= Carrier density in a graphene cone
{title2=$n=\operatorname{sgn}(E_F)E_F^2/[\pi(\hbar v_F)^2]$}
Assume zero <temperature>, the linear <Dirac cone>, and both real-<spin> states and both <electronic valleys>. Define $n$ as excess <Electron> number per unit <area>, negative for an excess of <electron holes>, relative to neutrality. A filled circular region of radius $k_F$ in <wave vector> space contains
$$
|n|=4\frac{\pi k_F^2}{(2\pi)^2}=\frac{k_F^2}{\pi},\qquad |E_F|=\hbar v_F k_F.
$$
Thus the excess <carrier density> grows quadratically with the <Fermi energy>, unlike the constant <density of states> of a parabolic two-dimensional band. Differentiating gives the positive <density of states> per unit <area>, $D(E)=2|E|/[\pi(\hbar v_F)^2]$. Both the geometry of occupied states and the fourfold degeneracy matter to the coefficient.
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