Realistic picture of helical edge states in HgTe quantum wells

Abstract

We propose a minimal effective two-dimensional Hamiltonian for HgTe/CdHgTe quantum wells (QWs) describing the side maxima of the first valence subband. By using the Hamiltonian, we explore the picture of helical edge states in tensile and compressively strained HgTe QWs. We show that both dispersion and probability density of the edge states can differ significantly from those predicted by the Bernevig-Hughes-Zhang (BHZ) model. Our results pave the way towards further theoretical investigations of HgTe-based quantum spin Hall insulators with direct and indirect band gaps beyond the BHZ model.The inverted HgTe/CdHgTe quantum well (QW) is the first two-dimensional (2D) system, in which a quantum spin Hall insulator (QSHI) state was theoretically predicted 1 and then experimentally observed 2-4 . The origin of the topologically nontrivial QSHI state is caused by inverted band structure of bulk HgTe, which leads to a peculiar confinement effect in HgTe/CdHgTe QWs. Specifically, in narrow QWs, the first electron-like subband E 1 lies above the first hole-like level H 1, and the system is characterized by normal band ordering with trivial insulator state. As the QW width d is varied (see Fig. 1a), the E 1 and H 1 subbands are crossed 5 , and the band structure mimics a linear dispersion of massless Dirac fermions 6 . When d exceeds the critical width d c , an inversion of the E 1 and H 1 levels drives the system in QSHI state with a pair of gapless helical edge states topologically protected due to time-reversal symmetry 1 .So far, theoretical description of the phase transition between trivial and QSHI states in HgTe QWs has been based on the Bernevig-Hughes-Zhang (BHZ) 2D model 1 . The latter is derived from the Kane Hamiltonian 7 , which includes Γ 6 , Γ 8 , Γ 7 bulk bands with the confinement effect. Within the representation defined by the basis states |E 1,+ , |H 1,+ , |E 1,-, |H 1,-, the effective 2D Hamiltonian has the form:

Massless Dirac fermions in III-V semiconductor quantum wells

“…There is indeed a good agreement for the conduction band, while the valence band is well described at small quasimomentum only. This is also typical for HgTe/CdTe QWs [16,17], in which the discrepancy for the valence band is explained by the effect of the remote subbands beyond the simplified Dirac-like Hamiltonian H D (k) [42].…”

Hybridization of topological surface states with a flat band

“…Equations(11) and(13) give the energy dispersion relations of the surface states. It is seen that the surface states at k = 0 exist if E tution of Eq.…”

“…Note that now the energy dispersions are calculated numerically on the basis of Eq. (4) and Eqs (11),(13). for the bulk and surface states, respectively.…”

“…(13) also results in a quadratic equation for k 2 z and k 2 x , which can be solved analytically. Such a quadratic equation gives the energy dispersion and energy contours of the surface states for the boundary parallel to the (010) crystallographic plane.Fig.…”

Температурная активация электронов проводимости в двойной квантовой яме HgTe/CdHgTe p-типа проводимости со слоями HgTe критической толщины

“…5). Величина энергетического зазора E 0 , вычисленная по формуле n = n 0 exp[−E 0 /(k B T )], составляет (28 ± 3) мэВ, что превышает расчетное расстояние от БМ до ближайшей подзоны проводимости 15 мэВ для данной структуры[1,7].…”