Electron-Electron Interaction. Andreas Wacker Mathematical Physics Lund University
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1 Electron-Electron Interaction Andreas Wacker Mathematical Physics Lund University
2 Hamiltonian for electron-electron interaction Two-particle operator with matrix element Fouriertrafo Spatial integrals disentangle From week 2: with k k' k+q k'-q Visualization by Feynman graph
3 Interaction in electron hole picture In electron-hole picture for a semiconductor (neglecting interaction with full valence band) Dynamics of polarization in the presence of a light field
4 Electron-electron interaction generates higher order expectation values New contribution by electron-electron interaction: contains terms No closed set of equations of motion for pk, nk
5 Hartree-Fock approximation decomposes two-particle expectation values Factorize into products of creation and annihilation operators (exact for Slater states) Spatial homogeneity: Only states with identical k survive
6 Semiconductor Bloch Equations in Hartree Fock approximation exchange shift with Coulomb field renormalization
7 Low electron and hole densities: Excitonic absorption below band gap Solution for ne=nh=0 with After Surge, Phys Rev 1962 Exciton line
8 Hartree-Fock approximation on the basis of operators Two particle operator Aim: Approximate by single particle operator Require if the states are identical or differ by
9 Hartree-Fock approximation provides one-particle operators pairs appear twice Replace pairs by expectation value - Electrons in external potential nonlocal
10 Hartree-Fock for homogeneous electron gas Expand: Homogeneity Fock term only for identical spins Vee -VHF= Vcorr correlation energy contains the corrections to Hartree Fock
11 Energy of free electron gas (jellium model) homogeneous electron gas with density n=n/v homogeneous positive background charge density en Total energy: Noninteracting: fill up to VHartree compensates homogeneous background VFock can be evaluated
12 Hartree-Fock approximation can explain metallic binding Energy per electron in HF approx: rsab is radius of volume occupied by one electron Minimum: -1.3 ev/electron at rs=4.823 n=1.43x1022/cm3 Alkali metals Li: n=4.7x1022/cm3, 1.63 ev (separation energy) Cs: n=0.9x1022/cm3, ev Fock term highly relevant for metallic binding
13 Correlation energy leads to corrections Result of many-particle perturbation theory: Hartree-Fock good for small rs (high densities) Correlation effects dominate for rs > Metals: n 10 /cm r s electrons in a Nanowire: Wigner Localization for long wire Kristinsdottir, Cremon, Nilsson, Xu, Samuelson, Linke, Wacker,Reimann, Physical Review B 2011
14 How density functional theory works in practice Electron-Electron interaction depends on density n(r) Hartree term is functional F{n(r)} Correction by Fock term and correlations: Common: Local Density Approximation: Effective single particle Hamiltonian Solve Selfconsistency Occupying N lowest levels:
15 The longitudinal dielectric function is related to induced charges provides longitudinal dielectric function is related to charge induced by the total electric potential
16 Electron density and Hamiltonian for the interaction Fouriertransformation with Electric potential containing external and internal Hartree field quasistatic
17 (restores homogeneity) ϕ=0 provides
18 The Lindhard formula for the dielectric function Induced charge density by the potential: Compare with
19 only partially filled band contribute Small q: ω 0:
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