Rashba splitting in n-type modulation-doped HgTe quantum wells with an inverted band structure

Rashba spin splitting has been observed in the first conduction subband of n-type modulation-doped HgTe quantum wells (QW’s) with an inverted band structure via an investigation of Shubnikov–de Haas oscillations in gated Hall bars. In accordance with calculations, no spin splitting was observed in the second conduction subband $\mathrm{}\left(E2\mathrm{}\right),$ but an obvious Rashba splitting is present in the first heavy-hole-like conduction subband $\mathrm{}\left(H1\mathrm{}\right)\mathrm{}$ that displays a large dependence on gate voltage. Self-consistent Hartree calculations of the band structure based on an $8\times 8\mathbf{k}\cdot \mathbf{p}$ model are compared with experiment, which enables us to understand and quantitatively describe the experimental results. It has been shown that the heavy-hole nature of the $H1\mathrm{}$ conduction subband greatly influences the spatial distribution of electrons in the QW and also enhances the Rashba spin splitting at large electron densities. These are unique features of type III heterostructures in the inverted band regime. The $\beta k_{\Vert }^3$ dispersion predicted by an analytical model is a good approximation of the self-consistent Hartree calculations for small values of the in-plane wave-vector $k_{\Vert }$ and has consequently been employed to describe the spin splitting of the $H1\mathrm{}$ conduction subband rather than the commonly used $\alpha k_{\Vert }$ dispersion for the conduction subband in type I heterojunctions. The relative magnitude of Rashba splitting in the $H1\mathrm{}$ and $E2\mathrm{}$ subbands as well as the splitting of the $H1\mathrm{}$ subband for different well widths are also presented.