The characterisation of the ‘‘X’’ crystal structure in the Stillinger-Weber potential

A recently observed crystal structure is characterised by reference to the calculated diffraction patterns. The ‘‘X’’ crystal is related to the b -Sn crystal structure via an atom displacement and a reduction in symmetry from tetragonal to orthorhombic. A further crystal polymorph is identiﬁed which retains the tetragonal symmetry of the b -Sn structure but involves a related atom displacement. The unit cells and space groups are reported. The two crystals are labelled as o-X and t-X respectively. The transforma- tion mechanisms from the b -Sn crystal structure to the two structures are identiﬁed. No routes to these crystals from other polymorphs are found.


Introduction
There has been enormous levels of experimental and theoretical effort devoted to probing the phase behaviour of systems dominated by local tetrahedral environments under ambient conditions. Systems such as Si and Ge appear homologous in that (broadly speaking) their phase diagrams map onto each other. Both have been studied extensively under pressure (see, for example, Refs. [ 24,3,12,19,10,14,16,5,29,17,7]) uncovering a rich tapestry of inter-related crystal structures. Computer simulation offers an alternative method for exploring the phase space both as a function of temperature and pressure (and hence mimicking direct experimental study) and as a function of the potential model itself (see, for example, Refs. [20,22,15]). The latter, although not experimentally reproducible, offers the potential to ''connect'' different (related) systems and to develop a fundamental understanding of the key interatomic interactions.
Whilst electronic structure methods [9] offer potentially the most accurate results (and, by construction, may include any major changes in electronic structure) relatively simple potential models, in which the system energy is expressed as a function of the atom positions, offer a flexible and efficient method for exploring the morphology space. Recent investigations have uncovered yet more complexity and richness in the phase diagrams of silicon and the related mono-water (mW) models [21]. These models employ a Stillinger-Weber (SW) potential in which the system energy is effectively expressed as the sum of 2-and 3-body terms [28].
The 3-body term acts to effectively bias the system against generating local coordination environments which deviate away from ideal tetrahedral angles. Altering the balance between the 2-and 3-body terms has been shown to give a good representation of Si [28], Ge [6] and even H 2 O [21]. 1 The result is that, under ambient conditions, the diamond crystal structure is energetically favourable. As the pressure is increased, for example, the balance between the 2-and 3-body terms changes and alternative crystal structures may become thermodynamically stable. Romano et al. [27] explored the relative energetics of $30 different crystal structures for both the Si and mW SW potentials. They showed how the SC16 crystal structure (space group Pa3 -#205) dominates the high pressure region of the phase diagrams.
Overall, therefore, a full understanding of the phase diagrams of systems such as Si and Ge is still potentially lacking.

The ''X'' crystal
In recent work the phase diagram for the SW potential was determined at positive pressures for a range of different balances between the 2-and 3-body interactions (different values of k) [2]. These investigations uncovered a potential new phase (termed ''X'' in the original paper 2 ) which formed spontaneously from a b-Sn  1 In the SW potential the magnitude of the 3-body interaction is controlled by a parameter usually identified as k. For example, models for Si, Ge and water use k ¼ 21, 20 and 23.15 respectively. 2 This is somewhat unfortunate in that, for silicon, Si-X is classified as FCC, which is thermodynamically stable at high pressures. Here we adopt the nomenclature t-X and o-X to denote crystals formed from the b-Sn crystal structure with tetragonal and orthorhombic symmetry respectively.

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Chemical Physics Letters j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c p l e t t crystal structure when simulated at k ¼ 20:08. The formed X-crystal appears stable at intermediate pressures (between 0.2 and 0.8 in reduced units) and low temperatures (between 0 and 0.02 in reduced units) where the b-Sn crystal structure was previously expected to dominate. Although the static structure factor was reported no attempt was made to characterize the polymorph further. In this Letter we will characterize the new crystal structure and establish its relationship with other known stable crystals including potential (atomistic) transformation mechanisms both into and out of the new structure.

Determination of the space group and primitive cell
A super cell was constructed from 5 Â 5 Â 10 primitive cells of the b-Sn crystal structure (giving a total of 1000 atoms) at the equilibrium cell lengths for p Ã ¼ 0:26 and T Ã ¼ 0:01 (corresponding to b ¼ a ¼ 4:88 Å and c=a ¼ 0:57). Molecular dynamics (MD) simulations were performed at constant stress (allowing the cell lengths and angles to vary) using the SW potential with k ¼ 20:08 using LAMMPS [25] at T Ã ¼ 0:01 and p Ã ¼ 0:26. The phase space point T Ã ¼ 0:01 and p Ã ¼ 0:26 corresponds to p ¼ 10 GPa and T ¼ 250 K for silicon and P ¼ 7:81 GPa and T ¼ 225 K for germanium. It should be noted that the value of the parameter k considered lies between values used previously to model germanium and silicon. Further work will be required to explore the model parameter space to establish the potential stable phase space range for related materials such as H 2 O and C. The state variables were controlled using Nosé-Hoover [23,13] thermostats and barostats [18] respectively. The initial atom velocities were selected from a Maxwell-Boltzmann distribution with each atom assigned a random velocity vector. Simulations were then performed starting from different sets of initial velocities.
At this state point the b-Sn crystal structure is seen to evolve to two new (related) structures. The initial b-Sn crystal structure is equilibrated over 50,000 time steps with time step of 0.383 fs (corresponding to $19.15 ps) with a further 100,000 time steps (38.3 ps) performed in the NVT ensemble. The energy of the final structure was minimised using the Polak-Ribiere version of the conjugate gradient method with an energy tolerance of 10 À4 and force tolerance of 0:97 Â 10 À6 (reduced units) with a maximum of 1000 iterations and 100,000 force/energy evaluations.
The final (energy minimised) configuration was used as the input into the spglib library [1,11] to determine the primitive cell and space group with tolerances in the cell lengths and atom positions of 0.2 Å. The space group of one of the new structures was found to be Pnma (#62) and is orthorhombic with basis The primitive cell is closely related to that for the b-Sn crystal structure (space group I4 1 /amd), which has fractional coordinates:  Fig. 2 shows the structure factor, SðkÞ, for key crystal structures; b-Sn, t-X and o-X. The total X-ray scattering function and structure factor are simply related via a (k-dependent) form factor function. Since we are primarily interested in the peak positions (rather than their intensities) then considering SðkÞ alone is appropriate. The Ashcroft-Langreth structure factors are generated from SðkÞ ¼ hA Ã ðkÞ Á AðkÞi, where the fourier components AðkÞ ¼ 1 ffiffi ffi N p P i e ikÁr i and fr i g are the atom positions. The X-ray diffraction pattern for the b-Sn crystal structure is well known (see, for example, references [14,16,12,19,10]) showing peaks corresponding to ½hkl ¼ ½200, ½101; ½220; ½221; ½301; ½112; ½400 and ½321. The sum h 2 þ k 2 þ l 2 is even as expected for this symmetry.

The diffraction pattern
For both the t-and o-distortions this constraint is relaxed and a number of additional peaks appear as highlighted. For example, Bragg peaks corresponding to ½120; ½111; ½201; ½221 and ½301 emerge. On reducing the symmetry from tetragonal to orthorhombic further peaks appear (also as indicated) as ½hkl -½khl. For example, the pairs {½200; ½020}, {½101; ½011} and {½120; ½210} become distinct. Most importantly, the peak positions for the o-X crystal structure agree with those obtained previously [2] (as shown in the figure inset). Fig. 3(a) shows the evolution of the simulation cell lengths as the b-Sn crystal structure transforms to either the tetragonal or orthorhombic X-forms. Fig. 3(b) shows the evolution of the system energy and volume for the same transformations mapped onto the (ideal) T ¼ 0 K energy/volume curves. The energy/volume curves were calculated by constructing a supercell from the appropriate  ¼ c) calculations are performed over a range of c=a and c=b ratios to identify the lowest energy structure. The unit cell volumes were varied by increasing the cell length in steps of 0.02 Å. The transformations from the b-Sn crystal structure originate at the same state point (as noted above) and proceed along analogous UV paths until U Ã $ À1:92 and V Ã $ 1:788 (both in reduced units). At this state point an effective ''bifurcation'' occurs with the system heading towards one of the two X forms and commensurate with the changes in cell lengths seen in Fig. 3(a). The pathway to either structure was found to be dependent upon the distribution of velocities and showed no systematic dependence upon the system cell size (number of unit cells). Table 1 lists the energies and volumes at the respective energy minima in the T ¼ 0 K energy/volume curves. The o-X and t-X crystals are $690 J mol À1 more energetically favourable than the b-Sn crystal structure.

Transformation mechanism
The key observation (consistent with that made in Ref. [2]) is that the o-X crystal structure is thermodynamically stable over the b-Sn crystal structure. Fig. 3(b) also highlights how it is also possible to thermallydrive the transformation from the t-to o-X forms. The t-X crystal  β-Sn is systematically heated from low temperature with a transformation to the orthorhombic form observed at T Ã $ 0:22.

Attempted transformations to the X phase
It is clear from the above discussion that the b-Sn crystal structure will, under appropriate conditions, transform to either a t-X or o-X structure depending on the atomistic detail of the starting con-figuration. To investigate potential transformation pathways to the X crystals from other crystal polymorphs, a range of alternative crystal structures (diamond, lonsdaleite, FCC, SC16, ST12 and BC8) were constructed and modelled at a point in phase space where the X crystal is expected to be thermodynamically stable, p Ã ¼ 0:26; T Ã ¼ 0:01. The diamond and lonsdaleite crystal structures differ in the stacking of the atomic layers, the former showing cubic stacking and the latter hexagonal stacking (see, for example, Refs. [8,26,4]). Simulations are performed using a SW potential with k ¼ 20:08. The only transformation to the X phase which was observed on the simulation time-scale was from the b-Sn crystal structure. The diamond, lonsdaleite, SC16, and ST12 crystal structures are all metastable at this state point. The BC8 crystal transformed to the SC16 crystal structure as seen previously [27].

Attempted transformations from the X phase
In addition to considering the transformation to the X phase potential transformations away from this polymorph are also considered. Ideal t-X and o-X crystals (both corresponding to 5 Â 5 Â 10 unit cells) are constructed and MD simulations are performed over a range of state points at which the X crystals would be anticipated as metastable and at which the diamond and SC16 crystal structures would be expected to be thermodynamically stable. Simulations are performed starting from the state point p Ã ¼ 0:26; T Ã ¼ 0:01 both along the isotherm and the isochore. As the system is heated along the isochore the transformations between the crystal structures t-X À! o-X À! b-Sn are observed at T Ã ¼ 0:0189 and 0:0298 respectively. In addition, both the t-X and o-X structures, pressurised along the isotherm, form the b-Sn crystal structure at p Ã ¼ 1:826 with the t-X structure undergoing a transformation to the o-X at p Ã ¼ 0:476. On de-pressurisation along the isotherm the o-X crystal transforms to the b-Sn crystal structure (at p Ã ¼ À0:15) whilst the t-X remains metastable to p Ã $ À0:20. The system shows the expected hysteresis which would likely be strongly system size dependent. Overall, therefore, both the o-X and t-X structures readily transform to the b-Sn crystal structure which remains metastable under the conditions tested.

Conclusions
In this letter the previously observed ''X'' crystal [2] has been fully characterised by reference to the diffraction patterns. The crystal structure is related to the b-Sn crystal structure via a simple atom displacement and a reduction in cell symmetry from tetragonal to orthorhombic (and is termed o-X here). A related distortion, in which the tetragonal cell symmetry is maintained, has also been observed (leading to the so-called t-X crystal structure). Finally, the accessibility of this crystal structure from potential crystalline polymorphs was considered with the route from the b-Sn crystal structure identified as unique. Table 1 Volumes and energies (expressed in reduced units) for the three key crystal structures (b-Sn, o-X and t-X). The corresponding values for the thermodynamically stable form at ambient conditions, the diamond crystal structure, are shown for comparison.  3. (a) The evolution of the simulation cell lengths on transformation from the b-Sn crystal structure to both the o-X and t-X forms. The original b-Sn crystals are heated from T = 10 K to 250 K at a pressure of p = 10 GPa using different distributions of initial velocities. After the phase transformations the system was cooled down to 10 K. (b) Potential energy evolution during the transformations from the b-Sn crystal structure to either the t-or o-X structures, and from the t-X to o-X as described in the text. For the t-to o-X transformation the same procedure as above was used with heating to T = 550 K required to drive the transformation. The T ¼ 0 K ideal energy/volume curves are overlaid to highlight the transformation pathways.