Spin-orbit coupling in wurtzite heterostructures

Effective spin-orbit (SO) Hamiltonians for conduction electrons in wurtzite\nheterostructures are lacking in the literature, in contrast to zincblende\nstructures. Here we address this issue by deriving such an effective\nHamiltonian valid for quantum wells, wires, and dots with arbitrary confining\npotentials and external magnetic fields. We start from an 8$\\times$8 Kane model\naccounting for the $s$--$p_z$ orbital mixing important to wurtzite structures,\nbut absent in zincblende, and apply both quasi-degenerate perturbation theory\n(L\\"owdin partitioning) and the folding down approach to derive an effective\n2$\\times$2 electron Hamiltonian. We obtain the usual $k$-linear Rashba term\narising from the structural inversion asymmetry of the wells and, differently\nfrom zincblende structures, a bulk Rashba-type term induced by the inversion\nasymmetry of the wurtzite lattice. We also find linear- and cubic-in-momentum\nDresselhaus contributions. Both the bulk Rashba-type term and the Dresselhaus\nterms originate exclusively from the admixture of $s$- and $p_z$-like states in\nwurtzites structures. Interestingly, in these systems the linear Rashba and the\nDresselhaus terms have the same symmetry and can in principle cancel each other\nout completely, thus making the spin a conserved quantity. We determine the\nintrasubband (intersubband) Rashba $\\alpha_\\nu$ ($\\eta$) and linear Dresselhaus\n$\\beta_\\nu$ ($\\Gamma$) SO strengths of GaN/AlGaN single and double wells with\none and two occupied subbands ($\\nu=1,2$). We believe our general effective\nHamiltonian for electrons in wurtzite heterostructures put forward here, should\nstimulate additional theoretical works on wurtzite quantum wells, wires, and\ndots with variously defined geometries and external magnetic fields.\n