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Dissipative quantum-creep studies in single-crystal La2CuO4+δ
PHYSICA ELSEVIER
Physica C 260 (1996) 283-289
Dissipative quantum-creep studies in single-crystal La 2CuO4+ 8 Lu Zhang a,*, J.Z. Liu b, I.C. Chang b, R.N. Shelton b a Department of Physics, California State University, Stanislaus, Turlock, CA 95382, USA b Department of Physics, University of California, Davis, CA 95616, USA
Received 9 October 1995; revisedmanuscriptreceived 2 February 1996
Abstract
Temperature-dependent magnetic relaxation was studied in a La2CuO4+ 8 crystal in the low-temperature range 2-10 K. The experimental data exhibited a non-vanishing magnetic relaxation in the low-temperature limit. These data could be well interpreted by a quantum-creep theory, suggesting the non-vanishing relaxation at low temperatures due to quantum tunneling of vortices. The effective Euclidean action determined as h / l d In M / d In t[ showed a quadratic temperature dependence. The quantitative analysis of the effective Euclidean action yields a classical activation energy, crossover temperature and zero-temperature quantum-tunneling rate.
1. Introduction
The magnetic properties of type-II superconductors in the mixed state are determined by the static and dynamic properties of vortices. Material defects and inhomogeneities acting as pinning centers can serve as effective traps for the vortices. In the static configuration, randomly arranged pinning centers cause a deformation of the flux-line lattice. The interaction between the flux-line lattice and material defects determines the pinning force. The critical state is established when the pinning force is balanced with the Lorentz force due to a density gradient in the vortices. However, the critical state characterized by a non-uniform vortex density is metastable and decays by flux motion in order to achieve a state with a more uniform vortex density.
* Corresponding author. Fax: + 1-20%667-3099.
The classical theories [1-4] concentrate on flux creep due to thermally activated vortex motion near the critical state. At finite temperature, thermal activation may allow flux lines to escape from their potential wells. This explains the decay of the critical state and strong temperature dependence of the critical current. The thermally activated flux creep leads to a slow change in the trapped field as a function of time, and this time dependence is observed to be logarithmic in experiments. The classical flux-creep theory indicates a vanishing magnetic relaxation rate at zero temperature. Extensive studies of magnetic relaxation in the high-T~ superconductors [5-8] have shown that relaxation is significantly larger (giant creep) than that observed in conventional superconductors. Most experiments showed a linear temperature dependence of the relaxation rate in a certain temperature region, which suggested a temperature-independent thermal activation energy. However, extrapolation of the measured relaxation rate yielded a finite value at low
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L. Zhang et a l . / Physica C 260 (1996) 283-289
temperatures [9-14], indicating that the rate does not vanish when T---, 0. The observation of relaxation at low temperatures cannot be explained within the framework of thermally activated flux-creep theories. At sufficiently low temperatures where the thermal processes are frozen out, some other mechanism such as the quantum tunneling of vortices is expected to dominate the relaxation process. Recently, Blatter and Geshkenbein [15] proposed a collective-pinning theory to describe flux motion, including quantum tunneling. These authors suggested a dissipative quantum creep of vortex motion and predicted a T 2 law for the effective Euclidean action in the quantum regime. Their theory predicted a large tunneling rate for the materials characterized by a small coherence length and a large normal-state resistivity. Magnetic-relaxation experiments at low temperatures were carded out in YBa2Cu3OT_ 8 and Bi2Sr2CaCu208 crystals by several groups [16-18]. They reported the existence of strong low-temperature relaxation. These materials are superconducting above 90 K and have structures of two layers of CuO planes. In order to investigate the mechanism of the low-temperature relaxation, the measured relaxation rates were used to compare with theoretical values predicted by quantum-creep theory. However, the absolute values of the quantum-creep rate from theoretical calculations depend on several parameters: the normal-state resistivity On, the coherence length ~, the anisotropy parameter E, and the ratio of the depalring to critical current density Jo/J~, which can be only determined from various experiments. These parameters usually vary from sample to sample in the same compound and could be different by 1-2 orders of magnitude. Quantitative analysis of relaxation experiments based on quantum-creep theory can be performed by studying the temperature-dependent relaxation rates. This can avoid the use of data from other experiments. In addition, some important parameters such as the classical activation energy, crossover temperature and zero-temperature quantum tunneling rate can be determined. The La2CuO4+ 8 compound is characterized by a large normal-state resistivity and a small coherence length, features which are expected to favor the
quantum tunneling of vortices. In addition, the La2CuO4+ 8 compound is composed of three elements and includes one CuO plane in a unit cell which has a much simpler structure compared with YBa2Cu307_8 and Bi2Sr2CaCu20 s. This material may be a good testing ground for our understanding of quantum-creep behavior. In this paper, we present new experimental data down to 2 K on a well-characterized crystal of La2CuO4+ 8. The results exhibited a strong non-vanishing magnetic relaxation rate at low temperatures. The finite dissipation of the quantum tunneling rate observed in La2CuO4+ 8 is reported. The effective Euclidean action at finite temperature was determined to be T 2 dependent, which was consistent with the thermal enhancement of the quantum creep predicted by quantum-creep theory [15].
2. Theoretical background The classical theories of thermally activated flux creep developed by Anderson et al. [1], Beasley et al. [3], and Campbell et al. [4] are used frequently to analyze the relaxation data of high-Tc superconductors. In these theories, the thermally activated fluxcreep rate for the magnetization in the critical state defined as R c = (1/Mo)dM/d In t can be obtained:
IL = kBr/V ,
(1)
where M 0 is the initial value of the magnetization M(t), Uc the classical activation energy, and ./co is the critical current if no thermal activation occurs (T = 0). The flux-creep studies for conventional superconductors can be well explained by the classical theories of thermally activated flux creep in the critical state. However, the rate of flux creep in high-Tc superconductors was observed to be large and had a strong temperature dependence. This behavior affects the current density at f'mite temperature significantly. It is generally accepted that the experiments of both magnetic and transport properties in high-Tc superconductors are performed well below the critical state, J 4~ Jc. The dependence of the activation bartier U on the current J was shown to be non-linear [19,20]. This non-linear dependence of activation
L. Zhang et al. / Physica C 260 (1996) 283-289 energy U ( J ) might be manifest in the thermally activated flux creep of vortices in high-Tc superconductors. Feigel'man et al. [21] and Vinokur et al. [22] proposed a collective-pinning model on the assumption of a power-law or a logarithmic current dependence for U(J). They developed the non-logarithmic behavior for the magnetic relaxation in type-II superconductors, which was consistent with the experimental observations in a long time period (up to 106 s) [14]. At zero temperature, the collective pinning model for thermally activated flux motion also predicted a zero relaxation rate. Our previous studies of flux creep on the high-T~ superconductors were carried out in YBa2Cu307_ 8, LuBa2CU3OT-8, Bi2Sr2CaCu208 and La2CuO,+ 8 systems [14]. The results indicated that the flux-creep rates were large and had a strong temperature dependence, which was consistent with the thermally activated flux-creep theory. However, all the relaxation experiments yielded a finite intercept of the fluxcreep rate by extrapolating linearly to zero temperature. This indicates that the rate R does not vanish when T ~ 0, which corresponds to a non-thermally activated flux creep, possibly due to quantum creep of vortices. Blatter and Geshkenbein [15] treated the quantum creep within the framework of the weak-collectivepinning theory. In high-Tc superconductors, weak collective pinning by point-like oxygen defects is the main pinning source. The vortex is pinned by the collective action of defects within a characteristic pinning length L c. The vortex pinning produces a potential barrier Uc against motion to an adjacent metastable state. In an elementary tunneling process, a vortex segment of length L c tunnels under the barrier U¢ to its neighboring state. The effective Euclidean action, S~ff, which determines the relaxation rate of quantum creep, Rq, in the limit of single vortex pinning and strong damping can be calculated as 1
h Lc
S~'e/h = qR-- = --'~ e 7'
(2)
where L c --~l~(jo/jc) I/2 is the collective pinning length near the critical state, p, the normal-state resistivity, ~ the coherence length, e the anisotropy
285
parameter, J0 and J¢ the depairing and critical current density, respectively. The quantum-creep rate is enhanced in a superconductor with short coherence length, high normal-state resistivity, weak collective pinning and strong anisotropy. According to quantum collective creep theory, the quantum tunneling can be thermally assisted, which yields an increase of the relaxation rate as temperature increases. The finite temperature correction to the Euclidean action in the limit of strong dissipation is shown to be [15] As~ff(r) = s~ff(T) - s~ff(o) -- -s~ff(o)
T
where t c is the characteristic tunneling time, which can be obtained approximately from tcU¢ .~. S~ff (0). Thus the Euclidean action at finite temperature is written as s~ff( T ) ~ S ~ f f ( 0 ) -
seff[o,3[ k B T I 2 E~ , ~ ' ~ c ] "
The crossover temperature Tqc As~ff(T)/S~ ff (0)= 1 is given as
(3) defined
by
Vc Tq¢ ~- k a S ~ f f ( o ) / h .
(4)
As Blatter and Geshkenbein mentioned in the quantum-creep theory [15], this crossover temperature can be used approximately to distinguish two regimes of vortex motion: quantum tunneling and thermal activation. At low temperatures, T < Tqc, the quantum tunneling is always dominant, and the creep rate due to quantum flux motion is expected to be temperature dependent.
3. E x p e r i m e n t a l details
A single crystal of La2CuO4 was grown in a CuO flux as described previously [23]. The crystal had a mass of 15.4 mg and dimensions of 2.5 × 1.2 × 0.8 mm 3 with the c-axis along the shortest dimension. Doping of this crystal can be obtained by introducing excess oxygen in interstitial sites to produce La2CuO4+ ~ (8 < 0.12). The crystal was annealed at 6 kbar 02 pressure at 600°C for 24 h. From previous
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L. Zhang et al. / Physica C 260 (1996) 283-289
work on similarly prepared crystals, the oxygen excess in the resulting La2CuO4+ 8 crystal was estimated to be 8 = 0.03. The superconducting transition temperature, as determined from DC susceptibility measurements, was 24 K. The magnetic properties were measured on a superconducting quantum interference device (SQUID) magnetometer [24]. The scan length, the distance which a specimen travels through a set of detection coils, was set at 3.0 cm. This relatively short distance minimized the magnetic variation where the specimen travels. Variation in field at this setting was estimated to be < 0.05%. An iterative regression mode was used to calculate the magnetization. In order to measure the magnetic relaxation in the fully penetrated sample, we first determined the penetration field H *, which can be taken to be roughly the field at which the peak in the M versus H curve occurs. The value of H * is measured as ~ 0.1 T at 2 K and decreases as temperature increases. At higher temperatures, the field easily penetrates into the sample from its sides and collapses at the center to form the fully penetrated state in a short time. Therefore, for all of the relaxation measurements, the applied field was chosen to be 0.4 T, to insure a well-defined penetrated state for the field oriented parallel to the c-axis of the crystal. All the magnetization measurements were made by first cooling the sample in zero field and then applying a field using the SQUID no overshoot mode to begin the magnetic-decay experiment. However, there may be some residual field left in the superconducting magnet (remnant field) even though the superconducting magnet is reset. To reduce the effect of the remnant field, the sample was first held 6 - 8 cm above the magnet and cooled down to a temperature well below the transition temperature to insure that the sample was in the superconducting state under zero field. Then the sample was slowly moved downward to the desired position. Any small remnant field would not affect the sample since it was well below He1, the lower critical field. After the temperature stabilized, there was another 120 s waiting time to allow the sample to reach a completely stable state. Then a magnetic field was carefully applied in a no overshoot mode. The fluctuation (variation) of field in this mode was estimated to be less than 0.5 G.
4. R e s u l t s a n d d i s c u s s i o n
The relaxation data were taken over a time period up to ~ 104 s at temperatures between 2 and 15 K. A normal-state baseline was observed to be temperature independent from T~ to 50 K, which was the background due to the sample holder and could be subtracted from the data. Fig. 1 gives the resultant magnetic-relaxation data of the LaECUO4+8 crystal for the field oriented parallel to the c-axis at H = 0.4 T in a log-log plot. An experimental uncertainty of ± 0.5% is estimated from the measurement. The solid line in the figure represents a linear fit between In M and In t, which is what would be expected based on extended thermally activated flux creep theory [22]. The temperature dependence of the logarithmic relaxation rate R = Id In M/d In t l is given in Fig. 2, which is obtained from the slope of each curve of Fig. 1. The relaxation rates increases more than a factor of 3 ( > 300%) between 6 and 10 K. At temperatures higher than 12 K, the relaxation appears to be negligible within the accuracy of the measurement ( < 0.5%). It could be expected that vortex-lattice melting occurs in this material above 12 K. Fig. 2 shows that the relaxation rate is linearly increasing with temperature above 6 K. The strong temperature dependence of the relaxation rates at high temperatures reflects the fact that giant flux creep is induced by thermally activated flux motion in the classical regime. The large values of the relaxation rate also
100 [] o
[] o
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• 2.1K [] 2.5K
+
o 2.9K •
o "~
x
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+ 3.7K
>~
A4K o 4.5K
LazCu04.
sC
× 5K
HllcH= I0 101
102
103
104
t (s) Fig. 1. Magnetic time relaxation of a La2CuO4+ S crystal at 0.4 T in a log-log scale. The solid lines mpmsem tmwer-law fits at different temperatures.
L. Zhang et al. / Physica C 260 (1996) 283-289
0.1
0.3
o.M
LasCuO4+ s Crystal __
287
H II c
0.2
.
.
.
.
.
.
.
-
H f f i 0 . 4 T / ,
s
•
'r~'
'
'
'"
±
0.1
Hilt H ; 0.4T 2
3
4
5
6
7
8
9
10
T (K) Fig. 2. The temperature dependence of the relaxation rate R = Id In M i d in t l for H = 0.4 T with field applied parallel to the c-axis over the temperature range of 2 - 1 0 K. The relaxation rate is saturated withIn 10% at low temperatures (T < 6 K). The lInear fit of the relaxation rate is shown between 7 and 10 K, which can be explained by thermally activated flux-creep theory. The temperature-Independent activation energy, Uc = 18 meV, is extracted from the lInear temperature dependence of relaxation rate.
indicate a small potential barrier against flux jumping to the adjacent metastable state. This may also introduce noticeable quantum tunneling of vortices under the barrier. The classical activation energy, Uc, is determined to be 18 meV by using the thermally activated flux-creep theory. At low temperatures, T < 6 K, we observe a saturation of the creep rates as shown in Fig. 2, which is about 0.08 within 10% scattering. This indicates that the rate R does not vanish when T - ~ 0, which corresponds to non-thermally activated flux creep. In the thermally activated flux-creep theories, magnetic relaxation of type-II superconductors should disappear at zero temperature. Our results exhibit a low-temperature magnetic relaxation in the LaECuO4+ ~ compound. In order to study the mechanism of quantum creep and provide reliable information from relaxation experiments, such as the potential barrier height and crossover temperature, a quantitative comparison between the experimental results and theoretical analysis is needed. In previous studies [16-18], the parameters used to calculate the theoretical relaxation rate, such as Pn, ~, ~, Jo/Jc, were determined from various experiments. However, these parameters usually vary from sample to sample in the same compound. The variation can be 1 - 2 orders of magnitude differ-
0.07
'
2
'
3
'
4
5
6
T (K) Fig. 3. The temperature dependence of the relaxation rate between 2 and 6 K with a quadratic fit presen~l by the solid line. The inset shows the deviations from the quadratic temperature dependence at high temperatures above 6 K.
ent. Therefore, a quantitative study of the quantum creep rate could be approached by analyzing the temperature dependence of relaxation rate instead of calculating an absolute value based on other experiments. The relaxation rates at low temperatures can be considered as temperature independent within a 10% range. However, the relaxation rates increase monotonically as temperature increases, similar to recent observation in YBa2Cu307_ ~ detwinned crystal [25]. The experimental relaxation rates in the temperature
16
LalCuO4. ~Crystal 14
10
2
,
t
,
3
4
5
6
T (K) Fig. 4. The temperature dependence of the effective Euclidean action S~f ( T ) / h at 0.4 T. The solid line represents a quadratic temperature dependence, which yields a classical activation energy (U~ = 15 meV), crossover temperature (Tqc = 7 K) and zerotemperature quantum-tunneling rate (R = 0.07).
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L. Zhang et a l . / Physica C 260 (1996) 283-289
range of 2-6 K is best fit with the quadratic function indicated by the solid line in Fig. 4. This is expected by the thermally enhanced quantumZcreep theory. Further scrutiny of the quadratic dependence reveals that the quality of fit is seen to deteriorate (within 5%) for the relaxation rate taken at temperatures above 6-7 K (see inset of Fig. 3). The deviation of a quadratic fit at high temperatures may be due to the effect of thermally activated flux creep. For small and intermediate field, the interaction between the vortices is small compared with the interaction of a single vortex with the pinning centers. This is the single-vortex regime. As the external field increased, the vortex separation decreases and interaction between neighboring vortices becomes more important. For the oxide superconductors it is expected that the single vortex regime extends to fields of the order of l0 T [15]. Our relaxation experiments on the L a 2 C u O 4 + 8 c r y s t a l w e r e p e r formed at 0.4 T. This is in the single-vortex regime and, therefore, the weak collective-pinning theory by Blatter and Geshkenbein [15] should be applicable in the present study. A quadratic temperature-dependent Euclidean action s[ff(T)/h = 1 / [ d In M/d In t[ is plotted in Fig. 4 where the solid line represents a quadratic fit. The zero Euclidean action, s[ff(o)/h, and classical activation energy, Uc, are determined using Eq. (3). The classical activation energy of 15 meV is obtained. This is in agreement with the result of 18 meV obtained in the classical regime. The zero Euclidean action is determined as 14, which is a unitless quality. A quantum-tunneling rate at T = 0 is calculated as large as 0.07. In contrast, the zero-temperature quantum-creep rates were estimated to be 0.01 in YBa2Cu3OT_ 8 [17] and 0.02 in Bi2SrECaCu208 [18]. The quantum-creep theory predicts that the quantum-creep rate is enhanced in a superconductor with weak collective pinning [16]. The relative larger value of the tunneling rate in the L a E C U O 4 + ~ compound reflects the flux pinning being weaker in one-layer superconductor than those with two layers of CuO planes. The crossover temperature is estimated as 7 K by using Eq. (4). At low temperatures, T <: 7 K, the flux creep due to quantum tunneling is predominant so that the creep rate can be quantitatively described by the quantum-creep theory. At T > 7 K, the flux motion due to thermal
LasCuO,+ 8 Crystal H I Ic
0
H = 0.4T
i
i
i
3
4
5
6
T (K) Fig. 5. The dissipative term of the effective Euclidean action. The solid line indicates a T 2 fitting to E,q. (3).
activation becomes important. The contribution of zero-temperature quantum tunneling can be subtracted from the total flux creep in the classical regime. The dissipative term of the effective Euclidean action, AS[ff(T)/h, is plotted in Fig. 5. The creep rate due to thermally assisted quantum tunneling follows a T 2 law, which is consistent with the quantum creep theory [15]. The flux-creep rate in the classical regime was observed to be large and to have a strong temperature-dependence in high-T~ superconductors. This behavior affects the current density at finite temperature significantly. It is generally accepted that the experiments of both magnetic and transport properties in bigh-Tc superconductors are performed well below the critical state, J "~ Jc. This leads to a field dependence of the magnetization and the magnetic relaxation in the classical regime [26]. The systematic study of the field dependence of quantum creep needs to be further investigated and will be presented in a future publication.
5. Conclusion Three questions have been addressed. The first is whether quantum flux creep occurs. The saturation of the creep rate at low temperatures reveals its existence in one-layer superconductor La2CuO4+ s. The second question is whether the quantum creep is thermally enhanced. A quadratic temperature dependence of the effective Euclidean action is obtained,
L. Zhang et al. / Physica C 260 (1996) 283-289
which is consistent with the quantum-creep theory of Blatter and Geshkenbein. Third, we observe the quantum-classical crossover around 6 K and the analysis of the temperature-dependent relaxation data based on quantum-creep theory yields a classical activation energy, crossover temperature and zerotemperature quantum-tunneling rate.
Acknowledgements This work is supported by the National Science Foundation under grant number DMR-94-03895 and by the U.S. Air Force Office of Scientific Research under grant number AFOSR-F49620-92-J-0514.
References [1] P.W. Anderson, Phys. Rev. Lett. 9 (1962) 309; P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. [2] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [3] M.R. Beasley, R. Labush and W.W. Webb, Phys. Rev. 181 (1969) 682. [4] A.M. Campbell and J.E. Evetts, Adv. Phys. 21 (1972) 199. [5] K.A. MEller, M. Takashige and J.G. Bednorz, Phys. Rev. Lett. 58 (1987) 1143. [6] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lett. 60 (1987) 2202. [7] Y. Yeshurun, A.P. Malozemoff, F.H. Holtzberg and T.R. Dinger, Phys. Rev. B 38 (1988) 11828. [8] A.P. Malozemoff, in: Physical Properties of High Temperature Superconductors, vol. 1, ed. D.M. Ginsberg (World Scientific, Singapore, 1989) p. 71. [9] A.C. Mota, A. Pollini, P. Visani, K.A. M~ller and J.G. Bednorz, Phys. Rev. B 36 (1987) 4011.
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[10] R. Griessen, J.G. Lensink, T.A.M. Schroeder and B. Dam, Cryogenics 30 (1990) 561. [11] A. Hamzic, L. Fruchter and I.A. Campbell, Nature (London) 345 (1990) 515. [12] L. Civale, A.D. Marwick, M.W. McElfresh, T.K. Worthington, A.P. Malozemoff, F.H. Holtzberg, J.R. Thompson and M.A. Kirk, Phys. Rev. Lett. 65 (1991) 1164. [13] H. Furukawa, K. Kawaguchi and M. Nakao, Adv. Supercond. vol. II, eds. T. Ishiguro and K. Kajimura (Springer, Heidelberg, 1990) p. 679. [14] J.Z. Liu, L. Zhang, M.D. Lan and R.N. Shelton, Phys. Rev. B 46 (1992) 9123; L. Zhang, J.Z. Liu, M.D. Lan and R.N. Shelton, Phys. Rev. B 50 (1994) 7092; idem, Physica C 235 (1994) 2897. [15] G. Blatter and V.B. Geshkenbein, Phys. Rev. B 47 (1993) 2725. [16] D. Prost, L. Fruchter, I.A. Cambeil, N. Motohira and M. Konczykowski, Phys. Rev. B 47 (1993) 3457. [17] L. Fruchter, A.P. Malozemoff, I.A. Campbell, J. Sanchez, M. Konczykowski, R. Griessen and F. Holtzberg, Phys. Rev B 43 (1991) 8709. [18] A.C. Mota, K. Aupke, A. Amann, T. Teruzzi, A. Pollini and P. Visani, J Low Tern. Phys. 95 (1994) 377. [19] E. Zeldov, N.M. Amer, G. Koren, A. Gupta, M.W. McE1fresh and E.J. Gambino, Appl. Phys. Lett. 56 (1990) 680. [20] M.P. Maley, J.O. Willis, H. Lessure and M.E. McHenry, Phys. Rev. B 42 (1990) 2639. [21] M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Phys. Rev. Lett. 63 (1989) 2303. [22] V.M. Vinokur, M.V. Feigel'man and V.B. Geshkenbein, Phys. Rev. Lett. 67 (1991) 915. [23] J.F. Shu, J. Akella, J.Z. Liu, H.K. Mao and L.W. Finger, Physica C 176 (1991) 503. [24] Quantum Design Inc., San Diego, CA 92121. [25] G.T. Seidler, T.F. Tosenbaum, K.M. Beauchamp, H.M. Jaeger, G.W. Crabtree, U. Welp and V.M. Vinokur, Phys. Rev. Lett. 74 (1995) 1442. [26] L. Zhang, J.Z. Liu, M.D. Lan, P. Klavins and R.N. Shelton, Phys. Rev. B 44 (1991) 10190.
Physica C 260 (1996) 283-289
Dissipative quantum-creep studies in single-crystal La 2CuO4+ 8 Lu Zhang a,*, J.Z. Liu b, I.C. Chang b, R.N. Shelton b a Department of Physics, California State University, Stanislaus, Turlock, CA 95382, USA b Department of Physics, University of California, Davis, CA 95616, USA
Received 9 October 1995; revisedmanuscriptreceived 2 February 1996
Abstract
Temperature-dependent magnetic relaxation was studied in a La2CuO4+ 8 crystal in the low-temperature range 2-10 K. The experimental data exhibited a non-vanishing magnetic relaxation in the low-temperature limit. These data could be well interpreted by a quantum-creep theory, suggesting the non-vanishing relaxation at low temperatures due to quantum tunneling of vortices. The effective Euclidean action determined as h / l d In M / d In t[ showed a quadratic temperature dependence. The quantitative analysis of the effective Euclidean action yields a classical activation energy, crossover temperature and zero-temperature quantum-tunneling rate.
1. Introduction
The magnetic properties of type-II superconductors in the mixed state are determined by the static and dynamic properties of vortices. Material defects and inhomogeneities acting as pinning centers can serve as effective traps for the vortices. In the static configuration, randomly arranged pinning centers cause a deformation of the flux-line lattice. The interaction between the flux-line lattice and material defects determines the pinning force. The critical state is established when the pinning force is balanced with the Lorentz force due to a density gradient in the vortices. However, the critical state characterized by a non-uniform vortex density is metastable and decays by flux motion in order to achieve a state with a more uniform vortex density.
* Corresponding author. Fax: + 1-20%667-3099.
The classical theories [1-4] concentrate on flux creep due to thermally activated vortex motion near the critical state. At finite temperature, thermal activation may allow flux lines to escape from their potential wells. This explains the decay of the critical state and strong temperature dependence of the critical current. The thermally activated flux creep leads to a slow change in the trapped field as a function of time, and this time dependence is observed to be logarithmic in experiments. The classical flux-creep theory indicates a vanishing magnetic relaxation rate at zero temperature. Extensive studies of magnetic relaxation in the high-T~ superconductors [5-8] have shown that relaxation is significantly larger (giant creep) than that observed in conventional superconductors. Most experiments showed a linear temperature dependence of the relaxation rate in a certain temperature region, which suggested a temperature-independent thermal activation energy. However, extrapolation of the measured relaxation rate yielded a finite value at low
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved PII S0921-4534(96)00131-1
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L. Zhang et a l . / Physica C 260 (1996) 283-289
temperatures [9-14], indicating that the rate does not vanish when T---, 0. The observation of relaxation at low temperatures cannot be explained within the framework of thermally activated flux-creep theories. At sufficiently low temperatures where the thermal processes are frozen out, some other mechanism such as the quantum tunneling of vortices is expected to dominate the relaxation process. Recently, Blatter and Geshkenbein [15] proposed a collective-pinning theory to describe flux motion, including quantum tunneling. These authors suggested a dissipative quantum creep of vortex motion and predicted a T 2 law for the effective Euclidean action in the quantum regime. Their theory predicted a large tunneling rate for the materials characterized by a small coherence length and a large normal-state resistivity. Magnetic-relaxation experiments at low temperatures were carded out in YBa2Cu3OT_ 8 and Bi2Sr2CaCu208 crystals by several groups [16-18]. They reported the existence of strong low-temperature relaxation. These materials are superconducting above 90 K and have structures of two layers of CuO planes. In order to investigate the mechanism of the low-temperature relaxation, the measured relaxation rates were used to compare with theoretical values predicted by quantum-creep theory. However, the absolute values of the quantum-creep rate from theoretical calculations depend on several parameters: the normal-state resistivity On, the coherence length ~, the anisotropy parameter E, and the ratio of the depalring to critical current density Jo/J~, which can be only determined from various experiments. These parameters usually vary from sample to sample in the same compound and could be different by 1-2 orders of magnitude. Quantitative analysis of relaxation experiments based on quantum-creep theory can be performed by studying the temperature-dependent relaxation rates. This can avoid the use of data from other experiments. In addition, some important parameters such as the classical activation energy, crossover temperature and zero-temperature quantum tunneling rate can be determined. The La2CuO4+ 8 compound is characterized by a large normal-state resistivity and a small coherence length, features which are expected to favor the
quantum tunneling of vortices. In addition, the La2CuO4+ 8 compound is composed of three elements and includes one CuO plane in a unit cell which has a much simpler structure compared with YBa2Cu307_8 and Bi2Sr2CaCu20 s. This material may be a good testing ground for our understanding of quantum-creep behavior. In this paper, we present new experimental data down to 2 K on a well-characterized crystal of La2CuO4+ 8. The results exhibited a strong non-vanishing magnetic relaxation rate at low temperatures. The finite dissipation of the quantum tunneling rate observed in La2CuO4+ 8 is reported. The effective Euclidean action at finite temperature was determined to be T 2 dependent, which was consistent with the thermal enhancement of the quantum creep predicted by quantum-creep theory [15].
2. Theoretical background The classical theories of thermally activated flux creep developed by Anderson et al. [1], Beasley et al. [3], and Campbell et al. [4] are used frequently to analyze the relaxation data of high-Tc superconductors. In these theories, the thermally activated fluxcreep rate for the magnetization in the critical state defined as R c = (1/Mo)dM/d In t can be obtained:
IL = kBr/V ,
(1)
where M 0 is the initial value of the magnetization M(t), Uc the classical activation energy, and ./co is the critical current if no thermal activation occurs (T = 0). The flux-creep studies for conventional superconductors can be well explained by the classical theories of thermally activated flux creep in the critical state. However, the rate of flux creep in high-Tc superconductors was observed to be large and had a strong temperature dependence. This behavior affects the current density at f'mite temperature significantly. It is generally accepted that the experiments of both magnetic and transport properties in high-Tc superconductors are performed well below the critical state, J 4~ Jc. The dependence of the activation bartier U on the current J was shown to be non-linear [19,20]. This non-linear dependence of activation
L. Zhang et al. / Physica C 260 (1996) 283-289 energy U ( J ) might be manifest in the thermally activated flux creep of vortices in high-Tc superconductors. Feigel'man et al. [21] and Vinokur et al. [22] proposed a collective-pinning model on the assumption of a power-law or a logarithmic current dependence for U(J). They developed the non-logarithmic behavior for the magnetic relaxation in type-II superconductors, which was consistent with the experimental observations in a long time period (up to 106 s) [14]. At zero temperature, the collective pinning model for thermally activated flux motion also predicted a zero relaxation rate. Our previous studies of flux creep on the high-T~ superconductors were carried out in YBa2Cu307_ 8, LuBa2CU3OT-8, Bi2Sr2CaCu208 and La2CuO,+ 8 systems [14]. The results indicated that the flux-creep rates were large and had a strong temperature dependence, which was consistent with the thermally activated flux-creep theory. However, all the relaxation experiments yielded a finite intercept of the fluxcreep rate by extrapolating linearly to zero temperature. This indicates that the rate R does not vanish when T ~ 0, which corresponds to a non-thermally activated flux creep, possibly due to quantum creep of vortices. Blatter and Geshkenbein [15] treated the quantum creep within the framework of the weak-collectivepinning theory. In high-Tc superconductors, weak collective pinning by point-like oxygen defects is the main pinning source. The vortex is pinned by the collective action of defects within a characteristic pinning length L c. The vortex pinning produces a potential barrier Uc against motion to an adjacent metastable state. In an elementary tunneling process, a vortex segment of length L c tunnels under the barrier U¢ to its neighboring state. The effective Euclidean action, S~ff, which determines the relaxation rate of quantum creep, Rq, in the limit of single vortex pinning and strong damping can be calculated as 1
h Lc
S~'e/h = qR-- = --'~ e 7'
(2)
where L c --~l~(jo/jc) I/2 is the collective pinning length near the critical state, p, the normal-state resistivity, ~ the coherence length, e the anisotropy
285
parameter, J0 and J¢ the depairing and critical current density, respectively. The quantum-creep rate is enhanced in a superconductor with short coherence length, high normal-state resistivity, weak collective pinning and strong anisotropy. According to quantum collective creep theory, the quantum tunneling can be thermally assisted, which yields an increase of the relaxation rate as temperature increases. The finite temperature correction to the Euclidean action in the limit of strong dissipation is shown to be [15] As~ff(r) = s~ff(T) - s~ff(o) -- -s~ff(o)
T
where t c is the characteristic tunneling time, which can be obtained approximately from tcU¢ .~. S~ff (0). Thus the Euclidean action at finite temperature is written as s~ff( T ) ~ S ~ f f ( 0 ) -
seff[o,3[ k B T I 2 E~ , ~ ' ~ c ] "
The crossover temperature Tqc As~ff(T)/S~ ff (0)= 1 is given as
(3) defined
by
Vc Tq¢ ~- k a S ~ f f ( o ) / h .
(4)
As Blatter and Geshkenbein mentioned in the quantum-creep theory [15], this crossover temperature can be used approximately to distinguish two regimes of vortex motion: quantum tunneling and thermal activation. At low temperatures, T < Tqc, the quantum tunneling is always dominant, and the creep rate due to quantum flux motion is expected to be temperature dependent.
3. E x p e r i m e n t a l details
A single crystal of La2CuO4 was grown in a CuO flux as described previously [23]. The crystal had a mass of 15.4 mg and dimensions of 2.5 × 1.2 × 0.8 mm 3 with the c-axis along the shortest dimension. Doping of this crystal can be obtained by introducing excess oxygen in interstitial sites to produce La2CuO4+ ~ (8 < 0.12). The crystal was annealed at 6 kbar 02 pressure at 600°C for 24 h. From previous
286
L. Zhang et al. / Physica C 260 (1996) 283-289
work on similarly prepared crystals, the oxygen excess in the resulting La2CuO4+ 8 crystal was estimated to be 8 = 0.03. The superconducting transition temperature, as determined from DC susceptibility measurements, was 24 K. The magnetic properties were measured on a superconducting quantum interference device (SQUID) magnetometer [24]. The scan length, the distance which a specimen travels through a set of detection coils, was set at 3.0 cm. This relatively short distance minimized the magnetic variation where the specimen travels. Variation in field at this setting was estimated to be < 0.05%. An iterative regression mode was used to calculate the magnetization. In order to measure the magnetic relaxation in the fully penetrated sample, we first determined the penetration field H *, which can be taken to be roughly the field at which the peak in the M versus H curve occurs. The value of H * is measured as ~ 0.1 T at 2 K and decreases as temperature increases. At higher temperatures, the field easily penetrates into the sample from its sides and collapses at the center to form the fully penetrated state in a short time. Therefore, for all of the relaxation measurements, the applied field was chosen to be 0.4 T, to insure a well-defined penetrated state for the field oriented parallel to the c-axis of the crystal. All the magnetization measurements were made by first cooling the sample in zero field and then applying a field using the SQUID no overshoot mode to begin the magnetic-decay experiment. However, there may be some residual field left in the superconducting magnet (remnant field) even though the superconducting magnet is reset. To reduce the effect of the remnant field, the sample was first held 6 - 8 cm above the magnet and cooled down to a temperature well below the transition temperature to insure that the sample was in the superconducting state under zero field. Then the sample was slowly moved downward to the desired position. Any small remnant field would not affect the sample since it was well below He1, the lower critical field. After the temperature stabilized, there was another 120 s waiting time to allow the sample to reach a completely stable state. Then a magnetic field was carefully applied in a no overshoot mode. The fluctuation (variation) of field in this mode was estimated to be less than 0.5 G.
4. R e s u l t s a n d d i s c u s s i o n
The relaxation data were taken over a time period up to ~ 104 s at temperatures between 2 and 15 K. A normal-state baseline was observed to be temperature independent from T~ to 50 K, which was the background due to the sample holder and could be subtracted from the data. Fig. 1 gives the resultant magnetic-relaxation data of the LaECUO4+8 crystal for the field oriented parallel to the c-axis at H = 0.4 T in a log-log plot. An experimental uncertainty of ± 0.5% is estimated from the measurement. The solid line in the figure represents a linear fit between In M and In t, which is what would be expected based on extended thermally activated flux creep theory [22]. The temperature dependence of the logarithmic relaxation rate R = Id In M/d In t l is given in Fig. 2, which is obtained from the slope of each curve of Fig. 1. The relaxation rates increases more than a factor of 3 ( > 300%) between 6 and 10 K. At temperatures higher than 12 K, the relaxation appears to be negligible within the accuracy of the measurement ( < 0.5%). It could be expected that vortex-lattice melting occurs in this material above 12 K. Fig. 2 shows that the relaxation rate is linearly increasing with temperature above 6 K. The strong temperature dependence of the relaxation rates at high temperatures reflects the fact that giant flux creep is induced by thermally activated flux motion in the classical regime. The large values of the relaxation rate also
100 [] o
[] o
• ,~
• 2.1K [] 2.5K
+
o 2.9K •
o "~
x
O x
3.3K
+ 3.7K
>~
A4K o 4.5K
LazCu04.
sC
× 5K
HllcH= I0 101
102
103
104
t (s) Fig. 1. Magnetic time relaxation of a La2CuO4+ S crystal at 0.4 T in a log-log scale. The solid lines mpmsem tmwer-law fits at different temperatures.
L. Zhang et al. / Physica C 260 (1996) 283-289
0.1
0.3
o.M
LasCuO4+ s Crystal __
287
H II c
0.2
.
.
.
.
.
.
.
-
H f f i 0 . 4 T / ,
s
•
'r~'
'
'
'"
±
0.1
Hilt H ; 0.4T 2
3
4
5
6
7
8
9
10
T (K) Fig. 2. The temperature dependence of the relaxation rate R = Id In M i d in t l for H = 0.4 T with field applied parallel to the c-axis over the temperature range of 2 - 1 0 K. The relaxation rate is saturated withIn 10% at low temperatures (T < 6 K). The lInear fit of the relaxation rate is shown between 7 and 10 K, which can be explained by thermally activated flux-creep theory. The temperature-Independent activation energy, Uc = 18 meV, is extracted from the lInear temperature dependence of relaxation rate.
indicate a small potential barrier against flux jumping to the adjacent metastable state. This may also introduce noticeable quantum tunneling of vortices under the barrier. The classical activation energy, Uc, is determined to be 18 meV by using the thermally activated flux-creep theory. At low temperatures, T < 6 K, we observe a saturation of the creep rates as shown in Fig. 2, which is about 0.08 within 10% scattering. This indicates that the rate R does not vanish when T - ~ 0, which corresponds to non-thermally activated flux creep. In the thermally activated flux-creep theories, magnetic relaxation of type-II superconductors should disappear at zero temperature. Our results exhibit a low-temperature magnetic relaxation in the LaECuO4+ ~ compound. In order to study the mechanism of quantum creep and provide reliable information from relaxation experiments, such as the potential barrier height and crossover temperature, a quantitative comparison between the experimental results and theoretical analysis is needed. In previous studies [16-18], the parameters used to calculate the theoretical relaxation rate, such as Pn, ~, ~, Jo/Jc, were determined from various experiments. However, these parameters usually vary from sample to sample in the same compound. The variation can be 1 - 2 orders of magnitude differ-
0.07
'
2
'
3
'
4
5
6
T (K) Fig. 3. The temperature dependence of the relaxation rate between 2 and 6 K with a quadratic fit presen~l by the solid line. The inset shows the deviations from the quadratic temperature dependence at high temperatures above 6 K.
ent. Therefore, a quantitative study of the quantum creep rate could be approached by analyzing the temperature dependence of relaxation rate instead of calculating an absolute value based on other experiments. The relaxation rates at low temperatures can be considered as temperature independent within a 10% range. However, the relaxation rates increase monotonically as temperature increases, similar to recent observation in YBa2Cu307_ ~ detwinned crystal [25]. The experimental relaxation rates in the temperature
16
LalCuO4. ~Crystal 14
10
2
,
t
,
3
4
5
6
T (K) Fig. 4. The temperature dependence of the effective Euclidean action S~f ( T ) / h at 0.4 T. The solid line represents a quadratic temperature dependence, which yields a classical activation energy (U~ = 15 meV), crossover temperature (Tqc = 7 K) and zerotemperature quantum-tunneling rate (R = 0.07).
288
L. Zhang et a l . / Physica C 260 (1996) 283-289
range of 2-6 K is best fit with the quadratic function indicated by the solid line in Fig. 4. This is expected by the thermally enhanced quantumZcreep theory. Further scrutiny of the quadratic dependence reveals that the quality of fit is seen to deteriorate (within 5%) for the relaxation rate taken at temperatures above 6-7 K (see inset of Fig. 3). The deviation of a quadratic fit at high temperatures may be due to the effect of thermally activated flux creep. For small and intermediate field, the interaction between the vortices is small compared with the interaction of a single vortex with the pinning centers. This is the single-vortex regime. As the external field increased, the vortex separation decreases and interaction between neighboring vortices becomes more important. For the oxide superconductors it is expected that the single vortex regime extends to fields of the order of l0 T [15]. Our relaxation experiments on the L a 2 C u O 4 + 8 c r y s t a l w e r e p e r formed at 0.4 T. This is in the single-vortex regime and, therefore, the weak collective-pinning theory by Blatter and Geshkenbein [15] should be applicable in the present study. A quadratic temperature-dependent Euclidean action s[ff(T)/h = 1 / [ d In M/d In t[ is plotted in Fig. 4 where the solid line represents a quadratic fit. The zero Euclidean action, s[ff(o)/h, and classical activation energy, Uc, are determined using Eq. (3). The classical activation energy of 15 meV is obtained. This is in agreement with the result of 18 meV obtained in the classical regime. The zero Euclidean action is determined as 14, which is a unitless quality. A quantum-tunneling rate at T = 0 is calculated as large as 0.07. In contrast, the zero-temperature quantum-creep rates were estimated to be 0.01 in YBa2Cu3OT_ 8 [17] and 0.02 in Bi2SrECaCu208 [18]. The quantum-creep theory predicts that the quantum-creep rate is enhanced in a superconductor with weak collective pinning [16]. The relative larger value of the tunneling rate in the L a E C U O 4 + ~ compound reflects the flux pinning being weaker in one-layer superconductor than those with two layers of CuO planes. The crossover temperature is estimated as 7 K by using Eq. (4). At low temperatures, T <: 7 K, the flux creep due to quantum tunneling is predominant so that the creep rate can be quantitatively described by the quantum-creep theory. At T > 7 K, the flux motion due to thermal
LasCuO,+ 8 Crystal H I Ic
0
H = 0.4T
i
i
i
3
4
5
6
T (K) Fig. 5. The dissipative term of the effective Euclidean action. The solid line indicates a T 2 fitting to E,q. (3).
activation becomes important. The contribution of zero-temperature quantum tunneling can be subtracted from the total flux creep in the classical regime. The dissipative term of the effective Euclidean action, AS[ff(T)/h, is plotted in Fig. 5. The creep rate due to thermally assisted quantum tunneling follows a T 2 law, which is consistent with the quantum creep theory [15]. The flux-creep rate in the classical regime was observed to be large and to have a strong temperature-dependence in high-T~ superconductors. This behavior affects the current density at finite temperature significantly. It is generally accepted that the experiments of both magnetic and transport properties in bigh-Tc superconductors are performed well below the critical state, J "~ Jc. This leads to a field dependence of the magnetization and the magnetic relaxation in the classical regime [26]. The systematic study of the field dependence of quantum creep needs to be further investigated and will be presented in a future publication.
5. Conclusion Three questions have been addressed. The first is whether quantum flux creep occurs. The saturation of the creep rate at low temperatures reveals its existence in one-layer superconductor La2CuO4+ s. The second question is whether the quantum creep is thermally enhanced. A quadratic temperature dependence of the effective Euclidean action is obtained,
L. Zhang et al. / Physica C 260 (1996) 283-289
which is consistent with the quantum-creep theory of Blatter and Geshkenbein. Third, we observe the quantum-classical crossover around 6 K and the analysis of the temperature-dependent relaxation data based on quantum-creep theory yields a classical activation energy, crossover temperature and zerotemperature quantum-tunneling rate.
Acknowledgements This work is supported by the National Science Foundation under grant number DMR-94-03895 and by the U.S. Air Force Office of Scientific Research under grant number AFOSR-F49620-92-J-0514.
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