138 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS
If we consider a volume of space in which charge transport and recombination is taking place,
we have the simple equality (see figure 3.26a) As a result of consideration of particle current,
Net Rate of particle flow=Particle flow rate due to current−
Particle loss rate due to recombination+Particle gain due to generation.Let us now collect the various terms in this continuity equation. Ifδnis the excess carrier density
in the region, the recombination rateRin the volumeA·Δxshown in figure 3.26 may be written
approximately as
R=δn
τn·A·Δx (3.9.1)whereτnis the electron recombination time per excess particle due to both the radiative and the
nonradiative components. The particle flow rate into the same volume due to the currentJnis
given by the difference of particle current coming into the region and the particle current leaving
the region, [
Jn(x)
(−e)
−
Jn(x+Δx)
(−e)]
A∼=
1
e∂Jn(x)
∂xΔx·A (3.9.2)IfGis the generation rate per unit volume, the generation rate in the volumeA·ΔxisGAΔx.
The rate of electron build up in the volumeA·Δxis then
A·Δx[
∂n(x, t)
∂t≡
∂δn
∂t=
1
e∂Jn(x)
∂x−
δn
τn]
(3.9.3)
whereδn/τnisU=G−R, the net recombination rate of electrons. We have similar terms for
holes, collecting the various terms we have, for the electrons and holes, the continuity equations
(note the sign difference in the particle current density for electrons and holes)
∂δn
∂t=
1
e∂Jn(x)
∂x−
δn
τn(3.9.4)
∂δp
∂t= −
1
e∂Jp(x)
∂x−
δp
τp(3.9.5)
Using these expressions, the the diffusion currents are
Jn(dif f)=eDn∂δn
∂x(3.9.6)
Jp(dif f)=−eDp∂δp
∂x(3.9.7)
We g e t
∂δn
∂t= Dn∂^2 δn
∂x^2−
δn
τn(3.9.8)
∂δp
∂t= Dp∂^2 δp
∂x^2−
δp
τp