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

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888 C. C. Tsuei and J.R. Kirtley

Fig. 16.18. Three-dimensional rendering of SQUID mi-

croscopy image of several types of vortex trapped in a

YBCO film on a tricrystal SrTiO

substrate designed to

generate the half-ﬂux quantum effect in a d

−y

2 supercon-

ductor

deed,the half-ﬂux quantum effect has been observed

directly with the SSM in tricrystal YBCO disks and

blanket films (Figs. 16.17(b) and (c) and Fig. 16.18),

in addition to rings (Fig. 16.17(a)).

Integer and Half-IntegerJosephson Vortices

The unpatterned (blanket) tricrystal sample geom-

etry has the advantage that no photolithographic

processing has to be done after the cuprate film

is deposited. It has the disadvantage that the total

ﬂux generated at the tricrystal point is more diffi-

cult to calibrate, compared with the ring geometry.

As noted in Sect. 16.2.2, superconducting rings in-

terrupted by Josephson weak links with large values

of ˇ

=2LI

/¥

have total magnetic ﬂuxes close

to N¥

or (N +1/2)¥

(N an integer), depending on

whether there is an intrinsic phase shift of 0 or 

in the ring respectively. In contrast, an unpatterned

tricrystal sample has only two allowed ﬂux states

¥ = ±¥

/2 atthe tricrystal pointin the frustratedge-

ometry: It is energetically favorable for higher quan-

tized ﬂux states (e.g. ¥ = ±3¥

/2, ±5¥

/2, etc.) to

split into a half-ﬂux quantum at the tricrystal point,

and some number of integer Josephson vortices at

a distance from the tricrystal point [121]. In fact,

Josephson vortices with ﬂux different from ±¥

have never been observed at the tricrystal point in a

frustrated tricrystal geometry.

Figure 16.19 compares images of a bulk vortex

(a), ¥

Josephson vortices trapped along the diago-

nal (b) and horizontal (c) grain boundaries, and a

/2 half-vortex trapped at the center of a -ring

thin-film tricrystal sample of YBCO [122]. Contour

lines have been placed on the data at 0.1, 0.3, 0.5, 0.7,

and 0.9 of the full scale amplitudes. Superposed on

panel (a) is a schematic of the 4 mdiameteroctag-

onal pickup loop used for these measurements. The

dots in Fig. 16.20 are cross-sections through the im-

ages of Fig.16.19as indicated by the contrastinglines

and letters in this figure.

To date all experiments on tricrystal samples have

been done with samples with film thicknesses com-

parable to the London penetration depth. It is possi-

bletoderiveexpressionsforthefieldsfromaJoseph-

son integer or half-integer vortex in samples with

thin films [123], but these expressions are cumber-

some, and all detailed analyses of experimental data

to date have used a simpler slab model.In this picture

the(thin-film) crystals making up thetricrystal point

are modeled as semi-infinite wedges meeting at the

tricrystal line x =0,y =0,z. The phase shift (r

)

across the i-th grain boundary is described by the

sine-Gordonequation(16.19) [100,101].Thesolution

to (16.19) fora h/4e half-vortex with (r

)=0, r

< 0

and (r

)=, r

> 0 was given in (16.20). The so-

lution to (16.19) which describes a h/2e Josephson

vortex ((r

) = 0 everywhere), centered about r

along a single grain boundary is

(r

)=4tan

−1

/

) . (16.24)

The general expression for the phase  at the tricrys-

tal point is complicated, if the grain boundaries have

different supercurrent densities, but the magnetic

ﬂux/unit length in the i-th branch of the vortex can

be written as:

d¥ (r

)

2

d

2

−4a



−r

/

1+a

−2r

/

(16.25)

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 889

Fig. 16.19. (a)–(d) High resolution images of several

types of vortex trapped in a YBCO film on a tricrys-

tal SrTiO

substrate designed to generate the half-ﬂux

quantum effect in a d

−y

2 superconductor

where in this case r

= 0 for all grain boundaries

at the tricrystal point, and r

is restricted to values

greater than 0. Inside the superconductors, the Lon-

don theory gives ∇

B = B/

. Using London the-

ory to describe the field inside a superconductor is

strictly valid only if no current ﬂows through the

sample boundary. This is a reasonable approxima-

tion if 

<< 

. Neglecting the derivative parallel

to the grain boundaries in the Laplacian leads to

, r

⊥





−r

/

1+a

−2r

/

−|r

i⊥

|/

, (16.26)

where r

i⊥

is the perpendicular distance from the i-th

grain boundary. A Josephson vortex with h/2e total

ﬂux in it, and with a single penetration depth, has

, r

i⊥

)=(¥

/2



)sech(r

/

−|r

i⊥

|/

(16.27)

For the tricrystal, the a

’s are normalization con-

stants determined by two conditions: First, the con-

dition that the magnetic induction at the tricrystal

point varies smoothly implies that a

/(

(1 + a

))

is the same for each grain boundary. Second, the

total ﬂux at the tricrystal point is given by ¥ =



(¥

/2)4tan

−1

)=¥

/2.

The magnetic inductions at a height z above the

surface are determined from those at the surface by

noting that for magnetic inductions derived from a

two-dimensional current distribution,

, k

, z)=exp(−



+ k

z)b

, k

, 0) ,

(16.28)

where b

is the two-dimensional Fourier transform

of the field B

(r), and k

and k

are the wavevectors

in the x and y directions respectively [124]. The ﬂux

is derived by integrating the induction over the area

of the pickup loop.The effective height of the pickup

loop above the sample surface is obtained by fitting

data from h/2e bulk vortices, which are resolution

limited since, except very close to T

,theLondon

penetration depth 

is much smaller than the size

of the pickup loop. In this limit the magnetic induc-

890 C. C. Tsuei and J.R. Kirtley

Fig. 16.20. (a)–(d)Thedots are cross-sections through the

vortex images of Fig. 16.19 as indicated by the contrasting

lines and letters. The lines arefitstothisdataasdescribed

in the text, assuming that the bulk vortex and Josephson

vortices along the grain boundaries have h/2e of total ﬂux

in them,and the vortex at the tricrystalpoint has h/4e total

ﬂux

tion in vacuum ata distancer away from the center of

a superconducting bulk vortex is given by [125–127]

B(r)=(¥

/2)r/ | r |

. (16.29)

The solid lines AA



and BB



in Fig. 16.20(a) are

fits to the data for a bulk vortex using the fields of

(16.29). The effective height determined from this fit

fixedthe height for the rest of themodeling.The solid

lines CC



− FF



in Figs.16.20(b) and (c) are fits of the

model above to the data,with 

= 150 nm,and using

the Josephson penetration depth 

as the only fitting

parameter. The lines GG



and HH



in Fig. 16.20(d)

are fits to the half-vortex at the tricrystal point, using

the three Josephson penetration depths as fitting pa-

rameters. This agreement between experiment and

modeling shows that there is approximately ¥

/2to-

tal ﬂux at the tricrystal point. For a tricrystal sample

in a geometry that is frustrated for a d-wave super-

conductor,if the sample is cooled repeatedly in fields

close to zero,the half-ﬂuxquantumJosephson vortex

is always present,either oriented with fields pointing

out of or into the sample.

A recent SSM ﬂux imaging tricrystal experiment

by Iguchi’s group [128] also observed thehalf-integer

ﬂux quantum effect (see Fig. 16.21),providing inde-

pendent phase-sensitive evidence for d-wave pairing

symmetry in YBCO.

The tricrystal magnetometry experiments di-

rectly image the half-integer ﬂux quantum effect,

in the ground state, with a tunable geometry to en-

sure that the pairing symmetry is being probed.

Fig. 16.21. (a) Three-dimensio-

nal rendering of the magnetic

fields above an unpatterned

YBCO film on a tricrystal sub-

strate with geometry designed

to be frustrated for a d-wave

superconductor. (b)Modeling

(solid lines) and cross-section

data (symbols)forah/2evor-

tex away from the grain bound-

aries,and a h/4e Josephson vor-

texat the tricrystal point.(After

Sugimoto et al. [128])

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 891

There are no complications associated with self-

field effects, since no currents are applied externally.

The tricrystal ring samples have very small sample

volumes, making ﬂux trapping highly unlikely. Any

ﬂux trapping to occur is imaged with high sensi-

tivity. These experiments do use grain boundary

junctions, which have moderately rough interfaces.

However, care was taken to account for this interface

roughness. Further, these experiments are extremely

reproducible: dozens of samples have been made and

measured in hundreds of cool downs. All tricrystal

or tetracrystal samples that had sufficiently high

supercurrent densities across the grain boundaries

(

< 50m) gave results (producing either 0 or

-rings) that were as expected for predominantly

d-wave pairing symmetry.

Evidence for Pure d- Wave Pairing Symmet ry

The phase-sensitive pairing symmetry tests de-

scribed above have yielded convincing evidence for

a predominantly d-wave order parameter in YBCO.

However, none of these experiments were able to un-

ambiguously eliminate the possibility of a d+s mixed

pair state. An admixture of s-wave and d-wave pair-

ing symmetries is allowed in orthorhombic YBCO.

The orthorhombic crystal structure of YBCO is de-

rived from the inequivalent lattice distortions along

the a, b directionintheCuO

planes. As a conse-

quence, the relevant group symmetry for the Cu-O

rectangular lattice is C

,insteadofC

as appro-

priate for the Cu-O square lattice planes in cuprates

such as LSCO, Hg-1201, Tl-2201, etc.. As discussed

in Sect. 16.1.2, the s-wave and d-wave spin-singlet

order parameter symmetries belong to the same ir-

reducible representation (A

) of the point group C

Hence,the d +s mixed pairing is an unavoidable con-

sequence of orthorhombicity [80,129]. The question

is: What is the amount of the s-wave pairing in the

mixedpair state?In a (d+s)-wavesuperconductor the

order parameter (k)variesasd(k

−k

)+s(k

)to

reﬂect theunderlying crystal lattice symmetry,where

s and d represent the size of the s-wave and d-wave

subcomponents in the admixture respectively. The

node lines ( = 0), given by k

= ±(

d+s

d−x

)

1/2

,ex-

ist only under the condition that the d-wave order

parameter is dominant (i.e. d/s > 1). The slope of

the node lines deviates from that of pure d-wave

= ±k

, the diagonals of the Brillouin zone, de-

pending on the degree of the s − d admixture.

The Pb-YBCO corner SQUID (or single Joseph-

son junction) interference experiments rely only on

a sign change of the order parameter between the

a and b faces of the YBCO single crystal, and there-

fore can not discriminate pure d-wave from d+s pair

statesaslongasd/s ≥ 1. In principle, the tricrystal

experiments with YBCO and Tl-2201 are capable of

locating the nodes on the Fermi surface of a (d+s)-

wave superconductor. However, this would require

a systematic series of tricrystal experiments based

on a knowledge of the bandstructure and a detailed

model describing the complex process of pair tun-

neling across a realistic grain boundary junction.In

the face of these difficulties in determining the de-

gree of s + d mixing in pairing, it is important to

demonstrate unambiguously the existence of a pure

−y

-wave cuprate superconductor. This is signifi-

cant in view of the substantial amount of theoretical

work [80,129] on the effect of s + d wave pairing on

the properties of cuprate superconductors, includ-

ing the origin of high temperature superconductiv-

ity. The demonstration of a pure d-wave pair state

can help to understand whether the coexistence of

d and s pairing channels is essential or just acci-

dental to high-temperature superconductivity. In the

following, phase-sensitive evidence for pure d-wave

pairing symmetry will be presented for two high-T

cuprate systems:

(a) Tetragonal Tl

CuO

6+ı

Stoichiometric Tl

CuO

6+ı

(Tl-2201) can be pre-

pared in the form of c-axis epitaxial films with a

tetragonal crystal structure. UnlikeYBCO,the tetrag-

onal Tl-2201 is intrinsically twin-free and has only

one CuO

layer per unit cell. Such a relatively sim-

ple lattice structure makes Tl-2201 a model system

for studying many normal-state and superconduct-

ing properties of the high-T

cuprates.

To demonstrate the existence of a pure d-

wave cuprate superconductor, a tetracrystal phase-

sensitive experiment, with tetragonal Tl-2201 [130],

adopted from a suggestion by Walker and Luettmer-

892 C. C. Tsuei and J.R. Kirtley

Fig. 16.22. (a) Tetracrystal geometry. (b) SSM

image of the crystal meeting point of a film

of Tl2201 epitaxially grown on a SrTiO

sub-

strate with the geometry of (a), cooled in nom-

inally zero field, and imaged at 4.2 K with an

8.2msquarepick-uploop.(c) Cross-sections

through a bulk Abrikosov vortex, and through

the half-vortex of (b), along the directions in-

dicated in (a). The dots are the experimental

data, and the solid lines are modeling, assum-

ing the Abrikosov vortex has h/2e ﬂux trapped

in it,and the vortex at the tetracrystal point has

h/4e ﬂux

Strathmann [80],was carriedout.Shown in Fig.16.22

is the geometrical configuration of a four-crystal

(100) SiTiO

substrate used in the experiment. The

zero misorientation angle at the grain boundary

MOM



makes the tetracrystal substrate effectively

a bicrystal made of two crystals rotated about the

normal (c-axis) to the substrate plane by /4 with

respect to each other. The built-in reﬂection sym-

metry with respect to the equal-partition line MOM



assures that,for a pair state with d

−y

or d

symme-

try, the pair tunneling currents across the two grain

boundaries OA and OB are equal in magnitude but

opposite in sign.

)

=−(I

)

. (16.30)

This conclusion can also be reached using the

general formula for the supercurrent of a Joseph-

son junction between two d-wave superconductors,

(16.16), [80]:

∝ C

2,2

cos(2

)cos(2

)

2,2

sin(2

) sin(2

)+··· (16.31)

By reﬂection symmetry,theangles 

and

are given

by:





+˛

−˛



, 



/4−˛

/4+˛



, (16.32)

for grain boundary junctions





respectively.

When this reﬂection symmetry operation is ap-

plied to (16.31), every term changes its sign, reduc-

ing to (16.30). Therefore, any superconducting loop

around the tetracrystal meeting point O will be in

a -ring configuration, giving rise to the half-ﬂux

quantum effect.On the other hand, for s-wave and g-

wave pairing symmetries, corresponding to A

and

irreducible representations respectively, there is

no sign reversal in I

. Standard integer ﬂux quan-

tization is expected in any superconducting loop

enclosing the wedge tip O. Shown in Fig. 16.22(b)

is the scanning SQUID image of an epitaxial Tl-

2201 film (T

= 83 K) on the tetracrystal STO sub-

strate (Fig. 16.22(a)). The tetragonal crystal struc-

ture of the single phase Tl-2201 epitaxial films used

in the experiment was confirmed by micro-Raman

spectroscopy, X-ray diffraction, bright-field imag-

ing transmission electron microscopy, selected-area

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 893

electron diffraction, and convergent-beam electron

diffraction [131,132].A detailed modeling [122,130]

of the image data (Fig. 16.22(c)) indicates the total

amount of ﬂux threading the wedge tip is indeed

/2. The fact that the half-ﬂux quantum is the only

vortex in a relatively large area suggests that it is

spontaneously generated, as expected. Since the con-

clusionaboutthesignchangeinthesupercurrentis

based on a rigorous model-independent symmetry

argument, the observation of the half-ﬂux quantum

effect represents strong phase-sensitive evidence for

d-wave pairing symmetry in Tl-2201.

(b) Orthorhombic Bi

CaCu

ı−x

The high-T

superconducting systems YBCO and Bi-

2212 are probably the most investigated members

of the cuprate family, owing to their importance in

fundamental studies and practical applications.Both

are characterized by an orthorhombic layered crystal

structure. The YBCO system derives its orthorhom-

bicity from the presence of Cu-O linear chains which

distort the CuO

square lattice, resulting in lattice

constants a = b. In the case of Bi-2212, an incom-

mensurate superlattice modulation in the BiO layers,

along the b direction, gives rise to unequal lattice

constants a and b in the CuO

planes [34,133]. An

important difference between the two cuprate sys-

tems is that the in-plane Cu-O bonds coincide with

the inequivalent a-andb axes in YBCO, but not in

Bi-2212. As for the case of YBCO, early pairing sym-

metry results on Bi-2212 using quasiparticle tunnel-

ing spectroscopy and ARPES produced controversial

results, supporting s-wave as much as d-wave sym-

metry. However, more recent ARPES studies on gap

anisotropy have yielded important evidence suggest-

ing d-wave pairing symmetry in Bi-2212 [53,54,134].

A phase-sensitive experiment is needed to determine

the pair order parameter symmetry unambiguously.

The results of a tricrystal ﬂux imaging experiment

are presented in Figs. 16.23(b) and (c). Shown in

Fig.16.23bis a scanning SQUID microscopy image of

a c-axis oriented Bi-2212 film deposited on a tricrys-

tal STO substrate designed for testing d-wave pair-

ing symmetry (the same as in Fig. 16.12). With the

tricrystal sample cooled in a field of 0.4Tandim-

aged at 4.2 K, many Abrikosov vortices are found

randomly distributed in the grains, while Joseph-

son vortices decorated the grain boundaries. At the

tricrystal point,a relatively weak vortex is always ob-

served. Detailed modeling of the SSM imaging data

shows that the vortex at the tricrystal point has a

total ﬂux of ¥

/2, while all others are single vor-

tices with the full ﬂux quantum ¥

(Fig. 16.23(d)).

To prove that the half-ﬂux quantum at the tricrystal

point is indeed generated spontaneously, the sample

was cooled to 4.2 K in a nominally zero ambient field

and imaged.The fact that only the half-ﬂux quantum

at the tricrystal point can be found(see Figs.16.23(c)

and (d)) provides the proof.

As shown in Fig. 16.24, the Brillouin zone for Bi-

2212 is rotated by exactly /4 with respect to that of

the orthorhombic YBCO. As a result, the in equiva-

lent axes a and b coincide with the node lines of the

anisotropic gap established by the ARPES measure-

ments [53,54,134]. From the viewpoint of the point

group symmetry operations, s-wave and d

−y

-wave

pair states correspondto two distinct irreducible rep-

resentations A

and B

, respectively. Therefore, Bi-

2212 can have either a pure s -wave or a pure d-wave

pairing symmetry.The resultsof the tricrystal exper-

iment and the ARPES measurements together argue

strongly in favor of a pure d-wave order parameter

symmetry for Bi-2212. This conclusion is also sup-

ported by the observation of zero [135,136] or nearly

zero [137] c-axis pair tunneling in Bi-2212.

16.3.4 Electron-Doped Cuprate Superconductors

The chemical composition of the electron-doped

cuprates [138] is given by Ln

2−x

CuO

4−y

Ln=Nd,Pr,Eu, or Sm; y0.04. Electron doping is ac-

complished by substituting the lanthanides Ln

and by removing oxygen. Of these compounds,

1.85

0.15

CuO

4−y

(NCCO) and Pr

1.85

0.15

CuO

4−y

(PCCO) have been the subjects of most studies. All

the cuprates are characterized by the Cu-O square

lattice layers in their crystal structures. However, the

electron doped ones are noted for the absence of the

apical oxygen atoms which form two-dimensional

arrays of Cu-O pyramids or octahedra with the CuO

planes in the hole-doped cuprates such as LCSO,

YBCO or Bi-2212. Superconductivity in electron-

894 C. C. Tsuei and J.R. Kirtley

Fig. 16.23. SSMdata on Bi2212.(a)Tricrystalge-

ometry,including polar plots of assumed d

−y

order parameters aligned along the crystalline

axes. (b) SSM image of 640 × 640m

area of

an epitaxial film of Bi2212 on a SrTiO

tricrys-

tal substrate with the geometry of (a), cooled

in a field of 0.4T, and imaged at 4.2 K with an

8.2msquarepick-uploop.(c) 400 × 400m

image of the same sample, cooled in nominal

zero field. (d) Comparison of horizontal cross-

sections through the images (b)and(c)with

modeling assuming that all vortices have ¥

ﬂux, except for at the tricrystal point, which has

/2 ﬂux spontaneously generated

Fig. 16.24. Brillouin zone for Bi2212. The reduced zone as a result of the incom-

mensurate super-lattice modulation with a period of 4.7b (b=5.41/AA) along the b

direction is outlined schematically. The orientation of the d

−y

2 -wave polar plots

is established by the ARPES data

doped cuprate systems occurs in a very narrow dop-

ing range (0.14 ≤x ≤0.17 for NCCO[138],and0.13 ≤

x ≤0.2 forPCCO [139,140]); inthe hole-doped coun-

terpart LSCO the range is broader (0.05 < x < 0.3).

The highest T

values in the hole-doped are over five

times those in the electron-doped cuprates. In opti-

mally doped YBCO and LSCO the in-plane resistiv-

ity exhibits a linear temperature dependence over a

wide range of temperature, with small or nearly zero

extrapolated values at T = 0. In NCCO and PCCO

(x =0.15), the in-plane resistivity is quadratic in

temperature with a large residual resistivity [141].

The results of ARPES have revealed CuO

-plane de-

rived ﬂat energy bands near the Fermi surface of

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 895

the high-T

hole-doped cuprates such as YBCO [142]

and Bi-2212 [143]; but not within 300 meV of the

Fermi surface of NCCO [144]. Other physical prop-

erties such as the Hall coefficient, thermopower, and

the pressure dependence of T

[145], are also dif-

ferent. For more details, reviews are available in the

literature [145,146].

In view of these remarkable differences between

the hole- and electron-doped cuprates,it is only logi-

cal to expect that the pairing symmetries of these two

classes of superconductors are also different. In fact,

based on the measurements of in-plane penetration

depth 

, tunneling characteristics, including the

absence of zero-bias conductance peaks (ZBCP), ...,

it was indeed widely believed that NCCO and PCCO

were s-wave superconductors [147]. In recent years,

a number of new experiments and new insight into

the data analysis have cast doubts about the s-wave

conclusion for the electron-doped superconductors.

After taking into account the effect of paramagnetic

ions [148], for example, a linear temperature

dependence of 

was found for NCCO [148–150],

expected for unconventional superconductors with

a line of nodes in the gap function. However, this

point remains controversial [151,152]. The absence

of a ZBCP in NCCO [151, 153, 154] was taken as

evidence for s-wave pairing symmetry, despite the

strong resemblance between the tunneling charac-

teristics of both types of cuprate superconductors.

Moreevidence against s-wavesymmetry comes from

the extremely small I

product (∼6V) observed

in c-axis pair tunneling in Pb/NCCO single crystal

junctions [155], almost three orders of magnitude

smaller than the 3 mV Ambegaokar–Baratoff limit

expected for an s-wave superconductor.

As in the case of the hole-doped cuprates,the con-

ﬂicting results from indirect symmetry studies with

NCCO and PCCO emphasize the need for a phase-

sensitive test. Tricrystal experiments are made dif-

ficult in these materials by the extremely small su-

percurrent density across the grain boundaries. For

example, the J

for a [100] tilt grain boundary junc-

tion with a 30

◦

misorientation in NCCO is about

1A/cm

at 4.2 K, five orders of magnitude smaller

than for YBCO [156,157]. This low critical current

density leads to a very long Josephson penetration

depth, making a half-ﬂux quantum Josephson vor-

tex at the tricrystal point difficult to observe. This

difficulty can be circumvented by making thick-film

tricrystal samples, provided that sample homogene-

ity and c-axis epitaxy can be maintained. Pulsed

laser-ablated c-axis oriented epitaxial films (∼1m

thick) of NCCO (T

∼22–25 K) and PCCO (T∼22–

23 K) have been developed for various tricrystal ex-

periments [158]. A J

of about 10A/cm

at 4.2 was

achieved with these thick films. Shown in Fig. 16.25a

is a SSM image of such a NCCO film deposited on

a tricrystal (100) STO substrate with the geometry

shown in Fig. 16.12 for testing d

−y

pairing sym-

metry regardless of whether the junction interface is

in the clean or dirty limit [87]. The Josephson vor-

tex centered at the tricrystal point spreads along the

grain boundaries for about 50m, a measure of the

Josephson penetration depth 

.Suchalong

re-

sults from the small J

(

∝ J

−1/2

). This in turn

results in small energy barriers I

/ between the

two degenerate half-ﬂux quantum states with fields

pointing in to,and out of,the sample.Therefore small

disturbances, such as an external field as small as

0.2 mG,or theinductivecoupling betweenthe sample

and the SQUID pick-up coil, can cause the Josephson

vortex at the tricrystal point to ﬂip to the opposite

field direction. Under any conditions of cooldown

and externally applied field there is always a vortex

at the tricrystal point. Figure 16.25a shows a three-

dimensional rendering of the SSM images obtained

by subtracting the two vortices with field directions

pointing into or out of the sample plane and dividing

by two. This procedure significantly reduces spuri-

ous signals in the SSM data. A detailed analysis of

the SSM data indicates that the Josephson vortex at

the tricrystal point has total ﬂux ¥

/2 (Figs.16.25(b)

and (c)) [158]. The value of 

from this analysis

was 48 m, corresponding to J

= c

/8ed

∼

20A/cm

(d=2

=0.5m), in agreement with a 4-

point probe measurement. Tricrystal experiments

with NCCO in two other configurations,designed to

be unfrustrated fora d-wave superconductor,didnot

show the half-ﬂux quantum effect. Identical results

were obtained in a series of tricrystal experiments

with PCCO [158].

896 C. C. Tsuei and J.R. Kirtley

Fig. 16.25. (a) Scanning SQUID microscope im-

age of the central area of a tricrystal sample of

NCCO in a -ring geometry for a d

−y

2 super-

conductor, cooled and imaged in zero applied

field. (b)and(c) are modeling of a bulk vortex

(b) and two cross sections through the half-

vortex (c). The dashed curve in (c)isthebest

fit assuming a total ﬂux of ¥

in the Joseph-

son vortex at the tricrystal point. The solid line,

assuming ¥

/2 of ﬂux, is a much better fit

In short, the tricrystal experiments described

above have provided strong phase-sensitive evidence

for d-wave pairing symmetry in the electron-doped

superconductors NCCO and PCCO. Recently, there

has been supporting evidence for a d-wave pair-

ing state in various electron-doped cuprate super-

conductors from ARPES measurements [159, 160],

temperature dependent penetration depth measure-

ments [149,150,161], measurements of complex con-

ductivity [162], the observation of zero bias con-

ductance peaks by point contact spectroscopy [163],

SQUID characteristics [164], and low-temperature

specific heat measurements [165].

16.3.5 Thin-Film SQUID Magnetometry

Mathai etal.[106]performed pairing symmetry tests

in YBCO by imaging the ﬂux in thin-film YBCO-Pb

(sample) SQUIDs using a scanning SQUID micro-

scope with a low-T

Nb-Pb (sensor) SQUID. In anal-

ogy with the experiments of Wollman et al.[104], the

sample SQUIDs had either a “corner” or “edge” con-

figuration (see, e.g. Figs. 16.8(a) and (b)): the “cor-

ner” SQUIDs should be -rings if YBCO is a d-wave

superconductor, while the “edge” SQUIDs should be

0-rings, independent of the superconducting pairing

symmetry.Thesample SQUIDs hada self-inductance

of about 75pH, and a ˇ

=2LI

/¥

∼ 1. This rela-

tively small ˇ

value meant that the spontaneous ﬂux

and circulating supercurrents at zero external field

(see Fig. (16.5)) generated by the -ring SQUIDs

were quite small. Further, there was a large mutual

inductance between the sample and sensor SQUIDs.

Therefore the inﬂuence of the Nb-Pb sensor SQUID

on the YBCO-Pb sample SQUID screening currents

had to be corrected for in these measurements.

Rather than directly measuring the amplitudes of

the spontaneous ﬂux at zero applied field, they mea-

sured the superconducting phase shift in their rings

by measuring the dependence of the screening cur-

rents in the YBCO-Pb sample SQUIDs as a function

of externally applied field, tracing out a number of

branches, each branch corresponding to a different

number of vortices in the YBCO-Pb sample SQUID.

16 Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors 897

The measurements were then repeated with the leads

to the Nb-Pb sensor SQUID and the external magnet

reversed, equivalent to a time-reversal operation. If

time-reversal symmetry invariance is satisfied, a 0-

ring YBCO-Pb sample SQUID in branch n will map

under time-reversal onto a branch n



with n = n−n



an even integer, while a -ring will have n odd.

Mathai et al. [106] found that the “corner” SQUIDs

had n odd while the “edge” SQUIDs had n even,

as expected for d-wave superconductivity. Further,

n was close to an integer for both types of sample

SQUIDs, implying that time-reversal invariance was

closely satisfied. These experiments had the advan-

tages that they directly imaging trapped ﬂux, so that

its inﬂuence could be avoided; and they allowed a

phase shift of  to be measured self-consistently us-

ing just the“corner” SQUID, without reference to the

“edge” SQUID.

Later work by Gim et al. [166] tested geometries

intermediatebetween the“corner”SQUID and“edge”

SQUID in an attempt to determine the momentum

dependence of the phase shift in the order param-

eter. In these SQUIDs one of the junction normals

was parallel to the YBCO b axis, while the other was

at an angle  relative to the a axis. If YBCO were a

pure d-wave superconductor, these SQUIDs would

be expected to have a 0-phase shift for 0<  <45

and a -phase shift for 45

◦

<  <90

◦

. The results

of these experiments were complicated by interface

roughness introduced by the etching process used.

However, the authors concluded that the angular de-

pendence of the phase of the order parameter follows

−y

symmetry,although it was necessary to under-

stand the details of the junction fabrication process

to reach this conclusion.

16.3.6 Spontaneous Magnetization in Facetted Grain

Boundaries

Although grain boundaries in cuprate superconduc-

tors often act like conventional Josephson weak links,

an exception is seen in [001] tiltboundaries in YBCO

with very large misorientation angles, in particular

in samples with the [100] or [010] axis oriented nor-

mal to the interface on one side, with the [110] axis

normal to the interface on the other side. Anoma-

lous dependences of the Josephson critical current

on applied magnetic field have been observed in

such junctions [167,168]. These anomalies can be

understood using a model which combines d-wave

superconductivity and facetting of the grain bound-

ary.In this case the sign of the component of the gap

function normal to the local boundary line can al-

ternate from facet to facet, producing local -rings.

In such a model the Josephson currents from adja-

cent facets tend to cancel, reducing or eliminating

the normally seen maximum in the junction criti-

cal current at zero applied magnetic field, in agree-

ment with experiment. An additional implication of

this model is that magnetic fields should be spon-

taneously generated in the grain boundaries, which

act as a linear array of -rings.Thisisinfactwhat

is observed [169].An example of this effect is shown

in Fig. 16.26. The upper panel shows a SQUID mi-

croscopy image of a 0-45

◦

[001] tilt grain boundary

in YBCO. There are 4 vortices in the grains to the

right of the image, but for the most part the im-

age is dominated by spontaneous magnetization, of

both signs, in the grain boundary. The lower panel

shows a cross-section through the image along the

grain boundary. The presence of this spontaneous

magnetization in these grain boundaries, and not in

samples with lower misorientation angles, provides

additional strong evidence for d-wave superconduc-

tivity.

Mints and co-workers [170–173] have proposed

that a linear array of -rings with spacing smaller

than a local Josephson penetration depth (given by



loc

=(c¥

/16



<| j

(x) |>)

1/2

,where

the London penetration depth of the grains, and

<| j

(x) |> is an average of the absolute values of the

critical current densities along the facet) will exhibit

pairs of Josephson vortices, each with ﬂux different

from ¥

, but with their ﬂuxes summing to ¥

.The

technology for producing junctions with controlled

0and-facets, and SQUIDs with controlled 0 and

-junctions is now being developed [174–176].

16.3.7 c-Axis Josephson Tunneling

The phase-sensitive pairing symmetry tests de-

scribed above have all made use of pair tunneling