Shockley diode equation
The Shockley diode equation, or the diode law, named after transistor co-inventor William Shockley of Bell Labs, models the exponential current–voltage (I–V) relationship of semiconductor diodes in moderate constant current forward bias or reverse bias: where The equation is called the Shockley ideal diode equation when the ideality factor
The ideality factor typically varies from 1 to 2 (though can in some cases be higher), depending on the fabrication process and semiconductor material.
The ideality factor was added to account for imperfect junctions observed in real transistors, mainly due to carrier recombination as charge carriers cross the depletion region.
Under reverse bias, the diode equation's exponential term is near 0, so the current is near the somewhat constant
Internal resistance causes "leveling off" of a real diode's I–V curve at high forward bias.
The Shockley equation doesn't model this, but adding a resistance in series will.
The reverse breakdown region (particularly of interest for Zener diodes) is not modeled by the Shockley equation.
The Shockley equation is a constant current (steady state) relationship, and thus doesn't account for the diode's transient response, which includes the influence of its internal junction and diffusion capacitance and reverse recovery time.
Shockley derives an equation for the voltage across a p-n junction in a long article published in 1949.
[3] He calls it "a theoretical rectification formula giving the maximum rectification", with a footnote referencing a paper by Carl Wagner, Physikalische Zeitschrift 32, pp.
To derive his equation for the voltage, Shockley argues that the total voltage drop can be divided into three parts: He shows that the first and the third of these can be expressed as a resistance times the current:
into the p material, and that the difference between the quasi-Fermi levels is constant in this layer at
The small current that flows under high reverse bias is then the result of thermal generation of electron–hole pairs in the layer.
The concentrations of electrons and holes in the layer is so small that recombination there is negligible.
In 1950, Shockley and coworkers published a short article describing a germanium diode that closely followed the ideal equation.
[4] In 1954, Bill Pfann and W. van Roosbroek (who were also of Bell Telephone Laboratories) reported that while Shockley's equation was applicable to certain germanium junctions, for many silicon junctions the current (under appreciable forward bias) was proportional to
[6] In 1981, Alexis de Vos and Herman Pauwels showed that a more careful analysis of the quantum mechanics of a junction, under certain assumptions, gives a current versus voltage characteristic of the form in which A is the cross-sectional area of the junction, and Fi is the number of incoming photons per unit area, per unit time, with energy over the band-gap energy, and Fo(V) is outgoing photons, given by[7] The factor of 2 multiplying the outgoing flux is needed because photons are emitted from both sides, but the incoming flux is assumed to come from just one side.
Although the analysis was done for photovoltaic cells under illumination, it applies also when the illumination is simply background thermal radiation, provided that a factor of 2 is then used for this incoming flux as well.
The analysis gives a more rigorous expression for ideal diodes in general, except that it assumes that the cell is thick enough that it can produce this flux of photons.
When the illumination is just background thermal radiation, the characteristic is Note that, in contrast to the Shockley law, the current goes to infinity as the voltage goes to the gap voltage hνg/q.
using a recent model[8] for 2D materials based Schottky diode.
