1. Concepts in High Temperature Superconductivity

3. 500 references
V. J. Emery, S. A. Kivelson, E.Arrigoni, I.Bindloss, E.Carlson, S.Chakravarty, L.Chayes, K.Fabricius, E.Fradkin, M.Granath, D-H.Lee, H-Q. Lin, U.Low, T.Lubensky, E.Manousakis, Z.Nussinov, V.Oganesyan, D.Orgad, L.Pryadko, M.Salkola, Z-X.Shen, J.Tranquada, O.Zachar

4. Superconductivity Fermions: Pauli exclusion principle
Bosons can condense
Stable phase of matter
Macroscopic Quantum Behavior
Pair wavefunction acts like “order parameter”

5. Superconductivity Fermions: Pauli exclusion principle
Bosons can condense
Stable phase of matter
Macroscopic Quantum Behavior
Pair wavefunction acts like “order parameter”

6. Conventional Superconductivity
John Bardeen
Leon Cooper
Bob Schrieffer
Conventional Superconductivity
BCS Theory
Instability of the metallic state

7. Simple Metals: The Fermi Gas
Fermi Surface
Free Electrons E ~ k2
Pauli exclusion principle
Fill to Fermi level
Adiabatically turn on temperature, most states unaffected
Very dilute gas of excitations: quasiparticles
KE most important
Pauli exclusion principle
fill to Fermi level
Fermi sea
Stable except for …

8. Simple Metals: The Fermi Liquid
Fermi Surface
Pauli exclusion principle
Fill to Fermi level
Adiabatically turn on temperature, most states unaffected
Very dilute gas of excitations: quasiparticles
Fermi Gas + Interactions  Fermi Liquid
Kinetic Energy
Dominant
Quasiparticles = Dressed Electrons

9. Retardation is Essential
Cooper Pairing
Fermi Liquid Unstable to Pairing
Pair electrons near Fermi Surface
Phonon mediated
Retardation is Essential
to overcome
Coulomb repulsion

10. Of A Tranquil Fermi Sea—
BCS Haiku:
Instability
Of A Tranquil Fermi Sea—
Broken Symmetry

11. High Temperature Superconductors
HgCuO
YBCO7
LSCO

12. High Temperature Superconductors
Copper Oxygen Planes
Other Layers
Layered structure  quasi-2D system
Layered structure  quasi-2D system

13. High Temperature Superconductors
Copper-Oxygen Planes Important
“Undoped” is half-filled
Antiferromagnet
Naive band theory fails
Strongly correlated
Oxygen
Copper

14. High Temperature Superconductors
Dope with holes
Superconducts at certain dopings T AF SC x
Oxygen
Copper

15. Mysteries of High Temperature Superconductivity
Ceramic! (Brittle)
“Non-Fermi liquid” normal state
Magnetism nearby (antiferromagnetism)
Make your own (robust)
Pseudogap
Phase ordering transition

16. Two Energy Scales in a Superconductor
Two component order parameter
Amplitude
Pairing
Gap
Phase
Phase Coherence
Superfluid Density

17. BCS is a mean field theory in which pairing precipitates order42 38
Y-123 (ud) 62
104
Bi-2212 (od)
140 90
116
Y-123 (op) 55
Y-123 (od) 60 95
220
Bi-2212 (op) 83
275
Bi-2212 (ud) 25 26
Tl-2201 (od)
160 80 91
122
Tl-2201 (op)
130
133
435
Hg-1223 (op)
108
290
Hg-1212 (op)
180 96
192
Hg-1201 (op)
100 20
LSCO (od) 54 58
LSCO (op) 47 30 75
LSCO (ud)
Material
Phase Fluctuations
Important in Cuprates
Emery, Kivelson, Nature, 374, 434 (1995)
EC, Kivelson, Emery, Manousakis, PRL 83, 612 (1999)
Material Pb
7.9
7.2
6X105
Nb3Sn
18.7
17.8
2X104
UBe13
0.8
0.9
102
BaKBiO
17.4 26
5X102
K3C60 26 20
102
MgB2 15 39
1.4X103

18. Tc and the Energy Scales
superconductivity AF x
BCS: ~ 1000 Tc
HTSC: ~ Tc underdoped

19. Mysteries of High Temperature Superconductivity
Ceramic! (Brittle)
“Non-Fermi liquid” normal state
Magnetism nearby (antiferromagnetism)
Make your own (robust)
Pseudogap
Phase ordering transition
BCS to the rescue?

20. There is no room for retardation in the cuprates
BCS:
Cuprates:
How do we get a high pairing scale
despite the strong Coulomb repulsion?

21. A Fermi Surface Instability Requires a Fermi Surface!
How do we get superconductivity
from a non-Fermi liquid?

22. Strong Correlation Fermi Liquid k-space structure
Kinetic energy is minimized
Pairing is potential energy driven
Real space structure
Kinetic energy is highly frustrated
System works to relieve KE frustration

23. Doped Antiferromagnets
Hole Motion is Frustrated

24. Doped Antiferromagnets
Compromise # 1: Phase Separation
Relieves some KE frustration
Pure AF
Hole
Rich
Like Salt Crystallizing From Salt Water,
The Precipitate (AF) is Pure

25. Coulomb Frustrated Phase Separation
Long range homogeneity
Short range phase separation
Compromise # 2: mesoscale structure
Patches interleave
quasi-1D structure – stripes ?
Hole
Rich
Poor

26. Stripes Rivers of charge between antiferromagnetic strips
Electronic structure becomes effectively 1D

27. Competition often produces stripes
Ferromagnetic garnet film
Faraday effect
Period ~ m
Ferrofluid confined between two glass plates
Period ~ 1cm
Ferromagnetic garnet film
Period ~ m
Block copolymers
Period ~ 4X10-8 m

28. What’s so special about 1D?
canal
John Scott Russell's Soliton Wave Re-created
On Wednesday 12 July 1995, an international gathering of scientists witnessed a re-creation of the famous 1834 'first' sighting of a soliton or solitary wave on the Union Canal near Edinburgh. They were attending a conference on nonlinear waves in physics and biology at Heriot-Watt University, near the canal. The occasion was part of a ceremony to name a new aqueduct after John Scott Russell, the Scottish scientist who made the original observation. The aqueduct carries the Union Canal over the Edinburgh City Bypass.
In 1834, John Scott Russell was observing a boat being drawn along 'rapidly' by a pair of horses. When the boat suddenly stopped Scott Russell noticed that the bow wave continued forward "at great velocity, assuming the form of a large solitary elevation, a well-defined heap of water which continued its course along the channel apparently without change of form or diminution of speed". Intrigued, the young scientist followed the wave on horseback as it rolled on at about eight or nine miles an hour, but after a chase of one or two miles he lost it.
What’s so special about 1D?

29. 1D: Spin-Charge Separation
spin excitation
charge excitation
point out topological, 1D special, point-like, and higher D different.

30. 1D: “Bosonization transformation”
Expresses fermions as kinks in boson fields:
spin soliton
electron = +
charge soliton
Hamiltonian of interacting fermions
becomes Hamiltonian of non-interacting bosons
Spin Charge Separation
Electron No Longer Exists!
Non-Fermi Liquid
(Luttinger Liquid)

31. Advantages of a quasi-1D Superconductor
non-Fermi liquid
strongly correlated
controlled calculations

32. Kinetic Energy Driven Pairing?
 Proximity Effect
superconductor
metal
Individually, free energies minimized
Metal pairs (at a cost!) to minimize kinetic energy across the barrier

33. Spin Charge Separation
1D spin-charge separation
Pair spins only
Avoid Coulomb Repulsion!

34. Spin Gap Proximity Effect
Kinetic energy driven pairing in a quasi-1D superconductor
Metallic charge stripe acquires spin gap through
communication with gapped environment

35. Spin Gap Proximity Effect
Kinetic energy driven pairing in a quasi-1D superconductor
kF = k’F
Conserve momentum and energy
Single particle tunneling is irrelevant kF
k’F

36. Spin Gap Proximity Effect
Kinetic energy driven pairing in a quasi-1D superconductor
kF = k’F
Conserve momentum and energy
Single particle tunneling is irrelevant
But pairs of zero total momentum
can tunnel at low energy
 pairs form to reduce kinetic energy
Step 1: Pairing kF
k’F

37. 1D is Special Spin Gap = CDW Gap = Superconducting Gap Which will win?
CDW stronger for repulsive interactions (Kc<1)
Don’t Make a High Temperature Insulator!

38. Stripe Fluctuations Step 2: Phase Coherence Frustrated Favorable
Discourage CDW
Stripe fluctuations
Encourage SC

39. Inherent Competition  think local stripe order
superconductivity T
Josephson
coupling
Pairing
 think local stripe order
 think “stripe phonon”
stripe
fluctuations
Static Stripes
Good pairing
Bad phase coherence
Fluctuating Stripes
Bad pairing
Good phase coherence

40. Behavior of a Quasi-1D Superconductor
Treat rivers of charge as 1D objects

41. Behavior of a Quasi-1D Superconductor
High Temperature Effective Dimension = 1
Spin charge separation on the Rivers of Charge
Electron dissolves
Non-Fermi Liquid (Luttinger Liquid)
Intermediate Temperature Effective Dimension = 1
Rivers of charge acquire spin gap from local environment
Pseudogap
Non-Fermi Liquid (Luther-Emery Liquid)
Low Temperature Effective Dimension = 3
1D system cannot order  Dimensional Crossover
Pair tunneling between rivers produces phase coherence
Electron recombines

42. Dimensional Crossover and the Quasiparticle
Chains coupled in superconducting state
c o n f i n e m e n t s c e
Superconducting order parameter switches sign across each soliton

43. Dimensional Crossover and the Quasiparticle
c o n f i n e m e n t s c
Bound state of spin and charge
 Electron is stable in superconducting state

44. Crossover Diagram of a Quasi-1D Superconductor
Do/2
1D Luttinger Liquid
1D Luther-Emery
Liquid
3D Fermi
Liquid T
Pseudogap
3D Superconductor
Gap
t (increases with x?)
Crossover Diagram of a Quasi-1D Superconductor

45. Is mesoscale structure necessary for high Tc superconductivity?
Mesoscale structure can enhance pairing (spin-gap)
May be necessary to get pairing from repulsion
Always bad for phase ordering

46. Conclusions Strongly correlated  Kinetic Energy driven order
Formation of mesoscale structure
Formation of pairs by proximity effect
Phase coherence by pair tunneling between stripes
Quasi-1D Superconductor
(Controlled calculations)
Non-Fermi liquid normal state
Pseudogap state
Strong pairing from repulsive interactions
Inherent competition between stripes and superconductivity
Dimensional crossover to superconducting state
Strong correlation leads to all sorts of new physics.
I focused on 3 that can occur.
“Controlled” is anthropomorphic