Phys560Notes-9 - Screening (Electron-Electron Interaction)...

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Screening (Electron-Electron Interaction) Classical picture  Classical metal 0 in E The internal field is zero. Electrons are free to move; a layer of charge ( screening charge ) accumulates at the surface. The external field is screened. Quantum picture: The field penetrates a short distance. Both the screening charge density and the internal field show damped oscillations (1D Friedel oscillations ). Problem: A solid is subjected to an external (applied) potential   , ext q ; find the response  , total q and , total q . Applications: (1) ext from an ion core – crystal potential for band calculations. (2) ext from an electron – electron-electron interaction; independent electron approximation is good if the net interaction is weak. (3) ext from a charged (ionic) impurity – scattering, relaxation time, transport properties. Definition: dielectric function     ,, , total ext qq q ext E Consistent with  , Dq q Eq                   33 44 4 4 11 , , 22 1 2 1 2 it ii t t te d q d e d q d drd t t e e dqd ed q d t d r d t qr qrr Dr q Eq qE r r       3 , tt t t d r d t r r Er , rr : nonlocal & noninstantaneous Cf: in classical E&M, is often treated as a constant. Computing : Let  total .   2 4  ext ind Assume    exp i .      2 4 4 ext ind ext ind ind q
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    2 , 4 ,1 , ind q q q q Computation scheme: Schrodinger  equation   ind Thomas-Fermi theory : semiclassical model, 0, long wavelength limit  .      2 2 2 eE m , where ~ constant in a small macroscopic volume about r . Approximate local solution:  22 ; 2 k Ee m kr r r treated as a parameter. For 0,     3 3 11 4 exp 1 2 o nd k k kT m r For     3 3 4 exp 1 2 o o n k ne k ek T m rr     2 o ind o n en n e r r Fourier transform: 2 ,0 o ind n e qq , treating o n (slowly varying) as a constant.
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Phys560Notes-9 - Screening (Electron-Electron Interaction)...

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