1. 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,
II. “all-electronic” superconductivity: what can we say without invoking a specific microscopic “model”?

2. Superconductivity: theory up to 1957
A. Pre-Meissner
Discovery of “superconductivity” (= zero resistance): Kamerlingh Onnes & Holst, 1911 (Hg at 4.3K)
: 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.

3. 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

4. ↑ ↑ 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

5. 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!

6. Post-Meissner microscopics : 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**

7. 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.

8. 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 vs (central plane of) Hg

9. ()
potential en of
cond.n e-,s in field
of static lattice I

10. 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

11. 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

12. 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?

13. 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, (2003)

15. ferropnictides?

16. 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)

17. 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%