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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 nf4lo, 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,
