deletions | additions       diff --git a/section_Density_of_states_In__.tex b/section_Density_of_states_In__.tex  index 180e24c..af2b5f4 100644  --- a/section_Density_of_states_In__.tex  +++ b/section_Density_of_states_In__.tex 
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= \sin^2 k(N+1)+\sin k N^2 -\cos k \sin kN\sin K(N+1) +\frac{V_g}{t} \sin kN\sin K(N+1)   \end{multline}  Now we simplify the following term  \begin{equation} \begin{multline}  \sin ^2 k(N+1)+\sin k N^2 -2\cos k \sin kN\sin K(N+1) \\  =\sin k(N+1) \left[ \sin k(N+1)-2 \cos k \sin kN \right]+\sin k N^2 \end{equation} \end{multline}  then we split the $2\cos k$ in two contributions (we do not consider the last term $\sin k N^2$ for these simplifications)  \begin{equation}  \sin k(N+1) \left[ \sin k(N+1)- \cos k \sin kN \right] =\sin k(N+1) \left[ \sin kN \cos k +\cos kN \sin k- \cos k \sin kN \right]  
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\end{equation}  Finally the density of states coincides with the one in \cite{Park_2013}. Using the fact that we are using $k$ and $q$ wavevectors in units of $a$, the lattice spacing, we have  \begin{equation}  \rho(E)=\frac{\sin q a\sin^ka}{(\pi a t)(\left t)\left  (\sin^2(ka)+\frac{eV_g}{t}\sin k(N+1)a \right) \sin kNa} \end{equation}