Unformatted text preview: steady state excess minority carrier concentration and the QFL’s Fn and Fp . Assume spatially uniform conditions, and approach the problem as follows. 5a) 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: 0 = Dn
−
+ GL dx 2
τn Δn
Simplify for spatially uniform conditions: 0 = 0 −
+ GL τn
So the simplified MDE equation is: Δn
−
+ GL = 0 τn 5b) Specify the initial and boundary conditions, as appropriate for this problem. Solution: Since there is no time dependence, there is no initial condition. Since there is no spatial dependence, there are no boundary conditions. 5c) Solve the problem. Solution: In this case the solution is trivial: Δn = GLτ n = 10 20 × 10 −6 = 1014 cm 3 ECE 606 6 Spring 2013 Mark Lundstrom 2/24/2013 Now compute the QFLs: Since we are doped p type and in low level injection: (E −F ) p ≈ p0 = N A = ni e i p k BT ⎛N ⎞
⎛ 1017 ⎞
Fp = Ei − k BT ln ⎜ A ⎟ = Ei − 0.026 ln ⎜ 10 ⎟ = Ei − 0.41 eV ⎝ 10 ⎠
⎝ ni ⎠ n ≈ Δn >> n0 = ni e( Fn − Ei ) k BT ⎛ Δn ⎞
⎛ 1014 ⎞
Fn = Ei + k BT ln ⎜ ⎟ = Ei + 0.026 ln ⎜ 10 ⎟ = Ei + 0.24 eV ⎝ 10 ⎠
⎝ ni ⎠ 5d) Provide a sketch of the solution, and explain it in words. Solution: The excess carrier density is just constant, independent of position. So are the QFL’s, but they split because we are not in equilibrium. 6) The hole QFL is essentially where the equilibrium Fermi level was, because the hole co...
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 Fall '08
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