Lecture 10: BCS theory --- Thermodynamics, electrodynamics, and the coherence factors Lecture 9: BCS theory --- Self-consistent solution and quasiparticles Next time Today Discussion the BCS theory in four parts: Clues to the mechanism and
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Presentation Transcript
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Lecture 10: BCS theory --- Thermodynamics, electrodynamics, and the coherence factors Lecture 9: BCS theory --- Self-consistent solution and quasiparticles Next time Today Discussion the BCS theory in four parts:
Clues to the mechanism and the Cooper instability problem
Attractive interaction and the BCS wavefunction and ground state
Self-consistent solution and quasiparticles
Thermodynamics, electrodynamics, and the coherence factors<br>
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BCS ground state: BCS (reduced) H: where and Last time Variational calculation to minimize Found Today to determine the ground state<br>
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Self-Consistent solutions In the normal state these vanish (phases are random) deviation : Diagonalize by a linear canonical transformation (assume fluctuations small) Key feature of the BCS state is electron correlations instantaneous time-averaged<br>
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Bogoliubov-Valatin Transformation --- obey anti-commutation relations INVERT Diagonalization condition: spin degrees of freedom (mix electron spins) Fermi operators and keep only terms of form Leads to:<br>
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Multiply by (+ solution is stable) Phases of related: for energy to be real, Solve by quadradic formula: Solve diagonalization condition to get ground state Solution: must be real BCS chose have the same phase factor<br>
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First term: Normal State KE PE Condensation energy DIFFERENCE Second term: excitation numbers of Bogoliubov quasiparticles Ground state Result of the diagonalization: Superconducting State = excitation energy,<br>
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Quasiparticle operators Define ground state by: Define quasiparticle by:<br>
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Normal state – Excitation picture describe excited states by addition of “quasi-particles” GROUNDSTATE EXCITATIONS OCCUPATION PROBABLILITY VELOCITY: ENERGY: CHARGE: CURRENT:<br>
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SUPERCONDUCTING STATE EXCITATIONS OCCUPATION PROBABILITY Energy: electron GROUND STATE ground state<br>
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Charge: electron GS Again, shows that qp’s are not independent of condensate.
QP’s have mixed electron hole character<br>
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Density of states: map over States below gap (NORMAL STATE) pushed up to peak at gap edge Tunneling and transport are probes of this.<br>
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BCS Model GROUND STATE QUASIPARTICLES Self-consistency: Temperature dependence
Thermodynamics phenomenological models
External perturbations coherence factors (selection, modes) mp Next time:<br>
