# High-Frequency Limits of Graphene Field-Effect Transistors with Velocity Saturation
## 4 Simple Model of GFETs
### Dirac Pinch-Off Model  
In a remote gate transistor, for example using a global silicon gate, the gate uniformly controls the charge of the semiconducting channel. However, considering a local gate transistor, where local means that the quantum capacitance \(C_{Q}=2e^{2}E_{F}/(\pi\hbar^{2}v_{F}^{2})\) and the geometrical capacitance per unit area \(C_{g}=\epsilon_{0}\epsilon_{r}/t_{BN}\) are on the same order of magnitude, the charge in the channel is not set by the gate voltage only and it can be influenced by the electrostatic potential in the channel. Applying a voltage on the drain electrode results in a linear electrochemical potential along the channel if we assume diffusive conduction, which is the case at large bias due to HPhP scattering. We write the electrochemical balance at the position x in the channel:  
\[\Delta E_{F}(x)-eV_{gr}(x)=-eV_{ds}^{0}\times\frac{x}{L} \tag{4}\]  
Figure 3: **(a)** Scattering length associated with the impurity scattering (black solid line) responsible for the low field mobility; here \(\mu=10^{4}\) cm\({}^{2}\)/V/s, and with the HPhP scattering (dashed lines) for different in plane electric fields \(\mathcal{E}=V_{ds}^{0}/L\). The green thick line is the total scattering length at the largest bias obtained by the Matthiessen’s law. **(b)** Carrier drift velocity simulated using \(v_{d}=V_{ds}^{0}/(RneW)\) with R from Equation (1), and the measured drift velocity at large doping (\(n=5\times 10^{12}\) cm\({}^{-2}\)) in black circles.  
where \(V_{gr}(x)\) is the electrostatic potential of graphene. This equation is combined with the electrostatic equilibrium \(n(x)e=C_{g}(V_{gr}(x)-V_{g})\), which allows calculating \(\Delta E_{F}(x)\):  
\[\Delta E_{F}=sign(\epsilon_{W})\times\epsilon_{c}\left(\sqrt{1+2\frac{|\epsilon _{W}|}{\epsilon_{c}}}-1\right)\,, \tag{5}\]  
with \(\epsilon_{c}=C_{g}\frac{\pi\hbar^{2}v_{g}^{2}}{2\epsilon^{2}}\) and \(\epsilon_{W}(x)=eV_{g}-eV_{ds}^{0}\times\frac{x}{L}\). It results in an inhomogeneous \(\Delta E_{F}\) within the channel. This effect is sketched in Figure 4a, and the calculated \(\Delta E_{F}(x)\) is shown in Figure 4b for increasing \(V_{ds}^{0}\). The corresponding carrier density in the channel as a function of \(V_{g}\) and \(x\) is shown in the color plot of Figure 4c. Starting from an n-doped channel and increasing \(V_{ds}^{0}\) leads to a decrease of the carrier density on the drain side. Figure 4b shows that the neutrality can be reached and that for larger \(V_{ds}^{0}\) there is even a localized neutral point whose location is shifted toward the source electrode. At the onset of neutrality on the drain side, increasing \(V_{ds}^{0}\) induces an increase of the resistance, which amplifies the sublinear behavior of the I-V characteristic and results in enhanced current saturation. This is the Dirac pinch-off. On the contrary, starting from a p-doped channel, an increase of \(V_{ds}^{0}\) leads to an increase of the carrier density on the drain side which contributes to decreasing the channel resistance. This explains the observed asymmetric shape of the high bias resistance curve in Figure 1f.