Mechanisms for pressure-induced crystal-crystal transition, amorphization, and devitrification of SnI₄

Pressure-induced poly(a)morphism (PIA) in solids is the results of abrupt structural phase transitions from a crystalline form to an amorphous structure by compression. This phenomenon has been observed in a wide variety of solids but the operative mechanisms are still lacking. There are evidences suggesting that PIA may be even related to density- and entropy-driven liquid-liquid phase transition and the negative thermal expansion in polyhedral structures. PIA has also found practical applications in the industrial production of amorphous materials. Inelastic x-ray scattering measurements of the atomic dynamics of a number of glasses have revealed evidence which suggests a possible connection between disordered and crystalline phases in medium range order.^1,2 More recently, it was further demonstrated that a lanthanide metallic glass can be ordered by compression.³ This astonishing observation opens a new dimension in glass science as it suggests there may be an inherent “hidden” order in disordered solids that can be amplified by pressure. It is important to establish a conceptual framework to understand and relate these structural transformations in order to develop better control over these processes.⁴ 

Although many examples on the transformations from crystal to amorphous are well documented, conflicting explanations abound. One such concept suggests that the crystalline phase of ice has crossed over the extrapolated melting line by pressurization at low temperature during the pseudomelting of ice Ih.^5,6 This conjecture suggests a thermodynamic connection between the amorphous state(s) and the liquid. The prevalent yet controversial two-liquid model for water was developed as a consequence of this conjecture.⁷ Another mechanism is based on the elastic stability of the crystalline framework where Born stability conditions apply.^8,9 Supporting high pressure measurements of the elastic constants in α-quartz¹⁰ and ice Ih¹¹ shows that a softening of the shear moduli occurs near the phase transition. Similar examples are the conversion of ice Ih to ice VII¹² and the observation of a cubic phase from amorphous SnI₄.¹³ Recently, the true identity of the amorphous structure has been challenged with improved diffraction experiments. Although several materials such as α-SiO₂¹⁴ and AlPO₄¹⁵ were thought to be amorphous, they were found to bypass the PIA under quasi-hydrostatic compression conditions and transformed directly to dense crystalline forms. This absence of the amorphous is further highlighted by recent single crystal diffraction study on coesite showing that under compression, the crystal transformed into four co-existing crystalline phases before finally recombining into a single post-stishovite structure without amorphization.¹⁶ Therefore, it is reasonable to speculate on whether or not short to intermediate range orders are preserved in the disordered structure. Ice Ih and SnI₄ are two pressure-amorphized solids where no crystalline form has been found in lieu of the disordered structure, and the nature of the amorphous phases remains an open question. Perhaps, the cleanest system to study is SnI₄. The consecutive high pressure transformations’ (crystal → amorphous → crystal) sequence has been established by a number of experimental techniques.^14–21 

Here, we clarify the structure of the amorphous phase and clarify the mechanism of the amorphous-to-crystalline transformation at high pressure in SnI₄. We use synchrotron nuclear resonant inelastic x-ray scattering experiments on SnI₄ at high pressure along with ab initio electronic structure calculations. We propose a consistent picture in which to rationalize structures of various high pressure SnI₄ polymorphs, and their associated transformation pathways. We further suggest a connection between the recrystallization of the amorphous structure and that observed in bulk metallic glasses.³ 

SnI₄ offers several advantages for the characterization of PIA transformation. It is a solid at room temperature and has a fairly high melting point making it comparatively easy to handle (compare to Ice Ih). The solid is quite compressible and hydrostatic pressure transmitting medium was found to have little effect on the transition. Both Sn and I are heavy atoms and high resolution diffraction patterns over a wide pressure range up to 154 GPa have been obtained.^13,17–20 A reversible crystal to amorphous transition in SnI₄ above 10 GPa at room temperature was reported in 1985.¹⁷ A subsequent study refined the transition pressure to 15 GPa,¹³ Prior an intermediate metallic crystalline phase (CP-2) was found at 7.2 GPa although no crystal structure was given.¹³ Recent diffraction experiments show that the formation of the CP-2 phase is dependent on the treatment of the sample. Direct amorphization can be achieved from the ambient (CP-1) crystalline structure only after multiple compression-decompression cycles at high temperature.¹⁸ Pseudomelting has been suggested to be responsible for the transformation.⁶ However, SnI₄ has a positive slope at the melting line at low pressure and there are presently no data close to the phase transition pressure that could be used to test this possibility.

Several structural models for the amorphous structure have been proposed. A random distribution of dimerized SnI₄ molecules was suggested from Raman scattering.²¹ Analysis of Mossbauer data²² favors a model consists of polymerized randomly oriented SnI₂ + I₂ dissociated molecules. X-ray absorption fine structure spectroscopy²³ suggested a random network of deformed tetrahedral units. More recent x-ray diffraction studies^17,19 suggested the basic unit of amorphized SnI₄ is a tetrahedron composed of four iodine and substitutional tin atoms but not the penta-atomic tetrahedral molecule.

Starting with the ambient cubic $(Pa3̄)$ structure of SnI₄, NPT MD (N-particles, P-pressure, T-temperature, and MD-molecular dynamics) calculations were performed for 300 K at selected pressures from 0 to 70 GPa. It is found that the ambient structure is stable up to 12 GPa, whereupon it transformed into a crystalline structure. This structure has a P-1 space group and identified to be the CP-2 structure observed at 7.2 GPa¹³ in the experiment. Further compression led to the collapse of the crystalline framework into a disorder structure at 15 GPa and finally at 60 GPa, a new crystalline structure is formed. The high-pressure crystalline structure is consisted of substitutional disorder on the Sn and I atoms. At 63 GPa, the Sn–I and I–I distances are very similar at 2.967 and 3.020 Å, respectively, and the mean of the Sn–I and I–I distances is 2.994 Å. This structure has a formal space group symmetry I4/m. However, if the distinction between Sn and I is ignored, within a tolerance of ±0.13 Å a cubic Fm3m space group with a = 4.260 Å can be fit within acceptable parameters. The theoretical structure is in favorable agreement with the cubic structure (CP-3) with a unit cell of 4.248 ± 0.002 Å and next nearest neighbor distance of 3.003 ± 0.001 Å identified by diffraction experiment¹³ at the same pressure. Good agreement is also found between the diffraction pattern calculated from the atomic trajectory at 300 K over 10 ps and the experiment (Fig. 1). In summary, the structural sequences obtained from MD calculations reproduce the experimental trend as well as obtain quantitative agreement between the computed structures and transition pressures.

Formation conditions for the intermediate CP-2 phase were analyzed by performing metadynamics calculations at 1, 7, and 10 GPa and 300 K. It was found that the CP-2 structure can be formed at 7 GPa. The calculated diffraction pattern for static CP-2 optimized at 8 GPa is compared with the measured pattern at 9.3 GPa¹³ in Fig. 1. After broadening with an experimental resolution of 0.2°, the two diffraction patterns become almost indistinguishable. The electronic density of states of CP-2 depicted in Fig. 2 shows that it is metallic. Therefore, CP-2 is the result of an insulator-metal transition from the CP-1 phase. This is consistent with the observed increase in electrical conductivity after the phase transition.¹⁷ Phonon band structure calculations confirm the CP-2 structure is stable. The agreements between theory and experiment on the diffraction pattern and electrical properties support the predicted CP-2 configuration. CP-2 has a 3D network structure consisting of imperfect but fully six-coordinated Sn–I octahedra linked in a zigzag ribbon along the b crystallographic axis via edge-shared I atoms (Fig. 3(a)).

Bader charges and volumes were calculated, in order to understand the devitrification of the amorphous phase into a crystalline structure. The results show there is a substantial charge transfer from Sn (−0.8 e/atom) to I (+0.2 e/atom). As a result, the (Bader) atomic volume of I (20.2 ų) becomes larger than Sn (15.6 ų). The size comparison and the perfect alignment of the Sn and I atoms are illustrated in the contour plot of the valence charged density in the (100) plane of the cubic structure (Fig. 4(a)). The crystal framework resembles a substitutional alloys and apparently determined by the packing of the larger I atoms with the small Sn atoms in the substitutional sites. The atomic radius may be estimated from the Bader volume assuming a spherical charge distribution. A plot of the ratio of the atom radii α = r(Sn)/r(I) with pressure (Fig. 4(b)) revels an unusual trend. In the amorphous state, α decreases slightly from 0.770 at 35 GPa to 0.725 at 55 GPa. After the transformation to the FCC structure, α increased to 0.925. The mismatch of the atomic radii is significantly larger in the amorphous than in the crystalline structure. Dense packing of binary hard spheres with different radius ratios is a subject of practical interest and has been studied intensely.²⁴ Recently, a theoretical phase diagram relating densest packed crystalline structures of binary spheres to the radius ratio (α) and the mole fraction (X = N_s/(N_s + N_l), N_s,l are the number of small and large spheres, respectively) has been proposed.^25,26 For FCC, SnI₄ at 60 GPa, α = 0.925 and X = 0.20, according to the phase diagram Sn and I should be phase separated into mono-dispersed small and large Barlow-packed spheres. No phase separation was observed by the experiment. The stability of cubic crystals (simple, body-centered, and face-centered) relative to (amorphous) random closet packing (RCP) of binary mixtures of hard spheres has been studied by a theoretical model.²⁷ According to this model at α = 0.925 and X = 0.20 corresponding to the FCC structure of SnI₄, the closet packing is much favored over random packing. By way of comparison, at α = 1.33 and X = 2.0 corresponding to the amorphous structure at 43 GPa, the model predicted the stability of RCP and FCC packings is competitive. Therefore, the devitrification of amorphous of SnI₄ at high pressure seems to arise from comparable atomic radii in the crystalline structure.

The equation of states (EOS) and energetics of CP-1, CP-2, and the PIA phase are compared in Figs. 5(a) and 5(b). The calculated EOS for CP-1 over the pressure range 0–15 GPa is in good accord with the fitted isothermal third-order Eulerian finite-strain (Birch-Murnaghan) equation to the experimental data at 293 K.¹ It is noteworthy that the treatment of the samples has a large effect on the unit cell volume. The fitted EOS underestimated the volumes from 10 to 15 GPa for a number samples treated under different temperature conditions. The calculated enthalpy differences between the static CP-1 with the CP-2 and amorphous structures are compared in Fig. 5(b). The athermal (0 K) phase transition pressures from CP-1 and CP-2 to the amorphous structure are calculated to be 4.3 and 16.3 GPa, respectively. The latter is consistent with experiment but the calculated direct transition pressure from CP-1 to amorphous based on the static structures of 4.3 GPa seems a bit low. Note that the enthalpy of the amorphous structure is lower than the CP-2 phase. This does not violate the third law of thermodynamics as the amorphous state is only metastable and not a genuine thermodynamic stable phase. There should be a global energy minimum crystalline structure yet to be discovered. We found temperature has a significant effect on the structural transition.¹ Experimentally, it was found that on a sample produced by repeated compression-decompression cycles at 523 K and cooled back to 293 K, the CP-2 phase can be by-passed but the amorphization transition is still close to 6 GPa. It has been also been shown that the PIA transition in SnI₄ is accompanied by a large hysteresis,¹³ a hint on the potential importance of kinetic effects. Moreover, the amorphization transition pressure was found to vary between 12 and 19 GPa depending on the nature and history of the sample.

Phonon band structure calculations were performed for both the CP-1 and CP-2 structures. No imaginary branches were revealed at pressures near the phase transition. However, several zone center modes were found to soften at pressures higher than 10 GPa (see Fig. 6). The energy dispersions of phonon branches at the long wavelength limit are related to the elastic moduli. Calculations of elastic moduli for CP-1 show the shear modulus (G′ = C₁₁ − C₁₂ − 2P, P is the external stress)²⁸ becomes negative at 10.4 GPa (Fig. 6(c)), thus violating a Born’s stability conditions.²⁹ The structural instability signals the onset of the transition to the CP-2 which was predicted to occur at 12 GPa from NPT MD calculations. Subsequent to this transformation, the mechanical stability of the triclinic CP-2 structure is determined by the determinants of the principle minors of the elastic constant matrix. For this purpose, the principle minors were computed from the optimized CP-2 structures at 7, 10, 15, and 20 GPa. The elastic constants calculated at 0 K show the CP-2 structure is stable up to 20 GPa, apparent in disagreement with experiments. We suspect that temperature may be a contributing factor to the discrepancy. To investigate this effect, the elastic responses for the CP-2 structure were computed using a molecular dynamics based strain-stress method³⁰ at several pressures and 300 K. A 2 × 2 × 2 supercell was used. At a given pressure, an average structure was first determined from a NPT calculation. This structure is then fixed in a succeeding NVT-MD simulation over ∼12 ps where temporal stresses were calculated. The elastic constants were obtained by fitting the stresses to the fluctuating strains. Calculations of determinants of the principle minors show the Born stability conditions are violated at 10.8 GPa (Fig. 6(d)) at 300 K.²⁹ Therefore, we can conclude that the PIA is not initiated by metallization, since at 7 GPa, the insulating CP-1 has already transformed to metallic CP-2. The theoretical results show unambiguously that the PIA transition is indeed due to mechanical instabilities which led to the loss of long range order caused by the disordered polyhedral. This is the same mechanism found in all PIA phenomena that have been carefully characterized. It is noteworthy that transformation to the amorphous state from the CP-2 structure was found to be 25 GPa from NPT MD calculations at 300 K. The over-pressurization is indicative of kinetic hindrance.

Now, we examine the PIA structure obtained from MD calculations. A snapshot of the structure is shown in Fig. 3(b). Several important features are observed. There is no indication of SnI₄ units. The disordered structure is polymeric with mostly six-coordinated Sn atoms. The prominent local structure is a Sn atom bonded to 4 I atoms which are corner shared with other units in the crystal: the remaining two I atoms come from a pair of I⋯I (I₂) which is cross linked to another octahedral. The Sn–I bonds to the individual I atoms are ca. 2.93 Å which is shorter than the Sn–I bonds to the I₂ pair of 3.3 Å. Moreover, the I–I bond length of 2.75 Å that is comparable to 2.67 Å in a free I₂ molecule. In a way, the structure is similar to the model proposed from a Mössbauer study.³¹ The major difference is that the SnI₄ units in the present structure obtained from the MD calculations are severely distorted to accommodate interactions with the I₂ pair and, therefore, no genuine four coordinated Sn can be identified. From the apparent resemblance to of the features in the static structure factor, S(q), between amorphous SnI₄ and elemental nickel metallic glass, it was suggested that the PIA SnI₄ can be modeled by dense random packing of Sn and I atoms of equal-sized. In spite of the good accord of the calculated S(q) and pressure trend of disordered SnI₄ with experiments, the local coordination for the Sn and I atoms are different. All Sn atoms in the disordered structure are 6-coordinated but the coordination at I varied between 6 and 7. Moreover, as discussed above, the Bader volume of the I atom is always larger than Sn in the amorphous state.

The temperature sensitivity of the PIA transition pressure is surprising. For this purpose, we investigated the dynamics and thermodynamics of the Sn atoms using ¹¹⁹Sn nuclear resonant inelastic x-ray scattering spectroscopy.³² The theory and procedure for the extraction of the atom projected vibrational density of states (PVDOS) have been described elsewhere.³² Results of the experimental Sn PVDOS at several pressures are compared with the theoretical predictions in Figs. 7(a) and 7(b), respectively. A significant observation is that at 0 GPa, the vibrational bands from 20 to 35 meV assigned to I–Sn–I and Sn–I are very broad (vide supra). When the sample is compressed to 2 GPa, the Sn–I stretch mode at 27 meV sharpens considerably. The sharp feature persists with a slight shift to higher frequency as the pressure is increased to 8 GPa. This is in agreement with the calculated Sn VDOS of the CP1 phase. At 10.5 GPa and higher, the band at ∼27 meV has broaden significantly. Eventually, the distinctive Sn–I stretch vibration disappears and the VDOS becomes a broad distribution above 18 GPa, an indication on the loss of long range order.

The observation variation in the Sn PDOS with pressure is consistent with the theoretical calculations. At first glance, the broad distribution of phonon modes at ambient pressure and temperature is surprising. The calculated Sn PVDOS at 0 GPa shows the bend and stretch modes are clearly separated at 7.5 meV and 24 meV, respectively. The discrepancy between theory and experiment under ambient pressure suggests that the SnI₄ molecules may execute large amplitude librations leading to strong coupling between the stretch and bend vibrations. To explore this possibility, the Sn PVDOS was measured at 65 K. The result differs from the 300 K measurement dramatically. The vibrational features are sharpened significantly, and now, the agreement with the theoretical calculated VDOS at 0 GPa is significantly improved. It is likely that the interaction between SnI₄ molecules is overestimated by the Perdew-Burke-Ernzerhof (PBE) functional due to the neglect of van der Waals (vdW) interaction. This suggestion is confirmed with additional MD calculations including vdW interactions using the Grimme potential.³³ Indeed under ambient conditions, the intermolecular interaction is reduced resulting in hindered rotational of the SnI₄ and broadening of the Sn PVODS. A detailed comparison of the theoretical and experimental temperature dependence Sn PVDOS spectra will be presented elsewhere. At higher pressure, the van der Waals forces are expected to be less important and the calculated Sn PVDOS are now in reasonable agreement with experiment. When compressed to 5 GPa, the stretch frequency increases to 26 meV and the band becomes very sharp as observed in the experiment. The sharp Sn–I feature changed into a broad two peak pattern with lower vibrational frequencies (21 and 24 meV) at 7 and 10 GPa. The change is associated with the incipient transformation to CP-2. Above 15 GPa, no distinctive features were observed in the VDOS indicating the system has been amorphized. At 63 GPa, the signature bands for the bend and stretch modes have disappeared and the VDOS is just like a monoatomic crystal.

Relevant dynamics and thermodynamics parameters for SnI₄ obtained from the analysis of the nuclear resonant inelastic x-ray scattering (NRIXS) spectra³⁴ are shown in Fig. 7(c). The Mossbauer-Lamb factor (f_LM), which is related to the mean-square displacement of the Sn atom within the lifetime (25.7 ns) of the ¹¹⁹Sn nuclear excitation at 23.88 keV, shows an initial increase from 0.08 to 0.20 until ca. 8 GPa where it drops to 0.135 at 18 GPa and then an abrupt rise to 0.24 at 25 GPa. This trend collaborates with the CP-I → II structural transition at 7 GPa followed by eventual amorphization above 17 GPa. The initial rise is consistent with a more restricted Sn thermal motion in the compressed cubic structure. The interatomic force is expected to “soften” in metallic CP-2 leading to a drop in f_LM but increase again after the transformation to a denser amorphous structure at 20 GPa. The derived mean force constants (F_M) were almost constant from ambient pressure to 5 GPa but decreased slightly between 10 and 18 GPa and increased again at 25 GPa. The resilience (κ_r) is a compact way to express the temperature dependence of the atomic fluctuations.³⁴ This quantity can be calculated from the Sn PVDOS at a given temperature³⁵ and is sensitive to weak forces (hydrogen-bond, van der Waals interaction, etc.) rather than strong covalent bonding in a material. The pressure dependence of κ_r shows clearly the environment experienced by the Sn atoms stiffened from 0 to 5 GPa, then soften from 10 to 18 GPa in metallic CP-2 and increase again in the amorphous phase.

The structure and dynamics of SnI₄ at high pressure have been examined with electronic structure calculations and synchrotron NRIXS experiments. Our results provide a mechanism for the PIA mechanism in SnI₄ and demonstrate the effect of pressure on the interaction of electron distribution and atomic radii. We find that SnI₄ molecules are loosely bound at low pressure and show large amplitude rotational oscillations. The amorphous structure is found to be both metallic and polymeric. The observed structural transformations from CP-1 to CP-2 at 7.2 GPa are followed by amorphization at 15 GPa and the appearance of a cubic crystalline structure (CP-3) above 60 GPa is reproduced. Our calculations show the PIA transition is related to a structural instability that violates Born stability criteria.²⁹ The local structure Sn atoms are six-coordinated with a distorted SnI₄ interacting with a pair of I₂. Upon further compression of the amorphous state, the Bader atomic volume of I was found to be larger than Sn with increasing pressure. At 60 GPa, the trend is reversed, even though the size of the I is still larger than Sn, and the system transformed back to a cubic crystal. The structures and structural transformations with the associated I/Sn volume ratio are in accord with the prediction from a theoretical model for the packing of binary systems.²⁵ It is not difficult to extend a similar explanation to the pressure-induced transformation from Ce₃Al metallic glass to a cubic crystal.²² It has been reported that the atomic volume of the more abundant Ce is much larger than Al in the high pressure glass.³⁶ 

First-principles isobaric-isothermal (NPT) MD and metadynamics calculations were performed within the density functional theory using gradient corrected PBE functional³⁷ and the projector augmented wave potentials³⁸ to replace the atomic core potentials implemented in the plane wave basis Vienna Ab initio Simulation Package (VASP).^39 119Sn nuclear resonant inelastic x-ray scattering spectra³² were measured at sector 3-ID at the Advanced Phonon Source, Argonne National Laboratory. Details on the calculations, experimental procedure, and data processing are given in the supplementary material.⁴⁰ 

We would like to thank Dongzhou Zhang for help during one of the experiments. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357. Calculations were performed at Westgrid Computing Facilities and the Laboratory Computing Resource Center’s high-performance computing clusters, Blues and Fusion, at Argonne National Laboratory. John S. Tse and Hanyu Liu acknowledge the National Natural Science Foundation of China (Grant No. 11474126) and support from the University of Saskatchewan research computing group and the use of the HPC resources (Plato machine).