Current Resonances in Graphene with Time-Dependent Potential Barriers

A method is derived to solve the massless Dirac-Weyl equation describing electron transport in a monolayer of graphene with a scalar potential barrier $U(x,t)$, homogeneous in the $y$ direction, of arbitrary space and time dependence. Resonant enhancement of both electron backscattering and currents, across and along the barrier, is predicted when the modulation frequencies satisfy certain resonance conditions. These conditions resemble those for Shapiro steps of driven Josephson junctions. Surprisingly, we find a nonzero $y$ component of the current for carriers of zero momentum along the $y$-axis.

Figure 1

(a) Current $j_{0y}$ (10) perpendicular to $\mathit{k}$ versus time for $U^{(1)}(x,t)=U_{0}x\sin \omega t$, $k_{x}x=\pi /8$, and $U_0/{\omega }^2=1/\pi$ [dark gray (blue) line] and $U_0/{\omega }^2=1/2$ [light gray (red) line], assuming a valley-polarized situation ${\tau }_z=1$. For “Shapiro-step” conditions [Eq. (16)] periodic oscillations can be seen [light gray (red)], while away from this condition aperiodic oscillations occur [dark gray (blue)]. (b) Same as (a) but for potential $U^{(2)}(x,t)=U_0\cos (x/L)\cos \omega t$ with $x=\pi L/2$, $k_x=0$, $U_{0}L=0.1$, and frequencies $\omega =(\pi /2)(1/L)$ [light gray (red) line] and $\omega =1/L$ [dark gray (blue) line]. Both currents are aperiodic. For the matching condition $\omega =1/L$ [dark gray (blue)] a considerable enhancement followed by a saturation of the amplitude of the current oscillations occurs, while away from this resonance no enhancement is seen versus time. (c) Current $j_{1y}$ (12) for $U^{(1)}(x,t)$ at $x=0$, using $k_x=0$. At Shapiro resonance [$U_0/{\omega }^2=1/2$, dark gray (blue) line] pronounced current enhancement occurs, cf. Eqs. (17, 18), while away from the resonance [$U_0/{\omega }^2=3/\pi$, light gray (red) line] no enhancement is seen. (d) Time-dependent reflectivity $R(t)$, calculated by numerical integration of Eq. (17), solid blue line, and by using the approximation (19), red dashed line, near resonance for $k_x/\sqrt{U_0}=0$ and ${\omega }^2/U_0=0.49$. (e) $R(t)$ at resonance (dashed green line, $k_x/\sqrt{U_0}=0$, ${\omega }^2/U_0=1/2$) and near resonance (solid blue line, $k_x/\sqrt{U_0}=0$, ${\omega }^2/U_0=0.49$). At the resonance, $R(t)$ increases with time $\sim t^2$, in agreement with Eq. (19). (f) Time-averaged reflectivity $\overline{R}$ as a function of $k_x/\sqrt{U_0}$ for $\omega /\sqrt{U_0}=1$. Equidistant resonances occur at $k_x/\sqrt{U_0}=1-n/2$ (dashed blue vertical lines), cf. Eq. (18). One of the resonance peaks is well fitted by the resonance equation (20), as shown by short-dashed green line. (g) $\overline{R}$ as a function of the driving frequency $\omega /\sqrt{U_0}$ for $k_x/\sqrt{U_0}=0$: resonance peaks are clearly seen at Shapiro-step conditions (18) $\omega /\sqrt{U_0}=\sqrt{2/n}$, indicated as dashed magenta vertical lines.