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Tight-binding model

  1. Mar 19, 2010 #1

    I've been reading about the tight-binding model, and I have some questions. Lets say we have the Hamiltonian H for our lattice, and it satisfies

    Hψ = Eψ,

    where ψ is a vector containing the wavefunction for each atom in the lattice. When I solve the above equation e.g. numerically, I get the eigenvectors ψ. In my case the ψ's just contain numbers, but does this mean that the waverfuncion for each atom is constant?
  2. jcsd
  3. Mar 20, 2010 #2
    No, it doesn't mean the wavefunction is a constant.
    Say you have two atoms in the unit cell, and one orbital for each atom. Then your Hamiltonian is 2x2, and your wavefunction is a two component vector. The values of these components indicate the relative contribution of each atom. If [tex]\phi_i(r)[/tex] are your basis orbitals, then your wavefunction is

    [tex]u_k(r) = c_1 \phi_1(r) + c_2 \phi_2(r)[/tex]

    Note that this is the solution for the periodic part of the Bloch wavefunction, so there is a long range exp(ik*r) phase factor.
  4. Mar 20, 2010 #3
    Ok, so the two eigenvectors I get have the above form. Does each eigenvector (and corresponding eigenvalue) corresponding to a single particle state? I.e., eigenvector #1 is for the single particle state at atom #1 and eigenvector #2 is for the single particle state at atom #2?

    I really appreciate this. Thanks.
    Last edited: Mar 20, 2010
  5. Mar 21, 2010 #4
    Yes they refer to single particle states. But the second part is wrong.

    If you have two eigenvectors, they will be of form (a, b) and (-b, a), as required by orthogonality. The first state (a,b) will be a single particle state that has a contribution a from atom 1 and a contribution b from atom 2. You will not get "eigenvector #1 is for the single particle state at atom #1 and eigenvector #2 is for the single particle state at atom #2?" unless b = 0, which will only happen if your Hamiltonian is diagonal. Both eigenvectors will have contributions on both atoms for realistic tight binding models.
  6. Mar 21, 2010 #5
    Thank you. That clarified it for me.
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