- Email: [email protected]
Location of flux-induced vortex
Available online at www.sciencedirect.com
Physica C 468 (2008) 848–851 www.elsevier.com/locate/physc
Location of flux-induced vortex J. Berger a,*, A. Kanda b, R. Furugen b, Y. Ootuka b b
a Physics Department, Ort Braude College, P.O. Box 78, 21982 Karmiel, Israel Institute of Physics and TIMS, University of Tsukuba, Tsukuba 305-8571, Japan
Accepted 30 November 2007 Available online 8 March 2008
Abstract We have obtained experimental evidence for a vortex that mediates between adjacent fluxoid states in a mesoscopic superconducting ring with nonuniform width. We have obtained information about the path of this vortex. For small fluxoid numbers the vortex crosses the sample through the narrowest part and for large fluxoid numbers through the widest part. We review our predictions for critical points. Our results are in agreement with the existent theory. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.78.Na; 74.25.Dw; 74.25.Op; 74.20.De Keywords: Fluxoid transition; Mesoscopic ring; Vortex visualization; Critical point
1. Introduction It has been known for a long time that the order parameter can vanish at certain points for static situations in 1D multiply connected superconducting circuits, provided that they enclose a magnetic flux that inhibits superconductivity [1]. An indirect experimental evidence for this scenario is provided by [2]. Particular cases in which a node appears are a loop with an arm [3] and a loop with nonuniform width [4]. It has also been claimed that a node can appear in a uniform loop [5]. In Ref. [4], we studied the phase diagram in the temperature–flux plane for a 1D ring with nonuniform cross-sectional Harea wðhÞ. We Hdefined an eccentricity parameter b ¼ 2 wðhÞ cosðhÞ dh= wðhÞ dh, where the angle origin is chosen such that the integral of wðhÞ sinðhÞ is zero. As in the usual Little–Parks case, the phase diagram is periodic in the flux U with period U0 ¼ hc=2e and the temperature for the onset of superconductivity is depressed when U=U0 is not integer, with the strongest depression being *
Corresponding author. E-mail address: [email protected] (J. Berger). URL: http://brd4.ort.org.il/~jorge/information.htm (J. Berger).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.084
located at U ¼ U0 =2 (modulo U0 ). We denote by P1 this point of strongest depression in the temperature–flux plane. Assuming b 1 and using a perturbational approach, we obtained that the temperature of P1 is determined by n2 ðT Þ ¼ 4r21D =ð1 jbjÞ, where nðT Þ is the coherence length at temperature T and r1D the radius of the ring. The most outstanding feature found in [4] is that for U ¼ U0 =2 (modulo U0 ) the superconducting order parameter has a node, and this node can mediate between states with different fluxoid numbers, enabling a continuous transition between them. However, this node exists only for a limited range of temperature, close to the onset of superconductivity: there is a critical point in the ðU; T Þ plane (which we dub P2 ), such that below P2 the node does not appear and the transition becomes discontinuous. P2 is located along the line U ¼ U0 =2 and, using the same perturbational approach mentioned above, its temperature is determined by n2 ðT Þ ¼ 4r21D =ð1 þ 2 j b jÞ. About a decade ago some of us extended these results to multiply connected circuits with finite width. A theoretically interesting situation is that of a sample with a hole, which encloses flux within the hole but does not support magnetic field in the sample itself. If the sample is not ‘‘too” symmetric and it encloses an integer plus half
J. Berger et al. / Physica C 468 (2008) 848–851
number of quantum fluxes, then a nodal line (nodal surface in 3D) appears, which destroys the connectivity of the superconducting region [6,7]. We also studied the case of a sample with a hole in a uniform field. We still considered a situation close to cylindric symmetry, close to the onset of superconductivity, and such that the induced magnetic field has negligible influence. In this case, the flux enclosed by the sample is not naturally defined, and some arbitrary convention has to be adopted in its definition. Moreover, the phase diagram is no longer periodic in the flux. Nevertheless, the phase diagram and the transitions between fluxoid states still exhibit similar features to the 1D case. For small fluxoid numbers, the temperature T ðUÞ for the onset of superconductivity has minima between consecutive fluxoid states n and n þ 1, and we now denote by P1 the positions of these minima (P1 depends on n). For large fluxoid numbers, T ðUÞ becomes monotonic. If b 6¼ 0, there may be a vortex in the sample, which is a 2D version of the node encountered in the case of the 1D ring. Instead of being present for a sharp flux U ¼ ðn þ 1=2ÞU0 , it is present for a range of fluxes. Since the presence of the vortex is mainly determined by the flux enclosed by the hole, rather than by the field at the sample itself, we dubbed it a ‘‘flux-induced vortex” [8]. The position of this vortex is a function of the field and of the geometry of the sample. When the field reaches the lower edge of the appropriate range, the vortex forms at the outer boundary of the sample; as the field increases, the vortex position moves towards the inner boundary, until it finally reaches this boundary and disappears. In this way, the fluxoid number of the sample can change continuously; while the vortex is in the sample, the fluxoid numbers are different at the inner and at the outer boundary. Flux-induced vortices have several features that are qualitatively different from those of ‘‘standard” vortices. Here we will mention two features that seem surprising. The first is that the critical points P2 still exist (also P2 depends on n), i.e., the passage between n and n þ 1 is continuous for temperatures above that of P2 and discontinuous for temperatures below it. We found that [9] if the temperature is fixed at that of P2 , then the magnetic susceptibility as a function of the flux diverges quadratically when P2 is approached. The critical character of P2 looks surprising because, unlike the 1D loops that undergo fluxoid transitions for a sharp enclosed flux, in the present case the transition occurs over a finite range. The only effect produced by a small change in the flux is a change in the position of the vortex and one might therefore expect that the critical point should be smeared. What happens is that in the vicinity of P2 the ratio between the change in the position of the vortex and the change in flux diverges. Although no experiments have been intentionally designed to test this feature, there is indirect experimental evidence for it [10]. Another surprising feature is the following. For transitions between small fluxoid numbers, the vortex crosses
849
the sample through its narrowest part; however, for large fluxoid numbers, it crosses through the widest part. The distinction between ‘‘small” and ‘‘large” fluxoid numbers depends on the ratio between the typical linewidth and the typical radius of the sample; for larger ratios, smaller fluxoid numbers are required. In Refs. [8,9], this feature was proven using a perturbative approach; however, numerical studies for boundaries with squared shapes [9] or eccentric cylinders [11] indicate that this trend is a generic feature. In Ref. [9], we obtained explicit expressions for the positions of P1 and P2 (for arbitrary n). P1 was determined by first evaluating the temperature for the onset of superconductivity and then finding the local minima. P2 was determined by equations that are equivalent to the requirements that the first and the second derivatives of the flux with respect to the vortex position vanish. The expressions for P1 and P2 are somewhat lengthy, and we refer the interested reader to [9] or [12]. Unlike the 1D case, the flux at which P2 occurs is not the same as that of P1 , but, at least for small values of n, they are quite close. Here we report on the first experiment that detects the existence and trajectory of a vortex during a fluxoid transition in a mesoscopic asymmetric multiply connected sample. A more detailed report was published elsewhere [12]. 2. Experiment and Interpretation The local density of states (LDOS) at the Fermi surface is a decreasing function of the local strength of superconductivity. The LDOS can be mapped by means of the multiple-small-tunnel-junction method [13], in which several tunnel junctions are attached to a mesoscopic superconductor. The larger the superconducting gap under a given junction, the larger the voltage required at that junction in order to pass a predetermined amount of current (0.3 nA). In particular, the resistance R of a junction is larger when the region under it is superconducting than the resistance Rn of the same junction when this region is normal and, if there is a vortex under the junction, then R Rn . Fig. 1 shows a schematic view of the sample. Two normal-metal leads cover the narrowest and the widest parts of a superconducting Al ring with an eccentric hole (outer Cu leads tunnel junctions
drain Al ring
Fig. 1. Schematic view of the sample. Two Cu leads are connected to an Al asymmetric ring through highly resistive small tunnel junctions (shaded areas). An Al drain is directly connected to the ring.
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J. Berger et al. / Physica C 468 (2008) 848–851
radius ro ¼ 0:42 lm, inner radius ri ¼ 0:18 lm, displacement of the inner center a ¼ 0:10 lm, thickness d ¼ 30 nm), forming small Al/AlOx/Cu tunnel junctions (shaded areas). The ring is connected to an Al drain lead at the widest part. The superconducting coherence length nðT ¼ 0Þ was estimated to be ð0:15 0:04Þ lm from the residual resistance of Al films prepared in the same way. For the measurement we connected a current source to each Cu lead and measured the current–voltage characteristics of the two junctions simultaneously, as functions of the perpendicular magnetic field and the temperature. Fig. 2 is essentially a phase diagram. Above the lines, R is essentially equal to Rn , meaning that the sample is normal; below the lines, R is measurably larger than Rn , meaning that the sample is superconducting. Taking into account our experimental resolution, the adopted criterion for the onset of superconductivity was R ¼ 1:05 Rn ; we estimate that the true onset line is about 10 mK above this line. However, we see that the curves for the two junctions do not coincide. The conclusion is that for some values of the temperature and the field, in spite of the fact that the sample as a whole is already superconducting, the region under one of the junctions is ‘‘almost normal,” i.e., it contains a vortex. Moreover, we see that the relative positions of the curves are interchanged. For low fluxes the curve that corresponds to the junction in the narrow part of the sample has dips below the curve for the other junction, indicating that the vortex passes through the narrow part; for high flux, we find the opposite situation. For low fluxes we also see that the curves coincide unless the field is close to the transition region, for which a vortex is predicted to be present; for high fluxes the curve for the wide part is always lower, since in this case the range in which the vortex or its precursor exists is quite large.
Fig. 2. Temperature where the resistance ratio R=Rn reaches 1.05, for each of the junctions, as a function of the applied magnetic field. U is the flux pffiffiffiffiffiffiffiffi through a circle of radius ri ro . The rhombs are calculated P1 -points, which indicate the theoretical positions of the minima of the onset curve.
These findings are in agreement with those obtained from a numerical analysis using the Ginzburg–Landau model [11,12]: for the transition between fluxoid numbers 0 and 1 the vortex passes through the narrow part, for the transition between 1 and 2 the range of temperatures for which the vortex exists is too small to be detected, and for the transition between 2 and 3 the vortex passes through the wide part. The same qualitative conclusions are obtained using the expressions provided by [9]. Fig. 3 is the phase diagram for fluxoid states. For temperatures below P2 , when the flux increases (decreases) the system reaches a stability limit and decays discontinuously from the fluxoid state n to the state n þ 1 ðn 1Þ. This stability limit is denoted by the open (filled) squares in the figure. For temperatures above P2 the passage between n and n 1 is continuous and smooth, i.e. there is no phase transition, even of second order. We can nevertheless detect when the resistance at the appropriate junction is a minimum, meaning that the vortex is closest to the junction, and regard the field at which this occurs as the transition field. This ‘‘transition” line is marked with diamonds in the figure. The lengths of the diamonds lines are the same as the distances between the curves in Fig. 2. In Figs. 2 and 3, we show the positions of the minima of the onset curve, and the positions of the P2 critical points, according to the expressions obtained in [9]. In order to obtain these positions we assumed a zero-temperature coherence length nð0Þ ¼ 135 nm and we had to assume that the linewidth of the sample was narrower than measured by SEM. We could argue that the borders of the sample were oxidized, and the points we have plotted are based on the assumption that ri is effectively 30 nm larger than observed and ro is 50 nm smaller than observed. However, the expres-
Fig. 3. Transitions between fluxoid states. Open (blue) and closed (red) squares correspond to transition fields for increasing and decreasing magnetic fields, respectively, and the diamonds (green) to the presence of a vortex. The circles are calculated positions for P2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
J. Berger et al. / Physica C 468 (2008) 848–851
851
sions in [9] were obtained under the assumption that deviation from cylindrical shape is small ða ro Þ, whereas for our sample a 0:25ro , so that these results should not be taken too seriously.
necessarily increase the superconducting energy density in the wide part. Therefore, if the magnetic field and the width/radius ratio are large, it becomes energetically advantageous to form the vortex at the wide part.
3. Summary and discussion
References
It follows from the Ginzburg–Landau model that fluxoid transitions in an asymmetric mesoscopic sample with a hole can be mediated by a novel kind of vortex, provided that the temperature is in the appropriate range. This vortex crosses the sample through the narrowest part for small fluxes and through the widest part for large fluxes. We have performed an experiment that confirms these predictions. We would like to offer an intuitive explanation for the flux-dependence of the path followed by the mediating vortex. We expect that the order parameter ought to vanish at the place where the loss of condensation energy is lowest. This gives an advantage to the presence of a vortex in the narrow part, where the volume involved is smaller. On the other hand, the kinetic energy density is proportional 2 to j ði$ 2pA=U0 Þwj , where A is the magnetic potential and w the order parameter. In the narrow part, the sample is essentially a 1D wire and the rate of phase variation of w can adapt itself so that j ði$ 2pA=U0 Þw j is kept small. By contrast, in the wide part, if we regard the sample as being close to cylindrical symmetry, A will be proportional to r, whereas j $w j will be proportional to 1=r. This misfit will
[1] R.D. Parks, Science 146 (1964) 1429; For a short review see, e.g. J.I. Castro, A. Lo´pez, in: J. Berger, J. Rubinstein (Eds.), Connectivity and Superconductivity, Springer, Berlin, 2000. [2] V. Bruyndoncx, C. Strunk, V.V. Moshchalkov, C. Van Haesendonck, Y. Bruynseraede, Europhys. Lett. 36 (1996) 449. [3] P.-G. de Gennes, C.R. Acad. Sc. Paris 292 (1981) 279. [4] J. Berger, J. Rubinstein, SIAM J. Appl. Math. 58 (1998) 103; J. Berger, J. Rubinstein, Phys. Rev. Lett. 75 (1995) 320. [5] E.M. Horane, J.I. Castro, G.C. Buscaglia, A. Lo´pez, Phys. Rev. B 53 (1996) 9296. [6] J. Berger, J. Rubinstein, Commun. Math. Phys. 202 (1999) 621. [7] B. Helffer, M. Hoffman-Ostenhof, T. Hoffman-Ostenhof, M. Owen, Commun. Math. Phys. 202 (1999) 629. [8] J. Berger, J. Rubinstein, Phys. Rev. B 59 (1999) 8896. [9] J. Berger, in: J. Berger, J. Rubinstein (Eds.), Connectivity and Superconductivity, Springer, Berlin, 2000. [10] J. Berger, J. Rubinstein, Phys. Rev. B 56 (1997) 5124; X. Zhang, J.C. Price, Phys. Rev. B 55 (1997) 3128. [11] D.Y. Vodolazov, B.J. Baelus, F.M. Peeters, Phys. Rev. B 66 (2002) 054531. [12] A. Kanda, B.J. Baelus, D.Y. Vodolazov, J. Berger, R. Furugen, Y. Ootuka, F.M. Peeters, Phys. Rev. B. 76 (2007) 094519. [13] A. Kanda, B.J. Baelus, F.M. Peeters, K. Kadowaki, Y. Ootuka, Phys. Rev. Lett. 93 (2004) 257002.
Physica C 468 (2008) 848–851 www.elsevier.com/locate/physc
Location of flux-induced vortex J. Berger a,*, A. Kanda b, R. Furugen b, Y. Ootuka b b
a Physics Department, Ort Braude College, P.O. Box 78, 21982 Karmiel, Israel Institute of Physics and TIMS, University of Tsukuba, Tsukuba 305-8571, Japan
Accepted 30 November 2007 Available online 8 March 2008
Abstract We have obtained experimental evidence for a vortex that mediates between adjacent fluxoid states in a mesoscopic superconducting ring with nonuniform width. We have obtained information about the path of this vortex. For small fluxoid numbers the vortex crosses the sample through the narrowest part and for large fluxoid numbers through the widest part. We review our predictions for critical points. Our results are in agreement with the existent theory. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.78.Na; 74.25.Dw; 74.25.Op; 74.20.De Keywords: Fluxoid transition; Mesoscopic ring; Vortex visualization; Critical point
1. Introduction It has been known for a long time that the order parameter can vanish at certain points for static situations in 1D multiply connected superconducting circuits, provided that they enclose a magnetic flux that inhibits superconductivity [1]. An indirect experimental evidence for this scenario is provided by [2]. Particular cases in which a node appears are a loop with an arm [3] and a loop with nonuniform width [4]. It has also been claimed that a node can appear in a uniform loop [5]. In Ref. [4], we studied the phase diagram in the temperature–flux plane for a 1D ring with nonuniform cross-sectional Harea wðhÞ. We Hdefined an eccentricity parameter b ¼ 2 wðhÞ cosðhÞ dh= wðhÞ dh, where the angle origin is chosen such that the integral of wðhÞ sinðhÞ is zero. As in the usual Little–Parks case, the phase diagram is periodic in the flux U with period U0 ¼ hc=2e and the temperature for the onset of superconductivity is depressed when U=U0 is not integer, with the strongest depression being *
Corresponding author. E-mail address: [email protected] (J. Berger). URL: http://brd4.ort.org.il/~jorge/information.htm (J. Berger).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.084
located at U ¼ U0 =2 (modulo U0 ). We denote by P1 this point of strongest depression in the temperature–flux plane. Assuming b 1 and using a perturbational approach, we obtained that the temperature of P1 is determined by n2 ðT Þ ¼ 4r21D =ð1 jbjÞ, where nðT Þ is the coherence length at temperature T and r1D the radius of the ring. The most outstanding feature found in [4] is that for U ¼ U0 =2 (modulo U0 ) the superconducting order parameter has a node, and this node can mediate between states with different fluxoid numbers, enabling a continuous transition between them. However, this node exists only for a limited range of temperature, close to the onset of superconductivity: there is a critical point in the ðU; T Þ plane (which we dub P2 ), such that below P2 the node does not appear and the transition becomes discontinuous. P2 is located along the line U ¼ U0 =2 and, using the same perturbational approach mentioned above, its temperature is determined by n2 ðT Þ ¼ 4r21D =ð1 þ 2 j b jÞ. About a decade ago some of us extended these results to multiply connected circuits with finite width. A theoretically interesting situation is that of a sample with a hole, which encloses flux within the hole but does not support magnetic field in the sample itself. If the sample is not ‘‘too” symmetric and it encloses an integer plus half
J. Berger et al. / Physica C 468 (2008) 848–851
number of quantum fluxes, then a nodal line (nodal surface in 3D) appears, which destroys the connectivity of the superconducting region [6,7]. We also studied the case of a sample with a hole in a uniform field. We still considered a situation close to cylindric symmetry, close to the onset of superconductivity, and such that the induced magnetic field has negligible influence. In this case, the flux enclosed by the sample is not naturally defined, and some arbitrary convention has to be adopted in its definition. Moreover, the phase diagram is no longer periodic in the flux. Nevertheless, the phase diagram and the transitions between fluxoid states still exhibit similar features to the 1D case. For small fluxoid numbers, the temperature T ðUÞ for the onset of superconductivity has minima between consecutive fluxoid states n and n þ 1, and we now denote by P1 the positions of these minima (P1 depends on n). For large fluxoid numbers, T ðUÞ becomes monotonic. If b 6¼ 0, there may be a vortex in the sample, which is a 2D version of the node encountered in the case of the 1D ring. Instead of being present for a sharp flux U ¼ ðn þ 1=2ÞU0 , it is present for a range of fluxes. Since the presence of the vortex is mainly determined by the flux enclosed by the hole, rather than by the field at the sample itself, we dubbed it a ‘‘flux-induced vortex” [8]. The position of this vortex is a function of the field and of the geometry of the sample. When the field reaches the lower edge of the appropriate range, the vortex forms at the outer boundary of the sample; as the field increases, the vortex position moves towards the inner boundary, until it finally reaches this boundary and disappears. In this way, the fluxoid number of the sample can change continuously; while the vortex is in the sample, the fluxoid numbers are different at the inner and at the outer boundary. Flux-induced vortices have several features that are qualitatively different from those of ‘‘standard” vortices. Here we will mention two features that seem surprising. The first is that the critical points P2 still exist (also P2 depends on n), i.e., the passage between n and n þ 1 is continuous for temperatures above that of P2 and discontinuous for temperatures below it. We found that [9] if the temperature is fixed at that of P2 , then the magnetic susceptibility as a function of the flux diverges quadratically when P2 is approached. The critical character of P2 looks surprising because, unlike the 1D loops that undergo fluxoid transitions for a sharp enclosed flux, in the present case the transition occurs over a finite range. The only effect produced by a small change in the flux is a change in the position of the vortex and one might therefore expect that the critical point should be smeared. What happens is that in the vicinity of P2 the ratio between the change in the position of the vortex and the change in flux diverges. Although no experiments have been intentionally designed to test this feature, there is indirect experimental evidence for it [10]. Another surprising feature is the following. For transitions between small fluxoid numbers, the vortex crosses
849
the sample through its narrowest part; however, for large fluxoid numbers, it crosses through the widest part. The distinction between ‘‘small” and ‘‘large” fluxoid numbers depends on the ratio between the typical linewidth and the typical radius of the sample; for larger ratios, smaller fluxoid numbers are required. In Refs. [8,9], this feature was proven using a perturbative approach; however, numerical studies for boundaries with squared shapes [9] or eccentric cylinders [11] indicate that this trend is a generic feature. In Ref. [9], we obtained explicit expressions for the positions of P1 and P2 (for arbitrary n). P1 was determined by first evaluating the temperature for the onset of superconductivity and then finding the local minima. P2 was determined by equations that are equivalent to the requirements that the first and the second derivatives of the flux with respect to the vortex position vanish. The expressions for P1 and P2 are somewhat lengthy, and we refer the interested reader to [9] or [12]. Unlike the 1D case, the flux at which P2 occurs is not the same as that of P1 , but, at least for small values of n, they are quite close. Here we report on the first experiment that detects the existence and trajectory of a vortex during a fluxoid transition in a mesoscopic asymmetric multiply connected sample. A more detailed report was published elsewhere [12]. 2. Experiment and Interpretation The local density of states (LDOS) at the Fermi surface is a decreasing function of the local strength of superconductivity. The LDOS can be mapped by means of the multiple-small-tunnel-junction method [13], in which several tunnel junctions are attached to a mesoscopic superconductor. The larger the superconducting gap under a given junction, the larger the voltage required at that junction in order to pass a predetermined amount of current (0.3 nA). In particular, the resistance R of a junction is larger when the region under it is superconducting than the resistance Rn of the same junction when this region is normal and, if there is a vortex under the junction, then R Rn . Fig. 1 shows a schematic view of the sample. Two normal-metal leads cover the narrowest and the widest parts of a superconducting Al ring with an eccentric hole (outer Cu leads tunnel junctions
drain Al ring
Fig. 1. Schematic view of the sample. Two Cu leads are connected to an Al asymmetric ring through highly resistive small tunnel junctions (shaded areas). An Al drain is directly connected to the ring.
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J. Berger et al. / Physica C 468 (2008) 848–851
radius ro ¼ 0:42 lm, inner radius ri ¼ 0:18 lm, displacement of the inner center a ¼ 0:10 lm, thickness d ¼ 30 nm), forming small Al/AlOx/Cu tunnel junctions (shaded areas). The ring is connected to an Al drain lead at the widest part. The superconducting coherence length nðT ¼ 0Þ was estimated to be ð0:15 0:04Þ lm from the residual resistance of Al films prepared in the same way. For the measurement we connected a current source to each Cu lead and measured the current–voltage characteristics of the two junctions simultaneously, as functions of the perpendicular magnetic field and the temperature. Fig. 2 is essentially a phase diagram. Above the lines, R is essentially equal to Rn , meaning that the sample is normal; below the lines, R is measurably larger than Rn , meaning that the sample is superconducting. Taking into account our experimental resolution, the adopted criterion for the onset of superconductivity was R ¼ 1:05 Rn ; we estimate that the true onset line is about 10 mK above this line. However, we see that the curves for the two junctions do not coincide. The conclusion is that for some values of the temperature and the field, in spite of the fact that the sample as a whole is already superconducting, the region under one of the junctions is ‘‘almost normal,” i.e., it contains a vortex. Moreover, we see that the relative positions of the curves are interchanged. For low fluxes the curve that corresponds to the junction in the narrow part of the sample has dips below the curve for the other junction, indicating that the vortex passes through the narrow part; for high flux, we find the opposite situation. For low fluxes we also see that the curves coincide unless the field is close to the transition region, for which a vortex is predicted to be present; for high fluxes the curve for the wide part is always lower, since in this case the range in which the vortex or its precursor exists is quite large.
Fig. 2. Temperature where the resistance ratio R=Rn reaches 1.05, for each of the junctions, as a function of the applied magnetic field. U is the flux pffiffiffiffiffiffiffiffi through a circle of radius ri ro . The rhombs are calculated P1 -points, which indicate the theoretical positions of the minima of the onset curve.
These findings are in agreement with those obtained from a numerical analysis using the Ginzburg–Landau model [11,12]: for the transition between fluxoid numbers 0 and 1 the vortex passes through the narrow part, for the transition between 1 and 2 the range of temperatures for which the vortex exists is too small to be detected, and for the transition between 2 and 3 the vortex passes through the wide part. The same qualitative conclusions are obtained using the expressions provided by [9]. Fig. 3 is the phase diagram for fluxoid states. For temperatures below P2 , when the flux increases (decreases) the system reaches a stability limit and decays discontinuously from the fluxoid state n to the state n þ 1 ðn 1Þ. This stability limit is denoted by the open (filled) squares in the figure. For temperatures above P2 the passage between n and n 1 is continuous and smooth, i.e. there is no phase transition, even of second order. We can nevertheless detect when the resistance at the appropriate junction is a minimum, meaning that the vortex is closest to the junction, and regard the field at which this occurs as the transition field. This ‘‘transition” line is marked with diamonds in the figure. The lengths of the diamonds lines are the same as the distances between the curves in Fig. 2. In Figs. 2 and 3, we show the positions of the minima of the onset curve, and the positions of the P2 critical points, according to the expressions obtained in [9]. In order to obtain these positions we assumed a zero-temperature coherence length nð0Þ ¼ 135 nm and we had to assume that the linewidth of the sample was narrower than measured by SEM. We could argue that the borders of the sample were oxidized, and the points we have plotted are based on the assumption that ri is effectively 30 nm larger than observed and ro is 50 nm smaller than observed. However, the expres-
Fig. 3. Transitions between fluxoid states. Open (blue) and closed (red) squares correspond to transition fields for increasing and decreasing magnetic fields, respectively, and the diamonds (green) to the presence of a vortex. The circles are calculated positions for P2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
J. Berger et al. / Physica C 468 (2008) 848–851
851
sions in [9] were obtained under the assumption that deviation from cylindrical shape is small ða ro Þ, whereas for our sample a 0:25ro , so that these results should not be taken too seriously.
necessarily increase the superconducting energy density in the wide part. Therefore, if the magnetic field and the width/radius ratio are large, it becomes energetically advantageous to form the vortex at the wide part.
3. Summary and discussion
References
It follows from the Ginzburg–Landau model that fluxoid transitions in an asymmetric mesoscopic sample with a hole can be mediated by a novel kind of vortex, provided that the temperature is in the appropriate range. This vortex crosses the sample through the narrowest part for small fluxes and through the widest part for large fluxes. We have performed an experiment that confirms these predictions. We would like to offer an intuitive explanation for the flux-dependence of the path followed by the mediating vortex. We expect that the order parameter ought to vanish at the place where the loss of condensation energy is lowest. This gives an advantage to the presence of a vortex in the narrow part, where the volume involved is smaller. On the other hand, the kinetic energy density is proportional 2 to j ði$ 2pA=U0 Þwj , where A is the magnetic potential and w the order parameter. In the narrow part, the sample is essentially a 1D wire and the rate of phase variation of w can adapt itself so that j ði$ 2pA=U0 Þw j is kept small. By contrast, in the wide part, if we regard the sample as being close to cylindrical symmetry, A will be proportional to r, whereas j $w j will be proportional to 1=r. This misfit will
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