High-field transport in two-dimensional graphene

Transport of carriers in two-dimensional graphene at high electric fields is investigated by combining semianalytical and Monte Carlo methods. A semianalytical high-field transport model based on the high rate of optical phonon emission provides useful estimates of the saturation currents in graphene. For developing a more accurate picture, the nonequilibrium (hot) phonon effect and the role of electron-electron scattering were studied using Monte Carlo simulations. Monte Carlo simulations indicate that the hot phonon effect plays a dominant role in current saturation, and electron-electron scattering strongly thermalizes the hot carrier population in graphene. We also find that electron-electron scattering removes negative differential resistance in graphene. Transient phenomenon such as velocity overshoot can be used to speed up graphene-based high-speed electronic devices by shrinking the channel length below 80 nm if electrostatic control can be exercised in the absence of a band gap.

Figure 1

Scattering rates vs energy in 2D graphene: $T=300$ K; $n_e=7.7\times {10}^{12}$ cm${}^{-2}$. The dashed line is the sum scattering of acoustic phonon and optical phonons.

Figure 2

Current saturation model in 2D graphene. (a) The distribution function of streaming model at steady state. $f\left(\mathbf{k}\right)=\mathbf{\text{1}}$ at colored region; otherwise, $f\left(\mathbf{k}\right)=\mathbf{\text{0}}$. (b) The saturation velocity vs carrier concentration.

Figure 3

(a) Current density vs electric field at two carrier concentrations: $n1=7.5\times {10}^{12}$ cm${}^{-2}$ and $n2=3.0\times {10}^{12}$ cm${}^{-2}$. Open symbols: no hot phonon. Solid symbols: hot phonon (${\tau }_{\text{ph}}=1$ ps). Dot-dash lines consider hot phonon effect but not $e$-$e$ scatterings. (b) Saturation current vs carrier density. Solid black curve: the current saturation results of streaming model. Solid square: intrinsic graphene without hot phonon effect, $\mathcal{F}=50$ kV/cm. Open square: with hot phonon effect (${\tau }_{\text{ph}}=1$ ps), $\mathcal{F}=50$ kV/cm. All simulations are at 300 K.

Figure 4

(a) Electron distribution function along $k_x$ with $k_y$ = 0. Solid line considers hot phonon and $e$-$e$ scattering. (Dash line: turning off hot phonon effect; dash-dot line: turning off $e$-$e$ scattering in simulation.) (b) The phonon number distribution function in $\mathbf{q}$ space. Carrier density: $n=7.5\times {10}^{12}$ cm${}^{-2}$, electric field $\mathcal{F}=50$ kV/cm, phonon lifetime ${\tau }_{\text{ph}}=1$ ps, and environment temperature $T=300$ K.

Figure 5

(a) Electron drift velocity evolution with time under field. (b) Carrier transit time vs channel length. Optical phonon lifetime ${\tau }_{\text{ph}}=1$ ps, environment temperature $T=300$ K, and carrier density $n=7.5\times {10}^{12}$ cm${}^{-2}$.