Klein Backscattering and Fabry-Pérot Interference in Graphene Heterojunctions

We present a theory of quantum-coherent transport through a lateral $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ structure in graphene, which fully accounts for the interference of forward and backward scattering on the $p\mathrm{\text{−}}n$ interfaces. The backreflection amplitude changes sign at zero incidence angle because of the Klein phenomenon, adding a phase $\pi$ to the interference fringes. The contributions of the two $p\mathrm{\text{−}}n$ interfaces to the phase of the interference cancel with each other at zero magnetic field, but become imbalanced at a finite field. The resulting half-period shift in the Fabry-Pérot fringe pattern, induced by a relatively weak magnetic field, can provide a clear signature of Klein scattering in graphene. This effect is shown to be robust in the presence of spatially inhomogeneous potential of moderate strength.

Figure 1

Schematic of electron transmission through a $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ structure at zero (a) and finite (b) magnetic field $B$. The amplitude of backreflection changes sign at the incidence angle $\alpha =0$. At finite $B$, the angles of incidence on both $p\mathrm{\text{−}}n$ interfaces are of the same sign for the trajectories satisfying condition (2). This adds a phase of $\pi$ to the electron phase accumulated between reflections, resulting in a half-period shift of Fabry-Pérot interference fringes. (c) Weak spatial inhomogeneity does not alter the relative sign of incidence angles.

Figure 2

Transmission of the $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ structure (a) at zero magnetic field and (b) at a finite field of $B=0.2{\Phi }_0/x_{*}^2$. Electron momentum $p_y$ is measured in units of $p_{*}=v_F/{\epsilon }_{*}$. Note that at finite $B$ the fringe contrast is reversed across the parabola (7) (black line), with the maxima and minima of the fringe pattern interchanging. This behavior is in agreement with the Fabry-Pérot model (9), with the reflection amplitude vanishing on the parabola, and changing its sign.

Figure 3

Fringes in resistance (13) at $B=0$, plotted in the units of $R_{*}=(x_{*}/W)h/4e^2$ (blue line). Inset illustrates a $n^{2/3}$ scaling for the maxima and minima of $R$, which is consistent with the ${\epsilon }^{3/2}$ dependence of the WKB phase (12). Averaging over smooth potential fluctuations, described by a sum of a few harmonics, suppresses fringe contrast (red dashed line). Here we use Eq. (14) with $\sum _m|a_m|=3{\epsilon }_{*}$.

Figure 4

Resistance (13) of the $p\mathrm{\text{−}}n\mathrm{\text{−}}p$ structure as a function of magnetic field and potential depth. The quantity plotted is $\log R/R_{*}$, with $R_{*}=(x_{*}/W)h/4e^2$. The resistance minima at high $B$, marked by dashed lines, are shifted by approximately half a period relative to those at zero $B$. In the close-up of a few fringes (inset), arrows indicate the field-induced shift.