Energy spectrum of the valence band in HgTe quantum wells on the way from a two- to three-dimensional topological insulator

The magnetic field and temperature dependences of longitudinal magnetoresistance and the Hall effect have been measured in order to determine the energy spectrum of the valence band in HgTe quantum wells with the width $d_{\text{QW}}=20$–$200$ nm. The comparison of hole densities determined from the period of Shubnikov–de Haas oscillations and the Hall effect shows that states at the top of the valence band are doubly degenerate in the entire $d_{\text{QW}}$ range, and the cyclotron mass $m_h$ determined from the temperature dependence of the amplitude of Shubnikov–de Haas oscillation increases monotonically from $0.2{{m}_{0}}$ to $0.3{{m}_{0}}$ (${{m}_{0}}$ is the mass of the free electron) with increasing hole density $p$ from $2 \times {{10}^{{11}}}$ to $6 \times {{10}^{{11}}}$ cm$^{-2}$. The determined dependence has been compared to theoretical dependences $m_h(p, d_{\text{QW}})$ calculated within the four-band $\mathbf{k}P$ model. These calculations predict an approximate stepwise increase in ${{m}_{h}}$ owing to the pairwise merging of side extrema with increasing hole density, which should be observed at $p=(4{-}4.5)\times10^{11}$ and $4\times10^{10}\,$cm$^{-2}$ for $d_{\text{QW}}=20$ and $200$ nm, respectively. The experimental dependences are strongly inconsistent with this prediction. It has been shown that the inclusion of additional factors (electric field in the quantum well, strain) does not remove the contradiction between the experiment and theory. Consequently, it is doubtful that the mentioned $\mathbf{k}P$ calculations adequately describe the valence band at all $d_{\text{QW}}$ values.