Critical currents and current-voltage characteristics of high temperature superconducting ceramics

Applied Superconductivity

PII: SO964-1807(96)00012-9

Vol. 4, No. l/Z, pp. 61-78. 1996 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0964-1807196 $15.00 + 0.00

Copyright0

CRITICAL CURRENTS AND CURRENT-VOLTAGE CHARACTERISTICS OF HIGH TEMPERATURE SUPERCONDUCTING CERAMICS? E. Z. MEILIKHOV Kurchatov Institute, 123182 Moscow, Russia (Received 21 January 1993; in revised form 21 August 1993)

Abstract-We discuss problems related to the “arrangement” and properties of those elements of the granular structure of HTSC ceramics (with an emphasis on the ceramics of the YBa+&O composition) which define the critical currents and current-voltage characteristics of ceramic materials. The basis element of this sort are intergranular boundaries. In materials of practical interest their predominant role as “weak” sites (that is, areas with a low critical current density) is to lower critical current density of ceramics. Far modeling purposes ceramics can be regarded as Josephson media, that is as sets of superconducting grains interconnected by Josephson Junctions. This model provides a means of describing a good number of properties of superconducting ceramics without going into detail about the specific genesis and “arrangement” of weak intergranular links. In this way we discuss tbe critical current and current-voltage characteristics of HTSC-ceramics and their dependencies on temperature, magnetic fiel’d and pressure.

1. INTRODUCTION

The use of high-T, superconducting ceramics is the only possible way of making a variety of engineering applications of superconductivity practical. One of the basic obstacles on the path of realization of these possibilities is small (even in a zero magnetic field) critical currents of HTSC ceramics which are by 1O-l 000 times (depending on the production technique) weaker than those of single crystals and have a tendency of rapidly decreasing with increasing applied magnetic field. It obviously follows from the fact that the only fundamental distinction between HTSC ceramics and HTSC single crystals is the macrogranular structure inherent to the former, that the reason for “bad” superconducting characteristics of HTSC ceramics lies not so much in the properties of individual superconducting crystallites (granules or grains) as in the contacts between them, or so-called intergranular (intergrain) boundaries (GBs). Hence the way to increasing current-carrying abilities of HTSC ceramics runs through a better insight into and improvement (based on this better insight) of the properties of intergranular boundaries. The discussion of the role of GBs in the determination of superconducting properties of HTSC ceramics should be started with GB classification. Unfortunately, there exists no single physical parameter of sudh boundaries which might form the basis of a non-ambiguous classification scheme. Hence there are several different approaches, each of them proceeding from a concept of the decisive role of this or that property of a GB [l]. As the proper parameter one would use: (1) a misorientation angle of neighbor (separated only by a GB) grains [2]; (2) a GB plane orientation relative to (001) and (100) planes in which the anisotropic correlation length takes on extreme values [3]; (3) oxygen stolichiometry of near-to-boundary areas of grains [ 11; (4) probability of impurity segregation on a GB; (5) a tendency toward the formation of stacking faults and secondary or amorphous phases on a GB [4, 51.

t Presented at the Worl,dCongress on Superconductivity (3rd International Conference and Exhibition), 15-18 September 1992, Munich, Germany. 61

62

E. Z. MEILIKHOV

No matter which of the above-listed (as well as left beyond our attention) factors form the basis of “bad” superconducting behavior of any given HTSC ceramic, all of them ultimately lead to the formation of weak superconducting links between its individual grains. That is why, for modeling purposes, ceramics can be regarded as Josephson media, that is sets of superconducting grains interconnected by Josephson junctions. This model provides a means for describing a good number of properties of superconducting ceramics without going into the details of a specific genesis and “arrangement” of weak intergranular links. Although this kind of a model allows one to describe magnetic and transports properties of ceramics, for working out concrete recommendations as to purposeful modification of these properties, it is expedient to take advantage of certain ideas of physical mechanisms, responsible for a distinction between GB properties and bulk properties of ceramic superconducting crystallites.

2. PROPERTIES

OF INDIVIDUAL

INTERGRANULAR

CONTACTS

In Refs [2,7, lo] properties of individual “artificial” GBs in epitaxial laser ablated YBa2Cu30~ films (where the film plane coincides with the basal &plane, the film thickness being - 1 pm) on bi-crystalline strontium titanate substrates have been discussed. The GB orientation in the film is determined by the bi-crystalline boundary of the substrate. Critical currents and current-voltage characteristics of GBs with different misorientation have been studied. It has brought to light two universal relationships: (a) an average (over the area) density 02”) of critical current across the boundary is a function solely of angle 9 of the grain misorientation on this boundary; and (b) characteristic voltage v, = icrN, which is a product of critical current i, of an intergrain contact and its resistance rN in the normal state, depends only on the critical current density (j:“) of this contact. The very existence of these universal relationships is evidence of superconducting properties of GBs in YBazCu306 being intrinsic properties of the material. The first of the above-mentioned relationships is illustrated by Fig. 1, from which it follows that critical current density 12” rapidly falls off with increase in 9, diminishing by an order of magnitude at 9 - 10”; with further rise of 8, theJcGB fall-off slows down. In different works where the JFB(S)-dependence was used for calculations of the critical current of HTSC ceramics, different approximations were suggested for it. Thus, in Ref. [ 1 l] it has been shown that experimental dependence jTB(8) is satisfactorily approximated with the expression J:B/J:

sin 6 sin(9 + 6) ’

=

1.00 -

l ,.

: .

.D l

Q,u

m0,”

0.10 -

l

. l

l

.

.D l

0.01 0

20

10 6

l

30

a 40

(deg)

Fig. 1. Intergrain boundary critical current density J:” vs misorientation angle 9 between contacting grains of the YBa&u~O&lm at T=5 K [8]. Boundary type: 0 - [OOI] tilt boundary; a - [loo] tilt boundary; - [ 1001 twist boundary.

Critical currents and current-voltage

characteristics of high temperature superconducting ceramics

where ~2 is the critical current density approximation has been utilized: ,yB/JT

=

in a grain,

6 = 0,87 - lo-*.

- 1 + 1 1 +X& 1 + (K/2 - 9)/80

0,6

63

In Ref. [ 121 a different

1-

0.08,

where 9 = 4”. In Ref. [13], finally, experimental data are described by exponential dependence JfB/Jz

=

eXp(--g/8,-,),

8,

5”.

=

(3)

It should be borne in mind, however, that the aforementioned method of forming GBs in YBa2CusOa films based on the use of cubic substrates gives a possibility of producing boundaries with misorientation angle 0 < 9 < 45”. Hence the assertion of monotonic GB critical current density decrease with the rise of the misorientation angle should be treated with caution. Thus, in Refs [ 14, 151 it ‘has been experimentally settled that in YBa2Cu30a [OlO] twist boundary with 9 = 90” does not display qualities typical to weak links. (A rotation of &planes by 90” round common axis a or b takes place on a GB of this kind.) In order to clarify the nature of Josephson junctions it is important to measure temperature dependences i,(T) of their critical currents. The theory [16] predicts that near T, i,(T) a [1 -

T/T,l”‘,

(4)

where m = 1 for a tunnel SIS-junction and m = 2 for a SNS-junction. temperature dependence i,(T) has the form [ 171

In the latter case the

Aid’) ic(T) o<<,(T)sinh[d,/~,(T)J where A,(T) is an order parameter in a superconductor the normal metal intergrain layer width, 0,23


[

(5)

at a large distance from the contact, dN is

112

hVFNIN

(1,

-

0,16’L

lcT)’

<

rN -

“dirty” limit)

1

lr;

is a coherence length in the normal layer material of charge carriers, respectively), and

(6) (lN 3 tN - “clean” limit) (vFN,

1N

are Fermi velocity and a mean free path

c(T) = th2

(7)

is a factor accounting for an order parameter decrease at the “superconductor-normal metal” interface resultin,g from the proximity effect (5, is a coherence length in a superconductor and x0 is a characteristic length defining the suppression of the order parameter near the boundaries of an intergrain contact). According to different estimates, 5N = 10-100 A in YBazCusOs [ 181; in Refs [ 19, 201 N = 2080 A, as obtained from an exponential dependence of the SNS-contact critical current on the thickness of lthe normal interlayer. From estimates made for the x0/&(O) ratio in Ref. [21] by means of comparison of theoretical expression (5) with experimental temperature dependences of the ceramic critical current it follows that x0 % 50 A for intergranular contacts in YBa2Cu306. It should be kept in mind, however, that a direct utilization of equations (4) and (5) for clarifying the nature of a junction is hindered by the presence of strong fluctuation effects near T = T, [9, 18,231. A comparison of experimental dependences i,(T) with theory accounting for fluctuations reveals [9] that most GBs are Josephson junctions of the SNS- (or SNINS-) type and may be approximated with equation (5), which predicts an exponential dependence of the SNS-junction critical current on the contact width dN in “thick” junctions:

i,(T) a

$f$U)ev[-A] N

N

(8)

E. 2. MEILIKHOV

64

An analysis of magnetic field dependences i,(B) of the critical current of individual intergrain contacts not only conforms their Josephson nature but also allows one to make conclusions about spatial distribution J~“(x, y) of the critical current density over the contact area. The analysis is based on the well known expression for a narrow (L < 412,) Josephson junction [ 161: (9) which relates to the case when the magnetic field projection (L$,)on the junction plane (plane xy) is directed along the y-axis; 0 = B,,Ld is a magnetic flux piercing the contact, L is a contact dimension perpendicular to the magnetic field, d 2 dN + 21, is an effective contact thickness and (I+, = hc/2e s 2 a 10e7 Gs ecm* is a magnetic flux quantum. Applying equation (9) to the known function of i,(B) it is in principle possible to deduce distibutionjyB(x, y), although this reverse problem is incorrect so it is common practice to limit oneself to “picking out” (proceeding from certain physical arguments) such kind of distribution $“(x, y), which would provide the i,(B) nearing the experimental one. The reference points here may be known magnetic field dependences i,(B) for a rectangular contact (see Fig. 2) corresponding (a) to uniform distribution of the critical current density, $“(x, y) = Const:

(10) a so-called “Fraunhofer diffraction” dependence; (b) to strongly non-uniform distribution jTB(x) o> 1) with increased critical current density at edges, and (c) to randomly non-uniform 1D distributionJyB(x) ==I: +j,(x), where]: = j,(x) is a random function with a zero average value j,(x) = 0, correlation radius r expression M~I)

-AXkCx2> -13

= Lk -.&?I2&-lx,

1D U-like the contact Const, and defined by

-x21/h

and effective dispersion y* = (2r/L)(j, -_$)‘/@)‘. According to a dislocation model of an intergrain tilt boundary, it is a dislocation wall, that is set of edge dislocations lying in the boundary plane and spaced at distance [2, 41 a

D = 2 sin(9/2) ’

(11)

Here a is a value of the Burgers vector which, for a boundary between grains with a common caxis, practically coincides (because of a nearly tetragonal structure of YBa2Cu30a) with the lattice parameter in the basal plane. For small-angle boundaries between such grains D >> a, and the

Fig. 2. Magnetic field dependences i,(B) of the critical current of a narrow Josephson junction at different distributions of the local critical current density/TB(x). (1) Uniform distribution [see equation (7)]. (2) U-like distribution (see the text) with x = 10 1161. (3) Random distribution (see the text) with 2nr/L = 0.01 and y = 0.09 [35].

Critical currents and current-voltage characteristics of high temperature superconducting ceramics

65

lattices of neighbor grains are well adjoined everywhere but in the areas near dislocation nuclei; for large-angle boundaries these areas overlap, and the whole boundary becomes “bad”. For further clarification of the model under consideration the authors [25] calculated spatial contours of the constant magnitude (equal to 0.01) of one of the components of the relative deformation caused by a dislocation wall on a symmetric tilt GB. Results of the calculation carried out within the framework of the isotropic theory of elasticity [24] are plotted in Fig. 3. It can be seen that as the angle 8 increases the thickness of weakly deformed (E, < 0.01) near-to-boundary areas diminishes and at 9 E 10” it becomes smaller than the unit cell dimension (in the &plane ). The latter means that the structure required for the existence of superconductivity is destroyed along the complete interface. This “destroyed” near-to-boundary layer is spread as deep into the contacting grains as D/21c E a/2x8 - a and is a normal metal (or insulator). It is in this manner that a Josephson transition of the SNS- (SNINS- or SIS) types originates on the boundary. Nevertheless, the situation considered does not mean that any GB with a sufficiently large misorientation angle 8 is necessarily a weak link. It has already been noticed above that in YBapCusOs [OlO] twist boundary with 8 = 90” exhibits no properties peculiar to weak links. (a)

-a

-4

0

a

4

ala

(b) 16 14 . 12 10 .

1

I

’

-6

A

-2

0

2

4

6

x/a

(c)

-3

-2

-1

0

1

2

3

xla Fig. 3. Deformation field E, (maximum deformation component) on a symmetrical intergrain tilt boundary (J:=O) with an infinite dislocation wall [25]. Within the darkened areas su 3 0.01. Misorientation angle 9: (a) 2”, (b) S’, (c) 10”.

66

E. Z. MEILIKHOV

Besides, from experiment [14] it follows that no such properties are exhibited by the [OOl] twist boundary which corresponds to two contacting crystallites with a common c-axis, mutually rotated around the a- (or b-) axis by angle 9 = 14”. It means that the atomic structure of real GBs may (at least at certain misorientation angles of contacting grains) rearrange in such a way that it would not manifest properties of a weak link. 3. CALCULATIONS

OF HTSC CERAMICS CRITICAL (PERCOLATION MODEL)

CURRENT

Calculations of critical current density j,_, of HTSC ceramics (regarded as an assembly of superconducting grains being connected by weak (Josephson) links) is a complex problem. When solving this problem in a general case one should take into account not only the spread of the coupling energy, ar, over Josephson contacts but also the correlation of the order parameter phases in different grains. The task may be significantly simplified if we neglect the latter: in this case currents in the adjacent contacts may be regarded as mutually independent. Such a situation is realistic either at sufficiently high temperatures (T >, a,), when temperature fluctuations of the order parameter are large, or in a sufficiently strong magnetic field B 2 @,/a2 which results in strong “magnetic field” fluctuations of the order parameter. In this case the critical current calculation can be made on the basis of the percolation theory [26, 271 or the effective medium theory [28]. Any of these theories assumes the function f(i,) of the intergranular contact distribution in critical currents to be known. The form of this function may in principle be deduced on the basis of model concepts about the properties of HTSC ceramics intergranular contacts or experimentally. One of the procedures of deriving a j,, value has been suggested in Ref. [29] and used for concrete calculations in [30]. In essence it is as follows. Let us imagine that a cluster consisting of contacts with critical currents i, > i determines critical current density& of the system (which means that for current density equal to jcr all the contacts outside this cluster are in a resistive state). It is clear that the magnitude ofj,, is limited by the “worst” (i.e. with the smallest critical current i, = i) contacts and, consequently, is directly proportional to i. Besides, the current flowing through the cluster under consideration is proportional to the density of percolation paths in this cluster. The latter density itself is proportional to the conductivity of an equivalent network of resistive bonds, where P is a part of unbroken couplings. This conductivity is, according to the percolation theory, in direct proportion to (P - PC)‘, where t is a critical conductivity factor depending on the form and dimensionality of the network. Thus,

The actual critical current of the system is calculated by means of maximization of equation (12) relative to P and it corresponds to some “critical” cluster with P = PC,; at PC -c P < PC, the cluster, although being infinite, is very “scarce”, while for P > PC, the cluster is crowded with “bad” contacts. Thus, the problem comes down to determination of distribution function f(i,). To this end several methods have been suggested based on the results of different experiments: dependence of the contact critical current on the m&orientation angle between contacting grains [ 1 I], magnetic flux creep in a ceramic sample [12], a current-voltage characteristic of HTSC ceramics in the resistive state [30]. In Ref. [30] the distribution Iunction

(13) was used for explanation of experimentally observed power-law current-voltage characteristics of HTSC ceramics (see below). The distribution (13) may be obtained for Josephson junctions of the

Critical curre:ntsand current-voltage characteristics of high temperature superconducting ceramics

67

6 (deg) Fig. 4. The distribution F(9) of the intergrain boundaries over misorientstion angles 9 in the YBa+&Os-ceramics

[ 131. Straight line: F(9) c( exp(-9/(s)),

(9) = 6”.

SNS-type in which the critical current is described with equation (10) if we assume that their distributionf&&) over the thickness dN of the normal-metal intergranular layer is exponential: fd(dN) oc exp(-d,,/(d)) [30]. The magnitude and the temperature dependence of parameter n near T, here are governed by the relationship:

n=5No_] (4

(14)

’

where (d) is an a,verage thickness of intergranular layers. A different procedure for determiningf(i,) may be recommended for ceramics with a known F(8) function of GB distribution over misorientation angles 9. Then f(i,) = F[8(i,)]&/8$l-‘, where S(i,). Distribution F(9) may, in principle, be found from electron microscopy analysis of a large number of GBs. The results of such analysis for the melt-produced YBa$&O~-ceramics is given in Fig. 4 [ :I31. It can be easily seen that this distribution is close to exponential: F(g) a exp(-WW.

(15)

In this case the average misorientation angle (9) z 6”. But this value is typical only for the ceramics under analysis. Making use of the distribution found and dependence i,(9) in the form (3), we arrive again at equation (13) where

4.

CALCULATION

OF THE HTSC CERAMICS CRITICAL (A JOSEPHSON NETWORK MODEL)

CURRENT

Under conditions when the correlation of the order parameter phases in different granules of HTSC-ceramics (see above) cannot be neglected, its critical current and current-voltage characteristics become collective properties of a system which in this connection is called a “Josephson med.ium”. The latter is usually analyzed using the Josephson network model.

E. Z. MEILIKHOV

68

Randomly located sites of this network correspond to ceramic granules, and its couplings correlate with Josephson contacts with their inherent properties (critical current, current-voltage characteristic, etc.). The study of the properties of this sort of network relies on the use of the Josephson equation (17) (where r#~ is an order parameter phase in the kth granule and V, is its potential), the equation defining current ik, of the weak link between the kth and the Zth granules. ikl =

1% Sin(4k

-

41)

+

(vk

-

q;)lRk19

is the coupling critical current and Rkt is its resistance Kirchhoff’s equation (2%

(18)

in the normal state), and of the

T &I = 0,

(19)

added with relevant boundary conditions. When there is no dissipation in the system (i.e. at current below the critical one) vk = V,. The calculation of the system state (for a given applied current) comes down to the determination of all currents ikt, phases & and potentials &, for which this or other numeric method is required [31, 321. A numerical calculation of the critical current (and also of a current-voltage characteristic, see below) or a 2D system exhibiting an exponential dependence (3) of the critical current of individual Josephson junctions on the misorientation angle, 8, and a “cut” Gauss distribution of these angles (F(9 < 20”) cx exp(-82/2(92)), F(9 > 20”) = 0), has been carried out in Ref. [33]. The critical current of this kind of system rapidly diminishes with increase in magnitude of the parameter < a2 > (that is, with the extent of the F(9) distribution widening): for < 8, >‘i2= 5”, 10” and co the values obtained were Z,,/Z,“,E 0.2,O.l and 0.06, respectively, where c’ is the critical current at c g2 >= 0 (a single crystal). 5. EXTERNAL

EFFECTS

5.1. Magnetic field dependence

ON THE CRITICAL

CURRENT

OF HTSC

CERAMICS

of the critical current

As has been previously mentioned, the allowance for the magnetic field effect on the currentcarrying ability of an HTSC ceramic within the framework of analytical calculations of its critical current requires the insight into modified by the magnetic-field “reorganized” distribution function fB(icB) of intergranular contacts over their critical currents ice. The latter may be calculated if function icB(ic, II), describing variations in the critical current of a single contact in the magnetic field, is known. This function depends on four factors, namely, (1) the size, (2) the orientation, (3) the shape and (4) the spatial distribution of local density JFB(x,Y)of the critical current over the contact area. For calculations, the shape and distribution ofJFB(x,y) are usually taken to be identical for all the contacts. As far as the contact sizes and orientations are concerned, they are averaged assuming, for simplicity, that corresponding distributions are uniform (within certain limits of parametric variations).* In Refs [30, 351 this problem has been solved for contacts of a square form with the initial distribution function (13) for uniform distribution of the critical current density,ZFB(x, u) = Const. [see equation (lo)] and for randomly nonuniform distributionZzB(x), discussed earlier. The results demonstrating an evolution of distribution functionfn(i&) in the magnetic field are plotted in Figs 5(a) and (b). The magnetic field magnitude was characterized by dimensionless parameter b = 27&aB/@c, where a is an average contact size; initial (b = 0) power-law distribution function f(i,) complies with equation (13) for n = 2. In both cases the magnetic field shifts distribution function f&a) towards smaller critical currents, although in a strong magnetic field * As has been shown in Ref. [34], the actual form of the contact distributionover sizes and orientationsaffects but insignificantly the results of averaging.

Critical currents and current-voltage

characteristics

of high temperature

superconducting

ceramics

69

1 (b) 10

-$j 8 'C

1

”

‘Z

,m

+?

5

6 b=O

I

0.1

I

I I

2 5

0.01

2

I .5

SO.12

0.2

‘b = 0

5

I

0.4

0.6

0.8

1.0

iEBliU

i,,li,

Fig. 5. Evolution of functionfa(i,a) field action. (a) Uniform distribution distribution [35]; correlation radius function f(i,)

of contact distribution over critical currents i, under the magnetic of the critical current density [30]. (b) Randomly nonuniform 1D r = 0.01@/274 dispersion $ = 0.09. Initial (b=O) distribution is described with relation (20) with n = 2.

(b >> 1) the form of this function is quite different: for uniform/~B-distribution functionfn(icn) varies as the magnetic field increases gradually, and at small critical currents &(&a) = Const. while for a randomly nonuniform/, GB-distribution there is a magnetic field region (in Fig. 5(a) they are fields 3 < b < 300), where this functional variation is negligible (as compared to fields below and above these limits) and at small critical currents fn(i,..) o( i& Such differences result in different behavior of magnetic field dependences Z,,(b) of the ceramic critical current in both situations under consideration: in the first case Z,,(b >> 1) o( l/b, and in the second case function Z,,(b) has a plateau (see Figs 6(a) and (b)) t. Experiments [.38, 391 actually show that magnetic field dependences Z,,(b) for HTSC ceramics of different composition have a plateau in the region of sufficiently strong magnetic fieldst. Is this an argument for spatial nonuniformity of Josephson contacts of HTSC ceramics causing a random distribution of the local density of the critical current over the contact area? In Refs [38, 391 a different interpretation is suggested: every (or nearly every) Josephson contact is spatially nonuniform and contains a region of a “strong” link. It is these regions that provide a current transfer in strong magnetic fields lessening the local density of the critical current in the rest (“weak”) sections of each of the contacts. The evidence in favor of this interpretation, in the author’s opinion [39], is provided by the investigation results for magnetic field dependences Z,,(b) in grain-aligned YBa&uaOa-ceramics which are aggregates of crystallites with nearly parallel ab-planes (misalignment of their c-axes is - 4~10’). In the:se experiments the transport current was passed along the ab-plane, and the magnetic field perpendicular to the current was directed either along or across this plane. In both cases plateau Z,(b) could be observed; however, field B*, corresponding to its high-field boundary on the side of strong fields is much stronger in the first case: B*(ll ab) s 30 T and B*(ll C) ?z 7 T at T = 76 K. Assuming these value of the field to be close to values &(I] ab) and Bc2(ll c) of the upper critical field for YBa2CusOa, the authors [39] come to the conclusion that the critical current of a textured material in a strong magnetic field is limited by the material properties close

t The form of dependence I,,(b) in strong magnetic fields (b > 1) for a uniform/, ‘GB-distributiondepends on the contact form: I,(b >> 1) u I/b for square contacts, but I&b > 1) a l/d/’ for round contacts [36]. The allowance for contact anisotropy resulting from anisotropy of the London penetration depth produces but an insignificant effect on this result: calculation [37] provides 1 < ]filnl,/8lnbl -z 2. $ Initial dependences Z,(b) nre meant, which correspond to monotonic variation of the magnetic field from zero up to a preset value. Otherwise hysteresis phenomena may be observed resulting from a magnetic flux capture inside superconducting grains

WI.

70

E. Z. MEILIKHOV 1

10-l

s -6 c ‘d

10-Z

3

10-3

10-4

I-

10-I

1

10’

102

103

104

b Fig. 6. Magnetic field depedencies of the ceramics critical current. (1) Uniform distribution of the local critical current density ty[30] (2) Randomly nonuniform ID $B(x)-distribution [35]. Correlation radius 2nr/L = l/300, dispersion y2 = 4.10u4, initial (b = 0) distribution function f(i,) is described with relation (20) with n = 2. Dots: experimental data for the YBaaCu30a-ceramics (T, = 90K, T = 77 K) [38].

to those of a single crystal rather than weak links. In this connection two observations are appropriate. First, well-known values of B,2(11 ab) E 130 T and B,2(11 C) g 25 T (for T= 76 K) [40] are noticeably higher than the measured values of B* . Second, when making an estimate of the magnetic field within crystallites an intercrystallite “layers” (which in this case have the form of comparatively thin lamellas parallel to the &-plane) it is necessary to account for demagnetization factors which are different for different field directions. Then the difference of the measured boundary fields, B*(ll c) and B*(ll ab) may be due to the difference of these factors. Thus, the interpretation suggested in Ref. [39] is not well-grounded. Both hypotheses considered (randomly nonuniform contacts and contacts with a “strong” coupling) may be combined if we assume that in a non-textured ceramic “strong” intergrain links are absent and its behavior in the region of the plateau of the I,, vs magnetic field dependence is defined by randomly nonuniform weak links, while in a textured ceramic “strong” links appear owing to peculiarities of its structure. One of the possible reasons for this phenomenon consists in favorable structural conditions for the formation of the above-mentioned intercrystallite [ 1001 twist boundaries (a mutual rotation of contacting crystallites with respect to each other around the common c-axis) with the misorientation angles near 8 = 14” (and also, possibly, with other favourable misorientation angles), exhibiting the properties of a “strong” link [ 141. The part of such boundaries may be sufficient ( > 15) for the formation of continuous current paths in strong magnetic fields, when all the rest of the (weak) links turn out to be “destroyed”.$ These considerations might explain the results obtained in Ref. [39]. 6 Since there is no systematic study of boundaries of this hind it is highly possible that 9 = 14” is not the only misorientation angle ensuring the properties of a “strong” link. Besides, if we suppose, on the analogy of the results in Ref. [8] (relating to small-angle tilt boundaries) that the properties of a “strong” link are preserved for all considered boundaries with misorientation angles 9 = 14” f A9, where Ah9x 5”, then the part of boundaries with “strong” links can be evaluated as being equal to (2A9/45”) Y 0.2 (for this estimate no difference is made between axes e and 6, i.e. the properties of contacts with misorientatin angles 9 = 14 and 76” are considered identical).

Critical currents and current-voltage characteristics of high temperature superconducting ceramics

71

The analysis of experimental data suggests that the plateau of Z,,(b) dependence is observed only in ceramics with sufficiently high critical current density and it is absent in the case with low current densities. The meaning of this correlation is discussed in the section dealing with the ceramics current-voltage characteristics.

5.2. Temperature dependence

of the critical current

The temperature dependence of the critical current, Z,,(T), of HTSC ceramics is determined by two factors, namely, (1) by temperature dependence i,(T) of the critical current of individual intergrain Josephson contacts of the ceramics, and (2) by the increase of effective (i.e. participating in the near-critical current transfer) contacts upon temperature decrease. As it has been previously emphasized, the majority of intergranular contacts are of the SNS (or of the SNINS) type, hence the temperature dependence of their critical current near T, obeys equation (5) with m = 2. As far as the other two factors in question are concerned, their role can be assessed with the help of the analytical method of the ceramic critical current calculation discussed in the previous section. As has been demonstrated in Ref. [30], for distribution function f(i,) of the form (13) and for temperature dependence i, of the form (13) and for temperature dependence i,(T) o< (1 - T/T,)m exp(-d/r,) the magnitude of Z,, can be deduced from the expression: m

) K(T) = [t(n + l)]‘[t(n + 1) + l]-I”(“+‘)+~l(l - ZQl(n+l),

(20)

where t 2 1.5 and P, = 0.25 are percolation parameters for 3D medium, n + 1 = r,(T)/(d) [see equation (14)]. Flor an intergrain layer made of “clean” materials n(T) + 1 + [~N(Tc)/(d)lTc/T. Application of (:20) reveals that almost within the whole temperature range of T < T, the temperature dependence of the ceramic critical current is close to that of the critical current of its separate intergrain contacts.

5.3. Critical current dependence

on pressure

(uniform and axial compression)

5.3.1. Uniform compression. Numerous experiments prove the critical current of HTSC ceramics to increase under hydrostatic pressure [4148]. Relative variations of the critical current at P - 10 kbar goes as high as - 100% and depends on its magnitude at the zero pressure: for “bad” ceramics (with low critical current) relative variations are, as a rule, more numerous than for “good” ones. For interpretation of these results it is essential that the conclusion made in the previous section (concerning the temperature dependence of I,,) is equally valid for the critical current dependence on pressure: the latter is determined mainly by the variation of the critical current of individual intergrain contacts. Thus, a mechanism of pressure action on the properties of a single Josephson SNS contact should be analyzed [49]. Nonuniform pressure distribution over the contact area leads to a change of its thickness d = d(x, y) and, in accordance with equation (8), to nonuniform distribution of the current local density. For relatively small pressures, when the ceramic deformation is reversible, distribution ZT”(x, y) may be substantially nonuniform only for “bad” ceramics for which d > rN. In “good” ceramics (d < rFl) this distribution is always nearing the uniform one. For “good” celramics the pressure action is confined to a change of areas of intergranular contacts.1 In this case in a zero magnetic field [49] 213

icAP,0) =

( >

Wr0)2j,,(O, 0) =i,,(O, 0) 1+ i

1 Experiments reveal ,Pweak dependence of intrinsic superconducting parameters on pressure (for YBazCu30a, for instance, a[T/T,]/W = IO-*- IO-'kbar-‘).

1

proper of high temperature

(21)

superconductors

72

E. Z. MEILIKHOV

which correlates well with experimentally

observed dependences; and in a strong magnetic field

WI j,,(P,B) =j&~)

116

( > 1+;

(22)

9

which corresponds to a much looser dependence of the ceramic critical current on pressure. It results from the fact that the contact area increase, potentially favorable from the point of view of raising its critical current, simultaneously brings about the rise of the magnetic flux through a contact lateral surface, which in its turn furthers the critical current decrease [equation (1 O)]. 5.3.2. Uniuxiul compression. Under uniaxial compression the deformation e(4) of the medium is anisotropic. Both its value and sign depend on the angle 4 between the direction of compression and that of deformation [50]: e(4) = 7,

P$ = P[(l + v) cos* 4 - v].

(23)

In accordance with relation (23), uniaxial compression is accompanied with expansion in directions for which lx/2 - 41 < [v/(1 + v)]. It means that all the intergranular contacts may be divided into two types: critical currents of the contacts of the first type (their planes form an angle with the compression direction which does not exceed arccos[v/(l + v)]) decrease, and those of the second type, on the contrary, increase. The total result of the uniaxial compression effect on the ceramic critical current in this situation depends on the orientation of an average direction of the transport current relative to the compression direction, as well as of the “sinuosity” of current transfer is performed. If this sinuosity characterized by the average deviation 01)of current paths from the transport current direction is not large (if, to be exact, (cl) < (6~)” 1 [51]), the for the current flow (at an average) along the compression direction they are the compressed contacts which are critical and for the current perpendicular (in general) to the compression direction the leading role belongs to the compression direction the leading role belongs to expanded contacts. Correspondingly, the critical current density should increase under uniaxial pressure in the former case and decrease in the latter one. Calculations reveal the ratio of variations of the densities of these current (for small sinuosity of current paths) to equal Aj,‘, Aj!, G+-v (for the YBa2Cus0a ceramics v ?! 0.2). Experiments carried out with the YBa2Cus0a ceramics (with the critical current j,,(77 K) N 102 A/cm*) confirm in full this conclusion [44]. From this, in particular, it follows that the sinuosity of current paths in this ceramic is not large: (a) < 30”.

6. CALCULATION HTSC

OF CURRENT-VOLTAGE CERAMICS (PERCOLATION

CHARACTERISTIC MODEL)

OF

Calculation of a current-voltage characteristic (CVC) of superconducting ceramics at currents I exceeding critical current Z,, is complicated by the fact that resistive elements of such a system (intergrain contacts) are essentially nonlinear: voltage drop v on a SNS contact with critical current i, and normal resistance r, comply with the function [52] 0,

i < i,

v(i, i,) =

(24) ( r,(i* - iz)“*,

i > i,

where i is current flowing t&ough the contact. Since for nonlinear systems the superposition principle is invalid the utilization of well-developed percolation and effective medium methods is not applicable for the calculation of this system conductance. Particularly, for a considerable spread of parameters iCand rN determining the nonlinear conductance of separate elements of the system the CVC of the system may be strongly different from that of the constituent elements.

Critical currents and current-voltage

characteristics of high temperature superconducting ceramics

73

It can be easily seen from a simple and frequently used model which regards an HTSC ceramic as a set of 1D “threads” interconnected in parallel and composed of a great number of serial weak links [53]. In this case voltage Vat the ends of the “threads” with current i is V(i) oc

v(i, i,lf(i,)di,.

(25)

Normal contact resistance rN iS prOpOKtiOId to its thickness dN for “thick” COntaCtS[See equation (8)] is but insufficiently (logarithmically) dependent on i,. If we neglect this dependence and make use of distribution function (13) we shall obtain from equation (24) and (25) [30]: V(i) a

(i2 - i~)1’2f(ic)dic cc

(26)

from which it follows that a 1D chain of Josephson links possess power-law CVC at moderate currents, these characteristics turning linear with larger currents. The natural current scale here is the maximum critical current of couplings, io. An attractive feature of the model in question is, apart from its simplicity, the feasibility of restoring the form of the contact distribution functionf(i,) over critical currents on proceeding from the CVC [30]. For an approximate solution of this problem let us approximate the CVC of a single contact (24) with a step function 0

, i < i,

rNi

, i > i,

(i, i,) =

(27)

Then, instead of (46) we arrive at i

r(i) E V(i)/i cc o 1s

f(i,)dL

(28)

for nonlinear “thread” resistance, from which it follows

where R = Nr and Z = Ni are the resistance and current of a sample consisting of N parallel “threads”. An example of the application of equation (29) is given in Fig. 7, where distribution functionsf(i,) an d&(&u) (in magnetic fields B = 0 and 250 Gs, respectively) as calculated using are plotted for the YBa2Cu306 ceramic with low critical current this equation j,,(78 K) % 10 A/cm2. The comparison with Fig. 5 reveals that in this case the majority of contacts of the HTSC ceramic are the contacts with a uniform distribution of the critical current density. This simple model seems to provide an adequate description of HTSC ceramics only in direct proximity to T,, when the network of conducting paths is sufficiently sparse, and may be roughly presented as a set of parallel connected “threads”. On lowering the temperature, more and more intergranular contacts get involved in this network. its structure becomes still more complex and CVC calculations require a more general approach. The first approximation consists in letting the contact resistance in the normal state be identical while retaining the contact spread over critical currents [30]. Then an analogy with a percolation problem on the conductance in a random system made of resistive and superconducting bonds can be utilized [54]. If part P, of superconducting bonds of such a random lattice is smaller than the threshold value, PC, then its conductance X is finite and equals X cx (PC - P,)-“. (For a 3D cubic lattice PC = 0.25 and s = 0.7-0.9; for a 2D square lattice P, = 0.5 and s = 1.1-l .15.) The “distance” to the superconducting state transition point is determined in this problem by the difference (PC - P,). Its counterpart in CVC calculations is the part of broken links in the above deduced (see Section 3) critical cluster, increasing with the Z > Zcrgrowth, i.e. difference (1 - P), where P’ = Z”(Z) is a part of the contacts remaining in the superconducting state at Z > I,,. For the power-law

E. Z. MEILIKHOV

74

0.00 I

0.1

0.01

I

i=, (a.u.) Fig. 7. Distribution function of intergrain Josephson junctions over critical currents in the YEIa#&Oaceramics in the absence (0) and presence (m) of a magnetic field I3 = 25OGs[30].

distribution function (13) dependence P(Z) is expected to be exponential, too; besides, taking into consideration that P(I,,) = 1, we find P(Z) - 1 o( (Z - Zcr)“+‘. Then for Z > Zcr we obtain the following expression for the conductance of the system under analysis: E(Z) o< (T&‘[l - P(z)]-S = (rN)-l(z - zJSfn+l), where < rN > is an average contact resistance in the normal state. Thus, CVC of the system in question V(Z) o( Z/X(Z) complies with relation v(I) o( (?$Z(Z - Z,,)‘“+”

(30)

and at Z >> I,, it acquires a power-law form:

v o< zr’, p

= 1 + (n + 1)s.

(31)

Taking into account that s z 1 we find that, in conformity with the above deduced expression for CVC of 1D “threads”, V rx Zn+2 [see equation (26)]. 7. EXTERNAL

EFFECTS

OF THE CURRENT-VOLTAGE OF HTSC CERAMICS

CHARACTERISTICS

Within the framework of the above considered (Section 6.1) model, CVC dependence on temperature and magnetic field is derived from corresponding dependences of exponent p, critical current Zcrand average resistance (rN) in the normal State [See equation (30)]. Of special interest and importance are, undoubted, functions ,u(T) and p(B). The temperature dependence of the exponent fi may be found from relation (21) [51]: &“)_

1

(T/Tc)-1’2 =sTN(T)=S~N(~)X

(4

(4

(T/T,)-’

(“dirty” limit), (32) (“clean” limit)

According to (32), parameter ~ashould grow monotonically with the temperature increase. As to the magnetic field dependence of the CVC exponent p(B), it depends on the manner of the distribution functionfn(i,n) evolution in the magnetic field (see Fig. 5). In the case of getting into

Critical currents and current-voltage

characteristics

of high temperature

superconducting

75

ceramics

the contacts with a randomly nonuniform distribution of local critical current density /zB(x),the magnetic field produces a scarcely noticeable effect on the form of the distribution function which correlates with a weak dependence of p(B). For contacts with uniform local current density [J:“(X) = Const] ,the part of the distribution fimctionfn(i&), which determines the ceramic critical current Icr corresponds (in a sufficiently strong magnetic field) to a value of n = 0. It means [see equation (3 l)] that with the magnetic field increase ~1should decrease down to p = 1 + s. Power-law CVC of the form (31) have been more than once observed in experiments with HTSC ceramics of different composition [55-601. The analysis proves p(B = 0) to be always b 1 for ceramics with low current carrying ability (j&77 K) x 10 A/cm2 at T,= 80-90 K) [56, 57, 591, and p(B = 0) >> 1 for ceramics with sufficiently high critical current density (j,,(77 K) * lo2 A/cm2 [58, 591. Such behavior is naturally explained with the help of equation (55): the former case (cl % 1)) corresponds to a “thick” contacts where (d)/cN >> 1 and, consequently, critical currents are small [see equation (8)] and the latter case does so for “thin” contacts ((d)/tN < 1) with large critical currents. This approach also explains the difference in the form of magnetic field dependences p(B) in ceramics with low and high current carrying ability. It only requires taking into consideration the fact that a randomly nonuniform distribution of the local critical current density, J:“(X), may exist only in “thin” contacts sensitive to structural inhomogeneities of near-contact areas of ceramics superconducting granules. For “thick” contacts, on the contrary, spatial distributionJyB(x) of the local critical current density is practically uniform. It has already been discussed above why it should lead to different magnetic field dependences p(B). In Figs 8(a) and (b) temyture dependences ~(2”) are plotted for B&ceramics (phase 2212) with&(77 K) x 1O3A/cm [60] and GdBazCusOa-ceramics with j,,(77 K) % 1 A/cm2 [60] and GdBa$usOg-ceramics with j,,(77 K) x 1 A/cm2 [56] at B = 0.It can be easily seen that in the first case, in fact, p > 1, while in the second case p M 1. Estimations with equation (55) give us Wl& % 0.1 for the Bi-ceramics with considerable critical current density and (d)/& x 1 for the GdBa&usOa-ceramics with low critical current density. Additional arguments in favor of the above cited considerations are given in Fig. 9 where magnetic field dependences p(B)are plotted for the Bi-ceramics (phase 2212) with j,,(77 K) * lo3 A./cm2 [58] (upper curve) and the YBa$u306-ceramics with j,,(77 K) w 1 A/cm2 [57] (la’wer curve) at T = 50 and 77 K, respectively. Here, similar to the first case,

‘00- (b)

b.

‘O- t ,.

50 I’ WI

60

T (K)

Fig. 8. Temperature depeadences of the exponent p of an power-law current-voltage characteristic. (a) GdBa#+Oa-ceramics with j&77 K) e lo3 A/cm*, B = 0 [56]; dotted curve-fitting dependence /A- 1 oc TT4. (b) Bi-ceramics (phase 22 12) with j,(77 K) % lo3 A/cm’, B = 0 [60]; solid curvedependence Jo- 1 o( p.

70

76

E. Z. MEILIKHOV

m c

10'

IO2

103

101

B (G) Fig. 9. Magnetic field dependencies p(B) of the exponent of power-law current-voltage characteristics. Top curve.-Bi-ceramics (phase 2212) with j,,(77 K) m 10’ A/cm’, T = 50 K [58]; bottom curveYBazCu30a-ceramics with j,,(77 K) e 1 A/cm’, T = 77 K [57].

P $2:1 (and it is essentially dependent on the magnetic field) and to the second case, p >> 1 (and is

slightly dependent on the magnetic field up to B z 1 T). 8. CONCLUSIONS

The prospect of increasing the critical current density of bulk high temperature superconductors is tightly related to a solution of two problems, namely: elimination (or minimization) of “bad” intergrain weak links and creation of effective pinning centers in the bulk of superconducting grains. In this review we passed over the questions connected with a solution of the second problem (of vital importance!) and just tried to trace the interconnection between the properties of individual intergrain contacts of HTSC ceramics with macroscopic transport properties such as the critical current and the current-voltage characteristic. Revelation of this interconnection provides a better understanding of the physical essence of earlier described methods of raising the current carrying capacity of HTSC ceramics and to determine new promising perspectives of achieving this goal. Among these methods let us dwell on the following. Priority formation of intergrain contacts with a favorable “geometry “. Typical contacts of this type are small angle slope boundaries with the planes parallel to the c-axis. It is these contacts which have the highest (inherent in a single-crystal film) critical current density. It should be specially noted that the well-known method of ceramic texturing [61, 621 results in meeting only one of these requirements, for ceramics of this type usually consist of grains with (nearly) parallel c-axes but randomly oriented in the basal ab-planes. As a result, the larger part of intergrain contacts are either tilt boundaries (parallel to the c-axis) with a broad distribution of misorientation angles 9, or twist boundaries with planes perpendicular to the c-axis with a reduced critical current density. Hence, any further improvement of the properties of textured HTSC ceramics requires either a development of the production process with would ensure more complete ordering of individual grains (including their ordering in the basal plane) or, else, working out ceramics with “architecture” distinctions which would give the possibility achieving sufficiently high critical current densities even with relatively poor intergrain twist boundaries. The former has not yet given any visible result, while the latter is seemingly implemented in a number of liquid-phase methods of HTSC ceramics production. Creation of randomly nonuniform intergrain contacts. From the practical point of view, it is of primary importance (apart from raising an absolute value of the critical current density) to preserve high values of& in sufficiently strong magnetic fields. The only effective mechanism of limiting the rapid drop of the Josephson contact critical current with the rise of the magnetic field is the inherent (or induced in this or that way) random nonuniformity of the local critical current density JFB(x,y) which

Critical currents and current-voltage characteristics of high temperature superconducting ceramics

77

is seemingly exactly the reason for an observable plateau on magnetic field dependences j,,(B) for HTSC ceramics of different composition [35, 631. Ratio y of critical current densities in magnetic fields corresponding to this plateau and in the zero magnetic field is y % (ro/L)S2 for a 1D random function,/yB(x) [35#]and y % [G/q”2S for a 2D random fimctionjf’(x, r) [ 16,631 (S is a contact area). Here r,, and 6 are a correlation radius and m.r.s. relative fluctuation of this function, respectively. Hence one should aim for an increase of the 6 and r. values which are, naturally limited with values 6 g 1 and r. < L ( 1D nonuniformity) and P$ < S (2D nonuniformity). On the other hand, a magnetic field corresponding to the upper limit of the plateau of magnetic field dependence&(B) is inversely proportional to r, [ 351. It presupposes the necessity of a certain compromise in the choice of value r. (if, of course, we assume that there are technological possibilities for such a choice). Manufacturing of textured ceramics with a ‘ffavorable” architecture. An example of this kind of architecture is the ceramic with grains in the form of comparatively thin platelets (parallel to the abplane) arranged similarly to bricks in a wall. In this case the total contact area between the “bricks” (grains) belonging to different rows (contacts of A type) noticeably exceeds the total contact area between the grains forming a row (type B contacts). The ratio of these areas equals L/D >> 1, where 2L is a grain size (along the row) and D is its thickness. Hence, the lines of the current directed (on the average) along the rows will have the form of sinusoidal lines passing through type A contacts of larger area and by-passing type B contacts of smaller area. As a result the critical current density of the ceramics increases and becomes j,, x ~2;" > (L/D), where < ~2" > is an average (over the contact area) density of the critical current across the twist boundary (type A contacts). Thus, raising the L/D ratio one can increase substantially the current-carrying ability of the ceramics [6]. It should be borne in mind, however, that the efficiency of this method is limited by the 02”) decrease as the (contact length L rises because of the own magnetic field of the current: 0:“) cx 1,/L at L >> A, [16]. Hence any further improvements ofj,, will require a decrease of the grain thickness D. Finally, one more possibility of raising the critical current density consists of the purposeful variation of the intergrain boundary “chemistry”. An example of this approach is the investigation of Ag-doped ceramics (see Ref. [43] and the references therein). Small (several percent) Ag admixtures substantially increase (by several times) the critical current density of the YBa2Cu30a ceramics without changing its critical temperature. The mechanism of the phenomenon is still vague and needs ;Mher investigation.

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