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Unformatted text preview: nm wide near x = 0 where the lifetime is 0.1 nsec. Find the steady state excess minority carrier concentration and QFL’s vs. position. You may assume that the sample extends to x = +∞ . HINT: treat the thin layer at the surface as a boundary condition – do not try to resolve Δn ( x ) inside this thin layer. Approach the problem as follows. 7a) Simplify the Minority Carrier Diffusion Equation for this problem. Solution: ∂ Δn
d 2 Δn Δn
Begin with: = Dn
−
+ GL ∂t
dx 2
τn d 2 Δn Δn
Simplify for steady state conditions: 0 = Dn
−
+ GL dx 2
τn The simplified MDE equation is: d 2 Δn Δn
d 2 Δn Δn GL
Dn
−
+ GL = 0 − 2+
= 0 Ln = Dnτ n dx 2 τ n
dx 2
Ln Dn d 2 Δn Δn GL
− 2+
= 0 where Ln = Dnτ n is the minority carrier “diffusion dx 2
Ln Dn
length.” 7b) Specify the initial and boundary conditions, as appropriate for this problem. Solution: Since this is a steady state problem, there is no initial condition. As x → ∞ , we have a uniform semiconductor with a uniform generation rate. In a uniform semiconductor under illumination, Δn = GLτ n , so Δn ( x → ∞ ) = GLτ n ECE 606 9 Spring 2013 Mark Lundstrom 2/24/2013 At the surface, the total number of e h pairs recombining per cm2 per second is Δn ( 0 )
Δx
Δx cm/s is the “front RS =
Δx =
Δn ( 0 ) = S F Δn ( 0 ) cm 2 s 1 where S F =
τS
τS
τS
surface recombination velocity. Δx 10−6
SF =
= −10 = 104 cm/s τ S 10 In steady state, carriers must diffuse to the surface at the same rate that there are recombining there so that the excess minority carrier concentra...
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 Fall '08
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