The order parameter of high Tc superconductors; Experimental probes

~ Pergamon

Solid State Communications, Vol. 92, Nos I-2, pp. 53-62, 1994 Elsevier Science Ltd Printed in Great Britain 0038- ! 098(94)00489-7 0038-1098/94 $7.00+.00

THE ORDER PARAMETER OF HIGH Tc SUPERCONDUCTORS; EXPERIMENTAL PROBES R.C. Dynes Department of Physics, University of California, San Diego 9500 Gilman Drive, La Jolla, CA 92093-0319, U.S.A.

The high Tc superconductors have many characteristics which distinguish them from conventional superconductors. Several theories of the superconducting state conclude that the order parameter is of a higher symmetry than conventional. This article reviews the experiments designed to probe the symmetry of the superconducting state and what we can conclude from the experiments. Keywords: Superconductivity, High Tc

The original discovery of superconductivity in cop-

Any disorder or a reduction in carrier density (below the

per oxides by Bednorz and Mueller [I] came as a major surprise to the physics, chemistry and materials community.

optimal value) pushes the material toward the M-I transi-

After many experimental and theoretical studies which

pelling evidence that the antiferromagnetism persists in the

concluded that a superconducting transition temperature

form of spin wave fluctuations well into the metallic region

above ~ 30K was unlikely, the discovery of a superconduc-

of the M-I transition. In the intermediate region between

tor with a Tc of 93K motivated the community to reappraise

antiferromagnetism and metal, it has been suggested that

some of the basic assumptions that were the underpinnings

there is a spin glass phase. With such a rich set of charac-

of that conclusion.

teristics, it is not perhaps surprising that there is such a

tion; 3) the insulator is antiferromagnetic and there is com-

With time, the discovery of other

cuprates has resulted in a large number of superconducting

large variety of suggested mechanisms for the pairing.

compounds and a maximum Tc (under pressure) of greater than 150K.[2] In spite of the progress on the materials aspects of this phenomenon, there are widely different

Unlike in the case of an isotropic 3-dimensional electron-phonon mechanism, when several of these alterna-

views as to the pairing mechanism responsible for the

tives are pursued to their conclusion, it is found that they

superconductivity. In addition to the conventional electron-

can result in a pairing state with higher angular momentum.

phonon interaction, serious calculations exploring the

While the anisotropic electron-phonon or charge transfer

whole spectrum of interactions (excitons, spin fluctuations,

model results in an appropriately anisotropic s-symmetry

holes, spin bags, valence fluctuations, acoustic plasmons,

state, it is found that models that rely on the antiferromag-

marginal Fermi liquids, RVB....etc.).

netic fluctuations for the pairing result in dxz-yZ symmetry or a mixed s + i dxe-y2.[3] Furthermore, an even more

There are some observations which seem to be con-

exotic dx2-y2 -i dxy pairing has been suggested. The debate

sistent and generally observed with all the high Tc cuprates

over the symmetry of the pairing state; a property which can in principle be measured experimentally.

and it is these characteristics which must be somehow incorporated in any physically realistic model. These general characteristics are 1) The high Tc cuprates are all

There have been a variety of experiments to probe this symmetry and they generally fall into 3 categories; a)

structurally and electronically anisotropic. It is generally believed that the CuOz planes are the conducting planes

spectroscopies and excitations, b) Joscphson measurements,

and the locations of the pair density; 2) the high Tc cuprates are in the vicinity of a metal-insulator (M-I) transition.

3) magnetization and transport. As the situation is still controversial, and new and different experiments arc being 53

54

ORDER PARAMETER OF HIGH T¢ SUPERCONDUCTORS

Vol.920 Nos 1-2

attempted, this article will most likely be out-of-date by the

the particular topology of the symmetry and whether there

time it is published. Nevertheless, we will attempt in this

are lines or points of zero gap on the surface. The result of

article to discuss the various experiments to determine the

this difference is that the properties which depend on

symmetry of the pairing state and appraise their conclu-

quasiparticle density (heat capacity, scattering rate, relax-

sions. For a discussion of the theoretical basis for these

ation rate, screening length, etc.) will have a very different

various symmetries and how they evolve from the various

dependence compared with a conventional BCS superconductor.

pairing mechanisms, the reader is referred to the article by J.R. Schrieffer in this volume.[4] Here we discuss the experimental evidence and what can be concluded about

It is very important to point out, however, that the

the various proposed symmetries. Since several models for

observation of points or lines of A = 0 on the Fermi surface

the pairing make clear predictions of the symmetry, the

does not Drove higher symmetry. It is, in principle, possi-

importance of this determination should not be underesti-

ble to retain an s-symmetry (extended s-wave symmetry)

mated.

and have zero crossings on the Fermi surface. These zero 2. Spectroscopies and Excitations

crossings must retain the square rotational symmetry however and so rather than the four zero crossings expected in a

An important distinction between a conventional

dx2-y2, in rotation through 2~, we would expect a minimum

isotropic s-symmetry superconductor and one of a dx2-y2

of eight (four positive to negative, four negative to posi-

symmetry can be summarized in Fig. 1. Here we schemati-

tive). Consequently, gaplessness does not Drove a higher

cally show the symmetry of the order parameter and the

order symmetry but is consistent with it. In this context it

resultant density of excitations. In the dx2-y2 symmetry, the

is also worth pointing out that while these zero crossings

order parameter has zero crossings along the diagonals.

are in principle possible in conventional superconductors,

This results in lines on the Fermi surface where the gap is

we do not know of any where they are observed.

zero and hence the gaplessness in the density of excitations.

Consequently, observation of zero crossings in the order

This is in striking contrast to the conventional picture of

parameter, and a density of states of the type illustrated in

Fig. la where the density of states follows a BCS form

As

Ad

ky

ky

N(E)

IN(E)

Here A is the (isotropic) energy gap. Any spectroscopy that is capable of measuring this density of excitations should be able to distinguish between the two cases. Furthermore, because of the differences in the excitation spectrum in the two cases, the number of thermal excitations should be strikingly different. In the BCS form, the number of excited quasiparticles should be exponentially activated with an activation energy = ,5. As a result of this form for the quasiparticle density, measurable quantities that depend on the number of excitations will have an activated behavior in the case of a conventional superconductor. In contrast, however, with the excitation spectrum expected for a superconductor with dx2-y2 symmetry (Fig. lb) a quasiparticle can be excited with an incrementally small energy. This comes about because of the

(a) s Symmetry

(b) d×2_y2 Symmetry

zero crossings of the gap and so there are places on the Fermi surface where A = 0. Consequently the number of quasiparticles will have a power law dependence on temperature, not exponential, i.e. nqp a T n. n will depend on

Figure 1. A schematic representation of the order parameter and the resultant density of excitations for a) an isotropic s-symmetry, and b) a dx2-y2 symmetry.

55

ORDER PARAMETER OF HIGH "Pc SUPERCONDUCTORS

Vol.92, Nos 1-2

1.2

Fig. lb is certainly unusual and, independent of whether the symmetry is extended s or higher, the physics of such a

.

.

.

+

. . . .

t


superconductor would be different than conventional.

•--~ 1 . 0 D

There have been several experimental probes to

gC.9

study the excitation spectrum. While it is impossible to

~ 0.8

describe all of them in detail here, the conclusions derived

~

....YII~C~O]~I .

.

.

.

.

I

.

.

.

.

I

.

.

.

.

i

. . . .

I

.

.

.

.

i

I

I

I

I

. . . .

II

N

from several of them will be summarized here.

0.7

o

z 0.6

a) Quasiparticle Tunneling

. . . .

0"5-;50 -1oo

I

. . . .

-bO

0

i . . . .

50

100

150

Voltoge (meV)

In conventional superconductors, the technique of quasiparticle tunneling has been very effective in determining the nature of the excitation spectrum and measuring the

2

++,+i,~+~i,,,

I'

'+I""''~[++~'-l

energy gap A. With some limitations, a measurement of the conductance (o--~-)of a supereonducting-insulating-normal metal tunnel junction yields a direct measure of the density of excitations N(E).[5] This is because the conductance in the superconducting state can be written:

-.

+ expt/~(e+ eV)]}2 ]

u i= u

0.5i

(2)

I dl where p =~-~, ~--vl~+is the conductance in the normal state

10*/,LeakaQe

"" " ,

0 5

and V is the applied voltage on the tunnel junction.

-10

-5

0

Voltage

,I,,,,I.., 5

10

(mV)

Because of the success of the technique in conventional superconductors and the relatively straightforward

Figure 2.

interpretation of the measurements, there have been a large

related to the density of excitations extracted from tunnel-

Normalized conductance which is directly

number of tunneling studies in the cuprates with a variety

ing measurements a) tunneling conductance for YBCO

of results. The techniques range from conventional planar

from [7], b) tunneling conductance for Nd2.xCexCuO4 from

junctions,[6,7,8] to point contact spectroscopy,[9] to scan-

[14].

ning tunneling microscopy (STM),[10,11] to break junctions.[12,13] While it is impossible to describe all of the results on all of the compounds, there appears to be a gen-

the YBa2Cu3OT.s appearing the most gapless and others

eral consensus emerging. It seems that from tunneling

showing gaps.

studies, the class of cuprates offer a rather broad range of

somewhere between the two shown in Fig. 2. The newly

characteristics from something that appears to resemble a

discovered Hg based cuprates apparently have a significant

BCS superconductor with a well defined energy gap, in the

BCS energy gap.J9]

Several compounds show behavior

case of Nd2.xCe, CuO4,[ 14] to gapless behavior in the case of YBa2Cu.~OT.s.[6] By way of example we show two of

While there are issues associated with the fact that

the cases above. In Fig. 2a we show the tunneling density

tunneling is an intrinsically surface probe (to a depth ~ 9, of

of states

for

the coherence length), the evidence continues to accumu-

YBa2Cu3OT.+.[71 It is clearly gapless showing a large density of excitations at E = 0. In contrast, Fig. 2b shows

late to suggest that some of the cuprates are gapless, others

(normalized to the

normal state)

appear to have a BCS excitation spectrum. While the gap-

the conductance of a point contact tunnel junction in

less behavior is consistent with a dx2-y2 symmetry, the BCS

Nd2.xCe~CuO+ and a fit to a BCS density of states. While

shape is not and so if one were to argue that the symmetry

not perfect it certainly suggests this material is closer to

of YBa2Cu30~8 is dx2-y2 as evidenced by their gapless

having a conventional energy gap A. Other studies on these

behavior in the tunneling, there are other cuprates that are not.

and others of the cuprates show this range of behavior with

56

ORDER PARAMETER OF HIGH T c SUPERCONDUCTORS

Vol.92, Nos !-2

b) Penetration Depth Measurements 10 s The temperature dependence of the penetration depth ~,(T) depends on the quasiparticle density. As a result in a conventional BCS superconductor, the tempera-

10 e

,,10 s

ture dependence of ~.(T) at temperatures well below Tc is given by

10 °

k(T) = ;L(O) I +

(3)

10"

5 the usual activated result coming from the quasiparticle or

15

20

25

TemperaLure (K)

normal fluid density. However, in a system of the type shown in Fig. Ib,

10

1000

the penetration depth should have the temperature depenBOO

dence

~,(Z)-~~L(O)(l +k-~ln2)

(4)

where A is the maximum gap. It is also expected [15] that in the case of elastic

600 4O0 2OO

scattering (impurity and defect scattering) the temperature dependence of k(T) should cross over from T to T 2. For very strong scattering (XE A ~ 1) superconductivity should

mOO

0 0

disappear but in the case of intermediate scattering at low

|

20 40 60 Tempm,oture (K)

80

temperature ZE T ~ I, a crossover to T 2 dependence should Figure 3. The temperature dependence of the penetration depth ;L for a) Nd2.xCexCuO4 [16], and b) YBCO.[17]

result. As in the case described above for tunneling, a power law temperature dependence of ~, suggests gapless-

In the one case it has an exponential dependence, the other case a linear dependence.

ness consistent with a dx2-y2 symmetry, but it does not prove it. Also as in the case described for tunneling, measurements of ~,(T) apparently yield a variety of results

rately the electronic states of the oxygen and copper sites in

ranging again, from the activated behavior [16] of Eqn. 3 in

the CuO2 planes. From the extensive measurements by a

the case of Nd2.xCexCuO, to the linear behavior [17] of

variety of investigators,[ 19,20] it is apparent that the anti-

Eqn. 4 in YBa2Cu3OT.s. This contrast is illustrated in Fig.

ferromagnetic spin correlations remain in the normal state

3. The very impressive linear T dependence of ~,(T) in the

of the superconducting compounds although the properties

case of YBa2Cu3OT.s is consistent with the gaplessness, and

indicate the spectrum of these correlations becomes sub-

the exponential behavior in Nd2.~Ce,CuO, is consistent with

stantially broadened both in energy and in k with increasing

a well defined gap. Also, as in the case of tunneling mea-

carrier density (and increasing Tc). In the superconducting

surements, other compounds (or deliberately doped YBa2Cu3OT.s) show behavior somewhere between these two

state it is found that the relaxation behavior - - is very T similar for many of the cuprates and not exponentially acti-

extremes.

Studies of impurity doped YBa2Cu3OT.s [18]

vated as would be expected for a BCS superconductor with

show the crossover from T to dependence expected for

a well defined gap A. Furthermore, a coherence peak is not

strong scattering.

observed. A compilation of ~1- for several of the cuprates is shown in Fig. 4 following I~/.aoka [19] Further, the

c) NMR and NQR Studies

effect of various impurities has been studied and fired with

I

etal.

a density of states for a gapless superconductor. It is conNMR and NQR studies are capable of probing sepa-

cluded that the NMR experiments cannot be accounted for

ORDER PARAMETER OF HIGH T,. SUPERCONDUCTORS

Vol.92, Nos 1-2 .... I

........

I

C.u plane site

"

.

~

"

'

57

for the d-symmetry, the results are consistent with such a

' '~'"I

~00"*

picture. One must be careful with this conclusion, however, as much of the data which is claimed to support a dwave picture can also be explained equally well within the

-

:C

"~

!

context of a very anisotropic s-wave gap. With current resolution in these experiments, it will be difficult to resolve this issue and higher resolution measurements might help.

1~~

+p

;+

,...,

One of the strengths of the photoemission technique

%

is its ability to probe band structure off the energy shell. As an aside to the central issue of this paper, it is worthwhile

o.'b

noting that several studies of the energy dependence of the • TI28a~C,t~,¥

I

o

x

!

O

X•D o&

angular resolved photoemission have shown a large region

L S C O , Sr:15"/,

x Po~'pt¢.a)C,O~)

of fiat C u t 2 bands near EF and strong indications of Fermi

O

YBCO 7

surface nesting. These results support a picture of a large

•

Bi PoSrC,ICUO

density of states near the Fermi surface as a result of a van

13 "

x

Hove singularity.[23,24] It is this large density of states "z •++ • , 1•1° ..

t

(11

i ,|11||

1

,

• • i,.,{

10

t

TI

Tc

which is envisioned responsible for the linear o(T) and high Tc. While this is a tempting connection, recent studies seem to suggest the relationship is not straightforward.

Figure 4.

A compendium of measurements of I for Tt various high Tc superconductors from [ 19].

To summarize the photoemission experiments which address this issue, we can conclude that while the

by a conventional BCS model with an isotropic energy gap.

photoemission has difficulty determining if a true node

The lack of a clear exponential temperature dependence has

exists, it does demonstrate a gap anisotropy which closely

been interpreted and is consistent with a dx2-y2 symme• 1 try.J20] It would be interesting to study _--- on a system

mimics the depicted dx2-y2 dependence. Unfortunately, this technique cannot distinguish between a d-wave order

that shows BCS behavior in tunneling mehsurements or

parameter and a very anisotropic s-wave symmetry.

penetration depth studies. 3. Josephson Studies of the Phase of the Order Parameter d) Angular Resolved Photoemission In a superconductor, the current is related to the With high enough resolution, angular resolved

gradient in the phase J ~ JoVe. For a Josephson junction

photoemission is, in principle, able to determine the magni-

the simple relationship between current and phase persists

tude of the gap (Ak) on different parts of the Fermi surface.

and the current density across a Josephson barrier is given

There have been several investigations, mostly on

by

Bi2Si2CcaCu2Os.x [21,22] to investigate this problem and it

J = JcsinAO.

is generally concluded that there is substantial anisotropy in the energy gap. While the energy resolution of the tech-

where A¢ is the phase difference between the two super-

nique is still only comparable to the measured energy gap,

conductors (q~1-~).[25] In principle then, the Josephson

the data from at least two investigations clearly show that

effect is able to measure directly the phase difference

the energy gap has a strong anisotropy with a small value

between two superconductors and so able to address the

along the diagonal toward the point (x,7t) in the brillouin

idea that in a dx2-y2 symmetry, there is a ~ phase shift

zone. This is the expected behavior for a dx2-y2 symmetry

between the kx and ky directions of the order parameter.

and while the resolution is such that it is not able to mea-

Indeed, for a strictly dx2-y2 symmetry, one would expect

sure a low value for the gap, and the technique is not able

orthogonality with s-symmetry such that Josephson tunnel-

to determine whether the gap changes sign as is necessary

ing between a conventional s-wave superconductor and a d-

58

ORDER PARAMETER OF HIGH Tc SUPERCONDUCTORS I-V Characteristics of Pb/YBCO Junction

Vol.92, Nos 1-2

4~

1.3 K

E

-

<

.<

3oo

5

1o

i,°

2oo

_I m

1{]

lot

0 c) - I 0

//ACF 0.5

I t.O

I t.5

1 2.0

I 2.5

I 3.0

1 3.5

0 4.0

Flux at n~tmmc¢ minimum (~o) -20

-8

-6

-4

-2

0

2

4

6

8

Bias Voltage (mV)

Figure 6.

Phase shift of a corner SQUID between

YBCO and Pb as a function of measuring current. The Figure

tunneling

phase difference between the two junctions on each side of

between an s-symmetry superconductor (Pb) and

5. Demonstration of Josephson

the comer is determined by an extrapolation to zero current.

YBCO.[26]

Conventional SQUIDs should extrapolate to zero phase difference.

wave superconductor should be not allowed. Sun et al. have shown that in tunnel junctions between Pb and

of the data to zero measurement current implied a ~ phase

YBa2Cu3Ors a Josephson current is observed.t26] The I-V

shift in the perpendicular case relative to the parallel case.

characteristic of such a measurement is shown in Fig. 5

The results of one of their studies are shown in Fig. 6 where

where a clear Josephson current is seen. Magnetic field

we show the measured phase shift as a function of mea-

dependencies of this Josephson current confirm it to be of a

surement current and the results of the extrapolation to zero

conventional Josephson nature and shows direct wavefunc-

current for a corner junction. This data implies a phase

tion overlap between the pair state in Pb and that in YBCO.

shift for currents orthogonal to each other.

While an argument can be made that some coupling is expected since the YBCO is orthorhombic and not tetrago-

Similar measurements comparing the phase shift on

nal, we would expect that in highly twinned samples which

SQUIDs of a conventional (Nb-Nb SQUIDs) and high Tc

integrate over both in-plane orientations, the critical current

(Nb-YBCO SQUIDs) draw the same conclusions.t28]

should vanish. This doesn't happen and so implies that the

Both these SQUID measurements are seriously handi-

gap symmetry is not strictly dx2_y2. On the other hand, the

capped by the self field effects due to the loop inductances

magnitude of the Josephson current is not what would be

and the non-linear nature of the weak links.t29] These two

expected for a simple conventional Josephson tunnel junc-

factors make this linear extrapolation to zero current diffi-

tion. The measured IcR is only -10%-20% of the full

cult. Furthermore, it has been pointed out that flux pene-

expected value.

tration and trapping in a superconductor has very different characteristics in a comer as opposed to along an edge.t30]

On the other hand, Wohlman et al. [27] have stud-

To resolve whether this has any impact on this measure-

ied the magnetic field dependence of a d c SQUID (two YBCO-Au-Pb proximity weak links). By fabricating these

in conventional superconductors. Nevertheless, the data in

SQUIDs on a single crystal of YBCO in two separate con-

Fig. 6 shows a phase shift determined from this extrapola-

figurations, they were able to study the relative phase shift

tion that is consistent with it and inconsistent with zero.

ment it would be useful to study comer and edge SQUIDs

across the pair of junctions for different directions. In the two geometries studied the current paths for the two junc-

These SQUID results in contrast to the single

tions were arranged so that they were parallel (an edge

Josephson tunneling result, indicate a g phase shift for cur-

junction) or perpendicular (a comer junction). These mea-

rents orthogonal and support a dx2-y2 symmetry. Chaudhari

surements had the difficulty of a large self inductance of

and Lin [31] have studied the critical current between a

the SQUID loops causing phase shifts but an extrapolation

hexagonal superconducting grain and its surroundings.

Vol.92, Nos 1-2

ORDER PARAMETER OF HIGH T, SUPERCONDUCTORS

59

Because of the geometry, a dx2-y2 symmetry would result in

currents in the a-b plane. Secondly, it is the only tunneling

zero critical current if the Josephson coupling strength

experiment where the coupling between the two supercon-

across all six sides of the hexagon were equal. Interference

ductors is extremely weak. The only coupling between the

in the phase between currents flowing in each boundaries

pairs on either side is via tunneling with a transmission

would result in cancellation of the current. With the sys-

probability reduced by - 106. The wave functions at the

tematic elimination of each side of the hexagon, the ampli-

barrier are characteristic of the bulk. The weak link (or

tude of the supercurrent should oscillate. The observations,

proximity) experiments are a result of a depression of the

however, are not consistent with this picture and are consis-

order parameter in a way which is difficult to physically

tent with conventional symmetry and the current simply

characterize and it is not clear whether it is due to micro-

scaled with the number of sides engaged. These results

scopic inhomogeneities or spatial depression of the order

suggested a strong s-symmetry component, to the order parameter.

parameter.J37] It is interesting to note that most of the experiments

An interesting consequence of the dx2-y2 symmetry

which have used the Josephson effect to study the phase of

was pointed out by Sigrist and Rice.[32] In an attempt to

the order parameter have been on YBCO. In other mea-

understand an observation by Braunsch et al. [33] where an

surements described in Section 2, we point out that differ-

"inverse" Meissner effect was detected (a paramagnetic

ent cuprates have distinctively different characteristics.

Meissner effect), these authors pointed out that a grain

The results in that section indicated that the cuprates appar-

boundary could result in a 7t phase shift. If it were a part of

ently offered the broad range of behavior from the case of

a continuous loop of superconduc tor, in order to ensure flux

gapless in YBCO to that approaching BCS behavior in Nd2.

quantization a spontaneous current in the loop would be

,Ce, CuO4. It is important to extend these Josephson exper-

necessary. This currentg hwould generate a flux through the

iments to the other cuprates in this family in order to

loop corresponding to --~ thus accounting for the other n

determine which of the results described in this section are

phase shift and ensuring flux quantization. Early reports of

unique to YBCO and which are general.

an experiment by Tseui et al. [34] to fabricate such a structure with a loop and junction suggests observation of this effect. Actually, any odd number of junctions would result in a frustration and a spontaneously nucleated current. Results for loops where the loop inductance L is such that L Ic >> ~o suggest that for loops with even number of junctions, integral numbers of fluxoids are trapped ( n ~ ) where for odd numbers of junctions (n + L/2 ) ¢ o is trapped. Further studies are obviously necessary but this result suggests that it is possible to have a junction with a negative phase shift. While this result is interpreted as evidence for dx2-y2 symmetry, an alternative explanation in terms of extended s-symmetry with zero crossings in the order parameter is possible.[35] Furthermore, a prediction by Spivak and Kivelson [36] in which magnetic spin flip scattering during the tunnel process would result in a n phase shift originally motivated the inverse Meissner effect experiment.

4. Magnetization and Transport Yip and Sauls [38] recently showed that the angular dependence of the magnetic response of a crystal in the Meissner state is capable of detecting the presence of nodes in the superconducting energy gap. Transverse magnetization measurements [39] in LuBa2Cu3Ozs as a function of both T and applied H have been analyzed in terms of ~r Fourier components at ~-, ~ and 2~r. For a d-wave comer ponent to dominate, an angular period of ~- is necessary and the results show that this component consistent with dwave pairing is at best, very small. Furthermore, it is found that the amplitude of the 7

component can be easily

accounted for as being generated from a higher harmonic of signals at angular period x and 2n. The results of a Fourier analysis of the magnetization is given in Fig. 7. Here the Fourier components as a function of angle are given. The component at n is off scale on this figure and is 6 x 10-5. The range of theoretical predictions are shown as

There are at least two important distinctions

well and the data is below the lowest possible prediction.

between the Josephson tunneling experiment [26] and the

The conclusion from this work is that the results are consis-

grain boundary or weak link experiments.[27,28,31,34]

tent with a nodeless s-wave anisotropic pairing. This

Firstly, the tunneling experiment is along the c axis, i.e.

experiment apparently rules out exclusive d-wave pairing and concludes that there is a lower energy gap.

current in the c direction. The other experiments are for

60

ORDER PARAMETER OF HIGH Tc SUPERCONDUCTORS .

• i

•.,

| - .

• i , , ,

i , , ,

i

,..

i , ' ' 1 ~

150

•

,

,

0 +

o X

w

,

i

YPrBaCuO Damaged

'

'

Vol.92, Nos 1-2

'

. . . .

'

[

,

,

,

,

YBaCuO

100 +

0

0

0

0

4" 0

o

1

0

y

oooo

50

O

0

4-

o

O 4"

20

Figure 7.

100 ~(degrees)

180 °0

500 I000 1500 Resistivity (rnieroohm-cms)

2000

Amplitude of various Fourier harmonics of the Tc vs. residual resistivity [45] (p(T=O)) for ion

angular dependence of the magnetization of a LuBa2Cu3OT.s

Figure 8.

single crystal.[39] The x is a theoretical prediction for dx2-y2 symmetry and the bottom of the bar is the lowest

damaged YBCO and Yt.,PLBa2Cu30:~s. The line in the lower left corner is that expected for a dx2-y2 symmetry

possible value consistent with some portion of d-symmetry.

superconductor. [41,42]

The harmonic with period x is off scale here at 6 x 10.5 pairing. In fact, the data looks very much like earlier stud-

emu.

ies of the 2-dimensional superconducting-insulatingtransiIn view of the differences observed in the other experiments with different cuprates, it would be interesting

tion in conventional superconductors with disorder.J46,47,48]

In this case, the transition occurs near the

again to study other cuprates compounds. One of the most

loffe Regel condition [49] where kF,1.= I. Here ~ is the

significant differences between the s and d-wave pairing

mean free path and this condition corresponds to eft ~ - h

state is the sensitivity of that state to elastic scattering. It

rather than the expected Xre ~ h for d-symmetry. The dif-

was shown by Anderson many years ago [40] and has been

ference illustrated in Fig. 8 is the difference between

experimentally confirmed many times that elastic scattering

and EF (approximately a factor of 50-100). This result

does not destroy superconductivity. Instead, it reduces the

suggests that superconductivity in YBCO is more robust to

anisotropy and results in an isotropic superconductor. As

elastic scattering than would be expected from a d-wave

the scatter rate around the Fermi surface 1 becomes coroTE

pairing. It is important to note that this scattering rate is

parable to the mean energy gap, i.e.

implied from the residual resistivity p(T--O). Other factors (carrier density changes, etc.) would affect the slope of the p(T) curves but the extrapolated residual resistivity will

fE

allow a determination of the elastic scattering. A systematic study of other systems where the data can be analyzed

the anisotropy disappears and a sharp, single gap results X. For a d-wave superconductor, however, the average gap

so as to determine a p (T=0) reliably should be performed; especially other cuprates beyond YBCO.

= 0 and so elastic scattering destroys superconductivity on the same scale ~ r E = ~. Such considerations have been

5. Conclusions

calculated by various authors.[41,42] Studies of ion damaged YBCO [43] and Pr substituted Y~.~Pr~Ba2Cu3OT.a [44]

There are theoretical and numerical arguments that

have been analyzed in terms of a residual resistivity and

suggest the pairing state in the high Tc cuprates might be of

hence an elastic scattering rate. It is found that for these

dx2-y2 symmetry rather than anisotropic s. There are also

two data sets, the superconducting T¢ is dependent upon the residual resistivity in a way which is much less sensitive

arguments against this idea. Experimentally, there are a

than would be expected from dx2-y2 pairing.[45] The

from different perspectives.

results of this study are shown in Fig. 8. The band in the

Josephson, thermodynamic and spectroscopic studies on the

lower left corner of the figure is that expected for d-wave

variety of cuprates have certainly narrowed down the pos-

growing number of studies attempting to address this issue Transport, magnetization,

ORDER PARAMETER OF HIGH Tc SUPERCONDUCTORS

Vol.92, Nos 1-2

61

sibilities, but there is still no general consensus. Some

shift at some junction in the a-b plane in YBCO is corre-

measurements imply strong gap anisotropy or gap nodes,

lated with the gaplessness or not. Experiments on the angu-

others show almost full gap values. Josephson studies

lar dependence of the magnetization on single crystals of

show substantial overlap between a conventional s-wave

BSCCO and Nd2.xCe~CuO4 would also be revealing. In

superconductor, but also suggest a g phase shift at some

any event, the experiments to date have shown that super-

grain boundaries.

It appears that the superconducting

conductivity in the high Tc cuprates is not simply of the s-

cuprates have shown us the whole spectrum of behavior

symmetry BCS nature. Gaplessness appears to be intrinsic

expected. It seems clear that, while most measurements

in YBCO but this should not be confused with dx2-y2 sym-

have focused on YBCO with less on BSCCO and even

metry for the pairing state.

fewer on the others, drawing general conclusions might be too simple. Most of the evidence points to the possibility that YBCO is a gapless superconductor. This is consistent with dx2-y2 pairing but certainly does not prove it. It has Acknowledgments

been pointed out several times that an extended s-symmetry could encompass gaplessness, nodes in the gap, and phase shifts at grain boundaries. It would seem odd if some

The work described in this paper comes from many

of the cuprates had dx2-y2 pairing and others were conven-

people and it is impossible to acknowledge it all. My col-

tional s-wave as they all apparently accommodate antifer-

laborators continue to ask the difficult questions and work

romagnetism in their insulating state and probably spin

tirelessly with me. At UCSD I thank my collaborators

fluctuations in their conducting state.

M.B. Maple, I.K. Schuller, F. Hellman, A.G. Sun, D. Gajewski, S. Han and A. Katz. Other colleagues from

From an experimental perspective, it is of great

whom I am constantly learning include S. Kivelson, D.

value to study the differences that the different cuprates

Scalapino, C. Varma, J. Valles, Jr., K. Char, T.H. GebaUe,

reveal for the same experiment. For example, Josephson

P.W. Anderson, J.P. Carbotte and S. Chakravarty. This

experiments performed on Nd2 ~CexCuO~, where an energy

work was supported by NSF grant DMR9113631 and

gap is observed, would clarify whether the apparent ~ phase

AFOSR grant F4962092J0070.

REFERENCES 1.

J.G. Bednorz and K.A. Mueller, Z. Phys. B64, 189

11.

(1986). 2.

L. Gao et al., Phil Mag. Lett. 68, 345 (1993; A.

(1993). 12.

J. Hartge et al., J. Phys. Chem. solids 54, 1359

13.

H.J. Tao, F. Lu, G. Zhang and E.L. Wolf, Physica C

Schilling, M. Cantoni, J.D. Guo and H.R. Ott,

Nature 362, 56 (1993). 3.

(1993).

For a set of papers describing several (not all) of the various models, see J. Phys. Chem. Solids 54

224, 117 (1994). 14.

(1993). 4.

See J. R. Schrieffer, Solid State Commun. This See for example, R. Meservey and B. B. Schwartz

15.

P.J. Hirschfeld and N. Goldenfeld, Phys. Rev. B 48, 4219 (1993).

16.

in Superconductivity, Ed. R.D. Parks, Marcel 6.

N. Tralshawala et ai., Phys. Rev. B 44, 12102 (1991).

issue. 5.

T. Hasegawa et al., J. Phys. Chem. Solids 54, 1351

S.N. Mao et al., Appl. Phys. Lett. 64, 375 (1994); D.H. Wu et al., Phys. Rev. Lett. 70, 85 (1993).

Dekker, New York 1969.

17.

W.N. Hardy et al., Phys. Rev. Left. 70, 3999 (1993).

J. Geerk, G. Linker, O. Meyer, Q. Li, R.V. Wang,

18.

D. Achkir et ai., Phys. Rev. B 48, 13184 (1993).

and X.X. Xi, Physica C 162, 837 0989). 7.

J.M. Valles et al., Phys. Rev. B 44, 11986 ( 1991).

8.

J. Leseur et al., Physica C 191,325 (1992).

9.

J. Chen et al., Phys. Rev. B 49, 3683 (1994).

10.

H.L. Edwards, J.T. Markert and A.L. DeLozanne,

Phys. Rev. Lett. 69, 2967 (1992).

19.

Y. Kitaoka et al., J. Phys. Chem. Solids 54, 1385 (1993).

20.

C.P. Slichter et al., J. Phys. Chem. Solids 54, 1439 (1993).

21.

Z.Y. Shen et al., Phys. Rev. Lett. 70, 1553 (1993).

ORDER PARAMETER OF HIGH Tc SUPERCONDUCTORS

62 22.

J. Ma. C. Quitmann, R.J. Kelley, H. Berger, G. Margaritondo and M. Onellion, preprint.

36.

23.

K. Gofron et al., J. Phys. Chem. Solids 54, 1193

37. 38.

(1993). D.S. Dessau et al., Phys. Rev. Lett. 71, 2781 (1993).

25.

See for example M. Tinkham in Introduction to

39.

Superconductivity,

40.

27.

A. Millis, preprint. S.K. Yip and J.A. Sauls, Phys. Rev. Lett. 69, 2264 (1992). J. Braun et al., Phys. Rev. Lett. 72, 2632 (1994). P.W. Anderson, J. Phys. Chem. Solids 11, 26 (1959).

(1975). 26.

B.I. Spivak and S.A. Kivelson, Phys. Rev. 8 43, 3740 (1991).

24.

Krieger, Malabar, Florida

Vol.92, Nos 1-2

A.G. Sun et al., Phys. Rev. Lett. 72, 2267 (1994).

41.

D.A. Wohlman et al., Phys. Rev. Lett. 71, 2134

42.

(1993).

R.J. Radtke et al., Phys. Rev. B 48, 653 (1993). P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).

28.

D.A. Brawner and H.R. Ott, preprint.

43.

J.M. Valles, Jr. et al., PhysRev. B 39, 11599(1989).

29.

U. Kesin and S.A. Wolf, preprint

44.

M.B. Maple et al., J. Alloys And Compounds 181,

30.

R. Klein et al., preprint.

31.

P. Chaudhari and S.-Y. Lin, Phys. Rev. Lett. 72,

135 (1992). 45.

A.G. Sun, L.M. Paulius, D.A. Gajewski, and R.C.

M. Sigrist and T.M. Rice, J. Phys. Soc. Japan 61,

46.

R.C. Dynes et al., Phys. Rev. Lett. 57, 2195 (1986).

4283 (1992).

47.

J.M. Graybeal and M.A. Beasley, Phys. Rev. B 29,

Dynes, Phys. Rev. 8 in print.

1084 (1994). 32. 33.

4167 (1984).

W. Braunisch et al., Phys. Rev. Lett. 68, 1909 (1992).

48.

34.

C.C. Tseui et al., preprint.

49.

35.

L. Chen and A . M . S . Tremblay, J. Phys. Chem.

Solids $4, 1381 (1993).

Y. Liuetal.,Phys. Rev. Lett. 67,2068(1991). A.F. Ioffe and A.R. Regel, Prog. Semiconductors 4, 237 (1960).