An Atomistic Study of Phase Transition in Cubic Diamond Si Single Crystal T Subjected to Static Compression ⁎ Dipak Prasad, Nilanjan Mitra

Computational Materials Science 156 (2019) 232–240

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Computational Materials Science

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An atomistic study of phase transition in cubic diamond Si single crystal T subjected to static compression ⁎ Dipak Prasad, Nilanjan Mitra

Indian Institute of Technology Kharagpur, Kharagpur 721302, India

ARTICLE INFO ABSTRACT

Keywords: It is been widely experimentally reported that Si under static compression (typically in a Diamond Anvil Setup- Molecular dynamics DAC) undergoes different phase transitions. Even though numerous interatomic potentials are used fornu- Phase transition merical studies of Si under different loading conditions, the efficacy of different available interatomic potentials Hydrostatic and Uniaxial compressive loading in determining the phase transition behavior in a simulation environment similar to that of DAC has not been Cubic diamond single crystal Silicon probed in literature which this manuscript addresses. Hydrostatic compression of Silicon using seven different interatomic potentials demonstrates that Tersoff(T0) performed better as compared to other potentials with regards to demonstration of phase transition. Using this Tersoff(T0) interatomic potential, molecular dynamics simulation of cubic diamond single crystal silicon has been carried out along different directions under uniaxial stress condition to determine anisotropy of the samples, if any. -tin phase could be observed for the [001] direction loading whereas Imma along with -tin phase could be observed for [011] and [111] direction loading. Amorphization is also observed for [011] direction. The results obtained in the study are based on rigorous X-ray diffraction analysis. No strain rate effects could be observed for the uniaxial loading conditions.

1. Introduction potential(SW) [19,20] for their simulations. Interestingly none of these studies validate their used potential in its ability to correctly predict Silicon is one of the most important elements used in modern na- phase transformation behavior in Si. It should be noted that while de- nodevices. It has been reported through experimental observations that veloping a interatomic potential, many different considerations are Si undergoes phase transition under application of pressure [1–6]. The tested but their ability to perform phase transition at initial ambient experimentally reported phases of Si on application of high pressure in temperature and pressure conditions under application of load (as ob- compression are shown in the supplementary file (supplementary served in experimental conditions) is not investigated. Fig. 1). It should also be noted that on decompression Si does not return The static compression loading mechanisms applied in the above back to its same phase but produces other different phases as shown in literature also differs significantly. Some literature focus on nano-in- the supplementary file (supplementary Fig. 2). Most of these experi- dentation [10,7,12,9] and some on abrasion at nanoscale [11]; in both mental investigations use a Diamond Anvil Cell (DAC) for application of of these the resulting stress state is quite different compared to uniaxial pressure which typically applies a hydrostatic type loading situation on or hydrostatic stress type compressive loading condition carried out in the sample. The Si specimen generally chosen for investigation is either this manuscript. There are also studies which considers uniaxially amorphous or polycrystalline cubic diamond form. It has been reported compressed silicon nano-spheres [13], but since it is a nanosphere the that both of these forms of Si undergo phase transition, albeit at dif- boundary conditions and the resulting stress conditions are quite different pressures [7,8]. ferent from either uniaxial or hydrostatic type loading situation con- There has also been numerous numerical literature on molecular sidered in this study. The literature [1] even though is on uniaxial dynamic simulations of Si in which phase transition was demonstrated compression, focus on the amorphous form of Si and not on the cubic under different types of compression using different types of intera- diamond crystal phase, as is considered in this manuscript. It should be tomic potentials [1,9–11,7,12,13]. Some of these literature [13,7,9–12] noted that literature pertaining to static compression has only been considers different variations of the Tersoff potential (hereby referred considered in this manuscript. There are numerous studies on dynamic to as T0 [14], T1 [15], T2 [16], T3 [17] and T4(Erhart-Albe) [18] re- compression such as shock compression reporting phase transforma- spectively). Other literature considers Modified Stillinger Weber tions, but these studies have not been reviewed in this literature review

⁎ Corresponding author. E-mail address: [email protected] (N. Mitra). https://doi.org/10.1016/j.commatsci.2018.09.037 Received 9 September 2018; Accepted 16 September 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved. D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240 since it is beyond the scope of this study. pressure phases of cubic diamond Si) but it should be understood that The mechanism for identification of phase transformation ina DFT studies typically show the most stable phase of Si under a specified sample on application of static compression also varies from study to pressure and 0 K temperature. In DFT simulations, there are issue also study and none of these are confirmatory in nature. Some literature [7] with regards to the use of basis functions; whether plane wave hy- considers changes in the Gibbs free energy to demonstrate phase pothesis is being utilized or it is a linear combination of all atomic transformation. Even though change in Gibbs energy denotes phase orbitals and various other such considerations based on which the re- transformation in a confirmatory manner at T = 0 K where the entropy sults may vary significantly. It should be mentioned that there isno of the system can be neglected. However, at ambient temperature and study in open literature comparing different DFT methods for studying pressure conditions (situation in which typical experimental in- phase transition in Si. vestigations are carried out) the Gibbs free energy is not the total po- The primary objective of this study is to determine which intera- tential energy since the entropy of the system cannot be neglected. tomic potential (out of the available ones and being commonly utilized) Therefore, under ambient temperature and pressure conditions it is not works best to predict phase transformation in Si under hydrostatic possible to determine phase transition using Gibbs energy. Moreover, compression as reported from experimental observations. Apart from since in Si there are numerous phases which have been reported and that another objective of the study is to determine the orientational using Gibbs energy calculations it is not possible to determine which dependence of cubic diamond crystal of Si under uniaxial compressive phase (or if a combination of phases) are present in the material. Based pressure loading (typically observed in static compression experiments on these studies, determination of which phase has been made from involving universal testing machines and dynamic compression ex- purely coincidental evidence from experimental investigations re- periments involving Kolsky-bars). porting phase transitions at those pressure ranges, albeit under different In this regard, it should be noted that there has been no work in loading conditions. For the case of Radial distribution function (RDF) published literature which investigates the topic of orientational de- (used in literatures [1,9,11]) and Angular distribution functions (ADF) pendence of Si under isentropic/static pressure loads (not involving any phase change may be observed based on changes in the second and significant rise in temperature). Under shock loads, involving both rise third coordination shell pattern but identification of the specific phase in temperature and pressure, there are studies which focuses on the cannot be done with accuracy. Moreover it should also be realized that issue of orientational dependence [23,24]. It should be noted that if the phase exists in small amounts then its effect cannot be identified compression as a result of shock loading is significantly different using these methods. Coordination number (CN) has also been used in compared to the uniaxial stress type compressive loading condition numerous literature [10,12,11,9] to demonstrate phase transformation considered in this literature. Uniaxial strain conditions are re- and identify the phase. However, even though from a theoretical presentative of shock loading situations in high velocity impact ex- viewpoint CN is ideal for determination of a specific phase but its im- periments. Uniaxial stress type loading has been considered in this plementation in Lammps [21] requires specification of a cut-off radius. study apart from hydrostatic loading. Differences in between different Ideally speaking the cutoff radius should be the distance where thefirst types of compressive loading scenarios have been described in details in coordination shell in the RDF (after the peak) touches the x-axis. One literature [25]. should note that as a sample is loaded, this coordination shell shifts and thereby unless one changes the cutoff radius during simulation (based on amount of strain given to the sample), the results are bound to give 2. Simulation methodology higher coordination shell numbers which may not be entirely realistic to demonstrate a phase transformation. In this regard it should be noted Cubic diamond single crystal silicon with periodic boundary con- that this method of CN may be acceptable if the load is applied locally ditions in all orthogonal directions is created and equilibrated using (such as in the case of a nanoindentation) and phase transformation is isothermal-isobaric, NPT ensemble integration scheme (for 80 ps with expected to happen in small regions. However, it should be noted that time step of 1 fs using Nose-Hoover Thermostat and barostat algorithm) this method fails if there is a significant shift in the first coordination at ambient temperature and pressure conditions. The observed density shell peak distance and along with load the second coordination shell of the final equilibrated Si using different interatomic potentials are peak left shifts to the region of the initial first coordination shell peak. 2.32 ± 0.02 gm/cc in comparison to that of recorded experimental In the literatures on Si [10,12,11,9], the cutoff distance chosen by re- density of 2.329 gm/cc. Utilizing suitable interatomic potentials these searchers ranges from 2.6 to 3.5 Å. Given the above mentioned methods samples were subjected to hydrostatic and uniaxial (along different for phase identification, the best way to identify change in crystalline directions such as [001] representing high symmetry direction, [011] phase is through X-ray diffraction (XRD) methods. As a matter of fact, if and [111] representing low symmetry directions) compressive loading. there are multiple phases in the sample, XRD will show peaks corre- Non-equilibrium molecular dynamics(NEMD) simulation have been sponding to different phases. XRD has been done in this manuscript for performed on the samples to investigate the phase transition of single identification of the phases as cubic diamond single crystal is statically crystal cubic diamond silicon. For details of samples with regards to compressed hydrostatically as well as uniaxially along different direc- orientation, size and number of atoms used in simulation, please see tions. Table 1. It should be mentioned that there also exist DFT (Density Functional Simulation of the hydrostatic compression corresponds to experi- Theory) studies which focus on pressure induced phase transformation mental Diamond Anvil Cell compression. Hydrostatic pressure was [22,8] (in which energy calculations demonstrate existence of high ramped up by 1 GPa pressure in 20 ps followed by relaxation for 2 ps using NPT ensemble. The sample was loaded till the final pressure

Table 1 Necessary details of the initial configurations of samples.

Sample No. Orientations Size(Å) No. of atoms

X Y Z X Y Z

1. 100 010 001 271.869 271.869 2718.69 10,000,000 2 1¯ 00 01¯ 1 011 276.988 276.988 2769.88 10,575,360 3 2¯ 11 0 1 111 268.627 259.089 2902.43 10,053,120

233 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240 reaches 150 GPa. The effect of relaxation time was also investigated by varying it from 2 ps to 20 ps in which no differences in behavior was 1,… ,N= v2 i, j+ v3 i,, j k observed. i, j i,, j k (1) For uniaxial compression the samples are subjected to compression where loading under uniaxial stress condition. Uniaxial stress condition can be r attained by maintaining constant ambient pressure and temperature at ij v2 () rij = f2 surfaces along two orthogonal directions using isothermal-isobaric, (2) NPT ensemble integration scheme (using Nose-Hoover Thermostat and r rj r barostat algorithm), whereas loading is applied in the third orthogonal i k v3 ri,, r j r k = f2 ,, direction. Since the pressure is maintained in the two directions at (3) ambient conditions, it can be quite expected that stresses will not de- Here is depth of potential well and (finite distance where inter- velop but strains will develop in those two directions upon compres- particle potential is zero, is chosen such that f (21/6 ) vanishes. The two sion. Quasistatic type uniaxial stress simulations have been conducted 2 body interaction term f is selected as follows: in which the simulation box has been compressed in one direction 2 under NPT ensemble. Strain values have been ramped up by 0.01 in p q() r a 1 f = A(), Br r e r< a 20 ps followed by relaxation for 2 ps. The sample is loaded till the 2 0, r a (4) maximum stress in the uniaxial direction exceeds 20 GPa. (similar strategy for studying uniaxial behavior using MD simulation has been The term a refers to the cutoff distance which has also been utilized done in other literature [26,27]). In order to investigate the effect of to three body interaction term f3. Here is given by strain rate, similar uniaxial stress simulations are carried out along f (,,)(,,)(,,)(,,)ri r j r k= h r ij r ik jik + h rji r jk ijk + h rki r kj ikj different directions but with a strain rateof 1010 /sec (for 40 ps with time 3 (5) step of 1 fs). No relaxation is provided as the strains are ramped up in where jik is angle between rj and rk subtended at vertex i. Function h these simulations. depends on two cutoff , and previously defined cutoff a by the fol- Any molecular dynamic simulations depends strongly upon the in- lowing relation: teratomic potential chosen to define the molecule/atom. For the case of 2 1 1 1 [()()]rij a + rik a Si, numerous interatomic potentials have been used in literature (as h rij,, r ik jik = e cos jik + mentioned in Introduction section). These interatomic potentials were 3 (6) evaluated in their ability to demonstrate phase transformations as ob- For ideal tetrahedral h vanishes eventually resulting in a cubic served in hydrostatic compression of Si in DAC experiments. The po- diamond structure which has been found to be as most stable structure tentials being considered are Stillinger-Weber (SW [19,20]), various among well known simple structures in terms of lattice energy. For versions of the Tersoff potential (T0 [14], T1 [15], T2 [16], T3 [17] and development of interatomic potential, parameters have been fitted with T4 [18]), Modified Embedded Atom method (MEAM [28]) and En- lattice constant and cohesive energy of cubic diamond structure in a vironment Dependent Interatomic potential (EDIP [29]). It should be manner such that melting point and structure of liquid silicon is well noted that there exists a previous study demonstrating comparisons in described. The value of fitted parameters are given as follows between the different potentials for Si [30] in which the following potentials were compared: Pearson [31], BiswasHamann [32], Stillin- A= 7.04956, B = 0.6022456, p= 4, q = 0, gerWeber [19,20], Dodson [33], Tersoff2 [14] (same as T0 in our a= 1.80, =2.0951, = 2.1683, =21.0, =1.20. study) and Tersoff3 [15] (same as T1 in our study). The study [30] was primarily aimed at comparing the ability of potentials to evaluate structure of the clusters, the values of elastic constants, phonon fre- 2.1.2. Tersoff potential family quencies, formation energy of various intrinsic defects, properties of Tersoff is a cluster functional potential [14–18] in consisting of two various surfaces of Si and also pressure induced phase transformations. and three body potential alongwith the fact that with increase in However, in their study on pressure induced phase transformation (in number of neighbors bond order or bond energy decreases, is also taken which they mentioned Tersoff3 [T1 as per our convention] performs into account. The primary difference between the SW and the Tersoff better compared to other potentials in predicting transformation from potential is the consideration that energy per bond decreases suffi- cubic-diamond to -tin) no details have been provided with regards to ciently rapidly with increasing coordination. This consideration of bond the type of loading situation as well as on the method of detection of order results in development of nonlinear type functionals for these different phases. Thereby, there needs to be a detailed study ofthe class of potentials. It should be noted that the form of the Tersoff po- evaluation of the interatomic potentials for Si with regards to phase tentials remains the same, even though the parameters were changed transformation comparing the results obtained to that of previous re- numerous times resulting in different versions of the potential. The ported experimental observations. Moreover, new potentials currently interatomic potential is taken as being utilized and developed after previously discussed literature [30] 1 EEV=i = ij has been considered in this study for comparative evaluation. 2 i i j (7)

2.1. Review of interatomic potentials for Si Vij = fC ()[()()] rij a ij fR rij+ b ij fA rij (8)

where Vij is bond energy; f is repulsive pair potential which includes Since the manuscript deals with numerous interatomic potentials R the orthogonalization energy when atomic wavefunctions overlap; fA is comparing them in their ability to predict phase transformation under attractive pair potential associated with bonding. Here fAR, f are applied hydrostatic compressive loads, a brief description of the dif- modelled as per the morse potential as follows: ferent potentials have been provided in this manuscript (while detailed ()1r descriptions can be obtained from referred papers). fR () r= Ae

()2r fA () r= Be (9) 2.1.1. Stillinger-Weber potential Stillinger-Weber is a three body cluster potential [19,20] where The term fc is smooth cutoff function to limit the range of the po- potential energy is given by tential and is represented as follows:

234 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240

1, r< R D 1 1 ()r R E =()rij + U ()rij + Uf( rij ) f ( r ik ) g cos(jik ) fC () r = sin[ ], R D< r < R + D 2 2 2D ij i j i jk 0, r> R + D (10) (13) where the two parameters R and D has been chosen to include the first where jik is angle formed by neighboring atoms centered at atom i. neighbour shell but not the second one. The bond order parameter bij is First term ()rij accounts for pair potential where as second one re- assumed to be monotonically decreasing function of coordination of presents the embedded term i.e. energy required to fix a particular atom atoms i and j. It is given as follows: i in the surrounding electron density of atom j. It should be noted that is augmented by an angularly dependent term. The third term re- 1 2n presenting the three body interaction had been adopted from Stillinger- b = 1 + n n ij ij Weber potential [19] but with a little change. Typically the MEAM potential is a mixture of Stillinger-Weber(SW) as well as embedded , 3 3 atom forcefields(EAM). Here g, , U and f are represented as cubic [()]3 rij r ik ij = fC ()() rik g ijk e splines each with 10 degrees of freedom. Second neighbour cutoff for k i, j is 4.5Åwhereas first neighbour cutoff for and f are 3.5Å. These parameters had been determined by least square fitting of various c2 c2 g ( )= 1 + properties like ab initio force and energy data for atom clusters, liquid 2 2 2 d [d+ ( h cos ) ] (11) and amorphous systems, ab initio LDA energetics for vacancy and in- terstitial point defects, experimental elastic constants and phonon fre- Here ij represents effective coordination which counts the No.of quencies. other bonds to atom i besides the ij bond. The term ijk is bond angle between bonds ij and ik. The coefficient of repulsive pair potential aij is 2.1.4. Environment dependent interatomic potential given by As the name suggests, Environment-Dependent Interatomic poten- 1 tial [29] is highly dependent on neighbors of individual atoms. Here 2n n n single atom energy Ei is given by aij = 1 + ij

EVRZi = 2 ij, i + VRRZ3 ij,, ik i 3 3 j i j i k i, k> j [()]3 rij r ik (14) ij = fC () rik e k i, j (12) where VRZ2 (,)ij i is pairwise interaction between atoms i and j while

VRRZ3 (,,)ij ik i Here ij plays an identical role as in . However contribution is interaction between atoms at i, j and k centered at from is taken to be quite small. Here also parameters are fitted with atom i. Effective coordination number of atom i is given by lattice constant, cohesive energy, bulk modulus of diamond structure Zi = f() Rim and cohesive energy of bulk polytypes of silicon. There are different m i (15) version of the potentials in which the primary difference is in the change of the parameters, which has been represented here in Table 2. where f() Rim measures the contribution of neighbour m to the co- These potentials had been extensively used to study lattice dynamics, ordination of atom i in terms of separation function Rim in following microclusters, bulk point defects, liquid and amorphous state, surface manner reconstruction, epitaxial growth from vapour etc. It should be noted 1 if r< c that even though the initial form of Erhart-Albe (T4) potential is dif- f() r = exp if c< r < a ferent as per the initial publication [18] but these can be related to that ()1 x 3 of the Tersoff potential since it is a development of the Tersoff potential 0 if r> a (16) to closely match the elastic constants with experimental observations. The two body interaction term V2 (,) r Z includes repulsive and attractive interaction along with dependence on cutoff r = a. 2.1.3. Modified embedded atom potentials B The functional form for energy in MEAM interatomic potential [28] V2 r, Z= A p() Z exp is given by r r a (17) Here also weakening of attractive interaction with increase in co- Table 2 Parameters for different Tersoff potentials. ordination number is taken into account. Such dependence on coordination number can be represented by gaussian function Parameters T0 T1 T2/T3 Erhart-Albe(T4) Z2 p() Z= e (18) A(eV) 3264.7 1830.8 1830.8 1899.385 B(eV) 95.373 471.18 471.18 361.557 The three body interaction term depends on radial and angular 1 1(Å ) 3.2394 2.4799 2.4799 2.6155 factors 1 2 (Å ) 1.3258 1.7322 1.7322 1.666 0.0 0.0 0.0 0.0 VRRZ3 (,,)()()(,)ij ik i= gRgRhl ij ik ijk Z i (19) 0.33675 1.1 ×10 6 1.1 × 1.0 RRij. ik n 22.956 0.78734 0.78734 1.0 where lijk=cos ijk = 5 RRij ik c 4.8381 1.0039 ×10 1.0039 × 1.13681 The radial function is given by d 2.0417 16.218 16.218 0.63397 h 0.0000 −0.59826 −0.59826 −0.335 1 1.3258 1.7322 0 0 g() r= exp 3 (Å ) r a (20) R(Å) 3.0 2.85 2.85 2.90 D( ) 0.2 0.15 0.15 0.15 The dependence of angular function on effective coordination number z through two different function ()Z and w() Z is given by

235 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240

1+ (Z ) h l, z= H w() Z (21) Here h(l,z) is chosen as

Q()(()) Z l+ Z 2 2 h( l , Z )= [(1 e )()(())]+Q Z l + Z (22) The equation is chosen in such a manner that angular forces become weaker with increase in effective coordination number, thus representing a transition from covalent bonding to metallic bonding. The function ()Z controls the equilibrium angle of three body interaction as a function of coordination. Here is approximated in terms of four parameters and smoothly interpolated between special points (Z = 2,3,4,6) which helps it to model proper hybridization of atoms. Fig. 1. Pressure vs. VV/ 0 pattern for silicon compressed under hydrostatic u4 Z2 u 4 Z ()(Z= u1 + u 2 u 3 e e ) (23) condition using different potential compared with experimental studies. Here all the parameters like A,B, ,, , a, c, are fitted on ab initio results based on DFT/LDA for bulk properties like cohesive energy, lattice compression at certain strains and thereby eventually demonstrate constant of cubic diamond structure, formation energy of point defects, phase transition as reported from experimental observations. generalized stacking-fault energy surfaces and experimental elastic constants. • The purity of samples for experimental investigations is also of a Values of different fitted parameters are as follows: A = 7.982173 eV, concern. It should be noted that presence of initial voids and defects a = 3.121382 Å, = 1.4533108 eV, Q0 = 312.1341346, = 3.1083847, (point defects such as vacancies or interstitials as well a line defects B = 1.5075463 Å, c = 2.5609104 Å, = 1.1247945 Å, µ = 0.6966326, such as edge and screw dislocations) may result in more volumetric = 1.2085196, = 0.5774108, = 0.2523244, = 0.0070975. change. The experimental literature does not mention in details with regards to the presence of initial voids and/or defects thereby this 2.2. Elastic constant estimation may be a problem since in numerical simulations we are only considering pristine single crystals without any initial voids or defects. One of the primary mechanisms by which a interatomic potential is • Even though it is mentioned that the experimental samples are compared against another is whether the potential is able to capture the crystalline in nature, no specifications have been made with regards elastic constants at ambient temperature and pressure conditions. The to whether the samples are single crystalline or polycrystalline. If comparison has been provided in Table 3. Most of the elastic property they are polycrystalline then obviously there will be differences in values have been calculated previously and reported in literature which results since it is well known that presence of grain boundaries along has been referenced here following the potential name in Table 3. Ty- with texture of the material influences mechanical response of pically, this check is performed for any developed interatomic potential samples. to demonstrate that it can closely simulate the behavior of the material. • In our MD simulations we are considering pristine bulk materials The best potential observed to reproduce the elastic constants of Si is whereas in experimental investigations apart from existence of bulk the T4 potential. However, this comparison may not necessarily mean material there also exist free surfaces which may act as nucleation that the best potential based on elastic constant estimation will de- sites for high pressure phases. monstrate the best behavior for the material with regards to phase transformation subjected to hydrostatic compressive loading, as has The objective of the study was not to develop a new interatomic been demonstrated in this manuscript for Si. potential because as observed above there needs to be proper clarity with experimental observations prior to development of potentials to 3. Results and discussion replicate experimental observation. The objective of this study is to demonstrate the performance behavior of commonly utilized potentials Simulation of hydrostatic compression has been done on the sam- for Si subjected to static compression thereby demonstrating phase ples with different interatomic potentials and compared against ex- change. At different pressure ranges (as reported from experimental perimental observations (refer Fig. 1). It can be observed that none of observations) numerical XRD investigation is carried out to identify the simulations are able to achieve the volume compression as observed different phases present in the sample. in the experiments. It should be noted that sudden changes in volume might provide a possible indication that the crystal structure has un- 3.1. X-ray diffraction analysis dergone a change in phase. The reason for non-attainment of volume compression using simulations to match experimental observations can The Bragg planes corresponding to the atomic coordinates for the be either of the mentioned reasons: cubic diamond structure are {(111), (022), (131), (004), (133) etc.}. The structure of cubic diamond lattice is symmetric consisting of 8 • There is necessity of further improvements on the interatomic po- repeatable atoms (supplementary Fig. 1). It should be noted that the tentials, which needs to be properly tuned to attain volume lattice structures of -tin (supplementary Fig. 1) phase do not

Table 3 Comparative study of elastic constants for different potentials.

Elastic properties(GPa) Exp. [30] SW [30] EDIP [29] MEAM [28] T0 [30] T1 [30] TT2/ 3a T4 [18]

C11 167 161.6 175 165.4 121.7 142.5 142.54 167 C12 65 81.6 62 82.3 85.8 75.4 75.38 65 C11-C12 102 80 113 83.1 35.9 67.1 67.16 102 C44 81 60.3 71 71.7 10.3 69 69 72 B 99 108.3 99 110 98 98 97.8 99

a Calculated from lammps Elastic script.

236 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240 correspond to the simple tetragonal form of the cubic diamond phase crystal of cubic diamond of Si (which is in good agreement with stan- since it has numerous Bragg planes which are completely absent in the dard X-ray diffraction pattern [35]. It is quite expected that with change cubic diamond structure. The atomic fractional coordinates for the -tin in pressure these peak positions will change and also new peaks cor- structure are {(0,0,0), (0.5,0.5,0.5), (0.5,0.0.75), (0,0.5,0.25)}. In other responding to other different planes may originate signifying phase words it can be taken as (0,0,0), (0,0.5,0.25) X (0,0,0), (0.5,0.5,0.5) transition in the material. The primary challenge is in identification of i.e. placement of pair of atoms at (0,0,0) and (0,0.5,0.25) at two lattice peaks corresponding to different planes such that numerical relations position of body-centered structure. The Bragg planes corresponding to between the planes are satisfied with reasonable accuracy. Our study the atomic coordinates for the -tin structure are (200)/(020), (101)/ involves hydrostatic pressure ranges up to around 150 GPa (refer Fig. 1) (011), (121)/(211), (112), (220), (301)/(031), (013)/(103) etc. It in which the fidelity of the interatomic potentials in demonstrating can be noted that the (002) plane is absent for the -tin phase. The reported experimental phase transformations is examined. 1 orthorhombic Imma phase also has 4 repeatable atoms per unit cell Once the square of the inverse interplanar spacing (d2 ) have been with the coordinates as (0,0,0), (0.5,0.5,0.5), (0.5,0,0.886), calculated, relations needs to be developed to identify which phase is (0,0.5,0.386) (supplementary Fig. 1). Since the atomic coordinates of present within the material. It may be possible that multiple phases are the orthorhombic Imma phase are quite similar to that of the -tin present within the material. In general the interplanar spacing for or- 2 2 2 phase the Bragg planes corresponding to this phase are similar. The 1 =h +k + l h,, k l thogonal crystal system is given as d2 a2 b2 c2 where re- (002) plane which is not observed for the -tin phase is observed for presents the miller indices of the desired planes and a,, b c represents the Imma phase. Apart from that, standard splitting of various peaks the lattice parameters. For a tetragonal system with (a= b c) the (on conversion from a tetragonal to an orthorhombic conversion) of the 1 h2+ k 2 l2 interplanar spacing relation is expressed as 2 =2 + 2 . -tin phase is also observed in the Imma phase. The planes for the Imma d a c Based on known permissible planes for the -tin phase, relationships phase are (020), (200), (002), (011), (101), (121), (211), (112), can be established in between the planes to identify the lattice para- (220), (022), (202), (031), (301), (013) etc. Hexagonal close packed meters. The planes can be related as structure can be understood as a rhombohedral lattice with two atoms 1 1 1 1 1 1 1 1 2 = 2 2, 2 = 2 2, 2 = 2 + 2, 2 = at fractional coordinate of (0,0,0) and (0.3333,0.6667,0.5) (supple- d (220) d(200) d (400) d(220) d (121) d (200) d(101) d (301) 1 1 1 1 1 + , = + , = 2 + . mentary Fig. 1). The Bragg planes corresponding to HCP structure are d2(121) d 2 (301) d2(101) d 2 (231) d2 (121) ¯ ¯ ¯ ¯ (1010), (0002), (10 2), (202 0), (01 3), (1 0 ), (0111) etc. Finally Relationships are also established between the different known 1 1 higher pressure structure FCC (supplementary Fig. 1) consists of a cubic planes of the Imma phase as = + , d2 (020) d2 (101) unit cell with atom at fractional coordinate of (0,0,0), (0.5,0.5,0), 1 1 1 1 1 1 1 = + , = 4 , = + , (0.5,0,0.5) and (0,0.5,0.5) . Characteristics peaks appearing in XRD d2 (202) d2(002) d 2 (004) d2(002) d 2 (222) d2(002) d 2 (440) 1 1 pattern for FCC phase are (111), (200), (220), (113), (222), (004), = 4 , = 2.25 . d2(220) d 2 (303) (331), (024) . For primitive hexagonal crystal system interplanar spacing d is As per experimental investigations, compression of cubic diamond 4 h2+ hk + k 2 l2 given by following relation: = 2 + 2 . Typical planes ap- phase of Si results in phase transition to following phases -tin phase at 3 a c pearing in xrd of primitive hexagonal structures are (001), (100), pressure ranges of 11–13 GPa, Imma phase in between 13 and 16 GPa, (002) , (200), (003), (012), (202), ( 10), (011), (112) etc. Due to Primitive hexagonal phase between 16 and 36 GPa, SiVI in range of presence of the term (h2 + hk + k2) some interesting relationship be- 36–42 GPa, HCP(SiVII) between pressure range of 42–78 GPa and 1 1 tween planes can be observed as = , FCC(SiX) from 78 GPa to higher pressure (diagrammatic representation d2(1¯20) d 2 (110) 1 1 1 1 of which is presented in the supplementary Fig. 1). The parameter fit- = , = . These exactly equal relationship do not d2(1¯10) d 2(100) d 2(3¯20) d 2 (120) ting for most of the interatomic potentials is done based on DFT si- help us in prediction of phase, we need relation between three or more mulations which demonstrates a particular stable structure at certain phases. Some of such relations are as follows: volume compression [22], however at 0 K temperature. Typically the 1 =41 ,1 = 91 ,1 = 3 1 d2 d2 d 2 d2 d 2 d2 . Thus we observe that phases that are considered for the case of Si include cubic diamond, (002) (001) (003) (001) (110) (100) with reduction in symmetry of crystal system dependence of interplanar -tin, FCC and HCP phases (e.g. refer Fig. 1 in literature [22]). spacing on miller indices gets complicated, subsequently it becomes In an effort to determine the phase of the material after compres- difficult to frame proper relation among them. Typically permissible sion, XRD of the entire sample has been carried out (with help of planes depend on unit cell structure while interrelationship among USER-DIFFRACTION package as implemented in LAMMPS [34]). This them depends upon crystal system. Hexagonal close packed also be- diffraction algorithm uses 3D mesh of reciprocal lattice points builtona longs to such crystal system. Different characteristics planes for HCP rectangular grid with certain reciprocal space mesh size. Diffraction structure are (100), (002), (102), (101), (110), (103), (201), (004) intensity at each reciprocal lattice point is computed using structure etc. Some of the relationships between planes are as follows: factor and Lorentz polarization factor. The 3D reciprocal grid spacing 1 1 1 1 1 1 1 2= 2 + 2, 2 = 42 , 2 = 3 2 . has been suitably tested for convergence prior to being used for re- d(102) d(100) d(002) d(004) d(002) d(110) d (100) porting observations. For determination of lattice parameter and As FCC belongs to the most symmetric Cubic crystal system, it has identification of respective planes, matlab scripts have been written large number of such relationships. Some of the relationships between different known planes are as follows: using input as relative positions of atoms inside unit cell, lattice para- 1 1 1 1 1 1 =2 =4 , = + , = + meter and atomic scattering factor of silicon. Initially, trial lattice d2 (400) d2(200) d 2 (113) d2 (111) d2(220) d 2 (331) 1 1 1 1 parameters are determined approximately considering two or three , =3 , =5 . Similar interplanar relationships d2(111) d 2 (222) d2(200) d 2 (240) peaks, then these lattice parameters are used as input in an attempt to can be obtained for Cubic diamond structure. Some of the peaks can be reproduce the xrd pattern matching at least 5 peaks. The parameters c1, related to each other in following manner: c2, c3 chosen for these simulations are 1, 1 and 1. However different 1 1 1 1 1 2 =9 2, 2 =2 =4 2, 2 =8 =3 . smaller values up to 0.25, 0.25 and 0.25 were also tested in order to d (115) d(111) d (044) d(220) d (224) Based on the above relations, different phases can be identified by check for convergence. assigning respective values obtained from the XRD analysis to the From the XRD analysis one obtains the 2 angles which can even- 1 tually be converted to (inverse of the interplanar spacing) using square of interplanar spacings for particular planes. d In the hydrostatic loading scenario, T0 [14] interatomic potential governing Bragg’s law 2dsin = n with n = 1 and wavelength of demonstrates changes in phase to that of -tin and Imma phase at 1.541838 Å (monochromatic Cu K radiation). Here supplementary around 12 GPa (refer Fig. 2) as well as HCP and FCC phase at an applied Fig. 3 shows initial inverse interplanar spacings for pristine single pressure range of 80 GPa (refer Fig. 3). Partial amorphization is also

237 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240

Fig. 2. XRD pattern of silicon compressed under hydrostatic loading using T0 Fig. 4. XRD pattern of silicon compressed along [001] crystallographic direc- potential at 12 GPa. tion under quasistatic uniaxial loading condition using T0 potential.

Identification of the two structures can also be done with regards to bond-angle or common neighbour analysis, as shown in the inset of Fig. 3. However, it should be noted that similar type of identification cannot be done for -tin and Imma phase as these phases had not been standardized in any post-processing software such as Ovito [36]. It is being observed that even though Tersoff (T0) potential did not perform well with regards to prediction of elastic constants (Table 2) but it performs much better than other potentials to demonstrate phase transformation at two different pressure ranges where it’s prediction of phases matches with DAC based experimental observations. Based on this observation, the uniaxial study of compression (up to 20 GPa) has been carried out for single crystal Si in different directions using the T0 Fig. 3. XRD pattern of silicon compressed under hydrostatic loading using T0 interatomic potential to investigate if there is presence of anisotropy or potential at 83 GPa. direction dependence on compression loading. Fig. 4 demonstrate formation of a -tin structure along with possi- bilities of formation of a tetragonal diamond (TD) structure, which is observed in the samples. The other potentials demonstrate shifts in the probably a metastable intermediary phase between the cubic diamond XRD peaks but no phase transition is observed at any pressure ranges and the -tin phase, when the sample is uniaxially compressed along (see supplementary Figs. 5–10). Since the observation of phase transi- the [001] direction. The planes observed for the TD phase are similar to tion using the T0 interatomic potential matches with that of experi- that of the CD phase but a characteristic splitting of peaks are observed mental investigations in comparison to that of the other potentials, it [37] such as (022) peak of the CD structure is observed to split into can be justified that T0 potential is better suited for phase transition (220) and (202)/(022); similarly splitting is observed for the (131) study of the material. It should also be noted that the experimental CD to (311)/(131) and (113) planes; (004) into (400)/(040) and investigations reports presence of certain phases at certain pressure CD (004); (133) into (313)/(133) and (331). The characteristic planes ranges but it does not specify that only that phase exists in the sample; CD of -tin which are (200) and (101) are not observed for this phase. thereby it is quite probable that a combination of phases are present in Relations can be developed for this structure to obtain the lattice the sample as being observed in the numerical simulations. Different 1 1 1 1 1 parameters: 2 = + 2, 2 =22 , 2 = + planes observed for Imma phase are (101), (200), (020), (002), d (311) d(220) d (400) d(220) d (224) 1 1 1 1 1 1 , = +2 , = +2 . The Tersoff po- (301), (312). However some of the peaks also correspond to the -tin d2(004) d 2 (333) d2 (113) d2(220) d 2 (313) d2 (202) phase such as (020) and (301) of Imma are identical with (101) and tential (T0) demonstrates presence of -tin phase with lattice para- (301) peak of -tin phase respectively. Beside these (101) and (301) meters as 4.440 Å and 1.834 Å (c/a = 0.413) at 10.63 GPa along with peaks, -tin phase also shows characteristics peaks for (200), (220) co-existence of a TD phase with lattice parameters as 6.279 Å and and (501) planes. Å 1 1 1 3.256 (c/a = 0.519). It should be noted that on application of further In Fig. 2, interplanar relationship of 4 = + can be d2 (101) d2 (200) d2 (002) loads till around 25 GPa, no phase transformation to Imma is observed used for assignment of Imma peak whereas lattice parameters can be in the samples. In this regard it should be pointed out that previous MD calculated from position of (200), (020), (002) peaks. Similarly for simulation of shock compression (loading condition being significantly 1 1 1 -tin, = + and = is used to assign peaks d2 (301) d2 (220) d2 (101) different to the uniaxial loading condition being applied inthis and lattice parameters are determined using (200) and (101) peaks. manuscript) had predicted -tin phase with lattice parameters as Based on identification of both -tin and Imma phase we can mention 6.897 Å and 2.548 Å (c/a value of 0.369) at 13 GPa [38]. that the lattice structure of the sample exhibits the following structure XRD patterns (refer Fig. 5) for loading along [011] directions shows (refer Fig. 2): c/a = 0.466 for the -tin phase; c/a = 0.82 and b/ formation of both Imma as well as -tin phase in addition to that of the a = 0.953 for the Imma phase. TD phase. The observation of Imma and -tin phase for loading along In Fig. 3, higher pressure phase FCC was predicted by using fol- [011] direction and only -tin phase for loading along [001] direction, 1 1 lowing interplanar relationship: =4 and since it is a cubic shows that the behavior of phase transition in Si is anisotropic in d2 (222) d2 (111) nature. In this regard, it should be mentioned that there are simulation crystal, hence position of (111) peak is sufficient to determine the studies which reports formation of Imma phase upon different types of lattice parameter. Similarly lattice parameter for HCP can be de- loading conditions for Si like shock loading along [001] direction using termined using position of (101¯ 0) and (0002) peaks. Based on iden- modified Tersoff [18] interatomic potential [39]. It should also be tification of both the FCC and HCP phase, the lattice structure ofthe highlighted that partial amorphous nature of sample has also been sample is identified to exhibit the following structure(refer Fig. 3): demonstrated by the Tersoff potential(T0) in which corresponding b/a a = 3.541Åfor the FCC phase; c/a = 1.625 for the HCP phase.

238 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240

loading condition (similar to that of state of stress in Diamond Anvil cell setups). It was demonstrated that accurate prediction of elastic constants (in relation to that of experimental observations) with a interatomic potential does not necessarily mean good capability to demonstrate phase transformations (refer Table 2 and Figs. 1–3 along with supplementary Figs. 5–10). Tersoff(T0) [14] inspite of not showing better elastic constant properties demonstrates better prediction of phase transformation on hydrostatic loading in comparison to other potentials, whereas the potential which closely matches elastic constant estimations (T4) does not demonstrate phase transformation as observed experimentally. As observed from the study of uniaxial compression along [001], [011] and [111] direction, it can easily be justified that there is sig- Fig. 5. XRD pattern of silicon compressed along [011] crystallographic direc- nificant orientational dependence of the material subjected to pressure tion under quasistatic uniaxial loading condition using T0 forcefield. loads in demonstration of phase transformation in the material. On application of uniaxial pressure along different loading directions: directions [001] demonstrated formation of -tin phase whereas [011] and [111] directions demonstrated formations of -tin and Imma phase. Interestingly, a tetragonal diamond phase has been reported to be observed while loading along all the direction. Partial amorphization has also been observed using the Tersoff potential while loading along the [011] direction, but not for the other two directions. No change in change of rate of strain was observed for uniaxial loading along different direction.

Acknowledgments

The authors would like to acknowledge the computational facilities Fig. 6. XRD pattern of silicon compressed along [111] crystallographic direc- provided by the Centre for Theoretical Studies as well as facilities in tion under quasistatic uniaxial loading condition using T0 potential. Prof. Mitra’s computational lab at IIT Kharagpur.

Appendix A. Supplementary material and c/a ratios for Imma are 1.1477 and 1.093 respectively along with a c/a ratio of 0.46 for the -tin phase. Experimentally b/a and c/a value Supplementary data associated with this article can be found, in the for Imma had been predicted as 0.95 and 0.54 respectively [4], as ob- online version, at https://doi.org/10.1016/j.commatsci.2018.09.037. served from DAC (note that DAC typically means hydrostatic loading whereas uniaxial loading is being applied here). It should also be References pointed out that amorphization of silicon had been observed for other different simulation studies like nanoindentation [10,40], two-body [1] D. Daisenberger, M. Wilson, P.F. McMillan, R.Q. Cabrera, M.C. Wilding, D. Machon, and three-body abrasion [11]. High-pressure X-ray scattering and computer simulation studies of density-induced For the case of loading along [111] direction for Si, observation of polyamorphism in silicon, Phys. Rev. B 75 (22) (2007) 224118. -tin, Imma and a TD phases can be made (refer Fig. 6). Existence of all [2] S.J. Duclos, Y.K. Vohra, A.L. Ruoff, Experimental study of the crystal stability and equation of state of Si to 248 GPa, Phys. Rev. B 41 (17) (1990) 12021. of these possible structures is supported by sufficient number of in- [3] J. Hu, I. Spain, Phases of silicon at high pressure, Solid State Commun. 51 (5) dividual peak. At 14.63 GPa lattice parameter for Imma is observed as (1984) 263–266. 4.389 Å, 2.994 Å and 4.155 Å (b/a = 0.682, c/a = 0.947) whereas [4] M. McMahon, R. Nelmes, New high-pressure phase of Si, Phys. Rev. B 47 (13) lattice parameter observed for -tin phase are 4.155 Å and 1.911 Å (c/ (1993) 8337. [5] H. Olijnyk, S. Sikka, W. Holzapfel, Structural phase transitions in Si and Ge under a = 0.46). Quite different from behavior along the [011] direction, no pressures up to 50 GPa, Phys. Lett. A 103 (3) (1984) 137–140. amorphization is observed along this direction. [6] G. Voronin, C. Pantea, T. Zerda, L. Wang, Y. Zhao, In situ X-ray diffraction study of It should be noted that in these uniaxial simulations we were only silicon at pressures up to 15.5 GPa and temperatures up to 1073 K, Phys. Rev. B 68 (2) (2003) 020102. interested in observing directional anisotropy with regards to phase [7] V. Ivashchenko, P. Turchi, V. Shevchenko, Simulations of indentation-induced transition. Thereby, these samples were not extended up to high pres- phase transformations in crystalline and amorphous silicon, Phys. Rev. B 78 (3) sures beyond 20 GPa. It should also be noted that the uniaxial com- (2008) 035205. [8] M. Durandurdu, D. Drabold, High-pressure phases of amorphous and crystalline pression studies have also been carried out at a strain rate of around silicon, Phys. Rev. B 67 (21) (2003) 212101. 10 10 , thereby to observe if strain rate influences results. No significant [9] C. Sanz-Navarro, S. Kenny, R. Smith, Atomistic simulations of structural transfor- differences in behavior could be observed for high strain rate loading mations of silicon surfaces under nanoindentation, Nanotechnology 15 (5) (2004) 692. situation compared to that of the quasistatic loading. The XRD plots [10] D. Kim, S. Oh, Atomistic simulation of structural phase transformations in mono- 10 along three different directions for higher strain10 rate( /sec) uniaxial crystalline silicon induced by nanoindentation, Nanotechnology 17 (9) (2006) compressive loading have been given in the supplementary Figs. 11, 12 2259. [11] J. Sun, L. Fang, J. Han, Y. Han, H. Chen, K. Sun, Phase transformations of mono- and 13. Interestingly no significant differences are observed in the crystal silicon induced by two-body and three-body abrasion in nanoscale, Comput. lattice parameters with change in strain rate. Mater. Sci. 82 (2014) 140–150. [12] Y.-H. Lin, T.-C. Chen, P.-F. Yang, S.-R. Jian, Y.-S. Lai, Atomic-level simulations of nanoindentation-induced phase transformation in mono-crystalline silicon, Appl. 4. Conclusions Surf. Sci. 254 (5) (2007) 1415–1422. [13] L. Hale, X. Zhou, J. Zimmerman, N. Moody, R. Ballarini, W. Gerberich, Phase transformations, dislocations and hardening behavior in uniaxially compressed si- A comparative evaluation of different interatomic potentials have licon nanospheres, Comput. Mater. Sci. 50 (5) (2011) 1651–1660. been performed in this study with regards to their abilities in prediction [14] J. Tersoff, New empirical approach for the structure and energy of covalent systems, of phase transformation of single crystal Si subjected to hydrostatic type Phys. Rev. B 37 (12) (1988) 6991.

239 D. Prasad, N. Mitra Computational Materials Science 156 (2019) 232–240

[15] J. Tersoff, Empirical interatomic potential for silicon with improved elastic prop- A.F. Voter, J.D. Kress, Highly optimized empirical potential model of silicon, erties, Phys. Rev. B 38 (14) (1988) 9902. Modell. Simul. Mater. Sci. Eng. 8 (6) (2000) 825. [16] J. Tersoff, Modeling solid-state chemistry: interatomic potentials for multi- [29] J.F. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, S. Yip, Interatomic potential for component systems, Phys. Rev. B 39 (8) (1989) 5566. silicon defects and disordered phases, Phys. Rev. B 58 (5) (1998) 2539. [17] J. Tersoff, Chemical order in amorphous silicon carbide, Phys. Rev. B49(23) [30] H. Balamane, T. Halicioglu, W. Tiller, Comparative study of silicon empirical in- (1994) 16349. teratomic potentials, Phys. Rev. B 46 (4) (1992) 2250. [18] P. Erhart, K. Albe, Analytical potential for atomistic simulations of silicon, carbon, [31] E. Pearson, T. Takai, T. Halicioglu, W.A. Tiller, J. Cryst. Growth 70 (1984) 33. and silicon carbide, Phys. Rev. B 71 (3) (2005) 035211. [32] R. Biswas, D. Hamann, New classical models for silicon structural energies, Phys. [19] F.H. Stillinger, T.A. Weber, Computer simulation of local order in condensed phases Rev. B 36 (12) (1987) 6434. of silicon, Phys. Rev. B 31 (8) (1985) 5262. [33] B.W. Dodson, Development of a many-body tersoff-type potential for silicon, Phys. [20] F.H. Stillinger, T.A. Weber, Erratum: computer simulation of local order in con- Rev. B 35 (6) (1987) 2795. densed phases of silicon [phys. rev. b 31, 5262 (1985)], Phys. Rev. B 33 (1986) [34] S. Coleman, D. Spearot, L. Capolungo, Virtual diffraction analysis of ni [010] 1451. symmetric tilt grain boundaries, Modell. Simul. Mater. Sci. Eng. 21 (5) (2013) [21] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. 055020. Comput. Phys. 117 (1) (1995) 1–19. [35] M.C. Morris, H.F. McMurdie, E.H. Evans, B. Paretzkin, H.S. Parker, N.C. [22] M. Yin, M.L. Cohen, Microscopic theory of the phase transformation and lattice Panagiotopoulos, C.R. Hubbard, Standard X-ray diffraction powder patterns, 1981, dynamics of Si, Phys. Rev. Lett. 45 (12) (1980) 1004. pp. 2. [23] W. Gust, E. Royce, Axial yield strengths and two successive phase transition stresses [36] A. Stukowski, Visualization and analysis of atomistic simulation data with for crystalline silicon, J. Appl. Phys. 42 (5) (1971) 1897–1905. OVITO–the Open Visualization Tool, Modell. Simul. Mater. Sci. Eng. 18 (1) (2009) [24] S.J. Turneaure, Y. Gupta, Inelastic deformation and phase transformation of shock 015012. compressed silicon single crystals, Appl. Phys. Lett. 91 (20) (2007) 201913. [37] B.D. Culity, S. Stock, Elements of X-ray Diffraction, Addison-Weseley, Reading, MA, [25] M.A. Meyers, A mechanism for dislocation generation in shock-wave deformation, 1978. Scr. Metall. 12 (1) (1978) 21–26. [38] P. Stubley, A. Higginbotham, J. Wark, Simulations of the inelastic response of si- [26] L. Tao, R. Shahsavari, Diffusive, displacive deformations and local phase transfor- licon to shock compression, Comput. Mater. Sci. 128 (2017) 121–126. mation govern the mechanics of layered crystals: the case study of tobermorite, Sci. [39] G. Mogni, A. Higginbotham, K. Gaál-Nagy, N. Park, J.S. Wark, Molecular dynamics Rep. 7 (1) (2017) 5907. simulations of shock-compressed single-crystal silicon, Phys. Rev. B 89 (6) (2014) [27] P.K. Sarkar, N. Mitra, Gypsum under tensile loading:a molecular dynamics study, 064104. Geochim. Cosmochim. Acta. (2018) (submitted for publication). [40] W. Cheong, L. Zhang, Molecular dynamics simulation of phase transformations in [28] T.J. Lenosky, B. Sadigh, E. Alonso, V.V. Bulatov, T.D. de la Rubia, J. Kim, silicon monocrystals due to nano-indentation, Nanotechnology 11 (3) (2000) 173.

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