Low-field electrodynamics of high-Tc superconductors — theory and experiment

Physica C 174 (1991) 109-116 North-Holland

Low-field electrodynamics of high-Tc superconductors - theory and experiment S.L. Ginzburg, V.P. Khavronin, G.Yu. Logvinova and I.D. Luzyanin Leningrad Nuclear PhysicsInstitute, Gatchina, Leningrad 188350, USSR

J. Henmann, B. Lippold, H. B/Srnerand H. Schmiedel Karl Marx University,Section of Physics, Leipzig 7010, Germany Received 6 September 1990 Revised manuscript received 29 November 1990

We have investigated, theoretically and experimentally, the penetration of magnetic fields into ceramic HTSC for fields below the lower critical field of the grains. In the framework of a critical state theory in the low-field electrodynamics of a random Josephson medium, expressions for the harmonics of the induction B(t) as a function of external parameters were derived for two asymptotic cases - the case of the weak field and the case of the thin sample. The behavior of the magnetic induction B was investigated in AC and DC fields in a wide range of temperature and field amplitudes. Rather good agreement between experimental and theoretical results has been found. From the experimental results the values of number of quantities characterizing the low-field electrodynamics of HTSC and their temperature dependences as well as an explicit expression for the critical current densityJc(H, T) =Jo (T) H~ (T) / ( H2 + Hoz(T) ) were derived.

1. Introduction T h e present work is d e v o t e d to the study o f the penetration o f low magnetic fields much smaller than the first critical field o f the grains themselves into ceramic high t e m p e r a t u r e superconductors. This p h e n o m e n o n , called low-field e l e c t r o d y n a m i c s o f HTSC, has been related to the fact that the ceramic H T S C represents a r a n d o m Josephson m e d i u m which behaves as a type-II s u p e r c o n d u c t o r [ 1,2 ]. F o r such a s u p e r c o n d u c t o r b o t h H¢~ a n d the q u a n t i t y He, playing the role o f Hc2, are d e t e r m i n e d by the Josephson current density a n d the grain size. Generally, one experimentally observes values for Hcl o f the o r d e r o f 1 0 - 3 0 e [ 3 ] a n d for H e o f 10 Oe [ 4 ]. To describe the p e n e t r a t i o n o f such low magnetic fields usually the concept o f the critical state theory is applied. Let us consider a slab o f thickness d along the xaxis a n d infinite in the d i m e n s i o n s along the o t h e r two coordinates. T h e external magnetic field h is directed along the z-axis. In this case the critical state equation has the form:

d

•

=4nj~(h),

x

jc(h)=ot(h)/(l~efrlhl).

(1)

T h e phenomenological function Jc(h) is proportional to the pinning force ot ( h ) a n d is called the critical current density. Here/~ff is the effective diamagnetic p e r m e a b i l i t y [ 1,2]. T h e function j¢(h) is the key q u a n t i t y in the critical state theory which is usually cast in some specific form, e.g.:

(a)j¢(h)=jo, • I-Io

(c)A(h)=Jo]~,

no (b)j¢(h)='~Ho+ Ih-------~l' H~,

(d).~(h)=JoH~+h 2 ,

where Ho is some characteristic field. G e n e r a l l y speaking there does not exist a choice o f the function jc(h) based on theory. Therefore, the necessity o f the e x p e r i m e n t a l d e t e r m i n a t i o n o f the form o f i t ( h ) is quite evident. In the present work we p r o p o s e two i n d e p e n d e n t m e t h o d s o f the i m m e d i a t e d e t e r m i n a t i o n o f the explicit form o f func-

0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

1 10

S.L. Ginzburg et al. / Low-field electrodynamics of HTSCs

tion jc (h). We succeeded in determining an evident aspect ofj~(h) in agreement with case (d). The most essential feature of the critical state is irreversibility. This leads [1,2,5 ] to the occurrence of points of inflexion in the induction distribution B(x) and also to hysteresis of this quantity and its mean magnitude/~ in the sample. A detailed consideration of hysteresis has been performed, for instance, in ref. [2 ]. Let a DC field H and an AC one with amplitude ho be collinearly applied to the system under study, i.e. H(t)=H+hocos(tot), then the induction /~(t) may be expanded in a Fourier series:

cause when the condition d<< l is satisfied, the field change over the sample dimension is always less than either H or Ho. In this case jo(H(t) ) can always be expanded in terms of the AC field amplitude ho and can be approximated just by the zero term of this expansion. We note that the weak-field concept is applicable to a thick sample as well as to a thin one. For the weak-field case expressions for the harmonics will be [2,5]:

B( t) = a o / 2 + ~ [ancos nwt+bnsin noJt] .

b2k÷l =

The harmonic values a~ and b, depend on the relationships between the four fields ho, H, Ho and h2, two of them (ho, H ) being external and Ho and h2 being determined by the properties of the superconductor itself. Here the field h2 = 2ndjo is determined by the sample dimensions too. Although theoretically one may consider the situation having arbitrary relations between these fields, nevertheless we distinguish two simple asymptotical cases for which, as we understand, it is rather appropriate to compare with the results of the experiments. Usually, to describe low-field phenomena in ceramic HTSC, an explicit form of the phenomenological function j¢(H) is used and then from experiments the parameters of this function are determined. This approach is an integral one. However in the present work we employ a differential approach, namely, we choose two such asymptotic cases in which the measured quantities are defined by the value o f j ~ ( H ) , where H is the applied DC field. It allows one to determine directly the explicit form of j~(H) from the experimental data. When h0 << H, Ho, h2, we call this the case of weak field. Note that there can be any relationship between H, Ho and h2 in this asymptotic case. The other asymptotic case is determined by the conditions ho~h2(H) and h2(H)<> l. In the thin-sample case the magnetic field gradient can be assumed to be independent of x, be-

/~effh2

a l - 4njc(H)d'

a2k+l= 0

k>~ 1 ,

/2¢ffhg 8n2jc(H)d 1

× (k2_l/4)(k+3/2),

k=0,1,2,... ,

a2- 32nd dH b2k+,=

aEk=0

k>~2,

/2~frh3 ~ ' ~ - - ~ }d 1 6 n E d - ~

k × (kZ_l/4)(k2_9/4),

k=1,2,3 .....

(3)

One can show that for the case of a thin sample when ho is of the same order as h2, but ho, h2 << Ho the dependence of susceptibilities a2k+~/ho and b2k+~/ho on T a n d H has scaling character and is represented by a universal function of the dimensionless variable y=ho/h2(H). Here we present two examples of such dependences for Z'~',X~ and X~ on y: For y < 1 2 4nZ~ = 0 , 2 4nZ~ = - 15-~/2~f~y. For y > 1 2

1

4~X't' = ~/tefr(y - ~y2) , 32 ( y - 1 )2/2 4~Z~ = - ~ #err y4 ,

S.L. Ginzburg et al. I Low-field electrodynamicsof HTSCs

4~Z~ = -

2 /16 40 30 15~-/teffl---Tyky~-- ~-2 + - - - - 5 ) . Y

(4)

We see from (4) that these quantities are oscillating functions of y, for instance, the polynomial in X~ has two extremes (fig. 1 ). It should be noted that the oscillations of the even harmonics as a function of DC field were observed in ref. [6].

2. Experimental The aim of the experimental part of our paper was the study of the AC and DC magnetic properties of ceramic HTSC and the comparison of the data obtained with the theoretical conclusions. To realize experimentally the different ratios between ho, H, h2 and Ho including the asymptotic cases of the "weak field" and of the "thin sample", various studies were performed within a wide range of temperatures and external fields. As a result of these studies it was found that the theory considered above describes rather well the phenomena observed. Therefore, it was possible to determine a number of important parameters characterizing the penetration of the magnetic field into a superconductor, namely the analytical expression 4qL'~3

of the function j c ( H ) and the dependence of#eel on T. It was found that j c ( H ) for the two asymptotic cases of the weak field and of the thin sample has the same type. The data obtained also allowed to determine the temperature dependence ofjo, h2 and Ho. Experiments were carried out on ceramic Y - B a - C u O ( p ~ 5 g / c m 3) prepared according to the method described in ref. [7]. Dependent on the investigation method various shapes of samples were used. The measurements of the static susceptibility XDC were taken with a nonstandard SQUID magnetometer [ 8 ], using the UJ I 11 type sensor and a first order gradi0meter. The measurements ofx' and %' have been carried out at frequencies of 20 and 100 kHz while those of the higher harmonics c , = ( a 2 + b E) ,/2 were taken at the fundamental frequency of 20 kHz. The experiments with the AC field have been carried out for 10 -20e~
(a.u,)

c 3/h o

-2

1,2

10

(G/Oe)

8

1

b

10-3

0.8 0.0

111

/ -4

IO

0.4

0.2

0

-0.2

0.1

i ,

...........

I0-6 i l l l l

illl 1

10 y= h o / h =

100

1000

Fig. I. Dependenceof the third harmonic of susceptibility on the dimensionless variable y=h0/h2(H) derived from eqs. (4): -4~X; (curve 1 for y< 1 and curve 3 for y> 1); -4nX'3 (curve 2); 14~X31=4g [ (X~)2+ (X;) 2] ,/2 (curve 1 for y< l, curve 4 for y>l).

10-0

i aa

~.6

i

I

i

I

i

so

so,6

oo

oo.6

ol

ol.6

Temperature (K) Fig. 2. Temperature dependencesof c3(T)/ho, obtained in zero DC field H for different AC field amplitudes ho (Oe): 1.0 (a); 0.5 (b);0.2 (c);0.1 (d);0.05 (e);0.02 (f);0.01 (g).

112

S.L. Ginzburg et al. / Low-field electrodynamics of HTSCs

note that two maxima were also observed in the dependences c5(T) ~he and b~(T) ~he= 4gX" (T). Such behavior of harmonics is related to the fact that in reality the picture of the critical state considered above is complicated by the penetration of the field into the grains themselves, which happens near Tc for the fields considered. According to current ideas [ 10,11 ] the high temperature m a x i m u m is due to the field penetration into the grains and the low temperature maximum is due to the field penetration into the intergranular medium formed by the weak link network. The temperature dependence of the static susceptibility indicates the existence of two mechanisms for the field penetration into ceramic HTSC, too. The dependence ZDc(T) (fig. 3) exhibits a point of inflexion at 89 K that divides two temperature regions corresponding to the maxima in c3(T)/he. Let us now directly turn to the results that have been obtained in the weak-field case: he<< h2, He, H. In this asymptotic case one can compare the experimental data with the theoretical expectations in a simple way and thus determine such important parameters characterizing the field penetration into superconductors as Jo and the form e r i c ( H ) . From the field dependences obtained for al, bl, ¢3 and c5 at H = 0 for each temperature point the coefficients at he2 were calculated (fig. 4). It should be noted that the ratios a~/b~, b~/c3 etc. within the temperature interval from 78 to 89 K practically do not

4'lCXdc (G/C~)

-0.2

-0.4

-0.6

-0.8

-1

I

82

84

i

I

80 88 Temperature (K)

i

90

92

Fig. 3. Temperature dependence of the DC susceptibility (after zero field cooling) in a field of 0.40e for cylindrical sample.

2

2

2

81 /h0 b t / h o ,C2k*l/ho

(G/Oe 2)

al/bl

-1

10

-2

10

10-8

-4

10

3 2 -5

10

1 0 77 78 79 80 81 82 88 84 86 80 87 88 80 gO Temperature (K) i

i

i

i

i

i

i

i

i

i

i

,

Fig. 4. Temperature dependences of a=/ h 2 ( 1 ), b=/ h ~ (2), c3/ h 2 (3), c5/h2o (4) and ofa~/b~ for zero DC field H.

depend on temperature despite the change of their values over about four orders of magnitude. This once again confirms that in these experiments the weakfield case was realized. Indeed, #~ff(T)/jc(H, T) enters all the eqs. (3) for a~ and b2k+l. Then it is obvious that the ratios a~/bl, b~/c3 etc. must depend neither on #elf(T) and jc(H, T), nor on T consequently. We note that the obtained ratios of the harmonics are close to those calculated from eqs. (3) and practically coincide with the results of ref. [ 12 ]. The value of/zefr was determined from the high field asymptotic (h >> He, hE) of the hysteresis loop or of the magnetization curve obtained with the SQUID magnetometer. The dependence/z~f(T) obtained this way is shown in fig. 5. One can see that the strongest change of/zeff takes place in the vicinity of Tc. As was mentioned before, the function j¢ (H) is the basic quantity of critical state theory. To determine the evident aspect of this dependence the method suggested in ref. [ 12 ] was employed in this work. The essence of this matter is the following. According to eqs. (3) in the weak-field case the odd harmonics should be proportional to 1/j~(H) and the even ones to d ( l / j ~ ( H ) )/dH. Therefore, studying the harmonics dependences on the DC field at a fixed he value, it is possible to determine the evident aspect o f j c ( H ) in a self-consistent way. In the present study such experiments were carried out for the third

S.L. Ginzburget aL / Low-field electrodynamics of HTSCs

~et~ C 3 / h 2 "10

1

4

113

(G/Oo 2)

25

o.g

a) 0.8

20

0,7 o.o

15 0.5 0.4

10 0.3 0.2 o.1 o 79

L

I

L

L

t

I

81

83

85

87

8g

gl

Q3

0

I

Temperature(K)

0

Fig. 5. Temperature dependence of the effective permeability/Art as determined by the magnetization measurements. The solid line is only a guide for the eyes.

and second harmonics. The interval of the AC field amplitude h0 was selected in such a way that the square dependence of c3 on ho and the cubic one of c2 could be observed. As a result of these studies it was found that for the DC field changing from 0 to 2 . 5 0 e the amplitude of the third harmonic c3 (i.e. l / j c ( H ) ) was growing like H 2 (fig. 6 ( a ) ) . At the same time the amplitude of the second harmonic c2 (i.e. d(1/j~(H))/dH was linear dependent on H (fig. 6 ( b ) ) . From this was drawn the important conclusion that the dependence jc(H) has the form corresponding to the case (d) in eqs. (2). H 2( T )

jc(H, T)=jo(T) H2(T)+H2.

2

H2

c2/rl 3 ,105

I

(002)

6

(G/Oe 3)

50

40

30

20

10

0

(5)

I

0

I

I

I

I

I

0.5

1

1.5

2

2.5

3

H (Oe)

Using eq. (5) it was found that the characteristic field Ho changes from 3 to 1 . 8 0 e in the temperature interval 7 8 4 T~<86 K (see below fig. 11 ). So far as known to us this kind of dependence (5) was observed neither for'the classical superconductors nor for the ceramic HTSC yet. Reference [ 12 ] is the unique work that determined the explicit form of the function j~(H) in the low-field electrodynam-

Fig. 6. Dependencesofc3/h 2 on H 2 (a) and ofc2/h~) on H (b) at T=83.9 K and h0=0.25 Oe.

ics. However, the dependence j¢ ( H ) obtained in ref. [12] agreed with the mod~l (d) in eq. (2), i.e. jc(H) =joHo/( IHI + H o ) . The reason for the different forms of the function j¢(H) is not clear so far. We note that the dependence (d) had been observed

114

S.L. Ginzburget al. / Low-fieldelectrodynamicsof HTSCs

earlier in the high-field range for classical superconductors [ 13 ]. The next set of experiments was performed to study the behavior of the susceptibility in the case of the thin sample, i.e. when the condition d<>Ho. However, the maximum applicable field H was of the same order as Ho in our experiments. Therefore, we could carry out the experiments only for sufficiently low values ofjo, i.e. relatively close to T¢. As follows from eqs. (4) in the asymptotic case of the thin sample at h0 ~ h2 << Ho the dependence 4~1Z3 (Y) I must have a maximum (fig. 1 ). This maximum, characteristic for the case of the thin sample, is clearly visible in the temperature dependence of c3(T)/ho in fig. 2. The dependence of the modulus of the third harmonic c3 on the AC field amplitude ho was studied for various values of the DC field in the temperature range where the thin-sample case was expected to be realized. Typical dependences c3 (ho)/ho = 4n IZ3 (ho) I at different DC fields (for T = c o n s t ) are shown in fig. 7. As can be seen these dependences at some field amplitude ho= hr, axhave a maximum. At the same time hmax depends on H and on T.

We introduced the new dimensionless variable u=ho/hmax and recalculated the quantities of fig. 7 dependent on u (fig. 8). Figure 8 shows that all dependences nearly match each other, except for small field values. Thus, for large values of the DC field we found scaling behavior of the susceptibility. For small fields j¢(H) ~Jo and it turns out that in this case the condition 2ndjo = hE << H 0 is not satisfied. Indeed, the extrapolation of the Ho values obtained in the weakfield case into the temperature region just below T¢ gives the value H o ~ 1.50e. On the other hand from the same data forjo it follows that at T ~ 89.6 K the value for h 2 ~ 1 . 5 0 e is comparable with Ho, i.e. at H = 0 the thin-sample criterion 2rtdjo= hE << H o is not satisfied. However, with increasing H the value j~ ( H ) and consequently h2(H) is falling down and thus the sample may become a thin one. To verify this assumption we compare the dependencies c3(ho)/ho (fig. 9 ) obtained at various temperatures for the two values of the DC field H = 0 (curves 1-3) and H = 1.95 Oe (curves 4-8). One can see that in contrast to the case H = 0 the dependences for the higher value of the DC field are matching each other rather good. From our point of view this is a very convincing argument for the presence of the thin-sample case, where all susceptibilities are universal functions of just one variable u=ho/h . . . . Earlier, when discussing the dependences c3 (ho) / ho, it was supposed that h2=flh . . . . i.e. y=flu where fl is some coefficient. A more correct way may be to

cS/ho 0.035 08/ho

(G/Oe)

(a.u.)

]

0.0~

,o,8

0.025 0.6

0.02 0.015

o.

0.01 o, 0.006 0 0.01

|

*

'

'

'

,At

I

i

,

,

,

,

**

0,1 AC Field Amplitude h o (Oe)

Fig. 7. Dependences ofc2(ho)/ho on ho for H (Oe)=2.33 (a); 1.95 (b); 1.55 (c); 1.17 (d); 0.78 (e) and 0.0 (f) at T=89.6 K.

0

I

I

i

I

i

I

I

I

i

1

2

3

4

5

6

"/

8

9

,

lO

ho/h~.

Fig. 8. Dependencesofc3(ho)/ho on ho/hm~for H (Oe) =0 ( 1); 0.78 (2); 1.17 (3); 1.55 (4); 1.95 (5); 2.33 (6) at T=89.6 K.

S.L. Ginzburg et al. / Low-field electrodynamics of HTSCs

c3/ho

(hm,ax(O)

(a.u.)

115

- h ~ x ( H ) ) / h ~ x (H)

I

4.8 l

4.4 4

36

0.8

3.2

8-'8 1

0.6

2.8 2.4 2

0.4 1.0 1.2 0.2

0.8 0.4 I

2

3

4

I

I

I

i

i

5

6

7

O

9

i

0 I0

h o /hm= Fig. 9. Dependences of c3(ho)/ho on ho/hm=x for two values of the DC field H: H = 0 (curves 1-3) and H = 1.95 Oe (curves 4 8). Curves 1-3, 4-6 correspond to T=89.1, 89.3, 89.6, 89.1, 89.3, 89.6 K and curves 7-8 to T=89.8, 89.9 K, respectively.

o

i

0.4 0.8

i

i

i

1.2

1.8

2

i

i

i

i

2.4 2.8 3.2 3.6

i

4

i

i

i

8

H 2 (Oo 2 )

Fig. 10. Dependence of (hm,,(O)-hm=,(H))/hm=~(H) on H 2 where hm=,(0) refers to H = 0 and hm=,(H) to a finite value of H.

h2. H 0 (Oe)

determine the value fl from the dependences c3 (ho) / ho obtained in an overlap region. That means when the AC field amplitude is changing in a rather wide range it is possible to observe simultaneously both the maximum of these dependences and the weakfield case (i.e. c3och2). As it turned out the temperature overlap region where this situation occurs is extremely small ( ~ 0.4 K). Having determined in this case the coefficient at h E from the dependences c3 (ho) and then - taking into account #eel - also using the value Jo (from eq. ( 3 ) ) it is possible to estimate h2=2~djo. Assuming on the other hand h E = flhmax it was found that fl= 1, i.e. hE is nothing else than h . . . . According to the observed scaling behavior of the susceptibility and assuming that hm,~(H) = h2(H)=2~djc(H) the explicit form of the dependence j¢(H) in the temperature region where the sample behaves as a thin one can be determined. As can be seen from a typical example of the dependence (fig. 10), hm~ linearly depends on H 2. Thus in the thin-sample case the dependencejc(H) has the same form as in the weak-field case, i.e. jc ( H ) =jo H 2 / ( H 2 -I- H 2 ). The value Ho determined in this case is nearly equal to 1 . 3 0 e . Thus the quantity Jo was determined for two asymptotic cases: the case of the weak field and the case of the thin sample (fig. 11 ). However, it is con-

i

4.4 4.8 6.2 6.0

Jo (a/cm2)

IOO0

500

50 ¸

h2

100

10

0.5

1

0,05

0.1

0.01

,

t

J

=

,

J

I

i

i

J

J

i

i

0,008

78 79 80 81 82 80 84 86 80 87 88 89 90 01 92 Temperature (K)

Fig. 1 I. Temperature dependences of the characteristic fields h2= 2ndjo and Ho ( × - Ho; • and Q - h2 taken for the weak-field and for the thin-sample case, respectively). The value o f Ho at T ~ 90 K was taken from the dependence of h , , , (H). The Jo dependence refers to the right y-axis.

venient to consider not the dependence jo(T) itself but h2 (T) because in this case it is possible to imagine more vividly the relationships between the characteristic fields ho, H, hE and Ho and this way the conditions of realization of one or the other asymptotic case. Figure 11 shows, first, that the characteristic field Ho depends on temperature only very slightly. Second, at T = 89 K ]/2 is approximately equal to Ho and

116

S.L. Ginzburg et al. / Low-field electrodynamics o f HTSCs

consequently at H = 0 the thin-sample case in our studies could only be observed at high temperatures. Third, within the temperature range from 78 to 90 K the current density Jo (numerical values ofjo may be obtained by multiplication of h2 by 1/ (2~d) ~ 0.5 c m - ~ as shown in fig. 11 ) changes for almost four orders of magnitude and this change of Jo becomes more rapid near To. This explains the narrow temperature interval in which the case of the thin sample was observed. Thus, from the experimental results obtained in a wide range of temperatures and of external fields with application of different methods one can conclude that the low-field theory of the critical state is a proper tool to describe the penetration of the magnetic field into granular HTSC.

Acknowledgements This work was supported by the Scientific Council on the Problems of HTSC and carried out within the framework of Project No 2 "Shkala" of the State Program of the USSR "High-temperature superconductivity" and also of the HTSC program at the KarlMarx-University Leipzig, Germany. It is for the authors a pleasant duty to thank S.V. Maleev, A.I. Oko-

rokov, J. Schreiber, B.P. Toperverg, A.I. Sibilev and S.M. Bezrukov for constant interest, for helping to perform the experiments and having discussions of results.

References [ 1] J.R. Clem, Physica C 153-155 (1988) 50. [2] S.L. Ginzburg, Preprint LNPI-1615 (1990). [3] B. Loegel, D. Bolmont and A. Mehdaoui, Physica C 159 (1989) 816. [4 ] H. Dersch and G. Blatter, Phys. Rev. B 38 ( 1988 ) 11391. [5] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [6] K.-H. Muller, J.C. MacFadane and R. Driver, Physica C 158 (1989) 69. [ 7 ] H. B~rner, M. Wurlitzer, H.-C. Semmelhack, B. Lippold and R.-M. B6ttcher, Phys. Status Solidi (A) 116 (1989) 537. [ 8 ] B. Lippold, Proc. 16. Int. Sympos. Tieftemperaturphysik u. Kryoelektronik, Bad Biankenburg (1984) p. 133. [9] I.D. Luzyanin and V.P. Khavronin, Zh. Eksp. Teor. Fiz. 85 (1983) 1029 [Soy. Phys.-JETP 58 (1983) 599]; ibid., 87 (1984) 2129. [101 R.B. Goldfarb, A.F. Clark, A.I. Braginski and A.J. Panson, Cryogenics 27 ( 1987 ) 473. [111 V. Calzona, M.R. Cimberle, C. Ferdeghini, M. Puni and A.S. Ski, Physica C 157 (1989) 425. [12l I.D. Luzyanin, S.L. Ginzburg, V.P. Khavronin and G.Yu. Logvinova, Phys. Lett. A 141 (1989) 85. [13] Y.B. Kim, C.F. Hempstead and A.R. Strnad, Phys. Rev. 131 (1963) 2486.