Topological Insulator Magnetic Tunnel Junctions: Quantum Hall Effect and Fractional Charge via Folding

We provide a characterization of tunneling between coupled topological insulators in 2D and 3D under the influence of a ferromagnetic layer. We explore conditions for such systems to exhibit integer quantum Hall physics and localized fractional charge, also taking into account interaction effects for the 2D case. We show that the effects of tunneling are topologically equivalent to a certain deformation or folding of the sample geometry. Our key advance is the realization that the quantum Hall or fractional charge physics can appear in the presence of only a single magnet unlike previous proposals which involve magnetic domain walls on the surface or edges of topological insulators, respectively. We give illustrative topological folding arguments to prove our results and show that for the 2D case our results are robust even in the presence of interactions.

Figure 1

(a) Standard magnetic domain wall picture on the surface of a 3D TI (or edge of a 2D TI) which gives rise to a propagating chiral mode (bound $e/2$ charge). The inset illustrates the mass domain wall seen by the Dirac fermions on the surface (edge). (b) An illustration of the folding process. Note that the chiral modes ($e/2$ charge) are preserved for every step in the fold. (c) Completion of the folding and gluing process which shows that the final state is a domain wall between a ferromagnet and a tunneling region. The tunneling region is topologically equivalent to the bulk of the TI. (d) When the tunneling region is removed, leaving empty vacuum, the chiral modes ($e/2$ charge) are destroyed.

Figure 2

Quantum spin Hall state in an $H$-bar geometry. Edge states conform to the geometry and in the lower half travel around an inset ferromagnet whose magnetization direction is shown by the arrows. An $e/2$ charge is localized between the two spots at the corners of the ferromagnet. In the ideal case with this geometry there are $e/4$ charges localized at the location of each corner spot.