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Frequency, field, and temperature dependence of the AC penetration depth of a GdBa2Cu3O7−δ film in the mixed state
PHYSlCA ELSEVIER
PhysicaC250 (1995) 389-394
Frequency, field, and temperature dependence of the AC penetration depth of a GdBazCu3OT_ film in the mixed state M. Zeisberger *, Th. Klupsch, W. Michalke 1PHT Jena, PF 100 239, D-07702 Jena, Germany Received 8 May 1995; revised manuscript received 20 June 1995
Abstract
We report on a systematic mutual induction measurement of the complex AC penetration depth A of a sputtered high-quality GdBa2Cu307_ ~ film in the mixed state by a very small coil system arranged near the sample surface. The complex penetration depth A(B, T, to) for DC inductions B ~<0.65 T (perpendicular to the film), for temperatures 36 K ~
I. Introduction
The vortex dynamics can be investigated by analyzing the response with respect to small periodic, time dependent forces acting on the flux line lattice in the mixed state of a superconductor. These periodic forces can be generated by external AC currents, AC magnetic fields or mechanical vibrations of the superconductor in an applied DC magnetic field. This results in various AC techniques, e.g. AC transport [1-5], self and mutual inductance (AC susceptibility) [6-11], vibrating REED [12] and microwave surface impedance [13,14] measurements. These investigations allow to determine the elastic restoring force constant of the pinned vortices (Labusch parameter). In the following we use the
* Corresponding author.
definition of the Labusch parameter a as it is given in Ref. [6] (with the dimension N m -2). In Ref. [15], another definition of the Labusch parameter ( a L) is used where a L = a B / ~ 0 (~0 is the flux quantum, B is the DC induction in the sample). For an YBaECU307_ ~ single crystal the observed temperature dependence of ot was fitted by ot = or0(1 - t2) 2 [6] (where t = T/T* and T* = To). In Ref. [14], a theoretical estimation is given which results in the same temperature dependence and in a maximum absolute value (for YBa2Cu307_ ~) ot0 = 3 x 105 N m -2. The experimental values given in Ref. [14] agree with this estimation of c~0 but show deviations in the temperature dependence. In Refs. [6] and [13], a Labusch parameter a which is independent of B was observed. Other authors [3-5,12] reported on rather complicated dependences of a upon B. Frequency dependent measurements in the microwave range [13,14] have been used to determine the depinning frequency, i.e. the frequency where the viscous
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 3 7 7 - 0
M. Zeisberger et al. /Physica C 250 (1995) 389-394
390
forces acting on the flux lines become larger than the pinning forces. The results of the frequency dependent AC penetration depth measurements few degrees below T¢ by means of conventional AC susceptibility arrangements can be explained by the vortex glass theory and the scaling behaviour near the phase transition to the vortex liquid [9-11]. In this paper we report on measurements of the AC penetration depth A at lower temperatures (36-81 K) in the frequency range of 1-500 kHz, and for rather low applied DC inductions B e (0-0.65 T). These measurements should be interpreted in terms of the complex conductivity o- or the complex AC penetration depth )t 2 ---- i / ( / x o too-) of the superconducting sample at the angular frequency to, as it follows from the single vortex pinning assumption. This provided, the following expression for the AC penetration depth is given in Ref. [15] 1 -
A2 = A2 + A2
i/tor
1 + itor o '
(1)
with )t L a s the London penetration depth, r o as the flux flow relaxation time 7/
ro.
. O~L
Bc2~ 0
.
.
,
(2)
(1). For small frequencies (to << 1 / % ) , Eq. (1) can be approached to A2 -- A2L+ A2c(1 - i / t a r ) .
(6)
2. Experimental We have investigated a c-oriented GdBa2Cu 3film with a thickness D = 160 nm, a critical temperature T~ = 91.0 K, and a susceptometric transition width of 0.4 K, which was prepared by magnetron sputtering on a ZrO 2 substrate. Films of this type usually show critical current densities of about 106 A cm-2 (at T = 70 K, zero applied field, applying a 3 /xV/cm criterion). For further details see Ref. [24]. In our experiments we used two small coils arranged at the same sample side (Fig. 1) with B e perpendicular to the film. Such arrangements behave like completely axially symmetric arrangements with the azimuthal AC current flowing only in the (a, b)-direction if the film diameter is sufficiently larger than the coil diameter [17,18] because only the (a, b) components of the AC conductivity tensor are relevant. In Ref. [18], a theoretical analysis of this
07-8
/OnO~
r as the relaxation time of the thermally activated flux motion r = ro e x p ( V / k r )
coil
(4)
U is an effective activation energy which depends on the DC current Js which is induced in the sample by a sweep of B e. Pn is the extrapolated (for T < Tc) normal state resistivity which shows a linear temperature dependence for YBa2CtI30 7 [16]
p.( r) = p.( ro)r/ro.
rp2--'ra
I rpl--'rsl~
(3)
and Ac as the Campbell penetration depth B2 ~0 B A2 = - /x0 a L /x0 oz
I
(5)
This relation is assumed to be valid also for our GdBazCu307-8 sample. For microwave frequencies the two-fluid model has to be considered [20] so that A would become a more complicated expression than
film Ohick~ss D) Fig. 1. Arrangement of our experiments consisting of a primary coil (80 turns, rpl = 0.5 mm, rp2 = 1.7 mm, %1 = 0.41 mm, %2 = 1.25 mm), a secondary coil (10 turns, rsl = 0.5 mm, q2 = 1.7 ram, Z~l = 0.15 mm, Zs2 = 0.26 ram), and a thin film (thickness D = 0.16 mm, size 1 0 × 10 m m 2) showing a mutual inductance of L(~) = 680 nH without a sample and L(0) = 300 nH for an ideal diamagnetic sample (~t = 0). The calibration factor for the measurement of the penetration depth is f = 1.57 n i l / t z m and f / D = 9.81 nI-I//.~m2, respectively. The AC current of the primary coil was 1 mA which results in an AC induction of about 10 #T. B~ is along the symmetry axis.
M. Zeisberger et al. / Physica C 250 (1995) 389-394
arrangement is presented. Eq. (10) of Ref. [18] gives the calibration function L(A) for an elementary arrangement with two single loops (with the radii rp, r s and axial distances to the film surface zp, zs) and can be approximated by L(A) = L(0) + L, A c o t h ( D / A )
("°t
A<
0.05 0.04
(7)
•
61K
•
56K
•
51K
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bff
2/~0~p r s ( 2 b - h)
Fig. 2. DC field dependence of the real part of A A2 and linear fits.
b=zp, h = z s - Z p. (8)
The calibration factor of the two coil arrangement of Fig. 1 with 80 primary and 10 secondary turns, which results from superposition of elementary arrangements, is f - - 1.57 nH//~m. The limits of inductance of our coil system are L(0) = 300 nH and L(A ~ ~ ) = 680 nil. The experiments were done with a primary current of 1 mA which produces an AC magnetic induction of about 10 /~T. Relevant values of the penetration depth are I A I = 0.1-1 /~m and cause relative signal changes of [ L ( A ) L(O)]/L(O)-~-10-4-10 -2. Hence the absolute measurement of the complex A requires to measure L(A) and D and to calculate L(0) with a precision of 10 - 4 which is very hard to reach. The measurement of changes A(A 2) = AE(Be) - A2(0) of the squared penetration depth by changing B e and keeping the other experimental parameters constant is much easier. The constant part of mutual inductance L ( 0 ) + fA2L/D + D/3 can be eliminated in the experiment by a compensation circuit. Hence A(A 2) can be calculated from the changes of mutual inductance AL by means of
At(Be) = ( f l D ) A [ A2( Be)].
71K
66K
0
with the calibration factor
fel = ( r p _ r s ) E + ( 2 b _ h ) 2 ,
81K 76K
o
0.01
r s , Zp, Z s ) ,
a •
•
0.03
0.02
L(o) +L, 5- + -2 ' (ifl A I > D ,
391
(9)
In our experiments, we used an external compensation which allows to measure changes in the mutual inductance down to 0.03 nil. After setting the temperature and oscillator frequency and compensating the signal L(B e = 0), the change of mutual inductance was measured with
increasing DC induction (sweep rate 2 mT/s). The in-phase and out-of-phase signal measured by a lockin amplifier (PAR M5302) yield the real and imaginary part of A(A2). Figs. 2, 3, 4 show that A(A2) is found as an approximately linear function of B e in the range T~< 81 K and B e ~< 0.65 T. Deviations from linearity which indicate a decreasing Labusch parameter with B e are observed at higher temperatures. In order to examine whether the set-up operates in the linear AC regime the measurement was performed using three different primary currents (0.3, 1 and 3 mA). The signals AL(B e) were found to be the same for the three cases which proves the linearity of the AC conductivity. Sweeping Be, the vortex system turns into the "critical" state where a non-vanishing DC current
0.014
i
|
!
i
G
a aa f o a l
0.0120.01 0.008
a
0.006
i
an
aa
u
•
00
o
a
0.004 -a aaa Oe,o
0.002 -
°° o°°o o
a
20kHz
"
*
50kHz
-
o
l(X)kl"lz
.
•
200kI-Iz
• .
0 0
0.I
0.2
0.3
0.4
0.5
0.6
B:r Fig. 3. D C field dependence of the imaginary part of A A 2 at T = 71 K and linear fits.
M. Zeisberger et al. /Physica C 250 (1995) 389-394
392 0.08
|
!
i
,
i
i
o
!
i
i
i
60
70
80
1E+05 r,
n
0.06 •
A~
10kHz
o
0.04
o o°°
50kHz
1E404
l~gI-Iz
0.02
200kl-lz 0
."i~="~T- I 0.1 0.2 0.3
I 0.4
I 0.5
I 0.6
IE.~3
30
BIT
I 40
50
T/K
Fig. 4. D C field d e p e n d e n c e o f the i m a g i n a r y part o f A A2 at T = 81 K a n d linear fits.
90
Fig. 5. Temperature dependence of the Labusch parameter and fit to the function a = oto(l_- t2) 2.
density Js becomes superimposed on the AC current density induced by the driving loop. The parameter U has to be considered as a function of this DC current density Js [21-23]. The same physical arguments should also be considereded for a and r. Consequently the parameters we determined from the experimental data are U(js), a(js) and r(js). It should be noted that the AC measurements can be performed exactly at Js = 0 operating in the fieldcooled state. But compared with the former method there appear larger errors in the A measurement because the convenient principle of observing changes of A2 by a B e variation is not available. Any hysteretic space dependence of B due to the existence of a DC current can be neglected for thin films if Be >> ~o(DjJ~)In(R/D) [25], which is the case in our experiments. Hence B can be equated by B e. Moreover, measurements with an upward and downward swept B e with different sweep rates (0.510 m T / s ) were performed resulting in the same penetration depth as a function of B. Hence there is no hysteretic time dependence. Summarizing we can state that A(A2) is uniquely determined in our experiments.
Ref. [6] is also a good fit ( s 0 = (1.1 ± 0.2) × 105 N m -2, T * = Tc = 91.0 K) to our values of a calculated from the slopes of Re A[ A2(B)]. The estimated error of a o is related with the mean deviation of the experimental values from the optimum fit curve. We note that in the interval 61 K ~< T < 81 K, those parameters which enter into Im[AA2(B)]/A 2 are treated to be independent of B because significant deviations from the linear B dependence of Im A2 could not be observed. Taken as functions of to, the slopes of Im[AAE(B)] are presented in Fig. 6 and show a to-1 behaviour, which is in agreement with Eq. (6). The relaxation time r (Table 1) which is calculated from this to-1 fits is approximately the same for 61, 71 and 81 K, which is in disagreement with Eq. (3) unless U slightly increases with increas!
1E+00
!
s
~ o
'7, IE-01
n
81K
•
71K
o
61K
1E-02
3. Results 1E-03
The linear behaviour of A[A2(B)] results in a Labusch parameter a which is independent of B including any implicit B-dependence via Js. Fig. 5 shows that the function a = ct0[1 - (T/T* ) 2 ] 2 from
I
1E~
i
Ig~5
I
IE~
O)/S"l Fig. 6. S l o p e s o f I m A A 2 ( B ) as function o f a n g u l a r f r e q u e n c y and fits to the f u n c t i o n ; t ~ / o s r .
M. Zeisberger et al. / Physica C 250 (1995) 389-394 Table 1 Relaxation times ~'0 (theoretical estimation), ~- (determined from the measured imaginary parts of A[A2(B)], parameter C, and effective activation energy U (calculated from Eq. (3)). The estimated errors are related with the mean deviation of the experimental values from the optimum fit curves T (K) ~'o (S) 61 71 81
~"(s)
C = ln(~'/~" 0) U (meV)
1.2X10 - t l (1.4+0.4)×10 -5 14.0+0.3 1.3×10 -11 (1.4+0.4)×10 -5 13.9+0.3 1.9x10 - H (1.6+0.3)×10 -5 13.6+0.2
73+2 85+2 95+2
ing temperature, or ~'0 shows a much stronger temperature dependence than it would follow from (2) and (5). Table 1 shows the relaxation times and the activation energy. The ~'0 values were calculated by Eqs. (2) and (5) using the YBa2Cu307 values of Be2(0) and Pn from Ref. [16] and assuming Bc2(T) / B~2(0) = [1 - (T/Tc)2]/[1 + (T/Tc) 2 ] [20]. The r values derived from the measured imaginary parts of A[A2(B)] and the activation energies were formally calculated by Eq. (3). From Table 1 we see that U(T) = U( js(T), T)very well satisfies U = CkT with C---14. Such a behaviour is characteristic for the quasi-steady-state dynamic magnetic-moment (and is) behaviour in flux creep experiments [21,23], where C is in agreement with the value found from dynamic magnetic-momentum relaxation measurements for YBa2Cu307_ 8 films (B e = 1 T, d B J d t = 10 m T / s , 13 K ~< T ~< 35 K) [23]. Contrary to the theoretical predictions given in Refs. [21,23], we did not observe a B dependence of C. The reason is not quite clear.
4. Conclusions In this contribution we have demonstrated that the AC penetration depth measurements by means of a mutual inductance measurement can be successfully used for simultaneously determining the pinning relevant parameters a, % U. In particular this method was used to determine these parameters for a sputtered GdBa2Cu3OT_8 film. The investigations were performed in the temperature range of 36-81 K and in the frequency range of 1-500 kHz at low DC magnetic fields (0-0.65 T), showing a field dependence of A2 which is in agreement with the theoretical model [15] with a DC field independent Labusch
393
parameter a (OrL ¢XB). The observed order of magnitude and the temperature dependence of ot is the same as found for YBa2Cu307_ ~ single crystals [6]. The imaginary part of A2 shows a frequency dependence as predicted in Ref. [15] and a temperature dependence corresponding to the relation U = CkT which is characteristic for the quasi-steady-state behaviour under flux creep conditions sweeping the external induction.
Acknowledgements This work was supported through BMVI" contract No. 13N6100. The authors thank P. G6rnert for critical discussions.
References [1] R.A. Doyle, D. Kumar, P. Pullan, R. Gross and A.M. Campbell, in: Appl. Supercond., Vol. 1, ed. H.C. Freyhard (DGM, Obemrsel, 1993) p. 689. [2] R.A. Doyle, A.M. Campbell and R.E. Somekh, Phys. Rev. Lett. 71 (1993) 4241. [3] R.A. Doyle, W.S. Scow, J.D. Johnson, D.W. Walker, A.M. Campbell, T.B. Doyle, R.E. Somekh and J.E. Evetts, Proc. 7th 1WCC, Alpbach 1994; in: Critical Currents in Superconductors, ed. H.W. Weber (World Scientific, Singapore, 1994) p. 181. [4] E.J. Tomlinson, P. Przyslupski and J.E. Evetts, Cryogenics 33 (1993) 28. [5] W.S. Seow, R.A. Doyle, J.D. Johnson, D. Kumar, R. Somekh, D.C.J. Walker and A.M. Campbell, Physica C 241 (1995) 71. [6] Dong Ho Wu and S. Sridhar, Phys. Rev. Lett. 65 (1990) 2074. [7] J.D. Johnson, A.M. Campbell, R.A. Doyle, N. Athanassopoulou and J.R. Cooper, Proc. 7th IWCC, Alpbach 1994; in: Critical Currents in Superconductors, ed. H.W. Weber (World Scientific, Singapore, 1994) p. 185. [8] P.L. Gammel, J. Appl. Phys. 67 (1990) 4676. [9] J. Kfitzler, M. Kaufmann, G. Nakielski, R. Behr and W. Assmus, Phys. Rev. Lett. 72 (1994) 2081. [10] J. K/Jtzler, G. Nakielski, M. Baumann, R. Behr, F. Goerke and E.H. Brandt, Phys. Rev. B 50 (1994) 3384. [11] Y. Ando, H. Kubota, Y. Sato and I. Terasaki, Phys. Rev. B 50 (1994) 9680. [12] P. Esquinazi and M. Ziese, in: Appl. Supercond., Vol. 1, ed. H.C. Freyhard (DGM, Oberursel, 1993) p. 709. [13] S. Revenaz, D.E. Oates, D. Labb~-Lavigne, G. Dresselhaus and M.S. Dresselhaus, Phys. Rev. B 50 (1994) 1178. [14] M. Golosovsky, M. Tsindlekht, H. Chayet and D. Davidov, Phys. Rev. B 50 (1994) 470.
394
M. Zeisberger et al. / Physica C 250 (1995) 389-394
[15] E.H. Brandt, Phys. Rev. Lett. 67 (1991) 2219. [16] D.R. Harshman and A.P. Mills Jr., Phys. Rev. B 45 (1992) 10684. [17] A.B. Pippard, Supercond. Sci. Technol. 7 (1994) 696. [18] Th. Klupsch and M. Zeisberger, Physica C 244 (1995) 153. [19] C.M. Hagen and R. Griessen, Phys. Rev. Lett. 62 (1989) 2857. [20] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. [21] H.G. Schnack, R. Griessen, J.G. Lensink and Wen Hai-Hu, Phys. Rev. B 48 (1993) 13178.
[22] R. Griessen, Wen Hai-Hu, A.J.J. van Dalen, B. Dam, J. Rector, H.G. Schnack, S. Libbrecht, E. Osquiguil and Y. Bruynderaede, Phys. Rev. Lett. 72 (1994) 1910. [23] Hai-Hu Wen, H.G. Schnack, R. Griessen, B. Dam and J. Rector, Physica C 241 (1995) 353. [24] E. Steinbeig, H. Bruchlos, Th. Eick, M. Manzei, W. Michaike, T. Schiiler and K. Steenbeck, Vacuum 44 (1993) 1113. [25] Th. Schuster, H. Kuhn, E.H. Bran&, M. Indenbom, M.R. Koblischka and M. Konczykowski, Phys. Rev. B 50 (1994) 16684.
PhysicaC250 (1995) 389-394
Frequency, field, and temperature dependence of the AC penetration depth of a GdBazCu3OT_ film in the mixed state M. Zeisberger *, Th. Klupsch, W. Michalke 1PHT Jena, PF 100 239, D-07702 Jena, Germany Received 8 May 1995; revised manuscript received 20 June 1995
Abstract
We report on a systematic mutual induction measurement of the complex AC penetration depth A of a sputtered high-quality GdBa2Cu307_ ~ film in the mixed state by a very small coil system arranged near the sample surface. The complex penetration depth A(B, T, to) for DC inductions B ~<0.65 T (perpendicular to the film), for temperatures 36 K ~
I. Introduction
The vortex dynamics can be investigated by analyzing the response with respect to small periodic, time dependent forces acting on the flux line lattice in the mixed state of a superconductor. These periodic forces can be generated by external AC currents, AC magnetic fields or mechanical vibrations of the superconductor in an applied DC magnetic field. This results in various AC techniques, e.g. AC transport [1-5], self and mutual inductance (AC susceptibility) [6-11], vibrating REED [12] and microwave surface impedance [13,14] measurements. These investigations allow to determine the elastic restoring force constant of the pinned vortices (Labusch parameter). In the following we use the
* Corresponding author.
definition of the Labusch parameter a as it is given in Ref. [6] (with the dimension N m -2). In Ref. [15], another definition of the Labusch parameter ( a L) is used where a L = a B / ~ 0 (~0 is the flux quantum, B is the DC induction in the sample). For an YBaECU307_ ~ single crystal the observed temperature dependence of ot was fitted by ot = or0(1 - t2) 2 [6] (where t = T/T* and T* = To). In Ref. [14], a theoretical estimation is given which results in the same temperature dependence and in a maximum absolute value (for YBa2Cu307_ ~) ot0 = 3 x 105 N m -2. The experimental values given in Ref. [14] agree with this estimation of c~0 but show deviations in the temperature dependence. In Refs. [6] and [13], a Labusch parameter a which is independent of B was observed. Other authors [3-5,12] reported on rather complicated dependences of a upon B. Frequency dependent measurements in the microwave range [13,14] have been used to determine the depinning frequency, i.e. the frequency where the viscous
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 3 7 7 - 0
M. Zeisberger et al. /Physica C 250 (1995) 389-394
390
forces acting on the flux lines become larger than the pinning forces. The results of the frequency dependent AC penetration depth measurements few degrees below T¢ by means of conventional AC susceptibility arrangements can be explained by the vortex glass theory and the scaling behaviour near the phase transition to the vortex liquid [9-11]. In this paper we report on measurements of the AC penetration depth A at lower temperatures (36-81 K) in the frequency range of 1-500 kHz, and for rather low applied DC inductions B e (0-0.65 T). These measurements should be interpreted in terms of the complex conductivity o- or the complex AC penetration depth )t 2 ---- i / ( / x o too-) of the superconducting sample at the angular frequency to, as it follows from the single vortex pinning assumption. This provided, the following expression for the AC penetration depth is given in Ref. [15] 1 -
A2 = A2 + A2
i/tor
1 + itor o '
(1)
with )t L a s the London penetration depth, r o as the flux flow relaxation time 7/
ro.
. O~L
Bc2~ 0
.
.
,
(2)
(1). For small frequencies (to << 1 / % ) , Eq. (1) can be approached to A2 -- A2L+ A2c(1 - i / t a r ) .
(6)
2. Experimental We have investigated a c-oriented GdBa2Cu 3film with a thickness D = 160 nm, a critical temperature T~ = 91.0 K, and a susceptometric transition width of 0.4 K, which was prepared by magnetron sputtering on a ZrO 2 substrate. Films of this type usually show critical current densities of about 106 A cm-2 (at T = 70 K, zero applied field, applying a 3 /xV/cm criterion). For further details see Ref. [24]. In our experiments we used two small coils arranged at the same sample side (Fig. 1) with B e perpendicular to the film. Such arrangements behave like completely axially symmetric arrangements with the azimuthal AC current flowing only in the (a, b)-direction if the film diameter is sufficiently larger than the coil diameter [17,18] because only the (a, b) components of the AC conductivity tensor are relevant. In Ref. [18], a theoretical analysis of this
07-8
/OnO~
r as the relaxation time of the thermally activated flux motion r = ro e x p ( V / k r )
coil
(4)
U is an effective activation energy which depends on the DC current Js which is induced in the sample by a sweep of B e. Pn is the extrapolated (for T < Tc) normal state resistivity which shows a linear temperature dependence for YBa2CtI30 7 [16]
p.( r) = p.( ro)r/ro.
rp2--'ra
I rpl--'rsl~
(3)
and Ac as the Campbell penetration depth B2 ~0 B A2 = - /x0 a L /x0 oz
I
(5)
This relation is assumed to be valid also for our GdBazCu307-8 sample. For microwave frequencies the two-fluid model has to be considered [20] so that A would become a more complicated expression than
film Ohick~ss D) Fig. 1. Arrangement of our experiments consisting of a primary coil (80 turns, rpl = 0.5 mm, rp2 = 1.7 mm, %1 = 0.41 mm, %2 = 1.25 mm), a secondary coil (10 turns, rsl = 0.5 mm, q2 = 1.7 ram, Z~l = 0.15 mm, Zs2 = 0.26 ram), and a thin film (thickness D = 0.16 mm, size 1 0 × 10 m m 2) showing a mutual inductance of L(~) = 680 nH without a sample and L(0) = 300 nH for an ideal diamagnetic sample (~t = 0). The calibration factor for the measurement of the penetration depth is f = 1.57 n i l / t z m and f / D = 9.81 nI-I//.~m2, respectively. The AC current of the primary coil was 1 mA which results in an AC induction of about 10 #T. B~ is along the symmetry axis.
M. Zeisberger et al. / Physica C 250 (1995) 389-394
arrangement is presented. Eq. (10) of Ref. [18] gives the calibration function L(A) for an elementary arrangement with two single loops (with the radii rp, r s and axial distances to the film surface zp, zs) and can be approximated by L(A) = L(0) + L, A c o t h ( D / A )
("°t
A<
0.05 0.04
(7)
•
61K
•
56K
•
51K
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bff
2/~0~p r s ( 2 b - h)
Fig. 2. DC field dependence of the real part of A A2 and linear fits.
b=zp, h = z s - Z p. (8)
The calibration factor of the two coil arrangement of Fig. 1 with 80 primary and 10 secondary turns, which results from superposition of elementary arrangements, is f - - 1.57 nH//~m. The limits of inductance of our coil system are L(0) = 300 nH and L(A ~ ~ ) = 680 nil. The experiments were done with a primary current of 1 mA which produces an AC magnetic induction of about 10 /~T. Relevant values of the penetration depth are I A I = 0.1-1 /~m and cause relative signal changes of [ L ( A ) L(O)]/L(O)-~-10-4-10 -2. Hence the absolute measurement of the complex A requires to measure L(A) and D and to calculate L(0) with a precision of 10 - 4 which is very hard to reach. The measurement of changes A(A 2) = AE(Be) - A2(0) of the squared penetration depth by changing B e and keeping the other experimental parameters constant is much easier. The constant part of mutual inductance L ( 0 ) + fA2L/D + D/3 can be eliminated in the experiment by a compensation circuit. Hence A(A 2) can be calculated from the changes of mutual inductance AL by means of
At(Be) = ( f l D ) A [ A2( Be)].
71K
66K
0
with the calibration factor
fel = ( r p _ r s ) E + ( 2 b _ h ) 2 ,
81K 76K
o
0.01
r s , Zp, Z s ) ,
a •
•
0.03
0.02
L(o) +L, 5- + -2 ' (ifl A I > D ,
391
(9)
In our experiments, we used an external compensation which allows to measure changes in the mutual inductance down to 0.03 nil. After setting the temperature and oscillator frequency and compensating the signal L(B e = 0), the change of mutual inductance was measured with
increasing DC induction (sweep rate 2 mT/s). The in-phase and out-of-phase signal measured by a lockin amplifier (PAR M5302) yield the real and imaginary part of A(A2). Figs. 2, 3, 4 show that A(A2) is found as an approximately linear function of B e in the range T~< 81 K and B e ~< 0.65 T. Deviations from linearity which indicate a decreasing Labusch parameter with B e are observed at higher temperatures. In order to examine whether the set-up operates in the linear AC regime the measurement was performed using three different primary currents (0.3, 1 and 3 mA). The signals AL(B e) were found to be the same for the three cases which proves the linearity of the AC conductivity. Sweeping Be, the vortex system turns into the "critical" state where a non-vanishing DC current
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M. Zeisberger et al. /Physica C 250 (1995) 389-394
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90
Fig. 5. Temperature dependence of the Labusch parameter and fit to the function a = oto(l_- t2) 2.
density Js becomes superimposed on the AC current density induced by the driving loop. The parameter U has to be considered as a function of this DC current density Js [21-23]. The same physical arguments should also be considereded for a and r. Consequently the parameters we determined from the experimental data are U(js), a(js) and r(js). It should be noted that the AC measurements can be performed exactly at Js = 0 operating in the fieldcooled state. But compared with the former method there appear larger errors in the A measurement because the convenient principle of observing changes of A2 by a B e variation is not available. Any hysteretic space dependence of B due to the existence of a DC current can be neglected for thin films if Be >> ~o(DjJ~)In(R/D) [25], which is the case in our experiments. Hence B can be equated by B e. Moreover, measurements with an upward and downward swept B e with different sweep rates (0.510 m T / s ) were performed resulting in the same penetration depth as a function of B. Hence there is no hysteretic time dependence. Summarizing we can state that A(A2) is uniquely determined in our experiments.
Ref. [6] is also a good fit ( s 0 = (1.1 ± 0.2) × 105 N m -2, T * = Tc = 91.0 K) to our values of a calculated from the slopes of Re A[ A2(B)]. The estimated error of a o is related with the mean deviation of the experimental values from the optimum fit curve. We note that in the interval 61 K ~< T < 81 K, those parameters which enter into Im[AA2(B)]/A 2 are treated to be independent of B because significant deviations from the linear B dependence of Im A2 could not be observed. Taken as functions of to, the slopes of Im[AAE(B)] are presented in Fig. 6 and show a to-1 behaviour, which is in agreement with Eq. (6). The relaxation time r (Table 1) which is calculated from this to-1 fits is approximately the same for 61, 71 and 81 K, which is in disagreement with Eq. (3) unless U slightly increases with increas!
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The linear behaviour of A[A2(B)] results in a Labusch parameter a which is independent of B including any implicit B-dependence via Js. Fig. 5 shows that the function a = ct0[1 - (T/T* ) 2 ] 2 from
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M. Zeisberger et al. / Physica C 250 (1995) 389-394 Table 1 Relaxation times ~'0 (theoretical estimation), ~- (determined from the measured imaginary parts of A[A2(B)], parameter C, and effective activation energy U (calculated from Eq. (3)). The estimated errors are related with the mean deviation of the experimental values from the optimum fit curves T (K) ~'o (S) 61 71 81
~"(s)
C = ln(~'/~" 0) U (meV)
1.2X10 - t l (1.4+0.4)×10 -5 14.0+0.3 1.3×10 -11 (1.4+0.4)×10 -5 13.9+0.3 1.9x10 - H (1.6+0.3)×10 -5 13.6+0.2
73+2 85+2 95+2
ing temperature, or ~'0 shows a much stronger temperature dependence than it would follow from (2) and (5). Table 1 shows the relaxation times and the activation energy. The ~'0 values were calculated by Eqs. (2) and (5) using the YBa2Cu307 values of Be2(0) and Pn from Ref. [16] and assuming Bc2(T) / B~2(0) = [1 - (T/Tc)2]/[1 + (T/Tc) 2 ] [20]. The r values derived from the measured imaginary parts of A[A2(B)] and the activation energies were formally calculated by Eq. (3). From Table 1 we see that U(T) = U( js(T), T)very well satisfies U = CkT with C---14. Such a behaviour is characteristic for the quasi-steady-state dynamic magnetic-moment (and is) behaviour in flux creep experiments [21,23], where C is in agreement with the value found from dynamic magnetic-momentum relaxation measurements for YBa2Cu307_ 8 films (B e = 1 T, d B J d t = 10 m T / s , 13 K ~< T ~< 35 K) [23]. Contrary to the theoretical predictions given in Refs. [21,23], we did not observe a B dependence of C. The reason is not quite clear.
4. Conclusions In this contribution we have demonstrated that the AC penetration depth measurements by means of a mutual inductance measurement can be successfully used for simultaneously determining the pinning relevant parameters a, % U. In particular this method was used to determine these parameters for a sputtered GdBa2Cu3OT_8 film. The investigations were performed in the temperature range of 36-81 K and in the frequency range of 1-500 kHz at low DC magnetic fields (0-0.65 T), showing a field dependence of A2 which is in agreement with the theoretical model [15] with a DC field independent Labusch
393
parameter a (OrL ¢XB). The observed order of magnitude and the temperature dependence of ot is the same as found for YBa2Cu307_ ~ single crystals [6]. The imaginary part of A2 shows a frequency dependence as predicted in Ref. [15] and a temperature dependence corresponding to the relation U = CkT which is characteristic for the quasi-steady-state behaviour under flux creep conditions sweeping the external induction.
Acknowledgements This work was supported through BMVI" contract No. 13N6100. The authors thank P. G6rnert for critical discussions.
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