proportional to the band gap. After years of development [@Zawadzki2017], this analogy lead the discovery and understanding of graphene and topological insulators (TIs). The concept of TIs is discussed at the end of the current chapter (sec. \[sQSHE\_HgTe\]).

Other material systems which we will only briefly mention are IV-VI semiconductors, such as (Pb,Sn)Te and (Pb,Sn)Se. (Pb,Sn)Te and (Pb,Sn)Se crystallize in the cubic rock-salt structure, provided that Sn content is low enough. Their multivalley band structure consists of four elongated ellipsoidal valleys at the L-points of the first Brillouin zone, one for each equivalent \[111\] direction. The growth of QWs is usually done on (111) $\text{BaF}_2$ along the \[111\] crystallographic axis, which results in lifting the fourfold L-valley degeneracy. Therefore, the ground-state of the 2D subband originates from a single valley with the long axis oriented along the \[111\] growth direction (longitudinal valley), and the lowest states formed by another three valleys obliquely oriented to the growth direction at an angle of $70.53^\circ$ (oblique valleys) has higher energy than the longitudinal valley first excited state [@Yuan1994; @Abramof2001]. It is worth to mention that low temperature band gap value in PbTe is 0.18 eV and SnTe has an inverted band structure with 