Available at: ht tp: //www. ictp. t r i e s t e . it/~pub_of f IC/98/141

United Nations Educational Scientific and Cultural Organization

and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SECOND HARMONICS AND COMPENSATION EFFECT

IN CERAMIC SUPERCONDUCTORS

Mai Suan Li

Institute of Physics, Polish Academy of Sciences,

Al. Lotnikow 32/46, 02-668 Warsaw, Poland

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

The nonlinear ac susceptibility and the compensation effect observed in ceramic superconductors

which show the paramagnetic Meissner effect are studied by the Monte Carlo simulations on a three

dimensional lattice model of the Josephson array with finite self-conductance. Our study is based on

the possible existence of the chiral glass phase in these materials. In agreement with experiments, the

compensation effect is demonstrated to be present in rf-wave superconductors but not in the s-wave

ones.

MIRAMARE - TRIESTE

September 1998

One of the most fascinating discoveries in the condensed matter physics is the paramagnetic Meiss-

ner effect (PME) in certain ceramic superconductors [1,2]. The nature of the unusual paramagnetic

behaviour is related to the appearance of spontaneous suppercurrents (or orbital moments) [3]. The

latter appear due to the existence of 7r-junctions characterized by the negative Josephson couplings

[3,4]. The PME is succesfully reproduced in a single loop model [5] as well as in a model of interacting

junction-loops [6,7].

Recently Heinzel et al [8] have shown that the PME may be analyzed by the compensation technique

based on the measurement of the second harmonics of the magnetic ac susceptibility. Their key obser-

vation is that the so called compensation effect (CE) appears only in the samples which show the PME

but not in those which do not show the PME. Overall, this effect may be detected in the following

way. The sample is cooled in the external dc field down to a low temperature and then the field is

switched off. At the fixed low T the second harmonics are monitored applying the dc and ac fields to

the sample. Due to the presence of non-zero spontaneous orbital moments the remanent magnetization

or equivalently the internal field appears in the cooling process. If the direction of the external dc field

is indentical to that during the field cooled (FC) procedure, the induced shielding currents will reduce

the remanence. Consequently the absolute value of the second harmonics \\2\ is decreasing until at a

field Hdc = Hcom when the signal of the second harmonics is minimized. So the CE is a phenomenon

in which the external and internal fields are compensated and the second harmonics become zero.

The goal of this paper is to explain the CE by the Monte Carlo simulations. Our starting point

is based on the possible existence of the chiral glass phase [9] in which the remanence necessary for

observing the CE should occur in the cooling procedure. Such remanence phenomenon is similar to

what happens in spin glass. Furthermore, the PME related to the CE can also be observed in the

chiral glass phase [7,9,10]. There are several experimental results [11] which appear to corroborate the

existence of such a novel glassy phase in ceramic high-Tc superconductors.

In the chiral glass phase the frustration due to existence of 0- and 7r-junctions (0-junctions correspond

to positive Josephson contact energies) leads to non-zero supercurrents [9]. The internal field (or the

remanent magnetization) induced by the supercurrents in the cooling process from high temperatures

to the chiral glass phase may compensate the external dc field.

Using the XY model of the Josephson network with finite self-inductance we have shown that in the

FC regime the CE appears in the rf-wave superconductors which show the PME but not in the s-wave

samples. In the zero field cooled (ZFC) regime decreasing of the external dc field also gives rise to the

CE in the rf-wave ceramics. Both of these findings agree with the experimental data of Heinzel et al [8].

We model ceramic superconductors by a three-dimensional lattice model of Josephson-j unction array

with finite self-inductance. Neglecting the charging effects of the grain, we consider the following

Hamiltonian [6,1

J2 Jij cos(6i - 6j

where #8- is the phase of the condensate of the grain at the i-th site of a simple cubic lattice, A being the

fluctuating gauge potential at each link of the lattice, <f>o the flux quantum, Jij the Josephson coupling

between the i-th and j - th grains, C is the self-inductance of a loop (an elementary plaquette), while the

mutual inductance between different loops is neglected. The first sum is taken over all nearest-neighbor

pairs and the second sum is taken over all elementary plaquettes on the lattice. Fluctuating variables

to be summed over are the phase variables, 6i, at each site and the gauge variables, Aij, at each link.

$ p is the total magnetic flux threading through the p-th plaquette, whereas $^* is the flux due to an

external magnetic field applied along z-direction,

HS if p on the < xy > plane , ,0 otherwise , { '

where S denotes the area of an elementary plaquette. The external field H includes the dc and ac parts

and it is given by

H = Hde + Haecoa{ut) . (3)

It should be noted that the dc field is necessary to generate even harmonics.

In s-wave ceramics, the sign of the Josephson couplings is always positive (Jij > 0), while in rf-wave

ceramics it could be either positive (0-junction) or negative (?r-junction) depending on the relative

direction of the junction and the crystal grains on both sides [5]. In rf-wave superconductors, the sign of

Jij is expected to appear randomly since the spatial orientation of each crystal grain would be random.

Such coexistence of 0- and TT-J unctions gives rise to frustration even in zero external field in rf-wave

ceramics. By contrast, there is no frustration in zero external field in s-wave ceramics.

In the present paper we deal with two types of bond distributions, each intended to mimic d- and s-

wave ceramic superconductors. For the rf-wave ceramics the spin glass type bimodal (± J) distribution

of Jij is taken. In this case the chiral glass phase may occur at low temperatures [9]. For the s-

wave superconductors the interactions Jij are assumed to be 'ferromagnetic' and distributed uniformly

between 0 and 2J.

The ac linear susceptibilty of model (1) has been studied [7] by the Monte Carlo simulations. It

was found that the rf-wave model exhibits much stronger dissipation than the s-wave model in the

3

low frequency regime. Here we go beyond our previous calculations of the linear ac susceptibility [7].

We study the dependence of the second harmonics as a function of the dc field. In this way, we can

make a direct comparison with the CE observed in the experiments [8]. The second harmonics of a

similar Josephson network model with finite self-inductance were considered by Wolf and Majhofer

[12]. However, these authors dealt with the s-wave system in two dimensions and the CE has not been

studied. In this paper we are mainly interested in the CE in the frustrated rf-wave three-dimensional

system.

C\2

X

FIG. 1. The dependence of |%2 on hdc for d- and s-wave superconductors at T = 0.1. The values of C arechosen to be equal 1, 3 and 10 and they are shown next to the curves. The results are averaged over 20 samples.

Dimensionless magnetization along the z-axis mormalized per plaquette, rh, is given by

rh = 2_] (®p ~ ^evxt) ,

pV p<xy>(4)

where the sum is taken over all Np plaquettes on the < xy > plane of the lattice. The real and imaginary

parts of the ac second order susceptibility x'2(w) and X2'(w) are calculted as

rh(t) cos(2u)t)d(uit) ,

m(t)sm(2ojt)d(ojt) , (5)

where t denotes the Monte Carlo time. The dimensionless ac field hac, dc field hdc and inductance C

are defined as follows

_ 2irHacS _ 2-irHdcSftac — 7 7 tide — 7 7

00 0o£ = (2n/<f>0)

2J£. (6)

The dependence of C on the parameters of the system such as the critical current and the typical size

of the grains is discussed in [7,10].

A Monte Carlo simulation based on the standard Metropolis updating has been employed. While

Monte Carlo simulations involve no real dynamics, one can still expect that they give useful information

on the long-time behavior of the system. We choose the gauge where the bond variables Aij along the

z-direction are fixed to be zero. The lattice studied are simple cubic with L x L x L sites and free

boundary conditions being adopted. In all calculations presented below we take L = 8 and w = 0.001.

The sample average is taken over 20-40 independent bond realizations. X'2,{u) has been estimated

following the procedure in [7,13]. Namely, at the beginnig of a given Monte Carlo run, we first switch

on the field (3). Then, after waiting for initial to Monte Carlo steps per spin (MCS), we start to

monitor the time variation of the magnetization, to is being chosen so that all transient phenomena can

be considered extinct. We set to to be 2 x 104 MCS. After passing the point t = to, fh(t) is averaged

over typically 200 periods, each period contains t? MCS (tx = 2TT/UJ). The real and imaginary parts

of the second order ac susceptibility are then extracted via Eq. (5). We set hac = 0.1, corresponding

to Pd 0.016 flux quantum per plaquette. Smaller value of hac turned out to leave the results almost

unchanged.

The dependence of \X2\, \X2\ = ^ (x ' 2 ) 2 + (x")2, on hdc at T = 0.1J is presented in Fig.l. For

small values of C the oscillation of |x21 is shown up. Such oscillation has been found for the two-

dimensional s-wave superconductors in Ref. [12] and its nature is related to the lattice periodicity. Our

new observation is that the oscillatory behavior is still present in rf-wave superconductors but to less

extent compared to the s-wave case. It is clear from Fig. 1 that \xrz\ does not decrease at large hdc

but gets saturated. This is an artifact of the assumption that the Josephson contact energies Jij are

field-independent. The field dependence of Jij should remove the saturation of \x2\ at strong dc fields

[12,14].

- 40.2 0.8 1.00.4 0.6

h d cFIG. 2. The second harmonics of d-wave superconductors obtained after field cooling in a dc field hdc = 1

from T = 0.7 to T = 0.1 and stepwise reducing hdc- The temperature step is set equal to 0.05. The inductanceis chosen to be equal to £ = 4. The arrows indicate the direction of the dc field changes. The results areaveraged over 40 samples and are qualitatively the same as those presented in Fig. 1 of Ref. [8].

In order to study the difference between rf-wave and s-wave samples through the CE we have to

consider the weak field region where the PME may be observed. For model (1) the PME in the rf-wave

case appears clearly for hdc < 1 [7]. So the largest hdc we take is 1. In this weak field regime there is

no periodicity of \\2\ versus hdc which may complicate the study of the CE. The chiral glass phase is

found to exist below a critical value of the inductance Cc where 5 < Cc < 7 [9]. One has to choose,

therefore, an C which is smaller than its critical value and in what follows we take C = 4.

In this paper we have focused on the system size L = 8, C = 4, w = 0.001, and T = 0.1. Our

preliminary studies show that the qualitative results do not depend on the choise of the parameters of

the system.

- s-wave, T=0.1J

1.0

FIG. 3. The second harmonics of s-wave superconductors obtained after field cooling in a dc field hdc = 1from T = 1.4 to T = 0.1 and stepwise reducing hdc- The temperature step is set equal to 0.1. The inductance istaken to be equal to £ = 4. The arrows indicate the direction of the dc field changes. The results are averagedover 20 samples.

Fig. 2 shows the dependence of second harmonics on hdc in the FC regime for the d- wave super-

conductors. Our calculations follow exactly the experimental procedure of Heinzel et al [8]. First the

system is cooled in the dc field hdc = 1 from T = 0.7 down to T = 0.1 which is below the paramagnet-

chiral glass transition temperature Tc Pd 0.17 [9]. The temperature step is chosen to be equal to 0.025.

At each temperature 2x l0 4 Monte Carlo steps are evolved. When the lowest temperature is reached

the dc field used in cooling is switched off and we apply the combined field given by Eq. (3). We

monitor the second harmonics reducing the dc field from hdc = 1 to zero stepwise by an amount of

Ahdc = 0.05. |x21 reaches minimum at the compensation field hcom = 0.7± 0.05. At this point, similar

to the experimental findings [8], the intersection of x'2 and x" is observed. This fact indicates that at

HCom the system is really in the compensated state. Furthermore, in accord with the experiments, at

the compensation point the real and imaginary parts should change their sign [8]. Our results show

that x'2 changes its sign roughly at hdc = hcom. A similar behavior is also displayed by x" but it is

harder to observe due to a smaller amplitude of x"-

6 -

4 -

2 -

1

-

;;

i

i1

d-wave,

*.° I .

n •D

1

ZFC

h

* .

V o

1 i

T=O.1J

1°com o

°° I

I •I

1

o

/I

:.i

i

io

o •o •

1

1o

—

-

1- 0 .5 0.0 0.5 1.0

FIG. 4. The dependence of |%2 on hdc obtained in the ZFC regime for the d- wave superconductors. Thesolid circles correspond to the case when the dc field is decreased. The open hexagons and squares correspond tothe increase of hdc from zero to 1 and to -0.5, respectively. The inductance is chosen to be equal to C = 4. Thedirections of changes of the dc field are marked by arrows. The results are averaged over 25 samples. The typicalerror bars are shown. The compensation field in the case when the fiels is decreased is about hcom = 0.15 ±0.05.

Fig. 3 shows the dependence of the second harmonics on hdc in FC regime for the s-wave supercon-

ductors. The calculations are carried out in the same way as in the d-wave case. A difference is that we

start to cool the system from T = 1.4 which is above the superconducting transition point Ts Pd 0.9 (Ts

is estimated from the maximum of the specific heat [7] for C = 4 and the results are not shown here).

The temperature step is set equal to 0.1. Obviously, \\2\ decreases with decreasing hdc monotonically.

Thus, there is no CE because the remanent magnetization does not appear in the cooling process. This

result is again in accord with the experimental data [8].

We now turn to the ZFC regime. The experiments [8] show that no CE can be expected if after the

ZFC procedure one increases the dc field. However, if the field is decreased a remanent magnetization

is developed and the CE appears [8]. The results of our simulations for the d-wave superconductors are

shown in Fig. 4. As in the FC regime the system is cooled from T = 0.7 to T = 0.1 but without the

external field. Then at T = 0.1 we apply the field given by Eq. (3) and study three cases. In one of

them hdc is decreased from hdc = 1 to -0.5. The values of \\2\ a r e represented by solid circles in Fig.

4. The CE is clearly seen at hcom = 0.15 ± 0.05. At this point the real and imaginary parts of the

second harmonics also intersect (the results are not shown). It is not surprising that the hcom in the

ZFC regime appears to be smaller than in the FC regime. Fig. 4 shows also the dependence of \\2\ ° n

the dc field when it changes from hdc = 0 to 1 (open hexagons) and to -0.5 (open squares). Obviously,

no CE is observed in this case. The results presented in Fig. 4 qualitatively agree with those shown in

Fig.2 of Ref. [8].

In conclusion we have shown that the CE may be explained at least qualitatively by using the chiral

glass picture of the rf-wave superconductors. The CE is shown to appear in the chiral glass phase in

which the PME is present but not in the s-wave samples without the PME. A study of this effect is

important because it would shed light on the problem of nature of pairing in high-Tc superconductors.

Namely, the CE confirms the existence of the spontaneous orbital moments which may be considered

as a sequence of dx2_y2 pairing symmetry [5,15].

Acknowledgments

We thank M. Cieplak for a critical reading of manuscript and H. Kawamura and D. Dominguez

for discussions. It is a pleasure to thank S. Shenoy for useful discussions and his warm hospitality

at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, where this work has

been completed. Financial support from the Polish agency KBN (Grant number 2P03B-025-13) is

acknowledged.

References

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[10] H. Kawamura and M.S. Li, J. Phys. Soc. Jpn 66, 2110 (1997).[11] L. Leylekian, M. Ocio and J. Hammann, Physica C 185-189, 2243 (1991); Physica B 194-196, 1865 (1994);

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M. Cieplak, J. Phys. (France) I 6, 1 (1996)[14] L. Ji, R. H. Sohn, G. C. Spalding, C. J. Lobb, and M. Tinkham, Phys. Rev. B 40, 10936 (1989)[15] See, for example, A. Mathai, Y. Gim, R.C. Black, A. Amar and F.C. Wellstood, Phys. Rev. Lett. 74 (1995)

4523; D.J. van Harlingen, Rev. Mod. Phys. 67, 515 (1995).

6

C\2

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4

0

d-wave, T=0.1J

ZFC

o Io

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I oo

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-0.5 0.0 0.5 1.0

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T=0.1J

10 15 20