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 q q q ext E Consistent with , , , D q q E q     3 3 4 4 3 3 4 3 3 4 1 1 , , , , 2 2 1 , , 2 1 , , 2 i t i t i t i t i i t t t e d q d e d q d d r dt t e e d q d e d q d t d r dt q r q r q r q r q r r D r D q q E q q E r q E r   3 , , , t t t t d r dt D r r r E r , t t r r : nonlocal & noninstantaneous Cf: in classical E&M, is often treated as a constant. Computing : Let total .    2 4 ext ind Assume exp i q r .     2 2 2 4 4 4 4 ext ind ext ind ind q q q
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