eliashberg theory of spin-fluctuation pairing in cuprate...
TRANSCRIPT
Andrey Chubukov
University of Wisconsin
Chernogolovka, October 9, 2010
Eliashberg theory of spin-fluctuation pairing in cuprate superconductors
Dear Sima and Seva:
Electron-doped
Hole-doped
High Tc cuprates
Electron-doped
Hole-doped
• Parent compounds are antiferromagnetically ordered
• Overdoped, but still superconducting materials, are Fermi liquids
• Superconducting state has d-wave symmetry
Why spin fluctuations?
Overdoped compounds are metals and Fermi liquids
Tl2Ba2CuO6+
Vignolle et al
Photoemission
Plate et al
Oscillations in resistance/magnetization
Areas are consistent with Luttinger count for electrons in a Fermi liquid (1+x)
X
G
Campuzano et al
2 cos )( 0
0
0
0
0
Shen, Dessau et al 93, Campuzano et al, 96
kF
The superconducting gap has d-wave symmetry
From the very early days of high-Tc
Scalapino et al, Monthoux,
Balatsky & Pines, Carbotte…
What if we replace phonons by antiferromagnetic spin fluctuations,
and perform BCS-type calculations?
(select spin channel)
=
p ,
'' p- ,
p ,
p- ,
'' p ,
p- ,
Analog of BCS
+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
tic)(paramagne
positive is q)-(kspin
No s-wave solution
Eqn. for asc gap
+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
) ,(at peaked is
q)-(k that Assume spin
Eqn. for asc gap
+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
gap d 2 2 yx
) ,(at peaked is
q)-(k that Assume spin
Eqn. for asc gap
Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
“Non-Fermi liquid” physics(most likely, Fermi liquid, but with small upper edge)
resistivity
T (T) ρ
22// )(
liquid, Fermi aIn
T
2T (T) ρ
liquid Fermi aIn
photoemission
)( ''
T. Valla et al
Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
• There is pseudogap behavior over a wide range of temperatures
Pseudogap
85 K
4.2 K
Bi2Sr2CaCu2O8
(Tc = 82 K)
500 0 -500 -1000 -1500 -2000 cm-1
-500 0 500 1000 1500 2000
Ch.Renner et al.PRL 80, 149 (1998)
dI/dV
H.Ding et alNature 382, 51 (1996)
170 K 85 K 10 K
300 K
STM
ARPES
IR:1/t
Raman
85 K
85 K
300 K
300 K
400 800 1200 1600 20000
1000
2000
tcm
-1
400 800 1200 1600
G.Blumberg et al.Science 278, 1427 (1997)
A.Puchkov et alPRL 77, 3212 (1996)
10 K
10 K
Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
• Parent compounds of cuprates are not just antiferromagnets,they are also Mott insulators.
• There is pseudogap behavior over a wide range of temperatures
Mott insulators
parent compounds(magnetic)
Resistivity
Ando et al
Mott insulators
Onose et al,
Zimmers et al
The same Mott gap of 1.7 eV in hole-doped and el-doped materials
Optical conductivity
What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)
• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required
• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –
one can study pairing using conventional theories
+ Small amount of holes
What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)
• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required
• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –
one can study pairing using conventional theories
+Hole
pockets
The crossover from small to large Fermi surface:
Hole
pockets Electron
pockets
A lot of fluctuations around these states, but Fermi liquid survives at T=0
Physical Review B 81, 140505(R) (2010)
Suchitra E. Sebastian, N. Harrison,
M. M. Altarawneh, Ruixing Liang, D. A. Bonn,
W. N. Hardy, and G. G. Lonzarich
Evidence for small Fermi pockets
Original observation:
N. Doiron-Leyraud, C. Proust, D.
LeBoeuf, J. Levallois,
J.-B. Bonnemaison, R. Liang,
D. A. Bonn, W. N. Hardy,
and L. Taillefer,
Nature 447, 565 (2007)
Evidence for small Fermi pockets
L. Taillefer, cond-mat. 0901.2313
d-wave
superconductor
An
tife
rrom
agnet
Incommensurate
antiferromagnetism
Evidence for antiferromagnetism
d-wave
superconductor
An
tife
rrom
agnet
Incommensurate
antiferromagnetism
Evidence for antiferromagnetism
Let’s take antiferromagnetism and change in Fermi
surface as the two ingredients of cuprate phase diagram
Quantum-criticalbehavior
Sachdev,…
Small Fermipockets
Question: can we get high Tc superconductivity?
Quantum-criticalbehaviorSmall
Fermipockets
Pairing mediated by strong antiferromagnetic spin fluctuations
An exchange of antiferromagnetic spin fluctuations
yields d-wave pairing
Weak coupling: just replace phonons by spin fluctuations
)/-(1
sfc e ~ T K) 300- (200 meV 30-20 ~ sf 100K ~ T 1 c
It is possible to explain Tc ~ 100 K within weak/moderate coupling theory
Fermi liquid
Weak couplingBCS
Inconsistent
Need to go beyond BCS
Eliashberg theory of spin fluctuation mediatedd-wave superconductivity
Search for “Eliashberg” yielded about 2500 papers only in PRB
Model: fermions coupled to their collective
bosonic fluctuations in the spin channel
(spin-fermion model)
Ingredients: low-energy fermions ( ) , spin excitations ( )
and spin-fermion interaction
Parameters: one overall energy scale one dimensionless coupling
g1
Fv/ g
1Fv g
Strong coupling: when ,
The scale g is still assumed to be smaller than the fermionic bandwidth
(mass renormalization m*/m = 1 + )
Why Eliashberg theory?
In bare theory, there is no difference between velocity of fermions and velocity of bosons – both are Fermi velocities, hence no Migdal theorem.
Once you dress up fermions AND bosons by self-energies, bosonsbecome Landau overdampred and hence slow compared to fermions
),k-(k v~ 0) (k, , ) ,(k FFF
, O(1) k)v()/ (k, ,1 )/ (k, F
Vertex corrections: g /g = O(1)
One needs large N to put theory under control [O(1) -> O(1/N)]
Logics:
• compute normal state fermionic and bosonic propagators(with self-energies included)
• compare with experiments, extract and g
• use the renormalized propagators and Eliashbergtheory for the pairing problem, see what Tc andfeedbacks from the pairing on electrons we get
At this stage, no free parameters.
Consider first doping region with no
Fermi surface reconstruction
Collective spin excitations can decay into particles
and holes and become Landau overdamped
Fermions acquire a finite damping due to interaction with Landau overdamped collective excitations
Normal state
sf/ i - 1
1 Im ~ ) ,( Im
2
2
sf
g
64
9 g/
2-1.5 ~ meV 30-20 ev,7.1~ sf g
Spin fluctuations in the normal state form a gapless continuum
Fong et al
1
2
Fermionic self-energy
Data: Kaminski et al
Nodal
MFL-like behavior
Anti-nodal
Fermi liquid is the upper boundary
of the Fermi liquid behaviorsf
sf
2
Conductivity in YBCO7
theory
experiment(Basov et al)
Quasiparticle dispersion from photoemission
Theory
Experimentpeak
hump
Fermi liquid
-2
sf Quantum-critical
behavior
upper boundary of a Fermi liquid
upper boundary of thestrong couping behavior
energy
g
The pairing problem
Two candidate scales
at vanishes
g ~ e ~ T -2)/ -(1
sfc
at finite
remains g, ~ Tc
2
g ~ sf
) (k, ) (k,
(if pairing is within a FL) (if pairing involves fermions outside of a FL)
Pairing in non-Fermi liquid regime is a new phenomenon
2/1 )( 2/1
L )( )(/ 1
,)g/|(| 1
1
|-| ||
)( T
4 - )(
2/12/12/1
kk
Equation for the pairing vertex has non-BCS form
This is strong coupling limit of Eliashberg theory
This is NOT BCS – summing up logarithms gives no pairing at all
0Tat g
)(
4/1
0
K) (200 at
g 0.025 T ins
Full solution (beyong logarithmic aproximation):the pairing instability exists at Tins ~ g
Fermi liquid pairing only, Tins ~ sf
T/g0.03
0.015
(same as the distance from a magnetic instability)
Abanov, A.C,Finkelstein
This problem is actually quite generic
1/2
) (log 0
2
1/4
1/3
Antiferromagnetic QCP
Ferromagnetic QCP
2kF QCP
Pairing by near-gapless phonons
3D QCP, Color superconductivity
1 0
0.7
1 Z=1 pairing problem
Abanov, A.C., Finkelstein, hot spots
Haslinger et al, Millis et al, Bedel et al…2/3 problem: gauge field, nematic ….
Krotkov et al, electron-doped
Allen, Dynes, Carbotte, Marsiglio, Scalapino, Combescot, Maksimov, Bulaevskii, Rainer, Dolgov, Golubov, …
Son, Schmalian, A.C….
pairing in the presence of SDW Moon, Sachdev
fermions with Dirac cone dispersion Metzner et al
2
- 1 ,
)g/|(| 1
1
|-| ||
)( T )(
-1
0.18 T Dc
Quantum-criticalbehavior
Pairing in the presence of Fermi surface reconstruction
Tc decreases due to the reduction in spin-fermion vertex
MoonSachdev
Numbers match – their EF for POCKETS is 10 times smaller than g
Pairing by spin waves,
relevant scale is J ~100 meV
Quantum-criticalbehavior
Story is far from completion:
• Linear in resistivity is NOT explained
Quantum-criticalbehavior
Theory: resistivity in the critical regime is linear in T only above some T0
• The nature of the pseudogap phase is not fully understood
It is natural to associate this phase with SDWprecursors (fluctuating pockets) , but
• There surely are superconducting fluctuations above Tc
• There are experimental evidences for potential discretesymmetry breaking in the pseudogap phase
Neutron scattering: breaking of rotational symmetry
Could be pre-emptive ordering of Ising degree of freedomassociated with incommensurate magnetic order [(Q,) or (,Q)], but this remains to be seen.
Conclusions
Recent experiments show that cuprates are more “conventional” than previously thought
• Coherent fermionic excitations are present at all dopings
• Long-range magnetic order extends up to optimal doping
This gives weight to the scenario that the pairing in the cuprates is mediated by near-critical spin fluctuations, and from this perspective is not that different from the pairing in heavy-fermion and organic superconductors
Quantum-criticalbehavior
• Artem Abanov (LANL/Texas)
• Sasha Finkelstein (Weizman/Texas A&M)
• Rob Haslinger (LANL)
• Dirk Morr (Illinois-Chicago)
• Joerg Schmalian (Iowa)
• Mike Norman (Argonne)
• Pavel Krotkov (Maryland)
• Ilya Eremin (Dresden)
• Karl Bennemann (Berlin)
• Oleg Tchernyshov (Baltimore)
• Boldizsar Janko (Notre Dame)
• David Pines (UC Davis)
• Philippe Monthoux (Edinburg)
• Matthias Eschrig (Karlsruhe)
• Tigran Sedrakyan (Maryland)
• A. Millis (Columbia)
• E. Abrahams (UCLA)
• S. Maiti (Wisconsin)
• D. Dhokarh (Wisconsin)
Collaborators
THANK YOU