We report on the observation of an unconventional structure of the quantum Hall effect (QHE) in a $p\text{−}\mathrm{type}$ $\mathrm{HgTe}/{\mathrm{Cd}}_x{\mathrm{Hg}}_{1-x}\mathrm{Te}$ double quantum well (DQW) consisting of two HgTe layers of critical thickness. The observed QHE is a reentrant function of magnetic field between two $i=2$ states (plateaus at ${\rho }_{yx}=h/i{e}^2$) separated by an intermediate $i=1$ state in the shape of a flat-top peak placed on the remarkably long $i=2$ plateau. This anomalous $i=1$ peak separates two different regimes: (i) a traditional QHE at relatively low fields corresponding to a small density of mobile holes $p_s$ and (ii) a high-field QHE with a 2-1 plateau-plateau transition corresponding to a much larger $p_s$. The high-field part is strongly sensitive to external influences such as gate voltages, in contrast to the low-field part, which is much less responsive. We explain the observed behavior by analyzing the calculated trajectories of the Fermi level $E_F\left(B\right)$ between hole-like and electron-like Landau levels (LLs). At low fields, $E_F$ is captured by the lateral maximum (LM) of the valence subband, and only holes in the center of the Brillouin zone contribute to QHE, while holes in the LM are inactive. In contrast, at fields above the reentry, $E_F$ rises significantly higher than LM, approaching the zero-mode LLs, and all holes come into play in QHE. At intermediate fields, the reentrance is caused by a combination of two factors in the specific energy spectrum of this DQW: (i) the superposition of an electron-like LL on hole-like LLs and (ii) the stabilizing influence of the LM reservoir on $E_F\left(B\right)$.

We present an experimental study of the double-quantum-well (DQW) system made of two-dimensional layers with inverted energy band spectrum: HgTe. The magnetotransport reveals a considerably larger overlap of the conduction and valence subbands than in known HgTe single quantum wells (QW), which may be regulated here by an applied gate voltage $V_g$. This large overlap manifests itself in a much higher critical field $B_c$ separating the range above it with a plain behavior of the Hall magnetoresistance ${\rho }_{xy}\left(B\right)$, where the quantum peculiarities shift linearly with $V_g$, and the range below with a complicated behavior. In the latter case, specific structures in ${\rho }_{xy}\left(B\right)$ are formed like a double-$N$-shaped ${\rho }_{xy}\left(B\right)$, reentrant sign-alternating quantum Hall effect with transitions into a zero-filling-factor state, etc., which are clearly manifested here due to better magnetic quantization at high fields, as compared to the features seen earlier in a single HgTe QW. The coexisting electrons and holes were found in the whole investigated range of positive and negative $V_g$ as revealed (i) from fits to the low-field $N$-shaped ${\rho }_{xy}\left(B\right)$, (ii) from the Fourier analysis of oscillations in ${\rho }_{xx}\left(B\right)$, and (iii) from a specific behavior of ${\rho }_{xy}\left(B\right)$ at high positive $V_g$. A peculiar feature here is that the found electron density $n$ remains almost constant in the whole range of investigated $V_g$ while the hole density $p$ drops down from the value a factor of 6 larger than $n$ at extreme negative $V_g$ to almost zero at extreme positive $V_g$ passing through the charge-neutrality point. We show that this difference between $n$ and $p$ stems from an order of magnitude larger density of states for holes in the lateral valence subband maxima than for electrons in the conduction subband minimum. We analyze our observations on the basis of a calculated picture of magnetic levels in a DQW and suggest that their specificity is due to (i) a nonmonotonic course of the valence subband magnetic levels and an oscillating behavior of the valence subband top versus field due to lateral maxima in the energy spectrum, (ii) a reduced gap between the lowest-electron and the highest-hole magnetic levels where the electron- and hole-type localized states are superposed, and (iii) a possible formation of the interlayer electron-hole excitons.

Magnetoresistivity features connected with the spin level coincidences under tilted fields in a ${\Gamma }_8$ conduction band of the HgTe quantum well were found to align along straight trajectories in a $(B_{\perp },B_{\left|\right|})$ plane between the field components perpendicular and parallel to the layer, meaning a linear spin polarization dependence on a magnetic field. Among the trajectories is a noticeable set of lines descending from a single point on the $B_{\left|\right|}$ axis, which is shown to yield a field of the full spin polarization of the electronic system, in agreement with the data on the electron redistribution between spin subbands obtained from Fourier transforms of oscillations along circle trajectories in the $(B_{\perp },B_{\left|\right|})$ plane and with the point on the magnetoresistivity under pure $B_{\left|\right|}$ separating a complicated weak field dependence from the monotonous one. The whole picture of coincidences is well described by the isotropic $g$ factor, although its value is twice as small as that obtained from oscillations under pure perpendicular fields. The discrepancy is attributed to different manifestations of spin polarization phenomena in the coincidences and within the exchange-enhanced spin gaps. In the quantum Hall range of $B_{\perp }$, the spin polarization manifests in anticrossings of magnetic levels, which were found to depend dramatically nonmonotonously on $B_{\perp }$.

The temperature and magnetic-field dependence of the components of the conductivity and resistivity tensors are calculated with the quantum corrections due to the weak localization effect and electron-electron interaction taken into account in the diffusion and ballistic regimes. The corrections to the conductivity from the weak localization and electron-electron interaction and also the influence of spin and oscillation effects are taken into account by a renormalization of the transport relaxation time of the electron momentum, with the result that the Drude conductivity becomes temperature dependent. A calculation of the components of the conductivity and resistivity tensors is carried out with the use of the theoretical values of the parameters of the theory of quantum corrections, which are determined solely by the values of the carrier density and mobility of a particular sample. The results of the calculation are compared with experimental results for two-dimensional $n\text{-}\mathrm{In}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ structures with double quantum wells. It is shown that taking only the quantum corrections into account with the theoretical values of the parameters does not permit even a qualitative description of experiment, and it is therefore necessary to take additional temperature-dependent contributions to the Drude conductivity into account.