Week7HWSolutionsV2

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Unformatted text preview: Sp p dx ⎝ ∂ E ⎠ ⎪ dx ⎪ h −∞ ⎩ ⎭ ( ( J px = σ Sp σ Sp 2q = h ( ) ) d Fp q ) dx 2 EV ⎛ ∂f ⎞ λ ( E ) MV ( E ) W ⎜ − 0 ⎟ dE ∫ ⎝ ∂E ⎠ −∞ ( ) ECE- 656 1 Fall 2013 Mark Lundstrom 10/5/13 2) In Lecture 15, we derived the drift- diffusion equation for a 2D n- type semiconductor with parabolic energy bands. Repeat the derivation for a 3D semiconductor with parabolic energy bands. Do not assume Maxwell- Boltzmann statistics. Solution: Begin with: dF q J nx = σ n (i) dx n = N CF 1/ 2 (η F ) 1 ⎛ 2 m*k BT ⎞ NC = ⎜ 4 ⎝ π 2 ⎟ ⎠ η F = ( Fn − EC ) k BT 3/ 2 Now find the gradient of the electrochemical potential: ⎧d ⎫ dη ⎧ dF dE ⎫ 1 dn = NC ⎨ F 1/ 2 η F ⎬ F = N CF −1/ 2 η F ⎨ n − C ⎬ dx dx ⎭ k BT ⎪ dη F ⎪ dx ⎩ dx ⎩ ⎭ () () F 1/ 2 (η F ) dFn 1 dn dEC dn dEC = k BT + = k BT + dx N CF −1/ 2 (η F ) dx dx dx dx N CF 1/ 2 (η F )F −1/ 2 (η F ) d ( Fn q ) dx = F 1/ 2 (η F ) k BT 1 dn d ( EC q ) + dx F −1/ 2 (η F ) q n dx (ii) (iii) Insert (ii) in (i) J nx = σ n ⎧ ⎫ dFn q...
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This document was uploaded on 01/15/2014.

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