Direct Evidence of Klein and Anti-Klein Tunneling of Graphitic Electrons in a Corbino Geometry

Transport measurement of electron optics in monolayer graphene $p\text{−}n$ junction devices has been traditionally studied with negative refraction and chiral transmission experiments in Hall bar magnetic focusing setups. We show direct signatures of Klein (monolayer) and anti-Klein (bilayer) tunneling with a circular “edgeless” Corbino geometry made out of gated graphene $p\text{−}n$ junctions. Noticeable in particular is the appearance of angular sweet spots (Brewster angles) in the magnetoconductance data of bilayer graphene, which minimizes head-on transmission, contrary to conventional Fresnel optics or monolayer graphene which show instead a sharpened collimation of transmission paths. The local maxima on the bilayer magnetoconductance plots migrate to higher fields with increasing doping density. These experimental results are in good agreement with detailed numerical simulations and analytical predictions.

Figure 1

(a)–(c) Klein or anti-Klein tunneling in layered graphene, compared with optical transmission. (a) Pseudospin evolution around the graphene Fermi surface, with red and blue lobes showing positive and negative lobes of dimer $p_z$ orbitals. Bilayer graphene (BLG) pseudospins rotate twice as fast as in monolayer graphene (MLG). Orange versus green branches label left-oriented pseudospin (i.e., antibonding dimer $p_z$ orbitals) versus right-oriented pseudospin (bonding). (b) Schematic of Klein and anti-Klein tunneling at MLG and BLG split gated $p\text{−}n$ junction (PNJ) at normal incidence. (c) Reflectivity $R$ (dashed line) and transmittivity $T$ (solid line) of in-plane versus perpendicular (blue $\parallel$ and red $\perp$) photons across a junction, dictated by Fresnel equations. Electron reflection at a graphene $p\text{−}n$ junction (red dashed line in MLG) shows KT at normal incidence, while in BLG (blue dashed line) it shows AKT at normal incidence as well as a Brewster angle set by ${\tan }^{-1}(V_2/V_1)$, here near 60°, where $R_{\mathrm{BLG}}$ vanishes. (d) Magnetoconductance plots (solid line experiment, parameters in the text, and dashed line simulation) for junctionless Corbino disk graphene structure that form the basis of our experiments. The cutoff fields $B_{c1}$ and $B_{c2}$ are where the electron cyclotron orbits narrowly miss the farthest and nearest access to the outer contact.

Figure 2

Electron optics in Corbino disk graphene (monolayer) $p\text{−}n$ junction. (a) Side view of the fabricated Corbino disk monolayer GPNJ device. (b) Experimental and (c) simulated conductance ($G$) color map between inner and outer contact versus magnetic field $B$ and carrier density $n_2$. (d) Top view schematic of Corbino disk monolayer GPNJ device with calculated ray tracing paths (for $B=0\mathrm{mT}$) for $n\text{−}{n}^{\prime }\text{−}n$ and $n\text{−}{p}^{\prime }\text{−}n$ conditions, with enlarged plots showing sharply collimated electron transmission for the latter. (e) $G\text{−}B$ plot for $n\text{−}{n}^{\prime }\text{−}n$ versus (f) $n\text{−}{p}^{\prime }\text{−}n$, plotted over a wider range of $B$ fields at $n_2={10}^{12}{\mathrm{cm}}^{-2}$. Simulations agree with experiments with two separately benchmarked, i.e., no adjustable parameters (see text). Shubnikov–de Haas oscillations inside a gated region must be separately accounted for in simulations, which focus here on low-bias Klein tunneling at the junction. The results show enhanced transmission with $n\text{−}{n}^{\prime }\text{−}n$ compared to $n\text{−}{p}^{\prime }\text{−}n$ due to Klein tunneling, as seen in (d) from the multiple transmitting paths across the junction. The $n\text{−}{p}^{\prime }\text{−}n$ peak is sharp due to the aggressive collimation from Klein tunneling that restricts transmission at $B=0$ to normal incidence. A full quantum simulation for a smaller structure with scaled magnetic fields is presented in the Supplemental Material [20].

Figure 3

Electron optics in Corbino disk graphene (bilayer) $p\text{−}n$ junction. (a) Side view of fabricated BLG device structure. (b) Top view and polar plot of angle-dependent transmission across bilayer graphene $p\text{−}n$ junction. (c) Ray tracing paths for $n\text{−}{p}^{\prime }\text{−}n$ device for $B=0\mathrm{T}$. Most of the electrons are filtered across the $p\text{−}n$ junction due to anti-Klein tunneling, compared to monolayer graphene [Fig. 2]. (d) Experimental versus (e) simulated $G\text{−}B$ plot, simulations including scattering (see text). A pronounced dip is seen at zero field in the magnetoconductance plot corresponding to quenching of the transmission at normal incidence, as expected for BLG AKT (Fig. 1, blue dashed line). The inset in (e) shows a simulation with minimal scattering, where the peaks are more pronounced. Arrows show analytical predictions [Eq. (7)] of the maximizing fields corresponding to the Brewster angle, migrating out to higher fields with increasing doping density. A full quantum simulation for a smaller structure with scaled magnetic fields is presented in the Supplemental Material [20].