Theory of invariants-based formulation of $\mathbf{k}\cdot \mathbf{p}$ Hamiltonians with application to strained zinc-blende crystals

Group theoretical methods and $\mathbf{k}\cdot \mathbf{p}$ theory are combined to determine spin-dependent contributions to the effective conduction band Hamiltonian. To obtain the constants in the effective Hamiltonian, in general all invariants of the Hamiltonian have to be determined. Hence, we present a systematic approach to keep track of all possible invariants and apply it to the $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian of crystals with zinc-blende symmetry, in order to obtain all possible contributions to effective quantities such as effective mass, g-factor and Dresselhaus constant. Further spin-dependent contributions to the effective Hamiltonian arise in the presence of strain. In particular, with regard to the constants $C_3$ and $D$ which describe spin-splitting linear in the components of $\mathbf{k}$ and $\mathit{\epsilon }$, considering all possible terms allowed by symmetry is crucial.