Sub-lattice pairing in Graphene

$$\textbf{k}=\frac{2\pi}{3a}\left(1,\frac{1}{\sqrt{3}}\right)$$$$\textbf{k}'=\frac{2\pi}{3a}\left(1,-\frac{1}{\sqrt{3}}\right)$$

$$V(\mathbf q,\omega)=\frac{e^2}{\epsilon_0}\frac{1}{|\mathbf \mathbf q|^2+k_{\mathrm{TF}}^2}\left(1+\frac{\omega_{\mathrm{ph}}^2}{\omega^2-\omega^2_{\mathrm{ph}}}\right)$$

$$\textbf{q}=\textbf{k}-\textbf{k}'=\frac{2\pi}{3a}(0,2\sqrt{3})$$