ee3161 spring10 hw2

# This has a solution that can be written in either of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 02/24/2010 for the course EE 3161 taught by Professor Prof.sang-­hyunoh during the Spring '10 term at University of Minnesota Crookston.

Ask a homework question - tutors are online