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Enhanced pinning of superconducting vortices at circular magnetic dots in the magnetic-vortex state
Physica C 470 (2010) 867–870
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Enhanced pinning of superconducting vortices at circular magnetic dots in the magnetic-vortex state T. Shapoval a,*, V. Metlushko b, M. Wolf a, V. Neu a, B. Holzapfel a, L. Schultz a a b
IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany Department of Electrical and Computer Engineering, University of Illinois at Chicago, IL 60612, USA
a r t i c l e
i n f o
Article history: Available online 21 February 2010 Keywords: Vortex pinning Magnetic dots Superconducting thin films Magnetic force microscopy
a b s t r a c t Low-temperature magnetic force microscopy (LT-MFM) was used to study the distribution of superconducting vortices in Nb above a square array of 1 lm-sized circular ferromagnetic dots in a magnetic-vortex state. The force that the MFM tip exerts on the individual vortex in the depinning process was used to estimate the spatial modulation of the pinning potential. It was found, that the superconducting vortices which are preferably located on top of the Py dots experience a pinning force, about 15 times stronger as compared to the pinning force in the pure Nb film. This strong pinning exceeds the repulsive interaction between the superconducting vortices and allows vortex clusters to be located above the dots. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Low temperature type-II superconductors (LTSC) are usually preferable candidates for basic studies of the vortex matter in thin films. Due to their rather high homogeneity, these films allow to probe the effect of an artificially modified pinning landscape on the vortex arrangement. Therefore, during the last decades artificial defects have been incorporated into a superconducting matrix in different manners: randomly, periodically, as holes or inclusions and, recently, magnetic defects have been used, which cause a variety of exciting phenomena. One of them is an interplay of superconductivity and magnetism. In ferromagnetic/superconducting (FM/SC) hybrid structures it was demonstrated that the vicinity of FM materials, formerly expected to be destructive for the superconductivity, can even lead to field-induced superconductivity and strongly enhanced pinning of vortices [1–5]. This counterintuitive interplay of magnetism and superconductivity is today a topic of high scientific interest, involving many phenomena needed to be investigated and explained. Low-temperature magnetic force microscopy (LT-MFM) is a powerful method to probe the local spatial variation of the pinning landscape, as it allows to combine the non-invasive imaging of individual vortices with the direct manipulation and depinning of vortices from their position [6]. The presence of the artificial pinning sites prohibits the formation of an ordered Abrikosov lattice [7], and leads to the forced organization of the flux lines [8].
* Corresponding author. E-mail address: [email protected] (T. Shapoval). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.071
In this paper LT-MFM was used to locally probe the enhancement of the pinning force of the superconducting vortices in the vicinity of magnetic dots in the magnetic-vortex state. 2. Experimental We have studied pinning of superconducting vortices in the following FM/SC hybrid structure: a square array of permalloy (Py = Ni80Fe20) dots with 1 lm diameter, 25 nm height and 2 lm periodicity prepared on a Si (1 0 0) substrate using standard e-beam lithography, e-beam evaporation, and lift-off processes followed by the deposition of a 100 nm thick superconducting niobium (Nb) film ðT c ¼ 8:32 KÞ on top of the Py dot array by sputter deposition [4]. A SEM image of this Py/Nb hybrid structure is shown in Fig. 1 (inset). Local studies of vortex distribution are performed by low-temperature magnetic force microscopy using a commercial scanning probe microscope (Omicron Cryogenic SFM) [9]. The MFM cantilever (Nanoworld MFMR) possesses a force constant of about 2.8 N/m and a resonance frequency of about 80 kHz. Image processing was done using the WSxM 4.0 software [10]. 3. Results and discussion 3.1. Magnetic-vortex state of Py dots To reach the magnetic-vortex configuration in the Py array, the sample was cooled in the microscope to 40 K. An in-plane field of +100 mT was applied along the positive y direction ðHy Þ to ensure full saturation of the dots. Then the sample was cooled to 14.6 K.
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T. Shapoval et al. / Physica C 470 (2010) 867–870
Magnetic moment, m (10-7 A*m2)
2
T = 5 K Nucleation
(a) (b)
(a)
(a) (c)
1
y x
(e) (a)
0
1 m (d) (a)
-1
Annihilation
-2 -100
-50
0
50
100
Applied in-plane field, Hy (mT) Fig. 1. Magnetic hysteresis loop, showing magnetic vortex behavior with vortex nucleation and annihilation fields. Within the inner loop, between 30 mT and +30 mT, the magnetization is reversible (vortex branch). The inset shows a SEM image of Py dots covered by a 100-nm thick Nb layer. The marked points on the loop correspond to the panels (a)–(e) in Fig. 2, respectively.
The MFM scan at a tip-sample distance of 75 nm shows four saturated Py dots at Hy ¼ þ100 mT (Fig. 2a). While decreasing the field the saturated state is maintained (Fig. 2b) until vortex nucleation occurs close to zero field (Fig. 2c). Decreasing the field to 25 mT ensures that in most of the dots a magnetic vortex has been nucleated which is located at the rim of the element but is not yet annihilated, such that the dot is largely magnetized in negative ydirection (Fig. 2d). Going back to zero along the vortex branch brings the dots into the radial symmetric magnetic-vortex state 0 imaged in Fig. 2e and e .
Magnetic vortices, generated in such a way, will have random polarity (i.e. out-of-plane magnetization components pointing randomly up or down) [11]. To set a defined polarity, a small positive out-of-plane field (+10 mT) was applied to the sample during the above-described field sequence. Thus, the magnetic-vortex core and the MFM tip, which is magnetized in positive z-direction, experience an attractive interaction that shows up as a dark con0 trast in the center of the dot (Fig. 2e ). The measured shift of the resonant frequency Df is proportional to the second derivative of the z component of the magnetic stray field of the sample at the set scanning distance from the surface [12]. In the magnetic-vortex state the overall stray field above the dot is strongly reduced. As a result, the signal measured by MFM for the magnetic-vortex state is almost 10 times less than for the saturated state (compare the scale bars 0 in Fig. 2e and e ). At the same time a strong enhancement of the out-of-plane stray field component is expected at the center of the dot. Due to the fact that the 100 nm thick Nb film covers the Py array, imaging took place far away (about 175 nm) from the surface of the Py dot. Thus, the magnetic-vortex core is visualized by MFM as a largely broadened, dark blue (attractive interaction) contrast. 3.2. Estimation of the local pinning force After reaching the magnetic-vortex state of the Py dots, as described above, the sample was cooled to a temperature below T c of the Nb film ðT ¼ 6:1 K ¼ 72%T c Þ in a perpendicular field Hz ¼ 1 mT, that is close to the second matching field for this hybrid structure [8]. Fig. 3a shows the MFM images of the distribution of the SC vortices in the vicinity of four magnetic dots: three of them being in the magnetic vortex state, the forth (lower right) maintained an in-plane domain configuration. One can clearly see two superconducting vortices per magnetic dot in the upper two elements (yellow contrast).
Fig. 2. To reach the magnetic-vortex state the in-plane field Hy was varied along the hysteresis loop starting from saturation at (a) 100 mT, through (b) 25 mT, (c) 0, and (d) 0 applying a negative field less than the magnetic-vortex annihilation field (25 mT) to the magnetic-vortex state at 0 mT (e) and (e ). Color bars give the measured Df signal. A small out-of-plane field of 10 mT was permanently applied to set the polarity of the magnetic vortex (attractive interaction). The scanning area was 4 lm 4 lm, the scanning distance 75 nm, T = 14.6 K. The white circle represents the location of the Py dot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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T. Shapoval et al. / Physica C 470 (2010) 867–870
h2 = 90 nm
(a)
(b)
h2 = 90 nm, 2nd scan
repulsive
1 m
1 m
f = 0.7 Hz attractive
h2 = 75 nm
(c)
(d)
h2 = 60 nm
repulsive
1 m
1 m
f = 1.0 Hz attractive
(e)
MFM tip h1
m
z
h2
Py dot
Si (100) substrate
1 m 100 nm
25 nm
Nb film
x
2 m
Fig. 3. (a)–(d) Distribution of the SC vortices in a Nb film in the presence of the Py dots in the magnetic-vortex state measured at T = 6.1 K and the second matching field. The SC vortices and the MFM tip exhibit a repulsive interaction. The distance between the tip and the surface (h2) decreases from panel (a), h2 ¼ 90 nm, to panel (d), h2 ¼ 60 nm. The arrows mark the depinned vortices. (e) Sketch of a cross-sectional cut of the FM/SC hybrid structure and the MFM tip scanning above the surface. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)
The effect of the cool-in field on the arrangement of the SC vortices has been studied in a previous paper [5]. In the following we concentrate on the vortex behavior for a frozen magnetic field equal to the second matching field. A first estimation of the pinning force is based on the fact that two SC vortices are situated close to each other on top of the Py dot rather than being organized in an ordered Abrikosov lattice. Consequently, the pinning force due to these artificial defects ðF p Þ is higher than the repulsive force between vortices F v—v . The repulsive force between two SC vortices in thin films with a thickness below the penetration depth k can be approximated in the Pearl model [13] as:
F v—v ¼
U20
pl0 a2
;
ð1Þ
where a is the distance between the vortices and U0 is the flux quantum [13]. For the second matching field the distance between two vortices in the Nb film pinned by the same Py dot was measured to be about a = 750 nm (Fig. 3a). Thus, the vortex–vortex repulsion force normalized by the Nb film thickness was estimated to be fv—v 19 0:3 pN=lm. Due to the long-range character of this interaction, the formation of SC vortex clusters in thin films is energetically unfavorable [13]. Consequently, the presence of a strong pinning potential is required to ensure the visualized distribution of SC vortices. As it was shown recently by Brandt [14], according to the finite-size of the film the real vortex–vortex interaction is weaker than the value calculated from the Pearl model for the infinite thin SC layer. While scanning with the MFM tip, an additional force that acts on the SC vortices arises. It is important to mention, that a noninvasive imaging of vortices by MFM is possible only if the vortices
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T. Shapoval et al. / Physica C 470 (2010) 867–870
are pinned. The tip–vortex interaction force can be accurately tuned during scanning by varying the tip-sample separation h [6]. If this force exceeds the pinning force of an individual vortex at a natural or artificial defect the vortex can be dragged from its position. This local tip–vortex interaction force can be estimated from the MFM scans using the monopole–monopole model [6]. From this model, the maximum lateral force that acts on the SC vortex from the MFM tip is:
maxðF lat Þ ¼ 0:38
~ U0 m ðh þ 1:27k þ dÞ2 ; 2p
ð2Þ
~ is the monopole moment per unit length of the tip and d is where m its position within the tip as sketched in Fig. 3e [6]. In order to probe this effect, MFM scans have been performed on the same sample position at different tip-sample distances in order to depin the SC vortices located on the Py dots and in the interstitial positions under the influence of the stray field of the tip (Fig. 3). As long as the tip-sample distance h2 (Fig. 3e) is larger than 90 nm, the vortices are not dragged by the tip. As soon as h2 reaches 90 nm (Fig. 3a), the interstitial vortex (marked by the arrow) is depinned and moved completely out of the scanned area. This is apparent from the second scan at the same distance (Fig. 3b), where this vortex between the Py dots is no longer visible. The pinning force normalized by the Nb film thickness for the interstitial SC vortex was estimated from Fig. 3a using the monopole–monopole model to be about 1.5 pN/lm. The presence of the 25 nm thick Py dots underneath the Nb film leads to a surface modulation of the SC film, as it is sketched in Fig. 3e. The AFM profile (image is not presented here) shows that the modulation h2 h1 30 nm is on the scale of the Py dot thickness. Consequently, the SC vortices imaged on top of the Py dots have a lower tip-sample distance ðh1 ¼ 60 nmÞ and experience a stronger lateral force from the MFM tip. Despite the decreased distance, the vortices on the Py dots are not dragged by the tip at h2 ¼ 90 nm and also at h2 ¼ 75 nm (Fig. 3c). Only when h2 decreases to 60 nm ðh1 ¼ 30 nmÞ and the lateral depinning force that acts additionally to the existing repulsive interaction, F v—v , reaches 2.3 pN/lm, the vortices on top of the Py dots also start to move (Fig. 3d). As a result, the total pinning force at the Py dots is estimated to be about 21 pN/ lm. This force is about 15 times stronger as compared to the pinning force in the pure Nb film estimated above. 4. Conclusions In summary, in the present study we have locally probed the modulation of the pinning force that acts on the superconducting
vortices in the vicinity of the inhomogeneous pinning landscape created by an ordered array of ferromagnetic dots in the magnetic-vortex state. The corresponding pinning forces were estimated from the dragging force that the magnetic tip exerts on the superconducting vortex during scanning. Our magnetic force microscopic imaging reveals that the superconducting vortices are preferably located at the edge of the Py dots, undergoing an about 15 times stronger pinning force at the Py dots compared to the pure Nb film. This pinning force overcomes the repulsive interaction between the superconducting vortices, leading to the formation of the vortex clusters on the top of the magnetic dots. Acknowledgements The work was partially funded by the European Community under the 6th Framework Programme Contract Number 516858: HIPERCHEM and by the U.S. NSF, Grant ECCS-0823813. T. Shapoval thanks the ESF for financing a conference visit. The authors acknowledge discussions with E. Reich and S. Haindl, thank D.S. Inosov and R. Schäfer for critical reading of the manuscript and U. Wolff for MFM tip preparation. References [1] M. Lange et al., Phys. Rev. Lett. 90 (2003) 197006. [2] A.Yu. Aladyshkin et al., Supercond. Sci. Technol. 22 (2009) 053001. [3] J.I. Martı´n et al., Phys. Rev. Lett. 79 (1997) 1929; D.J. Morgan et al., Phys. Rev. Lett. 80 (1998) 3614; M.J. Van Bael et al., Phys. Rev. Lett. 86 (2001) 155; J.E. Villegas et al., Phys. Rev. Lett. 99 (2007) 227001; J.E. Villegas et al., Science 302 (2003) 1188; Y. Jaccard et al., Phys. Rev. B 58 (1998) 8232; S. Erdin et al., Phys. Rev. 66 (2002) 014414; M.J. Van Bael et al., Phys. Rev. 68 (2003) 014509; A.V. Silhanek et al., Appl. Phys. Lett. 89 (2006) 182505; A.V. Silhanek et al., Appl. Phys. Lett. 90 (2007) 182501; S. Haindl et al., Supercond. Sci. Technol. 21 (2008) 045017. [4] A. Hoffmann et al., Phys. Rev. B. 77 (2008) 060506(R). [5] T. Shapoval et al., Phys. Rev. B 81 (2010) 092505. [6] E.W.J. Straver et al., Appl. Phys. Lett. 93 (2008) 172514; O.M. Auslaender et al., Nat. Phys. 5 (2009) 35. [7] A.A. Abrikosov, Sov. Phys. JEPT 5 (1957) 1174. [8] V.V. Moshchalkov et al., Phys. Rev. B 54 (1996) 7385. [9] T. Shapoval et al., Physica C 460–462 (2007) 732. [10] I. Horcas et al., Rev. Sci. Instrum. 78 (2007) 013705. [11] J.E. Villegas et al., Phys. Rev. B 77 (2008) 134510. [12] T.R. Albrecht et al., J. Appl. Phys. 69 (1991) 668. [13] J. Pearl, Appl. Phys. Lett. 5 (1964) 65. [14] E.H. Brandt, Phys. Rev. B. 79 (2009) 134526.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Enhanced pinning of superconducting vortices at circular magnetic dots in the magnetic-vortex state T. Shapoval a,*, V. Metlushko b, M. Wolf a, V. Neu a, B. Holzapfel a, L. Schultz a a b
IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany Department of Electrical and Computer Engineering, University of Illinois at Chicago, IL 60612, USA
a r t i c l e
i n f o
Article history: Available online 21 February 2010 Keywords: Vortex pinning Magnetic dots Superconducting thin films Magnetic force microscopy
a b s t r a c t Low-temperature magnetic force microscopy (LT-MFM) was used to study the distribution of superconducting vortices in Nb above a square array of 1 lm-sized circular ferromagnetic dots in a magnetic-vortex state. The force that the MFM tip exerts on the individual vortex in the depinning process was used to estimate the spatial modulation of the pinning potential. It was found, that the superconducting vortices which are preferably located on top of the Py dots experience a pinning force, about 15 times stronger as compared to the pinning force in the pure Nb film. This strong pinning exceeds the repulsive interaction between the superconducting vortices and allows vortex clusters to be located above the dots. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Low temperature type-II superconductors (LTSC) are usually preferable candidates for basic studies of the vortex matter in thin films. Due to their rather high homogeneity, these films allow to probe the effect of an artificially modified pinning landscape on the vortex arrangement. Therefore, during the last decades artificial defects have been incorporated into a superconducting matrix in different manners: randomly, periodically, as holes or inclusions and, recently, magnetic defects have been used, which cause a variety of exciting phenomena. One of them is an interplay of superconductivity and magnetism. In ferromagnetic/superconducting (FM/SC) hybrid structures it was demonstrated that the vicinity of FM materials, formerly expected to be destructive for the superconductivity, can even lead to field-induced superconductivity and strongly enhanced pinning of vortices [1–5]. This counterintuitive interplay of magnetism and superconductivity is today a topic of high scientific interest, involving many phenomena needed to be investigated and explained. Low-temperature magnetic force microscopy (LT-MFM) is a powerful method to probe the local spatial variation of the pinning landscape, as it allows to combine the non-invasive imaging of individual vortices with the direct manipulation and depinning of vortices from their position [6]. The presence of the artificial pinning sites prohibits the formation of an ordered Abrikosov lattice [7], and leads to the forced organization of the flux lines [8].
* Corresponding author. E-mail address: [email protected] (T. Shapoval). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.071
In this paper LT-MFM was used to locally probe the enhancement of the pinning force of the superconducting vortices in the vicinity of magnetic dots in the magnetic-vortex state. 2. Experimental We have studied pinning of superconducting vortices in the following FM/SC hybrid structure: a square array of permalloy (Py = Ni80Fe20) dots with 1 lm diameter, 25 nm height and 2 lm periodicity prepared on a Si (1 0 0) substrate using standard e-beam lithography, e-beam evaporation, and lift-off processes followed by the deposition of a 100 nm thick superconducting niobium (Nb) film ðT c ¼ 8:32 KÞ on top of the Py dot array by sputter deposition [4]. A SEM image of this Py/Nb hybrid structure is shown in Fig. 1 (inset). Local studies of vortex distribution are performed by low-temperature magnetic force microscopy using a commercial scanning probe microscope (Omicron Cryogenic SFM) [9]. The MFM cantilever (Nanoworld MFMR) possesses a force constant of about 2.8 N/m and a resonance frequency of about 80 kHz. Image processing was done using the WSxM 4.0 software [10]. 3. Results and discussion 3.1. Magnetic-vortex state of Py dots To reach the magnetic-vortex configuration in the Py array, the sample was cooled in the microscope to 40 K. An in-plane field of +100 mT was applied along the positive y direction ðHy Þ to ensure full saturation of the dots. Then the sample was cooled to 14.6 K.
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T. Shapoval et al. / Physica C 470 (2010) 867–870
Magnetic moment, m (10-7 A*m2)
2
T = 5 K Nucleation
(a) (b)
(a)
(a) (c)
1
y x
(e) (a)
0
1 m (d) (a)
-1
Annihilation
-2 -100
-50
0
50
100
Applied in-plane field, Hy (mT) Fig. 1. Magnetic hysteresis loop, showing magnetic vortex behavior with vortex nucleation and annihilation fields. Within the inner loop, between 30 mT and +30 mT, the magnetization is reversible (vortex branch). The inset shows a SEM image of Py dots covered by a 100-nm thick Nb layer. The marked points on the loop correspond to the panels (a)–(e) in Fig. 2, respectively.
The MFM scan at a tip-sample distance of 75 nm shows four saturated Py dots at Hy ¼ þ100 mT (Fig. 2a). While decreasing the field the saturated state is maintained (Fig. 2b) until vortex nucleation occurs close to zero field (Fig. 2c). Decreasing the field to 25 mT ensures that in most of the dots a magnetic vortex has been nucleated which is located at the rim of the element but is not yet annihilated, such that the dot is largely magnetized in negative ydirection (Fig. 2d). Going back to zero along the vortex branch brings the dots into the radial symmetric magnetic-vortex state 0 imaged in Fig. 2e and e .
Magnetic vortices, generated in such a way, will have random polarity (i.e. out-of-plane magnetization components pointing randomly up or down) [11]. To set a defined polarity, a small positive out-of-plane field (+10 mT) was applied to the sample during the above-described field sequence. Thus, the magnetic-vortex core and the MFM tip, which is magnetized in positive z-direction, experience an attractive interaction that shows up as a dark con0 trast in the center of the dot (Fig. 2e ). The measured shift of the resonant frequency Df is proportional to the second derivative of the z component of the magnetic stray field of the sample at the set scanning distance from the surface [12]. In the magnetic-vortex state the overall stray field above the dot is strongly reduced. As a result, the signal measured by MFM for the magnetic-vortex state is almost 10 times less than for the saturated state (compare the scale bars 0 in Fig. 2e and e ). At the same time a strong enhancement of the out-of-plane stray field component is expected at the center of the dot. Due to the fact that the 100 nm thick Nb film covers the Py array, imaging took place far away (about 175 nm) from the surface of the Py dot. Thus, the magnetic-vortex core is visualized by MFM as a largely broadened, dark blue (attractive interaction) contrast. 3.2. Estimation of the local pinning force After reaching the magnetic-vortex state of the Py dots, as described above, the sample was cooled to a temperature below T c of the Nb film ðT ¼ 6:1 K ¼ 72%T c Þ in a perpendicular field Hz ¼ 1 mT, that is close to the second matching field for this hybrid structure [8]. Fig. 3a shows the MFM images of the distribution of the SC vortices in the vicinity of four magnetic dots: three of them being in the magnetic vortex state, the forth (lower right) maintained an in-plane domain configuration. One can clearly see two superconducting vortices per magnetic dot in the upper two elements (yellow contrast).
Fig. 2. To reach the magnetic-vortex state the in-plane field Hy was varied along the hysteresis loop starting from saturation at (a) 100 mT, through (b) 25 mT, (c) 0, and (d) 0 applying a negative field less than the magnetic-vortex annihilation field (25 mT) to the magnetic-vortex state at 0 mT (e) and (e ). Color bars give the measured Df signal. A small out-of-plane field of 10 mT was permanently applied to set the polarity of the magnetic vortex (attractive interaction). The scanning area was 4 lm 4 lm, the scanning distance 75 nm, T = 14.6 K. The white circle represents the location of the Py dot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
869
T. Shapoval et al. / Physica C 470 (2010) 867–870
h2 = 90 nm
(a)
(b)
h2 = 90 nm, 2nd scan
repulsive
1 m
1 m
f = 0.7 Hz attractive
h2 = 75 nm
(c)
(d)
h2 = 60 nm
repulsive
1 m
1 m
f = 1.0 Hz attractive
(e)
MFM tip h1
m
z
h2
Py dot
Si (100) substrate
1 m 100 nm
25 nm
Nb film
x
2 m
Fig. 3. (a)–(d) Distribution of the SC vortices in a Nb film in the presence of the Py dots in the magnetic-vortex state measured at T = 6.1 K and the second matching field. The SC vortices and the MFM tip exhibit a repulsive interaction. The distance between the tip and the surface (h2) decreases from panel (a), h2 ¼ 90 nm, to panel (d), h2 ¼ 60 nm. The arrows mark the depinned vortices. (e) Sketch of a cross-sectional cut of the FM/SC hybrid structure and the MFM tip scanning above the surface. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)
The effect of the cool-in field on the arrangement of the SC vortices has been studied in a previous paper [5]. In the following we concentrate on the vortex behavior for a frozen magnetic field equal to the second matching field. A first estimation of the pinning force is based on the fact that two SC vortices are situated close to each other on top of the Py dot rather than being organized in an ordered Abrikosov lattice. Consequently, the pinning force due to these artificial defects ðF p Þ is higher than the repulsive force between vortices F v—v . The repulsive force between two SC vortices in thin films with a thickness below the penetration depth k can be approximated in the Pearl model [13] as:
F v—v ¼
U20
pl0 a2
;
ð1Þ
where a is the distance between the vortices and U0 is the flux quantum [13]. For the second matching field the distance between two vortices in the Nb film pinned by the same Py dot was measured to be about a = 750 nm (Fig. 3a). Thus, the vortex–vortex repulsion force normalized by the Nb film thickness was estimated to be fv—v 19 0:3 pN=lm. Due to the long-range character of this interaction, the formation of SC vortex clusters in thin films is energetically unfavorable [13]. Consequently, the presence of a strong pinning potential is required to ensure the visualized distribution of SC vortices. As it was shown recently by Brandt [14], according to the finite-size of the film the real vortex–vortex interaction is weaker than the value calculated from the Pearl model for the infinite thin SC layer. While scanning with the MFM tip, an additional force that acts on the SC vortices arises. It is important to mention, that a noninvasive imaging of vortices by MFM is possible only if the vortices
870
T. Shapoval et al. / Physica C 470 (2010) 867–870
are pinned. The tip–vortex interaction force can be accurately tuned during scanning by varying the tip-sample separation h [6]. If this force exceeds the pinning force of an individual vortex at a natural or artificial defect the vortex can be dragged from its position. This local tip–vortex interaction force can be estimated from the MFM scans using the monopole–monopole model [6]. From this model, the maximum lateral force that acts on the SC vortex from the MFM tip is:
maxðF lat Þ ¼ 0:38
~ U0 m ðh þ 1:27k þ dÞ2 ; 2p
ð2Þ
~ is the monopole moment per unit length of the tip and d is where m its position within the tip as sketched in Fig. 3e [6]. In order to probe this effect, MFM scans have been performed on the same sample position at different tip-sample distances in order to depin the SC vortices located on the Py dots and in the interstitial positions under the influence of the stray field of the tip (Fig. 3). As long as the tip-sample distance h2 (Fig. 3e) is larger than 90 nm, the vortices are not dragged by the tip. As soon as h2 reaches 90 nm (Fig. 3a), the interstitial vortex (marked by the arrow) is depinned and moved completely out of the scanned area. This is apparent from the second scan at the same distance (Fig. 3b), where this vortex between the Py dots is no longer visible. The pinning force normalized by the Nb film thickness for the interstitial SC vortex was estimated from Fig. 3a using the monopole–monopole model to be about 1.5 pN/lm. The presence of the 25 nm thick Py dots underneath the Nb film leads to a surface modulation of the SC film, as it is sketched in Fig. 3e. The AFM profile (image is not presented here) shows that the modulation h2 h1 30 nm is on the scale of the Py dot thickness. Consequently, the SC vortices imaged on top of the Py dots have a lower tip-sample distance ðh1 ¼ 60 nmÞ and experience a stronger lateral force from the MFM tip. Despite the decreased distance, the vortices on the Py dots are not dragged by the tip at h2 ¼ 90 nm and also at h2 ¼ 75 nm (Fig. 3c). Only when h2 decreases to 60 nm ðh1 ¼ 30 nmÞ and the lateral depinning force that acts additionally to the existing repulsive interaction, F v—v , reaches 2.3 pN/lm, the vortices on top of the Py dots also start to move (Fig. 3d). As a result, the total pinning force at the Py dots is estimated to be about 21 pN/ lm. This force is about 15 times stronger as compared to the pinning force in the pure Nb film estimated above. 4. Conclusions In summary, in the present study we have locally probed the modulation of the pinning force that acts on the superconducting
vortices in the vicinity of the inhomogeneous pinning landscape created by an ordered array of ferromagnetic dots in the magnetic-vortex state. The corresponding pinning forces were estimated from the dragging force that the magnetic tip exerts on the superconducting vortex during scanning. Our magnetic force microscopic imaging reveals that the superconducting vortices are preferably located at the edge of the Py dots, undergoing an about 15 times stronger pinning force at the Py dots compared to the pure Nb film. This pinning force overcomes the repulsive interaction between the superconducting vortices, leading to the formation of the vortex clusters on the top of the magnetic dots. Acknowledgements The work was partially funded by the European Community under the 6th Framework Programme Contract Number 516858: HIPERCHEM and by the U.S. NSF, Grant ECCS-0823813. T. Shapoval thanks the ESF for financing a conference visit. The authors acknowledge discussions with E. Reich and S. Haindl, thank D.S. Inosov and R. Schäfer for critical reading of the manuscript and U. Wolff for MFM tip preparation. References [1] M. Lange et al., Phys. Rev. Lett. 90 (2003) 197006. [2] A.Yu. Aladyshkin et al., Supercond. Sci. Technol. 22 (2009) 053001. [3] J.I. Martı´n et al., Phys. Rev. Lett. 79 (1997) 1929; D.J. Morgan et al., Phys. Rev. Lett. 80 (1998) 3614; M.J. Van Bael et al., Phys. Rev. Lett. 86 (2001) 155; J.E. Villegas et al., Phys. Rev. Lett. 99 (2007) 227001; J.E. Villegas et al., Science 302 (2003) 1188; Y. Jaccard et al., Phys. Rev. B 58 (1998) 8232; S. Erdin et al., Phys. Rev. 66 (2002) 014414; M.J. Van Bael et al., Phys. Rev. 68 (2003) 014509; A.V. Silhanek et al., Appl. Phys. Lett. 89 (2006) 182505; A.V. Silhanek et al., Appl. Phys. Lett. 90 (2007) 182501; S. Haindl et al., Supercond. Sci. Technol. 21 (2008) 045017. [4] A. Hoffmann et al., Phys. Rev. B. 77 (2008) 060506(R). [5] T. Shapoval et al., Phys. Rev. B 81 (2010) 092505. [6] E.W.J. Straver et al., Appl. Phys. Lett. 93 (2008) 172514; O.M. Auslaender et al., Nat. Phys. 5 (2009) 35. [7] A.A. Abrikosov, Sov. Phys. JEPT 5 (1957) 1174. [8] V.V. Moshchalkov et al., Phys. Rev. B 54 (1996) 7385. [9] T. Shapoval et al., Physica C 460–462 (2007) 732. [10] I. Horcas et al., Rev. Sci. Instrum. 78 (2007) 013705. [11] J.E. Villegas et al., Phys. Rev. B 77 (2008) 134510. [12] T.R. Albrecht et al., J. Appl. Phys. 69 (1991) 668. [13] J. Pearl, Appl. Phys. Lett. 5 (1964) 65. [14] E.H. Brandt, Phys. Rev. B. 79 (2009) 134526.