrapolation of the enthalpy difference. b Primitive cells of the two intermediate phases that exist in between αand β-GeSe. The blue and yellow spheres represent the Ge and Se atoms, respectively. The lattice constant, the angle between two lattice vectors, and the relative atomic displacement from the cell center are labeled with L, β, and δ, respectively. The dashed gray arrows indicate the [1 1 1] direction transition path. With increasing pressure, GeSe transforms from the α phase through two intermediate phases of R3m and Fm3m into the recently discovered β phase. Although the experimental phase transition pressures were not precisely reproduced (e.g., the ambient α phase is predicted to exist under a small negative pressure of approximately −1 GPa and the β phase is predicted to exist above 10 GPa, in contrast to the experimental 6 GPa 15 ), the correct transition order from the α to the β phase is predicted. The differences between the exact values of theoretical and experimental phase transition pressures may be related to the complex high-pressure experimental conditions and the exclusion of vibrational entropy in DFT calculations. The newly predicted R3m phase has a trigonal layered crystal structure of the GeTe prototype and transforms into the Fm3m phase with a rock-salt crystal structure at approximately 6 GPa. A similar pressure-induced displacive phase transition between R3m and Fm3m was also observed in another IV-VI compound, GeTe 30 , suggesting the robustness of our theoretical study.
In addition to the energetics, the lattice dynamics of the R3m and Fm3m phases are also investigated to examine the possibility of experimental synthesis. As shown in Fig. 2a, all phonon modes of the R3m phase have positive frequencies, indicating that this phase is dynamically stable. However, the Fm3m phase only becomes dynamically stable under a suitable hydrostatic pressure (see Fig. 2b). At 0 GPa, a large number of phonon modes of the Fm3m phase exhibit imaginary frequencies (see Fig. 2c), suggesting that this phase is dynamically unstable. These soft modes induce a phase transformation from the Fm3m phase to the R3m phase at low pressures. As the external pressure is increased, the soft phonon modes are gradually hardened, consistent with previous DFT calculations 31 . Therefore, the Fm3m phase only exists under suitable hydrostatic pressures.
To obtain more evidence for our newly predicted GeSe high-pressure phases, we have performed laser-heated DAC experiments. From the XRD pattern shown in Fig. 3, α-GeSe and MgO were unambiguously identified. Additionally, several extra diffraction peaks were observed, such as the two peaks indicated by arrows near 21°. By testing all of the competing crystal structures, we found that the R3m phase reasonably fits these peaks in the Rietveld refinement, suggesting that our theoretical prediction is robust. The refined lattice parameters and the molar fractions of the different phases are summarized in Table 1. It is seen that the experimental lattice parameters are in reasonable agreement with the DFT results. It is also noted that laser heating is critical for synthesizing the R3m phase of GeSe; the phase transition only occurs through melting and crystallization under pressure. Furthermore, a rhombohedral phase similar to our R3m  Fig. 3 Rietveld refinement of the GeSe sample recovered from 2 GPa and 1600 K. The black circles and red solid curves represent the experimental data and the simulated results, respectively. The purple, blue, green, and gray curves denote the contributions from α-GeSe, R3m-GeSe, MgO, and the differences between the experimental and the simulated results, respectively phase was recently observed in GeSe-AgSbSe 2 alloys 32 . As alloying GeSe with AgSbSe 2 may be regarded as an internal chemical pre-compression, the observation of this rhombohedral phase represents additional indirect evidence for our work.
To better understand the newly discovered highpressure phases of GeSe, their electronic band structures and density of states (DOS) were further studied. As shown in Fig. 4a, the R3m phase was found to be a semiconductor at 0 GPa with a direct band gap of approximately 0.49 eV near the high symmetry point, L. From the projected band structure and DOS, it was observed that the bottom of the conduction bands mainly consists of Ge 4p states, whereas the top of the valence bands is largely the Se 4p states. Most interestingly, the Fm3m phase has a gapless band structure at 8 GPa (see Fig. 4b). The gap is closed in the vicinity of the high symmetry point, L. Similar to the R3m phase, the electronic states of the Fm3m phase below the Fermi level are mostly attributed to the Se 4p states, with the main contribution to the electronic states above the Fermi level coming from the Ge 4p orbitals.
The R3m phase of GeSe has a similar crystal structure as α-GeTe and exhibits robust ferroelectricity [33][34][35] . The existence of the R3m phase makes GeSe distinctly different from most other IV-VI compounds. In contrast to the rock-salt crystal structure, the relative atomic displacement (RAD) of the Ge or Se atom along the [1 1 1] direction (see Fig. 1b) in the R3m phase results in a spontaneous polarization 36 . As a rule of thumb, the larger the RAD, the stronger the spontaneous polarization and the higher the ferroelectric transition temperature 37 . From DFT calculations (see Table 2), the RAD of R3m-GeSe is similar to that of α-GeTe implying a comparable ferroelectric transition temperature. Moreover, the ferroelectricity of R3m-GeSe is as robust as that of α-GeTe as demonstrated by their similar polarization densities. The ferroelectric nature of the newly discovered R3m-GeSe indicates potential applications in information storage and real-time optical processing 33,38 ; however, a more efficient synthesis technique still needs to be developed.
The electronic band structure of the newly discovered Fm3m phase of GeSe was further studied. As shown in Fig. 5a, where the band structure was calculated without considering spin-orbit coupling (SOC), Fm3m-GeSe has a gapless Dirac cone along the L-W path. After taking into account the SOC effects in our DFT calculations, we found that a band gap is opened, and the Dirac cone is broken consequently (see Fig. 5b). The Fm3m phase has an inverted band order around the L point, which cannot be broken by SOC, indicating that the Fm3m phase is different from that of the time-reversal Z 2 topological insulator. Furthermore, the band gap (E L ) at the high symmetry point, L, can be closed and reopened as the hydrostatic pressure increases from −8 to   Table 2 Lattice parameters (L and β), relative atomic displacements (δ), unit cell volumes (V), and the polarization densities (P) of R3m-GeSe and α-GeTe at 0 GPa Polarization was calculated using Berry phase method 8 GPa (see Fig. 5c). All these calculations imply a crystal-symmetry-driven nontrivial topological state. To explicitly define Fm3m-GeSe as a TCI, the mirror Chern number n M needs to be evaluated 12 . According to the k•p theory of band structure 39 , the n M of the mirror symmetry plane (110) of the Fm3m phase changes its value by two when the pressure is increased from −8 to 8 GPa. Because Fm3m-GeSe has no band inversion and is topologically trivial (n M = 0) when the external pressure is equal to −8 GPa, the n M is equal to −2 when a hydrostatic pressure of 8 GPa is applied. (The sign of n M has been discussed in detail in ref. 40 .) In addition, the (001) surface band structure is also calculated from a tight binding model, which is obtained utilizing the maximally localized Wannier functions 41,42 . A slab of 83 atomic layers was constructed to study the surface band structure. Figure 5d shows that a massless Dirac cone exists in the surface band structure along the Γ À X path, demonstrating that Fm3m-GeSe is a TCI under a hydrostatic pressure of 8 GPa.

Conclusions
The structural phase transitions of GeSe under hydrostatic pressures up to 16 GPa have been systematically studied from ab initio evolutionary structure searches. Two new phases with space groups of R3m and Fm3m are predicted to be stable in the pressure range between the α and the β phases. Laser-heated DAC experiments have been performed to provide evidence for the existence of the newly predicted R3m phase. It is found that R3m-GeSe has a layered crystal structure and exhibits robust ferroelectricity with a large spontaneous polarization. Under further compression, the R3m phase continuously transforms into the Fm3m phase. Phonon dispersion calculations show that this phase is dynamically stable at pressures above 8 GPa. Detailed electronic band structure studies reveal that the . The evolution of band gap at the high symmetry point, L, from DFT calculations with SOC (c). A negative gap represents that the bands are in an inverse order. The (001) surface band structure around X was calculated using the tight-binding method (d). The Fermi level is located at 0 eV. The size of the blue dots in panels (a) and (b) is proportional to the weight of the Ge 4s orbitals. The red curves in panel (d) represent the surface bands Fm3m phase is a TCI. The intriguing properties of the newly discovered GeSe phases further enrich our knowledge on the high-pressure behaviors of IV-VI compounds.