Temp

Problem 1

Consider a tight-binding Hamiltonian of 1D nanowire:

${\hat {H}}=\sum _{m}\cos(2\pi m\alpha )|m\rangle \langle m|+t\sum _{m}{\frac {1}{2}}\left(|m\rangle \langle m+1|+|m\rangle \langle m-1|\right)$,

where $\alpha=5/3$. The integer $m$ should be thought of as indexing sites along the chain of atoms. The ket $| m \rangle$ locates an electron on atom $m$ (e.g., $\langle x | m \rangle = \psi(x-m)$ is the wave function, or "orbital", which decays fast away from the position of an atom m).

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(a) What is the periodicity of the Hamiltonian?

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(b) Use Bloch theorem to reduce the eigenvalue problem of an infinite matrix $\hat{H}$ (obtained by representing the Hamiltonian in the basis of orbitals $|m\rangle$) to the solution of a small {\em finite} matrix equation [note that the size of this matrix will be equal to the periodicity of the Hamiltonian found in (a)].

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(c) Compute and plot the bands as a function of Bloch wave vector $k$ throughout the first Brillouin zone (this task will have to be carried our numerically).}