 Gate control of graphene carrier density

Consider an ideal uniform graphene sheet connected to an Electron reservoir, with an insulated gate. Let $c_g>0$ be geometric capacitance per unit area, $e>0$ the elementary charge, and $V_0$ the neutral-point offset. Neglect temperature, trapped electric charge and spatial inhomogeneity. A change in gate voltage pays for both the electrostatic potential drop, $en/c_g$, and the change in the sheet's Fermi energy, $E_F/e$. Here $E_F$ is measured relative to the band crossing; it is negative for excess electron holes.

Combine this relation with the carrier density in a graphene cone. Write $u=|E_F|/e$, measured in volts, $V=|V_g-V_0|$, and $\alpha =e^3/[\pi (\hslash v_F{)}^2{c}_g]$. ThenThe positive root is unique because $u+\alpha u^2$ is strictly increasing for $u\ge 0$. The often useful geometric approximation $n\simeq c_g(V_g-V_0)/e$ neglects the Fermi energy term. It overestimates $|n|$ in this model, especially near neutrality where relatively few states are available.

For a deliberately chosen model with $c_g=0.010F/{\mathrm{m}}^2$, $v_F=10^{6}m/s$, $V_0=0$ and $V_g=1\mathrm{V}$, the formulas give $E_F\simeq 0.252\mathrm{eV}$ and $n\simeq 4.67\times 10^{12}\mathrm{c}{\mathrm{m}}^{-2}$. The purely geometric value would be $6.24\times 10^{12}\mathrm{c}{\mathrm{m}}^{-2}$. These are chosen-model predictions, not measurements of the historical devices. Numerical conversion uses the exact Planck constant and elementary charge.^[ref]