Question

I have a metal-insulator-metal interface, the potential barrier by the insulator is 3ev, this is a capacitor, for reliable storage at 20c we need a tunelling coefficient of no more10^-30, whats the minimum length of the potential barrier.

Answer 1

That's 20 degrees Celsius?

Answer 2

it is sir

Answer 3

I feel as though this is considerably more complicated that your last question, so I'll ask how precisely you need this. If you calculate the rms velocity of electrons at that temperature, you obtain an average kinetic energy of ~3.8 eV. Would you like to make the assumption that all of the electrons have this energy and then calculate the necessary width of your barrier, or would you like to go into much more detail and consider the fermi-dirac energy distribution and density of state functions and all of that mess?

Answer 4

Well I used the T\[T=16*(E/V _{0})*(1-E/V_{0})e^-2kl\] and just solved for l, however according to my professor this method fails here, since its from a previous test and i got points deducted, the process wouldnt need fermi-dirac stadistics since this subject was seen before fermi-dirac distributions and all that

Answer 5

The transmission coefficient should be
\[T = \frac{1}{1+\frac{V_0^2 \sinh^2(kL)}{4E(V_0)}} \]
so \[\frac{V_0^2\sin^2(kL)}{4E(V_0-E)} = \frac{1}{T}-1\]
And with some algebra,
\[L =\frac{1}{k} \sinh^{-1}\left(\frac{4E(V_0-E)}{V_0^2}\left[\frac{1}{T} - 1\right] \right) = 9.94 \space \text{nm}\]
when you plug everything in. I recalculated and assumed that E is approximately equal to kT, or 2.53 eV, E = 3 eV, T= 10^-30.

At least, this is how my quantum mechanics classes treated rectangular potential barriers...

Answer 6

oops, make that a hyperbolic sine in the second line.. I'm sorry, it's 3:30 here, so I'm going to sleep but good luck on your problem.

Answer 7

yeah they hyperbolic sine is used there, i think the assumption of getting E=kt is what i ignored!!... thanks