  We propose a novel class of temperature-tunable semiconductor metamaterials
that exhibit negative refraction in the terahertz spectral range. These
metamaterials are based on doped semiconductor superlattices with ultrathin
barriers of about 1 nm thickness. Due to the tunnel transparency of the
barriers, layers of the superlattice cannot be considered as isolated and,
therefore, the classical homogenization approach is inapplicable. We develop a
theory of quantum homogenization which is based on the Kubo formula for
conductivity. The proposed approach takes into account the wave functions of
the carriers, their distribution function and energy spectrum. We show that the
components of the dielectric tensor of the semiconductor metamaterial can be
efficiently manipulated by external temperature and a topological transition
from the dielectric to hyperbolic regime of metamaterial can be observed at
room temperature. Using a GaAs/Al$_{0.3}$Ga$_{0.7}$As superlattice slab as an
example, we provide a numerical simulation of an experiment which shows that
the topological transition can be observed in the reflection spectrum from the
slab.
