# Energy of First Tunnelling Resonance

1. Mar 25, 2012

### Mr LoganC

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

2. Relevant equations
I have an equation for the tunnelling probability:
T(E)=exp$\left[-\sqrt{\frac{8m}{h^{2}}}\int(V(x)-E)^{1/2}dx\right]$
where the integral would go from $x_{1}$ 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 $x_{1}$ 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. Mar 27, 2012

### francesco85

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.