P-N junction formulas

Holds at thermal equilibrium with complete ionization and non-degenerate doping ($N_A,N_D$ well below the effective densities of states). $N_A$ and $N_D$ are the net acceptor and donor concentrations on the two sides. $V_{bi}$ is positive: the n-side sits $V_{bi}$ above the p-side. It cannot be read with a voltmeter across the diode, because the contact potentials at the metal contacts cancel it.

Equilibrium majority and minority concentrations on the two sides, with $n_{n0}\approx N_D$ and $p_{p0}\approx N_A$. This is the same result as above, written with the mass-action law.

For an abrupt (step) junction under the depletion approximation. It holds in reverse bias and in forward bias only while $V_A$ stays well below $V_{bi}$; the approximation fails as $V_A\to V_{bi}$. In reverse bias $V_A=-V_R$, so the bracket becomes $V_{bi}+V_R$ and $W$ grows. ${ϵ}_s={ϵ}_r{ϵ}_0$; for Si, ${ϵ}_r\approx 11.7$, giving ${ϵ}_s\approx 1.04\times 10^{-12}\text{F/cm}$.

Charge neutrality: the ionized charge on the two sides is equal and opposite. The more lightly doped side always holds most of $W$.

The peak is at the metallurgical junction ($x=0$). The field falls linearly to zero at each depletion edge, so the profile is a triangle whose area equals $V_{bi}-V_A$.

For a doping profile $N_D-N_A=ax$ that changes linearly through the junction, where $a$ is the grading constant in ${\text{cm}}^{-4}$. Here $W\propto (V_{bi}-V_A{)}^{1/3}$, compared with the square-root dependence of the step junction.

$A$ is the junction cross-sectional area and $W$ is the depletion width at the bias point. This holds for any doping profile as a small-signal capacitance.

$C_{j0}$ is the capacitance at zero bias. The grading exponent is $m=1/2$ for an abrupt junction and $m=1/3$ for a linearly graded one. In reverse bias $V_A=-V_R$, so the bracket is $1+V_R/V_{bi}$ and $C_j<C_{j0}$. The formula applies in reverse bias and at small forward bias only; it wrongly diverges as $V_A\to V_{bi}$.

$N_B$ is the doping of the lightly doped side. A plot of $1/C_j^2$ against $V_R$ is a straight line: its slope gives $N_B$, and its intercept on the voltage axis is at $V_R=-V_{bi}$.

Minority-carrier concentrations at the n-side and p-side depletion edges. Valid for low-level injection, meaning the injected minority density stays well below the majority density.

$I$ flows from p to n through the diode. $V_A>0$ is forward bias and $V_A<0$ is reverse bias. When $|V_A|\gg V_T$ in reverse bias, $I\to -I_0$. Assumes low-level injection and no generation or recombination inside the depletion region.

$\eta \approx 1$ when diffusion current dominates. $\eta$ approaches 2 when recombination in the depletion region dominates, which is typical of Si at low forward current. Use $\eta =1$ unless the question gives another value.

$D_p$ and $D_n$ are the minority-carrier diffusion coefficients, with $D=\mu V_T$ (Einstein relation). $L_p=\sqrt{D_p{\tau }_p}$ and $L_n=\sqrt{D_n{\tau }_n}$ are the diffusion lengths. This form holds when each neutral region is much longer than its diffusion length. In a short-base diode, replace $L_p$ and $L_n$ by the neutral-region widths.

$p_{n0}$ is the equilibrium hole concentration on the n-side and $n_{p0}$ the equilibrium electron concentration on the p-side. From the mass-action law, for non-degenerate doping.

$I$ is the DC forward current at the bias point, with $I\gg I_0$. For example, $r_d\approx 25.9\Omega$ at 1 mA with $\eta =1$ at 300 K.

Charge-control result for a one-sided long-base diode in forward bias ($I\gg I_0$). The stored charge is $Q=\tau I$, where $\tau$ is the minority-carrier lifetime on the lightly doped (injected) side. In a short-base diode, replace $\tau$ by the transit time $W_n^2/(2D_p)$, where $W_n$ is the neutral n-region width. A full small-signal analysis of a long-base diode at low frequency gives half this value, $\tau I/(2\eta V_T)$; use whichever form the question specifies.

$N_B$ is the doping of the lightly doped side. $E_{crit}$ is the critical field for avalanche, a few $\times 10^5$ V/cm in Si, and it changes only weakly with doping. $V_{bi}$ is neglected next to $V_{BR}$. So $V_{BR}\propto 1/N_B$: lighter doping gives a higher breakdown voltage.

$I_Z$ is the reverse current, taken positive, with $I_Z\ge I_{Z,min}$. $V_{Z0}$ is the knee voltage and $r_z$ the dynamic resistance. An ideal Zener has $r_z=0$.

Supply $V_S$ feeds the Zener in parallel with the load through a series resistor $R$. $R_{max}$ keeps the Zener in regulation in the worst case (lowest supply, highest load current). $R_{min}$ keeps the Zener current within its rating in the worst case (highest supply, lowest load current). Assumes $r_z=0$.

${\varphi }_M$ is the metal work function and $\chi$ the semiconductor's electron affinity. ${\varphi }_{Bn}$ is the barrier to electrons on n-type material and ${\varphi }_{Bp}$ the barrier to holes on p-type material. On n-type, the built-in potential of the contact is ${\varphi }_{Bn}-(E_C-E_F)$, in volts.

$A$ is the contact area, $A^{\ast \ast }$ the effective Richardson constant, and ${\varphi }_B$ the barrier height in volts (numerically equal to its value in eV). The prefactor $A{A}^{\ast \ast }{T}^2{e}^{-{\varphi }_B/V_T}$ is the saturation current, which replaces the diffusion-based $I_0$ of a p–n junction. For typical barrier heights it is several orders of magnitude larger than a comparable p–n junction's $I_0$.