Band inversion transition in HgTe nanowire grown along the [001] direction

The low-energy effective Hamiltonian of a cylindrical HgTe nanowire grown along the [001] crystallographic direction is constructed by using the perturbation theory. Both the anisotropic term and the bulk inversion asymmetry term of the Kane model are taken into account. Although the anisotropic term has converted the crossing between the E₁ and H₁ subbands into an anticrossing at kzR\!=\!0, the gap-closing-and-reopening transition in the subband structure can still occur at finite wave vectors kzR\!≈\!±0.24 for critical nanowire radius R\!≈\!3.45 nm. The bulk inversion asymmetry does not contribute to the low-energy effective Hamiltonian, i.e., there is no spin splitting in the E₁, H₁, and H₂ subbands for a [001] oriented cylindrical nanowire.