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2D problem of nearly free electron model

  1. May 16, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Find energies of states at ##(\frac{\pi}{a},0)##.
    (b) Find secular equation

    simon_15_4.png


    2. Relevant equations


    3. The attempt at a solution

    Part(a)

    In 1D, the secular equation for energy is:
    [tex]E = \epsilon_0 \pm \left| V(x,y) \right|[/tex]

    When represented in complex notation, the potential becomes
    [tex]V(x,y) = V_{10} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} + e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right] + V_{11} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} \right] \left[ e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right][/tex]

    [tex]E = \epsilon_0 \pm \sqrt{V_{10}^2 + V_{11}^2 } [/tex]

    Part(b)
    I know the central equation is given by
    [tex] \left(\epsilon_0 - E \right) C_{(k)} + \sum\limits_{G} U_G ~ C_{(k-G)} = 0 [/tex]

    How do I find the 4x4 matrix?
     
  2. jcsd
  3. May 21, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. May 23, 2015 #3
    bumpp
     
  5. May 31, 2015 #4
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