# Fowler Nordheim Tunnelling - Plz Help

1. Sep 20, 2006

### shreekumarc

Hi,

I wanted to know how the pre-exponential and exponential terms of the standard Fowler Nordheim Tunnelling is calculated to a certain value. So can anyone give the calculations of these two terms?

The FN Expression is given as

J = A.F^2.exp(-B/F)

Where J = Current Density, A = pre exponential term = q*mo/(16*pi*pi*h*mox*phi_b)
B = Exponential Term = (4/3)*[sqrt(2*mox)]*[q*phi_b]^(3/2)/(q*h_bar*F)

with due regards

Shre

2. Sep 20, 2006

### ZapperZ

Staff Emeritus
Er.. I'm not sure what you mean by the calculations of those parameters. If you mean as in how they are derived, that is going to be rather involved, since you have to start right from the transmission current density, and then invoking a series of simplification. The form that you have for the FD equation is valid at very low temperatures only.

If you mean how do we obtain numbers out of them, then this technically do not involve "calculations", but rather values that you need to obtain, such as the work function (your phi). However, more often than not, these values are "simplified" in experiments where if you plot ln(J/F^2) versus 1/F, you can obtain the value of the work function from the slope of your graph over a limited range of field strength.

FD model is quite "un-transparent" over the range of which it is valid. That's one annoying factor of this model, at least to me.

Zz.

3. Sep 20, 2006

### shreekumarc

Thank you so much for your answer. I am quite confused over the term "Current Density" - how can we match analytical values and simulated values when we are uncertain about the area of the conduction material (say Gate for instance)? That is the expression does not involve the area but simulations with EDA tools like Atlas, they need to be fed with the area in order to get the current density.

4. Sep 20, 2006

### ZapperZ

Staff Emeritus
That is the one part of Fowler-Nordheim model that is not part of the theory. You will have to decide what is the area of the field emitter. This is where you have to know probably via experiment (for example, you look at a particular surface via a microscope or an SEM and see if there are any obvious field emitter and estimate the surface area).

There are, I think, a few papers that give you calculations for the field-enhancement value that depends on geometry of the protrusions. However, I am not sure if this is what you are looking for. So maybe at the simplest stage, your conduction surface would be the area that you want if you can estimate that all the field currents come out of the surface. In practice, though, this is can be not the case.

Zz.

5. Sep 21, 2006

### shreekumarc

Okay...the standard expression is in Amps/Cm^2.Right? If Our area is say 10 um^2 - then we multiply the result with 10*1e-08 with the result to get the result in amps/um^2, which is the current density over an area of 10 um^2....is it right or am I acting foolish?

6. Sep 21, 2006

### ZapperZ

Staff Emeritus
That is usually how we calculate the current density. The main assumption here is that you do know the surface area of your emitter, which in many experiment, is nothing more than an estimate.

Zz.

7. Sep 21, 2006

### shreekumarc

Thank you once more Zapper......I am walking on the right path then. I was initially getting confused over the calculation of the pre-exponential and exponential terms because even if we leave the effective mass and the barrier height away, it was coming to someting remotely related to the values that are quoted in papers. I tried calculating by replacing values of plancks constant and electronic charge - but the values I got were shocking. May be units needs to be cared about.

Any light upon that Zapper?

Thanking you in advance once more.