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Interacting crossing vortex lattices in the presence of quenched disorder
Physica C 412–414 (2004) 372–378 www.elsevier.com/locate/physc
Interacting crossing vortex lattices in the presence of quenched disorder S.J. Bending
a,*
, A.N. Grigorenko a,b, I.A. Crisan a, D. Cole a, A.E. Koshelev c, John R. Clem d, T. Tamegai e,f, S. Ooi e,f
a Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK c Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011-3160, USA e Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8627, Japan f Japan Science and Technology Corporation, CREST, Tokyo, Japan b
d
Received 29 October 2003; accepted 6 January 2004 Available online 6 May 2004
Abstract We have used high resolution scanning Hall probe microscopy (SHPM) to study vortex structures in the interacting crossing lattices regime of Bi2 Sr2 CaCu2 O8þd (BSCCO) single crystals under independently applied Hc and Hk fields. At very low c-axis fields we observe a novel 1D vortex chain state where all pancake vortex (PV) stacks become trapped on underlying stacks of Josephson vortices (JVs). In this regime distortions of the JV lattice, induced by varying Hk , enable the indirect manipulation of PVs trapped on them. Preliminary results of experiments are presented in which we have attempted to realise a vortex ‘lens’ based on this vortex ‘pump’ principle. The existence of 1D vortex chains also explains many of the features observed in the magnetisation of BSCCO under strongly tilted magnetic fields. Finally we demonstrate how the presence of quenched disorder leads to indirect JV pinning via interactions with weakly pinned PV stacks and show how fragmentation of both PV and JV stacks can occur when stacks of JVs ‘decorated’ with PVs are forced abruptly through regions of disorder. 2004 Elsevier B.V. All rights reserved. PACS: 74.60.)w; 74.25.Ha; 74.72.)h Keywords: Layered superconductors; Vortices; Crossing lattices; Vortex pumps
1. Introduction Superconductivity in high temperature superconductors (HTS) is known to reside in weakly*
Corresponding author. Tel.: +44-1225-385173; fax: +441225-386110. E-mail address: [email protected] (S.J. Bending).
coupled CuO2 layers lying parallel to the a–b plane. Under c-axis magnetic fields vortices have their circulating supercurrents flowing within the CuO2 planes and are formally viewed as stacks of 2D pancake vortices [1] which interact to form well-ordered hexagonal Abrikosov lattices [2] in disorder-free samples. When the field is applied in the a–b plane Josephson vortices [1] arise whose
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.058
S.J. Bending et al. / Physica C 412–414 (2004) 372–378
‘cores’ reside in the spaces between CuO2 planes and whose circulating currents derive partly from weak Josephson coupling between them. This anisotropic current distribution leads to strongly anisotropic vortex–vortex interactions and a rhombic lattice whose unit cell is greatly stretched out in the a–b plane. In extremely anisotropic HTS uniformly tilted vortices, composed of a staircase of PVs linked by segments of JV, are unstable with respect to the formation of independent, perpendicular hexagonal PV and rhombic JV lattices [3]. Furthermore, small PV displacements driven by the underlying JV supercurrents lead to an attractive interaction between these two ‘crossing’ lattices [4,5] with rather profound consequences since the symmetries of the projected JV and PV lattices in any given direction are, in general, incommensurate. Thus a rich variety of broken symmetry phases can arise in the crossing lattices regime which remain largely unexplored.
2. Experimental method We have used high resolution scanning Hall probe microscopy (SHPM) to directly observe discrete PVs in Bi2 Sr2 CaCu2 O8þd (BSCCO) single crystals under independently applied Hc and Hk fields. The SHPM used is a modified commercial low temperature STM in which the usual tunnelling tip has been replaced by a microfabricated GaAs/AlGaAs heterostructure chip. Electron beam lithography and wet chemical etching were used to define a Hall probe in the two-dimensional electron gas (2DEG) at the intersection of two 300 nm wide wires approximately 5 lm from the corner of a deep mesa etch. The latter had been coated with a thin Au layer to act as an integrated STM tip. The sample is first approached towards the sensor until tunnelling is established and then retracted about 100 nm allowing rapid scanning. The Hall probe makes an angle of about 1 with the sample plane so that the STM tip is always the closest point to the surface, and the Hall sensor was about 300–400 nm above the sample in the images shown here. A more detailed description of the instrument is given elsewhere [6].
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3. Experimental results Fig. 1 shows a set of SHPM images of the magnetic induction just above the a–b face of an as-grown BSCCO single crystal (Tc ¼ 90 K) as the tilt angle, h, between the applied field and the caxis is progressively increased [7]. When h ¼ 0 (Fig. 1(a)) a well-ordered hexagonal PV lattice is observed in a field of 12 Oe at 81 K. However, as the in-plane field, Hk , is increased from zero ‘‘chain states’’ were observed, where domains of approximately sixfold PV lattice are separated by chains of PVs as illustrated in Fig. 1(b),(c). As has been observed previously [8] the chains are aligned along the direction of the in-plane field (indicated by an arrow) and the vortex separation within the chains is smaller than that of the Abrikosov lattice. In addition, the pronounced induced disorder within the lattice domains (see, for example, the indicated sevenfold (A) and fivefold (B) PV rings) points to the presence of frustration due to the incommensurate vortex chains. As the c-axis component of the applied field, Hc , is decreased below the ‘‘ordering’’ field [9], quenched disorder destroys the sixfold symmetry of the Abrikosov domains as shown in Fig. 1(d). However, as the tilt angle is increased still further we observe a new phase transition to another ordered state where all the PV stacks condense onto the vortex chains and the domains of Abrikosov lattice disappear. Fig. 1(e) illustrates this 1D vortex chain configuration, and it is evident that the PVs have assumed the projected quasi-one dimensional symmetry of the underlying JV lattice. The separation between PV chains could easily be reduced by increasing Hk at fixed Hc , as illustrated in Fig. 1(g), and did not depend measurably upon Hc or the temperature in the range 77–86 K. As the field is tilted even further towards the a–b plane the PV chains become unstable and a further transition occurs to a state formed of very low flux density 1D chains of ‘sublimed’ PV stacks (Fig. 1(f)). We speculate that the latter correspond to the decomposition of PV stacks into highly mobile PV segments linked by sections of JV. The vortex behaviour illustrated in Fig. 1 finds a natural explanation in terms of the crossing lattices picture [3–5]. The JV lattice corresponds to
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Fig. 1. SHPM images of: (a) the hexagonal Abrikosov lattice of pancake vortices obtained at Hc ¼ 12 Oe (Hk ¼ 0) at 81 K. (Grayscale spans 2.5 G.) The composite vortex lattice state at (b) Hc ¼ 14 Oe and Hk ¼ 32 Oe at 81 K (GS ¼ 1.7 G) and (c) Hc ¼ 10 Oe and Hk ¼ 32 Oe at 77 K (GS ¼ 1.9 G). (d) The composite lattice state below the ‘‘ordering’’ field at Hc ¼ 2 Oe and Hk ¼ 27 Oe at 81 K (GS ¼ 2.9 G). (e) The ‘‘isolated chain’’ state at Hc ¼ 0:5 Oe and Hk ¼ 35 Oe at 81 K (GS ¼ 2.5 G). (f) The ‘sublimed’ PV state at Hc < 0:2 Oe and Hk ¼ 38 Oe at 83 K (GS ¼ 0.4 G). (g) Graph illustrating the linear dependence of chain separation, c, as a function of 1=2 (Hk Insets show the ‘‘isolated chain state’’ at fields Hc ¼ 1:2 Oe and Hk ¼ 50 Oe at 81 K and a linescan of magnetic induction along the indicated dashed line.
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the penetration of Hk in the sample and in the simplest case, consists of JV stacks ffi with a lateral qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi separation of c ¼ 3cU0 =ð2Hk Þ, where c is the anisotropy parameter and U0 is the flux quantum [5]. For small c-axis fields, Hc , a JV core contains only one row of PVs and the energy of a PV located on a JV is reduced due to the relaxation of the PV structure in response to the JV supercurrents as illustrated in Fig. 2(a). As a result, there is a finite PV–JV attraction, which increases the PV density along JV stacks. The total number of PVs per unit area is governed by the c-axis field component, Hc , and when it becomes low enough all PVs can condense onto JV stacks. The transition field to an ‘‘isolated chain’’ state can be found by comparison of the PV–JV attraction energy [5] with the PV repulsion energy in the chains and is given by Hc U0 =ðkab c lnðc=c0 ÞÞ, where kab is the
Fig. 2. (a) Sketch of the displaced PV stack interacting with a JV stack. (b) PV distribution at Hc ¼ 2 Oe for Hk ¼ 11 Oe at 83 K (GS ¼ 3.2 G). (c) The same PV distribution at Hc ¼ 2 Oe after the field Hk is increased to 16.5 Oe at 83 K with a JV stack capturing the top-right PV chain (GS ¼ 3.3 G). (d) Linescans of magnetic induction along the directions indicated in (b),(c).
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penetration depth parallel to the a–b planes, c0 ¼ Ak2ab =ðcsÞ, A is a constant close to unity and s is the CuO2 plane separation. This means that the isolated chain state is stable when Hc , Hk satisfy the 2 following inequality; pffiffiffi H2c =H2k < Htr ðT ; Hk Þ, where Htr ðT ; Hk Þ ¼ 2U0 =ð 3ckab ln ðc=c0 ÞÞ depends only weakly upon Hk and is about 0.04 Oe at 77 K in our case. The distance between PVs trapped on a JV in an isolated chain is found to be d ¼ U0 =ðcHc Þ. All these features of crossing lattices agree remarkably well with our experimental observations. The chain separation, c, at 81 K is pffiffiffiffiffiffi plotted as a function of 1= Hk in Fig. 1(g). The excellent linear dependence allows us to directly evaluate the anisotropy parameter c ¼ 640 25, which is in reasonable agreement with other estimates [10]. Figs. 2(b) and (c) illustrate the elliptical form of the trapped PV stacks at elevated temperatures, predicted by the Koshelev theory [5], after displacements along the chains driven by the JV supercurrents. Fig. 2(b) shows the situation with a small in-plane field applied, while in Fig. 2(c) a JV stack has been aligned along the chain indicated with an arrow. Linescans along the indicated chain before and after application of the in-plane field are shown in Fig. 2(d) and clearly reveal the broadening of the PV stack along the chain when the JV stack is present. One of the most interesting and important experimental properties of the isolated chain configuration is that PVs remain trapped on JVs under changes of applied field over a wide studied temperature range of 77–88 K. This may be anticipated close to Tc because the PV–JV interaction is almost temperature independent [5] while the PV pinning energy decreases rapidly close to Tc . However, the coherent motion of pancakes and Josephson vortices at temperatures as low as 77 K is remarkable. This is important as it is known that the irreversible magnetisation of BSCCO single crystals can be suppressed by the application of inplane ac fields [11]. Figs. 3(a) and (b) show the increase of the pancake density inside a PV chain trapped on a JV stack at 81 K as the c-axis field is increased from zero at fixed Hk ¼ 35 Oe [7]. We find that the PV mobility along JV stacks appears to be considerably higher than in JV-free regions, and the presence of JVs allows PVs to overcome
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Fig. 3. Penetration of pancake vortices along 1D chains at 81 K as Hc is increased at fixed Hk ¼ 35 Oe. (a) Hc ¼ 2 Oe, (GS ¼ 3.2 G). (b) Hc ¼ 4 Oe, (GS ¼ 3.7 G). (c) Local magnetisation loop measured with the Hall probe 1 lm above the sample with and without an in-plane magnetic field.
only then propagate into the interchain spaces if the chain density becomes saturated, i.e. if Hc2 =Hk < Htr ðT ; Hk Þ. Conversely, if Hc is reduced, PVs exit by travelling along JVs to the sample edges. Figs. 4(a) and (b) show the dragging of the pancake chain by a JV stack at 83 K as Hk is reduced at fixed Hc ¼ 0:5 Oe [7]. During this coherent movement the JV stack is able to depin and drag with it two isolated PV stacks (A, B) which were originally pinned on the right hand side of the image. Pancake motion induced by the motion of JV stacks is unusual because these PV displacements are reversible and uniquely determined by the applied field Hk , in stark contrast to the customary situation where c-axis magnetic fields and/or in-plane currents determine the forces acting on PVs rather than vortex positions. We have been able to show [7], for example, that it is possible to demagnetise a sample containing trapped PV stacks by turning on a large in-plane field, to drive the system into the 1D vortex chain state, and then slowly removing it. As the JVs are gradually swept out of the sample they drag the PV stacks with them, leaving a fully demagnetised sample. Quenched disorder in our sample plays an important role in controlling the static and dynamic properties of pancake chains. It is well known that the disorder in BSCCO single crystals is often strongly anisotropic, leading to the formation of pinned PV chains and stripes along the
quenched disorder, effectively depinning them. Fig. 3(c) shows local magnetisation loops (lo Mloc versus Hc ) at constant Hk ¼ 35 Oe, measured with the scanning Hall probe fixed 1 lm above the same region of the sample, which reveal a dramatic suppression of the irreversibility as compared to measurements in the absence of JVs (Hk ¼ 0). The presence of Hk also slightly decreases the PV penetration field (Hp ). At fields just above Hp PVs preferentially enter the sample along JVs, where the superposition of Meissner and JV currents at the edges leads to a slight lowering of Bean–Livingston penetration barriers [12], and then enjoy a much higher mobility along the JV chains. PVs
Fig. 4. Images showing how a pancake vortex chain can be dragged by a JV stack at 83 K as Hk is reduced at fixed Hc ¼ 0:5 Oe. (a) Hk ¼ 33 Oe (GS ¼ 2.9 G). (b) Hk ¼ 23 Oe (GS ¼ 3.4 G). A, B are PV stacks that have been picked up by the chain as it moves.
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10 8
Hc=5Oe T=84K
6
∆B c /H c (%)
4 2 0 -2 -4 -6
Fig. 5. (a) Indirect pinning of JVs by PVs trapped in regions of quenched disorder running parallel to the crystallographic aaxis (indicated). (Hc ¼ 8:8 Oe, Hk ¼ 35 Oe, T ¼ 85 K, GS ¼ 2.2 G.) (b) A fork in the PV chain produced by the sudden motion of a JV stack containing partially pinned PVs. (Hc ¼ 2 Oe, Hk ¼ 38 Oe, T ¼ 81 K, GS ¼ 3.8 G).
crystallographic a-axis [13]. Remarkably, we find that JV stacks can be indirectly pinned by this disorder via the PVs trapped on them [14]. It is interesting to note that the pancake chains tend to be straight at low trapped PV densities and wavy at high PV densities due to the increasing mutual repulsion between trapped PVs. As a result indirect pinning of JVs by PVs is more pronounced at high PV densities. Fig. 5(a) shows a pronounced ‘kink’ in the uppermost JV due to indirect pinning along the crystalline a-axis (in the direction of arrow shown) along a segment in its centre comprising about a half of the visible length at 85K. When the in-plane field is varied quite rapidly we occasionally observe rare events where pinned PVs may split JV stacks into two (or more) sections, producing JV ‘forks’ as illustrated in Fig. 5(b) [14]. The observed forking of the pancake chain originated from a uniform isolated chain during its motion in response to a change in Hk and relaxed back to a single chain after a few minutes. Fig. 6 demonstrates how deformations of the rhombic JV lattice, upon varying the in-plane component of magnetic field, lead to the indirect manipulation of interacting PV stacks, and flux lenses based on this mechanism have been proposed [15]. To test this scenario we have mounted an as-grown BSCCO single crystal onto a linear GaAs/AlGaAs 2DEG Hall probe array (25 · 25 lm active area sensors with 150 lm separation) with low melting-temperature wax. After carefully
-8 -10 -150
-100
-500
0
50
100
150
H// (Oe)
Fig. 6. Measurement of flux focusing as a function of Hk with Hc ¼ 5 Oe at T ¼ 84 K.
aligning the in-plane field to the a–b crystallographic planes, Hk was swept around a ±150 Oe field cycle with a small fixed out-of-plane field in vortex lens geometry. Fig. 6 shows the results for the out-of-plane magnetic induction measured at one element of the Hall probe array near the center of the crystal under an in-plane field cycle at Hc ¼ 5 Oe and T ¼ 84 K. Although we do not yet fully understand these data there appears to be clear evidence for vortex lensing as Hk is increased in the positive sense from zero. This effect tends to saturate at high fields (jHk j > 40 Oe) when we presume that PVs start to spill out of the sample along the JVs. The small peak at jHk j < 10 Oe may reflect a transition back to a different vortex phase (possibly the tilted vortex lattice) for angles close to the c-axis. Under the conditions used in Fig. 6 the maximum lensing efficiency corresponds to 5% of the applied out-of-plane field, although much larger effects are to be expected if the edges of the samples are irradiated to prevent PVs spilling out along JVs [15].
4. Conclusions Our preliminary measurements demonstrate that the unique vortex ‘pump’ properties of pancake chains trapped on JVs could be exploited to create controlled non-uniform pancake vortex
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distributions by cycling an inhomogeneous in-plane field. In addition, we speculate that the ability to reversibly position PV stacks with high precision could find applications in single flux quantum logic. Our technique, whereby stacks of PVs are used to ‘decorate’ JVs, opens up exciting new possibilities for investigating JV dynamics in layered superconductors. Finally we note that these findings may be relevant in other fields of science where interacting objects with different symmetries coexist, e.g., for phase transitions in liquid crystals produced by molecules of different forms or for monopole condensation on strings in cosmological physics.
Acknowledgements We thank Dr. M.J.W. Dodgson for valuable discussions. The work was supported in the UK by EPSRC and MOD, in the USA by MARTECH, Florida State University and in Japan by the
Ministry of Education, Science, Sports and Culture.
References [1] J.R. Clem, Supercond. Sci. Technol. 11 (1998) 909. [2] A.A. Abrikosov, JETP 32 (1957) 1442. [3] L.N. Bulaevski, M. Levdij, V.G. Kogan, Phys. Rev. B 46 (1992) 366. [4] D.A. Huse, Phys. Rev. B 46 (1992) 8621. [5] A.E. Koshelev, Phys. Rev. Lett. 83 (1999) 187. [6] A. Oral, S.J. Bending, M. Henini, Appl. Phys. Lett. 69 (1996) 1324. [7] A.N. Grigorenko et al., Nature 414 (2001) 728. [8] C.A. Bolle et al., Phys. Rev. Lett. 66 (1991) 112; I.V. Grigorieva et al., PRB 51 (1995) 3765. [9] D.G. Grier et al., Phys. Rev. Lett. 66 (1991) 2270. [10] J.C. Martinez et al., Phys. Rev. Lett. 69 (1992) 2276. [11] N. Avraham et al., Nature 411 (2001) 451. [12] L. Burlachkov et al., Phys. Rev. B 50 (1994) 16770. [13] A. Oral et al., Phys. Rev. B 56 (1997) R14295. [14] A.N. Grigorenko et al., Phys. Rev. Lett. 89 (2002) 217003. [15] S. Savel’ev, F. Nori, Nature Mater. 1 (2002) 179.
Interacting crossing vortex lattices in the presence of quenched disorder S.J. Bending
a,*
, A.N. Grigorenko a,b, I.A. Crisan a, D. Cole a, A.E. Koshelev c, John R. Clem d, T. Tamegai e,f, S. Ooi e,f
a Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK c Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011-3160, USA e Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8627, Japan f Japan Science and Technology Corporation, CREST, Tokyo, Japan b
d
Received 29 October 2003; accepted 6 January 2004 Available online 6 May 2004
Abstract We have used high resolution scanning Hall probe microscopy (SHPM) to study vortex structures in the interacting crossing lattices regime of Bi2 Sr2 CaCu2 O8þd (BSCCO) single crystals under independently applied Hc and Hk fields. At very low c-axis fields we observe a novel 1D vortex chain state where all pancake vortex (PV) stacks become trapped on underlying stacks of Josephson vortices (JVs). In this regime distortions of the JV lattice, induced by varying Hk , enable the indirect manipulation of PVs trapped on them. Preliminary results of experiments are presented in which we have attempted to realise a vortex ‘lens’ based on this vortex ‘pump’ principle. The existence of 1D vortex chains also explains many of the features observed in the magnetisation of BSCCO under strongly tilted magnetic fields. Finally we demonstrate how the presence of quenched disorder leads to indirect JV pinning via interactions with weakly pinned PV stacks and show how fragmentation of both PV and JV stacks can occur when stacks of JVs ‘decorated’ with PVs are forced abruptly through regions of disorder. 2004 Elsevier B.V. All rights reserved. PACS: 74.60.)w; 74.25.Ha; 74.72.)h Keywords: Layered superconductors; Vortices; Crossing lattices; Vortex pumps
1. Introduction Superconductivity in high temperature superconductors (HTS) is known to reside in weakly*
Corresponding author. Tel.: +44-1225-385173; fax: +441225-386110. E-mail address: [email protected] (S.J. Bending).
coupled CuO2 layers lying parallel to the a–b plane. Under c-axis magnetic fields vortices have their circulating supercurrents flowing within the CuO2 planes and are formally viewed as stacks of 2D pancake vortices [1] which interact to form well-ordered hexagonal Abrikosov lattices [2] in disorder-free samples. When the field is applied in the a–b plane Josephson vortices [1] arise whose
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.058
S.J. Bending et al. / Physica C 412–414 (2004) 372–378
‘cores’ reside in the spaces between CuO2 planes and whose circulating currents derive partly from weak Josephson coupling between them. This anisotropic current distribution leads to strongly anisotropic vortex–vortex interactions and a rhombic lattice whose unit cell is greatly stretched out in the a–b plane. In extremely anisotropic HTS uniformly tilted vortices, composed of a staircase of PVs linked by segments of JV, are unstable with respect to the formation of independent, perpendicular hexagonal PV and rhombic JV lattices [3]. Furthermore, small PV displacements driven by the underlying JV supercurrents lead to an attractive interaction between these two ‘crossing’ lattices [4,5] with rather profound consequences since the symmetries of the projected JV and PV lattices in any given direction are, in general, incommensurate. Thus a rich variety of broken symmetry phases can arise in the crossing lattices regime which remain largely unexplored.
2. Experimental method We have used high resolution scanning Hall probe microscopy (SHPM) to directly observe discrete PVs in Bi2 Sr2 CaCu2 O8þd (BSCCO) single crystals under independently applied Hc and Hk fields. The SHPM used is a modified commercial low temperature STM in which the usual tunnelling tip has been replaced by a microfabricated GaAs/AlGaAs heterostructure chip. Electron beam lithography and wet chemical etching were used to define a Hall probe in the two-dimensional electron gas (2DEG) at the intersection of two 300 nm wide wires approximately 5 lm from the corner of a deep mesa etch. The latter had been coated with a thin Au layer to act as an integrated STM tip. The sample is first approached towards the sensor until tunnelling is established and then retracted about 100 nm allowing rapid scanning. The Hall probe makes an angle of about 1 with the sample plane so that the STM tip is always the closest point to the surface, and the Hall sensor was about 300–400 nm above the sample in the images shown here. A more detailed description of the instrument is given elsewhere [6].
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3. Experimental results Fig. 1 shows a set of SHPM images of the magnetic induction just above the a–b face of an as-grown BSCCO single crystal (Tc ¼ 90 K) as the tilt angle, h, between the applied field and the caxis is progressively increased [7]. When h ¼ 0 (Fig. 1(a)) a well-ordered hexagonal PV lattice is observed in a field of 12 Oe at 81 K. However, as the in-plane field, Hk , is increased from zero ‘‘chain states’’ were observed, where domains of approximately sixfold PV lattice are separated by chains of PVs as illustrated in Fig. 1(b),(c). As has been observed previously [8] the chains are aligned along the direction of the in-plane field (indicated by an arrow) and the vortex separation within the chains is smaller than that of the Abrikosov lattice. In addition, the pronounced induced disorder within the lattice domains (see, for example, the indicated sevenfold (A) and fivefold (B) PV rings) points to the presence of frustration due to the incommensurate vortex chains. As the c-axis component of the applied field, Hc , is decreased below the ‘‘ordering’’ field [9], quenched disorder destroys the sixfold symmetry of the Abrikosov domains as shown in Fig. 1(d). However, as the tilt angle is increased still further we observe a new phase transition to another ordered state where all the PV stacks condense onto the vortex chains and the domains of Abrikosov lattice disappear. Fig. 1(e) illustrates this 1D vortex chain configuration, and it is evident that the PVs have assumed the projected quasi-one dimensional symmetry of the underlying JV lattice. The separation between PV chains could easily be reduced by increasing Hk at fixed Hc , as illustrated in Fig. 1(g), and did not depend measurably upon Hc or the temperature in the range 77–86 K. As the field is tilted even further towards the a–b plane the PV chains become unstable and a further transition occurs to a state formed of very low flux density 1D chains of ‘sublimed’ PV stacks (Fig. 1(f)). We speculate that the latter correspond to the decomposition of PV stacks into highly mobile PV segments linked by sections of JV. The vortex behaviour illustrated in Fig. 1 finds a natural explanation in terms of the crossing lattices picture [3–5]. The JV lattice corresponds to
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Fig. 1. SHPM images of: (a) the hexagonal Abrikosov lattice of pancake vortices obtained at Hc ¼ 12 Oe (Hk ¼ 0) at 81 K. (Grayscale spans 2.5 G.) The composite vortex lattice state at (b) Hc ¼ 14 Oe and Hk ¼ 32 Oe at 81 K (GS ¼ 1.7 G) and (c) Hc ¼ 10 Oe and Hk ¼ 32 Oe at 77 K (GS ¼ 1.9 G). (d) The composite lattice state below the ‘‘ordering’’ field at Hc ¼ 2 Oe and Hk ¼ 27 Oe at 81 K (GS ¼ 2.9 G). (e) The ‘‘isolated chain’’ state at Hc ¼ 0:5 Oe and Hk ¼ 35 Oe at 81 K (GS ¼ 2.5 G). (f) The ‘sublimed’ PV state at Hc < 0:2 Oe and Hk ¼ 38 Oe at 83 K (GS ¼ 0.4 G). (g) Graph illustrating the linear dependence of chain separation, c, as a function of 1=2 (Hk Insets show the ‘‘isolated chain state’’ at fields Hc ¼ 1:2 Oe and Hk ¼ 50 Oe at 81 K and a linescan of magnetic induction along the indicated dashed line.
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the penetration of Hk in the sample and in the simplest case, consists of JV stacks ffi with a lateral qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi separation of c ¼ 3cU0 =ð2Hk Þ, where c is the anisotropy parameter and U0 is the flux quantum [5]. For small c-axis fields, Hc , a JV core contains only one row of PVs and the energy of a PV located on a JV is reduced due to the relaxation of the PV structure in response to the JV supercurrents as illustrated in Fig. 2(a). As a result, there is a finite PV–JV attraction, which increases the PV density along JV stacks. The total number of PVs per unit area is governed by the c-axis field component, Hc , and when it becomes low enough all PVs can condense onto JV stacks. The transition field to an ‘‘isolated chain’’ state can be found by comparison of the PV–JV attraction energy [5] with the PV repulsion energy in the chains and is given by Hc U0 =ðkab c lnðc=c0 ÞÞ, where kab is the
Fig. 2. (a) Sketch of the displaced PV stack interacting with a JV stack. (b) PV distribution at Hc ¼ 2 Oe for Hk ¼ 11 Oe at 83 K (GS ¼ 3.2 G). (c) The same PV distribution at Hc ¼ 2 Oe after the field Hk is increased to 16.5 Oe at 83 K with a JV stack capturing the top-right PV chain (GS ¼ 3.3 G). (d) Linescans of magnetic induction along the directions indicated in (b),(c).
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penetration depth parallel to the a–b planes, c0 ¼ Ak2ab =ðcsÞ, A is a constant close to unity and s is the CuO2 plane separation. This means that the isolated chain state is stable when Hc , Hk satisfy the 2 following inequality; pffiffiffi H2c =H2k < Htr ðT ; Hk Þ, where Htr ðT ; Hk Þ ¼ 2U0 =ð 3ckab ln ðc=c0 ÞÞ depends only weakly upon Hk and is about 0.04 Oe at 77 K in our case. The distance between PVs trapped on a JV in an isolated chain is found to be d ¼ U0 =ðcHc Þ. All these features of crossing lattices agree remarkably well with our experimental observations. The chain separation, c, at 81 K is pffiffiffiffiffiffi plotted as a function of 1= Hk in Fig. 1(g). The excellent linear dependence allows us to directly evaluate the anisotropy parameter c ¼ 640 25, which is in reasonable agreement with other estimates [10]. Figs. 2(b) and (c) illustrate the elliptical form of the trapped PV stacks at elevated temperatures, predicted by the Koshelev theory [5], after displacements along the chains driven by the JV supercurrents. Fig. 2(b) shows the situation with a small in-plane field applied, while in Fig. 2(c) a JV stack has been aligned along the chain indicated with an arrow. Linescans along the indicated chain before and after application of the in-plane field are shown in Fig. 2(d) and clearly reveal the broadening of the PV stack along the chain when the JV stack is present. One of the most interesting and important experimental properties of the isolated chain configuration is that PVs remain trapped on JVs under changes of applied field over a wide studied temperature range of 77–88 K. This may be anticipated close to Tc because the PV–JV interaction is almost temperature independent [5] while the PV pinning energy decreases rapidly close to Tc . However, the coherent motion of pancakes and Josephson vortices at temperatures as low as 77 K is remarkable. This is important as it is known that the irreversible magnetisation of BSCCO single crystals can be suppressed by the application of inplane ac fields [11]. Figs. 3(a) and (b) show the increase of the pancake density inside a PV chain trapped on a JV stack at 81 K as the c-axis field is increased from zero at fixed Hk ¼ 35 Oe [7]. We find that the PV mobility along JV stacks appears to be considerably higher than in JV-free regions, and the presence of JVs allows PVs to overcome
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Fig. 3. Penetration of pancake vortices along 1D chains at 81 K as Hc is increased at fixed Hk ¼ 35 Oe. (a) Hc ¼ 2 Oe, (GS ¼ 3.2 G). (b) Hc ¼ 4 Oe, (GS ¼ 3.7 G). (c) Local magnetisation loop measured with the Hall probe 1 lm above the sample with and without an in-plane magnetic field.
only then propagate into the interchain spaces if the chain density becomes saturated, i.e. if Hc2 =Hk < Htr ðT ; Hk Þ. Conversely, if Hc is reduced, PVs exit by travelling along JVs to the sample edges. Figs. 4(a) and (b) show the dragging of the pancake chain by a JV stack at 83 K as Hk is reduced at fixed Hc ¼ 0:5 Oe [7]. During this coherent movement the JV stack is able to depin and drag with it two isolated PV stacks (A, B) which were originally pinned on the right hand side of the image. Pancake motion induced by the motion of JV stacks is unusual because these PV displacements are reversible and uniquely determined by the applied field Hk , in stark contrast to the customary situation where c-axis magnetic fields and/or in-plane currents determine the forces acting on PVs rather than vortex positions. We have been able to show [7], for example, that it is possible to demagnetise a sample containing trapped PV stacks by turning on a large in-plane field, to drive the system into the 1D vortex chain state, and then slowly removing it. As the JVs are gradually swept out of the sample they drag the PV stacks with them, leaving a fully demagnetised sample. Quenched disorder in our sample plays an important role in controlling the static and dynamic properties of pancake chains. It is well known that the disorder in BSCCO single crystals is often strongly anisotropic, leading to the formation of pinned PV chains and stripes along the
quenched disorder, effectively depinning them. Fig. 3(c) shows local magnetisation loops (lo Mloc versus Hc ) at constant Hk ¼ 35 Oe, measured with the scanning Hall probe fixed 1 lm above the same region of the sample, which reveal a dramatic suppression of the irreversibility as compared to measurements in the absence of JVs (Hk ¼ 0). The presence of Hk also slightly decreases the PV penetration field (Hp ). At fields just above Hp PVs preferentially enter the sample along JVs, where the superposition of Meissner and JV currents at the edges leads to a slight lowering of Bean–Livingston penetration barriers [12], and then enjoy a much higher mobility along the JV chains. PVs
Fig. 4. Images showing how a pancake vortex chain can be dragged by a JV stack at 83 K as Hk is reduced at fixed Hc ¼ 0:5 Oe. (a) Hk ¼ 33 Oe (GS ¼ 2.9 G). (b) Hk ¼ 23 Oe (GS ¼ 3.4 G). A, B are PV stacks that have been picked up by the chain as it moves.
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10 8
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6
∆B c /H c (%)
4 2 0 -2 -4 -6
Fig. 5. (a) Indirect pinning of JVs by PVs trapped in regions of quenched disorder running parallel to the crystallographic aaxis (indicated). (Hc ¼ 8:8 Oe, Hk ¼ 35 Oe, T ¼ 85 K, GS ¼ 2.2 G.) (b) A fork in the PV chain produced by the sudden motion of a JV stack containing partially pinned PVs. (Hc ¼ 2 Oe, Hk ¼ 38 Oe, T ¼ 81 K, GS ¼ 3.8 G).
crystallographic a-axis [13]. Remarkably, we find that JV stacks can be indirectly pinned by this disorder via the PVs trapped on them [14]. It is interesting to note that the pancake chains tend to be straight at low trapped PV densities and wavy at high PV densities due to the increasing mutual repulsion between trapped PVs. As a result indirect pinning of JVs by PVs is more pronounced at high PV densities. Fig. 5(a) shows a pronounced ‘kink’ in the uppermost JV due to indirect pinning along the crystalline a-axis (in the direction of arrow shown) along a segment in its centre comprising about a half of the visible length at 85K. When the in-plane field is varied quite rapidly we occasionally observe rare events where pinned PVs may split JV stacks into two (or more) sections, producing JV ‘forks’ as illustrated in Fig. 5(b) [14]. The observed forking of the pancake chain originated from a uniform isolated chain during its motion in response to a change in Hk and relaxed back to a single chain after a few minutes. Fig. 6 demonstrates how deformations of the rhombic JV lattice, upon varying the in-plane component of magnetic field, lead to the indirect manipulation of interacting PV stacks, and flux lenses based on this mechanism have been proposed [15]. To test this scenario we have mounted an as-grown BSCCO single crystal onto a linear GaAs/AlGaAs 2DEG Hall probe array (25 · 25 lm active area sensors with 150 lm separation) with low melting-temperature wax. After carefully
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0
50
100
150
H// (Oe)
Fig. 6. Measurement of flux focusing as a function of Hk with Hc ¼ 5 Oe at T ¼ 84 K.
aligning the in-plane field to the a–b crystallographic planes, Hk was swept around a ±150 Oe field cycle with a small fixed out-of-plane field in vortex lens geometry. Fig. 6 shows the results for the out-of-plane magnetic induction measured at one element of the Hall probe array near the center of the crystal under an in-plane field cycle at Hc ¼ 5 Oe and T ¼ 84 K. Although we do not yet fully understand these data there appears to be clear evidence for vortex lensing as Hk is increased in the positive sense from zero. This effect tends to saturate at high fields (jHk j > 40 Oe) when we presume that PVs start to spill out of the sample along the JVs. The small peak at jHk j < 10 Oe may reflect a transition back to a different vortex phase (possibly the tilted vortex lattice) for angles close to the c-axis. Under the conditions used in Fig. 6 the maximum lensing efficiency corresponds to 5% of the applied out-of-plane field, although much larger effects are to be expected if the edges of the samples are irradiated to prevent PVs spilling out along JVs [15].
4. Conclusions Our preliminary measurements demonstrate that the unique vortex ‘pump’ properties of pancake chains trapped on JVs could be exploited to create controlled non-uniform pancake vortex
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distributions by cycling an inhomogeneous in-plane field. In addition, we speculate that the ability to reversibly position PV stacks with high precision could find applications in single flux quantum logic. Our technique, whereby stacks of PVs are used to ‘decorate’ JVs, opens up exciting new possibilities for investigating JV dynamics in layered superconductors. Finally we note that these findings may be relevant in other fields of science where interacting objects with different symmetries coexist, e.g., for phase transitions in liquid crystals produced by molecules of different forms or for monopole condensation on strings in cosmological physics.
Acknowledgements We thank Dr. M.J.W. Dodgson for valuable discussions. The work was supported in the UK by EPSRC and MOD, in the USA by MARTECH, Florida State University and in Japan by the
Ministry of Education, Science, Sports and Culture.
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