Hi everybody,(adsbygoogle = window.adsbygoogle || []).push({});

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) [tex]J_{FN}=\alpha F^2 \exp{\frac{-\beta}{F}}[/tex]

where [tex]\alpha[/tex] and [tex]\beta[/tex] are the so-called pre-exponential and exponential Fowler-Nordheim parameters, and F the electrical field across the tunnel oxide.

[tex]\alpha[/tex] and [tex]\beta[/tex] depend on the potential barrier height [tex]\Phi_{0}[/tex] and the ratio of effectives masses (in the oxide conduction band and in the silicon conduction band) in the following way:

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

and

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

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 [tex]J_{FN}[/tex] 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) [tex]J_{FN}=\frac{q}{m} \int_{0}^{E_{m}} n(E) f(E) T(E) dE[/tex]

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) [tex]n(E)=\frac{2\pi m^*}{h^3}[/tex]

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 [tex]q\phi_{0}-E[/tex] and electric field F) can be derived easily in solving the steady-state one-dimensional Schrödinger equation, giving:

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

From, this, the question is:

How does one go from:

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

to (E.1)?

THANKS.

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# Derivation of Fowler-Nordheim current density

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