Week7HW Solutions

4 1013 cm 3 4 dx x0 l dn l 78 5 10 ece 606 16

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Unformatted text preview: 2 Δn Δn Begin with: = Dn − + GL ∂t dx 2 τn d 2 Δn Δn Simplify for steady- state: 0 = Dn − + GL dx 2 τn Let’s treat the generation in a thin surface layer as a boundary condition, GL = 0 The simplified MDE equation is: d 2 Δn Δn d 2 Δn Δn d 2 Δn Δn Ln = Dnτ n Dn − = 0 2 − = 0 − 2 = 0 dx 2 τn dx Dnτ n dx 2 Ln d 2 Δn Δn − 2 = 0 where Ln = Dnτ n is the minority carrier diffusion length. dx 2 Ln 8b) 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 expect all of the minority carriers to have recombined, so: Δn ( x → ∞ ) = 0 At the surface, the total number of e- h pairs generation per cm2 per second is GS = GL Δx = 102410−6 = 1018 cm -2s-1 . In steady- state, these must diffuse away at the same rate that they are generated, so ECE- 606 13 Spring 2013 Mark Lundstrom − Dn 2/24/2013 d Δn = GS dx x =0 8c) Solve the problem. Solution: d 2 Δn Δn − 2 = 0 solutions is Δn ( x ) = Ae− x / Ln + Be+ x / Ln dx 2 Ln To satisfy the first boundary condition in 8b): B = 0. Now consider the second: Dn GS d Δn 1018 − Dn →+ A = GS → A = = = 3.6 × 1014 cm -3 −4 dx x=0 Ln ( Dn Ln ) 7.8 27.9 × 10 Δn ( x ) = ( ) GS e− x / Ln = 3.6 × 1014 e− x / Ln ( Dn Ln ) 8d) Provide a sketch of the solution, and explain it in words. Solution: ECE- 606 Electrons are generated at the surface and diffuse into the bulk, so the concentration is high at the surface and approaches zero several diffusion lengths into the bulk. 14 Spring 2013 Mark Lundstrom 2/24/2013 Deep...
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This note was uploaded on 01/15/2014 for the course ECE 606 taught by Professor Staff during the Fall '08 term at Purdue.

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