Energy-gap--controlled current oscillations in graphene under periodic driving

Abstract

We investigate the impact of an induced mass term $Δ$ on the current density in graphene subjected to a space- and time-dependent periodic potential $U(x,t)$. By solving the Dirac equation and deriving both the quasi-energy spectrum and the corresponding eigenspinors, we obtain explicit analytical expressions for the current density, which exhibits a clear dependence on $Δ$. We show that $Δ$ acts as a tunable control parameter that governs the amplitude, sign, and resonance structure of Josephson-like current oscillations. For normal incidence and a purely time-periodic potential, our results reveal that the oscillations within the energy gap gradually diminish as the mass term $Δ$ increases. This suppression leads to a weakening of the Josephson-like effect typically observed in such systems. When the potential $U(x,t)$ is periodic in both space and time, the behavior becomes more complex. The current density can take either positive or negative values depending on the magnitude of the induced gap, and it generally decreases over time. As a result, the resonance phenomena--prominent at lower gap values--become progressively less significant as $Δ$ increases. These findings underscore the tunable nature of light-matter interactions and quantum transport in gapped graphene, suggesting potential applications in terahertz (THz) nanoelectronic devices and optically controlled quantum switches.

The paper asks how a finite Dirac mass $\Delta$ changes oscillatory currents in periodically driven graphene, extending a gapless-graphene transport treatment. It considers the single-particle Hamiltonian $H=v_F(\tau_z\sigma_xp_x+\sigma_yp_y)+U(x,t)\mathbb I+\Delta\sigma_z$, with potentials $U=(U_0/L)x\sin\omega t$ and $U=U_0\cos(x/L)\cos\omega t$. The analysis uses characteristic trajectories, an expansion in transverse momentum $k_y$, Bessel-function expansions, and numerical plots of the resulting current expressions. At normal incidence, the derived transverse current has a sinusoidal phase dependence and an amplitude proportional to $\sqrt{|\varepsilon^2-\delta^2|}$, where $\varepsilon=E/(\hbar v_F)$ and $\delta=\Delta/(\hbar v_F)$. For $E>\Delta$, increasing the gap therefore suppresses this amplitude, which vanishes at $E=\Delta$. For the spatially linear drive, the identified Shapiro-like condition is $\omega^2=2U_0v_F/(L\hbar n)$, with positive integer $n$. Finite-$k_y$ corrections introduce additional resonances and secular growth at resonance. For the space–time-periodic potential, a matching resonance occurs at $\omega=v_F/L$. Parameter plots illustrate changes in current sign and oscillation structure. These are non-superconducting, “Josephson-like” currents rather than Cooper-pair transport. The stated scope is coherent ballistic transport, requiring $E_F\tau/\hbar\gg1$, $\omega\tau\gg1$, and $\Delta\tau/\hbar\gg1$. Disorder and temperature are expected to reduce visibility. The first-order $k_y$ treatment is acknowledged to require higher-order or nonperturbative analysis at sufficiently long times near resonance; experimental accessibility is discussed rather than demonstrated.