= Gate control of graphene carrier density
{title2=$V_g-V_0=en/c_g+E_F/e$}
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 $\alpha=e^3/[\pi(\hbar v_F)^2c_g]$. Then
$$
V=u+\alpha u^2,\qquad u=\frac{2V}{1+\sqrt{1+4\alpha V}},\qquad
n=\operatorname{sgn}(V_g-V_0)\frac{e^2u^2}{\pi(\hbar v_F)^2}.
$$
The positive root is unique because $u+\alpha u^2$ is strictly increasing for $u\ge0$. 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 $c_g=0.010\,\mathrm{F/m^2}$, $v_F=10^6\,\mathrm{m/s}$, $V_0=0$ and $V_g=1\,\mathrm V$, the formulas give \b[$E_F\simeq0.252\,\mathrm{eV}$ and $n\simeq4.67\times10^{12}\,\mathrm{cm^{-2}}$]. The purely geometric value would be $6.24\times10^{12}\,\mathrm{cm^{-2}}$. These are chosen-model predictions, not measurements of the historical devices. Numerical conversion uses the exact <Planck constant> and <elementary charge>.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[Exact h and e entries; material parameters in this example are stipulated]{ref}
\Image[/graphene-gate-control-and-quantum-capacitance.png]
{title=Gate control changes graphene's carrier population and differential capacitance}
{description=Original functional device sketch and calculated zero-temperature curves for the explicitly chosen model. Signed $n$ counts excess electrons; negative $n$ counts holes. The dashed density curve neglects the change in Fermi energy. These are not historical transport data.}
{source=graphene-gate-control-and-quantum-capacitance.py}
Back to article page
