Effective Hamiltonian, energy spectrum, and phase transition induced by in-plane magnetic field in symmetric HgTe quantum wells

The effective $6\times 6$ matrix Hamiltonian for two-dimensional states in HgTe/CdTe quantum wells is derived. The use of the extended basis (in contrast to the previously studied $4\times 4$ matrix Hamiltonian) allows us to describe quantum wells with arbitrary orientation of interfaces and investigate the influence of in-plane magnetic field. The Hamiltonian is applied to calculation of energy spectra of both two-dimensional subbands and edge states in the range of well widths corresponding to the topological insulator phase. It is found that the in-plane magnetic field opens a gap in the edge-state spectrum, and the increase in the field causes disappearance of one of the edge-state branches. A strong (about 10 T) magnetic field induces a phase transition from the gapped state to a gapless two-dimensional state, with energy spectrum similar to that of bulk HgTe. An analytical expression for the transition field through the parameters of the effective Hamiltonian is obtained.

Figure 1

Subband energies in symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells as functions of the well width $a$. Thick solid and dashed lines correspond to interface orientations [001] and [013], respectively, and show the first and the second heavy-hole subbands ($h0$ and $h1$) with energies ${\varepsilon }_0$ and ${\varepsilon }_1$ as well as the interfacelike subband ($e$) with energy ${\varepsilon }_e$, which originates from mixing of ${\Gamma }_6$ band states and light-hole states of the ${\Gamma }_8$ band. Thin solid lines show the energies of the subbands formed as a result of hybridization of $e$ and $h1$ states in [013]-grown wells at zero in-plane magnetic field.

Figure 2

Parameters entering nondiagonal part of the effective Hamiltonian for symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells. Solid lines correspond to interface orientation [001] and dashed lines to low-symmetry orientations.

Figure 3

Parameters entering the diagonal part of the effective Hamiltonian for symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells with interface orientations [001] and [013]. The absolute value of $b_e$ (thick line) is much larger than those of $b_1$ and $b_2$ (thin lines). Dashed lines show two principal contributions to $b_e$ coming from the sums over remote $e$ states [the second term in Eq. (20)] and remote $h$ states [combination of the third and the fourth terms in Eq. (20)].

Figure 4

2D state (solid line) and edge-state (dashed line) spectra in symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells of widths $a=7.5$ and $8.5$ nm for interface orientation [013]. The inset shows the magnified part of the plot where anticrossing of the edge-state branches occurs.

Figure 5

2D state (solid line) and edge-state (dashed line) spectra in symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells of width $a=7.5$ nm for interface orientation [001]. The 2D subbands with different spin orientation $s$ are shown by different colors. A nonzero in-plane magnetic field induces a gap in the edge-state spectrum. Further increase in the field removes the edge-state branch below this gap.

Figure 6

2D state (solid line) and edge-state (dashed line) spectra in symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells of width $a=7.5$ nm for interface orientation [001]. In contrast to Fig. 5, only the high-energy part is shown. Strong in-plane magnetic field transforms the electronic system into a gapless 2D state, similar to bulk HgTe. The edge state does not disappear.

Figure 7

Low-energy part of electronic spectrum for symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum well of width $a=8$ nm for interface orientation [013] under in-plane magnetic field of $B=2$ T. Solid and dashed lines show 2D states for [100] ($Oy$) and [03$\overline{1}$] ($Ox$) directions of the vector $\mathbf{B}$. Edge-state spectrum is shown only for $Oy$ direction (dotted line).

Figure 8

Transition field $B_c$ calculated according to Eq. (42) for symmetric HgTe/Cd${}_{0.7}$Hg${}_{0.3}$Te quantum wells with different interface orientations. Bold lines stand for high-symmetry interfaces [001] and [111]. Solid and dashed lines for low-symmetry interfaces correspond to $Oy$ and $Ox$ directions of the vector $\mathbf{B}$, respectively.