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Activation energies in superconducting high temperature ceramics
MATERIALS SCIENCE & ENGINEERING ELSEVIER
B
Materials Science and Engineering B34 (1995) 132 137
Activation energies in superconducting high temperature ceramics A. Mehdaoui, D. Berling, D. Bolmont, B. Loegel Laboraloire de Physique et de Spectroscopic Electronique, URA 143.5, Fa¢'ult~; des Sciences et Techniques, Unirersit[ de Haute Alsace, 4 rtw des ./?bres Lumii, re, 68093 Mulhouse COdex, France
Received 11 October 1994
Abstract
Low field a.c. susceptibilities of H g B a 2 C a 2 C u 3 O s + a (Hg-1223) and Bi2Sr_,CaCu2Os+,, (Bi-2212) have been studied. The loss peak of Z", relevant to intergranular pinning, shifts to higher temperatures for increasing frequencies. The results can be fitted with an Arrhenius law, leading to activation energies E~, ~ 1 eV, but the hopping frequencies fi~ are not realistic. More realistic values can be obtained by introducing a Fulcher law or a power law which is also characteristic of strongly interacting systems. K e y w o r d s : Superconductivity perovskites: High t e m p e r a t u r e superconductivity; Meissner effect; Magnetic m e a s u r e m e n t s
I. Introduction A.c. susceptibility provides a powerful tool for characterizing high temperature superconductors (HTSCs) and has been extensively used because of the contactless measurement possibilities. A.c. susceptibility provides simultaneously the real part Z' (diamagnetic shielding) and the imaginary part Z" of the complex susceptibility Z = Z' + iz"- The imaginary part Z" of the a.c. susceptibility is relevant to the energy losses arising from the hopping of vortices between pinning sites. This measuring technique can thus be used for characterizing the Meissner effect and for determining the irreversibility line and the same technique can moreover give the activation energy involved in the above mentioned hopping processes. The hopping of vortices results in a non-linear relationship between the induction B and the magnetic field H, and the energy losses are proportional to the area A of the B - H loop. As the penetration depth 2L increases with temperature for T < ice, whereas pinning strength weakens with increasing temperature for T > Tp, Z" exhibits a peak at some temperature Tp at which A reaches also its highest value. Increasing the area of a B H loop leads also to an increasing distortion of the a.c. signal taken at the secondary of a Hartshorn bridge and actually the strongest distortions are observed for T = Tp [1].
The high sensitivity of Tp to a.c. and d.c. magnetic field amplitudes has been extensively used to investigate the irreversibility lines for HTSCs, both in the (/1..... T) plane and in the (Hat, T) plane. The frequency dependence of Tp gives also information about the pinning mechanisms. It is the aim of this paper to present some experimental results obtained for Hg-1223 and Bi-2212 HTSCs and these are discussed in the framework of a simple critical state model. The measurements under consideration here are done with zero d.c. magnetic field and the a.c. magnetic fields are of small amplitude and low frequency. In a first approximation we assume therefore that we are in a linear regime allowing thus the use of Kim- Anderson's formulation in the following discussion.
2. Pinning potential and activation energy Applying an a.c. field involves hopping through the Lorentz force F acting on the vortices. Assuming that either the field or thermally induced hopping frequency f for intergranular Josephson vortices is given by Anderson's formula (taking into account forward and backward hopping), this leads to
j ,i,le,p ( 0921-5107/95/S09.50 t; 1995
Elsevier Science S.A. All rights reserved S S D I 0921-5107(95)01249-4
A. Mehdaoui et al. / Materials" Science and Engineering B34 (1995) 132 137
-- exp
3. Experimental details
kT
where J0 is a characteristic hopping frequency bro ~ 10v H z - 1 0 ~3 Hz), r is the pinning potential range and V is the flux bundle volume. The intergranular pinning potential U(T, H) can be taken as [2]
U(T, H) = Uo(T)Ho/2(IHI + Ho/2)
(2)
assuming that U(T, H) is proportional to the intergranular Josephson current and where [3]
Uo(T) = Uo(0)(1 - T/Tc) ~
(3)
(:~ = 1 for superconductor-insulator-superconductor grain boundary junctions and ~ = 2 for superconduct o r - n o r m a l - superconductor grain boundary j unctions) and where Ho is a parameter including the flux quantum 4~0 and the average grain size. The critical state in type II superconductors is reached when the Lorentz force F balances the pinning force Fp. Assuming large enough Lorentz force density and therefore neglecting the backward hopping contribution in Eq. (1), Fp can be taken as
Fp = (1/Vr)/[U(T, H) + k T In(f/f,)]
(4)
Assuming Bean's model where the applied field enters the centre of a cylindrical sample for T = Tp, Eq. (4) can also be written in the form of an Arrhenius law:
f /J; = exp[(Fp Vr - U ( T, g ) / k Tp]
(5)
Arrhenius laws have already been reported for the frequency dependence of the susceptibility maxima in diluted magnetic systems (for an extensive discussion of a.c. susceptibility in dilute magnetic systems see Ref. [4] and for a detailed discussion of Arrhenius-like behaviour in spin glasses and HTSCs see Ref. [5]) and in amorphous superconducting (RE)2Co amorphous alloys (RE = rare earth) [6]: the frequency-induced shift, (Sf=ATp/Af is, for example, 0.06K decade ~ in Er65Co35 [6] and 0.23 K decade ~ in Cuo.954Mno.o46 [7]. The peak temperature can be fitted in both cases with an Arrhenius law:
fifo = exp( - Ea/kT)
(6)
where Ea is the activation energy involved in the transition processes (e.g. spin flip in spin glass and vortex hopping in (RE)2Co). Comparing Eqs. (5) and (6), the activation energy is given by
E~ = U(T, H)/Vr)Fp
133
(7)
and leads for example for Cu~ x Mn,. to activation energies E, ~ 5 x 10 3 eV [7]. We will discuss below the results obtained from similar analyses for Hg- and Bi-based superconducting compounds.
The zone-melted Bi-based samples used in this study were fabricated in the laboratory of the department of Physics of the University of Technology at Eindhoven (Netherlands). Solid state reaction was achieved by using Bi203, CaCo3, Sr(OH)2, 8H20 and CuO powders. The sintered pellets (50 h at 870 °C in 02 atmosphere) of nominal composition Bi2Sr2CaCu2Oy (Bi-2212) were subsequently zone melted; the molten zone travelling speed was about 3 cm h-~ (for more details see Ref. [8]). The zone-melted samples exhibit a density d = 5.4 g cm 3 which is 2.5 times higher than that of the sintered samples. X-ray diffraction (XRD) spectra show that the samples are mainly single phased (c = 3.06 nm). The standard a.c. four-probe method gives To(p = 0 ) = 90 K and a transition width (between 10% and 90'}/0) A T c = 2 K . The critical currents Jc were 102A cm 2 at 50 K and 104A cm -2 at 20 K [8]. The sintered Hg-based samples were fabricated in the Laboratory for Crystallography at Grenoble (France) by high pressure and high temperature solid state reactions. A precursor with nominal composition Ba2Ca2Cu 3 is first prepared and HgO is subsequently added. The mixture is ground and placed in a gold capsule adapted for high pressure synthesis. The high pressure high temperature reaction is carried out in the following manner: the pressure is first increased to 18 kbar in 20 min, the temperature is then raised to 880 °C in 40 min and temperature and pressure were kept constant for 1.5 h. After treatment, pressure and temperature were decreased to normal in 20 min (for more details see Refs. [9,10]). Lattice parameters and impurity concentration were checked with a Guinier focusing camera. The reflections of the nominal HgBa2Ca2Cu3Os+ ~ (Hg-1223) were indexed on a tetragonal cell with lattice parameters a = 0.38536 nm, c = 1.5796 nm. The purity of the Hg-1223 phase is at least 95%, the main impurities corresponding to the Bi-1234 phase. The resistive transition exhibits a sharp drop between 130K and 135 K and preliminary a.c. susceptibility measurements reveal the occurrence of diamagnetic shielding at T = 133 K. The samples used in the present work all have typical sizes of a few cubic millimetres. Our susceptibility measurements were achieved on a Hartshorn bridge using a.c. fields 0.01 Oe < hac < 30 Oe in the frequency range 33 Hz < f < 12 000 Hz. Details of susceptibility measurements have already been published elsewhere [11]. Figs. 1 and 2 show examples of the results we obtained for a sintered Hg-1223 sample at a.c. fields of 5, 10, 20 and 30 Oe for frequencies f = 330 Hz and f = 1100 Hz. We observe a frequency-induced shift for Tp which is much higher than those reported for spin glasses [7] or amorphous low temperature superconductors [6] and
A. Mehdaoui et al. ,' Materials Science and Engineering B34 (1995) 132 137
134
X
T (K)
" (arb. units)
P
135
130
125
.... I .... I,~lr~l~L~l~l~l~ 100
105
110
115
120
125
120 130
135
140
T (K)
Fig. 1. The imaginary part Z" of the susceptibility for Hg-1223 at 330Hz for h~,~=5Oe (O), h~,~= 10Oe ( I ) , h~,~=20Oe ( t ) and
h~,, = 30 Oe (A).
our results for the whole frequency range are summarized in Fig. 3. The experimental results are restricted to lower frequencies for increasing a.c. fields, owing to the increasing impedance of the coils with frequency. Fig. 4 displays plots on a semilogarithmic scale of frequency vs. Tpl, showing that the results agree with an Arrhenius-like behaviour, allowing therefore an estimation of the activation energy E, for each applied a.c. field. The same qualitative features are obtained for zone-melted Bi-2212 polycrystals. Fig. 5 summarizes the results obtained for both Hg-1223 sintered polycrystal and for Bi-2212 zone-melted polycrystal. The activation energies obtained in both cases increase from some tenths of an electronvolt in the highest a.c. field up to some electronvolts in the low field limit. Our results are also in agreement with those obtained on superconducting high temperature polycrystalline sintered ceramics [2,12,13] where Miiller and coworkers obtain pinning potentials in the electronvolt range for
X " (arb. units) ''~1""1''~'
,,I 100
J''''l'~''l""l''
'1 '
. . . . I,rrlllllElllrlllrll
105
110
115
120
'~
irlll 125
130
135
14.0 T
(~)
Fig. 2. The imaginary part Z" of the susceptibility for Hg-1223 at l l 0 0 H z for h ~ ¢ = 5 O e (O), h ~ = 10Oe (m), h , ~ = 2 0 O e (!l,) and h~,~.= 30 Oe ( A ).
115
110
0
2ooo
400o
6ooo
soo0
1oo00 1200o f (Hz)
Fig. 3. The peak temperature Tp vs. frequency for several amplitudes of the a.c. field ha~: O, h,~ = 0.10e; II, h~ = 10e; ~, h,~ - 50e; A, hd~= 10 Oe; ~), h~,~= 25 Oe. The lines are guides to the eye.
lead-doped Bi-based HTSCs (Tc(p = 0) = 103 K) for Tc ~ 0.8 in applied a.c. fields ha~ of some oersteds.
T/
4. D i s c u s s i o n
Fitting the experimental results for HTSCs with an Arrhenius law (6) leads usually to activation energies Ea in the electronvolt range, which may be considered as acceptable, but, as was already pointed out for spin glasses [7] and for lead-doped Bi-based HTSCs [12], such an analysis gives also unrealistic f j values for HTSCs. We obtain 1022 Hz < f j < 101~° Hz for Bi-2212, and 1017Hz
A. Mehdaoui et al.
Materials Seience and Engineering B34 (1995) 132 137
f (Hz)
135
f (Hz)
10 5
10 4 ' ' ' ' 1
10 4
h ae =0.I
o.
Oe
....
J ....
I
~
....
I'''
hac=l Oe
103 10 3
,I,,,[,,,I,,,IJ,,I,,
102 7.60 10 -3
7.66 10"3
7.72 10 .3
7.78 10"3
T-1 (K-X)
102 7.70 10 "3
f (Hz)
7.82 10 .3
7.94 10 "3
T "1 (K"l)
f (I-Iz)
104
10 4 '
h a¢ =$
I
'
I
'
I
'
I
'
I
~
Oe
'
I
ha¢ =10
•
103
I
'
I
'
I
'
Oe
103
zl,,I,,l,,I,,I,,I,,I,,I,,I, 102 7.80 10 .3 7.59 10 "3 7.98 10 "3
102
8.07 10 .3
(T "1) (K"1)
~ I r I ,I
7.85 10 .3
,
8.00 10 -3
B J I,I
, 1 1 1 , I J
8.15 10 -3
8.30 10.3
T "1 (K"t)
Fig. 4. Arrhenius plots of f vs. 1/Tp for Hg-1223 for different a.c. fields h.~.
by
f / f * = [T/(T-- T*)I ~
(8)
Such a law applies well to spin glasses and other glasses [15] where the T* values are close to the experimentally determined critical temperatures and where the f * values are of physical significance (106 Hz < f * < 1014 Hz). The values determined experimentally for the exponent ~ are 4 < ~ < 11. Introducing a Fulcher law replaced Eq. (6) by
f/Jo = e x p [ - E , / k ( T -
To)]
(9)
Ea (eV) 3.5
....
'"'[
' ''"'"1
' ''"'"1
' ''"'"1
, ,,r,
3 2.5 2 1.5 -
II B
1 0.5 ......
0 10.3
I
10.2
, ,,~m,I
10"1
BBAA , ,, ..... I , , ...... I ~ l r l 10° 101
lO: h ac (Oe)
Fig. 5. Activation energies E, (eV) vs. applied field h,c for Hg-1223 (A) and Bi-2212 (B). The analysis was achieved by using an Arrhenius law.
where To (which characterizes the temperature where an order builds up) approaches Tp as the strength of the interaction responsible for ordering increases. Such a fit leads for example in C u M n spin glasses to activation energies E, similar to those obtained from an Arrhenius law, but gives also more realistic values off0 ~ ~ 1013 Hz) [7]. We therefore have also analysed our experimental results in the framework of the two above-mentioned phenomenological models. (a) Power law. Fitting our results with a power law leads to the results illustrated by Figs. 6 and 7 which give J* vs. T* for various applied a.c. fields. Assuming an average value of ~ = 9 (similar to those deduced from experimental results of Ref. [15]) we obtain 105 Hz < f * < 1011 Hz for Hg-1223 and 10 9 Hz < f * < 1013 Hz for Bi-2212. Tables 1 and 2 give the corresponding values obtained for T* in different applied a.c. fields. The zero field limits are T*(h,c--+O) = 127.1 K for Hg-1223 and T*(h,~--+O) = 88.2 K for Bi-2212. (b) Fulcher law. In such an analysis we first set the frequency range by assuming that realistic values for f0 correspond to those of the hopping processes introduced in Anderson's formulation (10 7 Hz < J ; < 1013 Hz). With such a hypothesis we obtain average values of To = 100 K + 20 K for Hg-1223 (corresponding to To/T c = 0.74) and To = 70 K 4- 10 K for Bi-2212 (corresponding to To/T c = 0.77). Tables 1 and 2 give also the values obtained for To in different applied a.c. fields when a Fulcher law is introduced. The zero field limits obtained in that way are To(h,~--+O)= 128 K for
136
A. Mehdaoui et al. / Materkds Science and Engineering B34 (1995) 132 137
f* ( H z )
5. C o n c l u s i o n I
I
I
1015 _
t
1013
1011
---q
10 9 ~107 r'-
4
? 10 5 b-
I,
I
I
I
40
50
60
70
80
....
90
T* (K)
100
Fig. 6. Hopping frequency .1" vs. characteristic temperature T* obtained for Bi-2212 from a fit with a power law. The curves are displayed for the following a.c. fields hd~ (Oe) from top to bottom: 0.01, 0.05, 0.1, 0.5, 1, 5, 10.
Hg-1223 and To(h,~--+O) = 85 K for Bi-2212. The corresponding activation energies are strongly reduced in such an analysis: Fig. 8 shows the temperature and a.c. field dependences of the activation energy when a Fulcher law (8) is introduced. Taking the abovementioned To values this leads roughly (at constant a.c. field) to a decrease by a factor 20 for both Hg-1223 (To = 100 K) and Bi-2212 (To = 70 K) and Fig. 9 gives the variation in E~ vs. h,~ for both systems. We notice that the activation energies obtained for both compounds are of similar magnitude whatever the phenomenological model is. This implies that the intergranular coupling strengths for sintered Hg-1223 and zone-melted Bi-2212 are also equivalent. f* ( H z )
1015
I
~ ~ ~ l l
1013 1011 10 9 10 7
105 20
40
60
80
T* (K) 100 120 140
Fig. 7. Hopping frequency j * vs. characteristic temperature T* obtained for Hg-1223 from a fit with a power law. The curves are displayed for the following a.c. fields h,,~ (Oe) from top to bottom: 0.5, 0.1, 0.5, 1, 2.5, 5, 7.5, 10.
In conclusion we have studied the low field imaginary part Z" of the a.c. susceptibility for sintered HgBa2Ca~Cu~Os +,~ (Hg-1223) and zone-melted Bi~Sr~CaCu20~ + ,~ (Bi-2212) polycrystals by varying the a.c. field amplitude and frequency. The Arrhenius-like behaviour observed for the frequency-dependent Z" peak temperature allows the determination of the activation energy E~, involved in the hopping processes of the vortices (with a characteristic frequency f 0 . The values obtained for E,, are similar for Hg-1223 and Bi-2212: we typically obtain E~ ~. 0.5 eV for h,c ~ 10 Oe and E~, ~ 2 3 eV for h~,~~ 10 2 0 e . The fact that E~, for sintered Hg-1223 polycrystals is as high as in zone-melted Bi2212 polycrystals suggests that Hg-1223 polycrystals obtained otherwise than by sintering could exhibit strongly increased activation energies. This is also in agreement with the strong shift of the irreversibility line observed, in Hg-1223 polycrystals, toward higher fields, when compared with sintered 123 polycrystals [11]. The magnitude of E~, obtained in both compounds gives also evidence for strong interaction among vortices if compared with similar analysis relevant to for example spin flip in spin glasses or vortex hopping in classical amorphous superconductors. However, the above-discussed analysis leads always to unrealistic values for f~. We have therefore introduced alternative analyses of the experimental results by introducing either a power law or a Fulcher law which appears to be more suitable for strongly interacting systems. Both analyses lead to more realistic values for the characteristic hopping frequencies J* or f~ which range in these cases between j * ~ 1 0 S H z and J * ~ 1 0 t 3 H z for a power law and between f~ ~ 107 Hz andf~ ~ 10 ~3 Hz for a Fulcher law. Both analyses lead to characteristic temperatures T* or To decreasing with increasing a.c. field. The choice between power law and Fulcher law is difficult to make, as each analysis has its own advantage. The parameter T* introduced by a power law appears to be closer to T~ and less field dependent than the parameter To introduced by a Fulcher law. However, the analysis introducing a Fulcher law reaches directly the activation energy E~. The activation energies obtained in this case are much lower than those deduced from a classical Arrhenius plot: we typically obtain E~,~ 2 x 10 2eV for h ~ , ~ 1 0 O e and E ~ 0 . 1 0.2eV for h~,~ ~ 10 2 0 e . We notice that the low values obtained for E~ by introducing a Fulcher law are also closer to the values expected in classical superconductors [3] where the upper limit should be around E,,/k ~ 100 K. The good agreement of the data with these phenomenological models confirms the fact that HTSCs can be considered as strongly interacting spin-glass-like systems.
A. Mehdaoui et al. ,; Materials Seienee and Engineering B34 (1995) 132 137
137
Table 1 The characteristic temperatures T* and Tc~ vs. applied field ha~ for a Bi-2212 polycrystal h~ (Oe) T* (K) T., (K)
0.01 88.0 _+ 0.4 84_+2
0.05 85.0 _+ 0.5 78_+4
0. l 83.3 -+ 0.7 75+5
0.5 81.2 -+ 0.8 76_+6
1 79.2 _+ 0.8 67_+7
5 76.0 + 1.0 65+5
10 71.8 _+ 1.3 52_+ 12
Table 2 The characteristic temperatures T* and To vs. applied field ha~ for an Hg-1223 polycrystal
h~,,. (Oe) T* (K) To (K)
Ea
0.05 121.8_+ 1.0 122_+3
0.1 116.5_+ 1.3 116_+4
0.5 112.2_+ 1.7 111 _+4
1 107.0_+2.0 107_+7
2.5 99.1 _+ 1.9 1004- 10
(eV)
Ea
101
5 96.5_+2.7 90-+ 10
7.5 84.0_+3.0 70-+ 10
10 76.8_+3.3 50_+ 10
(eV)
2.0 10"1
'
' '"'"l
i
i i~lnlj
1.5 10 "1
10-1 1.0 10"1
1 0 .3 5.0
10"2 A m
i
10 .5
,
-50
,
,
[
0
,
,
,
,
I
5o
,
,
,
,
I
100
,
,
,
o.o,o
150
T O (K)
........
10 .3
,
10 .2
,.
10 "1
10 0
.-,
101
102h=
(Oe)
Fig. 8. Activation energy E,~ vs. characteristic temperature To obtained for Hg-1223 when introducing a Fulcher law. The curves are displayed for the following a.c. fields h,c(Oe) from top to bottom: 0.05, 0.1, 0.5, 1, 2.5, 5, 7.5, 10.
Fig. 9. Activation energies E, (eV) vs. applied field h,~ for Hg-1223 ( l l ) and Bi-2212 (I1,). The analysis was achieved by using a Fulcher law.
References
[7] J.L. Tholence, Solid State Commun., 35 (1980) 113. [8] J. Emmen et al., J. Less-Common Met., 151 (1989) 63. [9] O. Chmaissem, Q. Huang, S.N. Putilin, M. Marezio and A. Santoro, Physiea C, 213 (1993) 259. [10] O. Chmaissem, Q. Huang, E.V. Antipov, S.N. Putilin, M. Marezio, S.M. Loureiro, J.J. Capponi, J.L. Tholence and A. Santoro, Physiea C, 217 (1993) 265. [11] D. Berling, E.V. Antipov, J.J. Capponi, M.F. Gorius, B. Loegel, A. Mehdaoui and J.L. Tholence, Physica C, 225 (1994) 212. [12] K.H. Miiller, Physica C, 168 (1990) 585. [13] N. Savvides, A. Katsaros, C. Andrikis and K.H. Mfiller, Physiea C, 197 (1992) 267. [14] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys., 49 (1977) 435. [15] J. Souletie and J.L. Tholence, Phys. Rev. B, 32 (1985) 516.
[l] A. Mehdaoui, D. Bolmont and B. Loegel, Mater. Sei. Eng. B, 18 (1993) 141. [2] K.H. Mfiller, M. Nikolo and R. Driver, Phys. Rev. B, 43 (1993) 7976. [3] P.G. de Gennes, in Supereonduetivity q/' Metals and Alloys, Benjamin, New York, 1966. [4] G. Williams, in R.A. Hein, T.L. Francavilla and H. Liebenberg (eds.), Magnetic Susceptibility q/' Supereonduetors and other Spin Systems, Plenum, New York, 1991. [5] J.L. Tholence, in R.A. Hein, T.L. Francavilla and H. Liebenberg (eds.), Magnetie Suseeptibility qf Supereonduetors and other Spin Systems, Plenum, New York, 1991. [6] A. Berrada, J. Durand, T. Mizoguchi, J.I. Budnick, B. Loegel, J.C. Ousset, S. Askenazy and H.J. GiJntherodt, Proe. 4th Int. Conf on Rapidly Quenched Metals, Sendai, 1981.
B
Materials Science and Engineering B34 (1995) 132 137
Activation energies in superconducting high temperature ceramics A. Mehdaoui, D. Berling, D. Bolmont, B. Loegel Laboraloire de Physique et de Spectroscopic Electronique, URA 143.5, Fa¢'ult~; des Sciences et Techniques, Unirersit[ de Haute Alsace, 4 rtw des ./?bres Lumii, re, 68093 Mulhouse COdex, France
Received 11 October 1994
Abstract
Low field a.c. susceptibilities of H g B a 2 C a 2 C u 3 O s + a (Hg-1223) and Bi2Sr_,CaCu2Os+,, (Bi-2212) have been studied. The loss peak of Z", relevant to intergranular pinning, shifts to higher temperatures for increasing frequencies. The results can be fitted with an Arrhenius law, leading to activation energies E~, ~ 1 eV, but the hopping frequencies fi~ are not realistic. More realistic values can be obtained by introducing a Fulcher law or a power law which is also characteristic of strongly interacting systems. K e y w o r d s : Superconductivity perovskites: High t e m p e r a t u r e superconductivity; Meissner effect; Magnetic m e a s u r e m e n t s
I. Introduction A.c. susceptibility provides a powerful tool for characterizing high temperature superconductors (HTSCs) and has been extensively used because of the contactless measurement possibilities. A.c. susceptibility provides simultaneously the real part Z' (diamagnetic shielding) and the imaginary part Z" of the complex susceptibility Z = Z' + iz"- The imaginary part Z" of the a.c. susceptibility is relevant to the energy losses arising from the hopping of vortices between pinning sites. This measuring technique can thus be used for characterizing the Meissner effect and for determining the irreversibility line and the same technique can moreover give the activation energy involved in the above mentioned hopping processes. The hopping of vortices results in a non-linear relationship between the induction B and the magnetic field H, and the energy losses are proportional to the area A of the B - H loop. As the penetration depth 2L increases with temperature for T < ice, whereas pinning strength weakens with increasing temperature for T > Tp, Z" exhibits a peak at some temperature Tp at which A reaches also its highest value. Increasing the area of a B H loop leads also to an increasing distortion of the a.c. signal taken at the secondary of a Hartshorn bridge and actually the strongest distortions are observed for T = Tp [1].
The high sensitivity of Tp to a.c. and d.c. magnetic field amplitudes has been extensively used to investigate the irreversibility lines for HTSCs, both in the (/1..... T) plane and in the (Hat, T) plane. The frequency dependence of Tp gives also information about the pinning mechanisms. It is the aim of this paper to present some experimental results obtained for Hg-1223 and Bi-2212 HTSCs and these are discussed in the framework of a simple critical state model. The measurements under consideration here are done with zero d.c. magnetic field and the a.c. magnetic fields are of small amplitude and low frequency. In a first approximation we assume therefore that we are in a linear regime allowing thus the use of Kim- Anderson's formulation in the following discussion.
2. Pinning potential and activation energy Applying an a.c. field involves hopping through the Lorentz force F acting on the vortices. Assuming that either the field or thermally induced hopping frequency f for intergranular Josephson vortices is given by Anderson's formula (taking into account forward and backward hopping), this leads to
j ,i,le,p ( 0921-5107/95/S09.50 t; 1995
Elsevier Science S.A. All rights reserved S S D I 0921-5107(95)01249-4
A. Mehdaoui et al. / Materials" Science and Engineering B34 (1995) 132 137
-- exp
3. Experimental details
kT
where J0 is a characteristic hopping frequency bro ~ 10v H z - 1 0 ~3 Hz), r is the pinning potential range and V is the flux bundle volume. The intergranular pinning potential U(T, H) can be taken as [2]
U(T, H) = Uo(T)Ho/2(IHI + Ho/2)
(2)
assuming that U(T, H) is proportional to the intergranular Josephson current and where [3]
Uo(T) = Uo(0)(1 - T/Tc) ~
(3)
(:~ = 1 for superconductor-insulator-superconductor grain boundary junctions and ~ = 2 for superconduct o r - n o r m a l - superconductor grain boundary j unctions) and where Ho is a parameter including the flux quantum 4~0 and the average grain size. The critical state in type II superconductors is reached when the Lorentz force F balances the pinning force Fp. Assuming large enough Lorentz force density and therefore neglecting the backward hopping contribution in Eq. (1), Fp can be taken as
Fp = (1/Vr)/[U(T, H) + k T In(f/f,)]
(4)
Assuming Bean's model where the applied field enters the centre of a cylindrical sample for T = Tp, Eq. (4) can also be written in the form of an Arrhenius law:
f /J; = exp[(Fp Vr - U ( T, g ) / k Tp]
(5)
Arrhenius laws have already been reported for the frequency dependence of the susceptibility maxima in diluted magnetic systems (for an extensive discussion of a.c. susceptibility in dilute magnetic systems see Ref. [4] and for a detailed discussion of Arrhenius-like behaviour in spin glasses and HTSCs see Ref. [5]) and in amorphous superconducting (RE)2Co amorphous alloys (RE = rare earth) [6]: the frequency-induced shift, (Sf=ATp/Af is, for example, 0.06K decade ~ in Er65Co35 [6] and 0.23 K decade ~ in Cuo.954Mno.o46 [7]. The peak temperature can be fitted in both cases with an Arrhenius law:
fifo = exp( - Ea/kT)
(6)
where Ea is the activation energy involved in the transition processes (e.g. spin flip in spin glass and vortex hopping in (RE)2Co). Comparing Eqs. (5) and (6), the activation energy is given by
E~ = U(T, H)/Vr)Fp
133
(7)
and leads for example for Cu~ x Mn,. to activation energies E, ~ 5 x 10 3 eV [7]. We will discuss below the results obtained from similar analyses for Hg- and Bi-based superconducting compounds.
The zone-melted Bi-based samples used in this study were fabricated in the laboratory of the department of Physics of the University of Technology at Eindhoven (Netherlands). Solid state reaction was achieved by using Bi203, CaCo3, Sr(OH)2, 8H20 and CuO powders. The sintered pellets (50 h at 870 °C in 02 atmosphere) of nominal composition Bi2Sr2CaCu2Oy (Bi-2212) were subsequently zone melted; the molten zone travelling speed was about 3 cm h-~ (for more details see Ref. [8]). The zone-melted samples exhibit a density d = 5.4 g cm 3 which is 2.5 times higher than that of the sintered samples. X-ray diffraction (XRD) spectra show that the samples are mainly single phased (c = 3.06 nm). The standard a.c. four-probe method gives To(p = 0 ) = 90 K and a transition width (between 10% and 90'}/0) A T c = 2 K . The critical currents Jc were 102A cm 2 at 50 K and 104A cm -2 at 20 K [8]. The sintered Hg-based samples were fabricated in the Laboratory for Crystallography at Grenoble (France) by high pressure and high temperature solid state reactions. A precursor with nominal composition Ba2Ca2Cu 3 is first prepared and HgO is subsequently added. The mixture is ground and placed in a gold capsule adapted for high pressure synthesis. The high pressure high temperature reaction is carried out in the following manner: the pressure is first increased to 18 kbar in 20 min, the temperature is then raised to 880 °C in 40 min and temperature and pressure were kept constant for 1.5 h. After treatment, pressure and temperature were decreased to normal in 20 min (for more details see Refs. [9,10]). Lattice parameters and impurity concentration were checked with a Guinier focusing camera. The reflections of the nominal HgBa2Ca2Cu3Os+ ~ (Hg-1223) were indexed on a tetragonal cell with lattice parameters a = 0.38536 nm, c = 1.5796 nm. The purity of the Hg-1223 phase is at least 95%, the main impurities corresponding to the Bi-1234 phase. The resistive transition exhibits a sharp drop between 130K and 135 K and preliminary a.c. susceptibility measurements reveal the occurrence of diamagnetic shielding at T = 133 K. The samples used in the present work all have typical sizes of a few cubic millimetres. Our susceptibility measurements were achieved on a Hartshorn bridge using a.c. fields 0.01 Oe < hac < 30 Oe in the frequency range 33 Hz < f < 12 000 Hz. Details of susceptibility measurements have already been published elsewhere [11]. Figs. 1 and 2 show examples of the results we obtained for a sintered Hg-1223 sample at a.c. fields of 5, 10, 20 and 30 Oe for frequencies f = 330 Hz and f = 1100 Hz. We observe a frequency-induced shift for Tp which is much higher than those reported for spin glasses [7] or amorphous low temperature superconductors [6] and
A. Mehdaoui et al. ,' Materials Science and Engineering B34 (1995) 132 137
134
X
T (K)
" (arb. units)
P
135
130
125
.... I .... I,~lr~l~L~l~l~l~ 100
105
110
115
120
125
120 130
135
140
T (K)
Fig. 1. The imaginary part Z" of the susceptibility for Hg-1223 at 330Hz for h~,~=5Oe (O), h~,~= 10Oe ( I ) , h~,~=20Oe ( t ) and
h~,, = 30 Oe (A).
our results for the whole frequency range are summarized in Fig. 3. The experimental results are restricted to lower frequencies for increasing a.c. fields, owing to the increasing impedance of the coils with frequency. Fig. 4 displays plots on a semilogarithmic scale of frequency vs. Tpl, showing that the results agree with an Arrhenius-like behaviour, allowing therefore an estimation of the activation energy E, for each applied a.c. field. The same qualitative features are obtained for zone-melted Bi-2212 polycrystals. Fig. 5 summarizes the results obtained for both Hg-1223 sintered polycrystal and for Bi-2212 zone-melted polycrystal. The activation energies obtained in both cases increase from some tenths of an electronvolt in the highest a.c. field up to some electronvolts in the low field limit. Our results are also in agreement with those obtained on superconducting high temperature polycrystalline sintered ceramics [2,12,13] where Miiller and coworkers obtain pinning potentials in the electronvolt range for
X " (arb. units) ''~1""1''~'
,,I 100
J''''l'~''l""l''
'1 '
. . . . I,rrlllllElllrlllrll
105
110
115
120
'~
irlll 125
130
135
14.0 T
(~)
Fig. 2. The imaginary part Z" of the susceptibility for Hg-1223 at l l 0 0 H z for h ~ ¢ = 5 O e (O), h ~ = 10Oe (m), h , ~ = 2 0 O e (!l,) and h~,~.= 30 Oe ( A ).
115
110
0
2ooo
400o
6ooo
soo0
1oo00 1200o f (Hz)
Fig. 3. The peak temperature Tp vs. frequency for several amplitudes of the a.c. field ha~: O, h,~ = 0.10e; II, h~ = 10e; ~, h,~ - 50e; A, hd~= 10 Oe; ~), h~,~= 25 Oe. The lines are guides to the eye.
lead-doped Bi-based HTSCs (Tc(p = 0) = 103 K) for Tc ~ 0.8 in applied a.c. fields ha~ of some oersteds.
T/
4. D i s c u s s i o n
Fitting the experimental results for HTSCs with an Arrhenius law (6) leads usually to activation energies Ea in the electronvolt range, which may be considered as acceptable, but, as was already pointed out for spin glasses [7] and for lead-doped Bi-based HTSCs [12], such an analysis gives also unrealistic f j values for HTSCs. We obtain 1022 Hz < f j < 101~° Hz for Bi-2212, and 1017Hz
A. Mehdaoui et al.
Materials Seience and Engineering B34 (1995) 132 137
f (Hz)
135
f (Hz)
10 5
10 4 ' ' ' ' 1
10 4
h ae =0.I
o.
Oe
....
J ....
I
~
....
I'''
hac=l Oe
103 10 3
,I,,,[,,,I,,,IJ,,I,,
102 7.60 10 -3
7.66 10"3
7.72 10 .3
7.78 10"3
T-1 (K-X)
102 7.70 10 "3
f (Hz)
7.82 10 .3
7.94 10 "3
T "1 (K"l)
f (I-Iz)
104
10 4 '
h a¢ =$
I
'
I
'
I
'
I
'
I
~
Oe
'
I
ha¢ =10
•
103
I
'
I
'
I
'
Oe
103
zl,,I,,l,,I,,I,,I,,I,,I,,I, 102 7.80 10 .3 7.59 10 "3 7.98 10 "3
102
8.07 10 .3
(T "1) (K"1)
~ I r I ,I
7.85 10 .3
,
8.00 10 -3
B J I,I
, 1 1 1 , I J
8.15 10 -3
8.30 10.3
T "1 (K"t)
Fig. 4. Arrhenius plots of f vs. 1/Tp for Hg-1223 for different a.c. fields h.~.
by
f / f * = [T/(T-- T*)I ~
(8)
Such a law applies well to spin glasses and other glasses [15] where the T* values are close to the experimentally determined critical temperatures and where the f * values are of physical significance (106 Hz < f * < 1014 Hz). The values determined experimentally for the exponent ~ are 4 < ~ < 11. Introducing a Fulcher law replaced Eq. (6) by
f/Jo = e x p [ - E , / k ( T -
To)]
(9)
Ea (eV) 3.5
....
'"'[
' ''"'"1
' ''"'"1
' ''"'"1
, ,,r,
3 2.5 2 1.5 -
II B
1 0.5 ......
0 10.3
I
10.2
, ,,~m,I
10"1
BBAA , ,, ..... I , , ...... I ~ l r l 10° 101
lO: h ac (Oe)
Fig. 5. Activation energies E, (eV) vs. applied field h,c for Hg-1223 (A) and Bi-2212 (B). The analysis was achieved by using an Arrhenius law.
where To (which characterizes the temperature where an order builds up) approaches Tp as the strength of the interaction responsible for ordering increases. Such a fit leads for example in C u M n spin glasses to activation energies E, similar to those obtained from an Arrhenius law, but gives also more realistic values off0 ~ ~ 1013 Hz) [7]. We therefore have also analysed our experimental results in the framework of the two above-mentioned phenomenological models. (a) Power law. Fitting our results with a power law leads to the results illustrated by Figs. 6 and 7 which give J* vs. T* for various applied a.c. fields. Assuming an average value of ~ = 9 (similar to those deduced from experimental results of Ref. [15]) we obtain 105 Hz < f * < 1011 Hz for Hg-1223 and 10 9 Hz < f * < 1013 Hz for Bi-2212. Tables 1 and 2 give the corresponding values obtained for T* in different applied a.c. fields. The zero field limits are T*(h,c--+O) = 127.1 K for Hg-1223 and T*(h,~--+O) = 88.2 K for Bi-2212. (b) Fulcher law. In such an analysis we first set the frequency range by assuming that realistic values for f0 correspond to those of the hopping processes introduced in Anderson's formulation (10 7 Hz < J ; < 1013 Hz). With such a hypothesis we obtain average values of To = 100 K + 20 K for Hg-1223 (corresponding to To/T c = 0.74) and To = 70 K 4- 10 K for Bi-2212 (corresponding to To/T c = 0.77). Tables 1 and 2 give also the values obtained for To in different applied a.c. fields when a Fulcher law is introduced. The zero field limits obtained in that way are To(h,~--+O)= 128 K for
136
A. Mehdaoui et al. / Materkds Science and Engineering B34 (1995) 132 137
f* ( H z )
5. C o n c l u s i o n I
I
I
1015 _
t
1013
1011
---q
10 9 ~107 r'-
4
? 10 5 b-
I,
I
I
I
40
50
60
70
80
....
90
T* (K)
100
Fig. 6. Hopping frequency .1" vs. characteristic temperature T* obtained for Bi-2212 from a fit with a power law. The curves are displayed for the following a.c. fields hd~ (Oe) from top to bottom: 0.01, 0.05, 0.1, 0.5, 1, 5, 10.
Hg-1223 and To(h,~--+O) = 85 K for Bi-2212. The corresponding activation energies are strongly reduced in such an analysis: Fig. 8 shows the temperature and a.c. field dependences of the activation energy when a Fulcher law (8) is introduced. Taking the abovementioned To values this leads roughly (at constant a.c. field) to a decrease by a factor 20 for both Hg-1223 (To = 100 K) and Bi-2212 (To = 70 K) and Fig. 9 gives the variation in E~ vs. h,~ for both systems. We notice that the activation energies obtained for both compounds are of similar magnitude whatever the phenomenological model is. This implies that the intergranular coupling strengths for sintered Hg-1223 and zone-melted Bi-2212 are also equivalent. f* ( H z )
1015
I
~ ~ ~ l l
1013 1011 10 9 10 7
105 20
40
60
80
T* (K) 100 120 140
Fig. 7. Hopping frequency j * vs. characteristic temperature T* obtained for Hg-1223 from a fit with a power law. The curves are displayed for the following a.c. fields h,,~ (Oe) from top to bottom: 0.5, 0.1, 0.5, 1, 2.5, 5, 7.5, 10.
In conclusion we have studied the low field imaginary part Z" of the a.c. susceptibility for sintered HgBa2Ca~Cu~Os +,~ (Hg-1223) and zone-melted Bi~Sr~CaCu20~ + ,~ (Bi-2212) polycrystals by varying the a.c. field amplitude and frequency. The Arrhenius-like behaviour observed for the frequency-dependent Z" peak temperature allows the determination of the activation energy E~, involved in the hopping processes of the vortices (with a characteristic frequency f 0 . The values obtained for E,, are similar for Hg-1223 and Bi-2212: we typically obtain E~ ~. 0.5 eV for h,c ~ 10 Oe and E~, ~ 2 3 eV for h~,~~ 10 2 0 e . The fact that E~, for sintered Hg-1223 polycrystals is as high as in zone-melted Bi2212 polycrystals suggests that Hg-1223 polycrystals obtained otherwise than by sintering could exhibit strongly increased activation energies. This is also in agreement with the strong shift of the irreversibility line observed, in Hg-1223 polycrystals, toward higher fields, when compared with sintered 123 polycrystals [11]. The magnitude of E~, obtained in both compounds gives also evidence for strong interaction among vortices if compared with similar analysis relevant to for example spin flip in spin glasses or vortex hopping in classical amorphous superconductors. However, the above-discussed analysis leads always to unrealistic values for f~. We have therefore introduced alternative analyses of the experimental results by introducing either a power law or a Fulcher law which appears to be more suitable for strongly interacting systems. Both analyses lead to more realistic values for the characteristic hopping frequencies J* or f~ which range in these cases between j * ~ 1 0 S H z and J * ~ 1 0 t 3 H z for a power law and between f~ ~ 107 Hz andf~ ~ 10 ~3 Hz for a Fulcher law. Both analyses lead to characteristic temperatures T* or To decreasing with increasing a.c. field. The choice between power law and Fulcher law is difficult to make, as each analysis has its own advantage. The parameter T* introduced by a power law appears to be closer to T~ and less field dependent than the parameter To introduced by a Fulcher law. However, the analysis introducing a Fulcher law reaches directly the activation energy E~. The activation energies obtained in this case are much lower than those deduced from a classical Arrhenius plot: we typically obtain E~,~ 2 x 10 2eV for h ~ , ~ 1 0 O e and E ~ 0 . 1 0.2eV for h~,~ ~ 10 2 0 e . We notice that the low values obtained for E~ by introducing a Fulcher law are also closer to the values expected in classical superconductors [3] where the upper limit should be around E,,/k ~ 100 K. The good agreement of the data with these phenomenological models confirms the fact that HTSCs can be considered as strongly interacting spin-glass-like systems.
A. Mehdaoui et al. ,; Materials Seienee and Engineering B34 (1995) 132 137
137
Table 1 The characteristic temperatures T* and Tc~ vs. applied field ha~ for a Bi-2212 polycrystal h~ (Oe) T* (K) T., (K)
0.01 88.0 _+ 0.4 84_+2
0.05 85.0 _+ 0.5 78_+4
0. l 83.3 -+ 0.7 75+5
0.5 81.2 -+ 0.8 76_+6
1 79.2 _+ 0.8 67_+7
5 76.0 + 1.0 65+5
10 71.8 _+ 1.3 52_+ 12
Table 2 The characteristic temperatures T* and To vs. applied field ha~ for an Hg-1223 polycrystal
h~,,. (Oe) T* (K) To (K)
Ea
0.05 121.8_+ 1.0 122_+3
0.1 116.5_+ 1.3 116_+4
0.5 112.2_+ 1.7 111 _+4
1 107.0_+2.0 107_+7
2.5 99.1 _+ 1.9 1004- 10
(eV)
Ea
101
5 96.5_+2.7 90-+ 10
7.5 84.0_+3.0 70-+ 10
10 76.8_+3.3 50_+ 10
(eV)
2.0 10"1
'
' '"'"l
i
i i~lnlj
1.5 10 "1
10-1 1.0 10"1
1 0 .3 5.0
10"2 A m
i
10 .5
,
-50
,
,
[
0
,
,
,
,
I
5o
,
,
,
,
I
100
,
,
,
o.o,o
150
T O (K)
........
10 .3
,
10 .2
,.
10 "1
10 0
.-,
101
102h=
(Oe)
Fig. 8. Activation energy E,~ vs. characteristic temperature To obtained for Hg-1223 when introducing a Fulcher law. The curves are displayed for the following a.c. fields h,c(Oe) from top to bottom: 0.05, 0.1, 0.5, 1, 2.5, 5, 7.5, 10.
Fig. 9. Activation energies E, (eV) vs. applied field h,~ for Hg-1223 ( l l ) and Bi-2212 (I1,). The analysis was achieved by using a Fulcher law.
References
[7] J.L. Tholence, Solid State Commun., 35 (1980) 113. [8] J. Emmen et al., J. Less-Common Met., 151 (1989) 63. [9] O. Chmaissem, Q. Huang, S.N. Putilin, M. Marezio and A. Santoro, Physiea C, 213 (1993) 259. [10] O. Chmaissem, Q. Huang, E.V. Antipov, S.N. Putilin, M. Marezio, S.M. Loureiro, J.J. Capponi, J.L. Tholence and A. Santoro, Physiea C, 217 (1993) 265. [11] D. Berling, E.V. Antipov, J.J. Capponi, M.F. Gorius, B. Loegel, A. Mehdaoui and J.L. Tholence, Physica C, 225 (1994) 212. [12] K.H. Miiller, Physica C, 168 (1990) 585. [13] N. Savvides, A. Katsaros, C. Andrikis and K.H. Mfiller, Physiea C, 197 (1992) 267. [14] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys., 49 (1977) 435. [15] J. Souletie and J.L. Tholence, Phys. Rev. B, 32 (1985) 516.
[l] A. Mehdaoui, D. Bolmont and B. Loegel, Mater. Sei. Eng. B, 18 (1993) 141. [2] K.H. Mfiller, M. Nikolo and R. Driver, Phys. Rev. B, 43 (1993) 7976. [3] P.G. de Gennes, in Supereonduetivity q/' Metals and Alloys, Benjamin, New York, 1966. [4] G. Williams, in R.A. Hein, T.L. Francavilla and H. Liebenberg (eds.), Magnetic Susceptibility q/' Supereonduetors and other Spin Systems, Plenum, New York, 1991. [5] J.L. Tholence, in R.A. Hein, T.L. Francavilla and H. Liebenberg (eds.), Magnetie Suseeptibility qf Supereonduetors and other Spin Systems, Plenum, New York, 1991. [6] A. Berrada, J. Durand, T. Mizoguchi, J.I. Budnick, B. Loegel, J.C. Ousset, S. Askenazy and H.J. GiJntherodt, Proe. 4th Int. Conf on Rapidly Quenched Metals, Sendai, 1981.