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EE3161 Semiconductor Devices SangHyun Oh 29 UNIVERSITY OF MINNESOTA Differential Equation Solution
A key equation we will be concerned with is d2 y 0 = L2 2 − y dx
corresponding to minority carrier diffusion in steady state without light. This has a solution that can be written in either of two equivalent forms:
y (x) = A1 ex/L + A2 e−x/L subject to the boundary conditions at x=0 and x=∞
y (x) = B1 cosh(x/L) + B2 sinh(x/L)
EE3161 Semiconductor Devices SangHyun Oh 30 UNIVERSITY OF MINNESOTA Notation
n = carrier concentration under arbitrary conditions. n0 = equilibrium carrier concentration ∆n = n  n0 = excess carrier concentration GL=external light that generates electronhole pairs Carriers being examined ∆pn (x)
which region
EE3161 Semiconductor Devices SangHyun Oh where in region
31 UNIVERSITY OF MINNESOTA Sample Prob 1
hν
ntype Sample Problem #1
Uniform illumination
Minority Carrier Diffusion Eq Example Low level injection Examples QuasiFermi Level ND = 1015 cm−3 , τp = 10−6 s For t ≤ 0− , GL = 0 At t ≥ 0+ , GL = 1017 EHP·cm−3 s−1 Find ∆pn (t ) for t ≥ 0+ . Find ∆pn(t) for t>0. 9 / 20 EE3161 Semiconductor Devices SangHyun Oh 32 UNIVERSITY OF MINNESOTA Sample Problem #1
Write down the governing equation:
∂ ∆ pn ∂ 2 ∆ pn ∆ pn = Dp − + GL 2 ∂t ∂x τp What is the boundary condition at t=0?
∆pn (t)t=0 = 0 Since a uniformly doped sample is illuminated uniformly, the spatial derivative is zero:
Differential equation:
EE3161 Semiconductor Devices SangHyun Oh d ∆ pn ∆ pn + = GL dt τp
33 UNIVERSITY OF MINNESOTA Sample Problem #1
Solve differential equation:
∆pn (t) = GL τp + Ae−t/τp Apply the boundary condition:
A = −GL τp Solution:
∆pn (t) = GL τp (1 − e−t/τp ) EE3161 Semiconductor Devices SangHyun Oh 34 UNIVERSITY OF MINNESOTA Sample Problem #2 0 x A uniformly doped semiinﬁnite bar of silicon. Excess holes are created at x=0. The wavelength of the illumination is such that no light penetrates into the interior of the bar. Determine Δpn(x)
EE3161 Semiconductor Devices SangHyun Oh 35 UNIVERSITY OF MINNESOTA Sample Problem #2
Under steady state conditions with GL...
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This note was uploaded on 02/24/2010 for the course EE 3161 taught by Professor Prof.sanghyunoh during the Spring '10 term at University of Minnesota Crookston.
 Spring '10
 Prof.SangHyunOh

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