PHYSICA
Physica C 216 ( 1993 ) 315324 NorthHolland
The temperature dependence of the irreversibility line of
Bi2SrECaCuEOxsuperconductors D. H u , V.A.M. B r a b e r s , J . H . P . M . E m m e n a n d W . J . M . de J o n g e Department of Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Received 22 July 1993 Revised manuscript received 24 August 1993
The irreversibility line has been determined down to 15 K for both a B i 2 S r 2 C a C u 2 0 x (Bi2212) single crystal and a meltprocessed polycrystaUine sample. It is shown that the scaling law H(T,) oc ( 1 TJTc)" is only valid close to To. A kink is observed at about 30 K for the single crystal in a semilogarithmic plot of the irreversibility line H(Tr), similar to the kink found in the temperature dependence of the critical current density Jc~(T). Theoretically, a thermally activated flux depinning model valid for arbitrary current dependence of the effective pinning potential is developed for the irreversibility line. This yields that the irreversibility line can be described as H( T, ) oc ( 1  Tr/T¢)"/T m over the whole temperature region by using an empirical field dependence of the pinning potential U(H) ocH 1/,,,. The experimental data are analyzed in the light of the theoretical result.
1. I n t r o d u c t i o n
Uo( T, H)oc ( 1  T / T c ) L 5 1 H .
One of the interesting magnetic properties of highTc superconductors is the existence of the irreversibility line between He1 and He2 in the H  T plane, below which the magnetic properties become irreversible. The first observation of the irreversibility line in highT~ superconductors, which was determined from fieldcooled ( F C ) and zerofieldcooled ( Z F C ) magnetization measurements, was reported by Miiller et al. [ 1 ]. They found that the irreversibility line obeys a scaling law: H(T~)=H~(1TJT~)
~
(1)
with an exponent n ~ 1.5. It was suggested that the irrevcrsibility line is due to the superconducting glass state arising from the granularity of the polycrystalline sample. However, the same behaviour was observed in single crystals of YBa2Cu3Ox (Y123), from which Yeshurun and Malozemoff [2 ] concluded that the irreversibility line is a depinning line due to thermally activated flux creep in these materials. By a scaling argument, they obtained [ 2 ] the temperature and field dependence of the activation energy Uo ( T, H ) as
(2)
Applying relation (2) to the traditional A n d e r s o n K i m flux creep model, which assumes a linear current dependence of the effective pinning potential, Ucff(j/jc) = Uo( T, H ) ( 1 J/Jc), gives the scaling law ( 1 ) with n = 1.5 at T close to T~. The existence of a boundary line in the H  T plane has also been studied by other experimental techniques, such as mechanical measurements [3,4], I V characteristics [ 5 ] and field dependence of the resistivity [6,7]. In addition to the model of flux depinning, a fluxlattice melting [8,9 ] and a transition from the vortexglass and vortexliquid state [ 10] have been proposed for the boundary line. These theories predict the same scaling law ( 1 ) near T~, but with an exponent n = 2 and ~, respectively (recently a q u a n t u m melting model [11] has predicted n = 1.45). These explanations, however, are still controversial [4,12,13 ]. A detailed comparison of these boundary lines determined by different techniques suggested that they were essentially the same [ 14 ]. Experimentally the exponent n is found to range from 1.2 to 2.3 for different types of samples [ 1417]. For the irreversibility line most of the theories and experiments mentioned above focused on the scaling
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316
D. Hu et al. / Temperature dependence of irreversibility line
law behaviour (1) near To. Recent experiments, however, have revealed quite different behaviour at lower temperatures. De Rango et al. [ 18 ] found that for a Bi2_xPbxSr2Ca2Cu3Oy bulk polycrystalline sample the irreversibility line follows an exponential law H i ~ ( T ) ~ e x p (  T r / T o ) at low temperatures. They supposed that this temperature dependence is caused by the magnetic breakdown of the superconductivity induced by the proximity effect in the normal layers between the CuO layers. In contrast, Gupta et al. [4] suggested that this exponential behaviour of the irreversibility line is a depinning line originating from a logarithmic field dependence of the pinning potential, U ( H ) oc  I n H. This exponential behaviour was also found in Y123 and Bi2212 films [19]. For Bi2212 and T12212 single crystals the irreversibility line has also been fitted with Hi~ ~ T; m [ 15,20 ], as well as with Hi~ ~ exp (  T J To) at low temperatures, since it is usually difficult to distinguish these two laws by fitting the curves [ 15 ]. For a series of YlxPrxBa2Cu306.97 polycrystalline samples it is evident that the irreversibility line deviates from the scaling law ( 1 ) below 0.6To [21 ]. The behaviour of the irreversibility line at low temperatures, however, still lacks a strong theoretical background. It is also found [ 18,19,21,22 ] that for Y123 superconductors the scaling law ( 1 ) holds for a wide temperature range, but is only valid for a narrow temperature range near Tc for the Bibased superconductors. In this paper, we present a detailed study of the irreversibility line for a Bi2212 single crystal and a Bi2212 meltprocessed polycrystalline sample. The reason for studying the polycrystalline sample, besides the single crystal, is that no weaklink behaviour of the field dependence of the critical current density was observed in these meltprocessed polycrystalline samples [23 ], in contrast to Y123 bulk polycrystalline samples. The irreversibility temperature is determined down to 15 K. In the semilogarithmic plot of H ( T r ) of the Bi2212 single crystal a kink is found at about 30 K, similar to the kink found in the temperature dependence of the critical current density Jc~(T), which indicates two different pinning regimes in the single crystals. Following the experimental results, we discuss the irreversibility line within a thermally activated flux creep model. Compared to the previous work in ref. [ 2 ], where a linear
j dependence of Ueff(j) is assumed, our approach is valid for any Ueff(j) relation. The behaviour of the irreversibility line at low temperatures, as well as at high temperatures, will be discussed and compared with the experimental results.
2. Experimental A Bi2212 single crystal with a lateral dimension of about 2 mm and thickness of about 0.1 mm was grown by a selfflux method described in detail in ref. [ 24 ]. This crystal was used for both the irreversibility line and critical current density measurements. Additional measurements of the critical current density were also performed on Bi2212 single crystals grown by the travellingsolventfloatingzone (TSFZ) method, by which single crystals as large as 10 X 3 X 0.3 mm 3 can be obtained [ 25 ]. A ringshaped Bi2212 polycrystalline sample with an outer diameter of 3.2 mm, a wall thickness of 0.8 mm and a height of 0.5 mm was prepared by a partialmelting process with the heating temperature up to about 905°C [23 ]. From the onset of the Meissner effect in an applied magnetic field of 20 Oe, the superconducting transition temperature T¢ was determined to be 89 K for the single crystals and 91 K for the polycrystalline sample. The irreversibility line was determined by measuring the zerofieldcooled (ZFC) and fieldcooled (FC) magnetization on a vibrating sample magnetometer (VSM) in a flow cryostat. The applied magnetic field ranged from 20 Oe up to 80 kOe. For all FC and ZFC curves we chose the same sweep rate of about 2 K/min. The temperature dependence of the critical current density Jet was obtained as follows: first, the sample was cooled in zero field to 4.2 K and a high magnetic field was applied for about 1 min and then switched off. The applied magnetic field was chosen to ensure a full critical state in the sample after switching off the field. The temperature dependence of the magnetization was recorded while heating the sample at a rate of 2 K/min. Although during the temperature sweep there was a relaxation effect of the magnetization, we found that the temperature effect dominated the changing of the magnetization. During the temperature sweep, the change of the
D. Hu et aL / Temperature dependence of irreversibility line
sample induces a voltage on the sample itself, i.e. Voc AM~At = S A M ~ A T where S = AT/At is the sweep rate, which can be considered as the voltage criterion of the critical current density. Our sweep rate corresponds to a voltage criterion of about 0.1 IxV/cm for determining the critical current density of these samples. The critical current density J~r was calculated from the magnetization using the Bean criticalstate model [26] for both single crystals and polycrystalline samples, since non weaklink behaviour is found in the meltprocessed polycrystalline samples [23].
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As shown in fig. 1, the irreversibility point is determined from the ZFC and FC magnetic moment curves, i.e. the lowest temperature where ~m= m ~ f ¢  m f ~ 0. To determine the irreversibility temperature, we always expand the curves around the irreversible point with a total scale of the magnetic moment of about 2 × 10 6 A m 2 in order to have the same criterion for different magnetic fields. In fig. 2 (a) the irreversibility line is plotted against ( 1  Tff To) to check the validity of the scalinglaw behaviour
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Fig. 2. The irreversibility line of the Bi2212 superconductors plotted with (a) H ( T r ) vs. ( 1  Tr/Tc), the dashed lines are the fits of the scaling law (see text); (b) H ( T , ) vs. Tr. Note the kink at about 30 K in the irreversibility line of the single crystal.
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40
50
60
70
80
90
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Fig. 1. The ZFC and FC magnetic m o m e n t (Hffi 100 Oe) of the Bi2212 single crystal and the determination of the irreversibility temperature T,.
of eq. (1). As shown in the figure with the dashed lines, the scaling law is only valid for T, near To, which is at T above 0.65Tc for the polycrystaUine sample and at T above 0.8To for the single crystal. This fit gives the exponent n to be about 2.2 for the polycrystalline sample and about 1.2 for the single crystal. The reason why the scaling law holds for a wider temperature region for the polycrystalline sample than the single crystal will be discussed in the next section. At low temperatures, the scaling law ( 1 ) does not apply to either the single crystal or the polycrystalline sample. In fig. 2 (b) H ( T r ) is plotted against Tr for both the single crystal and the polycrystalline sample. The irreversibility field of the polycrystalline sample is larger than that of the single
318
D. Hu et al. / Temperature dependence of irreversibility line
crystal, indicating an improved pinning in the polycrystalline sample. The irreversibility line of the single crystal exhibits a kink at about 30 K, marking two different regions. A similar kink at about 30 K in the irreversibility line of Bi2212 single crystals has been reported in ref. [ 16,17 ]. In fig. 3 the temperature dependence of the critical current density Jcr is shown for both single crystal and polycrystalline samples, which has also been observed from magnetic hysteresis measurements in a previous study [27 ]. At T> 20 K the critical current density of the polycrystalline sample is higher than that of the single crystal, again indicating that there are more large pinning centres in the polycrystalline sample. The critical current density of the single crystals exhibits a kink at 30 K. A similar kink was also observed for the Bi2212 single crystals grown by TSFZ techniques, which means that this behaviour observed for the single crystals is not changed by the difference between the two crystal growth methods. The temperature dependences of the irreversibility line in fig. 2 (b) and of the critical current density in fig. 3 show an interesting similarity, particularly the kink for the single crystal, which strongly suggests a correlation between the driving physical mechanisms underlying these two quantities. The
correlation between Jcr(T) and H(Tr) has also been observed in other experiments: for a series of Y123 samples it was found [ 22 ] that a higher value of the irreversibility field Hi~(T) corresponds to a larger value of the magnetic hysteresis width AM(T) (AMocJcr); by introducing pinning through irradiation damage, it was found that both J=(T) and H(T~) were enhanced [15,28]. Such a correlation implies a common origin for both J¢r(T) and H(Tr). In the highTo superconductors the temperature dependence of JCr(T) is dominated by the thermally activated flux motion [27,2932], which favours the hypothesis that the irreversibility line is a depinning line. The kink at 30 K in J¢r(T) of the single crystal has been discussed in more detail in a previous paper [27]. In order to obtain more insight into the connection between the kinks in J ~ ( T ) and H(Tr), we show some FC and ZFC magnetization curves of the single crystal in fig. 4 with a semilogarithmic plot. For convenience, the pinning regime at T< 30 K is referred to as regime 1, and the pinning regime at T> 30 K is referred to as regime 2. In low magnetic fields ( H < 2000 Oe) the ZFC magnetization curves exhibit an "S" shape or a weak kink below the irreversibility temperature T~, which indicates a
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20
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50
60
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T(K) Fig; 3. The temperature dependenceof the critical eurrem density for both Bi2212 singlecrystaland polycrystallinesamples. Note the kink at about 30 K in the curve of the singlecrystal.
D. Hu et aL / Temperature dependence ofirreversibility line
319
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T(K) Fig. 4. Some ZFC and FC magnetic moment curves of the Bi2212 single crystal plotted as log [m I vs. T for H= 300, 600 and 2000 Oe. The arrows indicate the irreversibilitytemperatures Tr (because of the small differenceof the FC curves, Tr cannot be clearlyseen in this plot for H 300 Oe and H= 600 Oe). When H< 2000 Oe the ZFC magnetization exhibits an "S" shape or a weak kink below Tr shown by the dashed line; when H~ 2000 Oe no "S" shape or kink is observedat T< Tr in the ZFC magnetization. smooth transition from pinning regime 1 to pinning regime 2. The transition is shown by the dashed lines in the figure. The irreversibility temperature, in this case, falls in the pinning regime 2. When the applied magnetic field is larger than 2 kOe, no "S" shape is observed below the irreversibility temperature Tr, which implies that the pinning regime 2 is destroyed by the magnetic field and only the pinning regime 1 exists. So, in high magnetic fields the irreversibility temperature falls in the pinning regime 1. Since these two pinning regimes have quite different origins, a kink forms in the irreversibility line of the Bi2212 single crystal. The origin of these two different pinning mechanisms, however, is still not clear and is controversial. In ref. [ 16 ] it is argued that, since the
flux density is small in a low magnetic field, the pinning regime 2, which is observed at high temperatures and in low magnetic fields, is due to singlevortex pinning; and the pinning regime 1 is due to collectivepinning behaviour of correlated vortices. However, a detailed study of the magnetic relaxation in the Bi2212 single crystals [27] has shown that the decay rate of the magnetization is much smaller in regime 2 than in regime 1, indicating that the effective pinning potential is much larger in regime 2 ( ~ 8 0 meV) than in regime 1 (,,, 15 meV). Therefore, we suggest that the pinning regime 1 is dominated by the 3D single vortex pinning of point defects with the pinning energy of the order of 10 meV; and the pinning regime 2 is the collectivepinning re
320
D. Hu et al. I Temperature dependence of irreversibility line
gime of the vortices for which the effective pinning energy can be enhanced. Here, we will not discuss this problem in more detail.
4. Thermally activated flux depinning model As mentioned in the introduction, the irreversibility line has been discussed in several different models. Here, in view of the evidence discussed above, we will consider the irreversibility line as a depinning line and consequently focus our attention on the thermally activated flux creep model. The depinning line due to thermally activated flux creep was first explored by Yeshurun and Malozemoff [2]. However, there are several reasons why it is necessary to improve on their approach. First, in ref. [ 2 ] the AndersonKim model, which assumes that Ucff(l'/j¢) = Uo( T, H ) ( 1  j / j c ) , was used to discuss the irreversibility line. However, recent experimental and theoretical studies of the magnetic relaxation have demonstrated that this linear j dependence of Uefr(j) is only valid for j ~ j c [ 5,10,13,27,3032 ], and different j dependences of Ueff(j) have been proposed (see, for instance, refs. [ 3032 ] ). Therefore, the AndersonKim model cannot be used to discuss the irreversibility line which occurs at j ~ 0 due to thermally activated flux depinning. Secondly, as discussed in the introduction, recent experimental results have revealed a quite different behaviour at low temperatures [ 1521 ], in comparison to the scaling law ( 1 ) near T¢; the irreversibility line at low temperatures has to be considered in more detail. Finally, in ref. [ 2 ] it was postulated that U(H) ocH  1, but the experimental results show that U(H)oc H  1/,, with m ranging from 1 to 4 for different types of samples [3335,3739]. In the following approach we will take these three aspects into consideration, namely, we will discuss the irreversibility line in the whole temperature region within the framework of the flux creep model without any specific assumption of the j dependence of Ucff(j).
4.1. The model We start from the flux creep equation which gives the dissipative electric field as [36 ]
E = Eo e  vat (r,nj)/kr,
(3)
where Eo = Bdto, B is the magnetic induction, d the hopping distance of the vortices, and o) the attempt frequency of the vortices; Ueff(T, H, j ) is the effective pinning potential which depends on the temperature, magnetic field and the current density. Following the procedure in ref. [ 31 ], we assume that
Ue~( T, H,j/j¢) = Uob( T/T~)g(H/Ho)f(j/j¢) =b( T/T¢)g(H/Ho) Uetr.o(j/j¢) ,
(4)
where b(T/T¢), g(H/Ho) describe the temperature and field dependence of Uo(T, H), respectively; Uorr.o(j/jc) is the currentdependent effective pinning potential; and j¢(T, H ) is the critical current density in the absence of any flux creep. A combination ofeqs. (3) and (4) gives
k T In (E/Eo) U~ff.o(J/jc)= b ( T / T ¢ ) g ( H / H o ) "
(5)
The depinning line is determined when the irreversible magnetic moment drops below the threshold of the detection of the equipment. However, the cflteflon for the depinning line depends not only on the current density in the sample but also on the size of the sample and the sensitivity of the equipment. Given the size of the sample and the sensitivity of the equipment, the criterion of the depinning line is j = e ~ 0. Since the irreversibility line is well below H¢2 (except at T ~ 0 K), j¢ can be considered to be field independent. The temperature dependence of j¢(T), however, has an influence on the value of Uefr,o(j/j¢). For this we assume that U,ff.o[e/ j¢ (T) ] = Uc a (T/T¢). The separation of this temperature factor a(T/Tc) is justified by the result of the collective pinning theory, in which U(j/ j¢) oc (j¢j) ~ is predicted for vanishing j [32]. Therefore, we have
kTrln(E/Eo) =U¢a(TdT¢) . b(TJT¢)g(H/Ho)
(6)
The value of U¢ could be much larger than Uo, since in the collective pinning theory Ueff,o (J/j¢) grows as the current density decreases. From eq. (6) it follows that
kTr In (EIEo) g ( H I H o ) = Uca(TrlTc)b(TJTo)"
(7 )
D. Hu et al. I Temperaturedependenceof irreversibility line Here, Eo is considered to be temperature and field independent since it is contained in the logarithmic term and the value of In (E/Eo) is typically between 15 and 25. For the field dependence of the pinning potential, the following empirical relation is introduced: [ H I I/m
g(H/Ho)
=LHooJ
'
(8)
which is justified by the fact that this kind of field dependence has been observed experimentally with rn ranging from 1 to 4 [3335,3739] and supported by the theoretical postulations for m = 1 [2,40] and m = 2 [41]. Inserting eq. (8) into (7), we obtain H(T~,E)
(9)
r uc ]~ra(TffT~)b(TffTc).] ~ = ~ ° L k In(Eo/E)J L Tr J " T
The term ln(EolE)m in eq. (9) describes the sweeprate dependence of the irreversibility line, since Eoc AMi~/At~ AMzf¢/At=SAMzf¢/AT, where S = AT~At is the sweep rate of the ZFC magnetization. The second term in eq. (9) contains the temperature dependence of the irreversibility line. At low temperatures a ( T~ To) ~ 1 and b ( T~ T~) ~ 1, and eq. (9) is simplified into the power law of T: H(T~, E) = H a ( E ) T r m ,
(10)
with Ha(E) =Ho[ U d k ln(Eo/E) ]m. The power law (10) has been used to fit the experimental results in the literature [ 15,20]. This power law of eq. (10), however, does not imply any new mechanism and is a natural consequence of the thermally activated flux creep model. Because, in the flux creep model, the important factor is the thermal energy k T which is not related to T~, the relevant physical quantities should be a function of T, and not ( 1  T~ T~), except when T is approaching T¢ where the GinsbergLandau parameters 2(T/T¢) and ~(T/T¢) become strongly temperature dependent. Thus, the scaling law (1), which shows that H(T~) is only a function of (1  T r / T c ) , can be valid only when Tis close T¢; and the irreversibility line should deviate from the scaling law ( 1 ) at low temperatures and has to be a function of T, not of T~ T¢.
321
The deviation of the scaling law has been mostly observed in Bibased superconductors, as mentioned in the introduction. For most of the Y123 samples the irreversibility line has been fitted with the scaling law ( 1 ). This, however, does not mean that the scaling law ( 1 ) is valid over the whole temperature region for the Y123 samples, because it is difficult to obtain the irreversibility line of the Y123 samples at low temperatures. Even in a magnetic induction of 5 T, the irreversibility temperature is above 70 K for the Y123 single crystals [3,22], which is still close to To. However, by depressing Tc through doping Pr into the Y123 system, the deviation of the scaling law (1) of the irreversibility line was observed [ 21 ]. Another reason why it is more difficult to observe the deviation of the scaling law ( 1 ) for Y123 single crystals than for Bi2212 single crystals is related to the different field dependence of the pinning potential, which will be discussed later. Now we consider the situation near To. Let 6t= 1Tr/Tc. When Tr,Tc or 6t<< 1, the factor of the temperature dependence of U(T) can be expressed as [33 ] b( TffT~)= ( 1  TffT~) p  ~t p .
( 1 la)
Since the temperature dependences of jc(T) and U(T) have a similar form [27,40] and in the collective pinning theory U(j/j~)oc ( j d j ) ~ for vanishingj, a(TffT~) can also be expressed as a power law of ( 1  Tff To) for T close to To, or a( Tff T~) = ( 1  Tff Tc) ~ ~t r .
(lib)
Let a=#+~,. Equation (9) can now be rewritten as ~t n H(T~) =H~  (1~t) m ~H~
6t n ( 1  m6t)
(n=am) , (fit<< 1) ,
where Hc=Ha/T'~. For m~t<< 1, the scaling law is obtained: H(T~) ~ H¢6t ~ =HcC1Tr/T¢) ~ .
(12)
The condition rn6t<< 1 for the validity of the scaling
322
D. Hu et al. I Temperaturedependenceof irreversibilityline
law eq. (12) is related to the exponent m of the field dependence of the pinning potential. The larger the value of m, the narrower the temperature range for the validity of the scaling law (12). This provides us with an explanation of why for some superconductors, such as Y123 superconductors, the scaling law (12) holds for a wider temperature range, and for other superconductors, such as Bi2212 single crystals and films, only for a very limited range. Because for Bi2212 single crystals the value of m is usually larger than 2 [33,35,38,39], and for Y123 single crystals the value of m is typically 1 [2,6,37 ]. Furthermore, the exponent of the scaling law (12) is n = otm, which is also related to the exponent m of the field dependence of the pinning potential. Since for different types of samples the field dependence of the pinning potential is different, a universal exponent n in the scaling law (12) is not expected in this model. In fact, experimentally, the value of n is found to vary from 1.2 to 2.3 [1417]. In order to describe the irreversibility line over the whole temperature region, we assume that b(Tr/T¢) and a ( T J T ¢ ) can be described by eqs. ( l l a ) and (1 lb) at low temperatures also. This leads to H(Tr, E ) = H a ( E )
( 1  T~/T~) n , T?
crystal samples by plotting log H versus log [ ( I  T~I To) / T 7/" ] for different values of m~ n in fig. 5. For low values of m / n , the curves show a positive cur= vature, and for high values of m / n the curves show a negative curvature. For the polycrystalline sample a straight line is observed at m / n ~ 0.75 with a slope of n ~ 2.0. The exponent m, therefore, is deduced to be 1.5, which means for the polycrystalline sample U(H) o c H  i/].s. For the single crystal, there are two different pinning regimes: one below 30 K and the other above 30 K. At T < 30 K, ( I  T/Tc) ~ I, and the data can be fitted with the power law of T F ml with m i ~ 4 . 1 . This exponent m , ~ 4 . 1 is in good
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which means that in a doublelogarithmic plot of H versus ( 1  T J T ~ ) / T m/n the data of the irreversibility line should be a straight line with a slope of n for the appropriate value of m / n . 4.2. Application o f the theoretical result to the experimental data In light of the result given in eq. (14), we have analyzed our data of the polycrystalline and single
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(13)
which combines the results of eq. (10) and eq. (12). It is better to fit the irreversibility line with eq. (13) over the whole temperature region rather than fitting the data with eq. (10) at low temperatures and with eq. (12) at T near T¢, since in the intermediate temperature region both eqs. (10) and (12) are effective. In order to fit the data with eq. ( 13 ), we rewrite eq. (13) as
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Fig. 5. log(H) vs. log[ ( 1 Tr/Tc)/T m/n] (see the text) for different values of m~n, (a) for the Bi2212 polyerystallinesample, (b) for the Bi2212 single crystal ( T> 30 K). The different values of m/n are shown in the figure. Note the curvatures of the curves with different values of m/n.
D. Hu et al. / Temperaturedependenceof irreversibilityline agreement with the experimental result of ref. [ 35 ], where U(H)ocH 1/4 was found for Bi2212 single crystals from AC susceptibility measurements in high magnetic fields. For T > 30 K we plotted the data as l o g H versus log[ ( 1  TffTc)/T~/~] for different values of m/n, as shown in fig. 5(b). A straight line with a slope of n = 1.26 is found for m/n = 1.75, from which the exponent m is found to be 2.2. This value of m = 2.2 for the single crystal is larger than the value of 1.5 for the polycrystalline sample. Therefore, the scaling law behaviour ( 1 ) at T close to T~ should be observed over a wider temperature range for the polycrystalline sample than for the single crystal, which is what has been observed in our experiments.
4.3. A remark about the field dependence of the pinning potential A final remark is made about the field dependence of the pinning potential U0 (H). By replacing the field dependence of eq. (8) by a logarithmic field dependence
g( H/ Ho) = In ( Ho/H) ,
(15)
the temperature dependence of the irreversibility line becomes
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kT~ln(Eo/f)
V o a ~ /
~ H o e  rr/T° (for T<< Tc) ,
]
323
5. Conclusion In conclusion, the irreversibility line is determined down to 15 K for both Bi2212 single crystal and polycrystaUine samples. A kink is found at about 30 K in the irreversibility line of the Bi2212 single crystal, similar to the kink found in the critical current density Jcr (T) of the single crystals. It is found that at T > 20 K both the irreversibility line and the critical current density are enhanced for the polycrystalline sample compared to that of the single crystal. The similarity between the temperature dependence of the irreversibility line and the critical current density in both Bi2212 single crystal and polycrystalline samples, together with the evidence of the correlation between the irreversibility line and the critical current density found by other groups, favours the hypothesis that the irreversibility line is a depinning line. Within the framework of the thermally activated flux creep, we argued that the scaling law H ( T r ) oc (1 Tr/T¢) n can only be valid near T~, and H ( T r ) should deviate from this scaling law and become only a function of T at low temperatures. Using an empirical field dependence of the pinning potential U(H) o c H  1/% we obtain H ( T r ) oc ( 1  Tr/ Tc)n/T~, which is in good agreement with our experimental results.
ro)l
(16)
in which To= Udkln(Eo/E). This exponential law (16) has been used to fit the irreversibility line at low temperatures by several groups [4,15,18 ]. For our data, eq. ( 16 ) can also fit the irreversibility lines of the single crystal at T < 30 K with To~ 5.3 and of the polycrystalline sample at T < 60 K with To ~ 11.5. Since, however, it is difficult to distinguish a power law of T  " from an exponential law of exp (  T~ To) by fitting the experimental data over the relatively narrow temperature range, and in addition the logarithmic field dependence (15) lacks strong experimental support, we prefer the powerlaw field dependence of eq. (8) which has been observed in several experiments [ 3335, 3739 ] and postulated in the theories for m = 1 [2,40] and 2 [41 ].
Acknowledgement
We thank Z.X. Zhao for providing some Bi2212 single crystals. Some of the experiments were performed when one of the authors ( D H ) was in ABB Corporate Research, Baden, Switzerland.
References
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