- Email: [email protected]
Non commensurate vortex lattices in a composite antidot lattice or dc current
Available online at www.sciencedirect.com
Physica C 468 (2008) 809–812 www.elsevier.com/locate/physc
Non commensurate vortex lattices in a composite antidot lattice or dc current G.R. Berdiyorov a,*, M.V. Milosˇevic´ a,b, Francßois M. Peeters a b
a Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK
Accepted 30 November 2007 Available online 29 February 2008
Abstract Within the phenomenological Ginzburg–Landau theory we investigate the effect of a composite antidot lattice and small dc applied current to the stability of commensurate and non commensurate vortex structures in perforated type-II superconducting samples in the weak pinning regime. We found that a composite antidot lattice, consisting of small and big antidots in the unit cell, considerably increases the probability to find square pinned vortex lattice as compared to a sample with a regular square array of antidots. An applied current also favors the square pinned vortex states. These results indicate that both the applied current and a composite pinning array distort the broad local minimum in the free energy which keeps the vortices away from the pinning centers. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Ha; 74.78.Na; 75.75.+a Keywords: Perforated superconductor; Weak pinning; Vortex state; Ginzburg–Landau theory
It is well known that the regular triangular vortex lattice has the lowest energy in defect free superconductors [1], which is due to the isotropic repulsive interaction between the vortices. A square lattice of pinning centers imposes its own symmetry on the vortex structure and leads to highly ordered configurations [2–4] resulting in strong enhancement of the critical currents (see e.g. Ref. [5] and references therein). However, if the vortex-pinning force in such a periodic square array is reduced, the vortex–vortex interaction starts to dominate over the pinning and the triangular Abrikosov lattice is recovered. Calculations within the London theory [6] show that depending on the strength and length scale of the pinning potential the triangular vortex lattice in which some vortices are pinned by the pinning sites and others are located between them may become the ground state. These kinds of partially pinned (PP) phases
*
Corresponding author. E-mail address: [email protected] (G.R. Berdiyorov).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.055
have been recently found experimentally in macroscopic Wigner crystals with a square pinning potential [7]. Recently, we studied stable vortex configurations in a superconducting film with a square array of antidots within the Ginzburg–Landau (GL) theory [8]. We found that in addition to the square pinned vortex structures at the matching fields, a variety of vortex states can be stabilized by decreasing the pinning strength of the antidots (i.e. decreasing their size), including: (i) the triangular vortex lattice where some vortices are pinned by the antidots and others are located between them; (ii) vortex chain structures, and (iii) a lattice of vortex clusters around the empty pinning centers. Although these PP vortex structures are obtained more frequently in field cooled experiments than the square pinned vortex lattice, they turned out not to be the lowest energy states, contrary to the results from a London approach. Namely, we found that the PP state reside in a broad local minimum of the GL free energy and are stable there in the obscure of fluctuations even though commensurable vortex lattices have lower energy.
810
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
Our objective in this paper is investigate the effect of a composite antidot lattice (CAL) [5] on the stability of the above mentioned square pinned and the PP vortex configurations in the weak pinning regime. The unit cell of this CAL consists of one big and three small circular holes (see Fig. 1). It was shown experimentally that such CAL has a strong influence to the stability of the different vortex states, and consequently to the critical parameters of the sample [5]. Together with direct imaging techniques [2], indirect methods like transport and dc magnetization measurements are used to study pinning properties of regular arrays of pinning centers. In the latter cases, in order to determine the critical current and pinning strength, the system is driven out of equilibrium by an applied dc current, which disturbs the vortex distribution. The effect of the applied current should be even more pronounced in the weak pinning regime, where even small change of the external parameters has a considerable influence to the vortex state. Therefore our objective here is also to investigate the effect of small dc current to the formation of stable vortex configurations in the superconducting film with a square array of identical holes. We consider a superconducting film of thickness d = 0.1n with a square array (period W = 6n) of big (radius R0) and small (radius R) circular holes (see Fig. 1) in the presence of a perpendicular uniform magnetic field H. For the given system we solved two nonlinear GL equations with periodic boundary conditions around the square simulation region [9]. At the same time, the boundary condition corresponding to zero normal current was used at the boundaries of the antidots. For details of the numerics we refer to Ref. [4]. In this paper we restrict ourselves to the case of the second matching field H = H2 and simulations are done for a 2 2 unit cell (see Fig. 1). In our recent work [8] we have shown that in the case of square antidot lattice with small radius of antidots deformed triangular vortex structures are stable. For the first matching field the PP state consists of alternating rows of pinned and unpinned vortices which restore the triangular symmetry of the vortex lattice and correspond to the vortex state found in the experiment [7] on metallic parti-
Fig. 1. Schematic view of the sample with a composite antidot lattice: a superconducting film (thickness d) with a square array (period W) of circular antidots. The unit cell of the antidot array consists of one big (radius R0) and three small (radius R) antidots.
cles and as was predicted by the London theory [6]. The totally pinned (TP) state in this case is the one with all the vortices pinned by the holes, i.e. the vortex lattice has the same highly symmetric structure as the pinning lattice. At the second matching field in the TP state all pinning centers are occupied and the remaining vortices are located at the interstitial region (Fig. 2a). The overall vortex lattice is a square but rotated over 45° with respect to the pinning array. The PP state (Fig. 2b) is the one with one hole occupied by a vortex and the adjacent one is empty. Three unpinned vortices form clusters around the empty holes and all the clusters are altered. The orientation of the vortex clusters is somewhat different from the one obtained in the experiment [7], where the clusters are oriented along the axis of the pinning array. However, vortex cluster orientation changes gradually to the experimentally obtained one with further increasing the distance between the antidots. The existence of both the TP and PP states for a given magnetic field is an indication of the stability of several vortex configurations. Although the PP states were observed more frequently than the TP states in the experiments with charged particles [7], it was concluded that these two states are degenerate, whereas in calculations within the London theory the PP states were found to be the ground state. In what follows we perform an analysis of these vortex states and study the effect of the CAL and dc current on the stability of these states for two different values of the effective GL parameter j* = k2/nd. For this purpose we conducted a ‘‘field cooled” simulations starting from a random initial vortex distribution, which gives us the possibility to study the statistics of the occurrence of the TP and the PP states. Fig. 3 shows that the PP state is obtained more frequently than the TP state for the weak pinning force. However, increasing the hole radius leads to a decrease of the probability to obtain the PP state. As we have shown before [8], although the PP states are obtained more frequently than the TP states, they do not have the lowest free energy. The presence of the CAL (squares and triangles in Fig. 3) decreases the probability of the PP state due to the
a
b
Fig. 2. Contour plots of the Cooper-pair density at the second matching field H = H2 (R = R0 = 0.5n, j* = k2/dn = 1) for the totally pinned (a) and the partially pinned (b) states. Gray/white regions correspond to high/ low density. Dashed black lines show the antidot lattice, white lines show the vortex lattice and black dots in (b) show antidots without a vortex.
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
811
a
b Fig. 4. The same as Fig. 3(a) but now in the presence of a dc current density j (in units of j0 = cHc2n/4pk2) for the case R0 = R and for different values of j: j = 0.05j0 (circles), j = 0.013j0 (squares) and j = 0.016j0 (triangles). The insets show the Cooper-pair density plots for j = 0.016j0 for the TP (1) and PP (2) states.
Fig. 3. Probability to find the totally pinned (TP) (solid curves) and partially pinned (PP) (dashed curves) vortex states at H = H2 as a function of the holes radius R for j* = 1 (a) and j* = 1 (b) and for different values of R0: R0 = R (circles), R0 = 1.5R (squares) and R0 = 2R (triangles).
stronger pinning of the vortices. We would like to mention that the probability of the TP state is larger in the case of the CAL as compared to the case of a regular antidot array with the same radius of the holes. The latter indicates that we promoted the evolution of the TP states by superposing a squarer lattice of larger holes on top of an existing regular array of antidots. The same behavior of the PP and TP states is found for smaller GL parameter j* (see Fig. 3b). In this case the probability to find the PP state considerably decreases as compared to the larger j* case due to the dominance of the vortex-pinning site interaction over the decreased vortex–vortex interaction. Next, we investigate the influence of applied dc current to the formation of the vortex states in our system. For simplicity we consider again the regular antidot lattice, i.e. R0 = R. The applied current j (in units of j0 = cHc2n/ 4pk2) is simulated by adding a constant to the vector potential [4]. We would like to mention that the current generated in the sample is much smaller than the critical one, so that only static displacement of vortices is allowed (see insets of Fig. 4). Fig. 4 shows the probability to realize the TP (solid curves) and PP (dashed curves) states at the second matching field when starting from a random distribution of vortices for three different values j. For small
drive currents the evolution of the states weakly depends on j (compare circles in Figs. 3a and 4). With increasing the applied current the probability to find the TP state increases for all considered radii of the antidots. In addition, the critical hole radius to obtain 100% TP state significantly decreases with increasing j. These results indicate that the energy barrier separating the TP and PP states decrease with increasing dc drive. The same dependence of the vortex states on the current is found for small j* with the difference that the probability of the PP states is suppressed compared to larger j* case. We explained previously [8] that the most probable vortex state is not necessarily the ground state as the pinning force of the antidots may be overwhelmed by the repulsive interaction between the vortices. In the case of the CAL these forces become anisotropic and decreases the compensation leading to an increase of the probability of square pinned states. The situation become even more complicated when there is an external current in the sample. In this case vortices are subject to three interactions (we neglect thermal effects), namely: the repulsive vortex–vortex interaction, the attractive interaction from the pinning sites, and the Lorentz force from the applied current. Surprisingly, although it is known that dc current eventually leads to destruction of the vortex lattice and superconductivity, we find that the resulting transverse Lorentz force induces lateral anisotropy and favors square, totally pinned vortex lattice. In conclusion, we have studied, within the Gingzburg– Landau formalism the effect of a composite antidot lattice and small dc drive to the totally pinned and partially pinned vortex configurations in a perforated superconducting film. We find that both the CAL and the dc current favor the square pinned vortex state over the partially pinned vortex structure. This indicates that the energy barrier separating these two stable states is
812
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
strongly affected by the anisotropy generated in the system in either way. Our future work will incorporate random defects in overall square antidot lattice which is expected to influence the vortex lattice formation process in superconductors with weak pinning centers and generally corresponds to experiments. Acknowledgements This work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IAP), and the ESF-AQDJJ network. M.V.M acknowledges support from EU Marie-Curie Intra-European Program and G.R.B acknowledges support from FWOVlaanderen.
References [1] W.H. Kleiner, L.M. Roth, S.H. Autler, Phys. Rev. A 133 (1964) 1226. [2] K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, V.V. Moshchalkov, Science 274 (1996) 1167. [3] C. Reichhardt, C.J. Olson, F. Nori, Phys. Rev. B 57 (1998) 7937. [4] G.R. Berdiyorov, M.V. Milosˇevic´, F.M. Peeters, Phys. Rev. Lett. 96 (2006) 207001. [5] A.V. Silhanek, L. Van Look, R. Jonckheere, B.Y. Zhu, S. Raedts, V.V. Moshchalkov, Phys. Rev. B 72 (2005) 014507. [6] W.V. Pogosov, A.L. Rakhmanov, V.V. Moshchalkov, Phys. Rev. B 67 (2003) 014532. [7] G. Coupier, M. Saint Jean, C. Guthmann, Phys. Rev. B 75 (2007) 224103. [8] G.R. Berdyorov, M.V. Milosˇevic´, F.M. Peeters, Phys. Rev. B 76 (2007) 134508. [9] M.M. Doria, J.E. Gubernatis, D. Rainer, Phys. Rev. B 39 (1989) 9573.
Physica C 468 (2008) 809–812 www.elsevier.com/locate/physc
Non commensurate vortex lattices in a composite antidot lattice or dc current G.R. Berdiyorov a,*, M.V. Milosˇevic´ a,b, Francßois M. Peeters a b
a Universiteit Antwerpen, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK
Accepted 30 November 2007 Available online 29 February 2008
Abstract Within the phenomenological Ginzburg–Landau theory we investigate the effect of a composite antidot lattice and small dc applied current to the stability of commensurate and non commensurate vortex structures in perforated type-II superconducting samples in the weak pinning regime. We found that a composite antidot lattice, consisting of small and big antidots in the unit cell, considerably increases the probability to find square pinned vortex lattice as compared to a sample with a regular square array of antidots. An applied current also favors the square pinned vortex states. These results indicate that both the applied current and a composite pinning array distort the broad local minimum in the free energy which keeps the vortices away from the pinning centers. Ó 2008 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Ha; 74.78.Na; 75.75.+a Keywords: Perforated superconductor; Weak pinning; Vortex state; Ginzburg–Landau theory
It is well known that the regular triangular vortex lattice has the lowest energy in defect free superconductors [1], which is due to the isotropic repulsive interaction between the vortices. A square lattice of pinning centers imposes its own symmetry on the vortex structure and leads to highly ordered configurations [2–4] resulting in strong enhancement of the critical currents (see e.g. Ref. [5] and references therein). However, if the vortex-pinning force in such a periodic square array is reduced, the vortex–vortex interaction starts to dominate over the pinning and the triangular Abrikosov lattice is recovered. Calculations within the London theory [6] show that depending on the strength and length scale of the pinning potential the triangular vortex lattice in which some vortices are pinned by the pinning sites and others are located between them may become the ground state. These kinds of partially pinned (PP) phases
*
Corresponding author. E-mail address: [email protected] (G.R. Berdiyorov).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.055
have been recently found experimentally in macroscopic Wigner crystals with a square pinning potential [7]. Recently, we studied stable vortex configurations in a superconducting film with a square array of antidots within the Ginzburg–Landau (GL) theory [8]. We found that in addition to the square pinned vortex structures at the matching fields, a variety of vortex states can be stabilized by decreasing the pinning strength of the antidots (i.e. decreasing their size), including: (i) the triangular vortex lattice where some vortices are pinned by the antidots and others are located between them; (ii) vortex chain structures, and (iii) a lattice of vortex clusters around the empty pinning centers. Although these PP vortex structures are obtained more frequently in field cooled experiments than the square pinned vortex lattice, they turned out not to be the lowest energy states, contrary to the results from a London approach. Namely, we found that the PP state reside in a broad local minimum of the GL free energy and are stable there in the obscure of fluctuations even though commensurable vortex lattices have lower energy.
810
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
Our objective in this paper is investigate the effect of a composite antidot lattice (CAL) [5] on the stability of the above mentioned square pinned and the PP vortex configurations in the weak pinning regime. The unit cell of this CAL consists of one big and three small circular holes (see Fig. 1). It was shown experimentally that such CAL has a strong influence to the stability of the different vortex states, and consequently to the critical parameters of the sample [5]. Together with direct imaging techniques [2], indirect methods like transport and dc magnetization measurements are used to study pinning properties of regular arrays of pinning centers. In the latter cases, in order to determine the critical current and pinning strength, the system is driven out of equilibrium by an applied dc current, which disturbs the vortex distribution. The effect of the applied current should be even more pronounced in the weak pinning regime, where even small change of the external parameters has a considerable influence to the vortex state. Therefore our objective here is also to investigate the effect of small dc current to the formation of stable vortex configurations in the superconducting film with a square array of identical holes. We consider a superconducting film of thickness d = 0.1n with a square array (period W = 6n) of big (radius R0) and small (radius R) circular holes (see Fig. 1) in the presence of a perpendicular uniform magnetic field H. For the given system we solved two nonlinear GL equations with periodic boundary conditions around the square simulation region [9]. At the same time, the boundary condition corresponding to zero normal current was used at the boundaries of the antidots. For details of the numerics we refer to Ref. [4]. In this paper we restrict ourselves to the case of the second matching field H = H2 and simulations are done for a 2 2 unit cell (see Fig. 1). In our recent work [8] we have shown that in the case of square antidot lattice with small radius of antidots deformed triangular vortex structures are stable. For the first matching field the PP state consists of alternating rows of pinned and unpinned vortices which restore the triangular symmetry of the vortex lattice and correspond to the vortex state found in the experiment [7] on metallic parti-
Fig. 1. Schematic view of the sample with a composite antidot lattice: a superconducting film (thickness d) with a square array (period W) of circular antidots. The unit cell of the antidot array consists of one big (radius R0) and three small (radius R) antidots.
cles and as was predicted by the London theory [6]. The totally pinned (TP) state in this case is the one with all the vortices pinned by the holes, i.e. the vortex lattice has the same highly symmetric structure as the pinning lattice. At the second matching field in the TP state all pinning centers are occupied and the remaining vortices are located at the interstitial region (Fig. 2a). The overall vortex lattice is a square but rotated over 45° with respect to the pinning array. The PP state (Fig. 2b) is the one with one hole occupied by a vortex and the adjacent one is empty. Three unpinned vortices form clusters around the empty holes and all the clusters are altered. The orientation of the vortex clusters is somewhat different from the one obtained in the experiment [7], where the clusters are oriented along the axis of the pinning array. However, vortex cluster orientation changes gradually to the experimentally obtained one with further increasing the distance between the antidots. The existence of both the TP and PP states for a given magnetic field is an indication of the stability of several vortex configurations. Although the PP states were observed more frequently than the TP states in the experiments with charged particles [7], it was concluded that these two states are degenerate, whereas in calculations within the London theory the PP states were found to be the ground state. In what follows we perform an analysis of these vortex states and study the effect of the CAL and dc current on the stability of these states for two different values of the effective GL parameter j* = k2/nd. For this purpose we conducted a ‘‘field cooled” simulations starting from a random initial vortex distribution, which gives us the possibility to study the statistics of the occurrence of the TP and the PP states. Fig. 3 shows that the PP state is obtained more frequently than the TP state for the weak pinning force. However, increasing the hole radius leads to a decrease of the probability to obtain the PP state. As we have shown before [8], although the PP states are obtained more frequently than the TP states, they do not have the lowest free energy. The presence of the CAL (squares and triangles in Fig. 3) decreases the probability of the PP state due to the
a
b
Fig. 2. Contour plots of the Cooper-pair density at the second matching field H = H2 (R = R0 = 0.5n, j* = k2/dn = 1) for the totally pinned (a) and the partially pinned (b) states. Gray/white regions correspond to high/ low density. Dashed black lines show the antidot lattice, white lines show the vortex lattice and black dots in (b) show antidots without a vortex.
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
811
a
b Fig. 4. The same as Fig. 3(a) but now in the presence of a dc current density j (in units of j0 = cHc2n/4pk2) for the case R0 = R and for different values of j: j = 0.05j0 (circles), j = 0.013j0 (squares) and j = 0.016j0 (triangles). The insets show the Cooper-pair density plots for j = 0.016j0 for the TP (1) and PP (2) states.
Fig. 3. Probability to find the totally pinned (TP) (solid curves) and partially pinned (PP) (dashed curves) vortex states at H = H2 as a function of the holes radius R for j* = 1 (a) and j* = 1 (b) and for different values of R0: R0 = R (circles), R0 = 1.5R (squares) and R0 = 2R (triangles).
stronger pinning of the vortices. We would like to mention that the probability of the TP state is larger in the case of the CAL as compared to the case of a regular antidot array with the same radius of the holes. The latter indicates that we promoted the evolution of the TP states by superposing a squarer lattice of larger holes on top of an existing regular array of antidots. The same behavior of the PP and TP states is found for smaller GL parameter j* (see Fig. 3b). In this case the probability to find the PP state considerably decreases as compared to the larger j* case due to the dominance of the vortex-pinning site interaction over the decreased vortex–vortex interaction. Next, we investigate the influence of applied dc current to the formation of the vortex states in our system. For simplicity we consider again the regular antidot lattice, i.e. R0 = R. The applied current j (in units of j0 = cHc2n/ 4pk2) is simulated by adding a constant to the vector potential [4]. We would like to mention that the current generated in the sample is much smaller than the critical one, so that only static displacement of vortices is allowed (see insets of Fig. 4). Fig. 4 shows the probability to realize the TP (solid curves) and PP (dashed curves) states at the second matching field when starting from a random distribution of vortices for three different values j. For small
drive currents the evolution of the states weakly depends on j (compare circles in Figs. 3a and 4). With increasing the applied current the probability to find the TP state increases for all considered radii of the antidots. In addition, the critical hole radius to obtain 100% TP state significantly decreases with increasing j. These results indicate that the energy barrier separating the TP and PP states decrease with increasing dc drive. The same dependence of the vortex states on the current is found for small j* with the difference that the probability of the PP states is suppressed compared to larger j* case. We explained previously [8] that the most probable vortex state is not necessarily the ground state as the pinning force of the antidots may be overwhelmed by the repulsive interaction between the vortices. In the case of the CAL these forces become anisotropic and decreases the compensation leading to an increase of the probability of square pinned states. The situation become even more complicated when there is an external current in the sample. In this case vortices are subject to three interactions (we neglect thermal effects), namely: the repulsive vortex–vortex interaction, the attractive interaction from the pinning sites, and the Lorentz force from the applied current. Surprisingly, although it is known that dc current eventually leads to destruction of the vortex lattice and superconductivity, we find that the resulting transverse Lorentz force induces lateral anisotropy and favors square, totally pinned vortex lattice. In conclusion, we have studied, within the Gingzburg– Landau formalism the effect of a composite antidot lattice and small dc drive to the totally pinned and partially pinned vortex configurations in a perforated superconducting film. We find that both the CAL and the dc current favor the square pinned vortex state over the partially pinned vortex structure. This indicates that the energy barrier separating these two stable states is
812
G.R. Berdiyorov et al. / Physica C 468 (2008) 809–812
strongly affected by the anisotropy generated in the system in either way. Our future work will incorporate random defects in overall square antidot lattice which is expected to influence the vortex lattice formation process in superconductors with weak pinning centers and generally corresponds to experiments. Acknowledgements This work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IAP), and the ESF-AQDJJ network. M.V.M acknowledges support from EU Marie-Curie Intra-European Program and G.R.B acknowledges support from FWOVlaanderen.
References [1] W.H. Kleiner, L.M. Roth, S.H. Autler, Phys. Rev. A 133 (1964) 1226. [2] K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, V.V. Moshchalkov, Science 274 (1996) 1167. [3] C. Reichhardt, C.J. Olson, F. Nori, Phys. Rev. B 57 (1998) 7937. [4] G.R. Berdiyorov, M.V. Milosˇevic´, F.M. Peeters, Phys. Rev. Lett. 96 (2006) 207001. [5] A.V. Silhanek, L. Van Look, R. Jonckheere, B.Y. Zhu, S. Raedts, V.V. Moshchalkov, Phys. Rev. B 72 (2005) 014507. [6] W.V. Pogosov, A.L. Rakhmanov, V.V. Moshchalkov, Phys. Rev. B 67 (2003) 014532. [7] G. Coupier, M. Saint Jean, C. Guthmann, Phys. Rev. B 75 (2007) 224103. [8] G.R. Berdyorov, M.V. Milosˇevic´, F.M. Peeters, Phys. Rev. B 76 (2007) 134508. [9] M.M. Doria, J.E. Gubernatis, D. Rainer, Phys. Rev. B 39 (1989) 9573.