Theory for electron and hole fine structure and Land\'e g-factors in lead chalcogenide nanowires

Using the atomistic tight-binding method in combination with symmetry analysis and extended effective mass theory we derive a phenomenological model for the fine structure of the ground electron and hole states in $[111]$-grown PbX, X=S,Se hexagonal and cylindrical nanowires. Projection of the tight-binding states to the basis of valley states enables the explicit parametrization of the valley mixing Hamiltonian, which is essential to obtain correct combinations of the valley states for electron (hole) ground levels analytically. The effective Hamiltonian of valley mixing allows to convert the intravalley components of the $g$-factors tensors to the Zeeman splittings of electron and hole levels in nanowires.Tensors of the intravalley electron (hole) Land\'e $g$-factors, parameters of the valley mixing Hamiltonian as well as the valley and anisotropic splitting energies and velocity matrix elements are also calculated.