Helical Quantum States in HgTe Quantum Dots with Inverted Band Structures

We investigate theoretically the electron states in HgTe quantum dots (QDs) with inverted band structures. In sharp contrast to conventional semiconductor quantum dots, the quantum states in the gap of the HgTe QD are fully spin-polarized and show ringlike density distributions near the boundary of the QD and spin-angular momentum locking. The persistent charge currents and magnetic moments, i.e., the Aharonov-Bohm effect, can be observed in such a QD structure. This feature offers us a practical way to detect these exotic ringlike edge states by using the SQUID technique.

Figure 1

(a) Schematic of a quantum dot formed in a HgTe heterostructure using the etching technique. (b) Schematic of electrical injection of electrons and holes into a HgTe QD (the shaded region) embedded in a $p\mathrm{\text{−}}n$ junction.

Figure 2

Energy spectra of the HgTe QDs in the band insulator ($M>0$) (a) and topological insulator ($M<0$) (b) cases as a function of the angular momentum $m$. The RESs are denoted by the red (rotating clockwise) and blue (rotating counterclockwise) squares. The red and blue circles are plotted from the analytical solutions. The inset is the enlarged circle region near the Dirac point. (c) The density distributions of the lowest conduction band electron states [marked by the black arrow in (b)] in the QD; (d) the same as (c), but for the RES [marked by the red arrow in (b)] near the Dirac point. The QD radius $R=55\mathrm{nm}$, and $M=-30\mathrm{meV}$.

Figure 3

(a) The energies of the up (red line) and down (black line) branches of the RESs vs the inverse of the radius of the QD $1/R$. The inset shows the dependence of the energies of the electron states in the bulk conduction (red dashed line) and valence (black dashed line) bands on the inverse square of the radius of the QD $1/R^2$, respectively. (b) The energy level spacing $\Delta E$ of the RESs vs the $1/R$; the inset shows the bulk energy gap of the QD vs $1/R^2$. The circles in (a) and (b) are obtained from the analytical expression of the edge states. (c) The spin projection $⟨S_z⟩$ for the ringlike edge states with the different angular momentum $m$. The inset is the schematic of the spin orientation of the RESs in the QD. $R=55\mathrm{nm}$, and $M=-30\mathrm{meV}$.

Figure 4

(a) The magnetic level fan in a HgTe QD with an inverted band structure. (b) The energy spectrum in the presence of magnetic fields for the TI QD. (c)–(f) The persistent charge current (CC) (solid line) and magnetic moment (MM) (dashed line) in a TI QD against the magnetic fields for a different electron number $N$. The persistent CC is in units of $2N{E}_0/{\Phi }_0$ ($2N*2nA$ for the hard-wall disk), where $E_0={\hslash }^2/[2m^{*}(R{)}^2]$. $R=55\mathrm{nm}$, and $M=-30\mathrm{meV}$. The magnetic moment is in units of $N{\mu }_B$, where ${\mu }_B=e\hslash /(2m_e)$. The red, green, blue, and orange curves indicate different electrons $N=1$, 2, 3, and 4 filled in the edge states [see (a)].