# Derivation of Fowler-Nordheim current density

1. Oct 20, 2004

### zeus_the_almighty

Hi everybody,
I study solid-state electronics, more precisely electron Fowler-Nordheim tunneling across the gate oxide of a Silicon-Oxide-Silicon capacitor.
I need help on the physics of electron tunneling across a triangular barrier.
It will be part of my PhD dissertation. I have been through many websites but couldn't find out any FULL, DETAILED, derivation of the so-called Fowler-Nordheim Current density. Looks like nobody knows exactly the math which lies behind!!!

Does anybody know how to derive the well-known formula of the Fowler-Nordheim current density resulting
from a triangular potential barrier, which is:

(E.1) $$J_{FN}=\alpha F^2 \exp{\frac{-\beta}{F}}$$

where $$\alpha$$ and $$\beta$$ are the so-called pre-exponential and exponential Fowler-Nordheim parameters, and F the electrical field across the tunnel oxide.
$$\alpha$$ and $$\beta$$ depend on the potential barrier height $$\Phi_{0}$$ and the ratio of effectives masses (in the oxide conduction band and in the silicon conduction band) in the following way:

(E.2) $$\alpha=\frac{q^3}{8\pi qh\Phi_{0}}\frac{m_{Si}}{m_{ox}}$$

and

(E.3) $$\beta=\frac{8\pi}{3qh}\sqrt{2m_{ox}}(q\Phi_{0})^\frac{3}{2}$$

I know part of the derivation but there are some missing steps.
Here are a few hints for those who may help me:
the current density $$J_{FN}$$ can be expressed as the product of:
i)
the number of electrons per unit area and time arriving at the Silicon/oxide interface,

and

ii)the tunneling probability T(E) for a triangular barrier.

From the above, and some straightforward homogeneity considerations, one finds out:

(E.4) $$J_{FN}=\frac{q}{m} \int_{0}^{E_{m}} n(E) f(E) T(E) dE$$

where n(E) is the density of states per unit energy,
f(E) the Fermi-Dirac function,
Em the highest energy of the electron gas.

Since we consider only cold emission, f(E)=1 .

Moreover, if the electrons are thought of as a free electron gas, the density of states per unit energy does not depend on the energy and is expressed (classicaly) as:

(E.5) $$n(E)=\frac{2\pi m^*}{h^3}$$

In addition to this, the tunneling probability, "seen" by an electron arriving at the Silicon/oxide interface with an energy "E", and resulting from a triangular barrier (whose height is $$q\phi_{0}-E$$ and electric field F) can be derived easily in solving the steady-state one-dimensional Schrödinger equation, giving:

(E.6) $$T(E)=\exp{(-\frac{8\pi}{3qh}\sqrt{2m_{ox}(q\Phi_{0}-E) }\frac{1}{F})}$$.

From, this, the question is:

How does one go from:

(E.7)$$J_{FN}=\frac{q}{m} \int_{0}^{E_{m}} \frac{2\pi m^*}{h^3} T(E) dE$$

to (E.1)?

THANKS.

2. Oct 20, 2004

### ZapperZ

Staff Emeritus
I don't have the paper here with me, but I could have sworn that the current density was derived in the original paper. Have you checked that?

There is a condensed derivation of that here...

Zz.

3. Oct 20, 2004

### zeus_the_almighty

I have already been checking this site.
The derivation described there only gives you
the tunneling probability T(E) but does NOT give the FULL derivation
of the current density $$J_{FN}$$.

The one thing which really puzzles me is:
where does $$F^2$$ come from in the expression of $$J_{FN}$$?

Help!

4. Oct 20, 2004

### ZapperZ

Staff Emeritus
Then you REALLY need to read the original Fowler-Nordheim paper since there are some "standard" symbols associated with most of these variables. F is typically defined as the "effective field", i.e. phi/L, where phi is the potential and "L" is the effective length of the triangular barrier.

If you can't get hold of FN's paper, you may want to check out Kevin Jensen's extension of this work. See

K.L. Jensen, J. Vac. Sci. Tech. B, v.21, p.1528 (2003).

Good luck!

Zz.

5. Oct 20, 2004

### zeus_the_almighty

Thank u Zz.
What are you doing?
Which field of physics?
I am finishing my PhD dissertation entitled:
"Modeling and Study of Tunnel oxide degradation of EEPROM Cells".
I will defend it on December 17th, 2004.
I've got a lot of work.

It's 7p.m. in Marseilles.
Gotta go home Now.

See u later on.

6. Oct 20, 2004

### ZapperZ

Staff Emeritus
My career in physics is well-documented in one of my journal entries on here, so there's no need to occupy more space than necessary to bore everyone else. You're welcome to browse it.

Zz.

7. Feb 9, 2010

### divB

I know I am six years too late ;-)

But I have the proof, I got it from my professor. He told me exactly the same that you can not find the derivation anywhere, it is kind of tricky and he sat a while until he derived it.

So if you should still need it just contact me.

8. Feb 23, 2010

### LydiaAC

Hello divB:
I am interested in the derivation.
LydiaAC

9. Apr 20, 2010

### sifangyou4

Hi diVB and LydiaAC,

Could any one of you send me a copy? Thanks a lot.

sifangyou4

10. Feb 22, 2012

### gjtrash

Hello, i'm studying they Fowler-Nordheim equations now and would love to get the information as well. hopefully every body's still around.

Thanks alot