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Probing the discrete motion of vortices with rf excitations
Physica C 470 (2010) 857–859
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Probing the discrete motion of vortices with rf excitations J. Van de Vondel a,*, A.V. Silhanek a, V.V. Moshchalkov a, B. Ilic b, J. Fields c, V. Metlushko c a
INPAC – Institute for Nanoscale Physics and Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium Cornell Nanofabrication Facility, School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA c Department of Electrical and Computer Engineering, University of Illinois, Chicago, IL 60607, USA b
a r t i c l e
i n f o
Article history: Available online 4 March 2010 Keywords: S/F hybrids Vortex dynamics Vortex pinning
a b s t r a c t In this work we present experimental results on the rectification of vortices in a superconductor/ferromagnet system under a high frequency drive. The two-dimensional pinning landscape, induced by the stray fields of the ferromagnetic template, has no intrinsic asymmetry. Nevertheless, an asymmetric potential is artificially induced by an applied dc bias. The experimental results unambiguously show a biased, discrete motion of the vortices in the periodic potential at frequencies above 10 MHz. This synchronized motion is very sensitive to the external applied field. Increasing temperature leads to a reduction of the pinning potential, which in turn results in a lower ac power needed to drive the vortex lattice. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction From the pioneering work of Lee et al. [1] a considerable amount of theoretical work has been done to understand the rectified motion of vortices in type-II superconductors [2,3]. Simultaneously, the possibility to create huge arrays of two-dimensional structures, using electron beam lithography, helped to verify and complement these new theoretical investigations by its experimental counterpart. This combined research revealed a wide variety of interesting properties related to the peculiar nature of the particles (i.e. vortices) in this type of ratchet system. A very strong density and temperature dependence of the ratchet efficiency [2,4], switching of the ratchet direction as a function of the applied driving force [5], the vortex density [6] and the polarity of the vortex [7], to name just a few. Besides the parallel progress made in both lines of research an unbridged gap between theory and experiment persists still today in the investigated frequency range of the applied driving force. If a vortex moves along a periodic pining potential a modulation of the vortex velocity is expected with a frequency given by:
f ¼
hv i ; d
ð1Þ
with hvi the average vortex velocity and d the periodicity of the pinning landscape along the direction of motion. This modulation resonates with an externally applied ac driving force leading to the well known Shapiro steps in the measured I(V)-curves [8,9]. This can, in a simplified way, be seen as the discrete motion of the vortex lattice in a periodic potential. * Corresponding author. E-mail address: [email protected] (J. Van de Vondel). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.079
In the case of vortex ratchets, depending on the frequency of the applied unbiased ac drive, similar discrete behavior can be present. At low driving frequencies the rectified motion during one cycle is a multitude of steps in the periodic pinning landscape. This results in a continuous rectification signal, as shown in [5,10]. However, if the frequency of the ac drive matches the same frequency as given in Eq. (1), with hvi now the average motion during the positive or negative cycle, a discretization of the vortex motion, and consequently of the rectification signal, should be present. Most of the theoretical investigation of vortex ratchets so far has been done based on molecular dynamics simulations of the Langevin equation of motion. In this simulations the frequency of the applied driving force is an easy accessible parameter. This allowed researchers to explore ratchets systems in the high frequency region. Within this approximation Zhu and coworkers [3] showed the presence of oscillations in the time averaged net velocity of the vortices versus the period of the applied ac drive. Recently, we demonstrated the presence of a very broad dynamic vortex phase in different types of dense ferromagnet/superconductor systems [11]. In [12] we artificially constructed a magnetic vortex ratchet by a superimposed dc drive. At high frequencies we are able to discretize the vortex motion at will by tuning applied driving force. In this work we will investigate in more detail this type of high frequency ratchet and check its dependence on the applied magnetic field and temperature.
2. Experimental details The ferromagnet/hybrid structure used in these experiments is described and characterized very thoroughly in [12]. As schematically shown in Fig. 1a, it consists of connected equilateral Py trian-
858
J. Van de Vondel et al. / Physica C 470 (2010) 857–859
(a)
(b)
(c)
Fig. 1. (a) Schematic drawing of the sample under investigation. The circles (discs) depict the exact location of the induced vortices (antivortices). (b and c) Give a schematic representation of the vortex motion at n1 and n2, as explained in Section 4. The red line (dotted purple) line shows the motion of a single vortex during the positive (negative) cycle (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.).
gles magnetized perpendicular to the chains of triangles deposited on top of a Al superconducting film (Tc = 1.315 K). If the triangles are magnetized perpendicular to the line connecting them, the present stray fields of the Py triangles creates two lines of vortices with opposite polarity, which have a very high mobility along this line. It was already shown in [12] that this structure gives rise to very pronounced Shapiro steps due to the so-called self-organized commensurability. This refers to the fact that both the induced vortices and the periodic pinning potential originate from the same magnetic template, which by default leads to perfect commensurability.
3. Determining the power input In this work we probe the high frequency dynamics of these inherently present vortices by applying an ac driving force perpendicular to the chains of triangles. In order to estimate the high frequency power going into the sample we will probe the shift in the dc critical current, using a 10 mV criterion, as a function of the applied ac frequency at 0.5 mT. The ac excitation is provided by a Rhode & Schwarz voltage signal generator, directly connected to the current leads of the sample through a high pass filter without any impedance matching. As shown in Fig. 2, the dc critical current changes with a rate of 4.2, 6.8, 10.5 and 13.4 lA/mV for 200, 150, 100 and 50 MHz, respectively. Since the total depinning force is a summation of both dc and ac components this change gives us
Fig. 3. Discrete motion of the vortex lattice at 1.1 K. Triangular (square) symbols shows Vdc (dVdc/dIac) as a function of the ac driving force (f = 150 MHz) with an applied dc bias current of 10 lA at zero applied field.
a good estimate for the voltage-to-current conversion of the high frequency voltage source at these specific frequencies. 4. Probing the discrete motion of vortices In order to probe the discrete dynamics of the vortex lattice we break the symmetry of the pinning landscape by applying a small dc tilt. As such, we create artificially a vortex ratchet and, at the same time, this allows us to measure the discrete motion without the need of time resolved measurements. The result for a dc tilt of 10 lA and an applied magnetic field of 0 mT is shown in Fig. 3, (blue)1 triangular symbols. The applied ac frequency is 150 MHz (similar results are obtained in a frequency range from 30 MHz up to 500 MHz) and the above obtained voltage-to-current conversion is used to determine the actual ac current through the sample. The response of the system can be divided in three different regions. At low ac driving force (Iac < 100 lA) the excitation is smaller than the pining barrier. The vortex oscillates inside the pinning potential and no net motion, i.e. no dc voltage, is detected. In the intermediate region the ac excitation is able to move the vortices and sharp peaks (label np) in the dc voltage readout are observed at equidistant steps (step size = 80 lA). Exactly at the peaks, np, the symmetric motion of the vortex lattice is broken. During the positive branch of the ac cycle the vortices tend to travel a distance of p d, while in the negative branch the vortices travel only a distance of (p 1) d. (As shown schematically in Fig. 1b and c for n1 and n2.) The dc voltage is a measurement of the unbalanced step size between both directions. The coherence of the rectification process is given by the height and sharpness of the peak. Another way to show the sharpness of this peaks, which we will use in the next section, is plotting dVdc/dIac, as shown in Fig. 3, (red) square symbols. A single peak in Vdc results in a positive and negative peak in the derivative. The location of the peak in Vdc appears exactly at the sign changes of the derivative. The sharpness of the peak in Vdc is directly reflected in the height of the amplitude of the dVdc/dIac oscillations. 5. Magnetic field and temperature dependence Fig. 4a shows the influence of an applied magnetic field on the discrete rectification. We visualized the peaks by plotting the
Fig. 2. Dependence of the dc critical current Idc as a function of the applied ac voltage for 200 MHz (triangle), 150 MHz (square), 100 MHz (star) and 50 MHz (circle) at a temperature of 1.1 K and 0.5 mT.
1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.
J. Van de Vondel et al. / Physica C 470 (2010) 857–859
859
These measurements clearly show a broad temperature region in which the discrete motion of the vortex lattice exists. This result suggests that the influence of temperature as a generator of thermal noise is negligible. Nevertheless, the magnetic pining potential, originating from the interaction between the dipole and the vortex, depends on the penetration length [13]. This explains the shift of np towards lower ac drives, i.e. a decrease of pinning, with increasing temperature. 6. Conclusion To summarize, we constructed artificially a ratchet device using a dc tilt. If the frequency of the ac drive exceeds ±10 MHz the distance traveled by a vortex during each positive and negative cycle is only one or a few unit cells. As a result, the average vortex motion is the result of a subtle balance between the amount of steps taken in both cycles. At specific values of the ac drive the unbalanced vortex motion results in sharp steps in the dc voltage. Moreover, we can attribute these peaks to a particular number of back and forth steps of the vortex lattice. In a next step we investigated the coherence of the motion on the applied magnetic field and temperature. These measurements showed a drastic decrease of the locked motion if a small magnetic field is applied. The influence of temperature is given by it changing the pinning potential, rather than inducing thermal noise. Acknowledgements
Fig. 4. (a) dVdc/dIac, as a function of the ac excitation and the applied field. (b) dVdc/ dIac as a function of the applied excitation and temperature.
derivative, dVdc/dIac, as a function of the ac excitation and the applied field. As seen from this plot, the discrete motion only persists very close to zero magnetic field. By applying a very small (<0.1 mT) positive or negative magnetic field the peaks are washed out. It was already shown in [12] that the dynamics of this type of superconductor/ferromagnet systems is ruled by the channel with the same polarity as the applied magnetic field. The channel with opposite polarity is immediately immobilized by the compensation of vortices via the externally induced vortices. This was clearly seen in a field independence of the dynamics of these vortex systems. Therefore this result shows that the compensation of vortices, immobilizing one of the two rows, does have a more subtle influence to the overall dynamics of the system. It will change the surrounding environment and as such it introduces random defects in the periodic potential. As a consequence, it destroys completely the ordered discrete motion of the vortex lattice. In a next step we check the influence of temperature on the discrete motion of vortices. In Fig. 4b the derivative dVdc/dIac, is plotted as a function of the ac excitation and the temperature at 0 mT.
This work was supported by the Methusalem Funding by the Flemish Government, NES–ESF program, the Belgian IAP, the Fund for Scientific Research – Flanders (F.W.O.–Vlaanderen). J.V.d.V. and A.V.S. are grateful for the support from the FWO-Vlaanderen. V.M. acknowledge funding support from US NSF, Grant ECCS-0823813. References [1] C.-S. Lee, B. Jankó, I. Derényi, A.-L. Barabási, Nature 400 (1999) 337. [2] J.F. Wambaugh, C. Reichhardt, C.J. Olson, F. Marchesoni, Franco Nori, Phys. Rev. Lett. 83 (1999) 5106. [3] B.Y. Zhu, F. Marchesoni, V.V. Moshchalkov, Franco Nori, Phys. Rev. B 68 (2003) 014514. [4] C.C. de Souza Silva, J. Van de Vondel, B.Y. Zhu, M. Morelle, V.V. Moshchalkov, Phys. Rev. B 73 (2006) 014507. [5] J.E. Villegas, S. Savelev, F. Nori, E.M. Gonzalez, J.V. Anguita, R. Garcia, J.L. Vicent, Science 302 (2003) 1188. [6] C.C. de Souza Silva, J. Van de Vondel, M. Morelle, V.V. Moshchalkov, Nature 400 (2006) 651. [7] C.C. de Souza Silva, A.V. Silhanek, J. Van de Vondel, W. Gillijns, V. Metlushko, B. Ilic, V.V. Moshchalkov, Phys. Rev. Lett. 98 (2007) 117005. [8] S. Shapiro, Phys. Rev. Lett. 11 (1963) 80. [9] L. Van Look, E. Rosseel, M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 60 (1999) R6998. [10] J. Van de Vondel, C.C. de Souza Silva, B.Y. Zhu, M. Morelle, V.V. Moshchalkov, Phys. Rev. Lett. 94 (2005) 057003. [11] A.V. Silhanek, J. Van de Vondel, A. Leo, G.W. Ataklti, W. Gillijns, V.V. Moshchalkov, Supercond. Sci. Technol. 22 (2008) 034002. [12] J. Van de Vondel, A.V. Silhanek, V. Metlushko, P. Vavassori, B. Ilic, V.V. Moshchalkov, Phys. Rev. B 79 (2009) 054527. [13] M.V. Milosevic, F.M. Peeters, Phys. Rev. B 68 (2003) 094510.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Probing the discrete motion of vortices with rf excitations J. Van de Vondel a,*, A.V. Silhanek a, V.V. Moshchalkov a, B. Ilic b, J. Fields c, V. Metlushko c a
INPAC – Institute for Nanoscale Physics and Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium Cornell Nanofabrication Facility, School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA c Department of Electrical and Computer Engineering, University of Illinois, Chicago, IL 60607, USA b
a r t i c l e
i n f o
Article history: Available online 4 March 2010 Keywords: S/F hybrids Vortex dynamics Vortex pinning
a b s t r a c t In this work we present experimental results on the rectification of vortices in a superconductor/ferromagnet system under a high frequency drive. The two-dimensional pinning landscape, induced by the stray fields of the ferromagnetic template, has no intrinsic asymmetry. Nevertheless, an asymmetric potential is artificially induced by an applied dc bias. The experimental results unambiguously show a biased, discrete motion of the vortices in the periodic potential at frequencies above 10 MHz. This synchronized motion is very sensitive to the external applied field. Increasing temperature leads to a reduction of the pinning potential, which in turn results in a lower ac power needed to drive the vortex lattice. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction From the pioneering work of Lee et al. [1] a considerable amount of theoretical work has been done to understand the rectified motion of vortices in type-II superconductors [2,3]. Simultaneously, the possibility to create huge arrays of two-dimensional structures, using electron beam lithography, helped to verify and complement these new theoretical investigations by its experimental counterpart. This combined research revealed a wide variety of interesting properties related to the peculiar nature of the particles (i.e. vortices) in this type of ratchet system. A very strong density and temperature dependence of the ratchet efficiency [2,4], switching of the ratchet direction as a function of the applied driving force [5], the vortex density [6] and the polarity of the vortex [7], to name just a few. Besides the parallel progress made in both lines of research an unbridged gap between theory and experiment persists still today in the investigated frequency range of the applied driving force. If a vortex moves along a periodic pining potential a modulation of the vortex velocity is expected with a frequency given by:
f ¼
hv i ; d
ð1Þ
with hvi the average vortex velocity and d the periodicity of the pinning landscape along the direction of motion. This modulation resonates with an externally applied ac driving force leading to the well known Shapiro steps in the measured I(V)-curves [8,9]. This can, in a simplified way, be seen as the discrete motion of the vortex lattice in a periodic potential. * Corresponding author. E-mail address: [email protected] (J. Van de Vondel). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.079
In the case of vortex ratchets, depending on the frequency of the applied unbiased ac drive, similar discrete behavior can be present. At low driving frequencies the rectified motion during one cycle is a multitude of steps in the periodic pinning landscape. This results in a continuous rectification signal, as shown in [5,10]. However, if the frequency of the ac drive matches the same frequency as given in Eq. (1), with hvi now the average motion during the positive or negative cycle, a discretization of the vortex motion, and consequently of the rectification signal, should be present. Most of the theoretical investigation of vortex ratchets so far has been done based on molecular dynamics simulations of the Langevin equation of motion. In this simulations the frequency of the applied driving force is an easy accessible parameter. This allowed researchers to explore ratchets systems in the high frequency region. Within this approximation Zhu and coworkers [3] showed the presence of oscillations in the time averaged net velocity of the vortices versus the period of the applied ac drive. Recently, we demonstrated the presence of a very broad dynamic vortex phase in different types of dense ferromagnet/superconductor systems [11]. In [12] we artificially constructed a magnetic vortex ratchet by a superimposed dc drive. At high frequencies we are able to discretize the vortex motion at will by tuning applied driving force. In this work we will investigate in more detail this type of high frequency ratchet and check its dependence on the applied magnetic field and temperature.
2. Experimental details The ferromagnet/hybrid structure used in these experiments is described and characterized very thoroughly in [12]. As schematically shown in Fig. 1a, it consists of connected equilateral Py trian-
858
J. Van de Vondel et al. / Physica C 470 (2010) 857–859
(a)
(b)
(c)
Fig. 1. (a) Schematic drawing of the sample under investigation. The circles (discs) depict the exact location of the induced vortices (antivortices). (b and c) Give a schematic representation of the vortex motion at n1 and n2, as explained in Section 4. The red line (dotted purple) line shows the motion of a single vortex during the positive (negative) cycle (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.).
gles magnetized perpendicular to the chains of triangles deposited on top of a Al superconducting film (Tc = 1.315 K). If the triangles are magnetized perpendicular to the line connecting them, the present stray fields of the Py triangles creates two lines of vortices with opposite polarity, which have a very high mobility along this line. It was already shown in [12] that this structure gives rise to very pronounced Shapiro steps due to the so-called self-organized commensurability. This refers to the fact that both the induced vortices and the periodic pinning potential originate from the same magnetic template, which by default leads to perfect commensurability.
3. Determining the power input In this work we probe the high frequency dynamics of these inherently present vortices by applying an ac driving force perpendicular to the chains of triangles. In order to estimate the high frequency power going into the sample we will probe the shift in the dc critical current, using a 10 mV criterion, as a function of the applied ac frequency at 0.5 mT. The ac excitation is provided by a Rhode & Schwarz voltage signal generator, directly connected to the current leads of the sample through a high pass filter without any impedance matching. As shown in Fig. 2, the dc critical current changes with a rate of 4.2, 6.8, 10.5 and 13.4 lA/mV for 200, 150, 100 and 50 MHz, respectively. Since the total depinning force is a summation of both dc and ac components this change gives us
Fig. 3. Discrete motion of the vortex lattice at 1.1 K. Triangular (square) symbols shows Vdc (dVdc/dIac) as a function of the ac driving force (f = 150 MHz) with an applied dc bias current of 10 lA at zero applied field.
a good estimate for the voltage-to-current conversion of the high frequency voltage source at these specific frequencies. 4. Probing the discrete motion of vortices In order to probe the discrete dynamics of the vortex lattice we break the symmetry of the pinning landscape by applying a small dc tilt. As such, we create artificially a vortex ratchet and, at the same time, this allows us to measure the discrete motion without the need of time resolved measurements. The result for a dc tilt of 10 lA and an applied magnetic field of 0 mT is shown in Fig. 3, (blue)1 triangular symbols. The applied ac frequency is 150 MHz (similar results are obtained in a frequency range from 30 MHz up to 500 MHz) and the above obtained voltage-to-current conversion is used to determine the actual ac current through the sample. The response of the system can be divided in three different regions. At low ac driving force (Iac < 100 lA) the excitation is smaller than the pining barrier. The vortex oscillates inside the pinning potential and no net motion, i.e. no dc voltage, is detected. In the intermediate region the ac excitation is able to move the vortices and sharp peaks (label np) in the dc voltage readout are observed at equidistant steps (step size = 80 lA). Exactly at the peaks, np, the symmetric motion of the vortex lattice is broken. During the positive branch of the ac cycle the vortices tend to travel a distance of p d, while in the negative branch the vortices travel only a distance of (p 1) d. (As shown schematically in Fig. 1b and c for n1 and n2.) The dc voltage is a measurement of the unbalanced step size between both directions. The coherence of the rectification process is given by the height and sharpness of the peak. Another way to show the sharpness of this peaks, which we will use in the next section, is plotting dVdc/dIac, as shown in Fig. 3, (red) square symbols. A single peak in Vdc results in a positive and negative peak in the derivative. The location of the peak in Vdc appears exactly at the sign changes of the derivative. The sharpness of the peak in Vdc is directly reflected in the height of the amplitude of the dVdc/dIac oscillations. 5. Magnetic field and temperature dependence Fig. 4a shows the influence of an applied magnetic field on the discrete rectification. We visualized the peaks by plotting the
Fig. 2. Dependence of the dc critical current Idc as a function of the applied ac voltage for 200 MHz (triangle), 150 MHz (square), 100 MHz (star) and 50 MHz (circle) at a temperature of 1.1 K and 0.5 mT.
1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.
J. Van de Vondel et al. / Physica C 470 (2010) 857–859
859
These measurements clearly show a broad temperature region in which the discrete motion of the vortex lattice exists. This result suggests that the influence of temperature as a generator of thermal noise is negligible. Nevertheless, the magnetic pining potential, originating from the interaction between the dipole and the vortex, depends on the penetration length [13]. This explains the shift of np towards lower ac drives, i.e. a decrease of pinning, with increasing temperature. 6. Conclusion To summarize, we constructed artificially a ratchet device using a dc tilt. If the frequency of the ac drive exceeds ±10 MHz the distance traveled by a vortex during each positive and negative cycle is only one or a few unit cells. As a result, the average vortex motion is the result of a subtle balance between the amount of steps taken in both cycles. At specific values of the ac drive the unbalanced vortex motion results in sharp steps in the dc voltage. Moreover, we can attribute these peaks to a particular number of back and forth steps of the vortex lattice. In a next step we investigated the coherence of the motion on the applied magnetic field and temperature. These measurements showed a drastic decrease of the locked motion if a small magnetic field is applied. The influence of temperature is given by it changing the pinning potential, rather than inducing thermal noise. Acknowledgements
Fig. 4. (a) dVdc/dIac, as a function of the ac excitation and the applied field. (b) dVdc/ dIac as a function of the applied excitation and temperature.
derivative, dVdc/dIac, as a function of the ac excitation and the applied field. As seen from this plot, the discrete motion only persists very close to zero magnetic field. By applying a very small (<0.1 mT) positive or negative magnetic field the peaks are washed out. It was already shown in [12] that the dynamics of this type of superconductor/ferromagnet systems is ruled by the channel with the same polarity as the applied magnetic field. The channel with opposite polarity is immediately immobilized by the compensation of vortices via the externally induced vortices. This was clearly seen in a field independence of the dynamics of these vortex systems. Therefore this result shows that the compensation of vortices, immobilizing one of the two rows, does have a more subtle influence to the overall dynamics of the system. It will change the surrounding environment and as such it introduces random defects in the periodic potential. As a consequence, it destroys completely the ordered discrete motion of the vortex lattice. In a next step we check the influence of temperature on the discrete motion of vortices. In Fig. 4b the derivative dVdc/dIac, is plotted as a function of the ac excitation and the temperature at 0 mT.
This work was supported by the Methusalem Funding by the Flemish Government, NES–ESF program, the Belgian IAP, the Fund for Scientific Research – Flanders (F.W.O.–Vlaanderen). J.V.d.V. and A.V.S. are grateful for the support from the FWO-Vlaanderen. V.M. acknowledge funding support from US NSF, Grant ECCS-0823813. References [1] C.-S. Lee, B. Jankó, I. Derényi, A.-L. Barabási, Nature 400 (1999) 337. [2] J.F. Wambaugh, C. Reichhardt, C.J. Olson, F. Marchesoni, Franco Nori, Phys. Rev. Lett. 83 (1999) 5106. [3] B.Y. Zhu, F. Marchesoni, V.V. Moshchalkov, Franco Nori, Phys. Rev. B 68 (2003) 014514. [4] C.C. de Souza Silva, J. Van de Vondel, B.Y. Zhu, M. Morelle, V.V. Moshchalkov, Phys. Rev. B 73 (2006) 014507. [5] J.E. Villegas, S. Savelev, F. Nori, E.M. Gonzalez, J.V. Anguita, R. Garcia, J.L. Vicent, Science 302 (2003) 1188. [6] C.C. de Souza Silva, J. Van de Vondel, M. Morelle, V.V. Moshchalkov, Nature 400 (2006) 651. [7] C.C. de Souza Silva, A.V. Silhanek, J. Van de Vondel, W. Gillijns, V. Metlushko, B. Ilic, V.V. Moshchalkov, Phys. Rev. Lett. 98 (2007) 117005. [8] S. Shapiro, Phys. Rev. Lett. 11 (1963) 80. [9] L. Van Look, E. Rosseel, M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 60 (1999) R6998. [10] J. Van de Vondel, C.C. de Souza Silva, B.Y. Zhu, M. Morelle, V.V. Moshchalkov, Phys. Rev. Lett. 94 (2005) 057003. [11] A.V. Silhanek, J. Van de Vondel, A. Leo, G.W. Ataklti, W. Gillijns, V.V. Moshchalkov, Supercond. Sci. Technol. 22 (2008) 034002. [12] J. Van de Vondel, A.V. Silhanek, V. Metlushko, P. Vavassori, B. Ilic, V.V. Moshchalkov, Phys. Rev. B 79 (2009) 054527. [13] M.V. Milosevic, F.M. Peeters, Phys. Rev. B 68 (2003) 094510.