Atomistic study of the pressure-induced phase-transition mechanism in GaAs by Möbius inversion potentials

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Journal of Physics and Chemistry of Solids 68 (2007) 445–457 www.elsevier.com/locate/jpcs

Atomistic study of the pressure-induced phase-transition mechanism in GaAs by Mo¨bius inversion potentials Jin Caia,, Nanxian Chena,b, Huaiyu Wangc,a a

Department of Physics, Tsinghua University, Beijing 100084, China Institute of Applied Physics, University of Science and Technology of Beijing, Beijing 100083, China c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China

b

Received 29 September 2006; received in revised form 28 November 2006; accepted 15 December 2006

Abstract In this work, the phase transition from zincblende (B3) to rocksalt (B1) structure in bulk GaAs is investigated by using Mo¨bius inversion potentials. A Cm transition path is proposed from the molecular static simulation results, even though it is different from previously proposed transition paths Imm2, R3m and P32. The present Cm path is quite close to the previously suggested Imm2 path according to the crystal cell geometries and activation enthalpies. By comparison, the activation enthalpies along the R3m and P32 path are relative high. Therefore, the Imm2 mechanism, as a simplified model of the Cm one, is suggested to describe the microscopic process of the B3–B1 phase transition of GaAs. In this way, we investigate the changes of the system features during the transition process characterized by Imm2 mechanism and obtain a concise picture for the common B3–B1 transition. All the calculated results are compared to relevant experimental observations and other calculations. r 2007 Elsevier Ltd. All rights reserved. Keywords: A. Semiconductors; D. Crystal structure; D. Phase transitions

1. Introduction Many materials exhibit amazing and interesting behaviors under appropriate pressure and temperature. In the last few decades, numerous high-pressure experiments have been performed in compound semiconductors because of their increasing importance in wide range of technological applications. Among these materials, GaAs is considered as a prototypical material, which shows especially complicated phases under high pressure. With the pressure increased, the experimentally observed phase transition sequence is GaAs-I, the zincblende structure (B3, space ¯ group F 43m), to GaAs-II (an orthorhombic phase with proposed space group Pmm2 [1] or Cmcm [2]), which can be seen as a distortion of the rocksalt structure (B1, space ¯ group Fm3m) [3–5]. The previously reported transition pressure was around 17 Gpa [1] and the latest result is 1271.5 Gpa [6]. When pressure is over 24 GPa, the Corresponding author. Tel./fax: +86 10 62772783.

E-mail address: [email protected] (J. Cai). 0022-3697/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.12.023

diffraction pattern suggests a further transformation from GaAs-II into GaAs-III (a body-centered orthorhombic structure, with space group Imm2) and a gradual transformation into GaAs-IV (a simple hexagonal structure) in the range 60–80 Gpa [1]. Also, Besson et al. [6] observed a reverse transition from the orthorhombic structure to B3 structure around 10 GPa in the course of pressure decreasing. In the present work, the effort will be concentrated on the GaAs-I–GaAs-II phase transition, and the GaAs-II phase (B1-like) is simplified as a perfect B1 structure. To understand structural phase transitions, there exist a variety of methods in which first-principle calculations have made significant contribution in the past two decades. One of the crucial tests of these methods is the prediction of transition pressures. Besides, microscopic details of the atomic displacements, or the real-time pictures during the transition process, are also essential for the global and kinetic description of phase transitions. Understanding the phase transition mechanism, or determining the transition path (TP) is also important for technological applications,

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J. Cai et al. / Journal of Physics and Chemistry of Solids 68 (2007) 445–457

which might be helpful to the control of transition processes in practice. Though the first-principle method can perform atomistic simulation, this method is applied to rather limit scale of size and time because of its great demand on CPU time and memories [7]. Simulations based on interatomic potentials turn into being an alternative method. Early in the development of interatomic potentials for semiconductors, the pair potential model is seldom considered as a possible scheme to describe the four-fold coordinated semiconductors, which involve angular correlation between the so-called covalent bonding and are typically represented by three-body interaction terms. In general, the presence of three-body potentials improves force constants and other second-order properties of covalent compounds and it has been successfully used to study the properties around the ground states. However, when the number of bonded interactions varies, e.g., in our concerned structural transition wherein lies bond-breaking or bond-forming, complication arises because three-body terms are biased towards a certain bond angle. In recent years, a pure pair potential model is proved to be likely to well represent the four-fold coordinated compound semiconductors and related properties by different research groups [8–15], though it is still not for element semiconductors. Such a pair potential model has shown its validity for the pressure-induced structure transition in several wurtzite (B4) compounds (GaN [10], CdSe [11,12], ZnO [13,14]): the B4–B1 phase transition. In our previous work [15], we have proposed a new kind of method to obtain interatomic potentials for compound semiconductors based on Chen–Mo¨bius lattice inversion technique [16,17]. In this scheme, both the equilibrium and a vast amount of non-equilibrium states (including B3 and B1 phases) are taken into account for extracting the interatomic potentials. It could be hopeful to describe the microscopic mechanism of the B3–B1 phase transition process since the potentials cover wide ranges of interatomic separations and bond angles. Taking GaAs as a typical system, the research objective of the present study is mainly threefold: (i) to show the potential studies in the structural phase transition of compound semiconductors by using pair penitential models, (ii) to make a comparative study of the previously proposed B3–B1 transition mechanisms from the energy point of view, and (iii) to investigate and reveal the microscopic process of the B3–B1 phase transition. This paper is organized as follows. Section 2 briefly describes how to obtain interatomic potentials from several cohesive energy curves. In Section 3, the functional forms of calculated potentials of GaAs are presented, and the static properties are calculated so as to test the validity of the potentials. In Section 4, a TP is obtained from the structure relaxation process under the appropriate pressures, and comparisons with previous work are carried out. Then, a detailed transition picture is described in Section 5. Finally, Section 6 is the summary of the present work.

2. Methodology In the early 1990s, a number-theoretic Mo¨bius inversion formula was applied creatively to some physical inverse problems [16–18] which resulted in the solution of a series of problems such as the capacity inverse problem [19], the field-ion microscopy image of Fe3Al analysis [20], the inverse problems in astrophysics field [21], and especially the lattice inversion to obtain interatomic potentials of rare-earth intermetallics [22,23], ionic compounds [24–27] and recently semiconductors [15] based on the ab initio total energy calculations. In this section, we briefly outline the way to obtain interatomic potentials as well as the atomistic study methods used in our work. 2.1. Lattice inversion method Formally, the crystal cohesive energy can be expressed approximately by pair potential F(x) as follows: EðxÞ ¼

1 1X r0 ðnÞFðb0 ðnÞxÞ, 2 n¼1

(1)

where x is the nearest-neighbor distance, r0(n) is the coordination number of nth neighbor and b0(n)x is the distance of nth neighbor, b0(1) ¼ 1. The series {b0(n)} can be extended into a multiplicatively closed semi-group {b(n)}. In {b(n)}, for any two integers m and n, there exists a sole integer k satisfying b(k) ¼ b(m)b(n). Then Eq. (1) can be written as EðxÞ ¼

1 1X rðnÞFðbðnÞxÞ, 2 n¼1

(2)

where ( rðnÞ ¼

r0 ðb1 0 ½bðnÞÞ if 0 if

bðnÞ 2 fb0 ðnÞg; bðnÞefb0 ðnÞg:

(3)

The pair potential from inversion can be written as FðxÞ ¼ 2

1 X

I ðnÞE ðbðnÞxÞ,

(4)

n¼1

where I(n) is determined by    X 1 bðkÞ IðnÞr b ¼ dk1 . bðnÞ bðnÞjbðkÞ

(5)

It is noted that I(n) is merely related to crystal geometrical structure, not to concrete element category, i.e. {I(n)} is uniquely determined by {r0(n)} and {b0(n)}. Eqs. (1) and (2) refer only to a homo-atomic crystal. And further with the multiple lattice construction [24], the partial cohesive energy of the hetero- or homo-atoms can be obtained respectively, from which the corresponding interatomic potentials are determined by the lattice inversion method.

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2.2. Acquisition of the cohesive energy curves From the previous section one sees that as long as the crystal cohesive energy is known, the pair potentials can be obtained. Hence the first step is to obtain the cohesive energy. It is calculated by first-principle method and is defined as E coh ðxÞ ¼ E tot ðxÞ  E tot ð1Þ,

(6)

where Etot(x) denotes the total energy at lattice parameter x, and Etot(N) can be taken as the energy of isolated atoms. The total energy of GaAs is calculated by use of the local density approximation (LDA) implemented in the CASTEP program [28]. In our calculation, the ultrasoft pseudopotentials for Ga and As are used. The crystal wave functions are expanded by plane-wave basis set with an energy cutoff of 330 eV. The k-mesh points in the irreducible Brillouin zone are generated with the k-points spacing 0.05/A˚ by the Monkhorst–Pack scheme [29]. The energy tolerance for the self-consistent field (SCF) convergence is 2  106 eV/atom. Ga 3d states have been included in the pseudopotential calculations since their energy closes to the 4s states (particularly to the As 4s states) [4]. To avoid the difficulty of LDA in representing accurately the energy of isolated atoms Etot(N), ab initio data are considered in a finite range of lattice constants. Nonlinear least-squares fitting of these ab initio data is used to acquire the total energy curve Etot(x), which is then extrapolated to infinite lattice constant to obtain the value Etot(N). In the fitting procedure, there are two adjustments by contrast with our previous work [15]. One is that lattice constant a ¼ 4.0–8.0 A˚ is considered in this work instead of the upper limit 16.0 A˚ in our previous work so as to improve the description of the cohesive energy. Considering that for GaAs, equilibrium lattice constant a of B3 phase is 5.65 A˚ and that of B1 phase is 5.32 A˚, the fitting range of interatomic spacing is still wide enough to describe the phase transition. Another adjustment is that a new function form without Coulombic term is chosen for the cohesive energy curve as well as the potential curve, see Eq. (1) below, which makes atomistic simulations timesaving since the Ewald summation was left out. The results achieved in our previous paper [15] can also be retrieved using the potentials obtained in this paper. 2.3. Atomistic study methods Energy minimization, or structure relaxation, is one of the most effective atomistic study methods to search the equilibrium or metastable states. It is a kind of molecular static simulation scheme with the temperature effect neglected. Using the interatomic potentials, the initial B3 phase is structural relaxed with elevated pressures and transformed into a new phase under appropriate pressure. The new phase is just the desired final B1 phase. It is expected that such a static simulation process can reveal a

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possible microscopic mechanism for the concerned phase transition. Though molecular dynamical simulations can produce more information comparable with experimental observations, the static simulation really can present a rather concise and clear picture for the microscopic process of the phase transition. The intermediate states along the TP are analyzed by constraint energy minimization with regard to a selected transition coordinate that changes from the initial B3 phase to the final B1 phase. The algorithm of energy minimization is a combination of two methods: a robust but less accurate algorithm (the Steepest Descent method) near the beginning of the calculation and a less robust but highly accurate one (Truncated Newton method) near the end of the run. 3. Interatomic potentials of GaAs and its validity test In the present work, we choose the Rahman–Stillinger– Lemberg (RSL2) function form [30] to fit the cohesive energy curves and the interatomic potential ones. The expression of RSL2 is of the following form: FðrÞ ¼ D0 egð1r=R0 Þ þ

a1 a1 a3 þ þ , 1 þ eb1 ðrc1 Þ 1 þ eb2 ðrc2 Þ 1 þ eb3 ðrc3 Þ

(7) which contains a repulsive exponential function, the first term. It is clear that such a potential model does not explicitly define the effect of Coulomb interaction or polarization. The parameters of the potentials for GaAs are listed in Table 1. The obtained potentials are called as Mo¨bius inversion potentials (MIPs). To test its validity, the MIPs are firstly applied to calculate the static properties of GaAs B3 and B1 structures, such as equilibrium lattice constant a0, cohesive energy Ecoh, bulk modulus B0 and elastic constants Cijs at zero temperature and pressure. The calculated static properties are listed in Table 2. Also listed in the table are the results from our pseudopotential calculations and from other references for comparison. Table 1 The parameters of the interatomic potentials of GaAs in the form of Eq. (1) Atom pair

Ga–As

As–As(Ga–Ga)

D0 (eV/atom) g R0(A˚)

1.995007 8.192512 1.898399

3.387571 3.173667 3.181883

a1 (eV/atom) b1(A˚1) c1 (A˚)

6.68156 1.972508 1.935195

2.88309 1.080723 4.604408

a2 (eV/atom) b2(A˚1) c2 (A˚)

6.67385 0.994309 2.300978

2.62290 1.448589 3.082734

a3 (eV/atom) b3(A˚1) c3 (A˚)

1.77432 0.950176 3.118602

2.296853 1.011169 4.713176

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Table 2 Static properties of GaAs B3 and B1 structures obtained by this work, other calculation and experiment Lattice constant

Cohesive energy

Bulk modulus

Elastic constants

a0(A˚)

Ecoh(eV)

B(GPa)

C11(GPa)

C12(GPa)

C44(GPa)

B3

a

This work This workb Other’sc Exp.d

5.580 5.567 5.653 5.653

3.616 3.636 3.355 3.355

75.2 78.1 73.3 74.8

99.54 117.6 123.6 118.1

63.0 51.5 48.2 53.2

58.8 61.8 39.4 59.4

B1

This worka This workb Other’sc

5.230 5.208 5.320

3.354 3.379 3.084

90.1 95.1 95.63

101.0

84.6

84.6

a

Calculated from our MIPs. LDA calculation. c Reference [31]: calculated from analytical bond-order potentials by fitting the experimental values. d Reference [32]. b

V/V0 0.6

-2.8

0.8

1.0

-3.0

Ecoh (eV/atom)

Overall agreement is obtained except that, for B3–GaAs phase, a relative low C11 leads to a soft shear modulus (C11C12)/2 compared to the experimental one. Besides the properties listed in Table 2, the 0 K Gibbs free energies G0, i.e., enthalpies H ¼ E þ PV of the B3 and B1 structures are also calculated at different pressures p. The pressure at which DHðpÞ ¼ H B3 ðpÞ  H B1 ðpÞ ¼ 0 is defined as thermodynamic transition pressure pt. The MIPcalculated DH versus p increasing from 0 to 30 GPa with a step of 0.1 GPa gives the value pt ¼ 12.370.1 GPa. As expected, it is in agreement with the result estimated by taking the common tangent between the energy versus volume curves for B3 and B1 structures, see Fig. 1. The ab initio total energy data are also shown in Fig. 1. It is seen that present MIP calculations are in good agreement with the pseudopotential calculations, and the similar common tangent construction gives the ab initio result pt ¼ 11.8 GPa. As comparison [36], the ab initio result of Garcia and Cohen was pt ¼ 11.7 GPa [4] and the experiment data of Besson was pt ¼ 1271.5 GPa [6]. All the above calculation results indicate that the present interatomic potentials well describe GaAs B3 and B1 phases and might be appropriate to investigate the phase transition between these two phases. The computations below are carried out by our obtained MIPs unless specified.

1.2

rocksalt (B1)

-3.2

-3.4

zincblende (B3) pt = 12.3 GPa

-3.6

-3.8 12

15

18

21

24

27

V(Å3/atom) Fig. 1. The transition pressure between B3 and B1 structures for GaAs crystal is estimated from a common tangent construction. Results are obtained using interatomic potentials (lines). Ab initio cohesive energies are plotted (dots) for comparison.

4. Investigation of phase TP Determination of the transition mechanism or TP is a challenging problem, especially on the experimental side, because it needs for real-time information of the microscopic process. The theoretical approach is very appealing in this field, as it may give information not accessible in experiments. In this section, we first look back three proposed TPs (R3m, Imm2, and P32 ) and the related techniques in the previous studies of B3–B1 phase transition by other researchers. Then, based on our molecular static simulations, a Cm path is obtained which is very close to the Imm2 one. We use Imm2 path to replace

Cm path for simplification in successive studies and calculations. Further comparison of activation enthalpies shows that Imm2 TP is energy favored over the other two. 4.1. Review of the study of B3–B1 TP Up to now, there mainly exist three proposed mechanisms for B3–B1 transition: R3m, Imm2, and P32 . Christy pointed out that the pathway for diffusionless phase transitions accompanied by atom shifts and shears could often be described within a common subgroup of both phases involved [37]. R3m is the first suggested

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symmetry of the intermediate phase for B3–B1 transformation [38,39]. In another study based on crystallographic and symmetry arguments without energy consideration, Sowa [40] obtained the Imm2 TP and later proposed an additional mechanism with P32 symmetry [41]. More recently, Hatch and Stokes et al. developed a systematic procedure for obtaining possible microscopic mechanisms for reconstructive phase transitions, which was implemented in the computer program COMSUBS [42]. However, the common subgroups are many and 925 possible atomistic TPs were obtained for the B3–B1 transition by COMSUBS [43]. Taking example for SiC, eight energetically favorable TPs were picked out and then a bilayer sliding model was proposed to explain the mechanism of the B3–B1 transition. Another important approach to investigate the transition mechanism is to directly perform atomistic simulation of the transition process. By analyzing the MD simulation results of SiC [44], Catti [45,46] proposed an orthorhombic intermediate state with space group of Pmm2 or more correctly Imm2 [47,48]. He also found that Imm2 TP was favored over the simpler R3m TP by further energy consideration using LCAO-DFT methods. It is seen from the above that a search for common subgroup can tell the possible pathways and energy consideration is necessary to pick out the most favored one. 4.2. Microscopic process during B3–B1 transition under pressure We first take a look at the structure changes of GaAs under pressures with the structure relaxation approach. Simulations are based on the GaAs rhombohedral primitive cell (see Fig. 2), in which, the structure parameter set (a, b, c, a, b, g) (x, y, z), is composed of the cell edges (a, b, c), cell angles (a, b, g), and the position of the interior atom (x, y, z), respectively. The initial structure is denoted by (ai, ai, ai, 601, 601, 601) (0.25, 0.25, 0.25). This structure is relaxed under external hydrostatic pressures. Although the initial cell has

Fig. 2. High-pressure transition of the rhombohedral unit cell with space group P1: two arrows indicate different cell-deformation modes (see text in detail). The tetrahedron is also traced out in the B3 cell.

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some symmetry of point group, during relaxation, we withdraw the symmetry and do not apply any other symmetry restriction except translation invariance. No more change other than volume constriction is observed until 29 GPa. While the pressure arrives at 29 GPa, the initial state has a local maximum enthalpy and hence turns into an unstable stable state. Small numerical errors lead to a net force and then let the system start descending in energy. Here, two kinds of changes in the cell structure can occur: one is to (af, af, af, 451, 601, 451) (0, 0.5, 0) and the other is to (af, af, af, 601, 601, 901) (0.5, 0.5, 0), see Fig. 2. It is worth noting that planes containing 601 bond angles are the closepacked ones in the initial cell, which partly remains in both of the two final cells. Another character is that both celldeformation modes lead to the same following changes: the nearest coordination from four-fold to six-fold without bond-breaking; the bond angle from 109.471 in tetrahedral structures to 901 in octahedral structures; space group from ¯ ¯ F 43m to Fm3m. These results indicate a transformation from B3 to B1 structure. The two cell-deformation modes shown in Fig. 2 can give a suggestion for some possible transition mechanism, though it happens at a higher pressure than the transition pressure. However, it is worth noting that these two different cell-deformation modes do not stand for two different transition mechanisms. In fact, they can be unifiedly described in a larger cell containing eight atoms, see Fig. 3. In this representation, the lattice parameters change from (ai, ai, ai, 901, 901, 901) to (af, af, cf, 901, 901, 70.531), which is the same for both cases sketched in Fig. 2. Based on the structure relaxation results described in the eight-atom representation, see Fig. 3, we investigate the intermediate states at the transition pressure pt along the following pathway ðai ; ai ; ai ; 90 ; 90 ; 90 Þ ! ða; b; c; a; b; gÞ 90 XgX70:53

¯ F 43m 



! ðaf ; af ; af ; 90 ; 90 ; 70:53 Þ , Fm3m;

Fig. 3. Transformations from B3 to B1 of the rhombohedral unit cell (groups of large atoms) figured out in eight-atom cell representation.

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where g varies linearly from 901 to 70.531. For each g value, the intermediate structure is obtained by optimizing the cell parameters a, b, c, a, b, and inner atomic positions to the minimum enthalpy. Thus, in some sense, we can take a look at an imaginary real-time transition process. Figs. 4(a) and 4(b) show the enthalpy and cell parameters as functions of cell angle g. It can be seen that the parameters a and b, as well as a and b, are always identical in the course of transition. There are some discontinuities in Fig. 4. That might be related to the fast movement of the atoms in the cell, which will be discussed again later (in Section 5.3). Note that the cell-deformation is very similar to the SiC supercell changes in the MD simulations [44], from which Imm2 TP was proposed to describe the B3–B1 phase transition. Here the structure relaxation result might also imply the similar transition mechanism in the GaAs case. Although the TP in Fig. 4 was calculated with only P1 symmetry restriction, the intermediate structures nevertheless appear to have some tendency towards higher symmetry structures. The primitive cell of each intermediate structure is four-atom one, see Fig. 5, which is also shown in the eight-atom representation. Most of the intermediate structures are orthorhombic lattices with the space group of Imm2. However, at a short stage across the energy barrier where a and b of the eight-atom cell slightly depart from 901, the corresponding primitive cells

Enthaplpy ΔH (eV/atom)

0.08

a

0.06

0.04

0.02

0.00

b

α β

90

Cell length (Å)

7 80

6

a b c

5

70

Cell angle (degree)

8

Fig. 5. Representative intermediate structure of the eight-atom GaAs cell during the transition from B3 to B1. The four-atom primitive cell is emphasized with large atoms in the eight-atom cell representation.

are monoclinic lattices with the space group of Cm, a supergroup of Imm2. We also point out that there are important connections between these two kinds of primitive cells. The Imm2 cell satisfies the orthorhombic restrictions a ¼ b ¼ g ¼ 901 and the point-group symmetry restrictions x ¼ 0 and y ¼ 0.5, see Fig. 6(a). If the cell angle a (or b) is allowed to deviate from 901 and the restriction x ¼ 0 (or y ¼ 0.5) is canceled, see Fig. 6(b), the spacegroup symmetry will decrease from Imm2 to Cm. In our results, the deviation of Cm from Imm2 is in some sense rather small. As shown in Fig. 4(b), the angles a and b of the eight-atom cell only change by a few degrees from 90 just past the transition state (TS) and quickly recover to 901. These small deviations are similar to the deviations from the ideal Imm2 path from first principle calculations by Miao and Lambrecht [49]. From the above results, strictly speaking, the GaAs B3–B1 phase transition proceeds along a path with the Cm symmetry based on our energy minimization calculations in this work. The present Cm path is quite close to the previous suggested Imm2 path according to their crystal cell geometries. Considering the Cm intermediate states mainly occur at a short stage just past the energy barrier and there is only a small deviation from Imm2, we will use Imm2 path to replace Cm path for simplification in successive studies and calculations. Thus, from a hypothetic static process, we have obtained a possible microscopic mechanism of GaAs B3–B1 transition.

4 60 90

85

80

75

4.3. Comparisons of enthalpies among possible pathways

70

γ (degree) Fig. 4. (a) Enthalpy changes along the TP coordinate g at pt. (b) Cell parameters change along the TP coordinate g at pt. The parameters a and b, as well as a and b are always identical.

Since in subsection 4.1, three possible GaAs B3–B1 transition pathways, R3m, Imm2, and P32 , were mentioned, it had better to determine which one is most possible. The method is to compare the activation

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Fig. 6. Sketch map of the unit cell of Imm2 and Cm structures and their connections. The lattice parameters (a, b, c, a, b, g) (x, y, z) are indicated respectively. (a) Imm2 unit cell (a ¼ b ¼ g ¼ 901, x ¼ 0, y ¼ 0.5) and (b) Cm unit cell (a ¼ g ¼ 901, y ¼ 0.5).

451

Fig. 7. Sketch map of the R3m transition path marked by dotted line, the rhombohedral unit cell labeling the internal atom positions of the two end phases B3 and B1.

enthalpies of them. The three TPs can be expressed in terms of the unit cell variables (a, b, c, a, b, g) and atomic fractional positions (x, y, z) as follows:Imm2 TP (four dimensional) ðai ; ai ; ci ; 90 ; 90 ; 90 Þ ! ða; b; c; 90 ; 90 ; 90 Þ ð0;0:5;0:25Þ

ð0;0:5;zÞð0:25pzp0:5Þ 





! ðaf ; cf ; cf ; 90 ; 90 ; 90 Þ , ð0;0:5;0:5Þ

R3m TP (three dimensional) ðai ; ai ; ai ; 60 ; 60 ; 60 Þ ! ða; a; a; a; a; aÞ ð0:25;0:25;0:25Þ

ðz;z;zÞð0:25pzp0:5Þ 

Fig. 8. Schematic representation of the Imm2 TP marking the kept closepacked plane during the transition: (a) B3 structure, (b) transition state and (c) B1 structure.



! ðaf ; af ; af ; 60 ; 60 ; 60 Þ , ð0:5;0:5;0:5Þ

and P32 TP (five dimensional) ðai ; ai ; ci ; 90 ; 90 ; 120 Þ ! ða; a; c; 90 ; 90 ; 120 Þ ð1=3;1=3;1=8Þ

ðx;y;zÞð1=3XxX0Þ 





! ðaf ; af ; cf ; 90 ; 90 ; 120 Þ . ð0;1=3;1=12Þ

The parameter z in the Imm2 or R3m TP, as well as x in the P32 TP, is chosen as the transition coordinate. The intermediate structures also define the corresponding dimensions of the transition configuration spaces [50], which are given in the parentheses following the TP names. Figs. 7–9 are the schematic representations of R3m, Imm2 and P32 TPs respectively. The P32 path is somewhat complicated and the interested reader may wish to consult Refs. [41] and [43]. Enthalpies of the intermediate state are minimized along the transition coordinates with respect to other free parameters. The enthalpy curves of the three different TPs at zero or equilibrium pressure are plotted in Fig. 10. On each pathway, the position of the TS, or ‘‘activated state’’ in the language of chemical kinetics, is a maximum along the path. Activation enthalpies of the phase transition in the forward (B3–B1) and backward (B1–B3) directions are readily estimated from these enthalpy curves by the enthalpy differences between the maximum and the two ends respectively. By inspection of Fig. 10, it is clear that R3m mechanism is not favorable from the view of energy compared with the

Fig. 9. Schematic representation of the structural variations along the P32 TP: (a) B3 structure, (b) transition state and (c) B1 structure. For each state, the top graph corresponds to the global view, and the bottom one corresponds to the projection.

other two pathways in our calculations. At zero pressure, activation enthalpies of different TPs are comparable. With pressure raised to pt ¼ 12.3 GPa, the activation enthalpy of R3m pathway only decreases slightly (from 0.296 to 0.283 eV/atom), which is still higher than that at zero pressure for the Imm2 case), whereas a dramatic drop from 0.265 to 0.072 eV/atom is observed in the Imm2 case. At equilibrium pressure pt, the activation enthalpy of the P32 pathway is 0.114 eV/atom, which is also rather lower than

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x 0.330 0.35

0.275

0.220

0.165

0.110

0.055

0.000 0.35

Imm2 (0GPa) Imm2 (12.3GPa)

0.30

ΔH (eV/atom)

0.25

0.30

R3m (0GPa) R3m (12.3GPa) P32 (12.3GPa)

0.20

0.20

0.20

0.15

0.15

0.10

0.10

0.05

0.05

0.00

0.00

0.25

0.30

0.35

0.40

0.45

0.50

Z Fig. 10. Enthalpies per atom of the intermediate Imm2(solid lines), R3m(dashed lines) and P32 (dotted line) phases of GaAs, corresponding to different B3–B1 TPs, are plotted against the chosen transition coordinates. Imm2 and R3m with transition coordinate z (bottom abscissa) and P32 with x (top abscissa).

that of the R3m pathway. Thus, Imm2 and P32 pathways are more favored than the R3m pathway for the B3–B1 GaAs phase transition in our calculations. Let us explain the reason by considering the changes of chemical bonding undergone along the pathways. In R3m TP there is one bond broken, while in Imm2 and P32 TPs there is not. Between P32 and Imm2 TPs, the activation enthalpy of the former is marginally higher than that of the latter. Sowa believed that P32 might be more possible than Imm2 pathway [41]. Our calculation does not confirm his viewpoint. Sowa has pointed out that P32 mechanism reveals some close-packed planes unchanged during the transition. It does too for the Imm2 case as shown in Fig. 8. On the other hand, Besson [6] has reported the observation of the grain boundary ð1¯ 1¯ 1Þ plane in the GaAs pressure-induced transition, which indirectly proves the close-packed plane remaining during the transition. In addition, the GaAs II samples are determined with the space group of Pmm2 and a further change into Imm2[1]. Therefore, we think that these experimental results might be regarded as support to the Imm2 mechanism. In conclusion, we suggest that Imm2 is most possible among the previously proposed transition mechanisms. In the next section, we investigate the details of this TP. 5. Investigation of Imm2 TP in GaAs In this section, the symmetries of intermediate states are restricted as Imm2, and lattice translational symmetry is

assumed to hold throughout the transformation. On the whole, the transformation is modeled as a cooperative movement of atoms in an ideal crystal. We are going to acquire detailed energetic and structural information of the Imm2 TP in GaAs. And based on a simple analysis of the calculation results, the drive mechanism of the phase transition will be discussed. As shown in Fig. 8, the Imm2 cell contains two GaAs formula units, with fractional coordinates of the Ga atoms being (0, 0, 0) and (0.5, 0.5, 0.5), and of As (0, 0.5, z) and (0.5, 0, z+0.5). The cell edges a, b, and c (or equivalently the volume V and the cell edge ratios b/a and b/c) are the free lattice parameters, and the internal coordinate z, ranging from 0.25 (B3) to 0.5 (B1), is assumed as transition coordinate of the Imm2 TP process. In the Imm2 representation, the cubic B3 and B1 pend ffiffiffi states are characterized by the constraints a ¼ b; c ¼ 2a; z ¼ 0:25, and b ¼ c; a ¼ pffiffiffi 2b; z ¼ 0:50, respectively. Their unit cells are related to the Imm2 one by the transformation     matrices 1 1¯ 0j1 1 0j0 0 1 (B3, cell I) and 0 1 1 0 1¯ 1 1 0 0 (B1, cell II), the rows of which represent the basis vector components of either I or II cell in terms of those of the orthorhombic Imm2 unit cell. Along the transformation path, cells I and II are distorted into a non-cubic geometry with only two equal edges and two right angles. Definitely speaking, their lattice constants are related to the orthorhombic Imm2 one according to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aI ¼ bI ¼ b2 þ a2 ; cI ¼ c; gI ¼ 2 arctanðb=aÞ, (8) aII ¼ bII ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ c2 ; cII ¼ a; gII ¼ 2 arctanðb=cÞ.

(9)

Therefore, the variation of Imm2 cell parameters along the TP can also tell how the cubic B3 structure deforms into a monoclinic cell along with the new cubic B1 structure coming into being. At given pressures, the intermediate structures are obtained by constraint leastenthalpy calculations, i.e., optimizing the enthalpy versus the cell edges (a, b, c) at each fixed z along the path. In the following, we discuss the changes of enthalpies and structures with the transition coordinate z. 5.1. Enthalpy changes at external pressures The enthalpies changing with the transition coordinate z at different pressures is plotted in Fig. 11. It is seen that B3 and B1 are two local minima at zero pressure, and B3 is much lower than B1. With increase of the pressure p, the B3 and B1 states undergo reverse changes, the first one rising from the minimum to the saddle point and the second changing from a shallow minimum to a deeper one. At the equilibrium transition pressure pt ¼ 12.3 GPa, B3 and B1 states share the same energy level with a barrier between them. When the pressure is up to 30 GPa, B3 state becomes the saddle point and the barrier disappears, and thus B3 phase can transform into B1 with no need of activation energy.

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of the discrepancy between the theoretical transition pressure (12.3 GPa) and the experiment transition pressure, which is around 17 GPa from most of the experimental measurements [36]. From the enthalpy curves in Fig. 11, the abscissa za of the maximum enthalpy point decreases with pressure. And the HðzÞ curve at pt ¼ 12.3 GPa has a rather asymmetrical shape, with the maximum at za ¼ 0.34 as the case of SiC(0.345) [45] and ZnS(0.337) [46]. The za value well supports the Landau-like model about B3–B1 transition proposed by Miao and Lambrecht [53]. 5.2. Structural changes along Imm2 TP The structural evolution of the Imm2 intermediate states could be defined by the dependence of the unit-cell edges a, b, and c upon the transition coordinate z. Equivalently, we adopt a representation based on the relative volume change DV with respect to B3 end phase, and on the b/a and b/c ratios, which are plotted as functions of z in Fig. 12 and Fig. 13, respectively, at different pressures. Fig. 12 shows the percentage of volume change with respect to the B3 phase versus the transition coordinate z along the Imm2 TP. The relative volume change DV/V0 at z ¼ 0.50 is about 17%, which is close to the experimental value 17.2%[1]. On the whole, the values of |DV/V0| decrease with increasing z except when the pressure is rather lower than pt and the state just departs from the initial one. On the other hand, they increase with the pressure in most part of the z range. However, it is interesting that, near the B1 phase (z ¼ 0.5), the higher the pressure, the smaller is the value of |DV/V0|. Fig. 13 shows the structure parameters versus the transition coordinate z at different pressures. Figs. 13(a) and (b) are the p=0GPa p=4GPa p=8GPa p=12.3GPa p=18GPa

0

ΔV/V0 (%)

-5

0

-5

-10

-10

Fig. 11. The enthalpy H changes along the internal coordinate z at different pressures p.

-15

-15

Noticeably, the activation enthalpy is reduced from 0.072 eV/atom (900 K) at the equilibrium transition pressure pt (12.3 GPa) to 0.025 eV/atom (300 K) at 18 GPa. Considering an Arrhenius factor expðDH=kB TÞ at room temperature, this implies a significant increase of the transition probability and might be an explanation

-20 0.25

0.30

0.35

0.40

0.45

-20 0.50

z Fig. 12. Percentage of volume changes (with respect to the B3 phase) along the Imm2 TP versus the transition coordinates z at different pressures.

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a

b

c

d

Fig. 13. Unit-cell ratios of the orthorhombic Imm2 intermediate state, and the g angle of the corresponding two end phases versus the transition coordinate z at different pressures. (a) Unit-cell ratios b/a; (b) unit-cell ratios b/c, (c) g in phase I unit (deformed-B3) and (d) g in phase II unit (deformed-B1).

b/a and b/c ratios between unit-cell edges of the orthorhombic Imm2 intermediate state, and Figs. 13(c) and (d) are the related angles gI and gII of the deformed cubic cells of the B3 (I) and B1 (II) structures. The behavior of the cell edge ratios is peculiar: the b/a curves are nearly linear, whereas there is a pronounced jump in the b/c case. The higher the pressure, the smoother the b/a curves, while conversely, the steeper the b/c curves. Accordingly, the angles gI in deformed-B3 cells decrease from 901 to 70.51 smoothly, whereas the angles gII start at 70.51 and jump to 901 near the inflection points. In fact, the inflection point exactly corresponds to the TS (or energy barrier, cf. Fig. 11). 5.3. Potential energy surface A phase transition process can be regarded as the concerted movement of the atoms of the system on a highdimensional potential energy surface (PES), which is helpful to get a coherent idea about the energy profiles and geometries during the transition process. Here we do not give the four-dimensional hypersurface H ¼ H(a, b, c, z) of Imm2 TP. Instead, the internal coordinate z and the cell parameter a are fixed, and the other two structure parameters b and c are fully relaxed. The PES is computed at the equilibrium pressure pt on a mesh of 59  59 points covering the initial and final structures and is shown as a contour plot in Fig. 14. We

can see that there is one saddle point near za ¼ 0.34 (TS) and two minima at z ¼ 0.25 (B3) and z ¼ 0.50 (B1). Start from the saddle point TS, and continuously trace in the steepest descent direction (gradient) to the two minima B3 and B1. In this way, we can obtain the so-called ideal TP (ITP) along with the system can ‘‘travel’’ from B3 over TS to B1. Some more details are that the coordinates z and a were normalized at first, that is, z and a were rewrote as (zzB3)/(zB1zB3) and (aaB3)/(aB1aB3), respectively and then spline interpolations and numerical derivatives were used in the calculation of gradient. The ITP is outlined in the contour graph (see Fig. 14) together with another two paths: one is called as the fixed position path (FPP) along which the enthalpy reaches the minimization at given z; the other is the fixed strain path (FSP) along which the enthalpy reaches the minimization at given a. It is clearly shown that ITP is always located between FPP and FSP [52] and these paths have one more common point TS besides the two end points B3 and B1. In fact, any path defined by different transition coordinates will always pass the B3, TS and B1 states. This implies the same activation enthalpies along the ITP, FPP and FSP. More interesting, they also determine most of the qualitative characters of the transition process, such as the global description of the changes of the system features during the transition process. That is to say, though the work in the above two subsections was performed along the FPP, the similar

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B1

0.50 -2 .014

0.45

-1 .956 -1 .790 -1 .885

-0 .8875

0.40 z

-2 .050 -2 .044

-2 .026 -2 .002

-1 .275 -1 .469 -1 .662

-1 .932

0.35 -1 .985

0.30

0.25

-2 .038

3.8

-2 .038

4.0

-1 .932 -1 .885

-1 .989

-2 .014 -2 .026 -2 .044 B3

-1 .989 -1 .985 -1 .980 TS -1 .956 -1 .980

-1.790 -2 .002

4.2

4.4

4.6

4.8

5.0

a (Å)

Fig. 14. The potential energy surface for Imm2-GaAs obtained by varying a and z with b and c optimized at pt (12.3 GPa). The solid line, dashed line and dashed–dotted line indicate ITP, FSP and FPP transition path, respectively. The unit of enthalpy is eV/atom.

behaviors for the B3–B1 transition in the description of Imm2 mechanism could be expected along the ITP. One should keep in mind that a represents cell parameter and z the relative position of atoms. Intuitively, the variation of a reflects the strain and the variation of z reflects the atom shift. Because the position of the point TS is at the lower right part in Fig. 14, a changes dominantly near the B3 region while z does near the B1 region. In other words, the cell deformation predominates at the first stage of the transition and then following with the atom shift. It can be interpreted as follows. The strain corresponds to the acoustic phonons, whereas the relative movement of the atom corresponds to the optic phonons [48,54]. The former have much lower excitation energy compared to the latter. Thus such a phase transition begins with the strain firstly and follows with the relative motion of the atoms, which is also an easy observation from Figs. 13(b) and (d). So, the strain can be considered, as the driving factor and the motion of the atoms is barely a response to it. By the way, there is a jump of z value past the TS along the FSP shown in Fig. 14, which was also observed in ab initio calculations for SiC [48,49]. As mentioned above, the atom-shift corresponds to the optic phonon mode. Also we pointed out in our previous work that ‘‘the optical branch seems overall lower than experimental data because no high frequency information is referred in our potentials’’ [15]. So, we wonder if the inaccurate optics phonon modes lead to the over sharp jump in the FSP here and the discontinuities shown in Fig. 4. 5.4. Discussions of Imm2 TP We have made a detailed investigation on the B3–B1 transition mechanism and given a possible and concise transition picture. Experimentally, the transformation is

455

observed at room temperature. The temperature effect is limited to energy differences kBT, which is much less than the calculated activate enthalpy, so it is reasonable to expect that neglecting the temperature effect will not bring significant influence on the physical picture given above. From our calculation, the transformation starts with the cell edges shortened or elongated in the orthorhombic Imm2 model. As argued by Limpijumnong and Lambrecht [35], this seems to imply that a combination of uniaxial and hydrostatic stress would facilitate the transition. There have been some examples of transitions under uniaxial and hydrostatic stress for the GaAs case. In Besson et al.’s [6] work, the transition in the ethanol–methanol mixture starts at pressure 13.6 GPa, lower than that in argon (15 GPa), which is related to the higher stress inhomogeneity in the former. Another study of GaAs with dynamic (shockwave) methods reports the complete transformation of GaAs-I–GaAs-II at 16.3 GPa [55] while a higher pressure (422 GPa) reported in most of other experiments [1,6] that is somewhat an evidence of the low transition pressure owing to the uniaxial stress. It is a general observation in reconstructive phase transitions that inhomogeneous stress will induce the phase transformation and, conversely, that hydrostatic conditions will conserve the low-pressure phase metastably further away from the equilibrium pressure. As a matter of fact, coherent epitaxial growth away from lattice-matched conditions can offer a uniaxialstress system. So, epitaxy-induced structural phase transformations might be a new technique to grow novel materials [56]. 6. Summary In this paper, the Mo¨bius inversion potentials for GaAs are extracted based on a great number of ab initio data, and checked by various static properties of B3 and B1 GaAs. Then the B3–B1 phase transition of bulk GaAs under appropriate high pressure is investigated as follows. Firstly, a Cm path for B3–B1 transition in GaAs is obtained from the static structure relaxation simulation, even though it is different from three previously proposed transition paths Imm2,R3m and P32 . The present Cm path is quite close to the previous suggested Imm2 path according to crystal cell geometry. And the very small deviation of Cm from Imm2 only happens in the region nearby the energy barrier. Besides, the activation enthalpy along the Imm2 path is preferable than that along R3m and P32 . Therefore, we use Imm2 path instead of Cm path for simplification in successive study and calculation. Secondly, more detailed investigation on the energetic and structural information of the intermediate states is taken along the Imm2 TP at different pressures, which shows a concise and coherent transition picture about the B3–B1 transition in GaAs. By analyzing the PES, we point out that the strain predominates at the first stage followed by atom shifts as a response to it, from which strain can be considered as the driving factor of the B3–B1 transition.

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The transition mechanism is also discussed based on our calculation results combining with available experimental observations. In conclusion, we have proposed a new approach to developing interatomic potentials for GaAs. Though its simplicity of the pair potential model, it produces various properties of GaAs, which agree with others’ work. As the important result of this work, the potentials also successfully give a good description of the B3–B1 microscopic mechanism in GaAs by using the static energy minimization approach. Combined with the previous efforts on the pair potential model for compound semiconductors [8–15], such a simple model is useful to understand those reconstructive phase transitions with large structural distortions and coordination changes for compound semiconductors. In future work, temperature factor should be considered further to explore the important role of defects in the kinetics of the transition, and many-body potentials might be expected as a small correction term for a possible improvement on the description of elastic constants and phonon dispersions which might have important implications for the transition studies.

Acknowledgments This work was supported by the National Nature Science Foundation of China (Grant No. 50531050) and the 973 Project in China (Grant No. 2006CB605100).

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