Gate-defined coupled quantum dots in topological insulators

Abstract

We consider electrostatically coupled quantum dots in topological insulators, otherwise confined and gapped by a magnetic texture. By numerically solving the (2 + 1) Dirac equation for the wave packet dynamics, we extract the energy spectrum of the coupled dots as a function of bias-controlled coupling and an external perpendicular magnetic field. We show that the tunneling energy can be controlled to a large extent by the electrostatic barrier potential. Particularly interesting is the coupling via Klein tunneling through a resonant valence state of the barrier. The effective three-level system nicely maps to a model Hamiltonian, from which we extract the Klein coupling between the confined conduction and valence dots levels. For large enough magnetic fields Klein tunneling can be completely blocked due to the enhanced localization of the degenerate Landau levels formed in the quantum dots. In topological insulators (TIs), according to the bulkboundary correspondence principle [1,2], topologically protected surface states are formed, which are robust against timereversal (TR) elastic perturbations. In the long-wavelength limit the two-dimensional (2D) electron states at the surfaces of three-dimensional (3D) TIs can be described as massless Dirac electrons with the peculiar property that the spin is locked to the momentum, thereby forming a helical electron gas. Charge and spin properties become strongly intertwined, opening new opportunities for spintronic [3,4] applications [5][6][7][8][9][10].To build functional nanostructures, such as quantum dots (QDs) or quantum point contacts, additional confinement of the Dirac electrons is needed. However, conventional electrostatic confinement in a massless Dirac system is ineffective due to Klein (interband) tunneling. In graphene this problem could be overcome by either mechanically cutting or etching QD islands out of graphene flakes [11][12][13] or by inducing a gap by an underlying substrate, which breaks the pseudospin symmetry [14,15]. Another promising idea to overcome the restrictions given by Klein tunneling is to use graphene strips or nanoribbons. An electrostatic confinement in such a system has been proposed in Ref. [16] by employing the transversal electron motion. Moreover, an effective spin exchange coupling of two gate-defined quantum dots becomes possible in a graphene nanoribbon by indirectly coupling the dots via the tunneling to a common continuum of delocalized states [17].In TIs a mass gap can be created by breaking the TR symmetry at the surface by applying a magnetic field. This could be achieved by proximity to a magnetic material [18,19], or by coating the surface randomly with magnetic impurities [20][21][22]. By modifying the magnetic texture of the deposited magnetic film, a spatially inhomogeneous mass term is induced, opening the possibility to define quantum dot (QD) regions [23], or waveguides formed along the magnetic domain wall regions [24]. Another interesting, possibly more feasible way of defining confinement regions, is to ind...

Blurring the Boundaries Between Topological and Nontopological Phenomena in Dots

“…1. This surprising result contrasts with the usual bulk-edge correspondence in TIs, as the non-topological dots here exhibit edge states with spin-angular-momentum locking similar to topological dots [14][15][16][17][18][19][20][21][22][23][24][25]. Interestingly, our quantum transport calculation shows that circulating currents [28,29] (Fig.…”

“…Following these pioneering works, a few other TI proposals [5][6][7][8][9][10][11] have been put forward with some experimental support [12,13]. More recently, topological QDs with cylindrical confinement have been investigated [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Their spectra feature discrete helical edge states protected against non-magnetic scattering and showing spinangular-momentum locking.…”

“…Their spectra feature discrete helical edge states protected against non-magnetic scattering and showing spinangular-momentum locking. These states are potentially important for spintronics [15,16], quantum computation and other quantum technologies [14,17,18].…”

“…We define our QDs by adding to Eq. (1) the in-plane cylindrical confinement [14][15][16][17][18][19][20][21][22][23][24][25][26]…”

Confinement and fermion doubling problem in Dirac-like Hamiltonians

“…This is a consequence of the constant Fermi velocity of the linear dispersion, which allows perfect matching of the injected and transmitted waves. Consequently, to attain confinement, one needs to either open a gap by breaking a symmetry in the outer region [21,[33][34][35][36][37][38], or invoke finite-size effects [39].…”