First-Principles Study on the Elastic Constants and Structural and Mechanical Properties of 30 ° Partial Dislocation in GaAs

The second-order elastic constants, third-order elastic constants, and the generalized-stacking-fault energy for semiconductor GaAs are investigated using the ﬁrst-principles calculations. The predictions of elastic constants are obtained from the coeﬃcients of the ﬁtted polynomials of the energy-strain functions. It is found that the nonlinear elastic eﬀects must be considered when the applied deformations are larger than approximately 1.5%. With the Lagrangian strains up to 6.4%, the terms included up to third order in energy expansion functions are suﬃcient. The elastic constants given in this work agree well with the previous results and experimental data except for C 144 . C 144 given by the present paper is a positive value, and the estimated 3GPa agrees well with the experimental result of 2 GPa. The research results can provide a reference for understanding the elasticity of GaAs. The gen-eralized-stacking-fault energy has been calculated without and with structural relaxation, respectively. The unstable stacking fault energy with structural relaxation is about two-thirds of that without relaxation. The dislocation width and Peierls stress for 30 ° partial in GaAs have been investigated based on the improved P - N theory. The dislocation width is very narrow (only about one-ﬁfth of Burgers vector b ), which is reasonable for covalent materials. The Peierls stress is about 4GPa, in good agreement with the experimental result of 2 ∼ 3GPa.


Introduction
III-V semiconductors receive the widespread attention due to the latent application prospect, such as photo-detectors, lasers, and light-emitting diodes [1][2][3][4][5]. Among the III-V direct bandgap semiconductors, GaAs is one of the most promising candidates due to its high electron mobility [6,7]. It has been widely used in many fields, such as solar cell fabrication [8], substrate materials [9,10], and efficient photovoltaic devices [11,12]. Recently, lots of experimental and theoretical research studies have been carried out to study the properties of GaAs. Elasticity and plasticity are two important mechanical properties of materials, and the study of these two properties plays an important role on understanding and utilization of materials. Consequently, the elastic and plastic properties of GaAs are also of particular interest. e elastic properties of solids can be well reflected by elastic constants. For single crystals, secondorder elastic constants (SOECs) and third-order elastic constants (TOECs) are both important parameters for modeling the mechanical response. In the process of plastic deformation of materials, dislocations play a key role. erefore, the study of the elastic constants (SOECs and TOECs) and the dislocation properties of GaAs is necessary. Many experimental and theoretical research studies have been performed on the elastic constants of semiconductor GaAs. Several methods such as ultrasonic wave propagation and optical interference are used to measure the SOECs and TOECs of GaAs [13][14][15]. Singh et al. have calculated the TOECs of GaAs using Keating theory [16]. First-principles calculations based on density functional theory are also widely used to investigate the elastic constants of GaAs [17][18][19].
ese results are very vital for understanding the elasticity of GaAs. However, there still exist argument for the magnitude and sign of the TOEC C 144 . e most fundamental quantity related to the plasticity of the crystal is the Peierls stress of dislocation. However, the research on the Peierls stress of dislocation in GaAs is very limited, no matter from experimental or theoretical method. Suzuki et al. and Kamimura et al. have estimated the Peierls stress of dislocations in GaAs by extrapolating the critical shear stress to the absolute zero temperature; the estimated Peierls stress is about 2∼3 GPa [20,21]. e Peierls stress of the 30°partial in GaAs estimated by Kamimura et al. based on the classical P-N model using ab-initio c-surface is about 23 GPa [22], which is one magnitude larger than the experimental result approximately [21]. Actually, the crystal has been treated as an elastic continuum body in the classical P-N model, while the discrete effect of lattice which is extremely important for the narrow dislocations in semiconductors is neglected [23][24][25]. e improved P-N theory which has fully considered the discrete effect of the crystal has been proposed by Wang [26][27][28]. e research results show that the agreement between theoretical prediction given by improved P-N theory and the numerical and experimental results can be significantly improved [29][30][31]. It has been established that the glide of dissociated dislocations on closed packed planes is responsible for the high temperature plasticity in zinc-blende crystals [21,32]. And, the 30°partial generally possesses higher Peierls stress compared with the 90°partial and controls the deformation [33]. And, as such, the improved P-N theory is applied for the 30°p artial in GaAs. e restoring force in the dislocation equation is generally given by the gradient of the generalized-stacking-fault energy (GSFE) as suggested by Christian and Vitek [34]. In this paper, the SOECs, TOECs, and GSFE along (1/2)〈112〉 on glide set of GaAs have been calculated by the first-principles calculations. Based on the calculated SOECs and GSFE, the core width and Peierls stress of 30°p artial in GaAs have been investigated by the improved P-N theory.

Computational Method.
e finite-strain continuum elasticity theory [35][36][37][38][39][40] is widely used to obtain the TOECs; the relation between the strain energy density Φ and the elastic constants can be expressed as follows [35,36]: where η is the Lagrangian strain tensor, and it can be defined as [37] where δ ij is an unit matrix; the deformation tensor J ij � (zx i ′ /zx j ) constructs the connection of initial configuration x j and strained configuration x i ′ at the equilibrium state.
e αth − order(α ≥ 2) elastic constants were defined by Brugger as [38] where I, J, and K are Voigt subscripts, and the Lagranian strain tensor η links this notation by Because of the cubic symmetry, there are three independent SOECs (C 11 , C 12 , and C 44 ) and six TOECs (C 111 , C 112 , C 123 , C 144 , C 155 , and C 456 ). Introducing six Lagrangian strain tensors in terms of a single parameter ζ to calculate the SOECs and TOECs, the strain energy per unit mass can be written as a polynomial of the strain parameter ζ [41], where the coefficients A 2 and A 3 are comprised by the SOECs and TOECs of the crystal. e applied strains η α with α � A to F and the corresponding coefficients A 2 and A 3 are listed in Table 1 [42].
To obtain accurate TOECs, ζ is changed from −0.064 to 0.064 with the step of 0.008. For covalent bonding materials, the maximal amplitude 0.064 of the deformations is enough to obtain the accurate TOECs. e elastic constants can be obtained from the least-square polynomial fitted from the strain-energy relation given by the first-principles totalenergy calculations.
We perform first-principles total-energy calculations based on the density functional theory (DFT), using the abinitio simulation package (VASP 4.6) [43][44][45]. In order to compare the performance of two different density functionals in the first-principles prediction of the lattice constant and SOECs, the local density approximation (LDA) and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional for generalized gradient approximation (GGA) are, respectively, used. A plane-wave basis set is employed within the framework of the projector augmented wave (PAW) method [46,47]. In the calculation, the wave functions are expanded as a plane wave basis with the cut-off energy of 550 eV. For the first Brillouin zone integrals, reciprocal space is represented by the Monkhorst-Pack special k-point scheme [48] with 21 × 21 × 21 grid meshes. e convergence of energy and force are set to 1.0 × 10 − 6 eV and 1.0 × 10 − 4 (eV/ � A), respectively. e equilibrium theoretical crystal structures are determined by minimizing the Hellmann-Feynman force on the atoms and stress on the unit cell.
e lattice constant and SOECs obtained with the LDA and GGA are presented in Table 2. e results show that the GGA overestimates, while the LDA underestimates the lattice constants, which is in agreement with [19] and [49]. And, it is obvious that the predictions of LDA are in better agreement with the experiment results, regardless of the lattice constant or the SOECs. Consequently, the calculation of the TOECs and the GSFE in the present paper is based on the LDA functional. 2 Advances in Materials Science and Engineering

Results and Discussion.
e strain energies from firstprinciples calculations and the fitted polynomials are shown in Figure 1. It is found that, for GaAs with the Lagrangian strain up to 6.4%, considering the terms up to third order in energy expansion is sufficient to obtain good consistency with the first-principles results. In addition, the strain energies with positive strains are always smaller than ones with negative strains; therefore, the values of TOECs are typically negative. In order to examine in which range of strains the third-order effects affect the properties of GaAs, the curves of the nonlinear elasticity comparison with the linear elasticity and the DFT results of a particular deformation η C are shown in Figure 2. One can see that when the applied deformations are larger than approximately 1.5%, the linear elasticity is not sufficient and the third-order effects must be considered. e calculated results for lattice constant and elastic constants are presented in Table 2. e results are all agreeing well with the previous results and experimental data except for C 144 . e TOEC C 144 given by the present paper is a positive value, and the estimated value of 3 GPa agrees well with the experimental result of 2 GPa given in [14]. e results given by the present paper can provide a reference for understanding the elasticity of GaAs.

First-Principles Calculations of GSFE.
e first-principles total-energy calculations were used to calculate the GSFE, and the process of optimization is the same as the calculation of elastic constants. While for the Brillouin zone, the calculations of GSFE employ 21 × 21 × 1 grid meshes. Because of the zinc-blende structure, there are two different slip planes (see Figure 3). e widely and closely spaced 111 { } planes, respectively, correspond to the shuffle set and glide set. For discussing the properties of 30°partial in GaAs, the GSFE along (1/2)〈112〉 on glide set has been calculated with a slab calculation. In order to simulate the process of stacking fault, we employ a slab consisting of 12 atomic layers in the 〈111〉 direction. e vacuum space of 15Å normal to 111 { } plane between the periodically repeated slab is chosen to avoid the interactions between two slabs. To obtain the reasonable results, we performed that the fluctuations of calculated results' vacuum gap of 15Å and 18Å are less than 0.1%. e GSFE was generated by a set of rigid shifts of the upper slab along (1/2)〈112〉 with respect to the lower slab. For the relaxed GSFE, all atoms of the system are allowed to move only in the 〈111〉 by minimization of the Hellmann-Feynman forces on each atom. e calculated GSFE is shown in Figure 4. e unstable stacking fault energy with relaxation is about two-thirds of that without relaxation.

Mechanical Properties of 30°Partial Dislocation.
e mechanical properties of 30°partial in GaAs have been studied theoretically based on the improved P-N model [28]. As done generally, the restoring force is given by the gradient of the GSFE as suggested by Christian and Vitek [34]; the dislocation equation given by the improved P-N model is  [16]. g Ref [17]. h Ref [18]. i Ref [19]. j Ref [49]. k Ref [50]. l Ref [51]. Table 1: e applied strains η α and the corresponding coefficients A 2 and A 3 in equation (5) as combinations of SOECs and TOECs for crystals with cubic symmetry.

Strain type
Advances in Materials Science and Engineering    Advances in Materials Science and Engineering e second-order derivative term proportional to β represents the correction from the discrete effect caused by the interaction among the atoms on the misfit plane. σ is the area of the primitive cell in the misfit plane, and u is the displacement field. e energy factor K is [53] K � μ where θ is the dislocation angle and μ and ] are the effective shear modulus and Poisson's ratio in 111 { } surface [53]. e discrete parameter β for zinc-blende structure crystals has been investigated based on a simple dynamics model [54]. For glide partials, the parameter β is where b is the Burgers vector; the fitting curves are shown in Figure 4, and the fitting parameters c, Δ 1 , and Δ 2 are listed in Table 3.
e core structure can be determined by solving dislocation equation (6) with the truncating method proposed by Wang, and the Peierls stress can be obtained from the maximum slope of the dislocation energy [54,55]. e calculated half width ξ and Peierls stress σ P of 30°partial in GaAs are listed in Table 3.
It is found that the core structure calculated from the relaxed GSFE is wider, and the Peierls stress is smaller. Because of bond flip appearing for large distortion, the results obtained from the relaxed GSFE is more reliable. e narrow core is reasonable for dislocations in covalent materials. Furthermore, it can be seen that the dislocations will be widened by the discrete effect, and consequently, the Peierls stress will be decreased. e discrete effect is especially important for the narrow dislocations in covalent crystals since it is related to the interaction among the atoms on the misfit plane. e estimated 4 GPa is in good agreement with the experimental results of 2∼3 GPa [20,21]. Furthermore, the Peierls stress of 30°p artial in GaAs given by the classical P-N model is 23 GPa [22], about larger by one order of magnitude than that given by improved P-N theory. In addition to negligence of the discrete effect, another possible reason for the difference is that the law of sinusoidal force is applied in [22], while the dimensionless parameters Δ 1 and Δ 2 have been introduced for modifying the sinusoidal force in the present paper.

Summary and Discussion
In this paper, the SOECs, TOECs, and GSFEs for semiconductor GaAs are investigated using the first-principles total-energy calculations. e predictions of SOECs and TOECs are obtained from the coefficients of the fitted polynomials of the energy-strain functions. It is found that the nonlinear elastic effects must be considered when the applied deformations are larger than approximately 1.5%. And, with the Lagrangian strains up to 6.4%, the terms included up to the third order in energy expansion functions are sufficient to obtain good agreement with our calculated results. e elastic constants given in this work agree well with the previous results and experimental data except C 144 . Regarding the disagreement in the magnitude and sign of the TOEC C 144 , the result given by the present paper is a positive value, and the magnitude of 3 GPa agrees well with the experiment result of 2 GPa given in [14]. Our result can provide a reference for understanding the elasticity of GaAs. e GSFE has been calculated, respectively, without and with structural relaxation. e unstable stacking fault energy Table 3: e fitting parameters of GSFE, half width ξ, and Peierls stress σ P of 30°partial in GaAs. ξ 0 and σ 0 P are the results without considering the discrete effect (β � 0). e half width and Peierls stress are, respectively, in units of Burgers vector b and GPa.  (9)), and the discrete points denote the results given by first-principles calculations.

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with relaxation is about two-thirds of that without relaxation. Based on the improved P-N theory, the dislocation width and Peierls stress of 30°partial in GaAs have been calculated. e dislocation is very narrow, and it is reasonable for covalent materials. Because of bond flip appearing for large distortion, the results obtained from the relaxed GSFE are more reliable. e estimated Peierls stress in this work is about 4 GPa; it is in good agreement with the experimental results of 2∼3 GPa [20,21]. Furthermore, according to the Peierls stresses given by the classical P-N model and improved P-N model, it is found that the discrete effect is very important and must be considered when investigating the narrow dislocations in covalent crystals.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.