We have studied the energetics, atomic, and electronic structure of Na and K point defects, as well as the (Na−Na), (K−K), and (Na−K) dumbbells in CuInSe 2 and CuIn 5 Se 8 solar cell materials by hybrid functional calculations. We found that although Na and K behaves somewhat similar; there is a qualitative difference between the inclusion of Na and K impurities. Namely, Na will be mostly incorporated into CuInSe 2 and CuIn 5 Se 8 either as an interstitial defect coordinated by cations, or two Na impurities will form (Na−Na) dumbbells in the Cu sublattice. In contrast to Na, K impurities are less likely to form interstitial defects. Instead, it is more preferable to accommodate K either as K Cu substitutional defect, or to form (K−K) dumbbells on Cu substitution positions. Our data show that all (Na−Na), (Na−K), and (K−K) dumbbells can form in both CuInSe 2 and CuIn 5 Se 8 . In the Cu-poor CuIn 5 Se 8 material the pristine Cu vacancies act as the most stable sites where Na and K can be inserted. The formation energy of Na-related defects is generally lower than the corresponding K-related defects, which would mean that if a defect site is already occupied by Na, then it is less likely that K is able to substitute Na during the postdeposition treatment. Regarding the electronic structure of the materials, Na and K point defects located in the Cu sublattice do not create deep defect levels in the gap, so they are not detrimental for the solar cell. In contrast, Se-related substitutional defects introduce defect levels in the gap, which act as charge traps, leading to severe degradation of the device efficiency. However, the formation energy of these Se-related defects are high so that they should have a low concentration in the material.

By using variational wave functions and quantum Monte Carlo techniques, we investigate the complete phase diagram of the Heisenberg model on the anisotropic triangular lattice, where two out of three bonds have super-exchange couplings J and the third one has instead J . This model interpolates between the square lattice and the isotropic triangular one, for J /J ≤ 1, and between the isotropic triangular lattice and a set of decoupled chains, for J/J ≤ 1. We consider all the fully-symmetric spin liquids that can be constructed with the fermionic projective-symmetry group classification [Y. Zhou and X.-G. Wen, arXiv:cond-mat/0210662] and we compare them with the spiral magnetic orders that can be accommodated on finite clusters. Our results show that, for J /J ≤ 1, the phase diagram is dominated by magnetic orderings, even though a spin-liquid state may be possible in a small parameter window, i.e., 0.7 J /J 0.8. In contrast, for J/J ≤ 1, a large spin-liquid region appears close to the limit of decoupled chains, i.e., for J/J 0.6, while magnetically ordered phases with spiral order are stabilized close to the isotropic point.

Using modified spin wave (MSW) method, we study the J1 − J2 Heisenberg model with first and second neighbor antiferromagnetic exchange interactions. For symmetric S = 1/2 model, with the same couplings for all the equivalent neighbors, we find three phase in terms of frustration parameter α = J2/J1: (1) a commensurate collinear ordering with staggered magnetization (Néel.I state) for 0 ≤ᾱ 0.207 , (2) a magnetically gapped disordered state for 0.207 ᾱ 0.369, preserving all the symmetries of the Hamiltonian and lattice, hence by definition is a quantum spin liquid (QSL) state and (3) a commensurate collinear ordering in which two out of three nearest neighbor magnetizations are antiparallel and the remaining pair are parallel (Néel.II state), for 0.396 ᾱ ≤ 1. We also explore the phase diagram of distorted J1 −J2 model with S = 1/2. Distortion is introduced as an inequality of one nearest neighbor coupling with the other two. This yields a richer phase diagram by the appearance of a new gapped QSL, a gapless QSL and also a valence bond crystal (VBC) phase in addition to the previously three phases found for undistorted model.