The glass state in weak-coupled superconductors

Physica C 153-155 (1988) 63 66 North-Holland, Amsterdam

THE GLASS STATE IN WEAK-COUPLED

SUPERCONDUCTORS

D. Stroud and C. Ebner Department of Physics,

The Ohio State University,

Columbus,

Ohio

~3210,

USA"

We review the work of the Ohio State group on the equilibrium and non-equilibrium properties of disordered granular superconductors in a magnetic field, pointing out the connections between the behavior of these materials and that of more conventional spin glasses, and considering briefly the possible connections to the behavior of high-Tc materials in a magnetic field. I.

£NTHODUCTION Glass-like behavior was discovered in the high-temperature superconductor LaBaCuO~ by MUller, Takashige, and Bednorz (i). Their work has been followed by other experimental studies of both the La- and the Y-based hightemperature superconductors, in which similar effects were observed (2). Theoretical studies (3) have interpreted these measurements in terms of a simple model of disordered granular superconductors, involving Josephson or proximity effect coupling between a random array of weak links (3-6). In this paper, we review our own work (3-4) on the equilibrium and non-equilibrium properties of disordered granular superconductors. We conclude with a brief discussion of the possible relevance of this model to the high-Tc materials. THE MODEL Our work is based on the following model for the Hamiltonian of a weakly coupled granular superconductor (3-7). H = ~ J(Rij ' B, Tl.cos(~i-~j-Aij) (I)

separation Rid between their centers, the magnetic field B, and the temperature T. In terms of this model, the Helmholtz free energy is given by (3) F = -kBT in Z Z=~d#lexp[_H(~l,...,~N)/kBT].

(4)

This model omits or oversimplifies a number of important effects, even in conventional isotropic low-temperature superconductors. These include (i) coupling of amplitude and phase fluctuations; (ii) charging energies of individual grains, which can cause zero-point phase fluctuations even in the ordered phase (8); (iii) dissipation arising from normal (one-electron) tunneling between grains (9); and (iv) screening currents, which may cause the local magnetic field to differ from the applied field (i0).

2.

~j

Aij : (2~/~o) I~i X.~

(2)

Here H is the Hamiltonian of a collection of weakly-coupled superconducting grains, the ith grain centered at Ri, in the presence of a magnetic

field B described by a vector

potential A; ~0 = hc/Ye is a flux quantum; • i is the phase of the superconducting order parameter on the ith grain; and J(Ri~ , B, T) is the strength of the coupling between grains i and J, which depends on the

"

Work supported by the U.S. National (D.S.) and DMR 84-04961 (C.E.).

0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

3.

EQUILIBRIUM PROPERTIES Some of the equilibrium properties of this model can be guessed without calculation. For example, at B = O, the model is equivalent to a classical xy Hamiltonian with disorder. In the absence of disorder, the Hamiltonian in three dimensions has a transition to a superconducting state with long-range phase-coherence and a non-zero order parameter , where the triangular brackets denote a thermodynamic average. In two dimensions, the ordered phase is of the kind first described by Kosterlitz and Thouless (ii). The presence of disorder in the form of random bond strengths is not expected to alter the universality class of these transitions. They should therefore persist even when the grains are in an amorphous configuration typical of a real granular superconductor.

Science Foundation,

Grants DMR 84-14257 and 87-18874

D. Stroud and C. Ebner / Glass state in weak-coupled superconductors

64

When a magnetic field is applied, the properties of the system change in a characteristic way. Namely, each phase factor Aij increases linearly with B, but the slope, dAi~/dB , is different for each bond. For J

sufficiently

large B, the A

's on bonds with ij

non-zero coupling will be distributed over a range which is wide compared to 2~. When this happens, we expect to have an xy 'spin' glass rather than an xy 'ferromagnet'. Of course, the 'spins' are actually the phases of the superconducting order parameter. In the phase glass regime, the ordered configuration, in both two or three dimensions, consists of phases which are frozen into a random arrangement. The field B 1 at which one moves into the phase glass regime is of order one flux quantum per plaquette, meanin E a typical closed loop of non-zero couplings 3(Ri~). The calculated phase diagram for a typical amorphous arrangement of grains, as a function of magnetic field, is shown in Fig. i. We have carried out Monte Carlo simulations of a model in which the J's are chosen so as to be typical of a collection of

---

TcCB) Anisotropy

0.5

--IO

0

c~

o rn

Pb spheres embedded in a Sn host (3). Since Sn is a superconductor with a lower transition temperature than Pb, this is an SNS phase glass. The couplings are thus strongly temperature dependent. The transition temperatures Tc(B) is determined by examining the spin wave stiffness constant tensor, defined by (4) I a2F 1 Yij : ~aAiaAj/ T

(5)

which measures the resistance of the phases to a twist applied to the boundaries of the sample. In the ordered phase, this tensor is non-zero. Fig. 1 also shows that Yij is anisotropic, i.e., not a multiple of the unit tensor. Several points should be noted about this phase diagram. First, the apparent Tc(B) flattens out and becomes nearly independent of B above a field of about four units (one "unit" in this calculation corresponds to about one flux quantum per typical plaquette of non-zero couplings). Above this field, the couplings are fully randomized, and further increasing the field does not change the statistical distribution of the couplings in the sample (though it does change the individual couplings). Second, there is no clear evidence in these simulations of a sharp transition between a high-fleld glassllke state Bnd a low-field "ferrocoherent" state within the ordered region of the phase diagram. But it must be emphasized that Shih et al did not explicitly look for this boundary in their simulations. Such a boundary was suggested in (6) and some simulational evidence for it has been reported by (3).

.--

c4.

0.3(~©

I

I

J

'I"-,

8,0

0.0

B FIGURE

1

Solid line: Calculated transition temperature T (B) for a model superconducting glass C

(here a dilute suspension of Pb spheres in Sn). Circles: anisotropy Yff/Yl - 1 at T/T

= 0.30. B = 1 corrsponds to one flux c,Pb quantum per average plaquette. T is the c,Pb transition temperature for Pb. Yff and YI are the principal

components

and perpendicular

to B.

of ~ parallel

NON-EQUILIBRIUM

PROPERTIES

The glass phase associated with eq. (i) has interesting tlme-dependent properties. This is not surprising, since for an amorphous arrangement of grains at strong fields, the Hamiltonian (i) is a very similar to an xy spin glass, which is believed not to have a true phase transition in two or three dimensions. Therefore, the apparently frozen phase observed at temperature below Tc(B) must, in reality, be only metastable, like more conventional glasses (12). Rather than persisting forever, it should slowly decay. At zero field, the ordering is "ferromagnetic" and therefore stable; the metastability should set in a some finite field. This metastability can be seen even in small clusters, such as a single loop of Josephson-coupled superconducting grains in a magnetic field (4). As a function of field, the lowest-energy configuration is a periodic, scalloped structure (Fig. 2). If the field is moved slowly past the cross-over point

D. Stroud and C. Ebner / Glass state in weak-coupled superconductors

65

su ce ,. Ill o

?

X

¢i. Ill

o

(c)

X

f

~

dc suscept.

c.,,.. 0

B/(%/S)

5

~'~GUHE 2 Internal energy E per loop at temperature T = 0 versus external magnetic field B, for (a) an assembly of superconducting loops of area S all oriented perpendicular to the field; (b) an assembly of loops of equal area S, randomly oriented; (c) an assembly of loops of random orientation and areas uniformly distrlbuted between O.IS and 2.OS. Curves are offset vertically. (After Per. 4)

(So, corresponding to a half-integer number of flux quanta in the loop), there will be a flux slip into a new phase configuration with lower energy. But if the field is changed sufficiently rapidly (as when a highfrequency s.c. magnetic field is superimposed on an underlying d.c. field), there will be no time for a flux slip, and the phases will remain in their metastable configuration. Such differences between a.c. and d.c. magnetic susceptibilities (Fig. 3) are characteristic of more conventional spin glasses. The magnetization M and susceptibility = (aM/aB) T can be calculated for larger clusters, and both reflect the existence of multiple competing low-lying energy states which cross frequently as the B-field is varied (see ref. ~ for detailed illustrations). DISCUSSION AND RELATION TO HIGH-T c MATERIALS In a typical experiment to study timedependence, a high-T c superconductor is cooled in zero field to below T c, Next, the f~e]d is ~ncreased to a maximum value, 5.

0

5

B/(~0/S) FIGURE 3 D.c. and a.c. susceptibilities for the loops of Figure 2(c). Vertical scale is in arbitrary units. (After Per. 4).

say Bo, then turned back off. When cycled through paths of this kinds in the B-T plane, high-T c superconductors typically exhibit irreversible behavior, which can be explained as follows. If the superconductor actually consists of a large collection of weak links, there are many low-lylng states of nearly equal energy. As the field is increased from zero, one of the excited state branches crosses the ground state branch and becomes the new ground state. Once the cross-over point is surpassed, the array will sit in a different configuration when the field is reduced. The magnetization will thus be history-dependent, as is observed experimentally. It will also depend on the rate of cycling. Many other similarities have been observed between the behavior of high-T c materials and the superconducting glass model (i). Besides the characteristically glassy Irreversibility, the measured magnetization and susceptibility curves are often similar to those obtained in the glass model, as is the shape of the S-N phase boundary in the 8-T plane, as deduced from either susceptibility o r resistivity measurements. Slow (usually logarithmic, or Kohlrausch-like subexponential) time decays of magnetization are invariably observed, both in single crystal and polycrystall~ne samples. Recently, Steinmann et al (13) have reported oscillations In the magnetoresistance above

66

D. Stroud and C. Ebner

/ Glass state in weak-coupled superconductors

Tc(B), suggesting weak-coupled superconducting grains with weak disorder in the plaquette areas. If this model applies to the high-T c materials, where does the disorder comes from? As noted by M~ller et al (i), the variation of T c with B suggest "grain" dimensions of order I000 A. If there is strong T c variation on this scale within the grain (arising, e.g., from spatial fluctuations in.oxygen concentration), then this could give rise to the apparent internal Josephson coupling seen experimentally. Other possibilities should also be investigated. Another possibility might be decoupling of the two-dlmensional layers within the single crystals, and consequent random orientation of these, as suggested by Morgenstern et al (3). A more detailed explanation of this nature has been put forward by Deutscher and Muller (14). The present model is obviously oversimplified in relation to the hlgh-T c materials. These are highly anisotropic, and the corresponding anisotropy in the coupling between grains should be taken into account. Furthermore, the order parameter in the new materials is not yet conclusively identified, though it seems to produce a conventional 2e/hc flux quantum. It is useful to develop a suitably generalized version of this model which will still explain many of the observed anomalies. ACKNOWLEDGMENTS We are very grateful to K.A. M~ller for valuable correspondence. We also thank numerous colleagues, especially D.L. Cox, G.S. Great, C. Jayaprakash, S. John, T. Lubensky, 8.R. Patton and W.Y. Shih for illuminatlnE discussions about the superconductinE glass phase.

REFERENCES (I) K.A. M~ller, M. Takashige, and J.G. Bednorz, Phys. Rev. Lett. 58 (198Z) 1143. (2) See the numerous articles in this ProceedinEs for extensive references. (3) O. Morgenstern , K.A. Muller, and J.G. Bednorz, Z. Phys. B-Condensed Matter 69 (1987) 33. (4) C. Ebner and D. Stroud, Phys. Rev. 831 (1985) 165. (5) W.Y. Shih, C. Ebner, and D. Stroud, Phys. Rev. B30 (1984), 134. (6) S. John and T.C. Lubensky, Phys. Rev. Lett. 59 (1985) 1014. (Z) S. Teitel and C. Jayaprakash, Phys. Rev. 827 (1983) 598. (8) R.S. Fishman and D. Stroud, Phys. Rev. B3Z, (1988) 1499 and references cited therein. (9) S. Chakravarty, G.-L. Ingold, S. Kivelson, and A. Luther, Phys. Rev. Lett. 56 (1986) 2303. (10) C. Lobb, D.W. Abraham, and M. Tinkham, Phys. Rev. 827 (1983) 150. (ii) J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (19Z3) 1181. (12) For a review of glassy dynamics, see Lecture Notes in Physics, Vol. 192: Heidelberg colloquium on glassy dynamics, eds. J.L. van Hemmen and I. Morgenstern (Springer, New York, 1987). (13) R. Steinmann, P. Lejay, J. Chaussy, and B. Pannetier, these proceedings. (l~) G. Deutscher and K.A. M~ller, Phys. Rev. Lett. 59 (1987) 1745.