Retro reflection of electrons at the interface of bilayer graphene and superconductor

Abstract

Electron reflection at an interface is a fundamental quantum transport phenomenon. The most famous electron reflection is the electron→hole Andreev reflection (AR) at a metal/superconductor interface. While AR can be either specular or retro-type, electron→electron reflection is limited to only the specular type. Here we show that electrons can undergo retro-reflection in bilayer graphene (BLG). The underlying mechanism for this previously unknown process is the anisotropic constant energy band contour of BLG. The electron group velocity is fully reversed upon reflection, causing electrons to be retro-reflected. Utilizing a BLG/superconductor junction (BLG/S) as a model structure, we show that the unique low energy quasiparticle nature of BLG results in two striking features: (1) AR is completely absent, making BLG/S 100% electron reflective; (2) electrons are valley-selectively focused upon retro-reflection. Our results suggest that BLG/S is a valley-selective Veselago electron focusing mirror which can be useful in valleytronic applications.

Figure 1. (a) Since electron group velocity component v_y is locked to the momentum component k_y, conservation of k_y allows only specular electron reflection (SER) to occur; (b) retro electron reflection (RER) requires v_y to be ‘unlocked' from k_y such that reversal of v_y can still occur while conserving k_y; (c) constant energy slice of a parabolic (or linear) energy spectrum.

The circular band contour allows only SER to occur; (d) constant energy slice of a hypothetical energy spectrum with boomerang-shaped band contour. The constant energy band contour of the incident states in k-space is convex while that of the reflected states is concave. The opposite band contours between incident and reflected states causes the sign reversal of v_y upon reflection. This results in RER.

Figure 2. (a) Bernal-stacked bilayer graphene lattice structure; (b) the energy spectrum contour plot in phase space, showing three distinct energy regime: Regime (A): ε_k < ε₀/4, Regime (B): ε₀/4 < ε_k < 10.9ε₀ and Regime (C): ε_k > 10.9ε₀.

Retro electron reflection occurs optimally in Regime (B) due to its ‘boomerang-like' anisotropic constant energy band contour in k-space. In high energy Regime (C), retro reflection is no longer possible as the band contour becomes parabolic-like; (c) Band contour of an constant energy slice in Regime (B). The green and blue arrows denotes incident and reflected electron direction of motion respectively. The opposite band contour between incident (convex) and reflected states (concave) causes v_y to reverse its direction upon reflection, leading to RER.

Figure 3. Group velocities of incident and reflected electrons.

(a)–(c) Group velocity v_x; (d)–(f) group velocity v_y; and (g)–(i) incident and reflection angles of an incident K valley electron at a given k_y. The energies are ε_k = 0.5ε₀, ε_k = 2ε₀ and ε_k = 5ε₀ respectively for column 1, 2 and 3. In (a)–(c), v_x is positive for incident electrons and negative for reflected electrons. The RER regime is enclosed between the dashed lines in (d)–(i). In (d)–(f), sign reversal of v_y corresponds to RER. In (g)–(i), the reflection angle of RER does not change sign since electron is reflected to the same side of the normal. At small ε_k, almost all of the permissible reflections are RER. As E_k increases, more permissible states becomes specular reflection states (which lies outside the RER windows) and the RER angle approaches 0°.

Figure 4. RER in BLG/S junction.

Green, blue and gray arrows indicate incident electron, retro-reflected electron and transmitted quasiparticle respectively. Transmission across the junction is forbidden due to the 2π Berry phase nature of bilayer graphene electron. The junction is hence 100% electron-reflective.

Figure 5. Schematic drawings of (a) Veselago lens with n_t = −n_i; and (b) Veselago mirror n_r ≈ −n_i.

In the Veselago lens, a ray emitted from a point source (denoted by yellow triangle) is focused at the transmitted side of the interface. For the Veselago mirror, n_r ≈ −n_i is chosen for better visual clarity. The retro-reflected ray is focused at the incident side of the interface.

Figure 6. The electron excitation spectrum of BLG/S at (a) a K valley and (b) a K′ valley with ε_F = 0.5ε₀ and E_k = ε₀.

The RER constant energy band contour of the K valley (red dashed curve) is significantly ‘smoother' than that of the K′ valley (green dashed curve). At large |k_y| (SER regime), K and K′ band contours become approximately identical; (c) Reflection angles of K (red curve) and K′(green curve) electrons. K electrons are predominantly focused via smaller angle (< 20°) than that of the K′ electrons (≈80°). (d) RER trajectory of electrons emitted from a point source situated at P (denoted by yellow triangle). Blue rays represent the incident electrons. The interface acts as a valley-selective dual-focus electron mirror with K electrons (red rays) being focused further away from the interface than the K′ electrons (green rays).