Basic Semiconductor Equations

Name	Formula	Related Formulae	Variables/Constants
Continuity equation	$\frac{\partial n}{\partial t} = \frac{1}{q} \frac{\partial J _{n}}{\partial x} + G_{n} - R_{n}$ $\frac{\partial p}{\partial t} = - \frac{1}{q} \frac{\partial J _{p}}{\partial x} + G_{p} - R_{p}$	$J_{n} = q μ_{n} n E + q D_{n} \frac{\partial n}{\partial x}$ $J_{p} = q μ_{p} pE - q D_{p} \frac{\partial p}{\partial x}$	$n : e l ec t r o n co n ce n t r a t i o n$ $q : e l e m e n t a r y c ha r g e$ $J_{n} : e l ec t r o n c u r r e n t d e n s i t y$ $G_{n} : e l ec t r o n g e n er a t i o n r a t e$ $R_{n} : e l ec t r o n r eco mbina t i o n r a t e$ $p : h o l e co n ce n t r a t i o n$ $J_{p} : h o l e c u r r e n t d e n s i t y$ $G_{p} : h o l e g e n er a t i o n r a t e$ $R_{p} : h o l e r eco mbina t i o n r a t e$
Drift-diffusion current density	$J_{n} = q μ_{n} n E + q D_{n} \frac{\partial n}{\partial x}$ $J_{p} = q μ_{p} pE - q D_{p} \frac{\partial p}{\partial x}$	$\frac{D _{n}}{μ _{n}} = \frac{k _{B} T}{q}$ $\frac{D _{p}}{μ _{p}} = \frac{k _{B} T}{q}$	$μ_{n} : e l ec t r o n m o bi l i t y$ $E : e l ec t r i c f i e l d$ $D_{n} : e l ec t r o n d i f f u s i o n coe f f i c i e n t$ $k_{B} : B o l t z mann co n s t an t$ $T : ab so l u t e t e m p er a t u r e$ $μ_{p} : h o l e m o bi l i t y$ $D_{p} : h o l e d i f f u s i o n coe f f i c i e n t$
Einstein relation	$\frac{D _{n}}{μ _{n}} = \frac{D _{p}}{μ _{p}} = \frac{k _{B} T}{q}$	$J_{n} = q μ_{n} n E + q D_{n} \frac{\partial n}{\partial x}$ $J_{p} = q μ_{p} pE - q D_{p} \frac{\partial p}{\partial x}$	$\frac{k _{B} T}{q} : t h er ma l v o l t a g e (\approx 0.0259 V a t 300 K)$ $k_{B} : B o l t z mann co n s t an t$
Poisson’s equation	$\frac{\partial ^{2} ψ}{\partial x ^{2}} = - \frac{ρ}{ε _{s}} = - \frac{q}{ε _{s}} (p - n + N_{D}^{+} - N_{A}^{-})$	$E = - \frac{\partial ψ}{\partial x}$	$ψ : e l ec t r os t a t i c p o t e n t ia l$ $ρ : s p a ce c ha r g e d e n s i t y$ $ε_{s} : se mi co n d u c t or p er mi tt i v i t y$ $N_{D}^{+} : i o ni z e d d o n or co n ce n t r a t i o n$ $N_{A}^{-} : i o ni z e d a cce pt or co n ce n t r a t i o n$
Law of mass action (equilibrium)	$n_{0} p_{0} = n_{i}^{2}$	$n_{i} = N_{C} N_{V} exp (- \frac{E _{g}}{2 k _{B} T})$	$n_{0} : e q u i l ib r i u m e l ec t r o n co n ce n t r a t i o n$ $p_{0} : e q u i l ib r i u m h o l e co n ce n t r a t i o n$ $n_{i} : in t r in s i c c a r r i er co n ce n t r a t i o n$ $k_{B} : B o l t z mann co n s t an t$
Intrinsic carrier concentration	$n_{i} = N_{C} N_{V} exp (- \frac{E _{g}}{2 k _{B} T})$	$N_{C} = 2 (\frac{2 π m _{n}^{} k _{B} T}{h ^{2}})^{3/2}$ $N_{V} = 2 (\frac{2 π m _{p}^{} k _{B} T}{h ^{2}})^{3/2}$	$N_{C} : e f f ec t i v e d e n s i t y o f s t a t es in co n d u c t i o n ban d$ $N_{V} : e f f ec t i v e d e n s i t y o f s t a t es in v a l e n ce ban d$ $E_{g} : ban d g a p e n er g y$ $h : P l an c k^{'} s co n s t an t$ $m_{n}^{} : e l ec t r o n e f f ec t i v e ma ss$ $m_{p}^{} : h o l e e f f ec t i v e ma ss$ $k_{B} : B o l t z mann co n s t an t$
Equilibrium carrier concentration (Boltzmann approx.)	$n_{0} = N_{C} exp (- \frac{E _{C} - E _{F}}{k _{B} T})$ $p_{0} = N_{V} exp (- \frac{E _{F} - E _{V}}{k _{B} T})$	$p_{0} = N_{V} exp (- \frac{E _{F} - E _{V}}{k _{B} T})$ $n_{0} = N_{C} exp (- \frac{E _{C} - E _{F}}{k _{B} T})$	$E_{C} : co n d u c t i o n ban d e d g e e n er g y$ $E_{F} : F er mi l e v e l$ $E_{V} : v a l e n ce ban d e d g e e n er g y$ $k_{B} : B o l t z mann co n s t an t$

Important Questions

Ques: What does the continuity equation physically represent?

The continuity equation is a statement of carrier conservation within a differential volume of semiconductor. It states that the net rate of change of carrier concentration at a point equals the net current flowing into that point, adjusted for carriers generated (by light, thermal excitation, impact ionization, etc.) and lost to recombination. It is not an independent physical law on its own but a bookkeeping identity that, combined with the drift-diffusion current relations and Poisson’s equation, forms the closed system of equations (the “basic semiconductor equations”) used to solve for carrier transport in a device. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007; D.A. Neamen, Semiconductor Physics and Devices, 4th ed., McGraw-Hill, 2012.

Ques: Why does the law of mass action only hold in thermal equilibrium?

$n_{0} p_{0} = n_{i}^{2}$ follows directly from the Boltzmann-approximated expressions for $n_{0}$ and $p_{0}$ , in which the electron and hole quasi-Fermi levels are equal (a single Fermi level $E_{F}$ ). Under non-equilibrium conditions such as current flow or optical excitation, electrons and holes are described by separate quasi-Fermi levels $E_{F n}$ and $E_{F p}$ , so the product $n p$ exceeds $n_{i}^{2}$ and the simple relation no longer applies; the more general form uses $n p = n_{i}^{2} exp (\frac{E _{F n} - E _{F p}}{k _{B} T})$ . Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

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Basic Semiconductor Equations

Important Questions

Ques: What does the continuity equation physically represent?

Ques: Why does the law of mass action only hold in thermal equilibrium?

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