Physics - Quantum Tunneling

The following is a brief overview of Quantum tunneling.

What is it?
Quantum Tunneling is the effect where a particle tunnels through a barrier it could not penetrate in classical physics.

Why does it happen?
Quantum physics tells us that matter exhibits both particle and wave-like behavior. Therefore if we apply this idea to an electron going up against an potential barrier, we can relate its behavior similarly to a more well known event, light traveling through a material such as glass. In this case, the brightness of the light passing through the surface is reduced and some is reflected back away from the observer behind the glass. With our electron, some of its probability function travels through the potential well, and some of it reflected, depending on the width of the energy barrier, which we can relate the opacity of an object.

Brief Overview of Formulation
The time independent Schrodinger equation is as follows

(-ħ²/2m)(d²Ψ(x)/dx²) + U(x)Ψ(x) = EΨ(x)

The full derivation will be done elsewhere but we can use these to get the solutions for traveling waves, which represents the probability function of a particle in motion. This represents our electron moving. Using the WKB approximation and a few other mathematical tricks such as series expansion, we arrive at the following equation for T the transmission coefficient or percentage chance at quantum tunneling.

The important part of this equations are, x1 and x2 which are the classical turning points for the energy barrier. We can therefore see that this equation depends primarily on two characteristics, the difference in energy between the particle/wave represented as E. V(x) which is the potential of the energy barrier given x, and the width of the barrier x where a higher energy difference results in a lower transmission coefficient, and a greater "width" again results in a lower transmission coefficient.

Relation to Electrical Engineering

This is a material science concept and when used in reference to tunneling in electrical components, we don't have to go into the full Feynman path integral method to just get an idea of how it works. We can get a good idea of the effects with the explanation above. Tunneling occurs with barriers the size of around 1-3 nanometers which will be extremely significant when dealing with finfets which promise to move the total size of the transistor to around the 10 nm range. Efforts must be made to maintain detection of the on and off state due to thermal noise and summation of inductance effects of nearby inductance effects. This means design rules must be made with greater spacing margins than might be otherwise done for normal mosfet design. 

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