This has a solution that can be written in either of

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Unformatted text preview: eglected) EE3161 Semiconductor Devices Sang-Hyun 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 Sang-Hyun 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 electron-hole pairs Carriers being examined ∆pn (x) which region EE3161 Semiconductor Devices Sang-Hyun Oh where in region 31 UNIVERSITY OF MINNESOTA Sample Prob 1 hν n-type Sample Problem #1 Uniform illumination Minority Carrier Diffusion Eq Example Low level injection Examples Quasi-Fermi 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 Sang-Hyun 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 Sang-Hyun 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 Sang-Hyun Oh 34 UNIVERSITY OF MINNESOTA Sample Problem #2 0 x A uniformly doped semi-infinite 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 Sang-Hyun Oh 35 UNIVERSITY OF MINNESOTA Sample Problem #2 Under steady state conditions with GL...
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