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22 A Spin Fluctuation Model for d-Wave Superconductivity

A.V. Chubukov Univ. of Wisconsin, Dept. of Physics, Madison, USA

D. Pines Los Alamos National Laboratory CNLS, USA

J. Schmalian Ames Laboratory, Dept. of Physics and Astronomy, USA

22.1 Introduction and Overview ...............................................................1349

22.2 Spin–Fermion Model .....................................................................1358

22.2.1PhysicalMotivationoftheSpin–FermionModel.......................................1358

22.2.2Weak-CouplingApproachtothePairingInstability....................................1361

22.3 Summary of Strong-Coupling Theory for Electron–Phonon Pairing ..........................1363

22.4 Strong-Coupling Approach to Spin–Fermion Interaction ....................................1366

22.4.1Normal-StateBehavior .............................................................1366

22.4.2 The d

−y

PairingInstabilityTemperature ............................................1371

22.4.3SuperconductingState..............................................................1372

22.5 Fingerprints of Spin–Fermion Pairing .....................................................1377

22.5.1ThePhysicalOriginoftheEffect.....................................................1377

22.5.2 Similarities and Discrepancies Between d-Wave and s-WaveSuperconductors............1379

22.5.3TheSpectralFunction..............................................................1380

22.5.4TheDensityofStates...............................................................1381

22.5.5SISTunneling......................................................................1384

22.5.6OpticalConductivityandRamanResponse...........................................1386

22.6 Comparison with the Experiments on Cuprates .............................................1389

22.6.1ParametersoftheModel............................................................1389

22.6.2TheNormalState ..................................................................1391

22.6.3TheSuperconductingState..........................................................1394

22.6.4ExperimentalFactsThatWeCannotYetDescribe......................................1398

22.6.5PhaseDiagram ....................................................................1399

22.7 Conclusions .............................................................................1403

22.8 Note Added .............................................................................1405

References..............................................................................1407

22.1 Introduction and Overview

The identification of the microscopic mechanisms

responsible for superconductivity and the nature of

thesuperconducting pairing state continuesto repre-

sent oneof the most exciting challengesin theoretical

physics [1]. In the so-called conventional supercon-

ductors, at frequencies less than or comparable to

the Debye frequency, the attractive phonon-induced

interaction between electrons wins out over the re-

pulsive screened Coulomb interaction [2] and brings

about superconductivity [3].The pairing ofelectrons

in the superconducting state is in an s-wave channel.

The primacy of phonon-induced interaction in con-

ventional superconductors has been demonstrated

with great clarity. The phonon density of states, ob-

1350 A.V. Chubukov, D. Pines, and J. Schmalian

tained by inelastic neutron scattering experiments,

and the spectrum of the bosons which mediate pair-

ing, as deduced from tunneling experiments, agree

very well in systems like Pb [4,5]. In addition to the

isotope effect [6,7],this comparison of two indepen-

dent experiments is generally considered to be a very

reliable proof of a phonon-mediated pairing state.

The analysis of the tunneling data relies heavily on

the existence of strong coupling effects in the quasi-

particle density of states and assumes the validity

of the Eliashberg approach to superconductivity [8].

Eliashberg theory for conventional superconductors

is extremely robust due to the decoupling of typical

electron and phonon time scales caused by the small

ratio of the velocity of sound and the Fermi velocity.

This smallness also implies that the Debye frequency

is much smaller than the Fermi energy, and hence

the quasiparticles that participate in the pairing are

low-energy quasiparticles, located in the near vicin-

ity of the Fermi surface. Landau [9] showed that the

low energy properties of a normal Fermi liquid are

characterized by a small number of parameters and

are independent of the details of the underlying lat-

tice Hamiltonian. The success of the BCS-Eliashberg

theory for conventional phonon mediated supercon-

ductors is almost unique for interacting many body

systems.

It is well known however that the pairing state

in the Bardeen–Cooper–Schrieffer theory does not

need to be caused by the interaction between elec-

trons and lattice vibrations.Generally there are two

distinct classes of theories of non-phonon pairing.

The first, and more conservative approach is to re-

place phonons by another collective bosonic excita-

tion of the solid.This approach successfully describes

the physics of superﬂuid

He [10], where the inter-

mediate bosons are failed ferromagnetic spin ﬂuctu-

ations (ferromagnetic paramagnons); a large value

of the ferromagnetic correlation length is not even

required in view of the Kohn–Luttinger effect [11].

This magnetically-mediated interaction causes pair-

ing in a state with angular momentum l =1(p-wave

pairing) and leads to a rich phase diagram and a

large class of new physical phenomena [10,12, 13].

Magnetically-mediated superconductivity has been

proposed for various organic and heavy fermion su-

perconductors by a number of authors [14–20]. A

second approach to unconventional pairing is more

phenomenological, and is based on the assumption

that the superconducting condensation energy is not

determined by the attractive interaction mediated by

some boson but ratherby the energy gain due to feed-

back effects associated with pairing. The latter may,

in principle, occur even for a purely repulsive pair-

ing interaction. In general, this approach assumes a

non-Fermi-liquid behavior in the normal state. Two

examples are the inter-layer tunneling model of An-

derson et al. [21,22] and the mid-infrared model of

Legett [23]. In both models, the expectation value of

the Hamiltonian is drastically different in the super-

conducting and the normal states, and the energy

gain due to pairing apparently cannot be traced back

to some boson-mediated attraction.

The two approaches to unconventional pairing

are not necessarily in contradiction with each other.

First, at strong coupling, the pairing interaction me-

diated by a low-energy bosonic mode is highly re-

tarded and is a complex function of frequency, so

it is not straightforward to determine whether it is

repulsive or attractive. Second the pairing obviously

changes the form of the fermionic self-energy and

hence affects the kinetic energy. And third, if the

bosonic mode is itself made out of fermions,then the

propagator of this mode also changes when the sys-

tem becomes superconducting. This change affects

the potential energy of the system. Which of the two

effects is larger depends on the details of the system

behavior, but in any case, there are clear similarities

between a strong-coupling theory which involves a

pairing boson, and a scenario based entirely on en-

ergy gain due to feedback from pairing.

In what follows we adopt the first approach and

investigate the role of antiferromagnetic spin ﬂuctu-

ations (antiferromagnetic paramagnons) as a possi-

ble cause for both anomalous normal state behavior

and unconventional i.e. non-s-wave superconduc-

tivity. This approach is chieﬂy motivated by the

physics of the high temperature cuprate supercon-

ductors which have been shown to exhibit both

highly anomalous normal state behavior and an un-

conventional pairing state with angular momentum

l =2(d-wave pairing) [24–26]. As may be seen in