Nv N Exp Qv Kt

Diode Constabulary Graph, shows relationship of voltage and current of an ideal diode

The Shockley diode equation or the diode police, named afterwards transistor co-inventor William Shockley of Bell Telephone Laboratories, gives the I–V (current-voltage) characteristic of an idealized diode in either frontwards or reverse bias (applied voltage):

${\displaystyle I=I_{\mathrm {S} }\left(due east^{\frac {V_{\text{D}}}{nV_{\text{T}}}}-ane\correct)}$

where

I is the diode current,

I _Southward is the contrary bias saturation current (or scale current),

V _D is the voltage across the diode,

V _T is the thermal voltage kT/q (Boltzmann abiding times temperature divided by electron charge), and

n is the ideality factor, also known every bit the quality factor or sometimes emission coefficient.

The equation is called the Shockley platonic diode equation when n, the ideality factor, is gear up equal to 1. The ideality cistron north typically varies from 1 to 2 (though tin can in some cases be higher), depending on the fabrication procedure and semiconductor material and is set equal to 1 for the instance of an "platonic" diode (thus the northward is sometimes omitted). The ideality factor was added to account for imperfect junctions every bit observed in real transistors. The factor mainly accounts for carrier recombination as the charge carriers cross the depletion region.

The thermal voltage Five _T is approximately 25.852mV at 300 K (27 °C; 80 °F). At an arbitrary temperature, information technology is a known constant defined by:

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

where m is the Boltzmann constant, T is the absolute temperature of the p–n junction, and q is the magnitude of charge of an electron (the elementary charge).

The reverse saturation electric current, I _S, is not constant for a given device, merely varies with temperature; unremarkably more significantly than V _T, so that V _D typically decreases as T increases.

The Shockley diode equation doesn't depict the "leveling off" of the I–V curve at high forrad bias due to internal resistance. This can exist 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 abiding (negative) reverse current value of −I_S . The reverse breakdown region is non modeled by the Shockley diode equation.

For fifty-fifty rather small forwards bias voltages the exponential is very large, since the thermal voltage is very small in comparing. The subtracted '1' in the diode equation is then negligible and the forrard diode electric current can be approximated past

${\displaystyle I=I_{\text{Southward}}e^{\frac {V_{\text{D}}}{nV_{\text{T}}}}}$

The apply of the diode equation in excursion problems is illustrated in the article on diode modeling.

Derivation [edit]

Shockley derives an equation for the voltage beyond a p-n junction in a long article published in 1949.^[1] Subsequently he gives a corresponding equation for current as a office of voltage under boosted assumptions, which is the equation nosotros phone call the Shockley ideal diode equation.^[2] 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 tin exist 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 betwixt 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 concluding.

He shows that the first and the third of these tin be expressed as a resistance times the electric current, R₁I. Every bit for the 2nd, the divergence between the quasi-Fermi levels at the junction, he says that we tin estimate the current flowing through the diode from this deviation. He points out that the current at the p last is all holes, whereas at the due north final information technology is all electrons, and the sum of these two is the abiding total current. So the total current is equal to the decrease in pigsty 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 charge per unit 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 ${\displaystyle \exp((\phi _{p}-\phi _{due north})/V_{\text{T}})}$ times the rate of generation. We then assume that well-nigh of the backlog recombination (or subtract in hole current) takes place in a layer going by i pigsty diffusion length (L_p ) into the n textile and one electron diffusion length (50_n ) into the p material, and that the difference between the quasi-Fermi levels is constant in this layer at V_J . Then nosotros notice that the total current, or the drop in hole current, is

${\displaystyle I=I_{s}\left[eastward^{\frac {V_{J}}{V_{\text{T}}}}-1\right]}$

where

${\displaystyle I_{s}=gq\left(L_{p}+L_{n}\right)}$

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

${\displaystyle V_{J}=V_{\text{T}}\ln \left(i+{\frac {I}{I_{south}}}\right)}$

and the total voltage drop is so

${\displaystyle V=IR_{1}+V_{\text{T}}\ln \left(1+{\frac {I}{I_{s}}}\correct).}$

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

The pocket-sized electric current that flows under high contrary bias is then the issue of thermal generation of electron-hole pairs in the layer. The electrons so menses to the northward terminal and the holes to the p concluding. The concentrations of electrons and holes in the layer is so minor that recombination there is negligible.

In 1950, Shockley and coworkers published a curt commodity describing a germanium diode that closely followed the ideal equation.^[3]

In 1954, Bill Pfann and W. van Roosbroek (who were also of Bell Phone Laboratories) reported that while Shockley's equation was applicable to certain germanium junctions, for many silicon junctions the current (nether observable frontwards bias) was proportional to ${\displaystyle e^{V_{J}/AV_{\text{T}}},}$ with A having a value as high as 2 or 3.^[4] This is the "ideality gene" chosen northward above.

In 1981, Alexis de Vos and Herman Pauwels showed that a more than careful analysis of the quantum mechanics of a junction, under certain assumptions, gives a current versus voltage characteristic of the form

${\displaystyle I(Five)=-qA\left[F_{i}-2F_{o}(V)\correct]}$

in which A is the cross-sectional surface area of the junction and F_i is the number of in-coming photons per unit of measurement surface area, per unit time, with energy over the ring-gap free energy, and F_o (5) is out-going photons, given by^[5]

${\displaystyle F_{o}(V)=\int _{\nu _{1000}}^{\infty }{\frac {1}{\exp \left({\frac {h\nu -qV}{kT_{c}}}\right)-one}}{\frac {two\pi \nu ^{two}}{c^{2}}}d\nu .}$

Where the lower limit is described subsequently. Although this analysis was done for photovoltaic cells under illumination, information technology applies as well when the illumination is simply groundwork thermal radiations. It gives a more than rigorous course of expression for ideal diodes in full general, except that it assumes that the jail cell is thick plenty that it tin can produce this flux of photons. When the illumination is just background thermal radiation, the characteristic is

${\displaystyle I(V)=2q\left[F_{o}(V)-F_{o}(0)\right]}$

Note that, in contrast to the Shockley police, the current goes to infinity as the voltage goes to the gap voltage hν_yard/q. This of form 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 I_s using a recent model ^[six] for 2D materials based Schottky diode.

References [edit]

1. ^ William Shockley (Jul 1949). "The Theory of p-n Junctions in Semiconductors and p-north Junction Transistors". The Bell Arrangement Technical Journal. 28 (iii): 435–489. doi:10.1002/j.1538-7305.1949.tb03645.x. . Equation 3.13 on page 454.
2. ^ Ibid. p. 456.
3. ^ F.S. Goucher; et al. (Dec 1950). "Theory and Experiment for a Germanium p-due north Junction". Physical Review. 81. doi:10.1103/PhysRev.81.637.2.
4. ^ W. G. Pfann; West. van Roosbroek (November 1954). "Radioactive and Photoelectric p‐n Junction Power Sources". Journal of Applied Physics. 25 (11): 1422–1434. Bibcode:1954JAP....25.1422P. doi:x.1063/1.1721579.
5. ^ A. De Vos and H. Pauwels (1981). "On the Thermodynamic Limit of Photovoltaic Free energy Conversion". Appl. Phys. 25 (2): 119–125. Bibcode:1981ApPhy..25..119D. doi:ten.1007/BF00901283. S2CID 119693148. . Appendix.
6. ^ Y. Due south. Ang, H. Y. Yang and 50. M. Ang (August 2018). "Universal scaling in nanoscale lateral Schottky heterostructures". Phys. Rev. Lett. 121: 056802.

Nv N Exp Qv Kt,

Source: https://en.wikipedia.org/wiki/Shockley_diode_equation

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