Exotic phase properties of dx2−y2-wave superconductors

ELSEVIER

Physica B 230-232 (1997) 806-810

Exotic phase properties of dx2_y2-Wavesuperconductors Manfred Sigrist Theoretische Physik, ETH-H6n99erberg, 8093 Ziirich, Switzerland

Abstract

Phase-sensitive test experiments for pairing symmetry of the high-temperature superconductors are reviewed. These experiments are discussed under the assumption of dxg_y2-wavepairing. The influence of the superconducting state on the c-axis Josephson effect is analyzed. Special grain boundaries between the YBCO films can be described as Josephson junctions with alternating segments with 0- and n-phase shifts. The occurrence of spontaneous flux pattern on these grain boundaries is discussed. Keywords: d-wave superconductivity; High temperature superconductivity; Josephson effect; Unconventional supercon-

ductivity

1. Brief review

In spite of three years of intense research the debate on the symmetry of the order parameter in high-temperature superconductors has not ended yet. Seemingly conflicting experimental results favor one or the other of two prominent candidates. Provided we accept that high-temperature superconductivity is caused by Cooper pairing we can classify these candidates by their symmetry: s-wave (including the so-called extended s-wave) and d-wave pairing, which belong to different representations of the tetragonal point group D4h. In these proceedings, I will analyze a number of experiments from the point of view that Cooper pairing in the d-wave channel is realized, at least in the systems investigated so far, YBa2Cu307_x (YBCO), BizSr2CaCuzO8 (BSCCO) and TlzBa2CuO6 (TBCO). Let us first briefly review some of the key experiments (for more detailed reviews see Ref. [1]). The term d-wave pairing implies that the pair wave function has the generic form IPd(k) ~ kx2 -- k2 (or cos kx cosky). Due to the difference of the sign along the

tWO main axes, the pair wave function (or order parameter) has an internal phase structure. Non-trivial phase structures are generally not found for s-wave pairing, although it is not excluded as the example of the extended s-wave state demonstrates (~'s(k)cx cos kx + cos ky). Clearly, however, both states are distinguished by their symmetry which enters into the geometry of devices designed to probe the order parameter symmetry directly. There is a class of probes which observes directly the phase structure. These tests are mainly based on the Josephson effect which probes the phase of the order parameter. The phase structure can be observed in multiply connected devices where the Josephson junctions are arranged in a special geometry. Note that the Josephson phase at junctions along the x- and y-axis are different by rt (as long as no current is flowing) in a dx2_y2-wave superconductor. Such geometrical properties have been used in two types of experiments. In the first type the phase shift ~ has been observed by the modification of interference patterns in a SQUID or in a single extended junction (Fraunhofer pattern) [2]. The second probe searches

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M. Sigr&t/PhyswaB230-232 (1997) 806 810 for a spontaneous current and modified flux quantization in a superconducting loop with an internal n-phase shift, which is a frustration effect. In both experiments the phase shift (n) was determined with very good accuracy for YBCO and recently also for TBCO [3]. They show a phase structure which is very compatible with the d-wave order parameter. As convincing as these experiments may look they have been contested by observation also based on interference effects. Let us assume that a dx2_y2-wave superconductor with perfect tetragonal symmetry. For such a system we would not expect Josephson tunneling along the c-axis into an s-wave superconductor. This is due to a selection rule which originates from the different behavior of the s-wave and d-wave order parameter under rotations around the c-axis. However, c-axis Josephson coupling has been observed in high-quality junctions between YBCO and Pb [4]. On the other hand, no Josephson effect appears between BSCCO and a conventional superconductor [5]. Both systems are not really tetragonal, but orthorhombically distorted so that the basis of the experiment, the selection rule, has to be revised. Although the selection rule does not apply for YBCO the situation is complicated by the fact that the samples show twinning. It was argued that twinning would yield destructive interference for YBCO suppressing the Josephson coupling below the values observed in the experiment, if the order parameter would have dominantly d-wave character. This raises the question whether this experiment provides a clearcut argument against d-wave pairing in YBCO or whether there are other ways to obtain a Josephson coupling of the observed magnitude (Section 2). Another test considers interference effects in a thin film of YBCO with a hexagonal YBCO-inclusion with crystal axes misaligned by 45 ° with the surroundings [6]. The purpose of this device is to test whether different Josephson phase shifts (expected for a d-wave superconductor) on the hexagon edges would cancel the net Josephson current. The measurement showed no sign of canceling which is clearly compatible with an s-wave order parameter. It was shown by Millis, however, that the simple interference argument is not valid, because the length of the edges is longer than the Josephson penetration depth 2j of the grain boundary so that supercurrents would not flow uniformly through the edges [7]. Actually, fluxes appear on the

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corners of the hexagon and compensate for most of the differences in the phase shifts on the edges through the induced gauge field A. Therefore, the interference effect is more complicated and certainly not completely destructive. It was, however, found that the magnitude of the critical current was measured to be about the same for all edges despite of the expectation that the structure of the d-wave pair wave function would cause large differences. This may be traced back to the quality of the interfaces (grain boundary) between the hexagon and the surrounding. Recent experimental resuits suggest considerable inhomogeneity of extended interfaces (Section 3).

2. Josephson effect along the c-axis Sun and coworkers measured the IcR-product of c-axis Josephson contacts between Pb and YBCO for various samples, twinned as well as twin-free (Ic is the Josephson critical current and R the junction normal resistance) [4]. Their result can be summarized to following points. The Ic R-product of twin-free samples is lower than the value expected from the AmbegaokarBaratoff theory using the measured values of the gap (under the assumption that YBCO is an s-wave superconductor). The reduction of the /~R-product in twinned samples is less than one would estimate for a d-wave superconductor in a orthorhombic crystal. As we mentioned initially the simple selection rule does not apply for YBCO, because it is orthorhombically distorted. Since the distortion is along the main axes with a strain e = e~ - eyy, the dx2_y2-wave and s-wave symmetry are not distinguished anymore, but belong both to the same irreducible representation of D2h. This leads to an effective admixture of an s-wave component to the "d-wave" superconducting state, i.e. d 5: s, where the sign is determined by the sign of e [8]. We may say that it is only the admixed s-wave component which yields the c-axis Josephson coupling which would explain the reduced icR-value. In this picture a YBCO sample with twins corresponds to an array of alternating (d + s)- and (d - s)-wave states. The Josephson phase shift is 0 for d + s and ~t for d - s. This alternation leads to destructive interference effects. In this case the length scale 2j, over which the Josephson phase can vary, is much longer than the size of the twin domains and

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even of the whole junction so that Millis' argument does not apply [7, 8]. The irregularity of the twin domain sizes implies that the net Josephson current is the sum of positive and negative contribution of random magnitude so that the random walk picture can be used to determine the total current. One finds that the IcR-product should be reduced by a factor N -1/2 where N is the number of twin domains (N ~ 104106). This would result in a reduction down to 1% or less while the actual measurements give values between 1 and 10% [4]. There are various ways to account for this discrepancy beside assuming pure s-wave pairing in YBCO. The interface normal vector of the YBCO superconductor may not be exactly parallel to the c-axis. This effect can, however, be reduced by careful sample preparation [4]. An alternative picture which has been proposed recently is based on a twist in the relative phase of the d- and s-wave component in the vicinity of the twin boundaries [8]. I will consider here this scenario and discuss the properties of the superconducting phase close to a twin boundary. As mentioned above the two types of twin domains are distinguished by the relative phase 7 of the d-wave (r/d) and the s-wave (r/s) order parameter. One domain corresponds to 7 = 0 and the other to y = ~. If we assume that the s-wave component changes sign by passing through zero at the twin boundary, then y switches discontinuously between 0 and zt there, while r/d as the dominant order parameter remains finite. By means of a Ginzburg-Landau theory one can demonstrate that a continuous variation of ~ can occur close to the twin boundary whereby both order parameter components, r/s and r/d remain finite [8]. This requires that qs and r/d add with complex coefficients instead to the real combination (d -4- s) assumed above. There are two degenerate states of this form, r/s + eiTr/d and q* + e-i~q~. Obviously, this state violates time-reversal symmetry. The Ginzburg-Landau theory shows that there is a critical temperature T* ( < T~) for the appearance of this state. For T > T* only the real-order parameter combination (strictly 7 = 0 or n) is realized at the twin boundary. By allowing the complex combination the superconductor gains condensation energy by keeping both order parameter components finite at the twin boundary (see Fig. 1). The continuous change of 7 has direct implications on the c-axis Josephson effect. Assuming that the

1.5

I~l

a)

1.0

0.s

"~N

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Irl~l

o 0.0

b)

1.0 \

\

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~" o.5 \ \ 0.0 -I0.0

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Fig. 1. Behavior of the order parameter at the twin boundary: (a) modulus of the order parameter components ~/d and r/s (arbitrary units); (b) relative phase V = ~ b s - ~bd. There is a Y-invariant (solid lines) and a 3"-violating state (dashed line). The units of the length x are of order of the coherence length.

phase of rid is essentially constant and the Josephson coupling occurs through the s-wave component, we may write for the local current-phase relation, J -Jc[sin ~ocos 7 - sin y cos ~p]. It is obvious that only the first term gives a contribution to the Josephson current, if y is restricted to 0 and re. The two values yield currents of opposite sign for a given phase difference ~o such that the destructive interference effect explained above reduces the total current by N -1/2 (random walk). The second term plays a role, however, if ~ assumes values between 0 and + ~ by varying continuously at the twin boundary. The Ginzburg-Landau theory shows that in a dense array of twin boundaries ~ is either restricted to the sector 0 ~<7 ~
114. Siyrist / Physica B 230-232 (1997) 806-810 not known. The existence of the time-reversal symmetry breaking twin boundary states is not established so far and has to be tested. Among the effects which occur in connection with these states, I would like to mention here one and refer to Ref. [8] for others. The twofold degeneracy allows the formation of domains on the twin boundary, which are separated by a line singularity. At these lines a phase winding of the order parameter occurs which leads to a magnetic flux smaller than the standard flux quantum ~0. Measurements using scanning SQUID microscopy, however, have so far not found evidence for this kind of fractional vortices [11]. In contrast to YBCO the orthorhombic distortion of BSCCO is due to a shear strain, e~-- 8xy. This distortion leaves the s- and dx2_y:-wave symmetry distinguishable. Therefore, no s-wave order parameter component is admixed in this case and the selection rule should hold. Indeed in the experiments no c-axis Josephson effect was observed despite of good quasiparticle tunneling properties of the junction [5].

3. Effects of faceting at grain boundary junctions Recently, the investigation of basal plane Josephson junctions between high-temperature superconductors has gained a lot of attention because of progress in sample preparation techniques and various exotic effects due to the unconventional order parameter symmetry. Simple symmetry arguments lead to the following generic Josephson current-phase relation between two dx2_y2-wave superconductors. J = J0 cos 20A COS20B sin ~p,

(1)

where 0A,B is the angle of the crystal x-axis with respect to the junction normal vector (for simplicity we assume a sinusoidal form) [9, 10]. Depending on 0A and 0B the coupling can have either sign which corresponds to phase shifts 0 or rt for positive and negative sign, respectively. An interesting situation should emerge if one of the angles, say 0A is close to 45 ° while 0B ~ 0. (Note, however, that Eq. (1) can be modified in this geometry, because various higher order contribution can become important.) If the interface between the two d-wave superconductors is not straight but shows faceting on a length scale much longer than the

Y X

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Fig. 2. Schematic view of a faceted asymmetric 45 ° boundary.

grain

coherence length, then one can find segments with alternating 0- and n-phase shifts along the interface (see Fig. 2) [12]. Although this situation is similar to the alternating phase shifts introduced by the twin domains for c-axis junctions the length scales are very different. Here the ratio of the local Josephson penetration depth 2j and the length of alternation d is by several orders of magnitude smaller than that of c-axis junctions, although still of order 102-103 . Therefore we can expect that the phase inhomogeneity of the junction should be experimentally observable. Indeed the measurements of the interference pattern in a magnetic field reveal a strong deviation from the usual Fraunhofer pattern indicating the irregularities of the phase on the interface [12]. An even more striking effect is the fluctuation of the Josephson phase ¢p. The spatial dependence of (p follows the Sine-Gordon equation d2cp/dx2 = 2j-2 sin(~p - c~(x)).

(2)

The first derivative of (p with respect to x is proportional to the local magnetic flux q~(x)= (q~o/2rc)dq~/dx. The variation of ~ creates a modulated potential landscape for ~p. If the length ).j were short compared to the segment length d, then ~pwould follow the modulations of e closely. At each border between a 0- and a x-segment ¢p would switch also by 1 -4- rt and would yield a local magnetic flux • = 4- ~ ~0-

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M. Sigrist/ Physica B 230-232 (1997) 806-810

In experiment most of the samples are thin films of YBCO which have grain boundaries with ,~j >>d. In this case q~ varies along the interface to take optimally advantage of the fluctuations of c~. Therefore, a wandering of ~0 can occur with a characteristic length scale L0 which depends on the grain boundary properties. This length characterizes the scaling behavior of (p and, thus the magnetic field distribution. The magnetic flux ~ accumulated over a length L along the junction is ((2x#(L)/#0) 2) = ((q~(x + L) - qg(x))2) ~

L/Lo (3)

for L >>L0. In a recent experiment on asymmetric 45 °grain boundaries (0A = 45 ° and 0B = 0) in YBCO films the appearance of spontaneous randomly distributed flux was observed by means of a scanning SQUID microscope. On grain boundaries with other angles OA.Bwhere the alternation of the intrinsic phase shift is unlikely, no such spontaneous fluxes were found. Therefore, the model of a phase-disordered Josephson junction gives an appropriate description of the faceted 45°-grain boundary. Although the length of the experimental grain boundaries are of order of 1 mm, it is not easy to find the scaling behavior of Eq. (5). There are various reasons: (1) correlations among the 0- and x-segments invalidate the assumption of complete randomness; and (2) external magnetic fields are not shielded completely and can give an additional contribution of additional magnetic flux on the grain boundary. Faceting is at present an unavoidable feature of grain boundaries between epitaxially grown YBCO films with different crystal orientations. The Josephson coupling is determined by the faceting angles and the distribution of 0- and x-segments. The grain boundaries of the hexagonal inclusion in the ChaudhariLin experiment are certainly faceted too [6]. We may explain their result that the critical Josephson current through the edges forming asymmetric 45°-grain boundaries is practically the same as for other edges with different orientations. The current density for this grain boundary is distributed essentially over the whole edge, while for the other edges currents flow only close to the corners due to screening effects. Thus, after all the total current flowing through each edge may be not so different from each other.

A number of experiments show that there are striking new phase properties in cuprate superconductors. The appearance of spontaneous magnetic flux and the modification of interference effects are some of the most exciting examples. These new aspects of superconductivity have certainly to be taken into account in future application of these superconductors, in particular in systems where the Josephson effect plays a role.

Acknowledgements These proceedings summarize works done with several collaborators from whom I benefited much in many stimulating discussions: K. Kuboki, P.A. Lee, A.J. Millis, T.M. Rice, J. Mannhardt, H. Hilgenkamp, B. Mayer, Ch. Gerber, J.R. Kirtley and K.A. Moler. I am also grateful to the Swiss Nationalfonds for financial support through the PROFIL-fellowship.

References [1] D.J. Scalapino, Phys. Rep. 250 (1995) 329; D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515. [2] D.A. Wollman et al., Phys. Rev. Lett, 71 (1993) 2134; D. Brawner and H.R. Ott, Phys. Rev. B 50 (1994) 6530; A. Mathai et al., Phys. Rev. Lett. 74 (1995) 4523; I. Iguchi and Z. Wen, Phys. Rev. 13 49 (1994) 12388; D.A. Wollman et al., Phys. Rev. Lett. 74 (1995) 797. [3] C.C. Tsuei et al., Phys. Rev. Lett. 73 (1994) 593; Science 271 (1996) 329. [4] A.G. Sun et al., Phys. Rev. Lett. 72 (1994) 2267; R. Kleiner et al., Phys. Rev. Lett. 76 (1996) 2161. [5] H.Z. Durusoy, D. Lew, L. Lombardo, A. Kapitulnik, T.H. Geballe and M.R. Beasley, preprint. [6] P. Chaudhari and S.Y. Lin, Phys. Rev. Lett. 72 (1994) 1048. [7] A.J. Millis, Phys. Rev. B 49 (1994) 15408. [8] M. Sigrist, K. Kuboki, P.A. Lee, A.J. Millis and T.M. Rice, Phys. Rev. B 53 (1996) 2835. [9] V.13. Geshkenbein and A.I. Larkin, Pis'rna Zh. Eksp. Teor. Fiz. 43 (1986) 306 [JETP Lett. 43 (1986) 395l; M. Sigrist and T.M. Rice, J. Phys. Soc. Japan 61 (1992) 4283. [10] For corrections to this form see M.B. Walker and J. LuettmerStrathmann, preprint. [11] K.A. Moler, J.R. Kirtley, R. Liang, D.A. Bonn and W.N. Hardy, Bull. Am. Phys. Soc. 41 (1996) 106. [12] J. Mannhardt, H. Hilgenkamp, B. Mayer, Ch. Gerber, J.R. Kirtley, K.A. Moler and M. Sigrist, Phys. Rev. Lett. 77 (1996) 2782, and references therein.