  We present the first experimental study of the double-quantum-well (DQW)
system made of 2D 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 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 where the
quantum peculiarities shift linearly with $V_g$ and the range below with a
complicated behavior. In the latter case the $N$-shaped and double-$N$-shaped
structures in the Hall magnetoresistance $\rho_{xy}(B)$ are observed with their
scale in field pronouncedly enlarged as compared to the pictures observed in an
analogous single QW. The coexisting electrons and holes were found in the whole
investigated range of positive and negative $V_g$ as revealed from fits to the
low-field $N$-shaped $\rho_{xy}(B)$ and from the Fourier analysis of
oscillations in $\rho_{xx}(B)$. 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 band maxima than for electrons in the
conduction band minimum. We interpret the observed reentrant sign-alternating
$\rho_{xy}(B)$ between electronic and hole conductivities and its zero
resistivity state in the quantum Hall range of fields on the basis of a
calculated picture of magnetic levels in a DQW.
