Shockley diode equation

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 doesn't model noise (such as Johnson–Nyquist noise from the internal resistance, or shot noise).

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.^[2] Later he gives a corresponding equation for current as a function of voltage under additional assumptions, which is the equation we call the Shockley ideal diode equation.^[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. 641–645 (1931).

To derive his equation for the voltage, Shockley argues that the total voltage drop can be divided into three parts:

• the drop of the quasi-Fermi level of holes from the level of the applied voltage at the p terminal to its value at the point where doping is neutral (which we may call the junction),
• the difference between the quasi-Fermi level of the holes at the junction and that of the electrons at the junction,
• the drop of the quasi-Fermi level of the electrons from the junction to the n terminal.

He shows that the first and the third of these can be expressed as a resistance times the current: ${\displaystyle I_{\text{D}}R_{1}.}$  As for the second, the difference between the quasi-Fermi levels at the junction, he says that we can estimate the current flowing through the diode from this difference. He points out that the current at the p terminal is all holes, whereas at the n terminal it is all electrons, and the sum of these two is the constant total current. So the total current is equal to the decrease in hole current from one side of the diode to the other. This decrease is due to an excess of recombination of electron-hole pairs over generation of electron-hole pairs. The rate of recombination is equal to the rate of generation when at equilibrium, that is, when the two quasi-Fermi levels are equal. But when the quasi-Fermi levels are not equal, then the recombination rate is ${\displaystyle e^{(\phi _{\text{p}}-\phi _{\text{n}})/V_{\text{T}}}}$  times the rate of generation. We then assume that most of the excess recombination (or decrease in hole current) takes place in a layer going by one hole diffusion length ${\displaystyle L_{\text{p}}}$  into the n material and one electron diffusion length ${\displaystyle L_{\text{n}}}$  into the p material, and that the difference between the quasi-Fermi levels is constant in this layer at ${\displaystyle V_{\text{J}}.}$  Then we find that the total current, or the drop in hole current, is

${\displaystyle I_{\text{D}}=I_{\text{S}}\left(e^{\frac {V_{\text{J}}}{V_{\text{T}}}}-1\right),}$ 

where

${\displaystyle I_{\text{S}}=gq(L_{\text{p}}+L_{\text{n}}),}$ 

and ${\displaystyle g}$  is the generation rate. We can solve for ${\displaystyle V_{\text{J}}}$  in terms of ${\displaystyle I_{\text{D}}}$ :

${\displaystyle V_{\text{J}}=V_{\text{T}}\ln \left(1+{\frac {I_{\text{D}}}{I_{\text{S}}}}\right),}$ 

and the total voltage drop is then

${\displaystyle V=I_{\text{D}}R_{1}+V_{\text{T}}\ln \left(1+{\frac {I_{\text{D}}}{I_{\text{S}}}}\right).}$ 

When we assume that ${\displaystyle R_{1}}$  is small, we obtain ${\displaystyle V=V_{\text{J}}}$  and the Shockley ideal diode equation.

The small current that flows under high reverse bias is then the result of thermal generation of electron–hole pairs in the layer. The electrons then flow to the n terminal, and the holes to the p terminal. 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 ${\displaystyle e^{V_{\text{J}}/AV_{\text{T}}},}$  with A having a value as high as 2 or 3.^[5] This is the ideality factor ${\displaystyle n}$  above.

Feynman gave a derivation using the Brownian ratchet in The Feynman Lectures on Physics I.46.^[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

${\displaystyle I_{\text{D}}(V)=-qA[F_{i}-2F_{o}(V)],}$ 

in which A is the cross-sectional area of the junction, and F_i is the number of incoming photons per unit area, per unit time, with energy over the band-gap energy, and F_o(V) is outgoing photons, given by^[7]

${\displaystyle F_{o}(V)=\int _{\nu _{g}}^{\infty }{\frac {1}{\exp \left({\frac {h\nu -qV}{kT_{c}}}\right)-1}}{\frac {2\pi \nu ^{2}}{c^{2}}}\,d\nu .}$ 

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

${\displaystyle I_{\text{D}}(V)=2q[F_{o}(V)-F_{o}(0)].}$ 

Note that, in contrast to the Shockley law, the current goes to infinity as the voltage goes to the gap voltage hν_g/q. This of course would require an infinite thickness to provide an infinite amount of recombination.

This equation was recently revised to account for the new temperature scaling in the revised current ${\displaystyle I_{\text{S}}}$  using a recent model^[8] for 2D materials based Schottky diode.