Lecture 7. The pseudogap
•Hole Doping and the Phase Diagram
•Strange Metals
•Experimental Probes
•Current Pseudogap Theori...
A Brief History
Sir Neville Mott described the dispersion
diagram (E vs.k) for disordered materials
(no lattice) in terms ...
HTSC
Doping and the Phase Diagram
Cuprates
High Tc Phase diagram
Plan
1. Overdoped – is it
`conventional’?
2. What is strange about the
strange metal?
3. Phenomenolo...
Eisaki et al, PRB 69,
064512 (2004)
With further increase of
layers, Tc does not go up
further. The inner planes
have less...
Pseudogap in
Other Superconducting Materials?
Experimental Probes
ARPES
(Angle Resolved Photoemission
Spectroscopy)
ARPES (Angle Resolved Photoemission
Spectroscopy)
Prototype HTSC Materials
Evolution of the Fermi surface with doping
Doiron-Leyraud et al
Nature (2007)
Tunneling Spectroscopy
Introduction to the phenomenology
of high temperature
superconductors
Patrick Lee and T. Senthil
2013-9-11 60
Eagles (1969): Pairing without superconductivity
ATHIC2008, TSUKUBA, JAPAN
In HTSC, the root of PG is not
qui...
A preliminary look: transport
Overdoped metal
• Does it have a Fermi surface? Size and shape?
Methods to detect – ARPES, deHaas-van Alphen
and related q...
Overdoped metal: Is there a Fermi
surface?
Quantum oscillations in Tl-2201
Tc = 10 K, B upto 60 T;
oscillations in both M and in
c-axis ρ
High Tc Phase diagram
Plan
1.Overdoped – is it
`conventional’?
2.What is strange about the
strange metal?
3.Theory interlu...
The strange metal: electrical transport
Linear-T resistivity near optimal doping
with nearly zero intercept.
Slope of resi...
Magnetotransport: Hall effect
ARPES: Fermi surface structure
Summary on strange metal
Strange metal: Power laws in many physical quantities;
Large Fermi surface but no Landau-like qua...
Why superconductivity?
Crucial Anderson insight:
Singlet valence bond between localized spins: A localized Cooper pair.
`P...
Cartoon understanding of phase diagram
Formation of singlet
valence bond
Coherence of hole motion
High Tc Phase diagram
Plan
1.Overdoped – is it
`conventional’?
2.What is strange about the
strange metal?
3.Theory interlu...
Tunneling
More on STM later!
Pseudogap state in ARPES
Evolution of pseudogap with doping
Summary of ARPES Fermi surface
evolution
1. Big antinodal gap – 50 meV or bigger
2. Gapless Fermi arcs near node that shri...
High field ground state: contrast
between under and over-doped
Scanning tunneling microscopy (STM)
(Credit: Jenny Hoffman website)
Tunnel electrons from metallic tip to surface of
syste...
STM: different measurement modes
http://hoffman.physics.harvard.edu/research/
STMmeas.php
STM in the cuprates at low-T: d-wave
gaps and spatial inhomogeneity
20 mV
)(r


Spatially averaged spectra;
consistent w...
Competing order and fluctuations
Apart from superconductivity, many other ordered or
nearly ordered (i.e short range order...
Carrier (hole)
concentration
d-wave
T*
BEC BCS
Tc Fermi
Liquids-wave
Superfluid
Pseudo
-gap
High Tc Cuprates Cold Fermi Ga...
High Tc SC in cuprates
• Highest known Tc (in K)
* electrons
• Repulsive interactions
• d-wave pairing
• near Mott transit...
BCS-BEC crossover and superfluidity in
atomic Fermi gases
Qijin Chen (陈启谨)
Zhejiang Institute of Modern Physics and
Depart...
Bosons vs fermions
BEC Fermi sea
EF = kBTF
spin 
spin 
T = 0
Pauli exclusion
Essence of Fermionic Superfluidity
fermions bosons
Attractive interactions turn fermions into
“composite bosons” (or Coope...
Remarkable Tuning Capability in Cold Gases via
Feshbach Resonance
.
molecules
→ ←
B>
Scattering length a
BCSBEC
Unitary l...
Crossover under control in cold Fermi
atoms (1st time possible)
Molecules of
fermionic atoms
BEC of bound
molecules
Cooper...
Overview of BCS theory
Fermi Gas
No excitation gap
BCS superconductor
2013-9-11 93
Theoretical History of Crossover:
• Leggett (1980) noted that BCS T=0 wavefunction could be
generalized to ar...
Zero T BCS-BEC crossover:
Tuning the attractive interaction
 Change of character:
fermionic  Bosonic
(Uc – critical coup...
Thermal excitations
 Pairs form without condensation  pseudogap.
 is natural measure of bosonic degrees of freedom.
Two...
Pseudogap seen in high TC superconductors!
Pseudogap (normal state gap) is very prominent.
BCS-BEC crossover physics is a ...
Other Theoretical Work at T  0
• Nozieres and Schmitt-Rink, 1985: T=Tc, inconsistent
• Most theoretical work in atomic Fe...
Two major schools of BCS-BEC crossover
theories
• BCS-Leggett extended to finite T.
– Leggett T=0 theory of BCS-BEC crosso...
BCS-BEC Crossover
Sa deMelo, Randeria & Engelbrecht, PRL (93), PRB (97)
Leggett (80) Nozieres & Schmitt-Rink (85)
Randeria...
Pseudogap Discoveries
Randeria group noted a pseudogap was
present in BCS-BEC crossover, as a
spin gap. (Varenna, ‘97):
“T...
Physical Picture of the
Pseudogap
.
Due to stronger- than- BCS attraction pairs form at T* and
condense at Tc .
Non-conden...
1. Spin singlets
2. Pre-formed pairs
3. Spin density wave
4. Charge density wave
5. d density wave
What is the Pseudogap D...
Lecture 7 pseudogap
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Lecture 7 pseudogap

  1. 1. Lecture 7. The pseudogap •Hole Doping and the Phase Diagram •Strange Metals •Experimental Probes •Current Pseudogap Theories •Pseudogap in BEC?
  2. 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.
  3. 3. HTSC Doping and the Phase Diagram Cuprates
  4. 4. 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
  5. 5. 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.
  6. 6. Pseudogap in Other Superconducting Materials?
  7. 7. Experimental Probes
  8. 8. ARPES (Angle Resolved Photoemission Spectroscopy)
  9. 9. ARPES (Angle Resolved Photoemission Spectroscopy)
  10. 10. Prototype HTSC Materials
  11. 11. Evolution of the Fermi surface with doping Doiron-Leyraud et al Nature (2007)
  12. 12. Tunneling Spectroscopy
  13. 13. Introduction to the phenomenology of high temperature superconductors Patrick Lee and T. Senthil
  14. 14. 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.
  15. 15. A preliminary look: transport
  16. 16. 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?
  17. 17. Overdoped metal: Is there a Fermi surface?
  18. 18. Quantum oscillations in Tl-2201 Tc = 10 K, B upto 60 T; oscillations in both M and in c-axis ρ
  19. 19. 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
  20. 20. 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
  21. 21. Magnetotransport: Hall effect
  22. 22. ARPES: Fermi surface structure
  23. 23. 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.
  24. 24. 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
  25. 25. Cartoon understanding of phase diagram Formation of singlet valence bond Coherence of hole motion
  26. 26. 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
  27. 27. Tunneling More on STM later!
  28. 28. Pseudogap state in ARPES
  29. 29. Evolution of pseudogap with doping
  30. 30. 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
  31. 31. High field ground state: contrast between under and over-doped
  32. 32. 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
  33. 33. STM: different measurement modes http://hoffman.physics.harvard.edu/research/ STMmeas.php
  34. 34. 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!
  35. 35. 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.
  36. 36. 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
  37. 37. 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
  38. 38. 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
  39. 39. Bosons vs fermions BEC Fermi sea EF = kBTF spin  spin  T = 0 Pauli exclusion
  40. 40. 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
  41. 41. Remarkable Tuning Capability in Cold Gases via Feshbach Resonance . molecules → ← B> Scattering length a BCSBEC Unitary limit a > 0 a < 0
  42. 42. 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
  43. 43. Overview of BCS theory Fermi Gas No excitation gap BCS superconductor
  44. 44. 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
  45. 45. 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
  46. 46. 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
  47. 47. 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
  48. 48. 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, …
  49. 49. 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
  50. 50. 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)
  51. 51. 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
  52. 52. 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
  53. 53. 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?

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