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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 containsThus 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, $D(E)=2|E|/[\pi (\hslash v_F{)}^2]$. Both the geometry of occupied states and the fourfold degeneracy matter to the coefficient.

At either displayed wave vector, the three complex hopping amplitudes add to zero: $1+2\cos (2\pi /3)=0$. Therefore the energy bands touch. There are two inequivalent electronic valleys; other equivalent corners are related by reciprocal-lattice translations. Setting $\mathbf{k}={\mathbf{K}}_{\tau }+\mathbf{q}$ and retaining first-order terms giveswhere ${\sigma }_x,{\sigma }_y$ are Pauli matrices. For example, at ${\mathbf{K}}_{+}$, $f\simeq -3b(q_x-i{q}_y)/2$. The matrix has the form of a massless two-dimensional Dirac Hamiltonian. Here $v_F$ is a material-model velocity, rather than the vacuum speed of light; this description does not change an Electron's vacuum rest mass. It describes the dispersion relation of electronic excitations near a band crossing.