The Properties of Shuffle Screw Dislocation in Semiconductors Silicon and Germanium

The width, Peierls barrier and stress for shuffle screw dislocation in Si and Ge have been calculated by the improved P-N theory. The calculated widths are about 0.6b, b is the Burgers vector. The Peierls barriers for shuffle screw dislocations in Si and Ge are respectively about 3.61~4.61meV/Å and 5.31~13.32meV/Å, Peierls stresses are 1.40~2.07 meV/Å and 1.93~3.29 meV/Å. Our calculated results may correspond to the metastable core of the shuffle screw dislocation which is centred on the bond between two atoms.


INTRODUCTION
Due to the technological importance, the mechanical properties of Si and Ge are widely investigated.The dislocations in Si and Ge can be present in the glide or shuffle set configurations due to the double lattice structure [1].In experiments, Rabier and Demenet showed that high external pressures on Si favor a pure shuffle dislocation population over the partialized glide set [2].In order to make better use of semiconductors Si and Ge, studying the structure and motion of shuffle dislocation is important.Several attempts have been made to determine the Peierls stress of the perfect shuffle screw dislocation.However, the screw dislocation has been less studied.Ab initio calculations predict the Peierls stress for Si at zero pressure to be 3.3±0.2GPa[3].Density functional theory gives the result 4.1±0.3GPa[4].
Besides the numerical methods, the analytical P-N theory [1,5,6] is generally used for studying the structure and motion properties of dislocations.However, because of treating the crystal as elastic continuum body, the classical P-N model becomes inaccurate for narrow dislocations increasingly [5,7,8].Recently, professor Wang have obtained the improved P-N equation which has relaxed the continuum approximation successfully [9][10][11].Compared to classical P-N theory, the improved P-N theory can remarkably improved the agreement between the theoretical prediction and the numerical result [12,13].
In this paper, the core structure, Peierls barrier and stress for shuffle screw dislocation in Si and Ge has been studied by the improved P-N equation.The outline is as follows: Sec 1, Introduction; Sec 2, Dislocation equation, core structure and Peierls barrier and stress.In Sec 3, the result and discussion.The last section is the conclusion.f

DISLOCATION EQUATION, CORE STRUCTURE AND PEIERLS BARRIER AND STRESS
According to the two-dimensional dislocation equation for straight dislocations obtained from the lattice dynamics and symmetry principle [11] and the method given in Ref. [5] and Ref. [13], the dislocation equation takes the following form where the displacement field u and the restoring force f(u) are defined along Burgers vector, σ is the area of primitive cell in the misfit plane.The discrete parameter β and energy factor K can be represented as [1,14] β = ( 3c 11 + 5c where is the dislocation angel, µ andν the effective elastic constants within{111} plane [1,5], 11 c and 12 c are the elastic constants, 0 a the lattice constant.The values for these constants are listed in Table 1.The f(u) in Eq. ( 1) is given by the gradient of the γ - surface [15] (u) The γ -surface of shuffle set for Si and Ge have been calculated by Kang and Cai [16], and it can be expressed as follows approximately [17]  in Table 2.The -surface given in Ref. [16] and by Eq. (3)   have been plotted in Fig. (1).The dislocation Eq. ( 1) can be solved by truncating method proposed by professor Wang and the trial solution possess the following form [14,18] where the parameter c is a constant that can be determined by the dislocation equation.
Substituting the solution Eq. ( 4) into Eq.( 1) and following the truncating method [18], it is found that parameter c should satisfy the algebraic equation  The core parameter c calculated from Eq. ( 6) and half width (the distance that u changes from 0 to b/4) are listed in Table 3.
The Peierls barrier and stress are obtained by calculating misfit energy only in the classical P-N theory.However, it has been shown that the contribution from strain energy is as important as that from the misfit energy [19].The total energy which including contribution from both misfit and strain energies should be evaluated to obtain the correct Peierls barrier and stress.For a dislocation with length L, the strain and misfit energies of dislocation per unit length are given by [14] where is the relative displacement for dislocation located at 0 x , a is the length of the primitive vector (period in direction of dislocation line).Just as shown in Fig. (2), sum is carried over the atoms located in the horizontal band in the misfit plane (the band width is a).According to Eq. ( 7) and Eq. ( 8), the total energy is  Due to discreteness of lattice, a dislocation cannot move unless the applied stress exceeds Peierls stress.Peierls stress is the minimum stress to move a dislocation, it can be obtained from the maximum slope of the total energy The calculated Peierls barriers and stresses are listed in Table 4.

RESULT AND DISCUSSION
The widths of screw dislocations in Si and Ge are about 0.6b.The higher unstable stacking fault energy and smaller discrete parameter is, the narrower dislocations are.
The Peierls barrier and stress ( p E and p σ ) calculated from SW potential are much lower than those calculated from three other potentials.Besides, the results given by classical P-N theory (E p (0) and σ p (0)) calculated from Baskes potential are also much lower than those calculated from LDA and GGA potentials.Our results indicate that SW and Baskes potential may be not so reliable models for Si and Ge.It is strange that the Peierls barriers and stresses for Si are a little smaller than those for Ge, though the unstable stacking fault energy for Ge is lower (see Fig. 1).The strange results may resulted from the shape and the fitting degree of the -surface.The Peierls barriers for shuffle screw dislocations in Si and Ge are respectively 3.61~4.61meV/Åand 5.31~13.32meV/Å,Peierls stresses are 1.40~2.07meV/ Å 3 (0.277~0.331GPa) and 1.93~3.29meV/Å 3(0.309~0.526GPa).It's worth to note that no matter the results obtained here, or those given in ref. [14], the Peierls stresses are all too small for shuffle screw dislocations in semiconductor Si and Ge (about one magnitude lower than the 3.3±0.2GPa[3] and 4.1±0.3GPa[4] for Si).It is indicated that two different core structures can be considered for the shuffle screw dislocations in Si: core A is centred in the 6-member ring of atoms [20], core B is centred on the bond between two atoms [21].Recent ab initio calculations show that core A is the ground state of a perfect screw dislocation in Si, while core B is metastable, with an energy 0.38eV/b higher than that of core A [3].The small Peierls barriers and stresses given in this paper may correspond to the metastable core B. Additional investigations have to be done to better understand the mechanical properties of the two different core structures and the influence of the shape and fitting degree on the calculation.

CONCLUSION
The core structure, Peierls barrier and stress for shuffle screw dislocation in semiconductors Si and Ge have been investigated.Our results indicate that compared with LDA and GGA, SW and Baskes potential are not so reliable models for Si and Ge.The Peierls barriers for shuffle screw dislocations in Si and Ge are respectively about 3.61~4.61meV/Åand 5.31~13.32meV/Å, Peierls stresses are 1.40~2.07meV/Å 3(0.277~0.331GPa) and 1.93~3.29meV/Å 3(0.309~0.526GPa).Our calculated results may correspond to the metastable core of the shuffle screw dislocation which is centred on the bond between two atoms.Besides, the strange results (Peierls barriers and stresses for Si are a little smaller than those for Ge) indicate that the shape and the fitting degree of the -surface may have great impact on the calculation.

1 Δ and 2 Δ
b and d are respectively the Burgers vector and the spacing between glide planes.are the modification factors to the sinusoidal-force law.For fitting the γ -surface given in Ref.[16], 1 Δ and 2 Δ have been listed classical P-N model when the discrete parameter β equals to zero.

Fig. ( 1 )
Fig. (1).Theγ-surface along ˂110˃direction of shuffle set for Si and Ge given by Kang et al.[16] and by Eq. (3), where the Burgers vector for Si and Ge is respectively b = 3.84Å and b = 4.00Å.

Fig. ( 2 ).
Fig. (2).Core structure of the shuffle screw dislocation.The black and white circles represent the atoms on the misfit planes that above and below the cut plane, respectively.

Table 2 . The modification factors Δ 1 and Δ 2 . SW and Baskes represent the
γ-surface has been calculated by Stillinger-Weber and MEAM-Baskes inter-atomic potential, respectively; LDA and GGA represent the γ-surface has been calculated by Vienna ab-initio simulation package (VASP) with the local density and the generalized gradient approximation, respectively.