Double-group formulation ofk·ptheory for cubic crystals

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Multiband $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ theory is often implemented with the one electron Schr\"odinger equation without spin (single group) as the unperturbed system. The effect of spin is taken into account by considering basis functions formed by a direct product between single-group eigenstates and spinor states (which give rise to the adapted double-group basis after a unitary transformation), with the spin-orbit interaction also treated as a perturbation. The $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ perturbation between these states is calculated using the single-group basis functions. This approach leads to a one-to-one link between occurrence of basis states in the single group with those under the double-group classification, placing constraints on the adapted double-group basis. This paper considers energy eigenstates which form the bases of irreducible representations (IRs) of the double group and derives the direct and remote (L\"owdin term) interaction matrices between the states using perturbation theory and symmetry properties of crystal lattice. The use of general double-group basis functions removes the constraints placed on the adapted double-group basis under the single-group formulation. Together with a change of paradigm in constructing atomic site wave functions using hybridized orbitals (rather than atomic orbitals), it allows direct contributions from $d$ and higher orbitals to the valence band with additional interaction matrices permitted by symmetry. A full description of interactions between states of ${\ensuremath{\Gamma}}_{8}^{\ifmmode\pm\else\textpm\fi{}}$ IRs require two linearly independent matrices and two scaling constants rather than the single matrix and scaling constant under single-group consideration. This formulation is developed from both perturbation theory and the method of invariant approach utilizing the Wigner-Eckart theorem and other group theoretical techniques for calculation of matrix elements. Crystals with diamond lattice are investigated first, with results for zincblende lattice obtained under the compatibility relation between the ${O}_{h}$ and the ${T}_{d}$ groups. We show that a unitary transformation of the ${\ensuremath{\Gamma}}_{8}^{\ensuremath{-}}$ basis of the ${O}_{h}$ group is required before they can be used in ${\ensuremath{\Gamma}}_{8}$ IR of the ${T}_{d}$ group. Consequently, existing data and optical transition selection rules shows that the symmetry assignment of the zone-center conduction band edge state should be ${\ensuremath{\Gamma}}_{6}^{\ensuremath{-}}({\ensuremath{\Gamma}}_{7})$ in Ge (GaAs and other semiconductors with zincblende lattice) with spin-orbit split-off band as origin. In addition to the new interaction matrix between states of ${\ensuremath{\Gamma}}_{8}^{\ifmmode\pm\else\textpm\fi{}}({\ensuremath{\Gamma}}_{8})$ IRs, the form of interband L\"owdin term between ${\ensuremath{\Gamma}}_{8}^{+}({\ensuremath{\Gamma}}_{8})$ and ${\ensuremath{\Gamma}}_{7}^{+}({\ensuremath{\Gamma}}_{7})$ in the Hamiltonian used in the literature is shown to be incorrect. A linear $k$ term between the degenerate valence band, different from those obtained previously, is shown to exist. It modifies the dispersion and density of state in the vicinity of $\ensuremath{\Gamma}$ point but does not lift the Krammer's degeneracy. When quantum well, wires, and dots are considered, operator ordering in the remote interaction emerges naturally by treating wave vector as an operator acting on the envelope functions. This differs from previous schemes based on single-group formulation and a new term, arising from interfacial symmetry breaking, is identified in the valence-band Hamiltonian coupling the degenerate heavy-hole states.