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Dynamics of transition from metastable disordered state to ordered state of vortex structure in 2H-NbSe2 single crystals
Physica C 436 (2006) 1–6 www.elsevier.com/locate/physc
Dynamics of transition from metastable disordered state to ordered state of vortex structure in 2H-NbSe2 single crystals P. Chowdhury, S.K. Gupta *, C.L. Prajapat, G. Yashwant, M.R. Singh, G. Ravikumar, J.V. Yakhmi, V.C. Sahni Technical Physics and Prototype Engineering Division, Mod. Labs, Bhabha Atomic Research Center, Mumbai, Maharashtra 400 085, India Received 7 August 2005; received in revised form 4 November 2005; accepted 28 December 2005 Available online 13 February 2006
Abstract Current driven transition from a highly pinned metastable disordered phase (DP) to a more ordered equilibrium phase (EP) of vortex structure has been investigated in the peak effect regime of weakly pinned type-II superconductor 2H-NbSe2. Critical current density (Jc) in DP shows a maximum at the onset of the peak effect (i.e. for applied field H = Hon), where Jc in the EP is observed to be minimum. Time needed for the transition depends exponentially on the transport current. A model to describe the kinetics of the transition is presented. Time dependence of voltage and the current dependence of relaxation time obtained from experiments are in good agreement with the model. Energy barrier (U0) characterizing the relaxation process extracted from the model also shows a peak at Hon. Peaks in Jc in the DP and U0 have been qualitatively understood in terms of the interplay between elastic and pinning forces. 2006 Elsevier B.V. All rights reserved. PACS: 74.25.Qt; 74.70.Ad; 74.25.Sv
1. Introduction Many applications of type-II superconductors depend on field dependence of critical current density Jc, which in turn depends on the nature of the vortex state (i.e. vortex lattice, vortex glass, Bragg glass or vortex liquid) and pinning centers. Transport properties such as resistivity, Jc and their magnetic field and temperature dependences have often been used to understand the nature of pinning and structural transitions in vortex state. In some of these studies, relaxation in resistivity and Jc have been investigated for this purpose. Altshuler et al. [1] have measured voltage relaxation in polycrystalline YBa2Cu3Ox (YBCO) samples to determine intra-granular critical current density and pinning energies. Ma et al. [2] have carried out a similar study in high Tc thin films to determine pinning properties. It may be noted that in these studies, the transients arise *
Corresponding author.Tel.: +9122 25593863; fax: +9122 25505296. E-mail address: [email protected] (S.K. Gupta).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.12.067
due to thermally activated flux creep, while the vortex structure presumably remains invariant during the relaxation process. In some other studies (discussed below), transients in transport properties caused by changes in the vortex structure have been investigated. These studies have been carried out in peak effect (PE) region i.e. for magnetic fields (H) close to Hc2 where weakly pinned type-II superconductors show an anomalous peak in Jc as function of H [3,4]. Vortex structure in the PE region is known to exhibit pronounced metastability and the application of a transport current drives the relaxation from a metastable state into the equilibrium state. As the interpretation of the results of these relaxation studies depend on the mechanism of the peak effect, we will briefly discuss mechanisms proposed in the literature. Structure of vortices penetrating a type-II superconductor depends on relative magnitudes of electromagnetic repulsion between them and pinning forces [4]. Due to random location of pinning centers, vortices form an
2
P. Chowdhury et al. / Physica C 436 (2006) 1–6
amorphous structure when pinning forces dominate. On the other hand, they form crystalline lattice when pinning forces dominate. An early model of PE by Pippard [3], that has been subsequently extended in other studies [4–8], attributes the PE to a structural transition from elastic vortex lattice (that is ordered over macroscopic dimensions) to a plastically deformed (disordered) vortex structure [6–8]. The transition occurs as a result of relative change in pinning and electromagnetic forces as the applied magnetic field is varied. The onset of this elastic to plastic structural transition occurs at magnetic field H = Hon where the critical current is minimum. As the magnetic field is increased (above Hon), vortex correlation volume reduces (and Jc increases) and attains its limiting value for H = Hp, where peak in Jc is observed. On the other hand, in the edge contamination model (ECM) proposed by Paltiel et al. [9], a first order transition (FOT) from ordered (OP) (at H < Hp) to a disordered phase (DP) (at H > Hp) occurs at magnetic field Hp corresponding to peak in critical current. An additional feature of this model is that for H < Hp, metastable disordered phase enters at the sample edges and dynamically coexists with the ordered equilibrium phase (EP) in the interior. Lifetime and annealing length (the distance over which the metastable disordered phase relaxes into the OP) of the DP diverge at Hp as free energies of DP and OP are presumably equal at Hp. Therefore, in this model Hon does not correspond to any transition in the vortex structure in contrast to that in Pippard model. We will now discuss some of the reported studies on transients in transport properties caused by changes in the vortex structure. Xiao et al. [10] have investigated the current induced voltage relaxation in the PE region of Fe doped 2H-NbSe2 single crystals. They observed that the application of current pulses drives a highly ordered vortex state formed upon zero field cooling (ZFC) to a relatively disordered equilibrium phase (EP). They find that the voltage relaxation has a stretched exponential time dependence exp(t/s)a where both s and a depend on the current. In another study, Xiao et al. [11] find that the field cooled (FC) state has a higher Jc (corresponding to highly disordered state). Application of a current drives this state towards the ordered equilibrium state (equivalent to ZFC state). Henderson et al. [12] have also studied relaxation of FC state (i.e. supercooled state obtained by field cooling from T > Tc) to more ordered state in 2H-NbSe2 crystals. Relaxation time (tr) of the disordered state was found to have a very strong dependence on current and was fitted to empirical relation tr / [(I I*)/I*]a or tr / exp[{I*(I I*)}1/2], where a is a constant and I* is critical current of FC state. They have also studied relaxation of a metastable ordered state (called superheated state obtained by heating equilibrium state from low temperatures to peak region) to more disordered state. The results have been interpreted in terms of Pippard model as that both superheating and supercooling occur across elastic to plastic flow boundary at Hon. Kim et al. [13] have investigated relaxation of FC
state to OP in MgB2 single crystals. They find qualitatively different voltage versus time characteristics for H < Hon and H > Hon. The voltage shows jumps in addition to smooth time dependence and in this regard the results are somewhat different from those reported on NbSe2. Above Hon, relaxation time was described by s ¼ s0 eI=I 0 with s0 diverging at Hon while I0 reduces to zero. Ravikumar [14] has presented a model for relaxation of metastable state to equilibrium state and has used Jc as a parameter that corresponds to the extent of lattice order. In that model, relations for time evolution of Jc and time constant for annealing process were postulated and the results obtained qualitatively described data obtained in some of the earlier studies [10]. It is seen that the relaxation results of different experimental studies have been fitted to empirical relations and a theoretical model to describe the results has not been presented. In the present study we have investigated relaxation over a wide field range and present a model to understand these results. While the results of the model are quantitatively different from the phenomenological model reported earlier [10], they are qualitatively similar in that relaxation time ! 1 as J ! Jc. 2. Experimental Transport measurements in four probe configuration were carried out using a 16T Oxford Instruments Cryomagnetic System. A pure 2H-NbSe2 single crystal of dimension 4 · 0.5 · 0.2 mm3 was mounted on the sample holder with the field parallel to the crystallographic c-axis. Low resistance contacts were made (to the crystal) using Ag0.1In0.9 solder. The measurements could be done at different angles (h) between magnetic field and the c-axis of the sample using appropriate sample holder. Measurements were performed near the liquid helium temperature of 4.2 K as temperature fluctuations were found to be minimum in this region. The sample temperature was controlled to within ±2 mK. DC current was applied using Keithley current source model 224 and voltage data were acquired through Keithley 2180 nano-voltmeter. A PC based automatic data acquisition system was used for measurements. Magnetization measurements (not shown here) on similar crystals for different angles h showed that the peak effect is most pronounced for 40 < h < 60 and therefore the measurements reported here were made at h = 50. We may add that optimum peak effect at angles other than 90 has also been observed in MgB2 single crystals [13]. I–V measurements were made at different applied magnetic fields under FC and ZFC conditions, as well as after relaxation of FC sample to equilibrium state by application of a large current. For these measurements, current was increased in steps of 2–3% each time and the voltage was measured. As reported by Xiao et al. [11], characteristics of EP were seen to be same as that of ZFC state. For study of relaxation, from metastable FC state to EP, a fixed
P. Chowdhury et al. / Physica C 436 (2006) 1–6
current was applied after field cooling the sample and the voltage developed across the sample was monitored. The measurements were repeated for different H and applied currents. 3. Results and discussion For study of the hysteresis and relaxation, I–V characteristics were repeatedly measured after field cooling the sample. During these measurements, the current was cycled between zero and maximum value Imax of 40.5 mA. Typical results are shown in Fig. 1. For first measurement, voltage during increasing current (data i1) is lower in comparison to that during decreasing current (data d1). On repeating I–Vs, initial voltage with increasing current (i2) was similar as that for curve d1 but deviated at higher currents. Similar results were observed on further cycling of the current. The equivalence of curves i2 and d1 at low currents shows memory effect [11], while the deviations at high currents, with voltage increasing for each subsequent measurement, indicates slow relaxation of the disordered state to ordered state. For sufficiently high value of maximum current, as shown in the inset of Fig. 1, for Imax = 70 mA, the I–Vs after first cycle were reversible except for small jumps that have been observed earlier and are attributed to plastic flow [4]. This indicates that equilibrium state is attained on application of sufficiently high current. The I–Vs of relaxed state shown in inset of Fig. 1 were seen to be same as those obtained in ZFC state (not shown here), indicating that the EP is nearly same as ZFC state as reported in earlier studies [11].
6 60
i4 d4
V (μV)
40
i2
2
i1
40
d2 i2
0
30 20
40
60
80
d1
I (mA)
i1 Imax = 40.5 mA H = 2.12 T T = 4.34 K
0
10
d3 i3
d1, d2
20
From these I–Vs, field dependent critical currents in the disordered (IFC) and the equilibrium states (IZFC) were obtained using voltage criteria of 500 nV. The field dependence of these currents is shown in Fig. 2. It is seen that IZFC exhibits peak effect with onset at a field of Hon = 2.08 T and a peak at Hp = 2.27 T. While the data in FC and ZFC states are similar to those reported in earlier studies [12,15], they differ in one important aspect. We observe for the first time, a peak in disordered state (IFC) at H = Hon. The reason that it has not been observed earlier could be that the measurements in earlier studies have been carried out for H//c-axis in contrast to the present case (measurements at h = 50). We believe that the enhanced peak effect in our experiment is due to reduced surface barrier and surface pinning effects for fields at an angle to c-axis. As the contribution to the critical current from surface effects is minimized, features intrinsic to peak effect are enhanced. Possible reason for the peak in IFC coinciding with minima in IZFC will be discussed later. We may add that the critical currents measured in this study correspond to slow ramp in current and as shown in earlier studies [11,15], actual critical currents in field cooled or disordered state may be much higher. This is also indicated by extrapolation of critical currents in disordered state, i.e. for H > Hp [9]. Relaxation of field cooled (DP) to equilibrium state (EP) was investigated by application of a fixed dc current to a FC sample and measurement of voltage (V) across the sample as function of time till it saturates. The measurements were repeated for different values of current. The sample was heated to T > Tc and cooled to desired temperature under applied field prior to each measurement. The results obtained at different fields (H < Hp) were qualitatively similar and typical results obtained at H = 2.10 T are shown in Fig. 3. Voltage V(t) was found to have a nearly logarithmic dependence on time as seen in Fig. 3 except for a different
15
20
25
30
35
40
Ic (mA)
V (μV)
4
Imax = 70 mA
3
Hp
20
IZFC IFC
45
I (mA) Fig. 1. Hysteresis and relaxation in I–V curves studied by repeatedly increasing and decreasing current for field cooled sample (with Imax = 40.5 mA). Here, i1and d1 is first set of increasing and decreasing current and i2 and d2 is second set, etc. It is seen that the relaxation is not complete even after four cycles of increasing and decreasing current. Inset shows similar plots with the maximum applied current of 70 mA. It is seen that sample relaxes to equilibrium state in first cycle in this case.
10
Hon
T = 4.34 K
1.5
1.8
2.1
2.4
Field (T) Fig. 2. Field dependence of critical current for field cooled (IFC) and zero field cooled (IZFC) sample at 4.34 K.
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P. Chowdhury et al. / Physica C 436 (2006) 1–6
If U0 is the activation energy needed to transform the disordered phase into ordered phase then
I = 63 mA 61 mA
60 mA 59 mA
40
dP ðtÞ ¼ P meðU 0 aJ Þ=kT ; dt
58 mA 56 mA 54 mA 52 mA
6
ln (τ)
20
B = 2.10 T T = 4.34 K 10 0.1
1
ln P ¼ ðAeaJ =kT Þt;
4 2 0
10
52
56
J (A/cm2)
100
60
64
1000
t (sec) Fig. 3. The voltage versus time characteristics showing relaxation of FC state to equilibrium state for different applied currents for H = 2.10 T. Arrows indicate relaxation time (s) when EP is attained. The inset shows the variation of relaxation time s (in seconds), with the currents density.
slope at the initial stage of the relaxation, which may be due to the initial transient state of the vortex configuration. The time (s) required for saturation of voltage i.e. relaxation of vortex lattice to EP showed an exponential dependence on current ðs ¼ s0 eI=I 0 Þ as shown in the inset of Fig. 3. In what follows, we present a simple model to understand these results. To understand the exponential dependence of the relaxation time on current (that is proportional to Lorentz force) as well as time dependence of voltage, we propose that Lorentz force assists the metastable disordered phase to overcome the energy barrier and transform into the EP. The transformation of a volume V of the metastable vortex phase into ordered phase involves Lorentz force aided relative movement (by average distance L) of vortices in this region. Typically, L could be of the order of or less than inter-vortex spacing. Let the superconducting material at any time have a fraction P(t) in disordered phase and (1 P) in equilibrium phase with P = 1 at t = 0 and P ! 0 as t ! 1. Let Jc1 and Jc2 be the critical current densities in disordered and ordered phases respectively with Jc1 > Jc2. As discussed earlier, the value of critical current density in disordered state Jc2 may be much higher than that measured by us (i.e. obtained from IFC) due to some relaxation occurring during the measurement. The critical current density of the material at time t is given by J c ðtÞ ¼ PJ c1 þ ð1 P ÞJ c2 .
ð4Þ
where A ¼ meU 0 =kT . Time constant s is determined using the criterion P = 0.001 implying nearly full conversion to the ordered phase and is given by AeaJ =kT s ¼ lnð0:001Þ 7. This may be simplified to a ln s ¼ J þ ðln 7 ln AÞ. kT
ð5Þ
Since A is constant at a given temperature, we expect a linear plot between ln s and J as observed (Fig. 3). Analysis of experimental data shows that ln A is much larger than ln 7 and the results are nearly independent of the criteria of P = 0.001 for final state. Voltage versus time relation (at fixed current) is obtained by rewriting Eq. (2) as ðE2 EÞ ¼ ðE2 E1 ÞP ðtÞ; where E1 = q(J Jc1) and E2 = q (J Jc2) are the initial and final electric fields. Using Eq. (4) we obtain lnðE2 EÞ ¼ lnðE2 E1 Þ bt;
ð6Þ
aJ/kT
where b = Ae is a constant. Eq. (6) gives the expected time dependence of the voltage. A typical plot of ln(E2 E) versus time is shown in Fig. 4. Here, E2 was obtained from Fig. 3 as the electric
-10
I = 54 mA B = 2.10 T
t=τ
-12
-14
ð1Þ
As the critical current density in disordered state is very large, vortex motion of the disordered phase may be extremely slow or the vortices (in DP) may be static for currents in the range applied to study the relaxation. Assuming flux flow model, E = q(J Jc) [14,16] we get EðtÞ ¼ qðJ J c2 Þ qP ðJ c1 J c2 Þ.
where aJ = VBLJ [17] is the work done by Lorentz force during the phase transformation of the correlated region of volume V and m is an attempt frequency [17]. Solution of Eq. (3) is given by
ln(E2-E)
V (μV)
30
ð3Þ
ð2Þ
-16 0
100
200
300
400
500
t (sec) Fig. 4. Plot showing agreement of the voltage versus time characteristics of Fig. 3 (for I = 54 mA) with Eq. (6). Here, ln(E2 E) is plotted as function of time with E2 and E in V/cm. Arrow shows saturation of voltage when equilibrium state is attained.
P. Chowdhury et al. / Physica C 436 (2006) 1–6
field corresponding to saturation voltage (shown by arrows in the figure). A linear behavior is seen before saturation is observed. Deviations at short time scales could arise from initial field transient or certain regions such as sample surface having different activation energy from that in the bulk. Fitting of ln s versus J curve (Fig. 3) to Eq. (5) yields A e33 sec1. Assuming a typical attempt frequency (m) of 108 Hz [17] and experimental value of A, the value of activation energy U0 is found to be 18.5 meV. We may note that the activation energy U0 in the present study is for relative movement of vortices for changing the vortex state. To a first approximation, it is expected to be of similar order as that needed for the thermally activated flux creep. The value measured in this study is in fact quite close to that estimated in flux creep model [17] and also measured in various studies for movement of flux bundles. In Fig. 5, we have plotted the field dependence of the parameter U0 and find that it has a maxima at Hon. While critical current in DP (JFC) and U0 shows a maxima at H = Hon, Jc in EP shows a minima at the same field. A peak in various parameters at H = Hon supports a structural transition at Hon as envisaged by Pippard model (transition from elastic to plastically deformed lattice). In terms of ECM, Hon corresponds to a field where contamination of DP has a measurable effect on Jc so that it starts increasing. As no actual transition occurs at this field (in this model), we would not expect peaks in other parameters at the same field. This conclusion is in agreement with many of the earlier studies [4,18] and results of Henderson et al. [12], where supercooling and superheating are observed on crossing Hon. We may add that while the present study supports existence of structural transition at Hon, it does not rule out the possibility of additional first order phase transition as envisaged by ECM and as
20
U0 (meV)
seen in a study by Banerjee et al. [18], both transitions could exist. We may also add that the relaxation from DP to EP as seen here is qualitatively different from that envisaged in ECM [9], where the life time of the disordered state as well as the relaxation length diverge at H = Hp, while they should tend to zero at Hon where the contamination from edges has negligible effect. In contrast, lifetime of metastable DP in the present study (for a given current) peaks at Hon and reduces near Hp as would be expected from field dependence of U0. Now we will provide a possible explanation for peak in JFC and U0 at Hon in terms of Pippard model. In terms of this model, inter-vortex repulsion and pinning forces dominate for H < Hon and H > Hon, respectively. At Hon, the two forces, one favoring elastic lattice and other plastic deformations in EP, are balanced. These two forces, in addition to the Lorentz force (due to applied current) help in structural reorganization at Hon. As the net result of these forces is minimum at Hon, relaxation rate from DP to EP is reduced, leading to peak in JFC and U0. 4. Conclusion Field dependence of critical currents in field cooled and zero field cooled states has been investigated. Field cooled state is found to have a peak in critical current at the onset of peak effect. Relaxation of the disordered state obtained by field cooling the sample to more ordered equilibrium state has been studied. A model to understand the current dependence of relaxation time and time dependence of voltage has been presented. The results support the presence of a transition in the vortex state at the onset of the peak effect in agreement with some of the earlier studies where a transition to plastically deformed vortex state at the onset of the peak effect has been proposed. References [1] [2] [3] [4] [5] [6] [7] [8]
18
16
14 [9]
12
10 1.9
5
[10]
2.0
2.1
2.2
2.3
Field (T) Fig. 5. Magnetic field dependence of activation energy U0. A peak in activation energy is seen for H = Hon.
[11] [12]
E. Altshuler, S. Garcia, J. Barroso, Physica C 177 (1991) 61. L.P. Ma, H.C. Li, R.L. Wang, L. Li, Physica C 291 (1997) 143. A.B. Pippard, Philos. Mag. 19 (1969) 217. M.J. Higgins, S. Bhattacharya, Physica C 257 (1996) 232. A.I. Larkin, Yu N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. R. Wo¨rdenweber, P.H. Kes, C.C. Tsuei, Phys. Rev. B 33 (1986) 3172. H.J. Jensen, A. Brass, A.J. Berlinsky, Phys. Rev. Lett. 60 (1988) 1676. M.C. Faleski, M.C. Marchetti, A.A. Middleton, Phys. Rev. B 54 (1996) 12427. Y. Paltiel, E. Zeldov, Y.N. Myasoedov, H. Shtrikman, S. Bhattacharya, M.J. Higgins, Z.L. Xiao, E.Y. Andrei, P.L. Gammel, D.J. Bishop, Nature 403 (2000) 398; Y. Paltiel, E. Zeldov, Y. Myasoedov, M.L. Rappaport, G. Jung, S. Bhattacharya, M.J. Higgins, Z.L. Xiao, E.Y. Andrei, P.L. Gammel, D.J. Bishop, Phys. Rev. Lett. 85 (2000) 3712. Z.L. Xiao, E.Y. Andrei, M.J. Higgins, Phys. Rev. Lett. 83 (1999) 1664. Z.L. Xiao, E.Y. Andei, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 85 (2000) 3265. W. Henderson, E.Y. Andrei, M.J. Higgins, S. Bhattacharya, Phys. Rev. Lett. 77 (1996) 2077.
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[13] H.J. Kim, H.S. Lee, B. Kang, P. Chowdhury, K.H. Kim, S.I. Lee, Phys. Rev. B 70 (2004) 132501. [14] G. Ravikumar, Phys. Rev. B 63 (2001) 224523. [15] Z.L. Xiao, E.Y. Andei, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 86 (2001) 2431. [16] T. Matsushita, B. Ni, Physica C 166 (1990) 423.
[17] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc., New York, 1996. [18] S.S. Banerjee, A.K. Grover, M.J. Higgins, Gutam I. Menon, P.K. Mishra, D. Pal, S. Ramakrishnan, T.V. Chandrasekhar Rao, G. Ravikumar, V.C. Sahni, S. Sarkar, C.V. Tomy, Physica C 355 (2001) 39.
Dynamics of transition from metastable disordered state to ordered state of vortex structure in 2H-NbSe2 single crystals P. Chowdhury, S.K. Gupta *, C.L. Prajapat, G. Yashwant, M.R. Singh, G. Ravikumar, J.V. Yakhmi, V.C. Sahni Technical Physics and Prototype Engineering Division, Mod. Labs, Bhabha Atomic Research Center, Mumbai, Maharashtra 400 085, India Received 7 August 2005; received in revised form 4 November 2005; accepted 28 December 2005 Available online 13 February 2006
Abstract Current driven transition from a highly pinned metastable disordered phase (DP) to a more ordered equilibrium phase (EP) of vortex structure has been investigated in the peak effect regime of weakly pinned type-II superconductor 2H-NbSe2. Critical current density (Jc) in DP shows a maximum at the onset of the peak effect (i.e. for applied field H = Hon), where Jc in the EP is observed to be minimum. Time needed for the transition depends exponentially on the transport current. A model to describe the kinetics of the transition is presented. Time dependence of voltage and the current dependence of relaxation time obtained from experiments are in good agreement with the model. Energy barrier (U0) characterizing the relaxation process extracted from the model also shows a peak at Hon. Peaks in Jc in the DP and U0 have been qualitatively understood in terms of the interplay between elastic and pinning forces. 2006 Elsevier B.V. All rights reserved. PACS: 74.25.Qt; 74.70.Ad; 74.25.Sv
1. Introduction Many applications of type-II superconductors depend on field dependence of critical current density Jc, which in turn depends on the nature of the vortex state (i.e. vortex lattice, vortex glass, Bragg glass or vortex liquid) and pinning centers. Transport properties such as resistivity, Jc and their magnetic field and temperature dependences have often been used to understand the nature of pinning and structural transitions in vortex state. In some of these studies, relaxation in resistivity and Jc have been investigated for this purpose. Altshuler et al. [1] have measured voltage relaxation in polycrystalline YBa2Cu3Ox (YBCO) samples to determine intra-granular critical current density and pinning energies. Ma et al. [2] have carried out a similar study in high Tc thin films to determine pinning properties. It may be noted that in these studies, the transients arise *
Corresponding author.Tel.: +9122 25593863; fax: +9122 25505296. E-mail address: [email protected] (S.K. Gupta).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.12.067
due to thermally activated flux creep, while the vortex structure presumably remains invariant during the relaxation process. In some other studies (discussed below), transients in transport properties caused by changes in the vortex structure have been investigated. These studies have been carried out in peak effect (PE) region i.e. for magnetic fields (H) close to Hc2 where weakly pinned type-II superconductors show an anomalous peak in Jc as function of H [3,4]. Vortex structure in the PE region is known to exhibit pronounced metastability and the application of a transport current drives the relaxation from a metastable state into the equilibrium state. As the interpretation of the results of these relaxation studies depend on the mechanism of the peak effect, we will briefly discuss mechanisms proposed in the literature. Structure of vortices penetrating a type-II superconductor depends on relative magnitudes of electromagnetic repulsion between them and pinning forces [4]. Due to random location of pinning centers, vortices form an
2
P. Chowdhury et al. / Physica C 436 (2006) 1–6
amorphous structure when pinning forces dominate. On the other hand, they form crystalline lattice when pinning forces dominate. An early model of PE by Pippard [3], that has been subsequently extended in other studies [4–8], attributes the PE to a structural transition from elastic vortex lattice (that is ordered over macroscopic dimensions) to a plastically deformed (disordered) vortex structure [6–8]. The transition occurs as a result of relative change in pinning and electromagnetic forces as the applied magnetic field is varied. The onset of this elastic to plastic structural transition occurs at magnetic field H = Hon where the critical current is minimum. As the magnetic field is increased (above Hon), vortex correlation volume reduces (and Jc increases) and attains its limiting value for H = Hp, where peak in Jc is observed. On the other hand, in the edge contamination model (ECM) proposed by Paltiel et al. [9], a first order transition (FOT) from ordered (OP) (at H < Hp) to a disordered phase (DP) (at H > Hp) occurs at magnetic field Hp corresponding to peak in critical current. An additional feature of this model is that for H < Hp, metastable disordered phase enters at the sample edges and dynamically coexists with the ordered equilibrium phase (EP) in the interior. Lifetime and annealing length (the distance over which the metastable disordered phase relaxes into the OP) of the DP diverge at Hp as free energies of DP and OP are presumably equal at Hp. Therefore, in this model Hon does not correspond to any transition in the vortex structure in contrast to that in Pippard model. We will now discuss some of the reported studies on transients in transport properties caused by changes in the vortex structure. Xiao et al. [10] have investigated the current induced voltage relaxation in the PE region of Fe doped 2H-NbSe2 single crystals. They observed that the application of current pulses drives a highly ordered vortex state formed upon zero field cooling (ZFC) to a relatively disordered equilibrium phase (EP). They find that the voltage relaxation has a stretched exponential time dependence exp(t/s)a where both s and a depend on the current. In another study, Xiao et al. [11] find that the field cooled (FC) state has a higher Jc (corresponding to highly disordered state). Application of a current drives this state towards the ordered equilibrium state (equivalent to ZFC state). Henderson et al. [12] have also studied relaxation of FC state (i.e. supercooled state obtained by field cooling from T > Tc) to more ordered state in 2H-NbSe2 crystals. Relaxation time (tr) of the disordered state was found to have a very strong dependence on current and was fitted to empirical relation tr / [(I I*)/I*]a or tr / exp[{I*(I I*)}1/2], where a is a constant and I* is critical current of FC state. They have also studied relaxation of a metastable ordered state (called superheated state obtained by heating equilibrium state from low temperatures to peak region) to more disordered state. The results have been interpreted in terms of Pippard model as that both superheating and supercooling occur across elastic to plastic flow boundary at Hon. Kim et al. [13] have investigated relaxation of FC
state to OP in MgB2 single crystals. They find qualitatively different voltage versus time characteristics for H < Hon and H > Hon. The voltage shows jumps in addition to smooth time dependence and in this regard the results are somewhat different from those reported on NbSe2. Above Hon, relaxation time was described by s ¼ s0 eI=I 0 with s0 diverging at Hon while I0 reduces to zero. Ravikumar [14] has presented a model for relaxation of metastable state to equilibrium state and has used Jc as a parameter that corresponds to the extent of lattice order. In that model, relations for time evolution of Jc and time constant for annealing process were postulated and the results obtained qualitatively described data obtained in some of the earlier studies [10]. It is seen that the relaxation results of different experimental studies have been fitted to empirical relations and a theoretical model to describe the results has not been presented. In the present study we have investigated relaxation over a wide field range and present a model to understand these results. While the results of the model are quantitatively different from the phenomenological model reported earlier [10], they are qualitatively similar in that relaxation time ! 1 as J ! Jc. 2. Experimental Transport measurements in four probe configuration were carried out using a 16T Oxford Instruments Cryomagnetic System. A pure 2H-NbSe2 single crystal of dimension 4 · 0.5 · 0.2 mm3 was mounted on the sample holder with the field parallel to the crystallographic c-axis. Low resistance contacts were made (to the crystal) using Ag0.1In0.9 solder. The measurements could be done at different angles (h) between magnetic field and the c-axis of the sample using appropriate sample holder. Measurements were performed near the liquid helium temperature of 4.2 K as temperature fluctuations were found to be minimum in this region. The sample temperature was controlled to within ±2 mK. DC current was applied using Keithley current source model 224 and voltage data were acquired through Keithley 2180 nano-voltmeter. A PC based automatic data acquisition system was used for measurements. Magnetization measurements (not shown here) on similar crystals for different angles h showed that the peak effect is most pronounced for 40 < h < 60 and therefore the measurements reported here were made at h = 50. We may add that optimum peak effect at angles other than 90 has also been observed in MgB2 single crystals [13]. I–V measurements were made at different applied magnetic fields under FC and ZFC conditions, as well as after relaxation of FC sample to equilibrium state by application of a large current. For these measurements, current was increased in steps of 2–3% each time and the voltage was measured. As reported by Xiao et al. [11], characteristics of EP were seen to be same as that of ZFC state. For study of relaxation, from metastable FC state to EP, a fixed
P. Chowdhury et al. / Physica C 436 (2006) 1–6
current was applied after field cooling the sample and the voltage developed across the sample was monitored. The measurements were repeated for different H and applied currents. 3. Results and discussion For study of the hysteresis and relaxation, I–V characteristics were repeatedly measured after field cooling the sample. During these measurements, the current was cycled between zero and maximum value Imax of 40.5 mA. Typical results are shown in Fig. 1. For first measurement, voltage during increasing current (data i1) is lower in comparison to that during decreasing current (data d1). On repeating I–Vs, initial voltage with increasing current (i2) was similar as that for curve d1 but deviated at higher currents. Similar results were observed on further cycling of the current. The equivalence of curves i2 and d1 at low currents shows memory effect [11], while the deviations at high currents, with voltage increasing for each subsequent measurement, indicates slow relaxation of the disordered state to ordered state. For sufficiently high value of maximum current, as shown in the inset of Fig. 1, for Imax = 70 mA, the I–Vs after first cycle were reversible except for small jumps that have been observed earlier and are attributed to plastic flow [4]. This indicates that equilibrium state is attained on application of sufficiently high current. The I–Vs of relaxed state shown in inset of Fig. 1 were seen to be same as those obtained in ZFC state (not shown here), indicating that the EP is nearly same as ZFC state as reported in earlier studies [11].
6 60
i4 d4
V (μV)
40
i2
2
i1
40
d2 i2
0
30 20
40
60
80
d1
I (mA)
i1 Imax = 40.5 mA H = 2.12 T T = 4.34 K
0
10
d3 i3
d1, d2
20
From these I–Vs, field dependent critical currents in the disordered (IFC) and the equilibrium states (IZFC) were obtained using voltage criteria of 500 nV. The field dependence of these currents is shown in Fig. 2. It is seen that IZFC exhibits peak effect with onset at a field of Hon = 2.08 T and a peak at Hp = 2.27 T. While the data in FC and ZFC states are similar to those reported in earlier studies [12,15], they differ in one important aspect. We observe for the first time, a peak in disordered state (IFC) at H = Hon. The reason that it has not been observed earlier could be that the measurements in earlier studies have been carried out for H//c-axis in contrast to the present case (measurements at h = 50). We believe that the enhanced peak effect in our experiment is due to reduced surface barrier and surface pinning effects for fields at an angle to c-axis. As the contribution to the critical current from surface effects is minimized, features intrinsic to peak effect are enhanced. Possible reason for the peak in IFC coinciding with minima in IZFC will be discussed later. We may add that the critical currents measured in this study correspond to slow ramp in current and as shown in earlier studies [11,15], actual critical currents in field cooled or disordered state may be much higher. This is also indicated by extrapolation of critical currents in disordered state, i.e. for H > Hp [9]. Relaxation of field cooled (DP) to equilibrium state (EP) was investigated by application of a fixed dc current to a FC sample and measurement of voltage (V) across the sample as function of time till it saturates. The measurements were repeated for different values of current. The sample was heated to T > Tc and cooled to desired temperature under applied field prior to each measurement. The results obtained at different fields (H < Hp) were qualitatively similar and typical results obtained at H = 2.10 T are shown in Fig. 3. Voltage V(t) was found to have a nearly logarithmic dependence on time as seen in Fig. 3 except for a different
15
20
25
30
35
40
Ic (mA)
V (μV)
4
Imax = 70 mA
3
Hp
20
IZFC IFC
45
I (mA) Fig. 1. Hysteresis and relaxation in I–V curves studied by repeatedly increasing and decreasing current for field cooled sample (with Imax = 40.5 mA). Here, i1and d1 is first set of increasing and decreasing current and i2 and d2 is second set, etc. It is seen that the relaxation is not complete even after four cycles of increasing and decreasing current. Inset shows similar plots with the maximum applied current of 70 mA. It is seen that sample relaxes to equilibrium state in first cycle in this case.
10
Hon
T = 4.34 K
1.5
1.8
2.1
2.4
Field (T) Fig. 2. Field dependence of critical current for field cooled (IFC) and zero field cooled (IZFC) sample at 4.34 K.
4
P. Chowdhury et al. / Physica C 436 (2006) 1–6
If U0 is the activation energy needed to transform the disordered phase into ordered phase then
I = 63 mA 61 mA
60 mA 59 mA
40
dP ðtÞ ¼ P meðU 0 aJ Þ=kT ; dt
58 mA 56 mA 54 mA 52 mA
6
ln (τ)
20
B = 2.10 T T = 4.34 K 10 0.1
1
ln P ¼ ðAeaJ =kT Þt;
4 2 0
10
52
56
J (A/cm2)
100
60
64
1000
t (sec) Fig. 3. The voltage versus time characteristics showing relaxation of FC state to equilibrium state for different applied currents for H = 2.10 T. Arrows indicate relaxation time (s) when EP is attained. The inset shows the variation of relaxation time s (in seconds), with the currents density.
slope at the initial stage of the relaxation, which may be due to the initial transient state of the vortex configuration. The time (s) required for saturation of voltage i.e. relaxation of vortex lattice to EP showed an exponential dependence on current ðs ¼ s0 eI=I 0 Þ as shown in the inset of Fig. 3. In what follows, we present a simple model to understand these results. To understand the exponential dependence of the relaxation time on current (that is proportional to Lorentz force) as well as time dependence of voltage, we propose that Lorentz force assists the metastable disordered phase to overcome the energy barrier and transform into the EP. The transformation of a volume V of the metastable vortex phase into ordered phase involves Lorentz force aided relative movement (by average distance L) of vortices in this region. Typically, L could be of the order of or less than inter-vortex spacing. Let the superconducting material at any time have a fraction P(t) in disordered phase and (1 P) in equilibrium phase with P = 1 at t = 0 and P ! 0 as t ! 1. Let Jc1 and Jc2 be the critical current densities in disordered and ordered phases respectively with Jc1 > Jc2. As discussed earlier, the value of critical current density in disordered state Jc2 may be much higher than that measured by us (i.e. obtained from IFC) due to some relaxation occurring during the measurement. The critical current density of the material at time t is given by J c ðtÞ ¼ PJ c1 þ ð1 P ÞJ c2 .
ð4Þ
where A ¼ meU 0 =kT . Time constant s is determined using the criterion P = 0.001 implying nearly full conversion to the ordered phase and is given by AeaJ =kT s ¼ lnð0:001Þ 7. This may be simplified to a ln s ¼ J þ ðln 7 ln AÞ. kT
ð5Þ
Since A is constant at a given temperature, we expect a linear plot between ln s and J as observed (Fig. 3). Analysis of experimental data shows that ln A is much larger than ln 7 and the results are nearly independent of the criteria of P = 0.001 for final state. Voltage versus time relation (at fixed current) is obtained by rewriting Eq. (2) as ðE2 EÞ ¼ ðE2 E1 ÞP ðtÞ; where E1 = q(J Jc1) and E2 = q (J Jc2) are the initial and final electric fields. Using Eq. (4) we obtain lnðE2 EÞ ¼ lnðE2 E1 Þ bt;
ð6Þ
aJ/kT
where b = Ae is a constant. Eq. (6) gives the expected time dependence of the voltage. A typical plot of ln(E2 E) versus time is shown in Fig. 4. Here, E2 was obtained from Fig. 3 as the electric
-10
I = 54 mA B = 2.10 T
t=τ
-12
-14
ð1Þ
As the critical current density in disordered state is very large, vortex motion of the disordered phase may be extremely slow or the vortices (in DP) may be static for currents in the range applied to study the relaxation. Assuming flux flow model, E = q(J Jc) [14,16] we get EðtÞ ¼ qðJ J c2 Þ qP ðJ c1 J c2 Þ.
where aJ = VBLJ [17] is the work done by Lorentz force during the phase transformation of the correlated region of volume V and m is an attempt frequency [17]. Solution of Eq. (3) is given by
ln(E2-E)
V (μV)
30
ð3Þ
ð2Þ
-16 0
100
200
300
400
500
t (sec) Fig. 4. Plot showing agreement of the voltage versus time characteristics of Fig. 3 (for I = 54 mA) with Eq. (6). Here, ln(E2 E) is plotted as function of time with E2 and E in V/cm. Arrow shows saturation of voltage when equilibrium state is attained.
P. Chowdhury et al. / Physica C 436 (2006) 1–6
field corresponding to saturation voltage (shown by arrows in the figure). A linear behavior is seen before saturation is observed. Deviations at short time scales could arise from initial field transient or certain regions such as sample surface having different activation energy from that in the bulk. Fitting of ln s versus J curve (Fig. 3) to Eq. (5) yields A e33 sec1. Assuming a typical attempt frequency (m) of 108 Hz [17] and experimental value of A, the value of activation energy U0 is found to be 18.5 meV. We may note that the activation energy U0 in the present study is for relative movement of vortices for changing the vortex state. To a first approximation, it is expected to be of similar order as that needed for the thermally activated flux creep. The value measured in this study is in fact quite close to that estimated in flux creep model [17] and also measured in various studies for movement of flux bundles. In Fig. 5, we have plotted the field dependence of the parameter U0 and find that it has a maxima at Hon. While critical current in DP (JFC) and U0 shows a maxima at H = Hon, Jc in EP shows a minima at the same field. A peak in various parameters at H = Hon supports a structural transition at Hon as envisaged by Pippard model (transition from elastic to plastically deformed lattice). In terms of ECM, Hon corresponds to a field where contamination of DP has a measurable effect on Jc so that it starts increasing. As no actual transition occurs at this field (in this model), we would not expect peaks in other parameters at the same field. This conclusion is in agreement with many of the earlier studies [4,18] and results of Henderson et al. [12], where supercooling and superheating are observed on crossing Hon. We may add that while the present study supports existence of structural transition at Hon, it does not rule out the possibility of additional first order phase transition as envisaged by ECM and as
20
U0 (meV)
seen in a study by Banerjee et al. [18], both transitions could exist. We may also add that the relaxation from DP to EP as seen here is qualitatively different from that envisaged in ECM [9], where the life time of the disordered state as well as the relaxation length diverge at H = Hp, while they should tend to zero at Hon where the contamination from edges has negligible effect. In contrast, lifetime of metastable DP in the present study (for a given current) peaks at Hon and reduces near Hp as would be expected from field dependence of U0. Now we will provide a possible explanation for peak in JFC and U0 at Hon in terms of Pippard model. In terms of this model, inter-vortex repulsion and pinning forces dominate for H < Hon and H > Hon, respectively. At Hon, the two forces, one favoring elastic lattice and other plastic deformations in EP, are balanced. These two forces, in addition to the Lorentz force (due to applied current) help in structural reorganization at Hon. As the net result of these forces is minimum at Hon, relaxation rate from DP to EP is reduced, leading to peak in JFC and U0. 4. Conclusion Field dependence of critical currents in field cooled and zero field cooled states has been investigated. Field cooled state is found to have a peak in critical current at the onset of peak effect. Relaxation of the disordered state obtained by field cooling the sample to more ordered equilibrium state has been studied. A model to understand the current dependence of relaxation time and time dependence of voltage has been presented. The results support the presence of a transition in the vortex state at the onset of the peak effect in agreement with some of the earlier studies where a transition to plastically deformed vortex state at the onset of the peak effect has been proposed. References [1] [2] [3] [4] [5] [6] [7] [8]
18
16
14 [9]
12
10 1.9
5
[10]
2.0
2.1
2.2
2.3
Field (T) Fig. 5. Magnetic field dependence of activation energy U0. A peak in activation energy is seen for H = Hon.
[11] [12]
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[13] H.J. Kim, H.S. Lee, B. Kang, P. Chowdhury, K.H. Kim, S.I. Lee, Phys. Rev. B 70 (2004) 132501. [14] G. Ravikumar, Phys. Rev. B 63 (2001) 224523. [15] Z.L. Xiao, E.Y. Andei, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 86 (2001) 2431. [16] T. Matsushita, B. Ni, Physica C 166 (1990) 423.
[17] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc., New York, 1996. [18] S.S. Banerjee, A.K. Grover, M.J. Higgins, Gutam I. Menon, P.K. Mishra, D. Pal, S. Ramakrishnan, T.V. Chandrasekhar Rao, G. Ravikumar, V.C. Sahni, S. Sarkar, C.V. Tomy, Physica C 355 (2001) 39.