Resonant second-harmonic generation in a ballistic graphene transistor with an ac-driven gate

We report a theoretical study of time-dependent transport in a ballistic graphene field effect transistor. We develop a model based on Floquet theory describing Dirac electron transmission through a harmonically driven potential barrier. Photon-assisted tunneling results in excitation of quasibound states at the barrier. Under resonance conditions, the excitation of the quasibound states leads to promotion of higher-order sidebands and, in particular, an enhanced second harmonic of the source-drain conductance. The resonances in the main transmission channel are of the Fano form, while they are of the Breit-Wigner form for sidebands. For weak ac drive strength $Z_1$, the dynamic Stark shift scales as $Z_1^4$, while the resonance broadens as $Z_1^2$. We discuss the possibility of utilizing the resonances in prospective ballistic high-frequency devices, in particular frequency doublers operating at high frequencies and low temperatures.

Figure 1

(a) A graphene field effect transistor, where the overall doping level is controlled by a back gate (BG), and the source (S)–drain (D) current is controlled by the top gate (TG) dc and ac signals. (b) The harmonic ac signal of frequency $\mathrm{\Omega }$ leads to inelastic scattering that under resonance conditions excites an otherwise unoccupied bound state in the top gate barrier potential at energy $E_b$. This leads to a Fano resonance in the transmission to $E_0$ due to interference between processes (1) and (2) and a Breit-Wigner resonance in the transmission to $E_2$ [process (4)]. Process (4) leads to higher-harmonic generation, in particular the $2\mathrm{\Omega }$ harmonic.

Figure 2

Energy and incidence angle dependence of transmission probabilities for (a) elastic scattering $T_0(E,\phi )$ and (b) inelastic scattering between energy $E$ and $E+2\mathrm{\Omega },T_2(E,\phi )$. The black regions to the left of the white dashed lines in (b) are regions where the second-sideband wave functions are evanescent waves decaying away from the barrier. The barrier strengths are $Z_0=0.4\pi$ and $Z_1=0.45$. Inset: Transmission probabilities for fixed $\phi =\pi /9$.

Figure 3

Zero-temperature source-drain dc linear conductance in the presence of ac drive of strength $Z_1=0.45$ on the top gate. Upper panel: Impact angle resolved dc conductance $G_0(E_F,\phi )$. Lower panel: Angle integrated time-average dc conductance $G_0\left(E_F\right)$ (black solid line). The green straight line is $G_0\left(E_F\right)$ in the static case, $Z_1=0$, for comparison. The dip-peak structures are related to the Fano and Breit-Wigner resonances in the elastic and inelastic transmission functions. The static barrier strength is $Z_0=0.4\pi$.

Figure 4

Zero-temperature source-drain linear conductances for sideband currents with $n=\pm 1$ and $n=\pm 2$ in the presence of an ac drive of strength $Z_1=0.45$ on the top gate. Upper panels: Impact angle resolved average conductances $G_n(E_F,\phi )$. Lower panels: Angle integrated real and imaginary parts of average conductances $G_n\left(E_F\right)$. The static barrier strength is $Z_0=0.4\pi$.

Figure 5

Dynamic Stark shift as illustrated by plotting $\text{det}\left[{\mathbf{M}}_{\mathbf{s}}\right]$ as a function of complex energy $E$, where the matrix ${\mathbf{M}}_{\mathbf{s}}$ is defined in Eq. (B16). The points $E_{\mathrm{zero}}$ in the complex energy plane where this function vanishes coincide with resonances in the Floquet scattering matrix. (a) Without the ac drive, $Z_1=0$, there is a true bound state and the determinant vanishes at $\mathrm{Re}\left[E\right]=E_b$ and $\mathrm{Im}\left[E\right]=0$. (b) and (c) Including the ac drive, $Z_1>0$, the minimum moves out into the complex plane. The dashed lines indicate the position of the bound state in the absence of the ac drive. The static barrier strength is $Z_0=0.4\pi$ and the impact angle is $\phi =\pi /9$.

Figure 6

(a) The dynamic Stark shift scales with the drive strength as $Z_1^4$ for small $Z_1$. (b) The broadening scales as $Z_1^2$. The static barrier strength is $Z_0=0.4\pi$ and the impact angle is $\phi =\pi /9$.

Figure 7

Energy and incidence angle dependence of transmission probabilities for inelastic scattering to the sidebands. The static barrier strength is $Z_0=0.4\pi$ and the ac drive strength is $Z_1=0.45$.

Figure 8

Energy dependence of transmission probabilities $T_0$ and $T_2$ for incidence angle $\phi =\pi /9,Z_0=0.4\pi$, and $Z_1=0.45$. The blue dotted line is the result for $T_0$ when neglecting scattering to the second sideband at $E_2$. In this case, the Fano resonance is fully developed (peak at unit transmission and dip at zero transmission).