In general, defects induce lattice distortions on both local- and long-range scales. In the specific case of point defects, the long-range elastic distortions can be completely described by means of the elastic dipole tensor (EDT). This tensor bridges the atomic structure of the defect with the elastic field it introduces in a material. Consequently, such mathematical entity also describes the interaction of point defects with strain or stress within linear elasticity. In order to understand the concept behind this tensor and how to calculate it, we analyze a truncated Taylor expansion of the energy E(n_d,ɛ) of a supercell, without entropy contributions, in terms of the strain (with components ɛ_ij) and the number of defects per supercell, n_d. If the entropy contributions are neglected, the expansion of the energy density reads as:

E(n_d,ɛ) = E₀+½ Ʃ_i,j,k,l C_i,j,k,lɛ_ijɛ_kl + n_d (E_d –Ʃ_i,j G_ijɛ_ij+...

Here, E₀ is the total energy of the non-strained defect-free system, Ed is the formation energy of a defect, C_ijkl are the components of the stiffness tensor, and G_ij is the elastic dipole tensor (EDT). G_ij describes the interaction of the defect with a strain field. The positive and negative EDT components imply emergence of respectively dilation and compression centers introduced by defect. The change in energy under strain, caused by formation of a defect is then given by

ΔE = – Ʃ_ijG_ijɛ_ij.

Therefore, if the elements of EDT and strain have the same (opposite) sign, the formation energy of the defect would decrease (increase). To compute the components of the EDT using the atomistic methods, we use the equation

G_ijkl = – V₀Δσ_ij,

where V₀ is the volume of the supercell containing one defect, and Δσ_ij is the difference between stress of the defective and pure cells. In a periodic supercell, an isolated defect is added and the atomic positions are relaxed and the stress induced by the defect is calculated. Then, the EDT can be obtained using the Eq. 3.The components of the stress tensor σ_ij and elastic dipole tensor of the Ga_In antisite in CuInSe₂ and for InGa antisite in CuGaSe2, we obtained are shown in Figure 1. As we can see, the diagonal components of the EDT of GaIn are negative. This is caused by the smaller effective ionic radius of Ga compared to that of In. Therefore, the formation energy of GaIn would decrease under a compressive strain. Reversely, for InGa, the diagonal components of EDT are positive and its formation energy would decrease under a tensile strain. A comparison between the EDT of Ga_In and In_Ga shows that under the same amount of strain but with an opposite sign, the formation energy of In_Ga in CuGaSe₂ would decrease more rapidly than that of the Ga_In in CuISe₂. To gain insights on how diffusion barrier changes under strain, we studied diffusion of In via Cu vacancy (V_Cu) in CuInSe₂ under various strain conditions along the (110) plane. Our results (as shown in Figure 3) indicate that the diffusion barrier decreases under compressive strain, however, the reduction is very small. To check influence of Na on diffusivity of In and Ga, we studied diffusion of In and Ga in CuInSe₂ in the presence of Na_int. Here, we considered the diffusion paths: In_Cu +V_Cu→V_Cu + In_Cu in CuInSe₂ and Ga_Cu + V_Cu →V_Cu + Ga_Cu in CuGaSe₂. In both cases, a Na_int was added to the next neighbour position of the V_Cu. Our calculations show that during the geometry optimization and in a barrierless recombination Na goes to the Cu vacancy. This is in contrast to Colombara’s postulation that presence of Na_int enhances In/Ga interdiffusion through V_Cu.