Nonlocal spin-entangled Andreev reflection, fractional charge, and current-phase relations in topological bilayer-exciton-condensate junctions

M. Veldhorst, M. Hoek, M. Snelder, H. Hilgenkamp, A. A. Golubov, and A. Brinkman

Phys. Rev. B 90, 035428 – Published 21 July 2014

Abstract

We study Andreev reflection and Josephson currents in topological bilayer exciton condensates (TECs). These systems can create 100% spin-entangled nonlocal currents with high amplitudes due to perfect nonlocal Andreev reflection. This Andreev reflection process can be gate tuned from a regime of purely retro reflection to purely specular reflection. We have studied the bound states in TEC–topological-insulator–TEC Josephson junctions and find a gapless dispersion for perpendicular incidence. The presence of a sharp transition in the supercurrent-phase relationship when the system is in equilibrium is a signature of fractional charge, which can be further revealed in ac measurements faster than relaxation processes via Landau-Zener processes.

Received 20 July 2012
Revised 8 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.035428

Authors & Affiliations

M. Veldhorst¹, M. Hoek², M. Snelder², H. Hilgenkamp^2,*, A. A. Golubov^2,†, and A. Brinkman²

¹Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, New South Wales 2052, Australia
²Faculty of Science and Technology and MESA + Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands

^*Also at Leiden Institute of Physics, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
^†Also at Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow district, Russia.

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Vol. 90, Iss. 3 — 15 July 2014

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Images

Figure 1
(a) TI-TEC heterostructure with individual gates to tune between $n$ - and $p$ -type surface states. The right side forms an exciton condensate due to the Coulomb interaction between $n$ and $p$ layers. Applying a voltage $V_{1}$ over the top surface states creates a nonlocal current $I_{2}$ through the bottom surface states. (b) Allowed transport processes in the device (the TI is considered here to be of $n n$ type); the arrows denote the spin normal to the interface, ${\hat{e}}_{x}$ , and the solid red (dotted blue) line indicates the top (bottom) surface. An incoming top surface electron (in) can be reflected $(r_{1})$ and Andreev reflected $(r_{2})$ as an electron. Transmission from the TI to the TEC occurs as quasiparticles with electronlike mass $(t_{1})$ and holelike mass $(t_{2})$ . Elastic cotunneling has a vanishing probability due to the large intrinsic TI bulk band gap.
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Figure 2
Exciton Andreev reflection. On the left we show the limit of specular Andreev reflection and on the right we show the limit of retro Andreev reflection. (a) Electrode configuration to obtain the specific configurations. The TI leads are of similar (opposite) type for specular (retro) reflection. (b) Conservation of parallel momentum, together with a group velocity pointing away from the interface, which results in specular reflection when the top and bottom leads are of the same charge type and retro reflection when the leads have opposite charge type. (c) Angle averaged tunneling coefficients and (d) $I V$ characteristics. Electrons with energy $| E | < M_{0}$ can only enter the exciton condensate by the exciton analog of Andreev reflection. For energies $| E | > M_{0}$ , also a quasiparticle current appears. The blue dashed line is the current through the same interface where the voltage is applied; the red solid line is the resulting nonlocal current at the other interface. At $| e V | < M_{0}$ the current is perfectly entangled in both scenarios.
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Figure 3
(a) Bilayer-exciton-condensate analogy of the Josephson junction. The arrows indicate the direction of the group velocity. The group velocity is in the same (opposite) direction as the momentum in the $n (p)$ -type branches. A possible exciton bound state is shown. (b) The bound state for perpendicular incidence (solid line), which is 4 $π$ periodic. A nonzero incidence angle results in the opening of a gap at finite length $(\frac{W}{ξ} = 0.1$ here) and momentum mismatch $(r_{k} = 0.1)$ , as shown for $θ_{T} = 0.1$ , 0.2, 0.3, 0.4, and $0.5 π$ in dashed lines. (c) The top and bottom TI layers with unequal Fermi densities, and thus the gap shifts from zero energy, here $E_{p, B} = 2.6 E_{p, T}, W = 0.1, θ_{T} = 0.2 π$ , and $r_{k} = 0.1$ .
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Figure 4
Exciton Josephson supercurrent-phase relationship in equilibrium. The limit of parallel incidence (normalized) results in a 2 $π$ periodicity due to the presence of a gap in the bound states (Fig. 3). However, a gapless dispersion for perpendicular incidence moves the maximum supercurrent to $ϕ = π$ and is the onset of a doubled periodicity and fractional charge. Relaxation causes a sharp transition around $ϕ = π$ , where the current switches between the two branches.
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