Single-parameter pumping in graphene

We propose a quantum pump mechanism based on the particular properties of graphene, namely chirality and bipolarity. The underlying physics is the excitation of evanescent modes entering a potential barrier from one lead, while those from the other lead do not reach the driving region. This induces a large nonequilibrium current with electrons stemming from a broad range of energies, in contrast to the narrow resonances that govern the corresponding effect in semiconductor heterostructures. Moreover, the pump mechanism in graphene turns out to be robust, with a simple parameter dependence, which is beneficial for applications. Numerical results from a Floquet scattering formalism are complemented with analytical solutions for small to moderate driving.

Figure 1

Sketch of a single-parameter graphene pump connected to two leads. The left half of the graphene ribbon is exposed to an ac gate voltage. Dispersion of (b) graphene and (c) a 2DEG, with longitudinal and transverse momenta $k$ and $q$, including the branches of evanescent modes. Photon absorption (and in the case of graphene also emission) may excite an evanescent mode to propagating.

Figure 2

Quasi-one-dimensional scattering potential. Profile of the pump, modeled as a barrier for which the left half (region $l$) experiences an oscillatory gate voltage. Regions $L$ and $R$ correspond to highly doped leads. Dotted lines denote sidebands to which the electron energy changes by absorption and emission of photons from the driving field.

Figure 3

Pump current $\overline{I}$ for graphene and 2DEG. Dashed lines mark the semiclassical result, which closely matches the numerical ones. The upper inset is a blowup of the 2DEG case. The lower inset depicts the differential current, demonstrating that the main contribution to the pump current arises from an energy range $\hslash \omega$ around the Dirac point, populated by modes that are evanescent in the barrier region.

Figure 4

Scaled net transmission $\Delta T/p$, where $p={(U/2\hslash \omega )}^2$, for (a) the graphene and (b) the 2DEG pump as a function of transverse wave number $q$ and the initial energy $\epsilon$ for weak driving $U\ll \hslash \omega$ at photon energy $\hslash \omega =2\mathrm{meV}\sim 500\mathrm{GHz}$. The dashed lines separate regions with propagating and evanescent modes under the barrier. (c) Cut at zero energy revealing the behavior in the evanescent region. The dashed line marks the static transmission $2T$ for graphene at energy $\hslash \omega$, while the dotted curve is its semiclassical approximation. The transverse wave number $q$ is scaled by $q_{\omega }$, which is defined by $q_{\omega }L=\hslash \omega /E_L^G$ for graphene, and ${\left(q_{\omega }L\right)}^2=\hslash \omega /E_L^N$ for the 2DEG.

Figure 5

Evanescent mode pump mechanism. Evanescent modes penetrating the barrier from the left absorb or emit photons such that they become propagating. The corresponding modes from the right lead decay before reaching the region with the ac gating. Solid arrows indicate strong population; open arrows mark negligible scattering channels.