- Email: [email protected]
Hall magnetometry of superconducting microstructures
Physica C 404 (2004) 44–49 www.elsevier.com/locate/physc
Hall magnetometry of superconducting microstructures J. Bekaert a
a,b
, M. Morelle
a,*
, W.V. Pogosov a, G. Borghs b, V.V. Moshchalkov
a
Laboratorium voor Vaste-Stoffysica en Magnetisme, Nanoscale superconductivity and Magnetism group, K.U.Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium b Interuniversity Micro-electronics Center, Kapeldreef 75, B-3001 Leuven, Belgium
Abstract We report experimental results of a superconducting Pb disk and triangle of the intermediate size, i.e. between the mesoscopic and the macroscopic regime. The magnetization is measured by Hall magnetometry. A comparison of the stability of different vortex configurations in the two different geometries is done. The experimental results for the disk are compared with theoretical predictions. There is a reasonable agreement between the theory and the experiment. 2004 Elsevier B.V. All rights reserved.
1. Introduction The superconducting properties of micron-sized disks have been intensively studied in the recent years. Most of these studies were concentrated on the properties of mesoscopic disks, which had sizes comparable with the coherence length nðT Þ. The phase boundary of these structures was found from transport measurements [1] and the superconducting properties deep in the superconducting state were studied using magnetization measurements [1,2]. While in disks with a radius close to the coherence length a giant vortex is formed, vortex molecules will be formed in superconductors of larger size. The confinement of the vortices in samples of intermediate size between the mesoscopic and macroscopic regime is still important. This means that the vortex configurations can be
*
Corresponding author. Fax: +3216327983. E-mail address: [email protected] (M. Morelle).
influenced by the circular geometry of the sample and differ from the Abrikosov vortex lattice structure found in bulk superconductors. While the disk geometry has been extensively studied during the last decade [1–6], only recently some other geometries have been investigated like triangles [7–10] squares [10–14], or more exotic structures [15,16]. The progress made in the growth of semiconductor heterostructures by molecular beam epitaxy (MBE), enabled the fabrication of Hall sensors and Hall magnetometers, which have been efficiently used for characterization of both ferromagnetic [17,18] and superconducting [2] materials. These sensors have also been applied in scanning Hall probe microscopy [18] where submicron probes with high sensitivity are used to map local field distributions in a non-invasive and quantitative way. With a magnetic or superconducting element deposited on the sensitive area of the sensor, the Hall voltage proportional to the perpendicular component of the local magnetic induction is
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.11.056
J. Bekaert et al. / Physica C 404 (2004) 44–49
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measured. This local induction can be in general considered as the averaged induction over the total sensing area of the probe [19]. In this paper, Hall magnetometry measurements of a micron-sized superconducting Pb disk and triangle are presented. The regime of intermediate size, where the coherence length nðT Þ is an order of magnitude smaller than the size of the element, has been investigated. The experimental magnetization curve of the disk is compared with a theoretical one.
2. Sample characteristics The micron Hall sensors were fabricated from GaAs/AlGaAs heterostructures with a twodimensional electron gas at a depth l ¼ 70 nm below the surface. The active area is patterned by optical lithography. After etching, the Hall cross junction is 1.5 lm wide, resulting in an effective electrical width of 1.4 lm due to depletion of carriers at the edges. At the low temperatures of our measurements, the carrier mobility is 650,000 cm2 /Vs and the 2D carrier concentration is 2.5 · 1011 cm2 , yielding a sensitivity of 0.25 lV per lAG. Our measurement noise corresponds to 0.05 G. The definition of the structures on the active area of the Hall probe was done by e-beam lithography, which requires a high accuracy of the alignment, followed by a lift-off process. The Pb film, deposited by molecular beam epitaxy, was covered with a Ge layer in order to prevent oxidation. The thickness s ¼ 40 nm was measured by X-ray spectroscopy. An area S of 2.5 lm2 for the disk and of 2.1 lm2 for the triangle was determined from SEM measurements (see Fig. 1). From a co-evaporated reference sample, we found a coherence length nð0Þ ¼ 40 nm. An ac current of 25 lA was sent through the Hall probe at a frequency of 27.7 Hz. No heating effects arising from this large current were observed during the measurements. The magnetization l0 M ¼ hBi l0 H was found from the difference between the Hall signal below and above the critical temperature Tc of the superconductor.
Fig. 1. SEM micrograph of the Hall magnetometers. A superconducting Pb disk and triangle are deposited on the sensitive area of the Hall probes.
3. Experimental results The experimental magnetization curves for the superconducting disk and triangle are presented in Figs. 2 and 3. Jumps are observed in the increasing and decreasing branches of the magnetization, corresponding to the stepwise penetration and the expulsion of vortices. The critical field is found to be higher for the triangle than for the disk, what is 2 expected since jWj is higher in the wedge [20]. For both structures, it can be seen in Figs. 2 and 3 that the magnetization curves for the decreasing magnetic field are not returning to zero at zero magnetic field. This means that at least one vortex is trapped and exits the superconductor only at a finite negative magnetic field. The position of the peaks as well for the increasing as for the decreasing branch is always very well reproduced when repeating the magnetization curve of the disk and the triangle. A paramagnetic magnetization has been observed for the two geometries with decreasing magnetic field. The origin of this observation is the formation of metastable states due to the Bean–Livingston surface barrier that keeps the vortices inside the superconductor. As is clearly seen from the experiment, the magnetization curves corresponding to the fixed number of vortices have the same slope. The physical reason of this feature is the fact that the
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J. Bekaert et al. / Physica C 404 (2004) 44–49 0.10
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Fig. 2. Experimental magnetization curves of the disk for the increasing (open squares) and the decreasing (open circles) magnetic field. The measurements are performed at T ¼ 0:6Tc .
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Fig. 3. Experimental magnetization curves of the triangle for the increasing (open squares) and the decreasing (open circles) magnetic field. The measurements are performed at T ¼ 0:6Tc .
sample sizes are much larger than the London penetration depth, and the magnetic flux of each
vortex is almost equal to the flux quantum U0 . Such a behavior of magnetization is typical for samples of the intermediate size (i.e. between the mesoscopic and the macroscopic scales). From the dependencies MðH Þ presented in Fig. 2, a certain quasi-periodicity of the background of the magnetization curves can be noticed in the increasing and decreasing branch. We attribute these features to the transitions between the phases with different shell structures for the vortex configuration. In order to study the vortex penetration and expulsion in the disk and the triangle, the magnetic field range DH between two jumps in the magnetization curve is presented in Fig. 4(a) for the penetration and in Fig. 4(b) for the expulsion of vortices. A clear difference between the two samples is observed. Sharp peaks are observed in the curve for the triangle for the flux penetration and expulsion, while these peaks are less pronounced for the disk. A higher DH value for a given vorticity L means that this vorticity is stable over a broad magnetic field range. The configurations with 6, 9 and 14 vortices seem to be stable for the increasing field in the triangle, while for the decreasing branch the numbers 7, 10 and 15 were found to be stable. We have to emphasize that the value of the vorticity is not exactly known since it is possible that some jumps were not resolved in the experimental magnetization curve MðH Þ. But notice that these numbers are very close to the vorticities L where the vortices can arrange in a triangular lattice keeping the C3 symmetry imposed by the boundary of the triangle. The vorticity L for these triangular symmetric configurations can be given by the numbers L ¼ 12 nðn þ 1Þ, with n an integer number, which gives among others the values L ¼ 6, 10 and 15 for n ¼ 3, 4 and 5. Baelus and Peeters [10] calculated the vortex configurations and their stabilities in a mesoscopic triangle. For the vorticity L ¼ 5 and 6, they found a vortex molecule consisting of 3 vortices forming a triangle and a giant vortex with vorticity 2 and 3 in the middle. Further increasing the vorticity leads to the growth of the giant vortex. In their calculation, they only found a stable configuration for L ¼ 3 explained by the fact that the vortex lattice tries to keep the same geometry as the sample. No other
J. Bekaert et al. / Physica C 404 (2004) 44–49
Disk Triangle Increasing field
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in the magnetization curve MðH Þ are observed. The formation of giant vortices can thus be excluded in this case for low vorticity and the observation of stable states in the triangle can be explained by the formation of a triangular lattice of vortices keeping the symmetry imposed by the confinement. It is likely that this triangular lattice is slightly deformed since the interaction of the vortices with the boundaries is strongly non-uniform. The stability of some states was less pronounced in the disk, which can be explained by the C1 symmetry of the disk. With this geometry, the difference between two configurations is expected to be much smaller.
14
L
4. Theory Disk Triangle Decreasing field
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Fig. 4. The range of magnetic field DH between two jumpsnormalized by the magnetic field H0!1 needed for the penetration of the first vortex. The black and gray bars in the figure represent the values calculated from the experimental magnetization curve of the disk and the triangle shown in Figs. 2 and 3, respectively (a) for the increasing, (b) for the decreasing magnetic field.
stable vortex state was found for higher vorticity of the triangle. It is worth mentioning that their calculations are done for the mesoscopic regime where only 13 vortices were present at the critical field Hc2 , while in this sample more than 100 jumps
Consider the superconducting disk with radius R and thickness s ðs RÞ, placed in the magnetic field. The magnetization is measured by a Hall sensor, situated at a distance l s below the disk. It follows from the Biot–Savart law that at l R the magnetization, measured by a Hall sensor, is approximately independent on the value of l, and we can assume that l ¼ 0. The area of the sensor is sufficiently larger than that of the disk. For this reason the sensor measures the total difference between the applied field and the local field in the plane of a disk. Therefore, we can map our problem on the problem of a two-dimensional disk (s < nðT Þ R) with the effective values of the 0 2 London penetration depth kðT Þ kðT Þ =s and 0 0 the Ginzburg–Landau parameter j ¼ kðT Þ =nðT Þ [21]. In order to describe the field of a vortex, we use the well known ClemÕs model [22]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0 ðr2 þ n2v Þ=k02 U0 h0 ðrÞ ¼ ; ð1Þ jnv K1 ðnv =k0 Þ 2pk0 where the parameter nv nðT Þ characterizes the vortex core size. If the distances between the neighboring vortices in a disk by far exceed the vortex core size we can expand the expression for the vortex field (1) in terms of ðnv =rÞ2 and take into
J. Bekaert et al. / Physica C 404 (2004) 44–49 0.25 0.20 0.15
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account only the first term of the expansion. In this case, the local magnetic field in the disk and the Gibbs energy of the disk can be calculated for any given arrangement of vortices by the method, proposed by Bobel in Ref. [23]. In a stable state, the vortices form a shell structure in the disk. We assume that vortices in each shell are situated in the vertices of an ideal polygon (but the sizes of polygons and the angles between them are not fixed). This approximation is rather accurate, if the number of vortices in the disk is not too large. The presence of the sample boundary results in a surface Bean–Livingston barrier, which prevents vortex entrance and exit. Therefore, the magnetization is different for the cases of increasing and decreasing fields, and can even be positive (paramagnetic response). For the penetration of each vortex we use the criterion of vanishing of the surface barrier at the distance nv from the surface. Following Ref. [23], we assume as a simplification of our model that the positions of preexisting vortices are not influenced by the vortex nucleation at the surface during the nucleation process. For the vortex exit we use an approximation, according to which the positions of other vortices remain the same, while a vortex is leaving the sample. The experimental and theoretical results at the increasing and decreasing fields are shown in Fig. 5. Here we chose the parameter j0 ¼ 2:89 (in the bulk we have j ¼ 1:14 for lead) in such a way as to fit the experimental and theoretical fields of the first vortex penetration. To obtain a reasonable agreement between the theoretical and experimental periodicities of the magnetization oscillations we had to assume that the effective radius of the disk is 18% smaller than the actual one. We believe that the physical reason is the oxidation of the sample edges, since lead oxidizes very easily, and the disk was protected by the Ge layer only from the top, not from the edges. We have calculated the magnetization curves up to the state with 17 vortices inside. There is a reasonable semiquantitative agreement between the theory and the experiment. For the decreasing field we have obtained theoretically the positive magnetization (paramagnetic Meissner effect). Thus, a paramagnetic response can be observed not only for the
0.05 0.00
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H (mT)
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Fig. 5. Experimental magnetization curves of the disk for the increasing (open squares) and the decreasing (open circles) magnetic field. The measurements are performed at T ¼ 0:6Tc . The theoretical curve is represented by solid lines.
mesoscopic samples, from which sizes are of the order of nðT Þ, but also for the intermediate size samples between the mesoscopic and the macroscopic scales. We attribute the discrepancies between the theory and the experiment to the effect of pinning, some deviations of the sample from the disk shape and inhomogeneity of the superconducting sample properties due to the oxidization.
5. Conclusion We have measured experimentally the magnetization MðH Þ of a micron-sized superconducting Pb disk and triangle. In particular, we focused on the case of intermediate size between the mesoscopic and macroscopic size and we found there the paramagnetic Meissner effect. The magnetization curve of the disk was compared with theoretical results and a semi-quantitative agreement was found. The transitions between the phases with different shell structures in the disk produce the superstructure of the background of the magnetization curve. A difference between the stability
J. Bekaert et al. / Physica C 404 (2004) 44–49
of the configurations with the same numbers of vortices in the triangle and the disk was observed. The paramagnetic response observed in the experiment could be reproduced in the calculations.
Acknowledgements This work has been supported by the Belgian IUAP, the Flemish FWO and the Research Fund K.U.Leuven GOA/2004/02 programmes, the DWTC and by the ESF programme VORTEX. J.B., M.M. and W.V.P. are postdoctoral fellows of the Research Council of the K.U.Leuven. The authors would like to thank S. Raedts for the help in the preparation of the sample.
References [1] O. Buisson, P. Gandit, R. Rammal, Y.Y. Wang, B. Pannetier, Phys. Lett. A 150 (1) (1990) 36. [2] A.K. Geim, S.V. Dubonos, J.G.S. Lok, I.V. Grigorieva, J.C. Maan, L. Theil Hansen, P.E. Lindelof, Appl. Phys. Lett. 71 (16) (1997) 2379. [3] V.V. Moshchalkov, X.G. Qiu, V. Bruyndoncx, Phys. Rev. B 55 (1997) 11793. [4] P.S. Deo, V.A. Schweigert, F.M. Peeters, A.K. Geim, Phys. Rev. Lett. 79 (23) (1997) 4653. [5] R. Benoist, W. Zwerger, Z. Phys. B 103 (1997) 377.
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[6] V.A. Schweigert, F.M. Peeters, Phys. Rev. B 57 (21) (1998) 13817. [7] M. Morelle, G. Teniers, L.F. Chibotaru, A. Ceulemans, V.V. Moshchalkov, Physica C 369 (2002) 351. [8] M. Morelle, Y. Bruynseraede, V.V. Moshchalkov, Phys. Stat. Sol. B 237 (2003) 365. [9] L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V.V. Moshchalkov, Phys. Rev. Lett. 86 (2001) 1323. [10] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [11] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature (London) 373 (1995) 319. [12] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (5) (1999) 4285. [13] H.T. Jadallah, J. Rubinstein, P. Sternberg, Phys. Rev. Lett. 82 (14) (1999) 2935. [14] L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V.V. Moshchalkov, Nature (London) 408 (2000) 833. [15] D.A. Dikin, V. Chandrasekhar, V.R. Misko, V.M. Fomin, J.T. Devreese, cond-mat/0206113. [16] V.R. Misko, V.M. Fomin, J.T. Devreese, Phys. Rev. B 64 (2001) 014517. [17] D. Schuh, J. Biberger, A. Bauer, W. Breuer, D. Weiss, IEEE Trans. Magn. 37 (2001) 2091. [18] J. Bekaert, D. Buntinx, C. Van Haesendonck, V. Moshchalkov, J. De Boeck, G. Borghs, V. Metlushko, Appl. Phys. Lett. 81 (2002) 3413. [19] F.M. Peeters, X.Q. Li, Appl. Phys. Lett. 72 (1998) 572. [20] A. Houghton, F.B. McLean, Phys. Rev. B 19 (3) (1965) 172. [21] J.J. Palacios, F.M. Peeters, B.J. Baelus, Phys. Rev. B 64 (2001) 134514. [22] J.R. Clem, J. Low Temp. Phys. 18 (1975) 427. [23] G. Bobel, Nuovo Cimento 38 (1965) 1740.
Hall magnetometry of superconducting microstructures J. Bekaert a
a,b
, M. Morelle
a,*
, W.V. Pogosov a, G. Borghs b, V.V. Moshchalkov
a
Laboratorium voor Vaste-Stoffysica en Magnetisme, Nanoscale superconductivity and Magnetism group, K.U.Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium b Interuniversity Micro-electronics Center, Kapeldreef 75, B-3001 Leuven, Belgium
Abstract We report experimental results of a superconducting Pb disk and triangle of the intermediate size, i.e. between the mesoscopic and the macroscopic regime. The magnetization is measured by Hall magnetometry. A comparison of the stability of different vortex configurations in the two different geometries is done. The experimental results for the disk are compared with theoretical predictions. There is a reasonable agreement between the theory and the experiment. 2004 Elsevier B.V. All rights reserved.
1. Introduction The superconducting properties of micron-sized disks have been intensively studied in the recent years. Most of these studies were concentrated on the properties of mesoscopic disks, which had sizes comparable with the coherence length nðT Þ. The phase boundary of these structures was found from transport measurements [1] and the superconducting properties deep in the superconducting state were studied using magnetization measurements [1,2]. While in disks with a radius close to the coherence length a giant vortex is formed, vortex molecules will be formed in superconductors of larger size. The confinement of the vortices in samples of intermediate size between the mesoscopic and macroscopic regime is still important. This means that the vortex configurations can be
*
Corresponding author. Fax: +3216327983. E-mail address: [email protected] (M. Morelle).
influenced by the circular geometry of the sample and differ from the Abrikosov vortex lattice structure found in bulk superconductors. While the disk geometry has been extensively studied during the last decade [1–6], only recently some other geometries have been investigated like triangles [7–10] squares [10–14], or more exotic structures [15,16]. The progress made in the growth of semiconductor heterostructures by molecular beam epitaxy (MBE), enabled the fabrication of Hall sensors and Hall magnetometers, which have been efficiently used for characterization of both ferromagnetic [17,18] and superconducting [2] materials. These sensors have also been applied in scanning Hall probe microscopy [18] where submicron probes with high sensitivity are used to map local field distributions in a non-invasive and quantitative way. With a magnetic or superconducting element deposited on the sensitive area of the sensor, the Hall voltage proportional to the perpendicular component of the local magnetic induction is
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.11.056
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measured. This local induction can be in general considered as the averaged induction over the total sensing area of the probe [19]. In this paper, Hall magnetometry measurements of a micron-sized superconducting Pb disk and triangle are presented. The regime of intermediate size, where the coherence length nðT Þ is an order of magnitude smaller than the size of the element, has been investigated. The experimental magnetization curve of the disk is compared with a theoretical one.
2. Sample characteristics The micron Hall sensors were fabricated from GaAs/AlGaAs heterostructures with a twodimensional electron gas at a depth l ¼ 70 nm below the surface. The active area is patterned by optical lithography. After etching, the Hall cross junction is 1.5 lm wide, resulting in an effective electrical width of 1.4 lm due to depletion of carriers at the edges. At the low temperatures of our measurements, the carrier mobility is 650,000 cm2 /Vs and the 2D carrier concentration is 2.5 · 1011 cm2 , yielding a sensitivity of 0.25 lV per lAG. Our measurement noise corresponds to 0.05 G. The definition of the structures on the active area of the Hall probe was done by e-beam lithography, which requires a high accuracy of the alignment, followed by a lift-off process. The Pb film, deposited by molecular beam epitaxy, was covered with a Ge layer in order to prevent oxidation. The thickness s ¼ 40 nm was measured by X-ray spectroscopy. An area S of 2.5 lm2 for the disk and of 2.1 lm2 for the triangle was determined from SEM measurements (see Fig. 1). From a co-evaporated reference sample, we found a coherence length nð0Þ ¼ 40 nm. An ac current of 25 lA was sent through the Hall probe at a frequency of 27.7 Hz. No heating effects arising from this large current were observed during the measurements. The magnetization l0 M ¼ hBi l0 H was found from the difference between the Hall signal below and above the critical temperature Tc of the superconductor.
Fig. 1. SEM micrograph of the Hall magnetometers. A superconducting Pb disk and triangle are deposited on the sensitive area of the Hall probes.
3. Experimental results The experimental magnetization curves for the superconducting disk and triangle are presented in Figs. 2 and 3. Jumps are observed in the increasing and decreasing branches of the magnetization, corresponding to the stepwise penetration and the expulsion of vortices. The critical field is found to be higher for the triangle than for the disk, what is 2 expected since jWj is higher in the wedge [20]. For both structures, it can be seen in Figs. 2 and 3 that the magnetization curves for the decreasing magnetic field are not returning to zero at zero magnetic field. This means that at least one vortex is trapped and exits the superconductor only at a finite negative magnetic field. The position of the peaks as well for the increasing as for the decreasing branch is always very well reproduced when repeating the magnetization curve of the disk and the triangle. A paramagnetic magnetization has been observed for the two geometries with decreasing magnetic field. The origin of this observation is the formation of metastable states due to the Bean–Livingston surface barrier that keeps the vortices inside the superconductor. As is clearly seen from the experiment, the magnetization curves corresponding to the fixed number of vortices have the same slope. The physical reason of this feature is the fact that the
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J. Bekaert et al. / Physica C 404 (2004) 44–49 0.10
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Fig. 3. Experimental magnetization curves of the triangle for the increasing (open squares) and the decreasing (open circles) magnetic field. The measurements are performed at T ¼ 0:6Tc .
sample sizes are much larger than the London penetration depth, and the magnetic flux of each
vortex is almost equal to the flux quantum U0 . Such a behavior of magnetization is typical for samples of the intermediate size (i.e. between the mesoscopic and the macroscopic scales). From the dependencies MðH Þ presented in Fig. 2, a certain quasi-periodicity of the background of the magnetization curves can be noticed in the increasing and decreasing branch. We attribute these features to the transitions between the phases with different shell structures for the vortex configuration. In order to study the vortex penetration and expulsion in the disk and the triangle, the magnetic field range DH between two jumps in the magnetization curve is presented in Fig. 4(a) for the penetration and in Fig. 4(b) for the expulsion of vortices. A clear difference between the two samples is observed. Sharp peaks are observed in the curve for the triangle for the flux penetration and expulsion, while these peaks are less pronounced for the disk. A higher DH value for a given vorticity L means that this vorticity is stable over a broad magnetic field range. The configurations with 6, 9 and 14 vortices seem to be stable for the increasing field in the triangle, while for the decreasing branch the numbers 7, 10 and 15 were found to be stable. We have to emphasize that the value of the vorticity is not exactly known since it is possible that some jumps were not resolved in the experimental magnetization curve MðH Þ. But notice that these numbers are very close to the vorticities L where the vortices can arrange in a triangular lattice keeping the C3 symmetry imposed by the boundary of the triangle. The vorticity L for these triangular symmetric configurations can be given by the numbers L ¼ 12 nðn þ 1Þ, with n an integer number, which gives among others the values L ¼ 6, 10 and 15 for n ¼ 3, 4 and 5. Baelus and Peeters [10] calculated the vortex configurations and their stabilities in a mesoscopic triangle. For the vorticity L ¼ 5 and 6, they found a vortex molecule consisting of 3 vortices forming a triangle and a giant vortex with vorticity 2 and 3 in the middle. Further increasing the vorticity leads to the growth of the giant vortex. In their calculation, they only found a stable configuration for L ¼ 3 explained by the fact that the vortex lattice tries to keep the same geometry as the sample. No other
J. Bekaert et al. / Physica C 404 (2004) 44–49
Disk Triangle Increasing field
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in the magnetization curve MðH Þ are observed. The formation of giant vortices can thus be excluded in this case for low vorticity and the observation of stable states in the triangle can be explained by the formation of a triangular lattice of vortices keeping the symmetry imposed by the confinement. It is likely that this triangular lattice is slightly deformed since the interaction of the vortices with the boundaries is strongly non-uniform. The stability of some states was less pronounced in the disk, which can be explained by the C1 symmetry of the disk. With this geometry, the difference between two configurations is expected to be much smaller.
14
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4. Theory Disk Triangle Decreasing field
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Fig. 4. The range of magnetic field DH between two jumpsnormalized by the magnetic field H0!1 needed for the penetration of the first vortex. The black and gray bars in the figure represent the values calculated from the experimental magnetization curve of the disk and the triangle shown in Figs. 2 and 3, respectively (a) for the increasing, (b) for the decreasing magnetic field.
stable vortex state was found for higher vorticity of the triangle. It is worth mentioning that their calculations are done for the mesoscopic regime where only 13 vortices were present at the critical field Hc2 , while in this sample more than 100 jumps
Consider the superconducting disk with radius R and thickness s ðs RÞ, placed in the magnetic field. The magnetization is measured by a Hall sensor, situated at a distance l s below the disk. It follows from the Biot–Savart law that at l R the magnetization, measured by a Hall sensor, is approximately independent on the value of l, and we can assume that l ¼ 0. The area of the sensor is sufficiently larger than that of the disk. For this reason the sensor measures the total difference between the applied field and the local field in the plane of a disk. Therefore, we can map our problem on the problem of a two-dimensional disk (s < nðT Þ R) with the effective values of the 0 2 London penetration depth kðT Þ kðT Þ =s and 0 0 the Ginzburg–Landau parameter j ¼ kðT Þ =nðT Þ [21]. In order to describe the field of a vortex, we use the well known ClemÕs model [22]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0 ðr2 þ n2v Þ=k02 U0 h0 ðrÞ ¼ ; ð1Þ jnv K1 ðnv =k0 Þ 2pk0 where the parameter nv nðT Þ characterizes the vortex core size. If the distances between the neighboring vortices in a disk by far exceed the vortex core size we can expand the expression for the vortex field (1) in terms of ðnv =rÞ2 and take into
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0.10
0
account only the first term of the expansion. In this case, the local magnetic field in the disk and the Gibbs energy of the disk can be calculated for any given arrangement of vortices by the method, proposed by Bobel in Ref. [23]. In a stable state, the vortices form a shell structure in the disk. We assume that vortices in each shell are situated in the vertices of an ideal polygon (but the sizes of polygons and the angles between them are not fixed). This approximation is rather accurate, if the number of vortices in the disk is not too large. The presence of the sample boundary results in a surface Bean–Livingston barrier, which prevents vortex entrance and exit. Therefore, the magnetization is different for the cases of increasing and decreasing fields, and can even be positive (paramagnetic response). For the penetration of each vortex we use the criterion of vanishing of the surface barrier at the distance nv from the surface. Following Ref. [23], we assume as a simplification of our model that the positions of preexisting vortices are not influenced by the vortex nucleation at the surface during the nucleation process. For the vortex exit we use an approximation, according to which the positions of other vortices remain the same, while a vortex is leaving the sample. The experimental and theoretical results at the increasing and decreasing fields are shown in Fig. 5. Here we chose the parameter j0 ¼ 2:89 (in the bulk we have j ¼ 1:14 for lead) in such a way as to fit the experimental and theoretical fields of the first vortex penetration. To obtain a reasonable agreement between the theoretical and experimental periodicities of the magnetization oscillations we had to assume that the effective radius of the disk is 18% smaller than the actual one. We believe that the physical reason is the oxidation of the sample edges, since lead oxidizes very easily, and the disk was protected by the Ge layer only from the top, not from the edges. We have calculated the magnetization curves up to the state with 17 vortices inside. There is a reasonable semiquantitative agreement between the theory and the experiment. For the decreasing field we have obtained theoretically the positive magnetization (paramagnetic Meissner effect). Thus, a paramagnetic response can be observed not only for the
0.05 0.00
-
48
-0.05 -0.10 -0.15 -0.20 0
10
20
30
40
H (mT)
0
Fig. 5. Experimental magnetization curves of the disk for the increasing (open squares) and the decreasing (open circles) magnetic field. The measurements are performed at T ¼ 0:6Tc . The theoretical curve is represented by solid lines.
mesoscopic samples, from which sizes are of the order of nðT Þ, but also for the intermediate size samples between the mesoscopic and the macroscopic scales. We attribute the discrepancies between the theory and the experiment to the effect of pinning, some deviations of the sample from the disk shape and inhomogeneity of the superconducting sample properties due to the oxidization.
5. Conclusion We have measured experimentally the magnetization MðH Þ of a micron-sized superconducting Pb disk and triangle. In particular, we focused on the case of intermediate size between the mesoscopic and macroscopic size and we found there the paramagnetic Meissner effect. The magnetization curve of the disk was compared with theoretical results and a semi-quantitative agreement was found. The transitions between the phases with different shell structures in the disk produce the superstructure of the background of the magnetization curve. A difference between the stability
J. Bekaert et al. / Physica C 404 (2004) 44–49
of the configurations with the same numbers of vortices in the triangle and the disk was observed. The paramagnetic response observed in the experiment could be reproduced in the calculations.
Acknowledgements This work has been supported by the Belgian IUAP, the Flemish FWO and the Research Fund K.U.Leuven GOA/2004/02 programmes, the DWTC and by the ESF programme VORTEX. J.B., M.M. and W.V.P. are postdoctoral fellows of the Research Council of the K.U.Leuven. The authors would like to thank S. Raedts for the help in the preparation of the sample.
References [1] O. Buisson, P. Gandit, R. Rammal, Y.Y. Wang, B. Pannetier, Phys. Lett. A 150 (1) (1990) 36. [2] A.K. Geim, S.V. Dubonos, J.G.S. Lok, I.V. Grigorieva, J.C. Maan, L. Theil Hansen, P.E. Lindelof, Appl. Phys. Lett. 71 (16) (1997) 2379. [3] V.V. Moshchalkov, X.G. Qiu, V. Bruyndoncx, Phys. Rev. B 55 (1997) 11793. [4] P.S. Deo, V.A. Schweigert, F.M. Peeters, A.K. Geim, Phys. Rev. Lett. 79 (23) (1997) 4653. [5] R. Benoist, W. Zwerger, Z. Phys. B 103 (1997) 377.
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[6] V.A. Schweigert, F.M. Peeters, Phys. Rev. B 57 (21) (1998) 13817. [7] M. Morelle, G. Teniers, L.F. Chibotaru, A. Ceulemans, V.V. Moshchalkov, Physica C 369 (2002) 351. [8] M. Morelle, Y. Bruynseraede, V.V. Moshchalkov, Phys. Stat. Sol. B 237 (2003) 365. [9] L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V.V. Moshchalkov, Phys. Rev. Lett. 86 (2001) 1323. [10] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [11] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature (London) 373 (1995) 319. [12] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (5) (1999) 4285. [13] H.T. Jadallah, J. Rubinstein, P. Sternberg, Phys. Rev. Lett. 82 (14) (1999) 2935. [14] L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V.V. Moshchalkov, Nature (London) 408 (2000) 833. [15] D.A. Dikin, V. Chandrasekhar, V.R. Misko, V.M. Fomin, J.T. Devreese, cond-mat/0206113. [16] V.R. Misko, V.M. Fomin, J.T. Devreese, Phys. Rev. B 64 (2001) 014517. [17] D. Schuh, J. Biberger, A. Bauer, W. Breuer, D. Weiss, IEEE Trans. Magn. 37 (2001) 2091. [18] J. Bekaert, D. Buntinx, C. Van Haesendonck, V. Moshchalkov, J. De Boeck, G. Borghs, V. Metlushko, Appl. Phys. Lett. 81 (2002) 3413. [19] F.M. Peeters, X.Q. Li, Appl. Phys. Lett. 72 (1998) 572. [20] A. Houghton, F.B. McLean, Phys. Rev. B 19 (3) (1965) 172. [21] J.J. Palacios, F.M. Peeters, B.J. Baelus, Phys. Rev. B 64 (2001) 134514. [22] J.R. Clem, J. Low Temp. Phys. 18 (1975) 427. [23] G. Bobel, Nuovo Cimento 38 (1965) 1740.