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**Symmetry of the Phase of the Order Parameter in**
**YBa2Cu3O7-δ**

Y. Gim, A. Mathai, R. Black, A. Amar, F. Wellstood

**To cite this version:**

Y. Gim, A. Mathai, R. Black, A. Amar, F. Wellstood. Symmetry of the Phase of the Order Parameter in YBa2Cu3O7-δ. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2299-2316.

�10.1051/jp1:1996219�. �jpa-00247314�

J. Phys. I France 6 (1996) 2299 -2316 DECEMBER 1996, PAGE 2299

Symmetry of trie Phase of trie Order Parameter in YBa2Cu307-à

Y. Gim (*), ^{A.} Mathai (**), R-C- ùlack _{(***),} _{A.} _{Amar} _{and} _{F-C-} _{Wellstood}

Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, MD 20742-4111, U-S-A-

(Received 7 àfay19§6, revised 12 August 1996, accepted 19 August 1.996)

PACS.74.50.+r Proximity elfects, weak Iinks, tunnehng phenomena, and Josephson eoEects PACS.74.20.Mn Non-conventional mechanisms (spm fluctuations, polarons and bipolarons,

resonating valence bond mortel, anion mechanism, marginal Fermi Iiquid, Luttinger hquid, etc.

PACS.74.72.Bk Y-based cuprates

Abstract. We first descnbe the basic idea behind phase-sensitive measurements to deter-

mine the symmetry of the order parameter in YBa2Cu307-à (YBCO). We then _{review} recent

expenmental work. While there _{are} still _{some} apparent disagreements, _{a} majonty _{consensus} _{ap-}

pears ^{to} have emerged that the order parameter in YBCO is consistent with

a d~2_y2 symmetry.

perhaps with _{an} added _{s-wave} component. TO further illustrat.e the nature of these experiments,

we discuss _{our} investigation of the angular dependence of the phase of the order parameter. ^{Us-}
ing _{a} 4.2 K scanning SQLÎID microecope, _{we} examined thin-film YBCO-Ag-Pbln SQUIDS and
measured trie phase of the order parameter _{m} twmned YBCO for 13 dilferent tunneling angles

in the _{a} b plane. We find that the painng symmetry ^{of} ^{the} ^{order} parameter ^{is} time-reversai
invariant and consistent with

a d~i_~2 symmetry, provided the junction surface

is not damaged
and the tunnehng angle _{is} weII-controlled. Front _{a} detailed analysis of the data, _{we} conclude
that there _{is} less than 2Sl of any component which breaks time-reversai symmetry. ^{This} ^{rules}

out states such _{as} d~i_~2+ is _{or} d~2_~2 + id~y.

1. Introduction

What is the physical mechanism which produces the high superconducting transition temper-

ature (T~) ^{in} YBa2Cu30y (YBCO) and other cuprates [Ii? At present, ^{this} ^{is} perhaps the

most important question _{m} superconductivity. In recent years, experimentahsts have tried to

find the _{answer} to this question by making detailed observations of the behavior of the Cooper

pairs. This has yielded much _{new} information about the mechanism. For example, compared to

pairs m low-T~ superconductors such _{as} Nb _{or} Pb, the pairs m high T~ superconductors can be

easily broken at temperatures ^{T} ^{which} are much less than T~. The _{presence} of such lo~v. energy
excitations _{is} revealed by Nuclear Magnetic Resonance experiments (NMR) [2-4j, _{microwave}
penetration depth measurements [5, 6], ^{and} Angle-resolved photoemission studies [î]. These
results cannot be readily explamed by conventional phonon-mediated _{pairing} (pure _{s-wave}

(* ^{Author} ^{for} correspondence je-mail: yggimfisquid.umd.eau)
(**) ^{Present} ^{address:} ^{Conductus} ^{Inc..} Sunnyvale, CA.

(***) Present address: Quantum Design, San Diego, CA.

@ Les Éditions _{de} _{Physique} _{1996}

la) ibi _{~Y}

+ +

k~

+ +

~

lc) Id)

b 8

~

p~ Pb

a

250 jm

Fig. 1. Schematic representation of order parameter ^{for} la) _{pure} _{s-wave,} (b) amsotropic _{s-wave,}

and (c) d~2_~2-wave ^{pairing.} (d) Schematic diagram of _{a} sample SQUID.

pairing), such _{as} used in the BCS mortel [8]. ^{As} a result, novel theories based _{on} anisotropic

s-wave pairing _{[9,}loi _{or} d~2-~2 (d-wave) pairing [11-13] have been developed.

The superconducting state predicted by the _{new} theories has _{many} features in _{common} with
the superconducting state of low-T~ superconductors. For example, in both _{cases} electrons

are paired and the motion of each pair ^{is} ^{descnbed} by _{a} macroscopic quantum ^{mechanical}

wave function _{or} order parameter. In general, the _{wave} function _{is} _{a} complex number with
both amplitude and phase. In the BCS mortel, the interaction is _{a} short range attraction
which arises from the exchange of phonons. The short range attraction leads to spherically
symmetric s-wave pairing; the amplitude and phase of the _{wave} function _{are} independent of

direction (see Fig. lai. By contrast, in anisotropic _{s-wave} and d-wave theories the magnitude

and phase of the pair wave function depend _{on} direction in momentum space (see Figs. lb
and c). In particular, in d~2_~2 theories, there is _{a} difference of _{~} between the phase of pairs
moving along the k~ direction and that of pairs moving along the ky ^{direction.} ^{The} ^{result} ^{is} an

order parameter with four lobes of altemating signs (see Fig. lc): this is just ^{the} ^{well-known}
d~>_~2 ^{state} ^{of} ^{atomic} physics.

Recently, there have been several phase-sensitive experiments which have sought to determme
whether the high-T~ superconductor YBCO has _{an} order parameter ^{with} an s symmetry, a

d~>_y2 symmetry, _{or} _{some} other symmetry. ^{While} ^{the} ^{details} ^{of} ^{these} experiments differ
greatly, ail of them _{use} Josephson junctions _{or} Superconducting QUantum Interference Devices

(SQUIDS) to gain information about the phase.

N°12 S"MMETRY OF THE ORDER PARAMETER IN YBa2Cu307-6 2301

2. Basic Idea of Phase Sensitive Experiments

At first sight, it may ^{not} be obvions how the phase of the order parameter can te measured.

However, _{more} than 30 years ago Josephson _{[14]} showed that if _{a} tunnel junction is used to

connect _{one} superconductor to aiiother, _{a} phase difference à between the superconductors leads

to the flow of _{a} supercurrent I between the two superconductors. This result is summanzed

m the well-known de Josephson relation:

I

= Io sin(à), (Ii

where Io _{is} the critical current of the junction. Since current flow is easily measured, _{we} _{can}
conclude that differences in the phase of the order parameter are directly measurable.

One subtlety ~vhich is not apparent ^{from} equation (1) is that the main contribution to
the tunneling current is from pairs which _{are} incident normal to the junction face. This
geometrical dependence is hidden in the parameter Io, which includes factors for the probability

of transmission through the tunnel barrier. Thus the tunnehng current I carries information
about the order paraineter along the direction normal to the tunnel junction ^{face.} A number

of receut experiments _{on} YBCO-Pb junctions have made direct _{use} of this effect to mfer

information about the order parameter ^{in} specific directions.

Many of the experiments on the syminetry of the order parameter use a de SQUIDS rather
than _{an} iiidividual junction. A de SQUID consists of two junctions connected in parallel to
form _{a} closed superconducting loop _{[15].} The phase-sensitive experiments follow _{a} general _{ap-}
proach first suggested by Geshkenbein _{et} ai. [16] to study the symmetry ^{of the} ^{order} parameter
in heavy-fermion superconductors. Later, Sigrist and Rice il îj proposed _{a} detailed geometry ^{in}

which the two junctions _{are} oriented _{so} _{as} to sample the order parameter ^{in} ^{dioEerent} ^{directions}

(see Fig. ld). One part of the SQUID loop is made from the conventional _{s-wave} super-
conductor Pb and the other part from YBCO. In this figure, the upper junction has _{a} face
oriented _{so} that its normal _{vector} points along the a-axis, while the lower junction has _{a} face
oriented with its normal vector at an angle f with respect to the a-axis. Pb is _{a} conventional

s-wave superconductor, _{so} that _{we} _{can} take the phase of the order parameter as just çipi m the

Pb at the upper junctiou and çip2 at the lower junction, independent of tunneling direction.

Similarly, _{m} the YBCO, _{we} _{can} wnte the phase in the _{Upper} junction _{as} çii m the direction
normal to the junction interface. Similarly, we con write the phase in the lower junction as çi2

m the direction normal to the junction interface. The phase difference _{across} the first junction

is thus ôi _{"} 41 4Pi ^{and} across the second junction 62 " 42 4P2.

If _{a} _{pair} travels _{once} around the SQUID loop it will develop _{a} total phase change of:

à = ôi 62 + ôd(f) + 2~4l/4lo, (2a)

where éd(f) is the intrinsic diiference in phase _{m} the YBCO between the two tunneling direc-
tions which diifer by angle ^{f,} 4l _{=} B-4 + LJ is the total magnetic flux _{m} the SQUID loop,
4lo _{=} h/(2e) is the flux quantum, B is the applied field, A is the pick-up _{area,} L is the loop
mductance, and J is the screening current. The first _{two} _{terms} _{are} just the phase drops _{across}

the two junctions, the third term takes mto account how the phase of the order parameter
depends _{on} direction _{m} YBCO, and the last term _{is} due to the AhaI.anov-Bohm eifect for _{a}

partiale of charge 2e. For the _{wave} function _{to} be single-valued, _{we} _{require} that the total phase

shift à

= 2n~, where _{n} _{is} _{an} integer. Thus _{we} _{can} write the flux-phase relation:

2~~ = ôi 62 + ôd(9) + 2~4l/4lo (2b)

It is interesting to note that the mtrinsic phase shift éd enters into equation (2b) in exactly

the _{same} way as an extra apphed flux of strength 4loôd/2~. Thus, _{a} phase difference oféd _{"} ~

is effectively equal to _{an} added flux of 4lo/2. This _{means} that it is essential to know the true
or absolute magnetic flux 4l in the SQUID in order to determine éd experimentally.

If YBCO is _{s-wave} symmetric (see Fig. la _{or} b), then éd

" 0 and _{one} gets ^{the} ^{well-known}

flux-phase relation for _{a} SQUID,

2n~ = ôi 62 + 2~4l/4lo. (2c)

When there is _{no} applied magnetic field B, we can use the de Josephson relation to write the
total flux in the SQUID _{as:}

4l

= LJ

= LIOsin[ôi)

" -LIO sin(62) (3)

It is easy ^{to} see that _{one} solution to these _{two} relations is ôi _{"} 62 " 0 which gives J

= 0 and

4l = 0. This is the minimum energy ^{state} of the system ^{and} ^{it} ^{has} no current flowing around
the loop at B

= 0.

On the other hand, if ~~BCO has _{a} d~2 _{-y2} symmetry, and _{oiie} junction connects to _{a} positive
lobe in the YBCO and the other connects to a negative lobe, then éd _{"} ~. Solving equations
(2b, 3) m this _{case} reveals that the phase ^{shift} tends to _{cause} a screening current J to flow
around the SQUID, just _{as} would _{occur} if _{a} flux of 4lo/2 _{were} applied. If the cntical current
of each junction is much larger than 4lo/(2L), then this results in _{a} spontaneous curreiit of

+4lo/(2L) which produces _{a} flux of ~4lo/2 in the SQUID loop. Essentially, the spontaneous

current tries to cancel the effective flux of 4lo/2.

The above basic ideas _{can} be made precise by solving the equations ^{of} motion for the SQUID.

Appendix A provides _{an} overview of such _{an} analysis. In particular, such _{an} analysis is required

to understand the behavior of the SQUID if the juiiction ^{critical} current is non large compared

to 4lo/(2L), to understand the effects of _{a} complex order parameter (which would break time-
reversal symmetry), an4 to understand the effects of _{a} bios current _{or} non-symmetric SQUID
parameters.

From these considerations, _{we} _{con} _{see} that in principle it should be _{a} simple matter to
determine whether the phase of the order parameter ^{in} ^{YBCO} depen4s on direction. One
builds _{a} SQUID with junctions oriented in smtable directions and tests for the presence or

ahsence of _{a} outrent flowing around the SQUID when there is _{no} magnetic ^{field} applied to the

SQUID.

3. Recent SQUID and Junction Experiments

The pioiieering measurements _{on} the symmetry ^{of the} ^{order} parameter ^{in} ^{YBCO} were made
by Wollman et ai. [18]. ^{In} ^{these} experiments, SQUIDS _{were} formed from junctions made by

depositing _{a} thin film of Au and then Pb film onto specific faces of single crystals of YBCO

(see Fig. 2a). In _{some} of these SQUIDS, _{one} junction was ou the a-axi~ face of the YBCO

crystal and the other _{was} _{on} the b-axis face. These "corner" _{or} _{a} b" SQUIDS _{are} sensitive
to the différence _{m} phase éd betweeu pairs traveling in the _{a} and b directions. For companson,

they also made "edge" or ^{"} a a" SQUIDS with both junctions _{on} the _{same} _{a-axis} face of the
YBCO.

To determine éd, Wollman et ai. biased a given SQUID with _{a} constant current and then
measured the voltage _{across} the SQUID _{as} _{a} function of the applied magnetic field. In _{a}

SQUID. the voltage oscillates periodically in the applied field, with _{a} period of 4lo. The
minimum voltages _{occuI.} at values of the flux where the phase differences _{across} the inactions,
ôi and 62, ailler by _{an} integral multiple of 2~. The fluxes _{at} which the minima _{occur} are thus

related to éd, the phase difference bet~v.een the _{a} and b-axis directions.

N°12 S'~MMETRY OF THE ORDER PARAMETER IN YBa2Cu307-6 2303

(a) (b)

YBCO

Pb a-axis

b-axis ~ ^{~~~~} c-axis

Pb

(c)

d @~

§ .,~ )

~fll,~

### ~fl

~ ~,

~'

b-axis YBCO

Î

a-mis

Fig. 2. Scheniatic diagram of (a) _{a} b SQUID in reference [17], (b c-axis junction _{in} reference [22j,
and (c) '~BCO rings _{on} _{a} tri-crystal _{m} reference [29j.

Unfortunately, the relationship between the voltage minima and the flux _{is} not _{as} simple _{as}

discussed in the previous section because the device _{is} biased into the finite voltage state: 1.e.

the m~asurements are taken with _{a} current I flowing through the SQUID loop. This current

con introduce additional magiietic flux into the loop. This happens if _{one} side of the SQUID
loop has _{a} different inductance thon trie other (mductance asymmetry) _{or} if trie junctions have
different critical currents (critical current asymmetry). To correct for this, ~vollman et ai.

recorded voltage minima for different bias currents and then hnearly extrapolated the results
to1= 0. In _{zero} magnetic field, they reported that the extrapolated minimum of the voltage
always occurred at _{a} flux of about 4lo/2 in the _{a} b SQUIDS and 0 for the _{a} _{a} SQUIDS.

This corresponds to a phase diiference bd _{=} ~ between the _{a} and b-axis directions and thus is
consistent with d~2-~> pairing. ^{Their} ^{results} ^{have} ^{been} supported by severai rater experiments
using Josephson junctions [19-21] and à~BCO-Nb SQUIDS _{[22].}

By _{way} of contrast, we next consider the Josephson junction experiments of Sun et ai. [23]. ^{In}
these experiments, tunnel junctions _{were} made by ^{first} depositing _{a} thin film of Ag (thickness

of about nm) onto the _{c-axis} face of _{a} crystal of YBCO. A film of Pb _{was} then deposited

on top of the Ag to complete the _{c-axis} junction (see Fig. 2b). For pure d~2-~2 symmetry,
the _{wave} function has _{a} node _{m} trie c-direction, and thus _{one} expects _{no} ^{critical} current. In

addition, _{even} if tunnehng _{anses} ^{from} _{pairs} which tunnel ~v.ithin _{a} small solid angle about the
c-axis, the four lobes of the d~2-~2 wave function would coiitribute equally, and with opposite
sign, so as to yield _{no} net current in _{zero} applied field. In fact, Sun _{et} ai. found _{a} _{maximum}

cntical current in _{zero} apphed fiera and weil-behaved modulation of the criticai current ~vith

apphed magnetic ^{fiera.} This result suggests that the junctions hâve _{a} umform cntical current

density, and that the pairing symmetry is s-wave or, ^{ai} the very least, _{is} trot pure d~2_~2.

Taken at face value, the experiments of Wollman et ai. and Sun et ai. yield apparently
contradictory results. This open up several possibilities, which _{we} _{now} examine. First, be-

cause YBCO is orthorhombic and non tetragonal, in fact, _{one} would _{not} expect a pure d~2_y2

symmetry [24]. Rather, _{one} _{expects} the order parameter to be distorted _{so} that _{one} lobe is

larger than the other, while maintaining a ~-shift between the _{a} and b-axis direction. In prin-
ciple this coula explain observed tunneling in c-axis junctions because then the positive and

negative lobes would not cancel exactly. However, _{some} of the samples which _{were} measured

were twinned. The presence of _{au} equal number of twinned regions lin which the YBCO a-axis
switched direction with the b-axis) would lead _{to} _{zero} _{average} current, _{even} in the _{preseuce} of

an orthorhombic distortion. This nuit _{current} result in twinned crystals would also holà if the
distortion _{~v.as} caused by _{a} combination of _{an} _{s} and d~2-y2 symmetry.

A second possibility has been suggested by Tanaka [25]; even with _{pure} d~>-y> symmetry,

one would expect to _{see} a second order tunneling current ont of the c-axis. In this _{case,} the

tunneling would follow _{a} sin(2p) current-phase relation rather than the usual sin(çi) dependence
given by equation (1). This dependence would produce measurable effects when the junction is

exposed to microwave fields. In particular, Shapiro _{steps} would _{appear} in the current-voltage

characteristics at voltages of [?1/2) f4lo instead of nf4lo, where _{n} is _{an} integer ^{and} f is the
microwave frequency. ^{This} possibility has been tested in _{a} _{recent} experiment by Kleiner et
ai. [26] using ^{YBCO-Pb} junctions. They find that ail the steps _{are} separated by _{n}f4lo, which

is consistent with first order tunneling.

Another possibility is that, in the experiment of Sun et ai., the tunneling ^{is} not happening ^{in}

the c-direction, but along the _{a} _{or} b _{axes} where _{a} step or other defect _{occurs} in the crystal face.

To produce _{a} well-behaved modulation of the critical current with applied field, the steps ^{would}
hâve to be closely spaced. We note that Sun et ai. has reported using _{an} AFM to examine the
flatness of the films and finds little evidence for such steps. ^{In} addition, the presence of steps
wou14 tend to mtroduce "corner junctions". If YBCO has _{a} d~2_y2 symmetry, ^{such} corner

junctions would produce _{a} distinctive flux modulation pattem ^{with} a minimum critical current
at _{zero} applied flux. ive _{note} that just such distinctive pattems hâve been observed by Iguchi

and Wen [19] ^{in} ^{Pb-YBCO} ^{c-axis} junctions and by Wollman et ai. in _{a-} b _{corner} junctions [21].

From the above observations, it is clear that the results of Sun _{et} ai. _{are} consistent with

s-wave pairing and _{s-wave} painng mixed with _{a} small d~2 _{-~2} component. But, _{a} _{pure} _{d~2} _{_~2-}

pairing symmetry ^{is} completely inconsistent with their results. For _{a} mixed symmetry of _{s}
and d~2_~2, ^{the} consistency depends _{on} whether ~~BCO is twinned _{or} untwinned. When the

YBCO is untwinned, d~>-~>-pairing mixed with a small _{s-wave} component would produce _{a}
finite junction critical current in _{zero} applied field. However, for twinned YBCO, the situation
is less clear and it appears thon averaging _{over} _{many} twins ~vould give zero'cntical _{current} _{if}

the d~>-y2-pairing _{component} _{was} suliiciently large. Since _{some} of their YBCO samples _{were}
heavily twinned, their results coula be explained by either _{s-wave} pairing or s-wave pairing

mixed ~vith _{a} small d~2_~2 component.

This might suggest ^{that} something is wrong in the SQUID and junction experiments [18-22].

In principle, there _{are} several effects which coula lead to an erroneous conclusion, including:

Iii trapped vortices in the superconductor which linked half of _{a} flux quantum into the _{a}
SQUID loops but _{not} the

a a SQUID loops, Iii) systematic errors m determiumg the magnetic
field _{or} in determmmg the relative field ai two different SQUIDS, (iii) problems with applying

a linear extrapolation technique _{to} _{a} non-linear device such _{a8} SQUID, and (iv the presence
of trapped flux in the Josephson junctions.

The possibility that the original ex~eriments might be _{m} _{error} lead _{to} _{a} second round of

expenmental work usmg different techniques. To avoid possible problems, it is necessary ^{to}
check for and elimmate _{any} trapped flux in the samples, _{measure} the magnetic field accurately,