Phase Transition and Elasticity of Gallium Arsenide under Pressure

Geometry optimization calculations were performed for some structural, elastic and mechanical properties of gallium arsenide (GaAs) under pressures up to 25 GPa. On contrast to previous works, a recent Stillinger-Weber type potential was used for the first time to elaborate the pressure dependence aspects of GaAs. B3→B1 phase transition pressure was determined as 17 GPa. Pressure dependence of density, typical cubic elastic constants, bulk, shear, and Young moduli, Poisson ratio, elastic velocities, anisotropy parameter, Kleinman parameter, elastic anisotropy degree, and stability conditions of GaAs were also evaluated. Dverall, our results are satisfactory and can be helpful for future investigations of GaAs under pressure.


Introduction
Elastic properties of materials under pressures provide better understanding of some physical knowledge comprising the interatomic forces, mechanical features, phase transitions and so forth.However, elastic property measurements under pressure are usually very difficult and the lack of experimental data can be compensated by computational methods 1 .
Apart from many experimental efforts 4,10 , Rino et al. 12 calculated phase transition pressure (P T ) as 22 GPa for GaAs with molecular dynamics in 2002.Later, Yu et al. 7 [7][8][9][10][11][12] have extreme motivation on P T , cubic elastic constants and bulk modulus.However, pressure depence of young modulus, shear modulus, elastic wave velocities, Poisson ratio, elastic anisotropy, Kleinman parameter, anisotropy degree are still lacking for GaAs except the analysis of Varshney et al. 11 .So, our objective is to elaborate these parameters by geometry optimization calculations.On contrast to previous methods and applied potentials and to the best of our knowledge, this is the first survey reporting the application of a recent and modified Stillinger-Weber type potential to GaAs under pressure.
The rest of the paper is organized as follows.After this introduction, a brief outline for geometry optimization and computational details are given in Section 2. Dur calculations results are presented and discussed in Section 3. We than summarize and conclude in Section 4.

Computational Details
Geometry optimization is an efficient and convenient method for both molecular dynamics and density functional theory to get a stable configuration for a molecule or periodic system through fast and inexpensive energy computations.An optimization procedure involves the repeated sampling of the potential energy surface until the potential energy reaches a minimum where all forces on all atoms are zero.All herein done geometry optimization calculations were performed with General Utility Lattice Program (GULP) code 4.0. [13,14]during the present work.GULP code allows wide-range property calculations for 3D periodic solids, 2D surfaces, and gas phase clusters by employing two-body, three-body, four-body, six-body, and many body (EAM) potentials 15,16 depending on the demands of research.Since Stillinger-Weber type potentials provide reliable results especially for semiconductors, we employed a recent and modified Stillinger-Weber type potential 17 to research for GaAs.The lattice constant for GaAs was set as a 0 = 5.66 Å which is slightly higher than the experimental lattice constant a exp = 5.65 Å. Dther details of applied Stillinger-Weber type potential and its modified parameterization procedure can be also found in Han and Bester 17 .Ot is also possible to optimize related structures at constant pressures (includes all internal and cell variables) and constant volume (unit cell remains frozen) with GULP [1,[13][14][15][16]18] . Consant pressure optimization was performed for GaAs in our survey.The geometry of cells was optimized by the Newton-Raphson method based on the Hessian matrix Phase Transition and Elasticity of Gallium Arsenide under Pressure calculated from the second derivatives and Hessian matrix was recursively updated during optimization by using the BFGS [1,[13][14][15][16]18] algorithm.After setting the preconditions for the B3 crystal structure of GaAs, we adopted multiple runs at zero Kelvin temperature and checked the pressure ranges starting from 0 GPa up to 25 GPa.

Results and Discussion
B3→B1 phase transition pressure of GaAs was determined from P-V plot as seen in Figure 1.The volume of GaAs has a sudden decrease at 17 GPa.This abrupt decrease in volume originates from the structural phase change associated with the B3→B1 phase transition.The obtained value for P T (17 GPa) not only agrees well with experimental value of 16.6 GPa, but also in the range of former theoretical data (See Table 1).
As pointed out by Douri and Ressak, OOO-V compounds are low-density materials which can they display structural phase transitions to a higher density phase under pressure 8 .On fact, as clear from Figure 2, density of GaAs increases with the increasing pressure between 0 GPa and 25 GPa and our results confirms the findings of Al-Douri and Reshak 8 .
Typical cubic elastic constants (C 11 ,C 12 and C 44 ) define the mechanical hardness and necessary for specifying the stability of the material.These constants derived from the total energy calculations represent the single crystal elastic properties.Dn the other hand, Voigt-Reuss-Hill approach is a confident scheme for elastic constants of polycrystalline materials 19 .To get the correct values of elastic constants and other connected parameters of GaAs, we considered the Voigt-Reuss-Hill values during calculations.Table 1 lists obtained elastic constants with existing experimental and theoretical data for the sake of comparison.Dur results are both consistent with experiments and prior theoretical data 6,[11][12][13][14] .Further, elastic constants show a smooth increment with increasing pressure in both B3 and B1 phases for the entire pressure range as in Figure 3. Besides, increment of C 11 is higher than C According to Born structural stability, elastic constants must satisfy the conditions: C 11 -C 12 > 0, C 11 > 0, C 44 > 0, C 11 + 2C 12 > 0, and cubic stability e.g.C 12 < B < C 11 [20]   .As another result, present C 11 , C 12 and C 44 values well support both structural and cubic stability conditions for GaAs.
Bulk modulus (B) offers much information about bonding strength of materials and it is a measure of resistance to external deformation 21,22 .Figure 4 displays the pressure behavior of three elastic moduli (B, G, and E) of GaAs for entire pressure scale.By the way, shear modulus, (G) defines the resistance to shape change caused by a shearing force 23,24 .Young modulus (E) is the resistance to uniaxial tensions and gives the stiffness degree i.e., the higher the value of E, the stiffer is the material 25,26 .From the prevalent physical equation of bulk modulus (B = ΔP/ΔV) one can expect an increment for B due its direct proportion to applied pressure.Thus, bulk modulus of GaAs show a linear increment in both B3 and B1 phases as expected in Figure 4.As well, B3 structure has a larger stiffness than B1 structure because of its higher young modulus values than B1.Numerical values of B, G, and E are further compared in Table 1.The calculated value of B is satisfactory with experiments, whereas present results of both G and E moduli underestimate the experimental data.Since ductility and brittleness play a key role during materials manufacturing, we also appraised the ductile (brittle) nature of GaAs under pressure.The adjectives brittle and ductile stand for the two separate mechanical behaviors of solids when they subjected to stress.Usually, brittle materials are not deformable or less deformable before fracture.Unlike brittle materials, ductile materials are deformable a lot before fracture.At this point, Pugh ratio is a determinative limit for ductile (brittle) behavior 27 and has a common use in literature.Of B/G ratio is about 1.75 and higher, the material is ductile, otherwise the material becomes brittle.Another reliable approach is Cauchy pressure, which is defined as C p = C 12 -C 44 .The negative (positive) values of the Cauchy pressure show the brittle (ductile) nature of the compounds 28 .So, we cross-checked the obtained results with these norms.
Figure 5 is a plot of the Pugh B/G ratio region of the GaAs up to 25 GPa.On Figure 5, Pugh ratio analysis proves the ductile nature of GaAs under pressure.After a cross-check Cauchy pressure values corroborates the same behavior.As another result, all values of the B/G are higher than 1.75 and increase with pressure, which mean that pressure can improve ductility.
Poisson ratio (ν) is the ratio between the transverse strain (e t ) and longitudinal strain (e l) in the elastic loading direction.Ot delivers detailed knowledge about the bonding force behavior in solids 15,29 .ν = 0.25 and ν = 0.5 values are the lower limit and upper limit for central force solids, respectively.Poisson ration begins with 0.36 and increases with applied pressure as in Figure 6.This result suggests that interatomic forces in the GaAs are mainly central forces.Moreover, 0.36 value for ν is close to experiments (See Table 1).
On solids, vibrational excitation starts from lowtemperature acoustic modes.Being dependent on this case, two typical elastic waves namely the longitudinal wave (V L ) and shear wave(V S ) arise 15 .V L has an uniform increment trend with the increasing pressure in B3 and B1 phases in Figure 7 whereas V S is increasing in the B3 phase and decreasing in the B1 phase.The current data of V L and V S well satisfy the experiments as can be seen from Table 1.
Kleinman parameter (ζ) for cubic materials describes the relative ease of bond bending to the bond stretching.Minimizing bond bending leads ζ = 0 and minimizing bond stretching leads to ζ = 1.On addition, Kleinman parameter links to elastic constants as + + [30] .Figure 8 displays the Kleinman parameter upon the pressure    increment.Under pressure, ζ increases with increasing pressure and it is found to be between 0.65 and 0.90.So, present Kleinman value 0.65 is better than those of other reports (See Table 1).
The elastic anisotropy of crystals has an important implication in engineering since it is correlated with the possibility to persuade microcracks in the materials.For that reason, the anisotropy factor  A C C C = − [31] has been evaluated for a deeper view on the elastic anisotropy of the GaAs.On the applied pressure range in Figure 9, the obtained anisotropy factors A are not equal to 1, which means the presence of elastic anisotropy in GaAs.
A convenient method of describing the degree of elastic anisotropy for a cubic crystal has been formulated as 32 : ( ) ( ) Degree of elastic anisotropy (A * ) in GaAs shows a monotonous increment as the pressure increases and the increment in B1 structure is higher than B3 structure.Ot is evident from Figure 10 that GaAs has a profound elastic anisotropy since the degree of elastic anisotropy for B3 structure is smaller than the B1 structure of the GaAs.
On summary, existing results of this survey show a fair agreement with the experiments, especially on the bulk modulus and phase transition pressure as obvious in Table 1.Although current findings slightly underestimate shear modulus and C 44 of GaAs, our other results are better than those of Varshney et al. 11 and fairly agree with experiments and other data.

Conclusions
Ot should be noted that the merit of this work is not only calculating the C 11 , C 12 and C 44 , B, P T data of GaAs under pressure but also explaining pressure dependence of other important data such as E, G, V L and V S , ν, ζ, A and A*.Application of a recent Stillinger-Weber type potential, which is originally, used to predict phonon and related properties of GaAs 17 well reproduce other mechanical, structural, and elastic features of GaAs under pressure.Finally, our detailed results can be useful for the lack of experimental data and inspire future theoretical works.

E (GPa)
63.0 85.5 (4) ν 0.36 0.31 (34) V L (km/s) 4.64 4.73 (4) 6.89 (11) V S (km/s) 2.33 3.34 (4) 3.87 (11) ζ 0.67 0.58 (17) 0.81 (11) 12 and C 44 .Physically, C 11 portrays the longitudinal elastic behavior whereas C 12 and C 44 explain the off diagonal and shear elastic characteristic of cubic crystals because of shearing, respectively.So, a longitudinal strain produces a change in volume without a change in shape.This volume change is related to pressure, thus reflects a larger change in C 11 .Contrarily, a transverse strain or shearing causes a change in shape without a change in volume.So, C 12 and C 44 are less sensitive to pressure than C 11 .

Figure 1 .
Figure 1.P-V plot of GaAs up to 25 GPa.

Figure 2 .
Figure 2. Pressure dependence of density in GaAs up to 25 GPa.

Figure 3 .
Figure 3.Typical cubic elastic constants C 11 , C 12 and C 44 of GaAs under pressure.

Figure 4 .
Figure 4. Pressure dependence of elastic moduli B, E, and G in GaAs.

Figure 5 .
Figure 5. B/G ratio versus pressure for GaAs.

Figure 6 .
Figure 6.Poisson ratio behavior under pressure for GaAs.

Figure 7 .
Figure 7. Pressure dependence of elastic wave velocities of (V L and V S ) GaAs up to 25 GPa.

Figure 8 .
Figure 8. Behavior of Kleinman parameter under pressure for GaAs.

Figure 10 .
Figure 10.Degree of anisotropy for GaAs between 0 GPa and 25 GPa.
stated P T as 16.3 GPa by density functional theory.Afterwards, Varshney and his collaborators conferred P T as 17 GPa in their pure theoretical paper 11 .Yu et al., Al-Douri and Reshak, Bueno et al., Christensen, Varshney et al. and Rino et al.

Table 1 .
Comparing some structural, elastic and mechanical parameters of GaAs with previous and present results.