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Evaluation of crossing energy between pancake and Josephson vortices in Bi2Sr2CaCu2O8+y
Physica C 412–414 (2004) 391–396 www.elsevier.com/locate/physc
Evaluation of crossing energy between pancake and Josephson vortices in Bi2Sr2CaCu2O8þy T. Tamegai *, M. Matsui, M. Tokunaga Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Received 29 October 2003; accepted 19 January 2004 Available online 7 May 2004
Abstract In highly anisotropic superconductors, magnetic fields tilted from c-axis give rise to a novel ground state of vortices, where pancake vortices (PVs) and Josephson vortices (JVs) coexist and attract to each other. We have evaluated the crossing energy between PVs and JVs experimentally by comparing it with the line energy of PV stacks in Bi2 Sr2 CaCu2 O8þy crystals with artificially introduced parallel grooves on the surface. The obtained value of the crossing energy is slightly larger than the theoretical estimate. 2004 Elsevier B.V. All rights reserved. PACS: 74.72.Hs; 74.60.Ge; 74.80.)g Keywords: Crossing energy; Pancake vortices; Josephson vortices; Bi2 Sr2 CaCu2 O8þy
1. Introduction In highly anisotropic superconductors like Bi2 Sr2 CaCu2 O8þy (BSCCO), a magnetic field tilted from c-axis does not tilt the flux lines. Instead, it produces a novel state of vortices called crossinglattices state, where pancake vortices (PVs) and Josephson vortices (JVs) coexist [1]. In the crossing-lattices state, PVs preferably locate on JVs due to their attractive interactions, which make rich variety of ground state arrangements of vortices and the nature of this state is extensively studied theoretically [2–5]. As a result of attractive inter-
*
Corresponding author. Tel.: +81-3-5841-6846; fax: +81-35841-8886. E-mail address: [email protected] (T. Tamegai).
actions, vortex states with vortex chains embedded in the vortex lattice, chain + lattice state, and those totally occupied by vortex chains, vortex chain state, are realized and observed by Bitter decorations [6–8] and scanning Hall probe microscopy [9]. The vortex chains are also visualized by magneto-optical method [10–12]. The attractive interactions between PVs and JVs opens a possibility of controlling one kind of vortices by driving another [13]. Although the presence of attractive interactions between PVs and JVs are now well established, its quantitative evaluation has not yet been undertaken. Here, we report the evaluation of the crossing energy by comparing it with the line energy of pancake vortex stacks in BSCCO crystals with artificially introduced parallel grooves on the surface.
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.059
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T. Tamegai et al. / Physica C 412–414 (2004) 391–396
2. Experimental BSCCO single crystals used in the present study are grown by the floating-zone method using an image furnace [14]. Crystals are cleaved to a thickness of about 30 lm, and cut into approximate dimensions of 1.0 · 1.0 mm2 . In order to introduce a spatial variation of the line energy, parallel grooves with various depths are fabricated on the cleaved surface of the crystals by Ar-ion milling. A typical example of the grooves with their widths 10 lm and spacings 20 lm is shown in Fig. 1(a). The groove pattern is defined by the conventional photolithography using thick photoresist. Overheating during the process is avoided by milling intermittently. The essential part of the experiment relies on how we detect the locations of PVs. For this purpose, we utilize differential magneto-optical (MO) technique [11,15,16] using an in-plane magnetization garnet film as an indicator. We put the indicator film on the opposite surface of the grooves to avoid spurious modulations of the local induction due to variation in the distance between the film and the sample surface. For this reason, the real image (Fig. 1(a)) and the MO image (Fig. 1(b)) are flipped to each other. Differential MO images are constructed by subtracting images taken at Hz ¼ Hz dHz =2 from Hzþ ¼ Hz þ dHz =2 with dHz 3 Oe using cooledCCD camera. Typically we average 500 differential images to attain the field resolution of a few tens of mG. Details of the experimental technique are described in Refs. [15,16]. Instead of a conventional He flow-type cryostat, we use a cryocooler-based MO observation system with reduced vibration level [17]. Magnetic field is applied using a small vector magnet system with rotatable horizontal field.
3. Results and discussion Fig. 1(b) shows differential MO images of BSCCO single crystal #1 (Fig. 1(a)) with 1.7 lmdeep grooves at 70 K. The in-plane field Hx ¼ 30 Oe is applied horizontally along the arrow. The out-of-plane field Hz ¼ 10 Oe, is modulated by
Fig. 1. (a) BSCCO single crystal #1 with artificially introduced deep grooves on the surface. Bright regions are etched by 1.7 lm by Ar-ion milling. (b) Differential MO image at T ¼ 70 K and Hx ¼ 30 Oe. Hz and dHz for this image are 10 and 3 Oe, respectively. The direction of the in-plane field is shown by the white arrow. The border between the regions with and without grooves is shown by dotted line. Vortices are seen along the grooves in the upper part of the sample. No vortices along the in-plane field are observed.
dHz ¼ 3 Oe for this differential image. Onedimensional stripe structures are observed in the region with grooves which is the right hand side of the dotted line. Comparison with the real image (Fig. 1(a)) shows that the stripe structures closely follow the direction of the grooves and the bright region corresponds to the bottom of the grooves. These facts indicate that the stripe structures are originated from vortices trapped in the grooves. No stripes along the in-plane field direction are observed. All these facts suggest that the line
T. Tamegai et al. / Physica C 412–414 (2004) 391–396
energy gain is larger than the crossing energy in this sample with deep grooves. At higher in-plane fields we lose images of stripes all over the sample, while at lower in-plane fields, the stripe direction remains the same.
393
Figure 2(a) shows the groove pattern of another BSCCO single crystal #2 with shallower grooves (0.38 lm) on the surface. Differential MO images of this crystal at 68 K and at various in-plane field values are shown in Fig. 2(b)–(f). Here, the
Fig. 2. (a) BSCCO single crystal #2 with artificially introduced shallow grooves on the surface. Bright regions are etched by 0.38 lm by Ar-ion milling. (b) Differential MO images at T ¼ 68 K and Hx ¼ ðbÞ 16.1, (c) 32.2, (d) 40, (e) 45.2, and (f) 60 Oe. Hz and dHz for these images are 8 and 4 Oe, respectively. The direction of the in-plane field is parallel to the grooves. Above Hx ¼ 40 Oe, vortex rows in addition to those on the grooves start to emerge between grooves.
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in-plane field is applied parallel to the grooves, which can be identified by trapped vortices on the left part of images. At Hx ¼ 16:1 Oe (Fig. 2(b)), PV chains attracted by JVs in the pristine (right) part is hardly seen except for a few marked by arrows. At Hx ¼ 32:2 Oe (Fig. 2(c)), many PV chains are observed in the pristine part and the spacing between the vortex chain is almost the same as the groove spacing, from which we estimate the anisotropy parameter for this crystal c ¼ 600 [11]. Up to Hx ¼ 40 Oe (Fig. 2(d)), stripe structures observed in the region with grooves are almost identical. However, when the in-plane field is increased to 45.2 Oe, additional structures start to emerge in the region with grooves as marked by
the arrow in Fig. 2(e). At higher Hx , there appear many irregular stripes in addition to those on the grooves (Fig. 2(f)). Some of these additional stripes can be traced to the region without grooves and identified as PV chains decorating JVs in the region with grooves. From this series of observations we may define a field where the crossing energy between PV stacks and JVs dominates over the line energy gain on the grooves as H 40 Oe in this crystal. If the crystal is uniform, the characteristic field H should be independent of the direction of Hx . Other differential MO images for the same crystal at Hx ¼ 32:2 Oe, which is lower than the above H , applied along different directions are shown in
Fig. 3. Differential MO images for BSCCO #2 at T ¼ 68 K and Hx ¼ 32:2 Oe for (a) 45 and (b) 90 between Hx and the grooves. Although Hx is lower than H defined for another in-plane field orientation, vortices are clearly seen both on the grooves and JVs. Differential MO images for BSCCO #2 at T ¼ 68 K and 15 between Hx and the grooves for Hx ¼ ðcÞ 21.1 and (d) 10.7 Oe. While vortices are observed along two different directions at Hx ¼ 21:1 Oe, vortices only on the grooves are seen at Hx ¼ 10:7 Oe.
T. Tamegai et al. / Physica C 412–414 (2004) 391–396
Fig. 3; (a) 45 and (b) 90 from the grooves. In both images, we can identify stripe structures in two directions, one along the grooves and the other along the in-plane field direction. Actually, the image at even lower Hx in Fig. 3(c) shows that the coexistence persists down to Hx ¼ 21:1 Oe. The stripe structure along Hx disappear only when PV chains becomes difficult to observe even in pristine part at 10.7 Oe (Fig. 3(d)). From the observations in Fig. 3 with Hx not parallel to the grooves, another estimate of H is about 20 Oe. Summarizing all these results for BSCCO #2, we estimate H ¼ 30 10 Oe for this crystal. The line energy gain, el , and the crossing energy, eX , per unit length are el ¼ ðU0 =4pkÞ2 lnðk=nÞ and eX ¼ pffiffiffi given by 1=2 ð 3cBx =2U0 Þ ð2:1U20 =ð4p2 c2 s lnð3:5cs=kÞÞÞ [1], respectively. Here, c is the anisotropy parameter, Bx is the in-plane magnetic induction, U0 is the flux quantum, k is penetration depth, n is coherence length and s is the layer spacing. Assuming kðT Þ ¼ 2 1=2 and kð0Þ=ð1 ðT =Tc Þ Þ with kð0Þ ¼ 2000 A 2 1=2 nðT Þ ¼ nð0Þ=ð1 ðT =Tc Þ Þ with nð0Þ ¼ 20 A, and using c ¼ 600, Tc ¼ 90 K, T ¼ 68 K, and s ¼ the ratio of the two energies for a sample with 15 A, a thickness D and a depth of p grooves d is calculated ffiffiffi as (eX DÞ= ðel dÞ ¼ 6:5 104 Bx ½G ðD= dÞ. Inserting D ¼ 16 lm and d ¼ 0:38 lm for BSCCO #2, and identifying Bx ¼ H ¼ 30 G, the ratio is calculated as (eX DÞ=ðel dÞ ¼ 0:15. The fact that this ratio is less than unity means that the experimentally obtained crossing energy is larger than the theoretical estimate by the inverse of this ratio. Finally, let us discuss the reason why we observe coexistence of stripes in two directions in a crystal. Obviously it is not caused by the inhomogeneities of the sample, since stripes in two directions coexist in the same region as clearly seen in Fig. 3(b). When the repulsive interactions between PV stacks are ignored, PVs will be trapped at locations with the lowest potential. At low Hx , the lowest potential is realized on the grooves and all PVs should be trapped there. However, as the number of trapped PVs on the grooves increases at higher Hz , the repulsive interaction between trapped PVs raise the potential on the grooves. At certain Hz , the potential on the grooves becomes larger than that on JVs and hence PVs start to move onto JVs. More quantitative evaluation of the crossing energy re-
395
quires to take into account the repulsive interactions between PVs which depend on the local induction on each part of the sample.
4. Summary We have evaluated the crossing energy of pancake and Josephson vortices in the crossing-lattices state of Bi2 Sr2 CaCu2 O8þy by comparing it with the line energy of pancake vortex stacks. The obtained value of the crossing energy is slightly larger than the theoretical estimate for a single pancake vortex stack crossing Josephson vortices. For more quantitative comparison, repulsive interactions between pancake vortices have to be taken into account.
Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology. References [1] A.E. Koshelev, Phys. Rev. Lett. 83 (1999) 187. [2] S.E. Savel’ev, J. Mirkovic, K. Kadowaki, Phys. Rev. B 64 (2001) 094521. [3] A. Buzdin, I. Baladie, Phys. Rev. Lett. 88 (2002) 147002. [4] M.J.W. Dodgson, Phys. Rev. B 66 (2002) 014509. [5] A.E. Koshelev, Phys. Rev. B 68 (2003) 094520. [6] C.A. Bolle, P.L. Gammel, D.G. Grier, C.A. Murray, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Phys. Rev. Lett. 66 (1991) 112. [7] I.V. Grigorieva, J.W. Steeds, G. Balakrishnan, D.M. Paul, Phys. Rev. B 51 (1995) 3765. [8] M. Tokunaga, T. Tamegai, Y. Fasano, F. de la Cruz, Phys. Rev. B 67 (2003) 134501. [9] A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, M. Henini, Nature 414 (2001) 728. [10] V.K. Vlasko-Vlasov, A. Koshelev, U. Welp, G.W. Crabtree, K. Kadowaki, Phys. Rev. B 66 (2002) 014523. [11] M. Tokunaga, M. Kobayashi, Y. Tokunaga, T. Tamegai, Phys. Rev. B 66 (2002) 060507. [12] T. Tamegai, M. Tokunaga, M. Matsui, M. Kobayashi, Y. Tokunaga, Physica C 392–396 (2003) 311. [13] S. Savel’ev, F. Nori, Nature Mater. 1 (2002) 179.
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[14] S. Ooi, T. Shibauchi, T. Tamegai, Physica C 302 (1998) 339. [15] A. Soibel, E. Zeldov, M. Rappaport, Y. Myasoedov, T. Tamegai, S. Ooi, M. Konczykowski, V.B. Geshkenbein, Nature 406 (2000) 282. [16] M. Yasugaki, M. Tokunaga, N. Kameda, T. Tamegai, Phys. Rev. B 65 (2002) 212502.
[17] T. Tamegai, M. Matsui, M. Yasugaki, Y. Tokunaga, M. Tokunaga, in: T.H. Johansen, D.V. Shantsev (Eds.), NATO SCIENCE SERIES II: Mathematics, Physics and Chemistry, vol. 142, Magneto-Optical Imaging, Kluwer Academic Publishers, Dordrecht, 2004, pp. 95– 102.
Evaluation of crossing energy between pancake and Josephson vortices in Bi2Sr2CaCu2O8þy T. Tamegai *, M. Matsui, M. Tokunaga Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Received 29 October 2003; accepted 19 January 2004 Available online 7 May 2004
Abstract In highly anisotropic superconductors, magnetic fields tilted from c-axis give rise to a novel ground state of vortices, where pancake vortices (PVs) and Josephson vortices (JVs) coexist and attract to each other. We have evaluated the crossing energy between PVs and JVs experimentally by comparing it with the line energy of PV stacks in Bi2 Sr2 CaCu2 O8þy crystals with artificially introduced parallel grooves on the surface. The obtained value of the crossing energy is slightly larger than the theoretical estimate. 2004 Elsevier B.V. All rights reserved. PACS: 74.72.Hs; 74.60.Ge; 74.80.)g Keywords: Crossing energy; Pancake vortices; Josephson vortices; Bi2 Sr2 CaCu2 O8þy
1. Introduction In highly anisotropic superconductors like Bi2 Sr2 CaCu2 O8þy (BSCCO), a magnetic field tilted from c-axis does not tilt the flux lines. Instead, it produces a novel state of vortices called crossinglattices state, where pancake vortices (PVs) and Josephson vortices (JVs) coexist [1]. In the crossing-lattices state, PVs preferably locate on JVs due to their attractive interactions, which make rich variety of ground state arrangements of vortices and the nature of this state is extensively studied theoretically [2–5]. As a result of attractive inter-
*
Corresponding author. Tel.: +81-3-5841-6846; fax: +81-35841-8886. E-mail address: [email protected] (T. Tamegai).
actions, vortex states with vortex chains embedded in the vortex lattice, chain + lattice state, and those totally occupied by vortex chains, vortex chain state, are realized and observed by Bitter decorations [6–8] and scanning Hall probe microscopy [9]. The vortex chains are also visualized by magneto-optical method [10–12]. The attractive interactions between PVs and JVs opens a possibility of controlling one kind of vortices by driving another [13]. Although the presence of attractive interactions between PVs and JVs are now well established, its quantitative evaluation has not yet been undertaken. Here, we report the evaluation of the crossing energy by comparing it with the line energy of pancake vortex stacks in BSCCO crystals with artificially introduced parallel grooves on the surface.
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.059
392
T. Tamegai et al. / Physica C 412–414 (2004) 391–396
2. Experimental BSCCO single crystals used in the present study are grown by the floating-zone method using an image furnace [14]. Crystals are cleaved to a thickness of about 30 lm, and cut into approximate dimensions of 1.0 · 1.0 mm2 . In order to introduce a spatial variation of the line energy, parallel grooves with various depths are fabricated on the cleaved surface of the crystals by Ar-ion milling. A typical example of the grooves with their widths 10 lm and spacings 20 lm is shown in Fig. 1(a). The groove pattern is defined by the conventional photolithography using thick photoresist. Overheating during the process is avoided by milling intermittently. The essential part of the experiment relies on how we detect the locations of PVs. For this purpose, we utilize differential magneto-optical (MO) technique [11,15,16] using an in-plane magnetization garnet film as an indicator. We put the indicator film on the opposite surface of the grooves to avoid spurious modulations of the local induction due to variation in the distance between the film and the sample surface. For this reason, the real image (Fig. 1(a)) and the MO image (Fig. 1(b)) are flipped to each other. Differential MO images are constructed by subtracting images taken at Hz ¼ Hz dHz =2 from Hzþ ¼ Hz þ dHz =2 with dHz 3 Oe using cooledCCD camera. Typically we average 500 differential images to attain the field resolution of a few tens of mG. Details of the experimental technique are described in Refs. [15,16]. Instead of a conventional He flow-type cryostat, we use a cryocooler-based MO observation system with reduced vibration level [17]. Magnetic field is applied using a small vector magnet system with rotatable horizontal field.
3. Results and discussion Fig. 1(b) shows differential MO images of BSCCO single crystal #1 (Fig. 1(a)) with 1.7 lmdeep grooves at 70 K. The in-plane field Hx ¼ 30 Oe is applied horizontally along the arrow. The out-of-plane field Hz ¼ 10 Oe, is modulated by
Fig. 1. (a) BSCCO single crystal #1 with artificially introduced deep grooves on the surface. Bright regions are etched by 1.7 lm by Ar-ion milling. (b) Differential MO image at T ¼ 70 K and Hx ¼ 30 Oe. Hz and dHz for this image are 10 and 3 Oe, respectively. The direction of the in-plane field is shown by the white arrow. The border between the regions with and without grooves is shown by dotted line. Vortices are seen along the grooves in the upper part of the sample. No vortices along the in-plane field are observed.
dHz ¼ 3 Oe for this differential image. Onedimensional stripe structures are observed in the region with grooves which is the right hand side of the dotted line. Comparison with the real image (Fig. 1(a)) shows that the stripe structures closely follow the direction of the grooves and the bright region corresponds to the bottom of the grooves. These facts indicate that the stripe structures are originated from vortices trapped in the grooves. No stripes along the in-plane field direction are observed. All these facts suggest that the line
T. Tamegai et al. / Physica C 412–414 (2004) 391–396
energy gain is larger than the crossing energy in this sample with deep grooves. At higher in-plane fields we lose images of stripes all over the sample, while at lower in-plane fields, the stripe direction remains the same.
393
Figure 2(a) shows the groove pattern of another BSCCO single crystal #2 with shallower grooves (0.38 lm) on the surface. Differential MO images of this crystal at 68 K and at various in-plane field values are shown in Fig. 2(b)–(f). Here, the
Fig. 2. (a) BSCCO single crystal #2 with artificially introduced shallow grooves on the surface. Bright regions are etched by 0.38 lm by Ar-ion milling. (b) Differential MO images at T ¼ 68 K and Hx ¼ ðbÞ 16.1, (c) 32.2, (d) 40, (e) 45.2, and (f) 60 Oe. Hz and dHz for these images are 8 and 4 Oe, respectively. The direction of the in-plane field is parallel to the grooves. Above Hx ¼ 40 Oe, vortex rows in addition to those on the grooves start to emerge between grooves.
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in-plane field is applied parallel to the grooves, which can be identified by trapped vortices on the left part of images. At Hx ¼ 16:1 Oe (Fig. 2(b)), PV chains attracted by JVs in the pristine (right) part is hardly seen except for a few marked by arrows. At Hx ¼ 32:2 Oe (Fig. 2(c)), many PV chains are observed in the pristine part and the spacing between the vortex chain is almost the same as the groove spacing, from which we estimate the anisotropy parameter for this crystal c ¼ 600 [11]. Up to Hx ¼ 40 Oe (Fig. 2(d)), stripe structures observed in the region with grooves are almost identical. However, when the in-plane field is increased to 45.2 Oe, additional structures start to emerge in the region with grooves as marked by
the arrow in Fig. 2(e). At higher Hx , there appear many irregular stripes in addition to those on the grooves (Fig. 2(f)). Some of these additional stripes can be traced to the region without grooves and identified as PV chains decorating JVs in the region with grooves. From this series of observations we may define a field where the crossing energy between PV stacks and JVs dominates over the line energy gain on the grooves as H 40 Oe in this crystal. If the crystal is uniform, the characteristic field H should be independent of the direction of Hx . Other differential MO images for the same crystal at Hx ¼ 32:2 Oe, which is lower than the above H , applied along different directions are shown in
Fig. 3. Differential MO images for BSCCO #2 at T ¼ 68 K and Hx ¼ 32:2 Oe for (a) 45 and (b) 90 between Hx and the grooves. Although Hx is lower than H defined for another in-plane field orientation, vortices are clearly seen both on the grooves and JVs. Differential MO images for BSCCO #2 at T ¼ 68 K and 15 between Hx and the grooves for Hx ¼ ðcÞ 21.1 and (d) 10.7 Oe. While vortices are observed along two different directions at Hx ¼ 21:1 Oe, vortices only on the grooves are seen at Hx ¼ 10:7 Oe.
T. Tamegai et al. / Physica C 412–414 (2004) 391–396
Fig. 3; (a) 45 and (b) 90 from the grooves. In both images, we can identify stripe structures in two directions, one along the grooves and the other along the in-plane field direction. Actually, the image at even lower Hx in Fig. 3(c) shows that the coexistence persists down to Hx ¼ 21:1 Oe. The stripe structure along Hx disappear only when PV chains becomes difficult to observe even in pristine part at 10.7 Oe (Fig. 3(d)). From the observations in Fig. 3 with Hx not parallel to the grooves, another estimate of H is about 20 Oe. Summarizing all these results for BSCCO #2, we estimate H ¼ 30 10 Oe for this crystal. The line energy gain, el , and the crossing energy, eX , per unit length are el ¼ ðU0 =4pkÞ2 lnðk=nÞ and eX ¼ pffiffiffi given by 1=2 ð 3cBx =2U0 Þ ð2:1U20 =ð4p2 c2 s lnð3:5cs=kÞÞÞ [1], respectively. Here, c is the anisotropy parameter, Bx is the in-plane magnetic induction, U0 is the flux quantum, k is penetration depth, n is coherence length and s is the layer spacing. Assuming kðT Þ ¼ 2 1=2 and kð0Þ=ð1 ðT =Tc Þ Þ with kð0Þ ¼ 2000 A 2 1=2 nðT Þ ¼ nð0Þ=ð1 ðT =Tc Þ Þ with nð0Þ ¼ 20 A, and using c ¼ 600, Tc ¼ 90 K, T ¼ 68 K, and s ¼ the ratio of the two energies for a sample with 15 A, a thickness D and a depth of p grooves d is calculated ffiffiffi as (eX DÞ= ðel dÞ ¼ 6:5 104 Bx ½G ðD= dÞ. Inserting D ¼ 16 lm and d ¼ 0:38 lm for BSCCO #2, and identifying Bx ¼ H ¼ 30 G, the ratio is calculated as (eX DÞ=ðel dÞ ¼ 0:15. The fact that this ratio is less than unity means that the experimentally obtained crossing energy is larger than the theoretical estimate by the inverse of this ratio. Finally, let us discuss the reason why we observe coexistence of stripes in two directions in a crystal. Obviously it is not caused by the inhomogeneities of the sample, since stripes in two directions coexist in the same region as clearly seen in Fig. 3(b). When the repulsive interactions between PV stacks are ignored, PVs will be trapped at locations with the lowest potential. At low Hx , the lowest potential is realized on the grooves and all PVs should be trapped there. However, as the number of trapped PVs on the grooves increases at higher Hz , the repulsive interaction between trapped PVs raise the potential on the grooves. At certain Hz , the potential on the grooves becomes larger than that on JVs and hence PVs start to move onto JVs. More quantitative evaluation of the crossing energy re-
395
quires to take into account the repulsive interactions between PVs which depend on the local induction on each part of the sample.
4. Summary We have evaluated the crossing energy of pancake and Josephson vortices in the crossing-lattices state of Bi2 Sr2 CaCu2 O8þy by comparing it with the line energy of pancake vortex stacks. The obtained value of the crossing energy is slightly larger than the theoretical estimate for a single pancake vortex stack crossing Josephson vortices. For more quantitative comparison, repulsive interactions between pancake vortices have to be taken into account.
Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology. References [1] A.E. Koshelev, Phys. Rev. Lett. 83 (1999) 187. [2] S.E. Savel’ev, J. Mirkovic, K. Kadowaki, Phys. Rev. B 64 (2001) 094521. [3] A. Buzdin, I. Baladie, Phys. Rev. Lett. 88 (2002) 147002. [4] M.J.W. Dodgson, Phys. Rev. B 66 (2002) 014509. [5] A.E. Koshelev, Phys. Rev. B 68 (2003) 094520. [6] C.A. Bolle, P.L. Gammel, D.G. Grier, C.A. Murray, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Phys. Rev. Lett. 66 (1991) 112. [7] I.V. Grigorieva, J.W. Steeds, G. Balakrishnan, D.M. Paul, Phys. Rev. B 51 (1995) 3765. [8] M. Tokunaga, T. Tamegai, Y. Fasano, F. de la Cruz, Phys. Rev. B 67 (2003) 134501. [9] A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, M. Henini, Nature 414 (2001) 728. [10] V.K. Vlasko-Vlasov, A. Koshelev, U. Welp, G.W. Crabtree, K. Kadowaki, Phys. Rev. B 66 (2002) 014523. [11] M. Tokunaga, M. Kobayashi, Y. Tokunaga, T. Tamegai, Phys. Rev. B 66 (2002) 060507. [12] T. Tamegai, M. Tokunaga, M. Matsui, M. Kobayashi, Y. Tokunaga, Physica C 392–396 (2003) 311. [13] S. Savel’ev, F. Nori, Nature Mater. 1 (2002) 179.
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[14] S. Ooi, T. Shibauchi, T. Tamegai, Physica C 302 (1998) 339. [15] A. Soibel, E. Zeldov, M. Rappaport, Y. Myasoedov, T. Tamegai, S. Ooi, M. Konczykowski, V.B. Geshkenbein, Nature 406 (2000) 282. [16] M. Yasugaki, M. Tokunaga, N. Kameda, T. Tamegai, Phys. Rev. B 65 (2002) 212502.
[17] T. Tamegai, M. Matsui, M. Yasugaki, Y. Tokunaga, M. Tokunaga, in: T.H. Johansen, D.V. Shantsev (Eds.), NATO SCIENCE SERIES II: Mathematics, Physics and Chemistry, vol. 142, Magneto-Optical Imaging, Kluwer Academic Publishers, Dordrecht, 2004, pp. 95– 102.