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Unformatted text preview: =0 for x<0, write down the governing equation:
d 2 ∆ pn ∆ pn Dp − =0 2 dx τp What are the boundary conditions at x=0 and ∞?
∆pn |x=0 = 1010 /cm3 ∆pn |x=∞ = 0 General solution of the differential equation:
∆pn (x) = Ae−x/LP + Bex/LP ∆pn (x) = ∆pn (0)e−x/LP
EE3161 Semiconductor Devices Sang-Hyun Oh LP ≡ DP τP 36 UNIVERSITY OF MINNESOTA Pierret 3.5.1. Diffusion Length Concept
As we see in Prob. #2, excess carriers diffuse from the point of injection, which is characterized by the exponential falloff with a characteristic decay length LN or LP:
∆pn (x) = ∆pn (0)e−x/LP LP ≡ DP τp LN ≡ DN τn
37 EE3161 Semiconductor Devices Sang-Hyun Oh UNIVERSITY OF MINNESOTA Fermi Level Meaning
The common energy at equilibrium Energy where f(E) = 1/2 The carrier concentration (equilibrium): n p EF = Ei + k T l n = Ei − k T l n ni ni Why do we need quasi-Fermi levels? Answer: To specify the carrier concentrations under nonequilibrium conditions, i.e. np≠ni2 EE3161 Semiconductor Devices Sang-Hyun Oh 38 UNIVERSITY OF MINNESOTA Quasi-Fermi Levels
If a semiconductor material is illuminated with intense light, # of electrons and holes are not those given by thermal equilibrium. Remember that we deﬁned Fermi level in a system under equilibrium. However, important devices such as a laser and a forwarded-biased p-n junction are NOT in equilibrium. In that case, how can we analyze the system?
EE3161 Semiconductor Devices Sang-Hyun Oh 39 UNIVERSITY OF MINNESOTA Quasi-Fermi Levels*
In that case, describe the electron and hole distributions as being Fermi-Dirac in form, but each having different “quasi-Fermi levels”. Deﬁne quasi-Fermi levels FN and FP to correspond to carrier concentrations even if there is not thermal equilibrium *Quasi-Fermi level is also sometimes known as “imrefs” (Fermi backwards). EE3161 Semiconductor Devices Sang-Hyun Oh 40 UNIVERSITY OF MINNESOTA Quasi-Fermi Levels
Equilibrium and Fermi level EF:
n0 = ni e(EF −Ei )/kT p0 = ni e(Ei −EF )/kT Nonquilibrium and deﬁnition of quasi-Fermi levels FN and FP:
n ≡ ni e(FN −Ei )/kT FN ≡ Ei + kT ln
EE3161 Semiconductor Devices Sang-Hyun Oh p ≡ ni e(Ei −FP )/kT
41 n ni FP ≡ Ei − kT ln p ni UNIVERSITY OF MINNESOTA...
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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.
- Spring '10