= Shockley diode equation
{c}
{title2=$I_D=I_S(e^{V_D/(nV_T)}-1)$}
{wiki}
For a <p-n junction> at fixed <temperature>, the <Shockley diode equation> models its <electric current> as
$$
I_D=I_S\left(\exp\frac{V_D}{nV_T}-1\right),\qquad V_T=\frac{k_BT}{q}.
$$
Here $I_S>0$ is the <diode saturation current>, $n>0$ the <diode ideality factor>, $V_T$ the <thermal voltage>, $k_B$ the <Boltzmann constant>, and $q>0$ the <elementary charge>.<The Junction Diode by W. Marshall Leach, Jr.>[The Junction Diode, §1.2, printed p. 4, equation (1.1) and parameter definitions]{ref}
For an original model example, choose $I_S=1\,\mathrm{nA}$, $n=2$, and $T=300\,\mathrm K$. Using $k_B=1.380649\times10^{-23}\,\mathrm{J/K}$ and $q=1.602176634\times10^{-19}\,\mathrm C$ gives $V_T=25.8520\,\mathrm{mV}$. Then $V_D=0.600\,\mathrm V$ gives $I_D=0.109591\,\mathrm{mA}$, whereas $V_D=0.800\,\mathrm V$ gives $I_D=5.24450\,\mathrm{mA}$. These are predictions of the declared model. There is no universal turn-on <electric potential difference>: solving the <Shockley diode equation> for the value needed at a chosen <electric current> gives
$$
V_D=nV_T\ln\left(1+\frac{I_D}{I_S}\right),\qquad I_D>-I_S.
$$
At $I_D\gg I_S$, increasing $V_D$ by $0.100\,\mathrm V$ multiplies the <electric current> by approximately $e^{0.100/(nV_T)}=6.91772$. At $V_D=-0.300\,\mathrm V$ the same model gives $I_D=-0.996979\,\mathrm{nA}$. Its reverse branch approaches $-I_S$; it cannot describe <diode reverse breakdown>.
A constructed extension adds internal series <electrical resistance> $R_s$ to a junction whose <electric potential difference> is $V_j$. Applying the <Kirchhoff cycle law> gives
$$
V_D=V_j+I_DR_s
=nV_T\ln\left(1+\frac{I_D}{I_S}\right)+I_DR_s.
$$
For $R_s=20\,\Omega$ and the same $I_S,n,T$, sustaining $I_D=5.24450\,\mathrm{mA}$ needs $V_D=0.904890\,\mathrm V$: $0.800\,\mathrm V$ across the junction plus $0.104890\,\mathrm V$ across its internal <electrical resistance>. The <derivative>
$$
\frac{dV_D}{dI_D}=\frac{nV_T}{I_D+I_S}+R_s
$$
shows why the <current-voltage characteristic> becomes nearly a <straight line> when the internal series <electrical resistance> dominates. This is still a fixed-<temperature> model; self-heating can change $I_S$, $n$ and $V_T$. Real <diodes> also have leakage and charge-storage effects omitted here.<The Junction Diode by W. Marshall Leach, Jr.>[The Junction Diode, §§1.3, 1.8 and 1.11: temperature dependence, charge storage and series bulk resistance]{ref}
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