Today Lecture 11 Electrodynamics nonequilibrium superconductivity and beyond

Today Lecture 11: Electrodynamics, non-equilibrium superconductivity, and beyond BCS 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. Temperature dependence, thermodynamics, and coherence factors 5. Electrodynamics, non-equilibrium superconductivity, and beyond BCS Next time Lecture 12: Quasiparticle tunneling --- theory

Last time Used the Bogoliubov quasiparticle operators: and resulting diagonalized Hamiltonian : ground state excitations to generate thermodynamic properties and the unique coherence factors that arise from pair correlations Today (1) Extend to electrodynamics to generate the effect of quasiparticles on the supercurrent (2) Consider non-equilibrium states induced by perturbations (3) Give a quick survey of phenomena beyond BCS superconductors

Electrodynamics • BCS: if odd diamagnetic (pairs) even paramagnetic (qp’s) equilibrium Fermi-Dirac distribution BUT THIS IS NOT TRUE

Linear Response Theory Fourier Transform neglect Type II coherence factors scattering long --- local regime creation (pairbreaking) annihilation (recombination)

“Doppler Shift”

no contribution from quasiparticles large contribution from quasiparticles

Physical picture Shifted Fermi sphere (all electrons contribute) qp’s lower energy by scattering to the backside of Fermi surface (some electron flow backwards) backflow term

BCS: BCS Kernal Real space: BCS coherence length

Impurities: Penetration depth: BCS does it all and got it (almost) all right !

Non-Equilibrium SC deviation from equilibrium Opposed by relaxation to equilibrium by scattering and diffusion Perturbations 1. 2. 3. 4. 5. Balance rates by the Boltzman equation Response – mostly like normal metal except : E-fields Temperature gradients EM fields Charge injection by transport or tunneling Phonon injection 5. Coherence factors --- effect on scattering events

OUT-OF-EQUILIBRIUM EVEN ODD CURRENT MODE (electric field) HEAT CURRENT MODE ODD (T gradient) No analogue for normal metals: ENERGY MODE GAP MODIFICATION EVEN (excess # of qp’s) CHARGE MODE CHARGE IMBALANCE ODD EVEN (excess charge in qp’s)

QP Relaxation 5 processes: Experiments/devices Dynamic – pulse perturbation, measure decay in time Diffusive – inject steady perturbation, measure spatial decay DO NOT NEGLECT PHONONS! Steady state – measure out-of-equilibrium distribution Kinetic equation (Boltzman equation) Equations are coupled by electron-phonon scattering and heating

Conventional (“classic”) superconductivity BCS theory: Bardeen, Cooper, Schrieffer (1957) § MECHANISM = attractive phonon-mediated electron-electron interaction Cooper pairing § GROUND STATE = superfluid pair condensate k'1 k 2 q e- k 1 e- k'2 = ns e i macroscopic phase coherence § EXCITATIONS = normal “quasiparticles” with an isotropic energy gap kz D(k) = D ky kx “s-wave” 23. 2 K Liquid He

BCS gap equation Weak coupling All of this changes if the pairing interaction is not isotropic electron-phonon scattering

High Temperature Superconductivity (1986) Alex Müller Georg Bednorz IBM Zurich Research Laboratory La 2 -x. Bax. Cu. O 4 YBa 2 Cu 3 O 7 -x Tc ~ 30 K Tc ~ 90 K

HTSC --- many unusual properties were discovered in the first few years 1. 2. 3. 4. 5. Ceramics --- oxide materials, not metals Tc was much higher than could be explained by conventional BCS Hc 2 was immeasurably large Layered materials --- low dimensional effects, strong in-plane correlations, anomalous vortices Unusual properties --- thermodynamics, transport, electrodynamics, … power laws instead of exponential implying that the energy gap was not fully formed 6. Unusual doping dependence --- complicated phase diagram T strange metal Mott insulator superconductor electron doping 7. UNCONVENTIONAL SUPERCONDUCTIVITY Not-so normal metal pseudogap superconductor hole doping

Unconventional superconductivity There was almost immediate evidence that the cuprates were “unconventional” What does “unconventional” mean? Not BCS • Mechanism other than phonon-mediated pairing • Symmetry not s-wave … exhibits anisotropy in phase and/or magnitude 1 st indication: UPt 3 (heavy fermion) two peaks in specific heat 1 st confirmation: YBa 2 Cu 3 O 7 -x (high-Tc superconductor) d-wave

Growing Family of Unconventional Superconductors Cuprate superconductors YBa 2 Cu 3 O 7 -x Tc = 95 K _ + 115 superconductors Organic superconductors Ce. Co. In 5 -(BEDT-TTF)2 Cu[N(CN)2]Br _ _ + + Tc = 11. 6 K _ _ + + “d-wave” Ruthenate superconductors Sr 2 Ru. O 4 Tc = 2. 3 K Tc = 1. 5 K px+ipy “anisotropic d-wave” Heavy Fermion superconductors UPt 3 Tc. A = 0. 50 Tc. B = 0. 45 K (kx 2 -ky 2) kz (kx+ iky)2 kz

Determining the Pairing Symmetry --- A Roadmap for Experimentalists Cuprate candidates - s vs. dx 2 -y 2 + + - Magnitude measurements probe quasiparticles --but can be masked or mimicked by impurities Phase measurements give a distinct signature --less susceptible to microscopic details Complex order parameter broken time-reversal symmetry phase shift 0,

Family tree of superconductors hydrogen sulphide fullerenes pnictides conventional carbon

Family Tree --- Iron Pnictides (1111) (122) (111) (11)

Phase diagrams Order parameter symmetry s 1111 family 122 family Ba(Fe 1 -x. Cox)2 As 2 Phase-shift of between different electron bands

What does this mean for the course? For quantum information science, not much. In general, unconventional SC is bad for qubits because of quasiparticles that are not suppressed by a fully-formed energy gap and because of vortices not pinned in Type II superconductors. Most qubits utilize conventional metallic s-wave superconductors such as Al and Nb. POSSIBLE EXCEPTION: p-wave superconductors may support Majorana fermion stateas that could offer topological protection from dephasing --- more on this later. For superconductor device physics, a lot. Quasiparticle tunneling is an excellent probe of order parameter/energy gap magnitude anisotropy Josephson tunneling is an excellent probe of order parameter/energy gap phase anisotropy and unconventional superconductivity.