Andreev bound states and current-phase relations in three-dimensional topological insulators

M. Snelder, M. Veldhorst, A. A. Golubov, and A. Brinkman

Phys. Rev. B 87, 104507 – Published 8 March 2013

Abstract

To guide the search for the Majorana fermion, we theoretically study superconductor/topological-insulator/superconductor (S/TI/S) junctions in an experimentally relevant regime. We find that the striking features present in these systems, including the doubled periodicity of the Andreev bound states (ABSs) due to tunneling via Majorana states, can still be present at high electron densities. We show that via the inclusion of magnetic layers, this 4 $π$ periodic ABS can still be observed in three-dimensional (3D) topological insulators, where finite angle incidence usually results in the opening of a gap at zero energy and hence results in a 2 $π$ periodic ABS. Furthermore, we study the Josephson-junction characteristics and find that the gap size can be controlled and decreased by tuning the magnetization direction and amplitude. These findings pave the way for designing experiments on S/3DTI/S junctions.

Received 8 October 2012

DOI:https://doi.org/10.1103/PhysRevB.87.104507

Authors & Affiliations

M. Snelder¹, M. Veldhorst^1,2, A. A. Golubov¹, and A. Brinkman¹

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

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 87, Iss. 10 — 1 March 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematic drawing of a topological insulator (TI) with a superconductor (S) and ferromagnet (F) on top. We consider the bound states at the surface of the TI with the proximity effect from both the superconductor and ferromagnet. (b) The energy dispersion at the TI surface (left) and at the S side (right). DP indicates the Dirac point. Here the case is shown without a F and with an incoming electron at the right interface.Reuse & Permissions
Figure 2
(a) Andreev bound states for different trajectories in a S/TI/S junction. A gap opens at finite angle. The arrows indicate which bound-state branches are connected to form a 4 $π$ periodicity. In these branches, a Majorana fermion is present at $ϕ = π$ . The legend applies to all of the figures. $μ_{TI} / Δ = 100, μ_{S} / Δ = 1000$ , and $L / L_{0} = 0.01$ where $L_{0} = v_{f} ℏ / μ_{TI}$ . (b) $μ_{TI} / Δ = 100, μ_{S} / Δ = 120$ , and $L / L_{0} = 0.01$ . (c) $μ_{TI} / Δ = 100, μ_{S} / Δ = 120$ , and $L / L_{0} = 0.1$ . (d) $μ_{TI} / Δ = 100, μ_{S} / Δ = 120, m_{z} / Δ = 60.0$ , and $L / L_{0} = 0.01$ . (e) Bound-state energy for different angles and phase difference $ϕ = π$ . Furthermore, $μ_{S} / Δ = 120, μ_{TI} / Δ = 100, m_{z} / Δ = 0$ . The energy is oscillating with length. $E_{gap}$ is defined as the distance from $E / Δ = 0$ to the minimum of the ABS. (f) Bound-state energy for fixed phase difference $ϕ = π$ and angle using the formula from Ref. 37.Reuse & Permissions
Figure 3
Bound-state energy vs the wave vector parallel to the interface for different phase differences. For all of the cases, a nonhelical Majorana quantum wire is seen at $ϕ = π$ . The legend is shown in (a) and holds also for the other two. (a) $μ_{TI} / Δ = 100, μ_{S} / Δ = 1000$ , and $L / L_{0} = 0.01$ . (b) $μ_{TI} / Δ = 100, μ_{S} / Δ = 110$ , and $L / L_{0} = 0.01$ . (c) $μ_{TI} / Δ = 100, μ_{S} / Δ = 1000, m_{z} / Δ = 60$ , and $L / L_{0} = 0.01$ .Reuse & Permissions
Figure 4
(a) Normalized Josephson supercurrent as a function of the phase difference between the superconductors. The numbers indicated in the inset correspond to the following parameters: (1) $μ_{s} / Δ = 1000, μ_{TI} / Δ = 100, m_{z} / Δ = 0, L / L_{0} = 0.01$ , (2) $μ_{s} / Δ = 120, μ_{TI} / Δ = 100, m_{z} / Δ = 0, L / L_{0} = 0.01$ , and (3) $μ_{s} / Δ = 120, μ_{TI} / Δ = 100, m_{z} / Δ = 60.0, L / L_{0} = 0.01$ . (b) Dependence of the normalized $I_{c} R_{N} e / Δ$ product on both the magnetization (dashed line) and the chemical potential of the superconductor (solid line) separately. $μ_{TI} / Δ$ is kept constant to 100. In the dependence of the magnetization, $μ_{S} / Δ$ is kept constant to 120. The temperature is $T / T_{c} = 0.01$ in (a) and (b). (c) Sketch of an Andreev bound state at nonzero angle of incidence. Due to magnetization, the gap of the ABS has become so small that through Zener tunneling, the electron in the lower branch can be promoted to the upper branch around $ϕ = π \pm n 2 π$ where $n$ is an integer. (d) The influence of the magnetization and $μ_{S} / Δ$ on the value of the gap. For both situations, $μ_{TI} / Δ = 100, θ = π / 3$ , and $L / L_{0} = 0.01$ .Reuse & Permissions

Physical Review B

covering condensed matter and materials physics