  Recently the Quantum Spin Hall effect (QSH) was theoretically predicted and
experimentally realized in a quantum wells based on binary semiconductor
HgTe[1-3]. QSH state and topological insulators are the new states of quantum
matter interesting both for fundamental condensed matter physics and material
science[1-11]. Many of Heusler compounds with C1b structure are ternary
semiconductors which are structurally and electronically related to the binary
semiconductors. The diversity of Heusler materials opens wide possibilities for
tuning the band gap and setting the desired band inversion by choosing
compounds with appropriate hybridization strength (by lattice parameter) and
the magnitude of spin-orbit coupling (by the atomic charge). Based on the
first-principle calculations we demonstrate that around fifty Heusler compounds
show the band inversion similar to HgTe. The topological state in these
zero-gap semiconductors can be created by applying strain or by designing an
appropriate quantum well structure, similar to the case of HgTe. Many of these
ternary zero-gap semiconductors (LnAuPb, LnPdBi, LnPtSb and LnPtBi) contain the
rare earth element Ln which can realize additional properties ranging from
superconductivity (e. g. LaPtBi[12]) to magnetism (e. g. GdPtBi[13]) and
heavy-fermion behavior (e. g. YbPtBi[14]). These properties can open new
research directions in realizing the quantized anomalous Hall effect and
topological superconductors.
