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Energy of First Tunnelling Resonance

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data
    COnsider electrons tunnelling through a 10eV barrier over a distance 0.001mm.
    Find E[itex]_{1}[/itex]. (The energy of the first tunnelling resonance.)


    2. Relevant equations
    I have an equation for the tunnelling probability:
    T(E)=exp[itex]\left[-\sqrt{\frac{8m}{h^{2}}}\int(V(x)-E)^{1/2}dx\right][/itex]
    where the integral would go from [itex]x_{1}[/itex] to 0.

    3. The attempt at a solution
    I would assume I use the above equation, where V(x) is given as the 10eV, m would be the mass of an electron (in eV?) and [itex]x_{1}[/itex] would be the distance of 0.001mm
    He said easiest way would be "numerically (i.e. using a spreadsheet, etc.)"
    So would I just set the tunnelling probability to 1 and solve for E? The first value being the first resonance energy?
     
  2. jcsd
  3. Mar 27, 2012 #2
    This is an exercise I have made some time ago; if I remember correctly, I think that the formula you have used is valid only when E<V; I think also that you should analyze what's going on for E>V and that in that case you can have resonances. I hope that what I remember is right :)
    Best,
    f.
     
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