Shockley equation

The Shockley equation , named after William B. Shockley , describes the current-voltage characteristic of a semiconductor diode .

According to Wagner it reads:

Characteristic curve of a 1N4001 diode (applies to 1N4001 to 1N4007)

${\displaystyle I_{\text{D}}=I_{\text{S}}(T)\,\left(\exp \left({\frac {U_{\text{F}}}{n\,U_{\text{T}}}}\right)-1\right)}$

With

• the current through the diode ${\displaystyle I_{\text{D}}}$
• the temperature-dependent saturation reverse current ( reverse current for short ) ${\displaystyle I_{\text{S}}(T)\approx {10^{-12}\dots 10^{-6}\;\mathrm {A} }}$
• the anode-cathode voltage or forward voltage ${\displaystyle U_{\text{F}}}$
• the emission coefficient ${\displaystyle n\approx 1\dots 2}$
• the temperature stress at 20 ° C ${\displaystyle U_{\text{T}}={\frac {k_{\mathrm {B} }\cdot T}{e}}\approx 25\;\mathrm {mV} }$
  • the absolute temperature ${\displaystyle T}$
  • the Boltzmann constant ${\displaystyle k_{\mathrm {B} }}$
  • the elementary charge . ${\displaystyle e}$

As the temperature rises, so does the current through the diode; Although the value of the exponential function decreases due to increasing temperature voltage, this is overcompensated by the strong increase in reverse current with temperature.

In the forward direction, i.e. for positive voltage , the exponential function increases strongly for values ​​of which are greater than . This gives a good approximation for the Shockley equation : ${\displaystyle U_{\text{F}}}$ ${\displaystyle U_{\text{F}}}$ ${\displaystyle n\ U_{\text{T}}}$

${\displaystyle I_{\text{D}}\approx I_{\text{S}}(T)\,\cdot \exp \left({\frac {U_{\text{F}}}{n\,U_{\text{T}}}}\right)}$

For this approximation deviates by less than 1% from the theoretical value, for by less than 1 ‰. As you can see from the characteristics, the actual voltage is significantly higher. ${\displaystyle U_{\text{F}}>n\cdot 120\,\mathrm {mV} }$ ${\displaystyle U_{\text{F}}>n\cdot 180\,\mathrm {mV} }$

The Shockley equation describes the large-signal behavior , i.e. the physically measurable quantities of a diode. In the small-signal behavior, the equation is approximated by a linear approximation in the vicinity of a selected operating point .

Individual evidence

1. ^ C. Wagner: Theory of Current Rectifiers . In: Phys. Z. band 32 , 1931, pp. 641-645 . (Ref. In: FS Goucher, GL Pearson, M. Sparks, GK Teal, W. Shockley: Theory and Experiment for a Germanium pn Junction . In: Physical Review . Volume 81 , no. 4 , January 15, 1951, p. 637 , doi : 10.1103 / PhysRev.81.637.2 . )
2. ↑ Ralf Kories, Heinz Schmidt-Walter: Pocket book of electrical engineering: Basics and electronics . Harri Deutsch Verlag, 2008, ISBN 978-3-8171-1830-4 , pp. 364 .