The ''Shockley diode equation'' or the ''diode law'', named after
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
co-inventor
William Shockley
William Bradford Shockley Jr. (February 13, 1910 – August 12, 1989) was an American physicist and inventor. He was the manager of a research group at Bell Labs that included John Bardeen and Walter Brattain. The three scientists were jointl ...
of
Bell Telephone Laboratories
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mult ...
, gives the I–V (current-voltage) characteristic of an idealized
diode
A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other.
A diod ...
in either forward or
reverse bias
Reverse or reversing may refer to:
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* ''Reverse'' (Eldritch album), 2001
* ''Reverse'' (2009 film), a Polish comedy-drama film
* ''Reverse'' (2019 film), an Iranian crime-drama film
* ''Reverse'' (Morandi album), 2005
* ''Reverse'' ...
(applied voltage):
:
where
:''I'' is the diode current,
:''I''
S is the reverse bias
saturation current (or scale current),
:''V''
D is the voltage across the diode,
:''V''
T is the
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
''kT''/''q'' (
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
times temperature divided by electron charge), and
:''n'' is the ''ideality factor'', also known as the ''quality factor'' or sometimes ''emission coefficient''.
The equation is called the ''Shockley ideal diode equation'' when ''n'', the ideality factor, is set equal to 1. The ideality factor ''n'' typically varies from 1 to 2 (though can in some cases be higher), depending on the fabrication process and semiconductor material and is set equal to 1 for the case of an "ideal" diode (thus the n is sometimes omitted). The ideality factor was added to account for imperfect junctions as observed in real transistors. The factor mainly accounts for
carrier recombination
In the solid-state physics of semiconductors, carrier generation and carrier recombination are processes by which mobile charge carriers ( electrons and electron holes) are created and eliminated. Carrier generation and recombination processes a ...
as the charge carriers cross the
depletion region.
The
thermal voltage
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
''V''
T is approximately 25.852mV at . At an arbitrary temperature, it is a known constant defined by:
:
where ''k'' is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, ''T'' is the absolute temperature of the p–n junction, and ''q'' is the magnitude of charge of an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
(the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
).
The reverse saturation current, ''I''
S, is not constant for a given device, but varies with temperature; usually more significantly than ''V''
T, so that ''V''
D typically decreases as ''T'' increases.
The Shockley diode equation doesn't describe the "leveling off" of the I–V curve at high forward bias due to internal resistance. This can be taken into account by adding a resistance in series.
Under ''reverse bias'' (when the n side is put at a more positive voltage than the p side) the exponential term in the diode equation is near zero and the current is near a constant (negative) reverse current value of −''I
S''. The reverse ''breakdown region'' is not modeled by the Shockley diode equation.
For even rather small ''forward bias'' voltages the exponential is very large, since the thermal voltage is very small in comparison. The subtracted '1' in the diode equation is then negligible and the forward diode current can be approximated by
:
The use of the diode equation in circuit problems is illustrated in the article on
diode modeling.
Derivation
Shockley derives an equation for the voltage across a
p-n junction in a long article published in 1949. 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. He calls it "a theoretical rectification formula giving the maximum rectification", with a footnote referencing a paper by
Carl Wagner
Carl Wilhelm Wagner (May 25, 1901 – December 10, 1977) was a German Physical chemist. He is best known for his pioneering work on Solid-state chemistry, where his work on oxidation rate theory, counter diffusion of ions and defect chemistry ...
, ''
Physikalische Zeitschrift
''Physikalische Zeitschrift'' (English: ''Physical Journal'') was a German scientific journal of physics published from 1899 to 1945 by S. Hirzel Verlag. In 1924, it merged with ''Jahrbuch der Radioaktivität und Elektronik''. From 1944 onwards, ...
'' 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
A quasi Fermi level (also called imref, which is "fermi" spelled backwards) is a term used in quantum mechanics and especially in solid state physics for the Fermi level (chemical potential of electrons) that describes the population of electrons ...
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, ''R
1I''. 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
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 (''L
p'') into the n material and one electron diffusion length (''L
n'') into the p material, and that the difference between the quasi-Fermi levels is constant in this layer at ''V
J''. Then we find that the total current, or the drop in hole current, is
: