Trions in a periodic potential

The group-theoretical classification of trion states (charged excitons X±) is presented. It is based on considerations of products of irreducible projective representations of the two-dimensional translation group. For a given BvK period N degeneracy of obtained states is N2. Trions X± consist of two identical particles (holes or electrons), so the symmetrization of states with respect to particles transposition is considered. There are N(N+1)/2 symmetric and N(N−1)/2 antisymmetric states. Completely antisymmetric states can be constructed by introducing antisymmetric and symmetric spin functions, respectively. Two symmetry-adapted bases are considered: The first is obtained from a direct conjugation of three representations, whereas in the second approach the states of an electrically neutral pair hole–electron are determined in the first step. The third possibility, a conjugation of representations corresponding to identical particles in the first step, is presented elsewhere.