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:

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

where

${\displaystyle I_{\text{D}}}$ is the diode current,

${\displaystyle I_{\text{S}}}$ is the reverse-bias saturation current (or scale current),

${\displaystyle V_{\text{D}}}$ is the voltage across the diode,

${\displaystyle V_{\text{T}}}$ is the thermal voltage, and

${\displaystyle n}$ is the ideality factor, also known as the quality factor, emission coefficient, or the material constant.

The equation is called the Shockley ideal diode equation when the ideality factor ${\displaystyle n}$ equals 1, thus ${\displaystyle n}$ is sometimes omitted. 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.

The thermal voltage ${\displaystyle V_{\text{T}}}$ is defined as:

${\displaystyle V_{\text{T}}={\frac {kT}{q}},}$

where

${\displaystyle k}$ is the Boltzmann constant,

${\displaystyle T}$ is the absolute temperature of the p–n junction, and

${\displaystyle q}$ is the elementary charge (the magnitude of an electron's charge).

For example, it is approximately 25.852 mV at 300 K (27 °C; 80 °F).

The reverse saturation current ${\displaystyle I_{\text{S}}}$ is not constant for a given device, but varies with temperature; usually more significantly than ${\displaystyle V_{\text{T}}}$, so that ${\displaystyle V_{\text{D}}}$ typically decreases as ${\displaystyle T}$ increases.

Under reverse bias, the diode equation's exponential term is near 0, so the current is near the somewhat constant ${\displaystyle -I_{\text{S}}}$ reverse current value (roughly a picoampere for silicon diodes or a microampere for germanium diodes,^[1] although this is obviously a function of size).

For moderate forward bias voltages the exponential becomes much larger than 1, since the thermal voltage is very small in comparison. The ${\displaystyle -1}$ in the diode equation is then negligible, so the forward diode current will approximate

${\displaystyle I_{\text{S}}e^{\frac {V_{\text{D}}}{nV_{\text{T}}}}.}$

The use of the diode equation in circuit problems is illustrated in the article on diode modeling.