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Irreversibility line and lock-in transition in Tl2Ba2CuO6 single crystals
Physica C 356 (2001) 115±121
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Irreversibility line and lock-in transition in Tl2Ba2CuO6 single crystals Gunadhor S. Okram a,b,*, Chieko Terakura a, Shinya Uji a, Haruyoshi Aoki b,c, Ming Xu d, D.G. Hinks e a
Cryogenics and Superconductivity Division, National Physical Laboratory, Dr. K.S. Krishnan Marg, New Delhi 110012, India b National Research Institute for Metals, 1-2-1 Sengen, Tsukuba, Ibaraki 305, Japan c Center for Low Temperature Science, Tohoku University, Sendai 980-77, Japan d The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA e Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA Received 20 September 2000; received in revised form 21 September 2000; accepted 11 January 2001
Abstract We investigated the irreversibility lines (Hirr ) and their correlation with lock-in transition on over-doped Tl2 Ba2 CuO2 single crystals using magneto-resistive measurement results. The lock-in transition is observed only above the irreversibility line Hirr? (t) for magnetic ®eld Hk c-axis, where t T =Tc is the reduced temperature. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction High temperature superconductors (HTSC) have very small coherence lengths n. This results in the rapid relaxation processes, and hence magnetisation of HTSC remains reversible over a wide temperature range below the superconducting transition temperature Tc [1]. This wide reversible region below the upper critical ®elds Hc2 in the H± T plane has a boundary known as the Ôirreversibility lineÕ below which the magnetisation shows
* Corresponding author. Address: Cryogenics and Superconductivity Division, National Physical Laboratory, Dr. K.S. Krishnan Road, New Delhi 110012, India. Tel.: +91-11-5786087; fax: +91-11-585-2678. E-mail address: [email protected] (G.S. Okram).
hysteresis [2]. The magnetic irreversibility implies persistent currents and hence zero-resistance. The irreversible behaviour is in fact the characteristic of the vortices concerned. Compared to the classic superconductors, the core structure (length scale n) of the HTSC is known to alter signi®cantly due to the presence of insulating or weakly superconducting layers separated by the CuO planes. Four types of vortex viz., pancake, tilted, combined and Josephson types may form depending on intensity of the magnetic ®eld H, anisotropy c and temperature T maintained [3,4]. They would form in order as the magnetic ®eld H changes its angle h from the c-axis (h 0) to the ab-plane (h 90°). Pancake, tilted and combined vortices have the usual normal cores but they are not rectilinear as Abrikosov vortex; they have structural dierences. The Josephson vortex has phase core
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as distinct from the normal core but the vortex is rectilinear. Thus, these vortices have dierent core structures and dynamics [4±8]. For the same H, c and T, they are in order of decreasing dissipation i.e., pancake vortex dissipates maximally and Josephson vortex dissipates the least. As a result, the irreversibility line for Hk c-axis lies lowest in the H±T plane [9]. Due to the dierence in the vortex dynamics, these vortices respond variously to an external perturbation [4]. An interesting aspect is the phenomenon of lock-in transition, which takes place at a critical lock-in angle hL . For h < hL , the ¯ux lines and magnetic ®eld induction run strictly parallel to the planes [10]. Indeed, the vortex dynamics and lockin phenomenon of these vortices can be studied through the results of magneto-resistivity measurements as a function of h choosing appropriate H, T and very small angular steps. Okram et al. has investigated this in some detail [4]. They found that the lock-in transition phenomenon is a consequence of the dierent vortex transitions, the complete lock-in transition being taken place at the onset of the Josephson vortices as the angle h ! 90°. These measurements were performed in the ¯ux-¯ow regime. Although pertinent from the de®nition, the lock-in transition phenomenon should take place in the irreversible region, it is of interest to see the experimental evidence. We report here the lock-in transition phenomenon being observed in the reversible regime (and hence above the irreversibility line of Hk c-axis in the H±T plane). 2. Experimental Single crystals of Tl2 Ba2 CuO6 (Tl2201) were grown by the contamination controlled ¯ux method [11]. They were of typically 0:4 0:3 0:075 mm3 size. The single crystals were well characterised for their structural, transport and other characteristics. They showed negligible dislocations and undetectable deviation from Tl2201 stoichiometry. The crystal on which the data presented here were obtained had superconducting transition Tc of 37 K and width of 4 K at H 0. Other details on this crystal were reported in Ref.
[4]. Good contacts (of typically 1 X) were made for the standard four-probe ac ab-plane qab (T) or c-axis qc (T) resistivity. A lock-in ampli®er was used for the purpose. The frequency chosen was as small as 7 Hz and it was in such a way not to interfere with the 50 Hz of the mains. The normal state I±V characteristic was linear. The current density used was as low as 28 mA/cm2 to avoid any heating or unwanted eects. The temperature T was measured using a carbon glass sensor. T was controlled using a Lakeshore temperature controller and it was stabilised to better than 3 mK. Resistivity was measured as the sample was cooling down slowly in the presence of a vertical magnetic ®eld H; H applied was up to 14 T. The direction of J was parallel to c-axis for qc (T) or parallel to ab plane for qab (T). Angular dependence of the thermally activated ¯ux-¯ow resistivity was measured for h 0±110°. This was done at constant temperature for various H values, and vice-versa. Angular step (Dh) chosen was as small as 0.01°; the other details of the experiment were described earlier [4].
3. Results and discussion Fig. 1 shows the results of temperature dependence resistivity qc (T) (and qab (T)) for the ®eld Hk c-axis (and ab plane) and for various H values. A rather fast suppression of Tc is seen as H increases for qc (T) when Hk c-axis (see Fig. 1a). The fast Tc suppression in this con®guration may be ascribed to the ¯ux ¯ow due to the pancake vortex [12±15]. Similar trend was found for qab (T) when Hk c-axis (see Fig. 1b). (It may be noted that the Tc at q 0 for zero ®eld in qab (T) is little higher than that in qc (T) as is often observed.) The Tc suppression in qc (T) as H increases is quite small when Hk ab plane (see Fig. 1c). This trend is consistent with the existence of Josephson vortices that have partially suppressed superconducting order parameter even at the phase core [16]. Thus, these results seemingly display the expected pancake vortex and Josephson vortex dynamics. The irreversibility ®elds Hirr (t) were determined from Fig. 1 by noting the temperature and corre-
G.S. Okram et al. / Physica C 356 (2001) 115±121
sponding H values at zero resistivity, where t T =Tc [17,18]. 1 The plot is shown in Fig. 2. When Hk c-axis, Hirr? (t) obtained independently from qc (T) and qab (T) almost coincide each other. These perhaps suggest that the ¯ux-¯ow behaviour for these con®gurations is identical. This happens so in spite of the fact that J and H are parallel for qc (T), and J and H are perpendicular for qab (T). They therefore seemingly imply that the Lorentz force is independent of the angle between H and J. This looks to be an intuitive example of such a situation wherein the classical angular dependence of the Lorentz force does not follow. These are however consistent with earlier reports [19±22]. In this situation, it seems to be reasonable to question the validity of the angular dependence of Lorentz force when applied to pancake vortex ¯ow. This interesting problem would require theoretical explanation. However, one interesting picture is that the types of vortices involved in these kinds of layered superconductors are distinctly dierent from the classic superconductors. In the former, the vortex line, unlike the continuous Abrikosov vortex, is not a rectilinear object. The ¯ux line is composed of the so-called pancake vortices directed along the c-axis joined by Josephson vortices whose axis threads through the junctions between the superconducting layers [15]. In the Josephson vortex, the vortex line is rectilinear as Abrikosov vortex [23]. Thus, one has to consider the corresponding vortex dynamics taking into accounts of these dierences also. Further, as seen from Fig. 2, when Hk ab plane, Hirrk (t) obtained from qc (T) data is quite higher than that of either of the two Hirr? (t) data. These results clearly show that the irreversible lines and their behaviour depend on the orientation of H. These irreversible lines must be related to the respective ¯ux dynamics of pancake vortex, and of Josephson vortex. Therefore, these features of irreversibility lines are indicative of vortex dynamics involved.
1 q 0 in fact was chosen at q 25 lX cm, which is equivalent to an electric ®eld of ~0.7 lV/cm at a current density of ~28 mA/cm2 .
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Fig. 1. Resistivity as a function of temperature at dierent magnetic ®elds H as indicated in each con®guration. (a) c-axis resistivity qc (T) for ®eld Hk c-axis, (b) ab plane resistivity qab (T) for ®eld Hk c-axis, and (c) c-axis resistivity qc (T) for ®eld Hk ab plane.
In fact, the vortex dynamics discussed above pertains to the special cases when the magnetic ®eld is parallel and perpendicular to the c-axis. The irreversibility ®elds at the same reduced temperature t for these two cases are remarkably different (Fig. 2). It would therefore be interesting to
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Fig. 2. Irreversibility ®eld Hirr (t) as determined from c-axis resistivity qc (T) for ®eld Hk c-axis (solid circle with line), ab plane resistivity qab (T) for ®eld Hk c-axis (solid diamond with line), and c-axis resistivity qc (T) for ®eld Hk ab plane (solid triangle with line).
see ``what happens to the ¯ux-¯ow resistivity above the ¯ux-¯ow onset as a function of angle h and how the lock-in transition features are related to the irreversibility lines?'' From the de®nition of the irreversibility ®elds (see also Fig. 2) and observation of lock-in transition in the ¯ux-¯ow regime [4], it seems apparent that the lock-in transition phenomenon can be observed only above the irreversibility line Hirr? (t) for Hk c-axis. In addition, one has to check the H and t values appropriately. To verify this, the resistivity qc h measurements were thus carried out at several H values at ®xed T and vice-versa; see Ref. [4] for details. Fig. 3a inset shows the vortex state qc h at 10 T for several values of T. For clarity in the peak-like features, h 0±70° has not been shown. A steplike feature near 95° is related to the change of angular step. There is dramatic change in the features of qc h as T varies from 20 to 38 K. At 28 K, as h increases, qc h remains almost constant at high value until 81° where qc h drops gradually to the dip at 89.65°. It then rises rapidly until the maximum at 89.85° and abruptly drops to the second minimum at 90°. There are therefore clearly four distinct regions of qc h. They are separated by a change in the slope of the resistivity qc h. We ascribe them to dierent types of vortices that may prevail as 10 T may be high enough to
Fig. 3. (a) Plot of qc h at 10 T for 28 K. Arrow indicates the angle de®ned for hLT1 at which qc h started dropping. Inset: Plots of qc h at 10 T for 20, 22, 24, 25, 26, 28, 30, 32, 34 and 38 K (bottom to top curves as indicated by an arrow as T increases). (b) qc h near 90° for 28 K to indicate its ®rst minimum (hLT2 ) and peak (hLT3 ) as indicated by arrows.
develop the combined vortex state as well. The region from 0° to 81° with comparatively large but very slowly increasing values of qc h should be related to the pancake vortex ¯ow because these vortices are considered to form at Hk c-axis, which is supposedly extended up to 81°. We assigned so because there is no change of qc h feature from 0°. The marginal enhancement of qc h with h may be due to presumably increasingly unstable pancake vortices as h increases. The region beyond 81° until 89.65° may be related to the development of tilted vortices for which the pinning has become better as h increases because the electromagnetic interactions between the planes and the Josephson coupling perhaps gradually enhance with h. From 89.65° to 89.85°, it is likely to be due to the development of combined vortices. The increase in the qc h can be understood from the fact that the pancake vortices arranged along the c-direction in the combined lattice are unstable with respect to small distortions caused by the Lorentz force induced by the in-plane currents of parallel vortices [3]. The eective resistivity is more or less comparable to that produced in the case of tilted vortices. However, beyond 89.85°,
G.S. Okram et al. / Physica C 356 (2001) 115±121
the pancake vortices can no longer persist and give way completely to the parallel (Josephson) vortices. At this, the applied magnetic ®eld (10 T) passes through the crystal with negligible disturbance to the superconducting plane CuO and the ®eld is completely locked-in due to intrinsic pinning [24]. One can thus realise how these features vary with temperature (Fig. 1a). It is also clear from the ®gure that the four regimes, especially associated with the dip and the peak in the qc h, are not exhibited in all the curves. These suggest that for 10 T, there is certain range of temperatures within which the four distinct features in qc h can be observed. Also, the data qc h < qcn (120 mX cm) for T < Tc indicates that these features in qc h are related to mixed state behaviour for all h, not related to the normal state. Notably, the values of qc h are contrary to the expectations from the usual Lorentz-force dependent ¯ux ¯ow. For this, qc h should be maximum for 90° and zero for 0°. This may be related to the type of vortices developed. In fact, the observed maximum and minimum in qc h happen to associate with pancake vortices and core-less Josephson vortices respectively, a situation that seems quite consistent; a subject matter discussed in separate publication [4]. We stress here that similar observations were made for qab h. The Lorentz force-related qc h paradox does not seem to be related to the twin planes that were found to be absent. This paradox seems to be related to the highly layered nature of the Tl2201 superconductor. Interestingly, such results have been reported in the literature (see for example Refs. [19±22]). The separation points of the presumably four types of vortex regimes assume importance for identifying the associated phase diagram. Drawing a tangent at the deviation point of (nearly linear) low angle part chooses the supposedly tilted vortex onset. This is de®ned as hLT1 (see Fig. 3) and indicates the transition from pancake to tilted vortex state. The dip is ascribed to the transition point from the tilted to combined lattice, and is de®ned as hLT2 (see Fig. 3b). Similarly, the peak, de®ned as hLT3 , is assumed to be the transition point from combined to Josephson vortex state (see Fig. 3b). We thus identify the three angles reliably as
119
hLT1 (T), hLT2 (T) and hLT3 (T) respectively for all T (Fig. 3) to study their behaviour and the vortex phase diagram. Further, we identi®ed similarly qc h data for 30 K recorded at several magnetic ®elds. These angles are represented by hLHY (H), where Y 1,2,3. Thus, the hLHY (t) and hLHY (H) graphs are plotted in Fig. 4a and b, where t is the reduced temperature T =Tc . Interestingly, hLT1 (t) increases nearly linearly with t suggesting perhaps the linear increment of the regime of pancake vortices to large h as t increases (see Fig. 4a). However, hLT2 (t) and hLT3 (t) are constant as t changes. These imply t independence of tiltedcombined and combined-Josephson vortex-state transitions. Further, the three angle-plots hLH1 (H), hLH2 (H) and hLH3 (H) are shown in Fig. 4b. Interestingly, hLH1 (H) increases initially very rapidly with increasing H. This trend is slows as H increases further. In contrast, hLH2 (H) and hLH3 (H)
Fig. 4. (a) The plots of hLTY (t), Y 1, 2, 3 (j, d, m respectively). Inset: hLTY (t), Y 2, 3 plots in expanded scale. (b) The plots of hLHY (H), Y 1, 2, 3 (r, . and ). Inset: expanded hLH2 (H) and hLH3 (H) plots.
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are large and yet vary (Fig. 4b, inset). Clearly, hLH2 (H) increases, remains constant, and then decreases with H. It is indicative of enhancement, attainment of its saturation and then reduction of the transition from tilted to combined vortex state. This trend together with the constancy of hLH3 (H) at 89.85° and its small increment beyond 10 T is in line with the favourable formation of combined lattice at high ®eld [3]. The behaviour of hLH3 (H) is indicative of the occurrence of transformation to Josephson vortex lattice closure to ab plane as H increases. This however is so beyond 10 T. Thus, the expected pancake regime is large while tilted vortex, combined vortex and Josephson vortex regimes are small. Also, the non-zero slopes of qc h for each vortex state may suggest non-equilibrium vortex state each with h. The transition features in qc h vary with temperature for 10 T (Fig. 3a). All the transition features of qc h are apparent for 24±30 K, not above or below these limits. Similar was the situation observed for 30 K at various ®eld values [4]. Thus, these lock-in transition features are displayed for speci®c values of H and T. Indeed, Fig. 4 contains valuable information on the dierent types of vortices that may be expected at various orientations of the (Tl2201) crystal and hence the vortex phase-diagram [4]. We show here the correlation among the loci of hLXY (t, H) and Hirr? (t). We consider the data point (0.75, 82°) at 10 T in Fig. 4a. This point corresponds to (0.75, 10 T) in Fig. 2, which lies above the Hirr? (t) line. Another data point (0.54, 68°) at 10 T in Fig. 4a corresponds to (0.54, 10 T) that also lies above the Hirr? (t) line (Fig. 2). Similarly, all other (hLTY (t), t) points in Fig. 4a lie above Hirr? (t) in Fig. 2. Similar argument can be applied to prove that the points (hLHY , H) on Fig. 4b lie above Hirr? (t) in Fig. 2. Hence, in magneto-resistive measurements, the lock-in transition can be observed above the irreversibility line Hirr? (t) for Hk c-axis.
4. Conclusion In conclusion, the irreversibility lines (Hirr ) and lock-in transition are correlated for over-doped
Tl2201 single crystals. The lock-in transition can be observed only above the irreversibility line Hirr? for Hk c-axis.
Acknowledgements One of us (GSO) acknowledges the ®nancial supports of the Science and Technology Agency (STA), Government of Japan as STA fellowship and the Council of Scienti®c and Industrial Research (CSIR) as Senior Research Associate (Pool Scheme) of Government of India.
References [1] M. Tinkham, in: Introduction to Superconductivity, McGraw-Hill, New York, 1996, p. 331. [2] K.A. Muller, M. Takashige, J.G. Bednorz, Phys. Rev. Lett. 58 (1987) 1143. [3] L.N. Bulaevskii, M. Ledvij, V.G. Kogan, Phys. Rev. B 46 (1992) 366. [4] G.S. Okram, H. Aoki, M. Xu, D.G. Hinks, Physica C, in press. [5] G. Blatter, M.V. Fielgel'man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [6] V.G. Kogan, L.J. Campbell, Phys. Rev. Lett. 62 (1989) 1552. [7] L.I. Glazman, A.E. Koshelev, Phys. Rev. B 43 (1991) 2835. [8] E. Sardella, Phys. Rev. B 45 (1992) 314. [9] G.S. Okram, H. Aoki, K. Nakamura, Solid State Commun. 110 (1999) 327. [10] D. Feinberg, C. Villard, Phys. Rev. Lett. 65 (1990) 919. [11] K.A. Moler, J.R. Kirtley, D.G. Hinks, T.W. Li, M. Xu, Science 279 (1998) 1193. [12] J.R. Clem, Phys. Rev. B 43 (1991) 7837. [13] J.R. Clem, Physica C 162±164 (1989) 1137. [14] J.R. Clem, et al., Phys. Rev. B 44 (1991) 2732. [15] S. Doniach, The phenomenology of ¯ux motion in high temperature superconductors, in: K.S. Bedell et al., (Eds.), Proceedings of the Los Alamos Symposium, AddisonWesley, Reading, MA, 1989, p.406. [16] G. Blatter, V.B. Geshkenbein, Phys. Rev. B 47 (1993) 2725. [17] Y. Yeshuran, A.P. Malezomo, Phys. Rev. Lett. 60 (1988) 2202. [18] L. Burlachkov, V.B. Geshkenbein, A.E. Koshelev, A.I. Larkin, V.M. Vinokur, Phys. Rev. B 50 (1994) 16770. [19] Y. Iye, S. Nakamura, T. Tamegai, T. Terashima, K. Yamamoto, Y. Bando, Physica C 166 (1990) 62.
G.S. Okram et al. / Physica C 356 (2001) 115±121 [20] Y. Iye, S. Nakamura, T. Tamegai, Physica C 159 (1989) 433. [21] Y. Iye, S. Nakamura, T. Tamegai, T. Terashima, K. Yamamoto, Y. Bando, Physica C 167 (1990) 278. [22] Y. Iye, et al., Physica C 174 (1991) 227.
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[23] A.A. Abrikosov, Zh. Eksperim. i. Teor. Fiz. 32 (1957) 1442, (Sov. Phys. JETP 5 (1957) 1174). [24] M. Tachiki, Takahashi, Solid State Commun. 70 (1989) 290.
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Irreversibility line and lock-in transition in Tl2Ba2CuO6 single crystals Gunadhor S. Okram a,b,*, Chieko Terakura a, Shinya Uji a, Haruyoshi Aoki b,c, Ming Xu d, D.G. Hinks e a
Cryogenics and Superconductivity Division, National Physical Laboratory, Dr. K.S. Krishnan Marg, New Delhi 110012, India b National Research Institute for Metals, 1-2-1 Sengen, Tsukuba, Ibaraki 305, Japan c Center for Low Temperature Science, Tohoku University, Sendai 980-77, Japan d The James Franck Institute, The University of Chicago, Chicago, IL 60637, USA e Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA Received 20 September 2000; received in revised form 21 September 2000; accepted 11 January 2001
Abstract We investigated the irreversibility lines (Hirr ) and their correlation with lock-in transition on over-doped Tl2 Ba2 CuO2 single crystals using magneto-resistive measurement results. The lock-in transition is observed only above the irreversibility line Hirr? (t) for magnetic ®eld Hk c-axis, where t T =Tc is the reduced temperature. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction High temperature superconductors (HTSC) have very small coherence lengths n. This results in the rapid relaxation processes, and hence magnetisation of HTSC remains reversible over a wide temperature range below the superconducting transition temperature Tc [1]. This wide reversible region below the upper critical ®elds Hc2 in the H± T plane has a boundary known as the Ôirreversibility lineÕ below which the magnetisation shows
* Corresponding author. Address: Cryogenics and Superconductivity Division, National Physical Laboratory, Dr. K.S. Krishnan Road, New Delhi 110012, India. Tel.: +91-11-5786087; fax: +91-11-585-2678. E-mail address: [email protected] (G.S. Okram).
hysteresis [2]. The magnetic irreversibility implies persistent currents and hence zero-resistance. The irreversible behaviour is in fact the characteristic of the vortices concerned. Compared to the classic superconductors, the core structure (length scale n) of the HTSC is known to alter signi®cantly due to the presence of insulating or weakly superconducting layers separated by the CuO planes. Four types of vortex viz., pancake, tilted, combined and Josephson types may form depending on intensity of the magnetic ®eld H, anisotropy c and temperature T maintained [3,4]. They would form in order as the magnetic ®eld H changes its angle h from the c-axis (h 0) to the ab-plane (h 90°). Pancake, tilted and combined vortices have the usual normal cores but they are not rectilinear as Abrikosov vortex; they have structural dierences. The Josephson vortex has phase core
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as distinct from the normal core but the vortex is rectilinear. Thus, these vortices have dierent core structures and dynamics [4±8]. For the same H, c and T, they are in order of decreasing dissipation i.e., pancake vortex dissipates maximally and Josephson vortex dissipates the least. As a result, the irreversibility line for Hk c-axis lies lowest in the H±T plane [9]. Due to the dierence in the vortex dynamics, these vortices respond variously to an external perturbation [4]. An interesting aspect is the phenomenon of lock-in transition, which takes place at a critical lock-in angle hL . For h < hL , the ¯ux lines and magnetic ®eld induction run strictly parallel to the planes [10]. Indeed, the vortex dynamics and lockin phenomenon of these vortices can be studied through the results of magneto-resistivity measurements as a function of h choosing appropriate H, T and very small angular steps. Okram et al. has investigated this in some detail [4]. They found that the lock-in transition phenomenon is a consequence of the dierent vortex transitions, the complete lock-in transition being taken place at the onset of the Josephson vortices as the angle h ! 90°. These measurements were performed in the ¯ux-¯ow regime. Although pertinent from the de®nition, the lock-in transition phenomenon should take place in the irreversible region, it is of interest to see the experimental evidence. We report here the lock-in transition phenomenon being observed in the reversible regime (and hence above the irreversibility line of Hk c-axis in the H±T plane). 2. Experimental Single crystals of Tl2 Ba2 CuO6 (Tl2201) were grown by the contamination controlled ¯ux method [11]. They were of typically 0:4 0:3 0:075 mm3 size. The single crystals were well characterised for their structural, transport and other characteristics. They showed negligible dislocations and undetectable deviation from Tl2201 stoichiometry. The crystal on which the data presented here were obtained had superconducting transition Tc of 37 K and width of 4 K at H 0. Other details on this crystal were reported in Ref.
[4]. Good contacts (of typically 1 X) were made for the standard four-probe ac ab-plane qab (T) or c-axis qc (T) resistivity. A lock-in ampli®er was used for the purpose. The frequency chosen was as small as 7 Hz and it was in such a way not to interfere with the 50 Hz of the mains. The normal state I±V characteristic was linear. The current density used was as low as 28 mA/cm2 to avoid any heating or unwanted eects. The temperature T was measured using a carbon glass sensor. T was controlled using a Lakeshore temperature controller and it was stabilised to better than 3 mK. Resistivity was measured as the sample was cooling down slowly in the presence of a vertical magnetic ®eld H; H applied was up to 14 T. The direction of J was parallel to c-axis for qc (T) or parallel to ab plane for qab (T). Angular dependence of the thermally activated ¯ux-¯ow resistivity was measured for h 0±110°. This was done at constant temperature for various H values, and vice-versa. Angular step (Dh) chosen was as small as 0.01°; the other details of the experiment were described earlier [4].
3. Results and discussion Fig. 1 shows the results of temperature dependence resistivity qc (T) (and qab (T)) for the ®eld Hk c-axis (and ab plane) and for various H values. A rather fast suppression of Tc is seen as H increases for qc (T) when Hk c-axis (see Fig. 1a). The fast Tc suppression in this con®guration may be ascribed to the ¯ux ¯ow due to the pancake vortex [12±15]. Similar trend was found for qab (T) when Hk c-axis (see Fig. 1b). (It may be noted that the Tc at q 0 for zero ®eld in qab (T) is little higher than that in qc (T) as is often observed.) The Tc suppression in qc (T) as H increases is quite small when Hk ab plane (see Fig. 1c). This trend is consistent with the existence of Josephson vortices that have partially suppressed superconducting order parameter even at the phase core [16]. Thus, these results seemingly display the expected pancake vortex and Josephson vortex dynamics. The irreversibility ®elds Hirr (t) were determined from Fig. 1 by noting the temperature and corre-
G.S. Okram et al. / Physica C 356 (2001) 115±121
sponding H values at zero resistivity, where t T =Tc [17,18]. 1 The plot is shown in Fig. 2. When Hk c-axis, Hirr? (t) obtained independently from qc (T) and qab (T) almost coincide each other. These perhaps suggest that the ¯ux-¯ow behaviour for these con®gurations is identical. This happens so in spite of the fact that J and H are parallel for qc (T), and J and H are perpendicular for qab (T). They therefore seemingly imply that the Lorentz force is independent of the angle between H and J. This looks to be an intuitive example of such a situation wherein the classical angular dependence of the Lorentz force does not follow. These are however consistent with earlier reports [19±22]. In this situation, it seems to be reasonable to question the validity of the angular dependence of Lorentz force when applied to pancake vortex ¯ow. This interesting problem would require theoretical explanation. However, one interesting picture is that the types of vortices involved in these kinds of layered superconductors are distinctly dierent from the classic superconductors. In the former, the vortex line, unlike the continuous Abrikosov vortex, is not a rectilinear object. The ¯ux line is composed of the so-called pancake vortices directed along the c-axis joined by Josephson vortices whose axis threads through the junctions between the superconducting layers [15]. In the Josephson vortex, the vortex line is rectilinear as Abrikosov vortex [23]. Thus, one has to consider the corresponding vortex dynamics taking into accounts of these dierences also. Further, as seen from Fig. 2, when Hk ab plane, Hirrk (t) obtained from qc (T) data is quite higher than that of either of the two Hirr? (t) data. These results clearly show that the irreversible lines and their behaviour depend on the orientation of H. These irreversible lines must be related to the respective ¯ux dynamics of pancake vortex, and of Josephson vortex. Therefore, these features of irreversibility lines are indicative of vortex dynamics involved.
1 q 0 in fact was chosen at q 25 lX cm, which is equivalent to an electric ®eld of ~0.7 lV/cm at a current density of ~28 mA/cm2 .
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Fig. 1. Resistivity as a function of temperature at dierent magnetic ®elds H as indicated in each con®guration. (a) c-axis resistivity qc (T) for ®eld Hk c-axis, (b) ab plane resistivity qab (T) for ®eld Hk c-axis, and (c) c-axis resistivity qc (T) for ®eld Hk ab plane.
In fact, the vortex dynamics discussed above pertains to the special cases when the magnetic ®eld is parallel and perpendicular to the c-axis. The irreversibility ®elds at the same reduced temperature t for these two cases are remarkably different (Fig. 2). It would therefore be interesting to
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Fig. 2. Irreversibility ®eld Hirr (t) as determined from c-axis resistivity qc (T) for ®eld Hk c-axis (solid circle with line), ab plane resistivity qab (T) for ®eld Hk c-axis (solid diamond with line), and c-axis resistivity qc (T) for ®eld Hk ab plane (solid triangle with line).
see ``what happens to the ¯ux-¯ow resistivity above the ¯ux-¯ow onset as a function of angle h and how the lock-in transition features are related to the irreversibility lines?'' From the de®nition of the irreversibility ®elds (see also Fig. 2) and observation of lock-in transition in the ¯ux-¯ow regime [4], it seems apparent that the lock-in transition phenomenon can be observed only above the irreversibility line Hirr? (t) for Hk c-axis. In addition, one has to check the H and t values appropriately. To verify this, the resistivity qc h measurements were thus carried out at several H values at ®xed T and vice-versa; see Ref. [4] for details. Fig. 3a inset shows the vortex state qc h at 10 T for several values of T. For clarity in the peak-like features, h 0±70° has not been shown. A steplike feature near 95° is related to the change of angular step. There is dramatic change in the features of qc h as T varies from 20 to 38 K. At 28 K, as h increases, qc h remains almost constant at high value until 81° where qc h drops gradually to the dip at 89.65°. It then rises rapidly until the maximum at 89.85° and abruptly drops to the second minimum at 90°. There are therefore clearly four distinct regions of qc h. They are separated by a change in the slope of the resistivity qc h. We ascribe them to dierent types of vortices that may prevail as 10 T may be high enough to
Fig. 3. (a) Plot of qc h at 10 T for 28 K. Arrow indicates the angle de®ned for hLT1 at which qc h started dropping. Inset: Plots of qc h at 10 T for 20, 22, 24, 25, 26, 28, 30, 32, 34 and 38 K (bottom to top curves as indicated by an arrow as T increases). (b) qc h near 90° for 28 K to indicate its ®rst minimum (hLT2 ) and peak (hLT3 ) as indicated by arrows.
develop the combined vortex state as well. The region from 0° to 81° with comparatively large but very slowly increasing values of qc h should be related to the pancake vortex ¯ow because these vortices are considered to form at Hk c-axis, which is supposedly extended up to 81°. We assigned so because there is no change of qc h feature from 0°. The marginal enhancement of qc h with h may be due to presumably increasingly unstable pancake vortices as h increases. The region beyond 81° until 89.65° may be related to the development of tilted vortices for which the pinning has become better as h increases because the electromagnetic interactions between the planes and the Josephson coupling perhaps gradually enhance with h. From 89.65° to 89.85°, it is likely to be due to the development of combined vortices. The increase in the qc h can be understood from the fact that the pancake vortices arranged along the c-direction in the combined lattice are unstable with respect to small distortions caused by the Lorentz force induced by the in-plane currents of parallel vortices [3]. The eective resistivity is more or less comparable to that produced in the case of tilted vortices. However, beyond 89.85°,
G.S. Okram et al. / Physica C 356 (2001) 115±121
the pancake vortices can no longer persist and give way completely to the parallel (Josephson) vortices. At this, the applied magnetic ®eld (10 T) passes through the crystal with negligible disturbance to the superconducting plane CuO and the ®eld is completely locked-in due to intrinsic pinning [24]. One can thus realise how these features vary with temperature (Fig. 1a). It is also clear from the ®gure that the four regimes, especially associated with the dip and the peak in the qc h, are not exhibited in all the curves. These suggest that for 10 T, there is certain range of temperatures within which the four distinct features in qc h can be observed. Also, the data qc h < qcn (120 mX cm) for T < Tc indicates that these features in qc h are related to mixed state behaviour for all h, not related to the normal state. Notably, the values of qc h are contrary to the expectations from the usual Lorentz-force dependent ¯ux ¯ow. For this, qc h should be maximum for 90° and zero for 0°. This may be related to the type of vortices developed. In fact, the observed maximum and minimum in qc h happen to associate with pancake vortices and core-less Josephson vortices respectively, a situation that seems quite consistent; a subject matter discussed in separate publication [4]. We stress here that similar observations were made for qab h. The Lorentz force-related qc h paradox does not seem to be related to the twin planes that were found to be absent. This paradox seems to be related to the highly layered nature of the Tl2201 superconductor. Interestingly, such results have been reported in the literature (see for example Refs. [19±22]). The separation points of the presumably four types of vortex regimes assume importance for identifying the associated phase diagram. Drawing a tangent at the deviation point of (nearly linear) low angle part chooses the supposedly tilted vortex onset. This is de®ned as hLT1 (see Fig. 3) and indicates the transition from pancake to tilted vortex state. The dip is ascribed to the transition point from the tilted to combined lattice, and is de®ned as hLT2 (see Fig. 3b). Similarly, the peak, de®ned as hLT3 , is assumed to be the transition point from combined to Josephson vortex state (see Fig. 3b). We thus identify the three angles reliably as
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hLT1 (T), hLT2 (T) and hLT3 (T) respectively for all T (Fig. 3) to study their behaviour and the vortex phase diagram. Further, we identi®ed similarly qc h data for 30 K recorded at several magnetic ®elds. These angles are represented by hLHY (H), where Y 1,2,3. Thus, the hLHY (t) and hLHY (H) graphs are plotted in Fig. 4a and b, where t is the reduced temperature T =Tc . Interestingly, hLT1 (t) increases nearly linearly with t suggesting perhaps the linear increment of the regime of pancake vortices to large h as t increases (see Fig. 4a). However, hLT2 (t) and hLT3 (t) are constant as t changes. These imply t independence of tiltedcombined and combined-Josephson vortex-state transitions. Further, the three angle-plots hLH1 (H), hLH2 (H) and hLH3 (H) are shown in Fig. 4b. Interestingly, hLH1 (H) increases initially very rapidly with increasing H. This trend is slows as H increases further. In contrast, hLH2 (H) and hLH3 (H)
Fig. 4. (a) The plots of hLTY (t), Y 1, 2, 3 (j, d, m respectively). Inset: hLTY (t), Y 2, 3 plots in expanded scale. (b) The plots of hLHY (H), Y 1, 2, 3 (r, . and ). Inset: expanded hLH2 (H) and hLH3 (H) plots.
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are large and yet vary (Fig. 4b, inset). Clearly, hLH2 (H) increases, remains constant, and then decreases with H. It is indicative of enhancement, attainment of its saturation and then reduction of the transition from tilted to combined vortex state. This trend together with the constancy of hLH3 (H) at 89.85° and its small increment beyond 10 T is in line with the favourable formation of combined lattice at high ®eld [3]. The behaviour of hLH3 (H) is indicative of the occurrence of transformation to Josephson vortex lattice closure to ab plane as H increases. This however is so beyond 10 T. Thus, the expected pancake regime is large while tilted vortex, combined vortex and Josephson vortex regimes are small. Also, the non-zero slopes of qc h for each vortex state may suggest non-equilibrium vortex state each with h. The transition features in qc h vary with temperature for 10 T (Fig. 3a). All the transition features of qc h are apparent for 24±30 K, not above or below these limits. Similar was the situation observed for 30 K at various ®eld values [4]. Thus, these lock-in transition features are displayed for speci®c values of H and T. Indeed, Fig. 4 contains valuable information on the dierent types of vortices that may be expected at various orientations of the (Tl2201) crystal and hence the vortex phase-diagram [4]. We show here the correlation among the loci of hLXY (t, H) and Hirr? (t). We consider the data point (0.75, 82°) at 10 T in Fig. 4a. This point corresponds to (0.75, 10 T) in Fig. 2, which lies above the Hirr? (t) line. Another data point (0.54, 68°) at 10 T in Fig. 4a corresponds to (0.54, 10 T) that also lies above the Hirr? (t) line (Fig. 2). Similarly, all other (hLTY (t), t) points in Fig. 4a lie above Hirr? (t) in Fig. 2. Similar argument can be applied to prove that the points (hLHY , H) on Fig. 4b lie above Hirr? (t) in Fig. 2. Hence, in magneto-resistive measurements, the lock-in transition can be observed above the irreversibility line Hirr? (t) for Hk c-axis.
4. Conclusion In conclusion, the irreversibility lines (Hirr ) and lock-in transition are correlated for over-doped
Tl2201 single crystals. The lock-in transition can be observed only above the irreversibility line Hirr? for Hk c-axis.
Acknowledgements One of us (GSO) acknowledges the ®nancial supports of the Science and Technology Agency (STA), Government of Japan as STA fellowship and the Council of Scienti®c and Industrial Research (CSIR) as Senior Research Associate (Pool Scheme) of Government of India.
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