How to Distinguish between Specular and Retroconfigurations for Andreev Reflection in Graphene Rings

Jörg Schelter, Björn Trauzettel, and Patrik Recher

Phys. Rev. Lett. 108, 106603 – Published 9 March 2012

Abstract

We numerically investigate Andreev reflection in a graphene ring with one normal conducting and one superconducting lead by solving the Bogoliubov–de Gennes equation within the Landauer-Büttiker formalism. By tuning chemical potential and bias voltage, it is possible to switch between regimes where electron and hole originate from the same band (retroconfiguration) or from different bands (specular configuration) of the graphene dispersion, respectively. We find that the dominant contributions to the Aharonov-Bohm conductance oscillations in the subgap transport are of period $h / 2 e$ in retroconfiguration and of period $h / e$ in specular configuration, confirming the predictions obtained from a qualitative analysis of interfering scattering paths. Because of the robustness against disorder and moderate changes to the system, this provides a clear signature to distinguish both types of Andreev reflection processes in graphene.

Received 19 October 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.106603

Authors & Affiliations

Jörg Schelter¹, Björn Trauzettel¹, and Patrik Recher^1,2

¹Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
²Institute for Mathematical Physics, TU Braunschweig, 38106 Braunschweig, Germany

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Vol. 108, Iss. 10 — 9 March 2012

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Images

Figure 1
(a) Device geometry showing a graphene ring structure that is penetrated by a magnetic flux $Φ$ measured in units of the flux quantum $Φ_{0}$ . At the interface with the superconductor (shaded region), electron-hole conversion may occur. (b) The gauge is chosen such that each of the eight individual electron (solid lines) and hole (dashed lines) paths picks up a phase $\pm Φ / 2$ as indicated. (c) Scattering paths for electrons injected from and holes leaving through the left normal conducting lead; only zeroth and first-order contributions are included, i.e., terms containing a single electron-hole conversion process and at most one additional round-trip of the electron or the hole. The paths are categorized according to the total phase that is picked up, and each path is associated with a corresponding amplitude, where first-order amplitudes are indicated by a prime.Reuse & Permissions
Figure 2
Schematics of the excitation spectrum (lower panel) and surfaces of constant excitation energy in $k$ space (upper panel) in the cases $E_{F} > ε > 0$ (retroconfiguration) and $0 < E_{F} < ε$ (specular configuration). Solid and dashed lines indicate electron- and holelike states, respectively, (hole) states originating from the valence band are shaded gray. The small arrows in the upper panel indicate the direction of propagation of the corresponding states. Electron-hole excitations are drawn assuming conservation of $k_{y}$ at the NS interface.Reuse & Permissions
Figure 3
Differential magnetoconductance for specular (black) and retro (gray) configuration for $E_{F}^{(r)} = 0.025 τ_{0} = ε^{(s)}$ , $E_{F}^{(s)} = 0.001 τ_{0} = ε^{(r)}$ , corresponding to 8 modes in the normal conducting lead, including all degeneracies (spin, valley, electron or hole). The high doping in the superconducting lead is chosen such that $E_{F} - U = 0.5 τ_{0}$ in both cases. Other parameter values are provided in the main text. The period of the dominant oscillation is $B_{0}^{(s)} \approx 1.8 \times 10^{- 6} a_{0}^{- 2} h / e$ in specular configuration and $B_{0}^{(r)} \approx 8.8 \times 10^{- 7} a_{0}^{- 2} h / e \approx 0.5 B_{0}^{(s)}$ in retroconfiguration. The weak beating pattern in retroconfiguration and the asymmetry in specular configuration arise due to minor contributions of contrary frequencies.Reuse & Permissions
Figure 4
Breakdown of the $h / e$ vs $h / 2 e$ signature for shifted positions of the NS interface, as explained in the text. Other parameters and color coding are chosen as in Fig. 3. For $d = 340 a_{0}$ (left), in specular configuration one observes oscillations of period $h / 2 e$ as in retroconfiguration. For $d = 490 a_{0}$ (right), contributions of specularly reflected holes in retroconfiguration become important, leading to the observation of additional $h / e$ oscillations. The value of the superconducting coherence length is $ξ = 50 a_{0}$ .Reuse & Permissions
Figure 5
Magnetoconductance of the system used in Fig. 3 with a smooth potential profile (inset) with $l = 90 a_{0}$ and bulk disorder of strength $σ = 0.01 τ_{0}$ as explained in the text. The $h / e$ vs $h / 2 e$ signature still persists. The color coding is the same as in Fig. 3.Reuse & Permissions

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