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## Main Question or Discussion Point

Hello, i am new in condensed matter physics and studying the article 'Engineering artificial graphene in a two-dimensional electron gas' (Gibertini et al., 2009). I read a lot about bloch theorem lately and i got the main idea but still i dont understand the component of the equation. In the article they used 2D muffin-tin potential with ##r=52.5nm## and the center to center distance between the disk is ##a=150nm## and band mass is ##m_b=0.067m## and ##V_0=1.0 meV##.

I'm trying to solve this equation numerically in Matlab. I created 2x100 matrices of ##\mathbf{G}## by the formula:

$$\mathbf{G}= l\mathbf{g_1}+p\mathbf{g_2}$$

with ##\mathbf{g_1}=\frac{2\pi(1,\sqrt 3)} {3a}## and ##\mathbf{g_2}=\frac{2\pi(1,-\sqrt 3)} {3a}##. But i don't know to how to formulize any other elements. For example i have tried to find the ##\mathbf{k}## with $$\mathbf{k}=\sum_{j=1}^2 \frac {m_j} {N_j} \mathbf{G_j}$$ However i didnt succeed.

How can i study to this topic and solve the equation?

$$\mathbf{G}= l\mathbf{g_1}+p\mathbf{g_2}$$

with ##\mathbf{g_1}=\frac{2\pi(1,\sqrt 3)} {3a}## and ##\mathbf{g_2}=\frac{2\pi(1,-\sqrt 3)} {3a}##. But i don't know to how to formulize any other elements. For example i have tried to find the ##\mathbf{k}## with $$\mathbf{k}=\sum_{j=1}^2 \frac {m_j} {N_j} \mathbf{G_j}$$ However i didnt succeed.

How can i study to this topic and solve the equation?