Scattering theory Handout

Scattering theory Handout - University of California,...

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University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Scattering theory Handout.fm Scattering theory (read Lundstrom 1.4, 1.5, 2.1, 2.2, See also, Schiff, Quantum Mechanics ) incident flux if particle goes thru d σ - an “event” occurs # of events / sec Elastic scattering - an event is defined as elastic scattering into a solid angle d Ω This is a ratio of fluxes. We can get this from quantum mechanics. incident beam d Ω R ds d Ω small solid ds R 2 ----- = small area, d σ I inc # particles passing unit area transverse to beam sec = I d σ = σ d # events/sec I ------------------------------ elastic = scatt # particles scattered into d Ω () sec I ------------------------------------------------------------------------------------- = σ d flux thru ds incident flux =
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- 93 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor incident state - plane wave scattered state far away from the scattering site: We will drop the Bloch functions for simplicity. For isotropic, parabolic bands, the Bloch function part of the wavefunctions will integrate to 1. Recall QM current density: Ψ inc Ψ scatt Ψ r 1 V ------- e ik r f θϕ , () e ikr r + = outgoing wave j 1 V --- h _ k m ------ = I j z ˆ h _ k m ⎝⎠ ⎛⎞ 1 V == Ψ 1 V f , e r r j ds R 2 d Ω h _ 2 im ------------ Ψ * r d d Ψ Ψ r d d Ψ * ⎜⎟ rR = = R 2 d Ω h _ k m f , 2 1 R 2 ----- [] 1 V = R r ˆ R 2 d Ω = σ d j j z ˆ --------------------- 1 V h _ k m f , 2 Ω d 1 V hk m -----------------------------------------
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- 94 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor First Born approximation (derivation can be found in most advanced quantum texts) Applications Ionized impurity scattering: Bare Coulomb potential (I’m using esu units here, while Lundstrom uses MKS) Screening In an electron gas, carriers screen the potential. Solve Poisson’s eqn, with density given by Maxwell-Boltzmann: k k ' Qk k ' = momentum transfer vector kk ' = elastic scattering: Q f Q () 2 m 4 π h _ 2 ----------- e iQ y Vy y 3 d = scattering potential 2 m 4 π h _ 2 V ˜
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This note was uploaded on 08/01/2008 for the course EE 230 taught by Professor Bokor during the Spring '08 term at Berkeley.

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Scattering theory Handout - University of California,...

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