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There’re 2 ways to calculate the subband structure of graphene nanoribbon using tight-binding model, for Zigzag chain:

The first one is to diagonize the Hamiltonian $H = H_0 + H_10*exp(i*k_x) + H_01*exp(-i*k_x)$ and obtains the eigenvalues.

The second method is just diagonize the Hamiltonian in Untitled-1.jpg.

The Hamiltonians derived by the methods above are slightly different, but they have the same graph of the dispersion relation.

The question is, when I change the hopping energy

P.S. The attached figures are my dispersion relation of armchair nanoribbon when φ=pi/4, number of atoms N=16 using those two methods respectively. I don’t know why the plot has been shifted horizontally by using the first method…

Thank you very much

The first one is to diagonize the Hamiltonian $H = H_0 + H_10*exp(i*k_x) + H_01*exp(-i*k_x)$ and obtains the eigenvalues.

The second method is just diagonize the Hamiltonian in Untitled-1.jpg.

The Hamiltonians derived by the methods above are slightly different, but they have the same graph of the dispersion relation.

The question is, when I change the hopping energy

*t*to t*exp(i*φ), here φ is any value you like, the graph plotted by the first method has been shifted horizontally while the graph of the second method remains the same. This is why it confuses me, I don’t know what’s wrong in here…P.S. The attached figures are my dispersion relation of armchair nanoribbon when φ=pi/4, number of atoms N=16 using those two methods respectively. I don’t know why the plot has been shifted horizontally by using the first method…

Thank you very much

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