Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
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908 C. C. Tsuei and J.R. Kirtley
a wide range in YBCO (Sect. 16.4). The results of
several phase-sensitive experiments (Sect. 16.4) also
consistently suggest that a subcomponent of the d-
wave order parameter in the form of d
x
2
−y
2
+ is or
d
x
2
−y
2
+ id
xy
, if it exists, is very small (less than a
few percent at most). The robustness of d-wave pair-
ing symmetry has been demonstrated also on the
surface of a Bi-2212 single crystal. In a recent c-
axis tunneling study of stacks of intrinsic Joseph-
son junctions formed on Bi-2212 single crystals, the
quasiparticle branch of the I-V characteristics indi-
cate that the surface layer has a suppressed T
c
,butits
d-wave parameter symmetry remains intact [288].
In short, the experimental evidence is that the d-
wave pair state is remarkably ubiquitous and robust,
despite the fragility suggested in several theoretical
studies. Such universal characteristics foretell that
the d-wave pairing in cuprates has an intrinsic com-
mon origin.
It is well established by theory [221, 289] and
experiment [290,291] that two-dimensional strong
Coulomb correlations in the CuO
2
layers play a fun-
damental role in the physics governing the supercon-
ducting and normal-state properties of the cuprates.
All cuprates are characterized by a strong on-site
Coulomb repulsion which results in the universally
observed Mott metal-insulator transition at near-
zero doping. The strong on-site Coulomb repulsion
rules out simple s-wave pairing symmetry. For pair-
ing interactions of range up to next nearest neigh-
bors, a strong Coulomb repulsion also rules out all
pair state candidates (see Sect. 16.1.2) other than
d
x
2
−y
2
pairing in models based on a van Hove sin-
gularity in the 2D band structure of the CuO
2
planes
[292]. For related work, see also [293–295]. Robust
pairing correlationsin the d-wave channel are found
in a variety of theoretical models treating pairing
between holes moving in a background of antifer-
romagnetic fluctuations [296–299]. Of particular in-
terest, long-range Coulomb repulsion interaction is
found to enhance d-wave pairing [300] and to pro-
duce d-wave pairing even in the presence of charge
stripes [301]. These works have provided a basis for
understanding why d-wave pairing persists in the
under-doped regime. In short, the pervasive nature
of the d-wave superconductivity in cuprates is a nat-
ural consequence of strong Coulomb correlations in
the CuO
2
planes.
16.6.2 Constraints on Pairing Interaction
Although the establishment of d-wave pairing sym-
metry in high-T
c
cuprate superconductors does not
necessarily specify a particular high-T
c
mechanism,
it does impose strong constraints on possible mod-
els for the high-T
c
mechanism. The pairing inter-
action responsible for superconductivity can be de-
fined with the aid of a generalized BCS Hamiltonian:
H =
k,
(k)c
†
k,
c
k,
+ H
pairing
, (16.35)
where (k) is the one-electron dispersion of the
quasiparticles in the CuO
2
band.The pairing interac-
tion H
pairing
must be compatible with the underlying
point group symmetry and can be expressed in terms
of the momentumtransfer k-k
=q (seeFig.16.34).For
scalar pairing interaction,
H
pairing
=
V(q)c
†
k+q,↑
c
†
−k−q,↓
c
−k,↓
c
k,↑
. (16.36)
The BCS gap equation can be expressed as:
(k)=−
V(k − k
)
(k
)
2E(k
)
tanh
$
E(k
)
2k
B
T
&
,
(16.37)
where the quasiparticle energy is defined as E(k)=
2
(k)+
2
(k).
For the case of d-wave pairing, the k-dependent
gap potential can be expressed as:
(k)=
0
(cos k
x
−cosk
y
) . (16.38)
To satisfy the conditions imposed by the d-wave
order parameter, (16.38), and the BCS gap equation,
(16.37),itisrequiredthatthepairinginteractionV(q)
is attractive (or less repulsive) for small momen-
tum transfer (q) and more repulsive for large q.For
thecaseofd-wave pairing, a constant can be added
to V(q) without affecting the BCS gap equation [2].
Hence, an all-repulsive V(q) can pair.A general form
of the pairing interaction of a d-wave superconduc-
tor is shown in Fig. 16.34.
d-wave pairing is consistent with a high-T
c
mech-
anism based on magnetic fluctuationmodels.In fact,
16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 909
Fig. 16.34. Schematic representation of the BCS pairing in-
teraction V(q), constrained by a d
x
2
−y
2 order parameter, as
a function of the momentum transfer q = k − k
the concept of d-wave pairing mediated by magnetic
fluctuations was proposed prior to the discovery of
high temperature superconductivity [302,303].This
work has stimulated much of the ensuing theoretical
work on high-T
c
mechanisms based on spin fluctua-
tions in strongly correlated cuprate superconductors
[296,297,304–309].The tremendous amount of work
developed in this field in conjunction with d-wave
pairing symmetry and HTS has been reviewed by
Scalapino [4] andbyChubokov,Pines,andSchmalian
in Chap. 22 of the present book.
The pairing interaction between charge carriers
and antiferromagnetic spin fluctuations can be ex-
pressed in terms of a q-dependent spin susceptibility
(q):
V(k − k
)=V(q)=g
2
(q) , (16.39)
where g is a measure of the coupling strength be-
tween charge carriers and spin fluctuations.Withthe
assumption that the cuprates can be considered as
a nearly antiferromagnetic Fermi-liquid system, the
spin susceptibilityfunction(q) is sharply peaked at
or near q =(, ). Such a repulsive pairing interac-
tion, when entered in the BCS gap equation (16.37),
always results in an order parameter with d
x
2
−y
2
sym-
metry, i.e., a gap with sign changes along the diago-
nals of the Brillouin zone.
One can calculate (q) from phenomenological
models to estimate the magnitude of T
c
.A(q) based
on NMR data on YBa
2
Cu
3
O
7−ı
predicts a T
c
of 100,
while a (q) determined from neutron scattering
data yields a T
c
≤ 20 K [310].
The electron-phonon pairing interaction, which
gives rise to conventional s-wave superconductivity,
does not naturally lead to a d
x
2
−y
2
gap symmetry.
The early observation of a vanishingly small iso-
tope effect in optimally doped high-T
c
cuprate su-
perconductors [311] supports a non-phonon high-
T
c
mechanism.However,recent systematic studies of
the isotope effects in several cuprate systems sug-
gest an important role for electron-phonon inter-
action in HTS [312,313]. The electron-phonon cou-
pling strength deduced from dynamical conductiv-
ity and dc resistivity data is too small to support
the HTS observed in cuprates. However, these con-
clusions are modified when one considers strong
Coulomb correlation effects. For example, it is sug-
gested that a strong electron-phonon interaction
renormalized by electronic correlations is respon-
sible for the apparently small values of electron-
phonon couplingstrength concluded from the trans-
port measurements [314–316]. The doping depen-
dence of the isotope effects can be understood on
the basis of electron-phonon interaction in the pres-
ence of strong Coulomb correlations [317–320].
All high-T
c
cuprate superconductors are charac-
terized by a relatively low carrier density. As a re-
sult, the ratio of k
B
T
c
to E
F
is in the range of 10
−1
–
10
−2
,whichis10
2
–10
3
times larger than that in
unconventional superconductors. The Migdal theo-
rem therefore holds for neither a spin-fluctuation
nor a phonon mediated high-T
c
mechanism [310,
319,321–323], and vertex corrections have to be in-
cluded in any HTS models. With the breakdown of
the Migdal approximation taken into account, d-
wave pairing symmetry has been found in the the-
ory of electron-phonon pairing interaction renor-
malized by strong Coulomb correlations [316]. In
particular, the pairing interaction V(q)hasamo-
mentum dependence shown in Fig. 16.34, i.e. a dom-
inant attractive electron-phonon forward scattering
in the small q regime, and a repulsive pairing in-
teraction for large q’s dominated by strong correla-
tions [314,316,324–327]. In other words, the on-site
Coulomb repulsion prevents pairing in the s-wave
channel while the electron-phonon coupling is en-
910 C. C. Tsuei and J.R. Kirtley
hanced in the small q regime, resulting in the next
available pair state: d
x
2
−y
2
superconductivity.Therole
of the electron-phonon interaction in HTS is also
emphasized in recent studies of angle-resolved pho-
toemission spectroscopy [328].
16.6.3 Nodal Excitations
In a conventional BCS superconductor, a minimum
energy of 2, the energy gap, is required to break a
pair to produce quasiparticle (qp) above the ground
state of the superfluid condensate. At low tempera-
tures (T T
c
), the number of thermally activated
quasiparticles is exponentially small, leading to ex-
ponentially reduced scattering processes. In an un-
conventional superconductor, however, low energy
quasiparticles can exist in significant numbers due
to the presence of nodes ( =0)inthek-dependent
gap function (k). These nodal quasiparticles are
responsible for the well-established power-law low-
temperature properties of the cuprate superconduc-
tors [51].
Recently, tremendous effort has been devoted to
the study of low energy nodal quasiparticles and qp
dynamics. With significant progress in both the the-
oretical and experimental fronts, this field has de-
veloped into an important area of d-wave supercon-
ductivity. In the following, some of the important
advances will be reviewed.
Superconductivity is assumed to prevail mostly in
theCuO
2
planesin thecupratesuperconductors,with
four nodes on the Fermi surface,located at k
x
= ±k
y
intheBrillouinzone(seeFig.16.35)ofthetwo-
dimensional Cu-O square lattice. The d-wave order
parameter can be written as (k)=
0
(k
2
x
− k
2
y
)/2. In
the vicinity of these nodes, the qp excitations spec-
trum can be approximated by a cone-like Dirac dis-
persion [329,330]:
E(k)=(v
2
F
k
2
1
+ v
2
k
2
)
1/2
, (16.40)
where k
1
, k
2
are local momentum variables at each
node; the Fermi velocity v
F
and the gap velocity v
arethecomponentsoftheqpgroupvelocitynor-
mal (parallel to k
1
) and perpendicular (parallel to
k
2
) to the Fermi surface respectively. The gap veloc-
ity v
owes its origin to the k dependence of the
Fig. 16.35. A schematic representation of the quasiparticle
dispersion E(k)atanodalpointintheBrillouinzoneofa
2D d-wave superconductor. Three other equivalent nodes
are not shown for clarity
gap, (k). The ratio v
F
/v
is an important measure
of the anisotropy of the low temperature qp state.
There are many important theoretical and experi-
mental studies of the nodal qp-dominated low en-
ergy physics in 2D d-wave superconductors. Exam-
ples include: qp transport [66,67,329–331], specific
heat [59, 61,62, 332], qp localization [333–340], qp
extended states [341,342], qp-qp interaction [343],
quantum critical phenomena [243,344], and vortex
state dynamics [345–349]. In particular, the domi-
nance of low energy nodal quasiparticles at low tem-
peratures has led to the prediction of novel thermo-
dynamicsandtransporteffects[329,332],whichwere
recently confirmed experimentally [59,66].
Low Temperature Specific Hea t
In the presence of a magnetic field H,theDiracdis-
persion (16.40) is modified, resulting in a Doppler
shift of the qp excitations outside the vortex core. In
the low temperature limit, T/T
c
√
H/H
c2
,Volovik
predicts that the electronic specific heatisdominated
by a term proportional to T
√
H/H
c2
. This effect was
observed in YBCO by Moler et al. [59], B. Revaz et
al.[62],andWright etal.[61].
These results represent important evidence in
support of d-wave pairing symmetry in YBCO. Fur-
ther, they are an important validation of the concept
of quasiparticles in an unconventional superconduc-
tor.
Universal Tra nsport Limit
In a d-wave superconductor,the presence of impuri-
ties generally induces a finite zero-energy density
16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 911
of quasiparticle states, concentrated at the nodes
on the Fermi surface [329,350]. It was predicted by
Lee [329] that the residual conductivity of a 2D d-
wave superconductor should be independent of im-
purity concentration. This universal transport limit
is a consequence of cancelation between the increase
in qp density of states and the decrease in qp mean
free path as impurity concentration is increased.
This novel effect can be generalized to include ther-
mal and spin conductivities [330,351]. Durst and
Lee [330] concluded that vertex corrections mod-
ify electrical conductivity, and Fermi-liquid correc-
tions renormalize both electrical and spin conductiv-
ities.Only thermal conductivity maintains its univer-
sal value obtained in the “bare bubble” calculations,
and is not affected by any of these corrections. Ther-
mal conductivity measurement is a well-established
probe of the bulk properties, and is free of surface
effects. The phonon contribution to thermal con-
ductivity can easily be separated from its electronic
counterpart. All these attributes make it an impor-
tant tool for studying the quasiparticle state at low
temperatures.
The residual thermal conductivity for a d-wave
cuprate superconductor is given by [330,351]:
T
=
k
2
B
3
n
d
v
F
v
+
v
v
F
, (16.41)
where n/d is the stacking density of the CuO
2
planes.
Taillefer et al. [66] reported the observation
of residual thermal conductivity in Zn-doped
YBa
2
Cu
3
O
6.9
. They found that the value of /T in
the zero temperature limit is independent of the Zn
concentration and in quantitative agreement with
the quasiclassical calculation for unconventional su-
perconductors [351]. Similar results were obtained
with Bi-2212 at optimal doping [352]. In the case
of Bi-2212, the value of the anisotropy parameter
v
F
/v
deduced from the thermal conductivity mea-
surement can be checked with the results of ARPES
measurements [53,353].An excellent agreement was
found [67]: For Bi-2212,
v
F
/v
= 19 (thermal conductivity results),
=20 (ARPESdata).
Similar measurements on over-doped Tl-2201
single crystals led to the conclusion that a large
residual thermal conductivity is observed and the
Weidemann-Franz law is verified [354].
The overall consistency between thermal trans-
port and ARPES data, and the agreement with the
d-wave theory, is quite remarkable. One can there-
fore conclude that the d-wave pair state in high-
T
c
cuprate superconductors is well-described by the
Fermi-liquid theory at low energy in the optimal
and over-doped regimes. However, exceptions to the
above have appeared in the literature. For exam-
ple, the universal thermal conduction was not ob-
served at low temperature in the electron-doped
PCCO at optimal doping [355], and the intrinsically
under-doped YBa
2
Cu
4
O
8
(T
c
= 80) [356].The resid-
ual thermal conductivity was recently observed in
LSCO cuprates over the entire range of supercon-
ducting doping phase, but there are indications that
the Wiedemann-Franz law is violated in the under-
doped regime in the LSCO system [357].
A clear deviation from the Wiedemann-Franz law
was found in the normal state of PCCO [355]. While
these results may indicate the breakdown of Fermi-
liquidtheory in cuprate superconductors,equallyin-
teresting alternatives are possible. For example, qp
localization can lead to a thermal insulator phase
within the d-wave condensate [337]. Another possi-
bility is via time reversal symmetry breaking in the
form of pair states such as d
x
2
−y
2
+ id
xy
or d
x
2
−y
2
+ is,
which are gapped [343,344].
16.6.4 Applicat ions of d-Wave Superconductivity
Several potential applications of d-wave supercon-
ductivity arise from the momentum dependence of
the sign of the pairing wavefunction. First, this en-
ables the design of -rings, with two degenerate
states that are identical, aside from reversing the sign
of time, at zero applied magnetic field.Although it is
possible to create degenerate flux states in a conven-
tional 0-ring by applying an external field equiva-
lent to ¥
0
/2 of magnetic flux, the two states in this
case are different because of the effects of Meissner
screening. These differences can have profound ef-
fects in, for example, attempts to observe long-range
912 C. C. Tsuei and J.R. Kirtley
antiferromagnetic coupling in arrays ofconventional
superconducting rings [358].
The possibility of degenerate,time-reversed states
has been used as a basis for a suggestion of com-
plementary Josephson logic (memory) circuits by
Terzioglu and Beasley [359], and in several sugges-
tions for qubits, the fundamental cell for quantum
computation [360, 361]. A qubit operates by super-
posing two degenerate quantumstates [362].In most
proposals for superconducting qubits these states
can have either distinct charge- or flux- states. The
proposal of Ioffe et al. [363] makes use of the pos-
sible deviations from the Josephson I
c
= I
0
sin
current–phase relationship in junctionswith d-wave
superconductors to produce a“quiet”qubit,in which
the information is expressed through the phase de-
gree of freedom. In the “quiet” qubit the phase-state
itself carries no charge and no current, and there-
fore has minimal interaction with states external to
the qubit,reducing dissipation [363].Several designs
take advantage of the degeneracy of the ground state
of a frustrated ring with d-wave superconductors
and the spontaneous currents and fluxes in equilib-
rium [364, 365]. Alternative concepts are based on
the occurrence of Andreev bound-states in different
types of Josephson junctions [366].
Macroscopic quantum tunneling and energy level
quantizationhave been observedin bi-epitaxial grain
boundary Josephson junctions [367,368]. Evidence
of macroscopic quantum tunneling in intrinsicjunc-
tions of Bi
2
Sr
2
CaCu
2
O
8+ı
have also been reported
[369,370].The demonstration of macroscopic quan-
tum effects in d-wave Josephson junctions has shown
that dissipation arising from low-lying excitations
of nodal quasiparticles does not necessarily have to
be a significant source of decoherence. Several theo-
retical studies on this topic exist [371–373]. Using a
novel YBCO/Nb ramp-type Josephson junction tech-
nology [217],fabricationof25,000-looparrays ona
single chip,and magnetic coupling between the half-
fluxons in the array have been demonstrated experi-
mentally [374]. Furthermore, flip-flopping fractional
quantum devices have been demonstrated using the
d-wave superconductorYBCO [375].All of these ad-
vances represent important steps towards the real-
ization of d-wave superconducting qubits.
16.7 Conclusions
A predominantly d
x
2
−y
2
-wave order parameter has
been established in the hole- and electron-doped
cuprate superconductors through a combination
of phase-sensitive and conventional techniques for
probing pairing symmetry. In particular, the phase-
sensitive techniques have demonstrated an amazing
robustness of the d-wave pairing symmetry, which
survives in the face of significant deviations from
optimal doping, through many classes of cuprate
compounds,from hole- to electron-doping, and over
a wide variation in temperature. Furthermore, the
phase-sensitive pairing symmetry tests have shown
that there cannot be more than a few percent of the
subdominant order parameter in mixed pair states
such as d
x
2
−y
2
+id
xy
ord
x
2
−y
2
+is.The resilience of the
d-wave pair stateisremarkable in view of the fact that
cupratesuperconductors are known to exhibit a great
variety in many of their normal-state and supercon-
ducting properties. The basic origin of the observed
universal d-wave pairing symmetry may stem from
the strong 2D Coulomb correlations, characteristic
of all cuprates. Furthermore, a HTS model has re-
cently been proposed that is decoupled from the on-
site Coulomb repulsion and based on a strong, local,
nonlinear interaction between electrons within the
Cu–O–Cu bond in the CuO
2
plane, and the oxygen
vibrational degree of freedom, termed the Fluctuat-
ing Bond Model [376], which captures the essence of
both the pseudogap and the superconducting states
of HTS.
Although the establishment of d-wave pairing
symmetry in the cuprates does not pin down a spe-
cific mechanism for high temperature superconduc-
tivity,itdoesimposewell-definedconstraintson pos-
sible HTS models. A pair state with d
x
2
−y
2
symme-
try occurs naturally in spin-fluctuation HTS models.
However, one can not rule out a HTS mechanism
based on electron-phonon interaction renormalized
with strong correlation effects.
Research on the low-lying quasiparticle excita-
tionsnearthenodesofthed-wave gap function has
led to new effects in the low temperature, low en-
ergy quasiparticle state, in support of d-wave pair-
ing symmetry. Further, these studies have produced
16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 913
strong evidence for the validity of the concept of
the quasiparticle in the context of BCS Fermi liquid
theory in some optimally and over-doped cuprates.
However, recent examples of exceptions to this ex-
istence proof for the Fermi-liquid theory remind us
vividly of the richness and complexity of the under-
lying physics in the cuprates.
Potential applications of d-wave superconductiv-
ity in areas of -SQUIDs and quantum computation,
while rich and exciting, are in the infancy of their
development, and are only mentioned briefly here.
Acknowledgments
The authors wish to thank A. Barone, B. Chesca, H.
Hilgenkamp, I. Iguchi, R. Koch, J. Mannhart, K.A.
Moler, D.M. Newns, H. Raffy, J.Z. Sun, M. Tachiki,
F. Tafuri, L. Taillefer, M.B. Walker, and J.Y.T Wei for
useful discussions during the preparation of this
manuscript.
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