1. Lecture 7. The pseudogap
•Hole Doping and the Phase Diagram
•Strange Metals
•Experimental Probes
•Current Pseudogap Theories
•Pseudogap in BEC?

2. A Brief History
Sir Neville Mott described the dispersion
diagram (E vs.k) for disordered materials
(no lattice) in terms of the density of
states.
He used the term pseudogap to describe
the energy region above the density of
valence “band” states, and below the
conduction “band” states. These states are
not describable as periodic Bloch waves.

6. HTSC
Doping and the Phase Diagram
Cuprates

9. High Tc Phase diagram
Plan
1. Overdoped – is it
`conventional’?
2. What is strange about the
strange metal?
3. Phenomenology of the
pseudogap
4. Transition to
superconductivity

11. Eisaki et al, PRB 69,
064512 (2004)
With further increase of
layers, Tc does not go up
further. The inner planes
have less hole and may
be AF ordered.

13. Pseudogap in
Other Superconducting Materials?

23. Experimental Probes

24. ARPES
(Angle Resolved Photoemission
Spectroscopy)

39. ARPES (Angle Resolved Photoemission
Spectroscopy)

43. Prototype HTSC Materials

47. Evolution of the Fermi surface with doping
Doiron-Leyraud et al
Nature (2007)

52. Tunneling Spectroscopy

59. Introduction to the phenomenology
of high temperature
superconductors
Patrick Lee and T. Senthil

60. 2013-9-11 60
Eagles (1969): Pairing without superconductivity
ATHIC2008, TSUKUBA, JAPAN
In HTSC, the root of PG is not
quite clear. The pre-formed boson
mechanism is only one explanation.

61. A preliminary look: transport

62. Overdoped metal
• Does it have a Fermi surface? Size and shape?
Methods to detect – ARPES, deHaas-van Alphen
and related quantum oscillations, other….. eg
Angle Dependant Magneto-Resistance
(ADMR)
• Is it really a Fermi liquid with Landau
quasiparticles?

63. Overdoped metal: Is there a Fermi
surface?

64. Quantum oscillations in Tl-2201
Tc = 10 K, B upto 60 T;
oscillations in both M and in
c-axis ρ

65. High Tc Phase diagram
Plan
1.Overdoped – is it
`conventional’?
2.What is strange about the
strange metal?
3.Theory interlude
4.Phenomenology of the
pseudogap
5. Transition to
superconductivity

66. The strange metal: electrical transport
Linear-T resistivity near optimal doping
with nearly zero intercept.
Slope of resistivity/layer roughly the
same (1.5 µΩ cm/K) for all materials.
Sheet resistance = ρ/d ~ (h/e^2) T/J
Bi-2201

67. Magnetotransport: Hall effect

68. ARPES: Fermi surface structure

69. Summary on strange metal
Strange metal: Power laws in many physical quantities;
Large Fermi surface but no Landau-like quasiparticles
Slow growth of antiferromagnetic spin correlations
Transition to superconductivity accompanied by appearance of
coherent quasiparticles and a sharp spin triplet `resonance’ mode.

70. Why superconductivity?
Crucial Anderson insight:
Singlet valence bond between localized spins: A localized Cooper pair.
`Pairing’ comes from superexchange due to a repulsive Hubbard interaction.
If spins were truly localized, Cooper pairs do not move => no superconductivity.
Nonzero doping: allow room for motion of valence bonds => superconductivity!
Hole picture: Coherent hole motion in valence bond sea

71. Cartoon understanding of phase diagram
Formation of singlet
valence bond
Coherence of hole motion

72. High Tc Phase diagram
Plan
1.Overdoped – is it
`conventional’?
2.What is strange about the
strange metal?
3.Theory interlude
4.Phenomenology of the
pseudogap
5. Transition to
superconductivity

73. Tunneling
More on STM later!

74. Pseudogap state in ARPES

75. Evolution of pseudogap with doping

76. Summary of ARPES Fermi surface
evolution
1. Big antinodal gap – 50 meV or bigger
2. Gapless Fermi arcs near node that shrink as T is
reduced;
possibly even extrapolate to 0 at T = 0.
3. Gap is apparently centered on large Fermi surface

77. High field ground state: contrast
between under and over-doped

78. Scanning tunneling microscopy (STM)
(Credit: Jenny Hoffman website)
Tunnel electrons from metallic tip to surface of
system of interest.
Tunneling current
s = tip-sample distance
Tip d.o.s System d.o.s
(Actually ``single particle d.o.s –
involves adding or removing an
electron from system).
Study tunneling density of states with sub-Angstrom spatial resolution

79. STM: different measurement modes
http://hoffman.physics.harvard.edu/research/
STMmeas.php

80. STM in the cuprates at low-T: d-wave
gaps and spatial inhomogeneity
20 mV
)(r


Spatially averaged spectra;
consistent with d-wave gap
But gap varies strongly on nm scale!
Pan, Hudson, Davis et al, 2001, 2002
20 meV
70 meV
Is the inhomogeneity just a surface property or does it affect bulk physics?
Note: ARPES probes the same surface!

81. Competing order and fluctuations
Apart from superconductivity, many other ordered or
nearly ordered (i.e short range ordered) states have
been reported in the underdoped cuprates.
Some prominent examples:
1. Antiferromagnetism/SDW/spin stripes
2. Charge order – charge stripes/CDW/checkerboard
3. Nematic order (breaking of lattice rotation symmetry
without breaking translation symmetry).
Implication/importance of these for
pseudogap/SC/strange metal not currently understood.

82. Carrier (hole)
concentration
d-wave
T*
BEC BCS
Tc Fermi
Liquids-wave
Superfluid
Pseudo
-gap
High Tc Cuprates Cold Fermi Gases
0 00.2
M. Randeria in
“Bose Einstein Condensation” (1995)
& Varenna Lectures (1997).
normal
Bose
gas
Strongly correlated non-Fermi-liquid
superconductors normal states
• low-energy pseudogap
• high-energy pseudogap
• strange metal: w/T scaling
Spin-Charge separartion?
T

83. High Tc SC in cuprates
• Highest known Tc (in K)
* electrons
• Repulsive interactions
• d-wave pairing
• near Mott transition
• competing orders: AFM,CDW
• repulsion U >> bandwidth
• x ~ 10 A
• Tc ~ rs << 
• Mean-field theory fails
• anomalous normal states
- strange metal & pseudogap
Breakdown of Fermi-liquid theory
Spin-charge separation?
BCS-BEC crossover
• Highest known Tc/Ef ~ 0.2
* cold Fermi atoms
• Attractive interactions
• s-wave pairing
• only pairing instability
• attraction > Ef
• x ~ 1/kf
• Tc ~ rs << 
• Mean-field theory fails
• pairing pseudogap

84. BCS-BEC crossover and superfluidity in
atomic Fermi gases
Qijin Chen (陈启谨)
Zhejiang Institute of Modern Physics and
Department of Physics, Zhejiang University
Cold Atom Workshop, KITPC 2009-10-19

88. Bosons vs fermions
BEC Fermi sea
EF = kBTF
spin 
spin 
T = 0
Pauli exclusion

89. Essence of Fermionic Superfluidity
fermions bosons
Attractive interactions turn fermions into
“composite bosons” (or Cooper pairs).
These are then driven by statistics to Bose condense.
Increased attraction
BCS-BEC Crossover

90. Remarkable Tuning Capability in Cold Gases via
Feshbach Resonance
.
molecules
→ ←
B>
Scattering length a
BCSBEC
Unitary limit
a > 0
a < 0

91. Crossover under control in cold Fermi
atoms (1st time possible)
Molecules of
fermionic atoms
BEC of bound
molecules
Cooper pairs
BCS superconductivity
Cooper pairs: correlated
momentum-space pairing
kF
Pseudogap /
unitary regime
hybridized Cooper pairs
and molecules
Magnetic Field

92. Overview of BCS theory
Fermi Gas
No excitation gap
BCS superconductor

93. 2013-9-11 93
Theoretical History of Crossover:
• Leggett (1980) noted that BCS T=0 wavefunction could be
generalized to arbitrary attraction: a smooth BCS-BEC crossover !
  2/12
2/1
1 kB
kB
k
N
N
v




BCS
  0exp 2/1
0



 kkk k ccNB 
BEC
Introduction to BCS-BEC Crossover
weak couplingstrong coupling
ATHIC2008, TSUKUBA, JAPAN

94. Zero T BCS-BEC crossover:
Tuning the attractive interaction
 Change of character:
fermionic  Bosonic
(Uc – critical coupling)
 Use ground state BCS-Leggett crossover wave
function:
Basis for :
• BdG theory (T=0)
• T=0 Gross-Pitaevskii theory in the BEC
• Unequal population theories
Simplicity and physical accessibility
Eagles and Leggett
Unitary

95. Thermal excitations
 Pairs form without condensation  pseudogap.
 is natural measure of bosonic degrees of freedom.
Two types of excitations
Except in BCS
BCS Unitary BEC
• Novel form of superfluidity
• Never seen before, except
possibly in high Tc

96. Pseudogap seen in high TC superconductors!
Pseudogap (normal state gap) is very prominent.
BCS-BEC crossover physics is a possible explanation.
High TcBCS
Introducing pseudogap into Fermi gases
Ding et al, Nature 1996

97. Other Theoretical Work at T  0
• Nozieres and Schmitt-Rink, 1985: T=Tc, inconsistent
• Most theoretical work in atomic Fermi gases:
– T=0 or strict mean-field calculation at finite T
– Follow NSR -- no noncondensed pairs in the gap equation.
– Many do not include the trap effect.
– Some based on BdG – no noncondensed pairs
• Other theories which address superfluid density: Report first
order transitions (Zwerger-Haussmann), or double valued
functions (Griffin) or breakdown of theory at Tc/2 (Strinati), or
artificial fixing with unphysical discontinuity at Tc (Hu and
Drumond)
• Stoof, Combesco, …

98. Two major schools of BCS-BEC crossover
theories
• BCS-Leggett extended to finite T.
– Leggett T=0 theory of BCS-BEC crossover in 1980.
– Ease and flexibility– can accommodate population imbalance, traps,
inhomogeneity.
– Pseudogap arises naturally.
– Better treatment of pair mass and fermion self-energy in the intermediate
crossover regime, whereas pairs are quasi-free.
• Nozieres Schmitt- Rink extended away from Tc.
(Strinati et al.).
– NSR theory of Tc in BCS-BEC crossover in 1985.
– Suffer from inconsistencies between equations. No pseudogap at Tc.
– Better treatment in the BEC limit at low T in terms of pair dispersion, but
encounters unphysical first order transition at Tc.
– Bad at intermediate (crossover) regime
• Ours can be regarded as BCS-Leggett-like, with proper inclusion of pairing
fluctuations. This is more readily applicable to cuprates

99. BCS-BEC Crossover
Sa deMelo, Randeria & Engelbrecht, PRL (93), PRB (97)
Leggett (80) Nozieres & Schmitt-Rink (85)
Randeria in “Bose-Einstein Condensation” (’95)
BCS regime:
• weak attraction
• cooperative
Cooper pairing
• pair size
BEC regime:
• strong attraction
• condensate of
tightly bound
molecules
• pair size
“attraction’’ 
Normal
Fermi liquid
Normal
Bose liquid
pseudogap
pairing
SC
Randeria,
Trivedi,
Moreo &
Scalettar,
PRL (‘92)

100. Pseudogap Discoveries
Randeria group noted a pseudogap was
present in BCS-BEC crossover, as a
spin gap. (Varenna, ‘97):
“There would be no pseudogap in the
charge channel.”
“The pseudogap phase was associated
with spin-charge separation”
Levin group was first to see pseudogap
in the spectral function.
This was a quasi-particle gap.
“No spin charge separation above Tc.”
The pseudogap would enter below Tc–
as pair excitations of condensate.
1992,
95,
97
1997

101. Physical Picture of the
Pseudogap
.
Due to stronger- than- BCS attraction pairs form at T* and
condense at Tc .
Non-condensed pairs appear below Tc as pair excitations of the
condensate.
Contrast with BCS
Crossover theory

102. 1. Spin singlets
2. Pre-formed pairs
3. Spin density wave
4. Charge density wave
5. d density wave
What is the Pseudogap Due to?
6. Orbital currents
7. Flux phase
8. Stripes/nematic
9. Valence bond solid/glass
10. Combination?