2D problem of nearly free electron model

  • #1
unscientific
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Homework Statement



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

simon_15_4.png

Homework Equations

The Attempt at a Solution



Part(a)[/B]
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?
 
  • #3
bumpp
 
  • #4
bumpp
 
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