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Anisotropic vortex pinning by a square array of rectangular antidots
Physica C 369 (2002) 113–117 www.elsevier.com/locate/physc
Anisotropic vortex pinning by a square array of rectangular antidots L. Van Look a,*, B.Y. Zhu a, R. Jonckheere b, V.V. Moshchalkov a a
Laboratorium voor Vaste-Stoffysica en Magnetisme, K.U. Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium b Inter-University Micro-Electronics Center (Imec vzw), Kapeldreef 75, B-3001 Leuven, Belgium
Abstract Using transport measurements, we investigate vortex pinning in thin superconducting films with a square array of rectangular antidots. In these films, we observe a pronounced anisotropy in the pinning properties. More specifically, the critical current vs. field curve, Ic ðH Þ, and the V ðIÞ curves depend strongly on the current direction. We demonstrate that this anisotropy is caused by an anisotropy in the vortex–vortex interaction, rather than in the single site pinning force, in agreement with the molecular dynamics simulations we have performed. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.76.Db; 74.60.Ge; 74.60.Jg; 74.25.Fy Keywords: Periodic pinning array; Anisotropic critical current density
1. Introduction Type-II superconductors (SCs) with a periodic pinning array (submicron holes or antidots, or magnetic dots) are good candidates to study the fundamentals of vortex pinning. In these systems, matching effects [1,2] occur at those magnetic fields generating a vortex density which ‘‘matches’’ the density of pinning sites. At these matching fields, the vortices form regular geometrical patterns, commensurate with the pinning array. This strongly reduces the vortex mobility and increases the critical current Ic .
*
Corresponding author. E-mail address: [email protected] (L. Van Look).
In this work, we study the pinning properties of a SC film with a square array of rectangular antidots by means of electrical transport measurements ðIc ðH Þ and V ðIÞÞ for different current directions. Molecular dynamics simulations are used to gain more insight in the mechanisms responsible for the observed anisotropy.
2. Experimental details ) film in a 5 5 We patterned the SC Pb(1500 A 2 mm cross-shaped geometry (see Fig. 1). The central part of the cross consists of two 300 lm wide strips containing the square array of rectangular antidots (dark gray area in Fig. 1(a)). A more detailed description of the sample preparation and characterization can be found in Ref. [3].
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 2 9 - 1
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Fig. 1. (a) Cross-shaped geometry of the sample, (b) schematic representation of a unit cell (1:5 1:5 lm2 ) of the antidot array and (c) atomic force micrograph of a 6 6 lm2 area of the antidot lattice.
Transport measurements are performed with the magnetic field perpendicular to the film surface.
Fig. 2. Critical current vs. magnetic field at T =Tc ¼ 0:995 measured with a current in the x- (open symbols) and the y-direction (filled symbols), respectively (Vc ¼ 100Rn lV/X).
the V ðIÞ transitions are very sharp, and are independent of the direction of the current. For H ¼ 0:4H1 , a large tail in the V ðIÞ curve is present for Ikx (open symbols), leading to a much broader
3. Results In Fig. 2, we show the critical current vs. field curves Ic ðH Þ, normalized to Ic0 Ic ðH ¼ 0Þ, at T =Tc ¼ 0:995. The field axis is given in units of the first matching field l0 H1 ¼ U0 =d 2 ¼ 9:2 Oe, with d ¼ 1:5 lm the period of the antidot lattice and U0 ¼ h=2e the superconducting flux quantum. At H1 , both Icx ðH Þ and Icy ðH Þ curves show a pronounced maximum up to the same normalized critical current value. In the field ranges between the integer matching fields, the critical current Icy ðH Þ is considerably enhanced compared to Icx ðH Þ. This increase is accompanied by the complete suppression of the rational matching peaks at Hp=q (p and q integers). For Icx ðH Þ, on the other hand, the rational matching peaks at H1=2 , H1=3 , H2=3 are clearly revealed. To investigate the origin of this qualitative difference between Icx ðH Þ and Icy ðH Þ, we have a closer look at the V ðIx Þ and V ðIy Þ curves for some selected field values (Fig. 3). For H ¼ 0 and H ¼ H1 ,
Fig. 3. Experimental V ðIÞ curves at selected magnetic field values at T =Tc ¼ 0:995. The full lines (open symbols) are the data for Iky (Ikx). The horizontal dotted line indicates the voltage criterion Vc used to define the critical current.
L. Van Look et al. / Physica C 369 (2002) 113–117
V ðIÞ transition for Ikx than for Iky. This behavior is typical for every magnetic field in between the matching fields. Summarizing the experimental observations, we have found a strong anisotropy in the critical current Ic ðH Þ and the V ðIÞ characteristics of the film with a square array of rectangular antidots. At the integer matching field H1 , both the critical current Ic ðH1 Þ and the V ðI; H1 Þ curve do not depend on the direction of the applied current––Ikx or Iky. However, in between the integer matching fields, we find broad V ðIx Þ transitions, a low Icx , and rational matching features in Icx ðH Þ for Ikx. For Iky, on the other hand, we see sharp V ðIy Þ transitions at every magnetic field, an overall high Icy , and no sign of rational matching features in Icy ðH Þ.
4. Discussion Neglecting the thermal noise force, the velocity of a vortex in a periodic pinning array (PPA) is determined by the superposition of three forces: the vortex–vortex interaction FVV , the vortex– antidot pinning force FP and the Lorentz force FL , directed perpendicular to the applied current. Precisely at the first matching field, all vortex– vortex interactions between the vortices trapped in the pinning sites cancel out. Therefore, at this field, the Lorentz force which is needed to create a nonzero vortex velocity, is only determined by the pinning force. This implies that the critical current at the first matching field is a measure of the single site pinning force of the individual antidots. Once the Lorentz force exceeds the pinning force, all vortices leave their pinning site simultaneously, resulting in a sharp V ðIÞ transition [4]. Since the Icx ðH1 Þ and Icy ðH1 Þ values and the V ðIx ; H1 Þ and V ðIy ; H1 Þ curves (Figs. 2 and 3) coincide, we conclude that the pinning force exerted by a rectangular antidot on a single pinned U0 vortex is isotropic along the two symmetry-axes of the rectangular antidot. This experimental observation is in agreement with predictions by Buzdin and Daumens [5] for an elliptic antidot, where the pinning force is only expected to be anisotropic when the ellipse is extremely elongated.
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The observed anisotropy in pinning properties can therefore not be attributed to an anisotropic single site pinning force of the rectangular antidots themselves. Instead, it should be associated with an anisotropic vortex–vortex interaction. Indeed, due to a smaller width of the SC strands between the antidots in the y-direction, we expect the y-component of the vortex–vortex interaction vectors to be considerably larger than their x-component. To understand how the anisotropic vortex– vortex interaction can give rise to the observed anisotropy in the pinning properties in between the matching fields, we examine the role of the vortex– vortex interaction in the depinning process of vortices in a regular pinning array. We have discussed earlier that at a matching field, the vortex–vortex interaction vectors are canceled out and depinning is governed by the single site pinning force. When the applied field is now slightly detuned from a matching field (integer or rational), the vortex lattice contains defects due to the incommensurability of the vortex array and the PPA. The vortex rows, parallel to FL , without such defects are called ‘‘commensurate’’ rows. In these commensurate vortex rows, the component of FVV parallel to FL will be zero for all vortices in the row. On the other hand, in vortex rows containing defects, the vortex–vortex interaction forces will not cancel out at all (FVV 6¼ 0). This FVV component in the direction of the Lorentz force FL will assist to the depinning of these ‘‘incommensurate’’ rows, i.e. depinning occurs when the sum of this FVV component and FL overcomes the single site pinning force. The commensurate rows stay pinned up to a higher Lorentz force FL [6]. If the vortex–vortex interaction is sufficiently large, the critical current Ic ðH Þ will therefore be substantially lower right before and after the matching fields, leading to pronounced matching peaks (integer and rational). When the current is applied in the x-direction (FL ky), the Icx ðH Þ curve shows pronounced maxima at the rational matching fields H1=2 , H3=2 , H1=3 and H2=3 and a much lower value in between these rational matching fields (see Fig. 2). This is a clear sign of a strong vortex–vortex interaction FVV component along the direction of the Lorentz force FL , in this case the y-direction. When a
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current Iy ky is applied (FL kx), we see a complete disappearance of the rational matching peaks and, instead, an overall high critical current Icy ðH Þ (Fig. 2). This indicates that the FVV component along the direction of the Lorentz force (in this case the x-direction) is much weaker than in the case Ix kx. The vortex–vortex interaction is apparently not able to provide the long-range order needed to generate the sparse geometrical patterns at the rational matching fields Hp=q . This implies that no commensurate rows in the direction of FL are formed, and all vortex rows are qualitatively equal (all are ‘‘incommensurate’’). All rows will depin approximately at the same driving current. This explains the sharp V ðIÞ transitions for all magnetic fields. The integer matching peaks are still present in Icy ðH Þ, since the commensurate vortex patterns at the integer matching fields Hn can be achieved even with a small interaction force present, due to a smaller vortex separation.
5. Molecular dynamics simulations Confirmation of the above-mentioned explanation can be found in the molecular dynamics simulations we have performed for square arrays (period d) of square pinning sites (side 0.4d), but with an anisotropic vortex–vortex interaction. The simulation model therefore contains the same main features as the experimental system: (i) an isotropic single site pinning force and (ii) an anisotropic vortex–vortex interaction. The overdamped equation of motion is used to calculate the average vortex velocity as a function of the applied Lorentz force FL , and to trace the vortex trajectories in the pinning array. The rectangular antidots are modeled as square potential wells with a Gaussian attractive pinning force at the edges but no pinning force at the interior of the pinning well. The anisotropic repulsive vortex– vortex interaction force is modeled by using an anisotropic penetration depth ky ¼ 2kx ¼ d. For more details about the calculation method, we refer to Refs. [3,7]. The obtained average vortex velocity as a function of the applied Lorentz force for H =H1 ¼ 0:58, i.e. a magnetic field in between matching fields, is
Fig. 4. Average vortex velocity as a function of the Lorentz force FL =FL0 at H =H1 ¼ 0:58 (FL0 is the critical depinning force at zero field). The open (filled) symbols show the result for a current along the x- (y-)axis.
shown in Fig. 4. We will compare these calculated hviðFL Þ curves at H =H1 ¼ 0:58 with the experimental V ðIÞ curves taken at H =H1 ¼ 0:4 (Fig. 3), which are typical for all magnetic fields in between matching fields. The two main features observed in the experimental curves are qualitatively reproduced. First, the onset of the vortex motion occurs at a lower driving force for Ikx ðFL kyÞ (open symbols) than for Iky ðFL kxÞ (filled symbols). Moreover, the V ðIÞ transition is clearly much broader for Ikx than for Ikx. Calculations of the vortex trajectories have shown [3] that the cause of the tail in the hviðFL Þ curve for FL ky lies in a partial vortex depinning, where the vortex rows with little order in the ydirection are the first ones to start moving. The commensurate rows remain pinned in this process. When the Lorentz force is increased further, the commensurate rows participate in the vortex motion. Due to the small vortex–vortex interaction in the x-direction, the vortex rows initiate their motion at the same time for FL kx. This corresponds to the sharp increase in the hviðFL Þ curve (filled symbols in Fig. 4). The results of these molecular dynamics simulations show that an array of isotropic pinning sites, combined with an anisotropic vortex–vortex interaction along the two principal axes of the PPA, can give rise to the same phenomena as observed in our pinning experiments on a square
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array of rectangular antidots. The low critical current and broad V ðIÞ transitions in between matching fields for Ikx (FL ky) are found to be due to the motion of the incommensurate vortex rows along the direction of the strong vortex–vortex interaction, in this case the y-direction. This result strengthens our interpretation that in our experiment, the rectangular antidots provide an isotropic pinning potential but induce an anisotropy in the vortex–vortex interaction.
6. Conclusion We have measured Ic ðH Þ and V ðI; H Þ curves of a SC film with a square array of rectangular antidots for two directions of the applied current. We find an overall high Ic ðH Þ with integer matching peaks but no rational matching features when the current is applied parallel to the long side of the antidots. This is a clear advantage compared to isotropic pinning arrays, where a large suppression of Ic ðH Þ is seen in between the rational matching fields. We attribute this effect to the anisotropic vortex–vortex interaction, which is stronger along the long side of the antidots. Molecular dynamics simulations on a square array of isotropic pinning sites, combined with an anisotropic vortex–vortex
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interaction, indeed show the same characteristic anisotropic features as observed in our measurements.
Acknowledgements This work was supported by the ESF ‘‘Vortex’’ Program, the Belgian Interuniversity Attraction Poles (IUAP), and the Flemish GOA and FWO Programs. We wish to thank M.J. Van Bael, Y. Bruynseraede, C. Reichhardt and C.J. Olson for helpful discussions.
References [1] M. Baert, V.V. Metlushko, R. Jonckheere, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. Lett. 74 (1995) 3269. [2] M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 59 (1999) 14674. [3] L. Van Look, B.Y. Zhu, R. Jonckheere, V.V. Moshchalkov, Phys. Rev. B, submitted for publication. [4] C. Reichhardt, N. Grønbech-Jensen, Phys. Rev. B 63 (2001) 54510. [5] A. Buzdin, M. Daumens, Physica C 294 (1998) 257. [6] C. Reichhardt, C.J. Olson, F. Nori, Phys. Rev. Lett. 78 (1997) 2648. [7] B.Y. Zhu, L. Van Look, V.V. Moshchalkov, B.R. Zhao, Z.X. Zhao, Phys. Rev. B 64 (2001) 12504.
Anisotropic vortex pinning by a square array of rectangular antidots L. Van Look a,*, B.Y. Zhu a, R. Jonckheere b, V.V. Moshchalkov a a
Laboratorium voor Vaste-Stoffysica en Magnetisme, K.U. Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium b Inter-University Micro-Electronics Center (Imec vzw), Kapeldreef 75, B-3001 Leuven, Belgium
Abstract Using transport measurements, we investigate vortex pinning in thin superconducting films with a square array of rectangular antidots. In these films, we observe a pronounced anisotropy in the pinning properties. More specifically, the critical current vs. field curve, Ic ðH Þ, and the V ðIÞ curves depend strongly on the current direction. We demonstrate that this anisotropy is caused by an anisotropy in the vortex–vortex interaction, rather than in the single site pinning force, in agreement with the molecular dynamics simulations we have performed. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.76.Db; 74.60.Ge; 74.60.Jg; 74.25.Fy Keywords: Periodic pinning array; Anisotropic critical current density
1. Introduction Type-II superconductors (SCs) with a periodic pinning array (submicron holes or antidots, or magnetic dots) are good candidates to study the fundamentals of vortex pinning. In these systems, matching effects [1,2] occur at those magnetic fields generating a vortex density which ‘‘matches’’ the density of pinning sites. At these matching fields, the vortices form regular geometrical patterns, commensurate with the pinning array. This strongly reduces the vortex mobility and increases the critical current Ic .
*
Corresponding author. E-mail address: [email protected] (L. Van Look).
In this work, we study the pinning properties of a SC film with a square array of rectangular antidots by means of electrical transport measurements ðIc ðH Þ and V ðIÞÞ for different current directions. Molecular dynamics simulations are used to gain more insight in the mechanisms responsible for the observed anisotropy.
2. Experimental details ) film in a 5 5 We patterned the SC Pb(1500 A 2 mm cross-shaped geometry (see Fig. 1). The central part of the cross consists of two 300 lm wide strips containing the square array of rectangular antidots (dark gray area in Fig. 1(a)). A more detailed description of the sample preparation and characterization can be found in Ref. [3].
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 2 9 - 1
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Fig. 1. (a) Cross-shaped geometry of the sample, (b) schematic representation of a unit cell (1:5 1:5 lm2 ) of the antidot array and (c) atomic force micrograph of a 6 6 lm2 area of the antidot lattice.
Transport measurements are performed with the magnetic field perpendicular to the film surface.
Fig. 2. Critical current vs. magnetic field at T =Tc ¼ 0:995 measured with a current in the x- (open symbols) and the y-direction (filled symbols), respectively (Vc ¼ 100Rn lV/X).
the V ðIÞ transitions are very sharp, and are independent of the direction of the current. For H ¼ 0:4H1 , a large tail in the V ðIÞ curve is present for Ikx (open symbols), leading to a much broader
3. Results In Fig. 2, we show the critical current vs. field curves Ic ðH Þ, normalized to Ic0 Ic ðH ¼ 0Þ, at T =Tc ¼ 0:995. The field axis is given in units of the first matching field l0 H1 ¼ U0 =d 2 ¼ 9:2 Oe, with d ¼ 1:5 lm the period of the antidot lattice and U0 ¼ h=2e the superconducting flux quantum. At H1 , both Icx ðH Þ and Icy ðH Þ curves show a pronounced maximum up to the same normalized critical current value. In the field ranges between the integer matching fields, the critical current Icy ðH Þ is considerably enhanced compared to Icx ðH Þ. This increase is accompanied by the complete suppression of the rational matching peaks at Hp=q (p and q integers). For Icx ðH Þ, on the other hand, the rational matching peaks at H1=2 , H1=3 , H2=3 are clearly revealed. To investigate the origin of this qualitative difference between Icx ðH Þ and Icy ðH Þ, we have a closer look at the V ðIx Þ and V ðIy Þ curves for some selected field values (Fig. 3). For H ¼ 0 and H ¼ H1 ,
Fig. 3. Experimental V ðIÞ curves at selected magnetic field values at T =Tc ¼ 0:995. The full lines (open symbols) are the data for Iky (Ikx). The horizontal dotted line indicates the voltage criterion Vc used to define the critical current.
L. Van Look et al. / Physica C 369 (2002) 113–117
V ðIÞ transition for Ikx than for Iky. This behavior is typical for every magnetic field in between the matching fields. Summarizing the experimental observations, we have found a strong anisotropy in the critical current Ic ðH Þ and the V ðIÞ characteristics of the film with a square array of rectangular antidots. At the integer matching field H1 , both the critical current Ic ðH1 Þ and the V ðI; H1 Þ curve do not depend on the direction of the applied current––Ikx or Iky. However, in between the integer matching fields, we find broad V ðIx Þ transitions, a low Icx , and rational matching features in Icx ðH Þ for Ikx. For Iky, on the other hand, we see sharp V ðIy Þ transitions at every magnetic field, an overall high Icy , and no sign of rational matching features in Icy ðH Þ.
4. Discussion Neglecting the thermal noise force, the velocity of a vortex in a periodic pinning array (PPA) is determined by the superposition of three forces: the vortex–vortex interaction FVV , the vortex– antidot pinning force FP and the Lorentz force FL , directed perpendicular to the applied current. Precisely at the first matching field, all vortex– vortex interactions between the vortices trapped in the pinning sites cancel out. Therefore, at this field, the Lorentz force which is needed to create a nonzero vortex velocity, is only determined by the pinning force. This implies that the critical current at the first matching field is a measure of the single site pinning force of the individual antidots. Once the Lorentz force exceeds the pinning force, all vortices leave their pinning site simultaneously, resulting in a sharp V ðIÞ transition [4]. Since the Icx ðH1 Þ and Icy ðH1 Þ values and the V ðIx ; H1 Þ and V ðIy ; H1 Þ curves (Figs. 2 and 3) coincide, we conclude that the pinning force exerted by a rectangular antidot on a single pinned U0 vortex is isotropic along the two symmetry-axes of the rectangular antidot. This experimental observation is in agreement with predictions by Buzdin and Daumens [5] for an elliptic antidot, where the pinning force is only expected to be anisotropic when the ellipse is extremely elongated.
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The observed anisotropy in pinning properties can therefore not be attributed to an anisotropic single site pinning force of the rectangular antidots themselves. Instead, it should be associated with an anisotropic vortex–vortex interaction. Indeed, due to a smaller width of the SC strands between the antidots in the y-direction, we expect the y-component of the vortex–vortex interaction vectors to be considerably larger than their x-component. To understand how the anisotropic vortex– vortex interaction can give rise to the observed anisotropy in the pinning properties in between the matching fields, we examine the role of the vortex– vortex interaction in the depinning process of vortices in a regular pinning array. We have discussed earlier that at a matching field, the vortex–vortex interaction vectors are canceled out and depinning is governed by the single site pinning force. When the applied field is now slightly detuned from a matching field (integer or rational), the vortex lattice contains defects due to the incommensurability of the vortex array and the PPA. The vortex rows, parallel to FL , without such defects are called ‘‘commensurate’’ rows. In these commensurate vortex rows, the component of FVV parallel to FL will be zero for all vortices in the row. On the other hand, in vortex rows containing defects, the vortex–vortex interaction forces will not cancel out at all (FVV 6¼ 0). This FVV component in the direction of the Lorentz force FL will assist to the depinning of these ‘‘incommensurate’’ rows, i.e. depinning occurs when the sum of this FVV component and FL overcomes the single site pinning force. The commensurate rows stay pinned up to a higher Lorentz force FL [6]. If the vortex–vortex interaction is sufficiently large, the critical current Ic ðH Þ will therefore be substantially lower right before and after the matching fields, leading to pronounced matching peaks (integer and rational). When the current is applied in the x-direction (FL ky), the Icx ðH Þ curve shows pronounced maxima at the rational matching fields H1=2 , H3=2 , H1=3 and H2=3 and a much lower value in between these rational matching fields (see Fig. 2). This is a clear sign of a strong vortex–vortex interaction FVV component along the direction of the Lorentz force FL , in this case the y-direction. When a
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current Iy ky is applied (FL kx), we see a complete disappearance of the rational matching peaks and, instead, an overall high critical current Icy ðH Þ (Fig. 2). This indicates that the FVV component along the direction of the Lorentz force (in this case the x-direction) is much weaker than in the case Ix kx. The vortex–vortex interaction is apparently not able to provide the long-range order needed to generate the sparse geometrical patterns at the rational matching fields Hp=q . This implies that no commensurate rows in the direction of FL are formed, and all vortex rows are qualitatively equal (all are ‘‘incommensurate’’). All rows will depin approximately at the same driving current. This explains the sharp V ðIÞ transitions for all magnetic fields. The integer matching peaks are still present in Icy ðH Þ, since the commensurate vortex patterns at the integer matching fields Hn can be achieved even with a small interaction force present, due to a smaller vortex separation.
5. Molecular dynamics simulations Confirmation of the above-mentioned explanation can be found in the molecular dynamics simulations we have performed for square arrays (period d) of square pinning sites (side 0.4d), but with an anisotropic vortex–vortex interaction. The simulation model therefore contains the same main features as the experimental system: (i) an isotropic single site pinning force and (ii) an anisotropic vortex–vortex interaction. The overdamped equation of motion is used to calculate the average vortex velocity as a function of the applied Lorentz force FL , and to trace the vortex trajectories in the pinning array. The rectangular antidots are modeled as square potential wells with a Gaussian attractive pinning force at the edges but no pinning force at the interior of the pinning well. The anisotropic repulsive vortex– vortex interaction force is modeled by using an anisotropic penetration depth ky ¼ 2kx ¼ d. For more details about the calculation method, we refer to Refs. [3,7]. The obtained average vortex velocity as a function of the applied Lorentz force for H =H1 ¼ 0:58, i.e. a magnetic field in between matching fields, is
Fig. 4. Average vortex velocity as a function of the Lorentz force FL =FL0 at H =H1 ¼ 0:58 (FL0 is the critical depinning force at zero field). The open (filled) symbols show the result for a current along the x- (y-)axis.
shown in Fig. 4. We will compare these calculated hviðFL Þ curves at H =H1 ¼ 0:58 with the experimental V ðIÞ curves taken at H =H1 ¼ 0:4 (Fig. 3), which are typical for all magnetic fields in between matching fields. The two main features observed in the experimental curves are qualitatively reproduced. First, the onset of the vortex motion occurs at a lower driving force for Ikx ðFL kyÞ (open symbols) than for Iky ðFL kxÞ (filled symbols). Moreover, the V ðIÞ transition is clearly much broader for Ikx than for Ikx. Calculations of the vortex trajectories have shown [3] that the cause of the tail in the hviðFL Þ curve for FL ky lies in a partial vortex depinning, where the vortex rows with little order in the ydirection are the first ones to start moving. The commensurate rows remain pinned in this process. When the Lorentz force is increased further, the commensurate rows participate in the vortex motion. Due to the small vortex–vortex interaction in the x-direction, the vortex rows initiate their motion at the same time for FL kx. This corresponds to the sharp increase in the hviðFL Þ curve (filled symbols in Fig. 4). The results of these molecular dynamics simulations show that an array of isotropic pinning sites, combined with an anisotropic vortex–vortex interaction along the two principal axes of the PPA, can give rise to the same phenomena as observed in our pinning experiments on a square
L. Van Look et al. / Physica C 369 (2002) 113–117
array of rectangular antidots. The low critical current and broad V ðIÞ transitions in between matching fields for Ikx (FL ky) are found to be due to the motion of the incommensurate vortex rows along the direction of the strong vortex–vortex interaction, in this case the y-direction. This result strengthens our interpretation that in our experiment, the rectangular antidots provide an isotropic pinning potential but induce an anisotropy in the vortex–vortex interaction.
6. Conclusion We have measured Ic ðH Þ and V ðI; H Þ curves of a SC film with a square array of rectangular antidots for two directions of the applied current. We find an overall high Ic ðH Þ with integer matching peaks but no rational matching features when the current is applied parallel to the long side of the antidots. This is a clear advantage compared to isotropic pinning arrays, where a large suppression of Ic ðH Þ is seen in between the rational matching fields. We attribute this effect to the anisotropic vortex–vortex interaction, which is stronger along the long side of the antidots. Molecular dynamics simulations on a square array of isotropic pinning sites, combined with an anisotropic vortex–vortex
117
interaction, indeed show the same characteristic anisotropic features as observed in our measurements.
Acknowledgements This work was supported by the ESF ‘‘Vortex’’ Program, the Belgian Interuniversity Attraction Poles (IUAP), and the Flemish GOA and FWO Programs. We wish to thank M.J. Van Bael, Y. Bruynseraede, C. Reichhardt and C.J. Olson for helpful discussions.
References [1] M. Baert, V.V. Metlushko, R. Jonckheere, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. Lett. 74 (1995) 3269. [2] M.J. Van Bael, K. Temst, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 59 (1999) 14674. [3] L. Van Look, B.Y. Zhu, R. Jonckheere, V.V. Moshchalkov, Phys. Rev. B, submitted for publication. [4] C. Reichhardt, N. Grønbech-Jensen, Phys. Rev. B 63 (2001) 54510. [5] A. Buzdin, M. Daumens, Physica C 294 (1998) 257. [6] C. Reichhardt, C.J. Olson, F. Nori, Phys. Rev. Lett. 78 (1997) 2648. [7] B.Y. Zhu, L. Van Look, V.V. Moshchalkov, B.R. Zhao, Z.X. Zhao, Phys. Rev. B 64 (2001) 12504.