100 YEARS of SUPERCONDUCTIVITY: ANCIENT HISTORY AND CURRENT CHALLENGES A.J. Leggett Dept. of Physics, University of Illinois at Urbana-Champaign Advanced Materials and Nanotechnology Conference, Wellington, NZ, 8 Feb. 2011 Theoretical work on superconductivity, 1911-1956 II. “all-electronic” superconductivity: what can we say without invoking a specific microscopic “model”?

Superconductivity: theory up to 1957 A. Pre-Meissner Discovery of “superconductivity” (= zero resistance): Kamerlingh Onnes & Holst, 1911 (Hg at 4.3K) 1911-1933: Superconductivity  zero resistance Theories: (Bloch, Bohr…): Epstein, Dorfman, Schachenmeier, Kronig, Frenkel, Landau… Landau, Frenkel: groundstate of superconductor characterized by spontaneous current elements, randomly oriented but aligned by externally imposed current (cf. domains in ferromagnetism) Bohr, Kronig, Frenkel: superconductivity must reflect correlated motion of electrons (but correlation crystalline…) Meissner review, 1932: are the “superconducting” electrons the same ones that carry current in the normal phase?  is superconductivity a bulk or a surface effect?  why do properties other than R show no discontinuity at Tc? In particular, why no jump in K?  maybe “only a small fraction of ordinary electrons superconducting at Tc?” thermal condy.

Meissner and Ochsenfeld, 1933:  T > Tc (H fixed) T < Tc superconductivity is (also) equilibrium effect! (“perfect diamagnetism”) contrast: I ← cannot be stable groundstate if  metastable effect (Bloch) Digression: which did K.O. see? Answer (with hindsight!) depends on whether or not Dj > p! cannot tell from data given in original paper. Hg D

↑ ↑ entropy specific heat Theory, post-Meissner: Ehrenfest, 1933: classification of phase transitions, superconductivity (and l–transition of 4He) is second-order (no discontinuity in S, but discontinuity in Cv) Landau, 1937 (?): idea of order parameter: OP for superconductivity is critical current. (Gorter and Casimir: fraction of superconducting electrons,  1 as T 0,  0 as T Tc) F. and H. London, 1935: for n electrons per u.v. not subject to collisions, ↑ ↑ entropy specific heat when accelerated, current (+ field) falls off in metal as exp –z/L (de Haas-Lorentz, 1925) Londons: drop time derivative! i.e. (+ Maxwell) In simply connected geometry, equiv to

Londons, cont.: why for a single particle, In normal metal, application of A deforms s.p.w.f’s by mixing in states of arb. low energy  second term cancelled. But if no such states exist,  only 2nd term survives. Then so, obtain(*). Why no low-energy states? Idea of energy gap (supported by expts. of late 40’s and early 50’s) (London, 1948: flux quantization in superconducting ring, with unit h/e) Ginzburg & Landau, 1950: order parameter of Landau theory is quantum-mechanical wave function (r) (“macroscopic wave function”): interaction with magnetic field just as for Schrodinger w.f., i.e. +Maxwell  2 char. lengths: (T) = length over which OP can be “bent” before en. exceeds condensn en. (T) = London penetration depth If eff. surface en. between S and N regions –ve. Abrikosov (1957): for vortex lattice forms!

Post-Meissner microscopics 1945-50: various attempts at microscopic theory (Frenkel, Landau, Born & Cheng…) Heisenberg, Koppe (1947): Coulomb force  electron wave-packets localized, move in correlated way; predicted energy gap with (assumed) exponential dependence on materials properties. Pippard, 1950: exptl. penetration depth much less dependent on magnetic field than in London theory and increases dramatically with alloying  nonlocal relation between J(r) and A(r), with characteristic length (“coherence length”) ~ 10-4 Å. Frohlich, 1950: indirect attraction between electrons due to exchange of virtual phonons. Bardeen & Pines, 1955: combined treatment of screened Coulomb repulsion and phonon-induced attraction  net interaction at low (w ≲ wD) frequencies may (or may not) be attractive. Schafroth, 1954: superconductivity results form BEC of electron pairs (cf. Ogg 1946) _____________ **1957: BCS theory**

WHICH ENERGY IS SAVED IN THE SUPERCONDUCTING* PHASE TRANSITION? A. DIRAC HAMILTONIAN (NR LIMIT): Consider competition between “best” normal GS and superconducting GS: Chester, Phys. Rev. 103, 1693 (1956): at zero pressure, *or any other.

e INTERMEDIATE-LEVEL DESCRIPTION: partition electrons into “core” + “conduction”, ignore phonons. Then, eff. Hamiltonian for condn electrons is high-freq. diel. const. (from ionic cores) with U(ri ) independent of  (?). If this is right, can compare 2 systems with same form of U(r) and carrier density but different . Hellman-Feynman: e Hence provided decreases in N  S transn, (assumption!) advantageous to have as strong a Coulomb repulsion as possible (“more to save”!) Ex: Hg-1201 vs (central plane of) Hg - 1223

() potential en of cond.n e-,s in field of static lattice I

HOW CAN PAIRING SAVE COULOMB ENERGY? [exact] Coulomb interaction (repulsive) bare density response function . ~min (kF,kTF)~1A-1 A. (typical for ) pertn-theoretic result to decrease must decrease but  gap should change sign (d-wave?) B. (typical for ) to decrease (may) increase and thus (possibly) increased correlations  increased screening  decrease of Coulomb energy! 10

ELIASHBERG vs. OVERSCREENING interaction fixed –k' k' ELIASHBERG k –k electrons have opposite momentum (and spin) REQUIRES ATTRACTION IN NORMAL PHASE interaction modified by pairing k3 k4 OVERSCREENING k1 k2 electrons have arbitrary momentum (and spin) NO ATTRACTION REQUIRED IN NORMAL PHASE

THE ROLE OF 2-DIMENSIONALITY As above, interplane spacing small q as important as large q. Hence, $64K question: In 2D-like HTS (cuprates, ferropnictides, organics…) is saving of Coulomb energy mainly at small q?

CONSTRAINTS ON SAVING OF COULOMB ENERGY AT SMALL q: Sum rules for “full” density response (q)* (any d)  for A = 0 (“jellium” model) no saving of Coulomb energy for q 0. Lattice is crucial! * M. Turlakov and AJL. Phys. Rev. B 67, 044517 (2003)

ferropnictides?

IF SAVING OF COULOMB ENERGY IS MAINLY IN LOW-q, MIR REGIME… NS must decrease –Im -1 in this regime i.e. *El-Azrak et al.,Phys.Rev.B 49,9846 (1994)

If this is right, what are good “ingredients” for enhancing Tc? 2-dimensionality (weak tunnelling contact between layers, but strong Coulomb contact) Strongest possible Coulomb interaction (intra-plane and interplane) Strong Umklapp effects wide and strong MIR peak (may come from strong AF-type fluctuations?) My bet on robust room temperature superconductivity: in my lifetime: ~ 10% in (some of) yours: > 50%