ls2_unit_7 - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 82 R. Victor Jones, April 4, 2000 VII. S EMICONDUCTOR P HOTONICS A. P RELIMINARIES : SEMICONDUCTOR BACKGROUND 29 T HE C RYSTAL H AMILTONIAN For an assembly of atoms the classical energy is the sum of the following: the kinetic energy of the nuclei; the potential energy of the nuclei in one another's electrostatic field; the kinetic energy of the electrons; the potential energy of the electrons in the field of the nuclei; the potential energy of the electrons in one another's electrostatic field; the magnetic energy associated with spin and orbital variables. Dividing the electrons into core and valence electrons and leaving out magnetic effects leads to the following expression for the crystal Hamiltonian: H = r p α 2 2 M α α + U r R α r R β ( ) α , β + r p l 2 2 m l l + V r r l r R α ( ) l , α + e 2 4 πε 0 r r l r r m l , m [ VII-0 ] where α and β label the ions, l and m label the electrons, r p is the momentum, M is an ionic mass, m is the mass of an electron, U r R α r R β ( ) is the interionic potential, and V r r l r R α ( ) is the valence-electron-ion potential. The quantum mechanics of the assembly is treated to a good approximation by taking the total wavefunction of the system as the product Ξ = Ψ r r ; r R ( ) Φ r R ( ) [ VII-1 ] 29 This discussion draws heavily on B. K. Ridley, Quantum Processes in Semiconductors (3rd edition), Clarendon Press (1993).
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 83 R. Victor Jones, April 4, 2000 where Φ r R ( ) = Φ r R α , r R β , r R δ , r R γ , K ( ) is the wavefunction of all the ions and Ψ r r ; r R ( ) = Ψ r r k , r r l , r r m , r r n , K ; r R α , r R β , r R δ , r R γ , K ( ) is the wavefunction of all the electrons at the instantaneous ionic positions. The Schrödinger equation is then written Ψ r r ; r R ( ) H lattice Φ r R ( ) + Φ r R ( ) H elec Ψ r r ; r R ( ) + H lattice Ψ r r ; r R ( ) Φ r R ( ) − Ψ r r ; r R ( ) H lattice Φ r R ( ) [ ] = E Ψ r r ; r R ( ) Φ r R ( ) [ VII-2 ] where the total Hamiltonian is parsed into two independent components -- viz. H lattice = r p α 2 2 M α α + U r R α r R β ( ) α , β . [ VII-3a ] H elec = r p l 2 2 m l l + V r r l r R α ( ) l , α + e 2 4 πε 0 r r l r r m l , m . [ VII-3b ] The essential assumption of the adiabatic approximation is that the bracketed term is negligible and that the global problem may be treated as two independent problems -- viz. H lattice Φ r R ( ) = E lattice Φ r R ( ) [ VII-4a ] H elec Ψ r r ; r R ( ) = E elec Ψ r r ; r R ( ) [ VII-4b ] As a further refinement, the electron problem must be parsed once more as H elec = H elec { } static + H elec { } dynamic
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 84 R. Victor Jones, April 4, 2000 where H elec { } static defines the problem of the many electron system interacting with the static ionic lattice and H elec { } dynamic incorporates the effects of the electron-phonon interaction.
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