Introduction to the phenomenology of high temperature superconductors

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Introduction to the phenomenology of high
temperature superconductors

• Patrick Lee and T. Senthil

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High Tc Phase diagram

• Plan
• Overdoped is it conventional?
• What is strange about the strange metal?
• Phenomenology of the pseudogap
• Transition to superconductivity

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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.
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A preliminary look transport
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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?

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ARPES (Angle Resolved Photoemission Spectroscopy)
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Overdoped metal Is there a Fermi surface?
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deHaas van Alphen, other quantum oscillations
classic Fermi surface determination methods
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Remarks on quantum oscillations
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Quantum oscillations in Tl-2201
Tc 10 K, B upto 60 T oscillations in both M
and in c-axis ?
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Thermal conductivity Wiedemann-Franz law
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Is the OD state really a Fermi liquid?
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High Tc Phase diagram

• Plan
• Overdoped is it conventional?
• What is strange about the strange metal?
• Theory interlude
• Phenomenology of the pseudogap
• 5. Transition to superconductivity

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The strange metal electrical transport
Linear-T resistivity near optimal doping with
nearly zero intercept. Slope of
resistivity/layer roughly the same (1.5 µO cm/K)
for all materials. Sheet resistance ?/d
(h/e2) T/J
Bi-2201
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Linear resistivity at very low-T
Tied to quantum criticality? Quantum critical
point second order phase transition at T 0
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Magnetotransport Hall effect
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Optical transport high frequency tail
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Optical transport low frequency peak
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Spin physics spin susceptibility and NMR
relaxation
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Dynamic spin correlations neutron scattering in
LSCO
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ARPES Fermi surface structure
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Analysis of ARPES data
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Absence of Landau quasiparticles
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Transition to SC onset of coherence
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Onset of coherence in transport
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Neutron resonance
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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.
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Brief theory interlude

• Some basic questions
• How does a
• metal emerge from a Mott insulator?
• 2. Why superconductivity?

Simple physical picture (Anderson1987)
Superexchange favors formation of singlet
valence bonds between localized spins. Doped
Mott insulator Hole motion in background of
valence bonds.
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Cartoon pictures
Large doping Hubbard-U not very effective in
blocking charge motion Expect large Fermi
surface with area set by 1-x. What happens as
doping is reduced to approach Mott insulator?
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Cartoon pictures
Low doping Most of the time most
electrons unable to hop to neighboring sites due
to Mott-blocking. If electrons stay localized
next to each other long enough, will develop
superexchange which will lock their spins into
singlets. Electron configuration changes at
long times conveniently view as motion of
holes in sea of singlets. Resulting state
metallic but with a spin gap due to valence bond
formation gt pseudogap metal.
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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 gt no
superconductivity. Nonzero doping allow room
for motion of valence bonds gt superconductivity!
Hole picture Coherent hole motion in valence
bond sea
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Fate of collection of valence bonds
Two general possibilities Valence bonds can
crystallize to form a solid (Valence Bond
Solid) OR Stay liquid to form a
Resonating Valence Bond Ongoing debates on
which one is more relevant but very formation of
valence bond crucial ingredient in much thinking
about cuprates.
VBS state (with doping a bond centered stripe)

RVB state quantum spin liquid
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Cartoon understanding of phase diagram
Formation of singlet valence bond
Coherence of hole motion
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Does valence bond formation provide a legitimate
theoretical route for superconductivity in a
repulsive doped Mott insulator?

• Many different kinds of studies
• 1d doped spin ladder
• Zero doping spin gapped insulator due to
  valence
• bond formation.
• Dope (power law) superconductor .
• 2. Quasi-1d Weakly coupled ladders
• 3. Inhomogenous 2d Checkerboard Hubbard model
• 4. Superconductivity in doped VBS Mott insulators
  (large-N methods) spontaneously generate
  weakly coupled ladders.
• 5. Superconductivity in doped spin liquid Mott
  insulators (i.e insulators with one electron per
  site)

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Superconductivity in doped spin liquids mean
field
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Superconductivity in doped spin liquids
variational wavefunctions
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Common features of superconductivity in doped
(paramagnetic) Mott insulators
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High Tc Phase diagram

• Plan
• Overdoped is it conventional?
• What is strange about the strange metal?
• Theory interlude
• Phenomenology of the pseudogap
• 5. Transition to superconductivity

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Pseudogap suppression of density of states in
many probes
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Tunneling
More on STM later!
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Out-of-plane transport
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In plane transport
YBCO6.75
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Pseudogap state in ARPES
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Quantifying the pseudogap in ARPES
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Evolution of pseudogap with doping
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T-dependence Gapless Fermi arcs
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Fermi arcs shrink with decreasing T
Extrapolate to 0 as T goes to 0?
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Summary of ARPES Fermi surface evolution

• Big antinodal gap 50 meV or bigger
• 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

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New mystery quantum oscillations in a magnetic
field at low T
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High field ground state contrast between under
and over-doped
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How do all this fit together?
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How to fit together?
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Arcs versus pockets

• Could it be that the arcs are really just one
  side of a closed pocket near the nodal region?

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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
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STM different measurement modes
http//hoffman.physics.harvard.edu/research/STMmea
s.php
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STM in the cuprates at low-T d-wave gaps and
spatial inhomogeneity
70 meV
Spatially averaged spectra consistent with
d-wave gap
20 meV
But gap varies strongly on nm scale!
Pan, Hudson, Davis et al, 2001, 2002
Is the inhomogeneity just a surface property or
does it affect bulk physics? Note ARPES probes
the same surface!
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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.

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Phase fluctuations above Tc Nernst/diamagnetism

• If Tc controlled by phase stiffness, might expect
  region with enhanced superconducting phase
  fluctuations in the normal state above Tc.
• Experiment Microwave conductivity
  (Corson,..Orenstein)
• Nernst effect and diamagnetism (Wang, Li,, Ong)
• (next few slides courtesy of Patrick, Lu Li)
• This fluctuations regime surely exists but does
  not extend all the way to T.

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Vortex Nernst effect
Vortices move in a temperature gradient Phase
slip generates Josephson voltage
Wang et al. PRB(2001)
EJ B x v
Nernst signal
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Diamagnetic signal need high-resolution
magnetometry !
Magnetization curve of type-II superconductors
Magnetization study

• Advantage
• Clear determination of Hc2 and Hc1
• Area Condensation energy U

• Difficulty
• In cuprates, Hc2 50-150 T
• M lt 1000 A/m ( 12 G)
• HARD to resolve with commercial SQUID
  magnetometers ( Hmax 5
  T or 7 T)

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Torque magnetometry
Torque on moment ? m B
Deflection of cantilever ? k f
Meff t / m0HVsin(f)
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Examples of Magnetization curves (I) M vs. T and
the onset temperature Tonset
Enhanced diamagnetic signals above TC
Meff t / m0HVsin(q)
Weakly linear orbital background DcorbH
Lu Li, Thesis
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Conclusion
Diamagnetism up to 130 K
Non-linear M-H
A universal SC fluctuation onset temperature vs x?
Bi2212
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Other order and fluctuations Antiferromagnetism
AF LRO disappears at very low doping but some
soft spin fluctuations persist to high doping.
Eg Neutron resonance in SC state seen in most
cuprates.
Resonance frequency decreases with Tc in
underdoped. Soft mode of AF LRO?
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Universal spin fluctuation spectrum of
superconducting cuprates
Yamada plot for LSCO
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Broken translation symmetry I charge stripes
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Broken translation symmetry in STM methods
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Broken translational symmetry in STM
bond-centered glass
Cu-O-Cu bond-centered Trans. symmetry breaking
C4?C2
Science 315, 1380 (2007) , Nature 454, 1072,
(2008)
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Checkerboards/CDW
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Checkerboard/CDW stronger in Bi-2201
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Electronic nematics
Break lattice rotation symmetry without breaking
translation symmetry
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Correlation with T
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Field induced magnetic ordering at low-T
Magnetic field stabilizes SDW order in favor of
superconductivity. Related to vortex core
checkerboard of Hoffman et al?
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Summary of some important underdoped phenomena
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A phenomenological synthesis
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Ong high field phase diagram
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Key assumption Electron coherence in a field
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Physics across Tc at zero field
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Modeling single particle incoherence
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Pseudogap and Fermi arcs
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Summary of synthesis
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Back to basic theory questions

• Many different kinds of studies
• 1d doped spin ladder
• Zero doping spin gapped insulator due to
  valence
• bond formation.
• Dope (power law) superconductor .
• 2. Quasi-1d Weakly coupled ladders
• 3. Inhomogenous 2d Checkerboard Hubbard model
• 4. Superconductivity in doped VBS Mott insulators
  (large-N methods) spontaneously generate
  weakly coupled ladders.
• 5. Superconductivity in doped spin liquid Mott
  insulators (i.e insulators with one electron per
  site)

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Refined basic theory questions

• Is superconductivity with gapless nodal
  excitations possible in a doped Mott insulator?
• Only currently known route is by doping a gapless
  spin liquid Mott insulator.
• Does this force us to a spin liquid based
  approach to cuprates?

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More questions

• More generally, large Fermi surface visible (at
  least at short time scales) already in
  underdoped.
• How should we understand the emergence of the
  large Fermi surface in a doped Mott insulator?

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Even more questions
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Last question