Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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898 C. C. Tsuei and J.R. Kirtley

Fig. 16.26. SQUID microscope image of an

asymmetric 0–45deg YBCO [001] tilt grain

boundary, cooled and imaged in nominal zero

field with a 4 m diameter SQUID pickup loop.

The polar plots indicate the orientation of an

assumed d

−y

2 gap function on the two sides

of the grain boundary

parallel to the CuO

planes. Several interesting ex-

periments have exploited Josephson tunneling along

the c-axis, perpendicular to the CuO

planes. As in

basal-plane Josephson tunneling, the magnitude of

the c-axis supercurrent can be estimated from (16.6).

There should be no c-axis pair tunneling between

an s-wave conventional superconductor and a pure

d-wave cuprate superconductor due to an exact can-

celation of the contributions from the positive and

negative lobes of the d-wave gap function. A finite

critical current (I

) is expected if the cuprate junc-

tion electrode is a d + s-mixed wave superconductor.

As a reference point, the value of I

at T = 0 K for a

tunnel junction between two identical s-wave super-

conductors is given by [177]:





(

+ 

)





− 



+ 



, (16.33)

where R

is the normal-state junction resistance, 

and 

are the BCS energy gaps of the junction elec-

trodes,and K is the complete elliptic integral.

The observation of a non-zero c-axis supercurrent

in a Pb/YBCO tunnel junction suggests the existence

of an s-wave component of the order parameter in

YBCO. Since second order tunneling [178] has been

ruled out by the absence of half-integer microwave

induced Shapiro steps in such c-axis tunneling ex-

periments [179], the ratio of the experimentally de-

termined value of I

to that predicted by (16.33)

can serve as a measure of the degree of the d + s

admixture in the pair state.

Effects of Twinning

Most of the c-axis tunneling experiments were done

with orthorhombic cuprates such as YBCO and Bi-

2212, which are usually twinned due to the in-plane

anisotropy in the CuO

layers, unless they are spe-

cially treated. For example, the twin boundaries in

YBCO run at 45

◦

with respect to the major crystallo-

graphic axes a and b. The two crystal lattice domains

on each side of a twin boundary are related via a mir-

ror reﬂection symmetry (see Fig. 16.27). In polycrys-

talline YBCO,the spacing between parallel twins can

be as small as 10 nm [34,180].Mostof theYBCO sam-

ples studied in phase-sensitive pairing symmetry ex-

periments were also twinned. However, the fact that

strong evidence for d-wave order parameter symme-

try has been invariably observed in phase-sensitive

experiments with a tetragonal cuprate superconduc-

tor such as Tl-2201 [130,181] and untwinned YBCO

samples [104, 110] make the complications arising

from twinning unimportant to the basic conclusion

of predominantly d-wave pairing symmetry in the

cuprate superconductors. On the other hand, twin-

ning isexpected to playa crucial rolein c-axis Joseph-

son tunneling, especially in determining the value of

. To understand the effect of twinning on these

experiments in such heavily twinned samples, it is

extremely important to know how the order param-

eter varies across a single twin boundary in a YBCO

single crystal. There are two alternatives in YBCO

[5,182,183]: (1) The d-wave component of the order

parameter changes sign across the twin boundary,

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 899

Fig. 16.27. Experimental geometry for c-axis tunneling

into a conventional superconductor (Pb) from single

crystal YBCO,with the junction spanning a single twin

boundary. YBCO is an orthorhombic superconductor

with an admixture of d-wave and s-wave pairing sym-

metries. The diagram shows the d

−y

2 components

with the same phase across the twin boundary, but the

s-wave component with opposite signs across the twin

boundary. There is therefore a change in sign of the

net c-axis coupling across the twin boundary. Adapted

from Kouznetsov et al. [184]

whereas the sign of the s-wave component remains

unchanged; or (2) the converse. The interpretation

of the symmetry experiments with twinned YBCO

single crystals or films depends on which alterna-

tive exists. The fact that a d-wave signature has been

consistently and reproducibly observed in both or-

thorhombic and tetragonal cuprates argues strongly

that the twin boundaries in YBCO have odd reﬂec-

tion symmetry [182]. The d-wave component of the

order parameter continues across the twin boundary

whereas the s-wave counterpart changes its sign.An

even-reﬂection twin boundary corresponds to a pre-

dominantly s-wave pair state, which is not observed

in any of the YBCO experiments.

Kouznetsov et al. [184], demonstrated the sign

reversal of the s-wave component across the twin

boundary in a study of a c-axis Josephson tunnel

junction that straddled a single YBCO twin bound-

ary.InFig. 16.27the polar plots of the gap functionin

the twins (top view) represent the mirror reﬂection

symmetry across the twin boundary (dotted line).

Only the s -wave component changes its sign across

the twin boundary. As a result, the signs of the c-

axis supercurrent switch across the twin boundary.

This is analogous to the Pb/YBCO corner junction

(Wollman et al. [104,105]. When a magnetic field

B is applied parallel to the twin boundary, the field

variation of the critical current I

(B) should show a

Fraunhofer pattern with the maximum critical cur-

rent peak shifted to the field value corresponding

to a half-ﬂux quantum. Furthermore, as the angle 

between the direction of B and the twin boundary

varies from 0 to 90

◦

, the Fraunhofer pattern I

(B)

should gradually recover its standard form. This is

as observed (Fig.16.28).These results provide strong

evidence for a d-wave dominant (d+s) mixed pairing

state in YBCO, and for a sign reversal of the s-wave

component across the twin boundary.Thisfinding is

consistent with the conclusions drawn from all the

phase-sensitive pairing symmetry experiments pre-

sented in the previous sections.

-Products

As discussed earlier, the I

-product of a c-axis

Josephson junction can in principle be used as a

measure of the strength of the s-wave component

of the d + s mixed order parameter. In practice, such

a determination is unreliable since I

products of

Pb/YBCO c-axis tunneljunctionsvary from∼1mVto

afew V.For example,the I

products are 1-1.3 mV

for untwinnedYBCO crystals[179]; 0.3–0.9mV [185]

or 0.1–0.3mV [179] for twinned crystals; and ∼10V

for heavily twinned thin films [179,186, 187]. Mi-

crostructural defects at junction interfaces such as

atomic steps and etch pits could give rise to ab

plane tunneling and contribute a random quantity

to the apparent c-axis tunneling current. A system-

atic study of the direction of tunneling in various

Pb/YBCO tunnel junctions seems to have ruled out

such a possibility. The observed large variation can

be dueto twinning.Thec-axistunneling experiments

of Kouznetsov et al. [184] has clearly demonstrated

that the d-wave component is dominant and contin-

900 C. C. Tsuei and J.R. Kirtley

Fig. 16.28. Josephson interference patterns from the

sample of Fig.16.27,with the magneticfield in the plane

of the junction, as a function of field angle relative to

the twin boundary. These characteristics are consistent

with the pairing phases as diagrammed in Fig. 16.27.

The inset is modeling. Adapted from Kouznetsov et

al.[184]

uous across the twin boundary in YBCO,whereas the

s-wave component changes its sign. Thus the magni-

tude of the I

products in twinned YBCO depends

sensitively on the size and relative distribution of

the d + s and d − s domains, as suggested by several

theoretical studies [80,182,188]. Indeed, this is con-

sistent with the observation that I

products in un-

twinned samples are invariably larger than those of

their untwinned counterparts. The nearly-zero I

products observed in heavily twinned thin film sam-

ples apparently can be attributed to the cancelation

arising from the nearly equal abundance of the two

types of twin domains. Obviously, one needs precise

knowledge of the relative population of the twin do-

mains to quantitatively understand the large scatter

in the I

data.Only data from untwinned junctions

is reliable for an estimateofthestrength of the s-wave

pairing symmetry in YBCO.

AcomparisonofI

values from untwinned

junctions with the Ambegaokar-Baratoff limit

(16.33) provides an estimate of as much as 20% s-

wave component in the d + s admixture for YBCO.

However, estimates from other measurements are

significantly smaller. For example, an upper bound

limit of 10% s-wave was concluded from a study of

the angular magnetic dependence of thermal con-

ductivity in a detwinned YBCO single crystal [68].

A quantitative data analysis of directional tunnel-

ing and Andreev-reﬂection measurements on YBCO

single crystals indicated a d

−y

-wave dominant gap

parameter with less than 5% s-wave component in

either the d + s or d + is scenarios [189,190].

Vanishingly small I

products (∼afewV)have

been observedin c-axis tunneling between Pb andBi-

2212 [137] or the electron-doped NCCO [155] single

crystals. A d + s mixed pair state is not allowed in

tetragonal NCCO or orthorhombic Bi2212,based on

group theory symmetry arguments (see Sects. 16.1.2

and 16.3.3).Phase-sensitive evidence from tricrystal

experiments (see Sect.16.3.3) with Bi-2212[191] and

NCCO [158] hasshown that bothcupratesare d-wave

superconductors. Thus it is logical to conclude that

the small observed I

product is derived from a

small s-wave component induced at twin-boundary

or junction interfaces, as suggested by several theo-

retical studies [192,193].

c-Axis Twist Junctions

In principle, one can test the order parameter sym-

metry by measuring the angular dependence of

the critical current I

() of (001) basal-plane grain

boundary junctions between two identical cuprate

superconductors with a twist angle  about the c-axis

direction (see the inset in Fig. 16.29). If the in-plane

momentum is conserved in the c-axis tunneling,

(16.6) leads, for the case of a pure d-wave supercon-

ductor, to

()=K()

cos2 , (16.34)

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 901

where K() is proportional to the -dependent tun-

neling matrix element squared, and 

is the value of

the d-wave gap in the a, b directions. The parame-

ter K() is not readily determined, but (16.34) pre-

dicts that I

≡ 0 for  = /4 for d

−y

-wave pair-

ing symmetry. There have been several theoretical

studies [194–203] and experiments [204,205] which

emphasized the importance of including incoherent

(momentum non-conserving) tunneling in the cal-

culation of I

(). It has been argued that the con-

dition I

( = /4) = 0 should always be satisfied

for a d-wave pair state, regardless of the amount of

coherence in the tunneling processes [203].

The critical currents of c-axis twist junctions have

been measured in the Bi-2212system.Although early

measurements reported a strong dependence of I

on  [207,208],the results of one group remarkably

showed no angular dependence [209–211]. This has

been taken as evidence against d-wave pairing sym-

metry in Bi-2212. However, these experiments were

done with macroscopic samples with junction areas

∼10

m

,andcriticalcurrentsof50-150mA[210].In

such experiments the c-axis tunneling currentcan be

limited by local heating, self-field, and bulk critical

current effects, masking the intrinsic -dependence

of I

. To clarify these issues, a recent experiment by

Tachiki et al. [206] has repeated the previous ex-

periment [210], with Bi-2212 single crystal whiskers.

The Bi-2212 whisker twist junctions have a junction

area of about 10

m

and critical currents one to

two orders of magnitude smaller than in the experi-

ments of Li et al. [210]. As shown in Fig. 16.29, I

()

is clearly a strong function of the twist angle ,ap-

proaching zero at  = /4. In sharp contrast with

the results obtained in a previous experiment [210],

the vanishingly small I

value observed at  = /4

is particularly interesting in view of the theoretical

prediction that I

should be zero for d-wave and

finite with no significant angular dependence near

 = /4 is expected for the case of an anisotropic

s-wave pair state [203]. Shown as a dotted line in

Fig. 16.29 is the | cos 2 | dependence expected for a

pure d-wave superconductor with no dependence of

the tunneling probabilities on . In-plane tilt grain

boundary junctions in the cuprates show an expo-

nential dependence of the critical current density on

Fig. 16.29.Measureddependenceof the critical current of c-

axis twist grain boundary junctions between single crystal

whiskers of Bi-2212 (solid symbols) [206]. The functional

forms | cos2 | (dotted lin e)ande

−90/ı

(dashed line)are

shown for comparison

misorientation angle J

(Ÿ)=C

−Ÿ/Ÿ

with Ÿ

∼ 5

◦

for the hole-doped cuprates, and Ÿ

∼ 2

◦

for the

electron-doped cuprates [156,157,212]. It is believed

thatthis exponential dependence is dominatedby the

strong dependence of the tunneling matrix elements

on misorientation angle, due to microstructural dis-

order effects [77,213]. The dashed line in Fig. 16.29

shows for comparison a fit of the data to an expo-

nential dependence of the critical current on twist

angle I

∼ e

(−90)/ı

,withı =9.3

◦

.The goodagree-

ment between this exponential dependence and ex-

periment makes it appear likely that the observed

angular dependence of I

can be explained in terms

of Fermi surface or -dependent tunneling matrix

element effects [203],in a d-wave superconductor,as

opposed to an anisotropic s-wave superconductor.

16.4 Angle-Resolved Determination of Gap

Anisotropy in YBCO

It is clear from the work described so far that an

admixture of d

−y

and s-wave order parameters

is well established in orthorhombic YBCO. Such an

anisotropic d + s pair state is required by the group

902 C. C. Tsuei and J.R. Kirtley

Fig. 16.30. (a) Schematic of two-junction YBCO/Nb rings used for angle-resolved determination of the gap symmetry

in optimally doped YBCO [214]. One ramp edge junction angle was held fixed from ring to ring relative to the YBCO

crystalline a or b axis while the other was varied. (b) Integrated spontaneous ﬂux generated by these rings, as determined

by scanning SQUID microscopy, for a sample with the first junction normal parallel to the majority twin a-axis direction,

as a function of the second junction normal angle () relative to this direction. The solid points are the data; the solid line

is a best fit using a predominant d

−y

2 component to the gap with a small s component. The dashed line shows much

worse fit results when a small imaginary component is added to the gap functional form

theoretic argument discussed in Sects. 16.1.2 and

16.3.3.The size of this gap anisotropy was measured

recently by locating the nodes in the in-plane gap

function for optimally doped YBCO [214] by mea-

suring the spontaneously generated magnetic ﬂux in

asetof0or-rings with varying junction angles.

This work was based on an experimental configura-

tion originallysuggested fortesting fortime-reversal

symmetry breaking (TRSB) in cuprate superconduc-

tors [215,216].A set of two-junctionYBCO/Nb rings

were fabricatedwith onejunctionnormalangle fixed

relative to a YBCO a or b crystallographic axis while

the second junction normal angle was rotated with

respect to the first in 5 degree or 0.5 degree incre-

ments (see Fig. 16.30) from ring to ring. Arrays of

suchthinfilmYBCO/Nbtwo-junctionringswerefab-

ricated using a ramp-type Josephson junction tech-

nology [217]. The magnetic ﬂux state of these rings,

cooled in various magnetic fields, were imaged at

4.2 K with a scanning SQUID microscope. The spon-

taneously generated ﬂux ¥ in the rings,when cooled

in zero field, alternated systematically as a function

of the second junction angle between nearly zero ﬂux

(N = 0) and the half-ﬂux quantum state ¥ = ¥

(N =1/2) in a manner consistent with a predomi-

nantly d

−y

pairing symmetry. The transitions be-

tween the N =0andN =1/2 states occur at angles

slightly different from the (2m+1)45

◦

expected for

apured-wave superconductor, and is in fact con-

sistent with an s/d ratio of 0.1±0.02 (Fig. 16.30). A

three-dimensional rendering of the SSM images of

the rings is presented in Fig.16.31,along with a polar

plot of the corresponding anisotropic d + s pairing

function.A s/d ratio of 0.1 corresponds to a gap mag-

nitude along the b-axis (along the Cu-O chains) di-

rection about20% larger than in the a-axis direction.

This is somewhat smaller than obtained from mea-

surements of the Josephson critical current density

as a function of junction angle in single junctions

produced using the same technology, [218] perhaps

because of incomplete detwinning of theYBCO films

used in the ring samples.

16.5 Universality of the d-Wave Pair State

In two-dimensional strongly correlated electron sys-

tems, especially near the quantum critical point,

phases and phase-transitions abound due to the

competition between various ordered states [219].

The pairing symmetry tests presented so far have

firmly established that the ground state of several

optimally-doped cuprates superconductors has an

order parameter with d

−y

pairing symmetry.How-

ever, there are many theoretical studies that suggest

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 903

Fig. 16.31. Three-dimensional rendering of

scanning SQUID microscope images of 72

YBCO/Nb two-junction rings with one junction

normal angle fixed at −22.5

◦

relative to the ma-

jority twin b-axis and the second junction nor-

mal angle varied in 5

◦

intervals [214]. The rings

were cooled and imaged in zero field. This re-

sulted in rings in either the N =0orN =1/2

ﬂux quantum state, with the transitions at an-

gles slightly different from (2m+1)45

◦

,m an in-

teger, because of a small s component added to

the predominantly d

−y

2 gap functional form

in optimally doped YBCO

that the d-wave pair state is not stable against var-

ious competing pairing symmetries, depending on

variations in doping, temperature, and time-reversal

symmetry breaking (TRSB) at surfaces and inter-

faces [220–229]. It is therefore important to exper-

imentally determine whether the d-wave state is ro-

bust to a change in these variables.These universality

issues can be investigate bydoing the phase-sensitive

symmetry tests with various cuprates systems at op-

timal doping, and for a given cupratesystem as func-

tions of composition and temperature. Possible evi-

dence for TRSB at the surfaces and interfaces of d-

wave superconductors should also be examined.

16.5.1 Optimally-Doped Cuprates

The bulk of pairing symmetry tests using phase-

sensitive methods was done in various cuprate sys-

tems at optimal doping. A complete list of the

cupratesuperconductors tested (with references) can

be found in a recent review [7].

Hole-doped Cuprates

Evidence for predominantly d-wave pairing has

been found in all the hole-doped cuprate super-

conductors successfully tested using phase-sensitive

methods, including YBCO [87, 104–106, 110, 117],

GdBa

7−ı

[230], Tl-2201 [130,181], and Bi-2212

[191]. Given a choice between the s-wave and d-

wave pairing channels, a number of numerical stud-

ies have concluded that gap symmetry depends on

band filling and certain band parameters. Based on

the t-t’-J model, with t



/t ≥ −0.4, the d-wave pair

state prevails in a doping range of the band cen-

tered around half-filling, and s-wave pairing sym-

metryismorestableintheunder-andover-doped

regimes [220,222,225].The phase-sensitive evidence

obtainedso far is generally consistent with these con-

clusions: the d-wave channel is favored near half-

filling where T

is optimized.

Electron-doped Cuprates

As described in Sect. 16.3.4, strong evidence has

recently been found for d

−y

symmetry in the

optimized electron-doped cuprate superconduc-

tors NCCO and PCCO from both tricrystal exper-

iments and conventional symmetry probing mea-

surements. It is quite remarkable that the d-wave

pair state predominates in both the electron-doped

and hole-doped cuprate superconductors in spite

of their significant differences in many of the su-

perconducting and normal state properties. On the

904 C. C. Tsuei and J.R. Kirtley

other hands, the observed universal d-wave pair-

ing in cuprate superconductors is not unexpected,

because the basic theoretical models for describ-

ing high-temperature superconductivity,such as the

one-band Hubbard model and its large-U limit, the

t-J model, are all intrinsically electron-hole sym-

metrical [231]. Electron-hole asymmetry can be in-

corporated into various t − t



− J models [232,233]

and three-band Hubbard models [234]. For example,

the differences in doping characteristics between the

electron- and hole-doped cuprates can be explained

in terms of band structureeffects based on a t −t



−J

model [233] or a three-band Hubbard model [235].

However, this feat and more can be accomplished

by taking into account the different dispersions for

the two types of charge carriers, and by using an ef-

fective one-band Hubbard model [236]. Therefore, it

appears that the apparent electron-hole asymmetry,

as manifested in many anomalous properties, stems

largely fromband structure effectsand is not relevant

in determining pairing symmetry. This finding and

the small in-plane coherence length (∼20Å) charac-

teristic to all cuprate superconductors, suggests that

the universal d-wave pairing occurs on a relatively

local scale.

16.5.2 Extended s-Wave Pair State

As mentioned in Sect. 16.2, there has been an enor-

mous amount of work on tests of the pairing sym-

metry in the cuprate superconductors using various

phase insensitive techniques [4, 5, 7]. Although this

work has resulted in a general consensus that the

pairing wavefunction has lines of nodes, these tech-

niques, with the exception of angle resolved pho-

toemission, are relatively insensitive to the momen-

tum dependence of the nodal structure. This makes

it difficult for them to distinguish, for example, be-

tween d

−y

symmetry (with four lines of nodes),

and extended s-wave symmetry (with eight lines of

nodes). There has recently been support for the lat-

ter symmetry from analysis of various phase insen-

sitive tests of pairing symmetry [237,238]. These au-

thors attempt to reconcile this conclusion with the

phase-sensitive experiments, which definitively fa-

vor d

−y

symmetry,by arguing that the cuprates can

have different bulk and surface symmetries [192].In

this view the phase-insensitive techniques probe the

bulk extended s-wave symmetry, while the phase-

sensitive techniques,which depend on surface sensi-

tive Josephson tunneling,probed-wavepairing sym-

metry.

This view seems untenable for several reasons.

First, the consistency of the tricrystal experiments

[7], both in that samples with a -ring geometry for

a d-wave superconductor always show the half-ﬂux

quantum effect,andthat the half-ﬂuxquantumeffect

can be turned on and off by changing the tricrystal

geometry in a way that is consistent with d

−y

pair-

ingsymmetry,makeitappear extremelyunlikelythat

these experiments are controlled by surface effects.

Second, the tricrystal experiments have shown that

the half-ﬂux quantum effect persists,with no change

in the total ﬂux within experimental accuracy, to

withina degree of T

[239].At these temperatures the

ab plane coherence length can be ten times the unit

cell dimension, meaning that these experiments are

not surface sensitive but rather probe the order pa-

rameter deeply into the bulk: nevertheless the results

strongly favor d-wave pairing symmetry.Further,al-

though the I

product forthegrainboundary junc-

tions used in the tricrystal experiments are relatively

small, experiments with -SQUIDs and facetted -

junctions in YBCO-Nb ramp junctions produce con-

vincing evidence for d-wave symmetry with junc-

tion I

products within a factor of four of the

Ambegaokar–Baratoff limit [174–176]. This means

that the superconducting gap is not significantly re-

duced by surface effects in these samples.

The tetracrystal experiments [130] have provided

strong evidence for pure d-wave pairing symmetry

in tetragonal Tl-2201. It also rules out anisotropic s-

wave pairing symmetry.The tetragonal crystal struc-

ture of the Tl-2201 epitaxial films used in this ex-

periment were checked and confirmed with three

techniques for microstructural analysis [130]. It is

well established that Tl-2201 can be fabricated with

either orthorhombic or tetragonal crystal structure

with identical superconducting properties (includ-

ing T

∼80 K) [240–242]. Furthermore, it has been

shown that there is no temperature-dependent struc-

tural phase transition in the temperature range

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 905

50(well below T

)–300 K in either form of Tl-2201.

This observation, as well as the results of neutron

scattering and x-ray diffraction, leads to the conclu-

sion that the orthorhombic to tetragonal structure

change is due to static compositional effects, and is

not driven by any thermal structural instability.

The combination of the observations above lead

us to conclude with confidence that the pairing sym-

metry in the cupratesuperconductors hasd

−y

sym-

metry, and not the extended s-wave symmetry.

16.5.3 The Effect of Doping

The competitionbetween pairing symmetries in var-

ious superconducting cuprate systems as a func-

tion of doping has been emphasized repeatedly

[219,220,222,223,225,243–246].In particular,a tran-

sition from a d to an s-wave pair state induced by a

change in doping has been suggested. For references,

see Sect. 16.4. The tricrystal pairing symmetry test is

especially well adapted for a systematic study of the

effect of doping on pairing symmetry.

Recently, we have carried out a series of tricrystal

experiments with a c-axis oriented Bi-2212 tricrys-

tal thin-film sample as a function of doping [247].

The systematic pairing symmetry study started with

an over-doped Bi-2212 tricrystal sample with a T

of 69 K. The doping level was then reduced by an-

nealing the sample in ﬂowing Argon at tempera-

tures between 350 C and 420 C for various lengths

of time to achieve an optimum T

of 82 K and a

of 40 K in the under-doped regime. The half-ﬂux

quantum effect at the tricrystal point was monitored

as a function of such annealing. The d-wave pairing

symmetry was found to persist throughout the entire

doping range studied so far. There are also similar

phase-sensitive evidences for d-wave pairing sym-

metry in both the under-doped and over-doped Tl-

2201 samples, overdoped Y

0.7

0.3

7−ı

,and

underdoped La

1.85

0.15

CuO

[247].A possible d

−y

to d

−y

+ s symmetry transition in Ca-doped YBCO

has been reported [248].

In addition to the tricrystal experiments, mea-

surements of low-temperature thermal conductiv-

ity [249,250] have provided strong evidence in sup-

port of a d-wave pair state that prevails throughout

the entire phase diagram from underdoped to over-

doped regimes for several cuprate systems.

16.5.4 Temperature Dependence

In view of the possible changes in pairing symmetry

as a function of temperature [222,223,225,251–253],

it is important to experimentally determine the tem-

perature dependence of the pairing symmetry in a

d-wave superconductor such as YBCO. It is diffi-

cult to study the temperature variation of the pair-

ing symmetry in an unconventional superconduc-

tor using conventionaltechniques,sincethey become

obscured by thermal excitations at higher tempera-

tures. Phase-sensitive pairing symmetry probes that

employ Josephson pair tunneling are in principle

immune to thermal broadening. In practice most

of these techniques also fail at higher temperatures

because they depend on tunneling into low-T

su-

perconductors [104–106,110]. Only phase-sensitive

symmetry tests, such as the tricrystal experiments,

which use all-cuprate grain boundary Josephson

junctions can be used for determining pairing sym-

metry atall temperatures below T

.Kirtley etal.[239]

used a variable sample temperature scanning SQUID

microscope [254] to measure the total spontaneous

ﬂux at the tricrystal point of a YBCO thin film in a

frustrated tricrystal geometry as a function of tem-

perature. With this microscope the tricrystal sam-

ple was heated to different temperatures while the

SQUID sensor was kept below its superconducting

temperature of ∼9 K. Shown in Fig. 16.32 are scan-

ning SQUID microscopy images of the d-wavesigna-

ture, the half-ﬂux quantum, at the tricrystal point at

selected temperatures.

These experiments showed that the half-integer

ﬂux quantum effect at the tricrystal meeting point

persists from T =0.5KthroughT

(∼90 K).Careful

modeling (Fig.16.33)showed that the totalﬂux in the

half-vortex was equal to ¥

/2 to within experimental

errors at all temperatures. This implies that d-wave

pairing symmetry dominates in this cuprate,with no

observable time-reversal symmetry breaking, over

the entire temperature range.

906 C. C. Tsuei and J.R. Kirtley

Fig. 16.32.Scanning SQUID microscope

images of the development with tem-

perature of the half-ﬂux quantum vor-

tex at the tricrystal point for a thin film

sample of YBCO in a geometry designed

to be a  -ring for a d-wave supercon-

ductor

Fig. 16.33. Best fit values for the total magnetic ﬂux spon-

taneously generated at the tricrystal point of a YBCO thin-

film sample in a tricrystal geometry designed to be frus-

trated fora d

−y

2 superconductor, as a function of temper-

ature. To within the experimental uncertainties, the total

ﬂux is ¥

/2from0.5KtoT

∼ 90 K. This indicates that

d-wave pairing symmetry dominates at all temperatures,

with little (if any) imaginary component, in YBCO

16.5.5 Time-Reversal Symmetry Breaking

It has been suggested that as temperature decreases

below a certain characteristic value, a pure d-wave

pair state is not stable against the formation of

mixed pair states such as d

−y

+ is or d

−y

+ id

[129, 251, 255–261]. Due to the coupling between

the two subcomponents of the order parameter, the

mixed pair state is fully-gapped, much like that in

an s-wave superconductor [251, 257]. The power-

law temperature dependence of observables such as

specific heat, thermal conductivity, and penetration

depth characteristic of d-wave superconductors re-

verts to the exponential behavior typical for s-wave

superconductors. Measurements of the penetration

depth of Zn substituted LSCO [262] shows that it

retains a T

dependence, indicative of d-wave pair-

ing in the presence of scattering, until T

is sup-

pressed to zero. This argues against an admixture

of s-wave pairing at any T

. There has been a report

of a temperature-dependent time reversal symme-

try broken pair state (i.e. d

−y

+ is state) in YBCO

near T

[263]. Furthermore, there is thermal trans-

port evidence suggesting thata phasetransition from

apured

−y

-wave state into time-reversal symmetry

broken states such as d + is or d

−y

+ id

can be

induced in Bi-2212 at low temperatures by magnetic

field [264] or by magnetic impurities [265]. How-

ever, there are new experiments [266–268] and cal-

culations [269] indicating the interpretation of the

original experiments are much more involved.

Further, it has been shown theoretically that bro-

ken time-reversal symmetry could occur locally in

a d

−y

superconductor at certain surfaces and in-

terfaces, or in the presence of magnetic impurities

[270–277]. Evidence for broken time reversal sym-

metry in YBCO has been observed from the splitting

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 907

of the zero bias conductance peak (ZBCP) induced

by the Andreev surface bound states [154,278,279],

and by a small spontaneous magnetization at the su-

perconducting transition [280].

A ZBCP in the quasiparticle tunneling spec-

trum, due to the formation of zero-energy Andreev-

bound states associated with sign changes across

the nodes of the k dependent order parameter, can

be used as a signature for d-wave pairing symme-

try [57,58,281,282].A split in the ZBCP is expected

in the presence of TRSB in mixed pair states such as

−y

+ is or d

−y

+ id

[283]. Recent theoretical

treatments have pointed out that there are possibili-

ties other than TRSB that can lead to ZBCP splitting

in pure d-wave superconductors[284,285].

In thetricrystal experiments withYBCO [119] and

Tl-2201 [181], the applied ﬂux needed to nullify the

SQUID difference signal in the three-junction ring

was always found to be within 3% of ¥

/2. Tricrystal

experiments with electron-doped NCCO and PCCO

at 4.2 [158] and YBCO at temperatures close to T

[239] have all demonstrated that the total amount of

ﬂux spontaneously generated at the tricrystal point

remains the same (within the experimental accu-

racy of ∼ 3%), regardless of which of the two time-

reversed half-ﬂux quantum states the tricrystal sam-

ple is in. All these observations imply that the gauge-

invariant phase shift as defined in (16.5) is very close

to , in turn suggesting that any out-of-phase com-

ponent of the order parameter must have an ampli-

tude less than a few percent of the primary compo-

nent. Such a mixed pair state (e.g. d + is) would lead

to a phase shift 0 < 

< ,and also a fractionalﬂux

quantum x¥

with 0 < x < 1/2 [80,193].

The observation of total ﬂux different from ¥

in the multi-crystal experiments in a ring geome-

try would not necessarily imply broken time reversal

symmetry. As seen in Fig. 16.5, if the LI

product is

not sufficiently largecomparedwith ¥

,then the total

ﬂux in a ring geometry will be less than ¥

/2. For ex-

ample, in a tricrystal experiment with Tl-2201 [181],

in which the I

L product was low, and the amount

of the spontaneously generated ﬂux through the ring

was indeed noticeably smaller than

(x ∼ 0.3).

However, multi-crystal experiments in a blanket film

geometry have always been done with the sample size

much larger than the Josephson penetration depth



. In this case, the total ﬂux in the Josephson vortex

at the multi-crystal point in a frustrated geometry

should be exactly ¥

/2. Mathai et al.[106] estimated

that the time-reversal symmetry breaking compo-

nent of the pairing order parameter was less than

5%, in agreement with the conclusions by Kirtley et

al.[119]from“magnetic titration”of a tricrystalring.

Recent studies on the magnetic field modulation of

critical currents in grain boundary junctions [286]

and-SQUIDs[176]haveproducedsimilarevidence

against TRSB at the junction interface.

Tafuri and Kirtley [287] have found evidence for

spontaneous magnetization, associated with the su-

perconducting transition, that appears to be corre-

lated with defects in some c-axis orientedYBCO thin

films.The association of spontaneous magnetization

with defects could help to explain why evidence for

time-reversal symmetry breaking is seen in some

samples, using some techniques, but not in others.

16.6 Implications of d-Wave Pairing

Symmetry

The implications of d

−y

-wave pairing symmetry

for understanding HTS and for its applications are

numerous. Due to space limitations, we shall brieﬂy

discuss only some of the topics here.

16.6.1 Universal d-Wave P air State in Cuprates

The phase-sensitive experiments, as presented in

Sects. 16.3 and 16.4 above, augmented with sev-

eral conventional order parameter symmetry prob-

ing techniques, have established the predominance

of d

−y

-wave pairing symmetry in both the hole-

and electron-doped cuprate superconductors. The

tricrystal and tetracrystal experiments have also

provided phase-sensitive evidence, based on group-

theoretic symmetry arguments, for the existence

of a pure d-wave state in Bi-2212 and Tl-2201

(Sect. 16.3.3). Furthermore, a series of tricrystal ex-

periments have also demonstrated that the d-wave

pair state in several high-T

cuprate superconduc-

tors persists in spite of deviations from optimal dop-

ing (Sect. 16.4), and variation in temperature over