Today Lecture 9 BCS theory Selfconsistent solution and

Today Lecture 9: BCS theory --- Self-consistent solution and quasiparticles Discussion the BCS theory in four parts: 1. Clues to the mechanism and the Cooper instability problem 2. Attractive interaction and the BCS wavefunction and ground state 3. Self-consistent solution and quasiparticles 4. Thermodynamics, electrodynamics, and the coherence factors Next time Lecture 10: BCS theory --- Thermodynamics, electrodynamics, and the coherence factors

Last time BCS ground state: BCS (reduced) H: Variational calculation to minimize Found where and Today to determine the ground state

Self-Consistent solutions : Key feature of the BCS state is electron correlations instantaneous timeaveraged In the normal state these vanish (phases are random) deviation (assume fluctuations small) neglect terms of Define to absorb parameter This is the “model” Hamiltonian: bilinear in and Diagonalize by a linear canonical transformation Is to be determined self-consistently after we find the ground state

Bogoliubov-Valatin Transformation INVERT Want to get rid of product states Diagonalization condition: Leads to: Fermi operators --- obey anti-commutation relations spin degrees of freedom (mix electron spins) and keep only terms of form

Solve diagonalization condition to get ground state Multiply by Solve by quadradic formula: Phases of BCS chose Solution: (+ solution is stable) related: for energy to be real, must be real have the same phase factor

Result of the diagonalization: First term: Normal State Superconducting State KE DIFFERENCE Condensation energy PE Ground state Second term: excitation numbers of Bogoliubov quasiparticles = excitation energy,

Quasiparticle operators 0 0 Define ground state by: 0 Define quasiparticle by: 0

Normal state – Excitation picture describe excited states by addition of “quasi-particles” EXCITATIONS GROUNDSTATE ENERGY: CHARGE: VELOCITY: CURRENT: holes OCCUPATION PROBABLILITY electrons

SUPERCONDUCTING STATE GROUND STATE EXCITATIONS OCCUPATION PROBABILITY minimum excitation energy Energy: electron ground state

electron GS Charge: electron –like qp’s Dependence of charge on charge as it accelerates means the qp changes qp’s not independent particles (many body system. ) hole-like qp’s Again, shows that qp’s are not independent of condensate. QP’s have mixed electron hole character

Density of states: map over States below gap (NORMAL STATE) pushed up to peak at gap edge Tunneling and transport are probes of this.

BCS Model GROUND STATE QUASIPARTICLES Self-consistency: mp Next time: • Temperature dependence • Thermodynamics phenomenological models • External perturbations coherence factors (selection, modes)