- Fowler-Nordheim equation
The Fowler-Nordheim equation in
solid state physics relates current, work andelectric field strength to determinefield emission . It has two parts: an equation for field emittedcurrent density , and the equation for total current. It is named afterRalph H. Fowler and Lothar W. Nordheim.The
current density flowing through a thin oxide layer due toFowler-Nordheim tunneling is a function of theelectric field across the oxide. The electric field is the voltage divided by the distance. This article describes how quickly current increases with voltage.:"V" = voltage (Volts):"t" = thickness of oxide (meters):"E" = "V"/"t" electric field (Volts per meter):"I" = current (Amperes):"A" = area of oxide (square meters):"J" = "I"/"A":"J" = current density (Amperes per square meter):"K"1 is a constant described in the reference:"K"2 is a second constant, also described in the reference
For the Fowler-Nordheim tunneling current density :
J=K_1 cdot E^2 expleft(-frac{K_2}{E} ight)
The point is that the current increases with the voltage squared multiplied by an exponential increase with inverse voltage. While the second factor, "E"², obviously increases rapidly with voltage, the third factor, the exponential, deserves another sentence. For people who are not familiar with exponentials of negative inverses, the following sentences are helpful.
Assume, temporarily, that "K"2 is normalized to be 1.
The factor e-1/"E" increases with "E". If "E" is near zero, the exponent is large, and exp(-large) is near zero.
If "E" is large, 1/"E" is small, and almost zero: exp(0) = 1
Therefore, exp(-1/"E") gets larger as "E" gets larger, since one is greater than zero. Ultimately, this third factor will maintain a value between zero and one.
Conclusion
"J " increases by two voltage-related factors that are multiplied: (exponentially with voltage) x (squared with voltage)
External links
* [http://www.fsr.ac.ma/MJCM/vol3-art10.pdf]
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