We study the geometrical confinement effect in Bi22Sr22CaCu22O8+δ8+δ mesoscopic vortex matter with edge-to-surface ratio of 7–12 %. Samples have in-plane square and circular edges, 30 μm widths, and ∼2 μm thickness. Direct vortex imaging reveals the compact planes of the structure align with the sample edge by introducing topological defects. The defect density is larger for circular than for square edges. Molecular dynamics simulations suggest that this density is not an out-of-equilibrium property but rather determined by the geometrical confinement.Fil: Cejas Bolecek, Néstor René. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (centro Atómico Bariloche). División Bajas Temperaturas; ArgentinaFil: Dolz, Moira Ines. Universidad Nacional de San Luis. Facultad de Cs.fisico Matematicas y Naturales. Departamento de Fisica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico San Luis. Instituto de Física Aplicada; ArgentinaFil: Kolton, Alejandro Benedykt. Comision Nacional de Energia Atomica. Gerencia D/area Invest y Aplicaciones No Nucleares. Gerencia de Fisica (cab). Grupo de Teoria de Solidos; ArgentinaFil: Pastoriza, Hernan. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (centro Atómico Bariloche). División Bajas Temperaturas; ArgentinaFil: Van Der Beek, C. J.. Ecole Polytechnique. Laboratoire des Solides Irradiés; FranciaFil: Knoczkykowski, M.. Ecole Polytechnique. Laboratoire des Solides Irradiés; FranciaFil: Menghini, M.. Katholikie Universiteit Leuven; BélgicaFil: Nieva, Gladys Leonor. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (centro Atómico Bariloche). División Bajas Temperaturas; ArgentinaFil: Fasano, Yanina. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (centro Atómico Bariloche). División Bajas Temperaturas; Argentin

Cejas Bolecek, Néstor René

Dolz, Moira Ines

Fasano, Yanina

Knoczkykowski, M.

Kolton, Alejandro Benedykt

Menghini, M.

Nieva, Gladys Leonor

Pastoriza, Hernan

Van Der Beek, C. J.

Journal of Low Temperature Physics

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Geometrical confinement effects in layered mesoscopic vortex-matterN. R. Cejas Bolecek,1 M. I. Dolz,2 A. Kolton,3 H. Pastoriza,1 C. J. vander Beek,4 M. Konczykowski,4 M. Menghini,5 G. Nieva,1 and Y. Fasano11Low Temperature Division, Centro Atómico Bariloche, CNEA, Argentina2Universidad Nacional de San Luis, San Luis, Argentina3Solid State Theory group, Centro Atómico Bariloche, CNEA, Argentina4Laboratoire des Solides Irradiés, Ecole Polytechnique, Palaiseau, France5Katholieke Universitat Leuven, Leuven, Belgium(Dated: September 26, 2014)We study geometrical confinement effects in Bi2Sr2CaCu2O8+δ mesoscopic vortex-matter withedge-to-surface ratio of 7 − 12 %. Samples have in-plane square and circular edges, 30µm widths,and ∼ 2µm thickness. Direct vortex imaging reveals the compact planes of the structure align withthe sample edge by introducing topological defects. The defects density is larger for circular thanfor square edges. Molecular dynamics simulations suggest this density is not an out-of-equilibriumproperty but rather determined by the geometrical confinement.PACS numbers:I. INTRODUCTIONUnderstanding the confinement effects introduced bysample geometry is crucial for characterizing the staticand dynamic properties of mesoscopic vortex mat-ter. This subject was actively investigated for low-temperature superconductors with dimensions compa-rable or smaller than coherence length or penetrationdepth, λ1–6. Mesoscopic vortex matter in these materi-als have structural properties strongly influenced by thegeometry of the specimens3, in contrast with results inmacroscopic samples for several compounds7–10. Con-finement effects are in competition with inter-vortex in-teraction that increases with field and temperature. Ma-terials with an important electronic anisotropy such aslayered high-Tc’s have quite a large value of λ and theninter-vortex interactions become more relevant.Due to the technical difficulties for fabricating micron-sized samples of layered high-Tc’s complex oxides, thereare few works in the literature investigating the ef-fect of confinement in vortex matter nucleated in thesematerials11. In this work we study this issue in theparadigmatic Bi2Sr2CaCu2O8+δ compound that presentsa rich vortex phase diagram governed by thermal fluctu-ations and extremely anisotropic magnetic properties. Inthis compound, the phase diagram of macroscopic as wellas mesoscopic21 vortex matter is dominated by a first-order transition12,13 between a solid phase at low tem-peratures and a liquid14 or decoupled gas15,16 of pancakevortices at high temperatures. The vortex solid phase ofmacroscopic samples presents quasi long-range positionalorder17.Here we report on the structural properties of themesoscopic vortex solid nucleated in Bi2Sr2CaCu2O8+δat low fields and with single-vortex resolution. We studyboth, experimentally and with simulations, the effectof confinement and inter-vortex interactions for sampleswith square and circular edges experimental.II. METHODSWe engineered micron-sized superconducting samplesfrom bulk Bi2Sr2CaCu2O8+δ crystals (Tc = 89 K). Wefabricated circular and square samples with typical di-mensions of 30µm by means of optical lithography andsubsequent physical ion-milling of the negative of thesamples18. Freestanding 2µm thick disks and cuboids areobtained after cleaving the towers resulting from milling.We directly imaged the solid vortex phase with single-vortex resolution by means of magnetic decoration exper-iments performed at 4.2 K after field-cooling19. In theseexperiments the evaporated magnetic nanoparticles landin the sample surface at the places where the gradient oflocal inductance is maximum, therefore decorating thevortex positions. The imaged structure corresponds tothe vortex solid frozen at the temperature at which pin-ning sets in, Tfreez ∼ Tirr7, of the order of 90 − 87 K forthe low-fields studied here20. Decreasing the sample sizedown to microns does not significantly affect the value ofTirr21.We also performed molecular dynamics simulationsof two-dimensional vortex matter in order to emulatethe experimentally-observed vortex matter structuralproperties22. We studied the case of 30µm diameterdisks and focused on the density of topological defectswhen varying the simulation cooling-rate.III. RESULTS AND DISCUSSIONFigure 1 shows snapshots of the mesoscopic vor-tex structure nucleated in the disk and cuboidBi2Sr2CaCu2O8+δ samples after field-cooling down to4.2 K at applied fields of 20 and 40 Oe. The local in-duction calculated as the number of vortices times theflux quantum yields B ∼ 0.75H, a reduction due to de-magnetizing effects (aspect ratio of the disks and cuboidsof ∼ 15). The direction of the compact planes of thearXiv:1409.7305v1  [cond-mat.supr-con]  25 Sep 20142vortex structure in micron-sized specimens is affected bythe confinement effect introduced by the edges of thesamples, particularly in the case of the outer shells ofvortices. This finding is in contrast to observations inmacroscopic samples7. The alignment is more evident inthe Delaunay triangulations of Fig. 2, a well known ge-ometrical algorithm that determines the first-neighborsfor every vortex in the structure17. First-neighbors vor-tices are bounded with lines and non-sixfold coordinatedones are highlighted in grey. In the case of the cuboidsamples, irrespective of the vortex density (447 vs. 1092),one of the compact planes of the structure is parallel tothe sample edge. For the disks, only a few outer shells ofvortices mimic the sample edges, the number dependinginversely with the vortex density. Towards the center ofthe sample, a rather ordered vortex crystallite is formedwith the compact planes having no register with the sam-ple edges. The transition between the orientation of theouter and inner shells is done via the plastic deformationsentailed by topological defects.For the vortex structure studied here these topologicaldefects are generally disclinations, namely vortices withfive or seven first-neighbors, and pairs of them or screw-dislocations associated to an extra plane of vortices. Forexample, isolated dislocations are observed in the mid-dle of the vortex structure nucleated in the cuboid at anFigure 1: Magnetic decoration images of the mesoscopicvortex matter nucleated in field-cooling processes in micron-sized Bi2Sr2CaCu2O8+δ samples. (a) Disk with 30µm di-ameter and (b) cuboid with 30µm sides length, all sampleswith 2µm thickness. Magnetic decorations were performed at4.2 K and at applied fields of 20 (left panel) and 40 Oe (rightpanel). The scale-bar corresponds to 10µm.applied field of 40 Oe. The density of non-sixfold coordi-nated vortices, ρdef , strongly depends on the local induc-tion. In the case of macroscopic Bi2Sr2CaCu2O8+δ vor-tex matter, ρdef decreases exponentially up to 20 Gaussand then saturates around 2% as shown in Fig. 3 (a).This is due to the enhancement of inter-vortex interac-tion on increasing field. This magnitude follows the sameB-evolution for mesoscopic vortex matter but is at least50% larger than for bulk samples. In addition, ρdef is al-ways larger in disks than in cuboids for roughly the samevortex density. This can be explained by considering thataligning a compact plane of vortices with the edges of acuboid does not imply to change the orientational orderof the structure whereas in order to do so in a disk thevortex planes have to bend.In the case of macroscopic samples, it has been provedthat the structure observed by means of field-cooling dec-orations at 4.2 K, and therefore its ρdef , is quite closeto the equilibrium17. The possibility of the increase onthe ρdef on decreasing the system size being an out-of-equilibrium phenomena can not be discarded. There-fore we performed molecular dynamics simulations of themesoscopic vortex matter nucleated in a 30µm disk witha density of 15 Gauss in order to test this possibility. Inparticular, we performed tests on the dependence of ρdefwith the cooling rate, inversely proportional to the timeallowed to the system to relax. First, we performed sim-ulations in a macroscopic sample in order to find thepinning magnitude that has to be considered in order toreproduce the observed ρdef . Then we used this magni-Figure 2: Delaunay triangulations of the mesoscopic vor-tex matter shown in Fig. 1. Disclinations are highlighted ingray; sixfold(non-sixfold)-coordinated vortices are indicatedin blue(red). The scale-bar corresponds to 10µm.3tude of pinning to perform simulations in micron-sizedsamples. The results of ρdef as a function of the coolingrate, see Fig. 3 (b), indicate that the observed experimen-tal values correspond to the case of large relaxation times.Therefore, we can ascertain that the amount of topolog-ical defects observed in the experiments in micron-sizedsamples is not an out-of-equilibrium feature and that theobserved structure is quite close to the equilibrium.Having this certainty in mind, we tried to estimatethe geometrical confinement energy induced by the sam-ple edges. In the case of bulk samples, the mean value ofthe inter-vortex interaction energy distribution is slightlyshifted upward with respect to the value for a perfectAbrikosov lattice with the same vortex density10. Sinceafter a field-cooling process vortices are close to equilib-rium, this shifting can only be accounted by the effect ofbulk pinning10. This can be expressed, by unit length,as < εint >b −εAbr = εbp, where b stands for the bulksample and < εint > is the mean value of a distributionof inter-vortex interaction energies, εAbr the value of theinter-vortex interaction energy in a perfect Abrikosov lat-tice (a delta-function), and εp the pinning energy. In thecase of mesoscopic vortex matter, an extra term entersinto the energy-balance, namely the confinement energyεconf and therefore < εint >meso −εAbr = εmesop + εconf ,where meso means for the case of mesoscopic vortex mat-ter and the energies are noted similarly as in the pre-vious case. Therefore, one can have access to an es-timation of the confinement energy in mesoscopic vor-tex matter just by assuming that the pinning magni-tude is the same irrespective of the sample size, and thenεconf =< εint >meso − < εint >b.The inter-vortex interaction energy per unit length de-pends on the inter-vortex distances rij , and for a vor-tex i has a value εiint =∑j 2ε0K0(rij/λ), with the sumover neighbor-vortices j, ε0 ∝ λ2 the vortex line ten-Figure 3: Density of topological defects (non-sixfold coordinated vortices) in mesoscopic and macroscopicBi2Sr2CaCu2O8+δ vortex matter. (a) Experimental data as afunction of applied field for the mesoscopic vortex structuresnucleated in the 30µm disk and cuboid, and macroscopic vor-tex matter. (b) Results from molecular dynamics simulationsas a function of the simulations relaxation time in the case ofa 30µm disk and B = 15 Gauss.sion, and K0 the zeroth-order modified Bessel function.In real cases this magnitude is spatially inhomogeneousdue to the elastic and plastic deformations of the struc-ture, and therefore there is a distribution of εiint withan almost-Gaussian shape10. Only in the case of anideal Abrikosov lattice this magnitude is space invari-ant and its distribution is a delta function. We haveperformed inter-vortex energy-distribution calculationsin vortex structures observed by magnetic decoration inthe macroscopic samples from which were engineered thedisks and cuboids. We also performed the same calcu-lations in the mesoscopic vortex matter nucleated in thedisks and cuboids at both applied fields. Irrespective ofthe field, the mean values of < εint >meso are alwayslarger than in the case of macroscopic vortex matter,what can be reasonably ascribed to the extra deforma-tions introduced by the larger amount of topological de-fects nucleated in the micron-sized samples. In accor-dance with this reasoning, the < εint >meso value is al-ways larger for the structure nucleated in the disks thanin the cuboids by a 6-9% on increasing field. Thereforethe estimated confinement energy is in the case of disksequal to 1 ± 0.1 × 10−8 erg/cm∼ 0.13ε0, larger than inthe case of cuboids, 0.8± 0.1× 10−8 erg/cm∼ 0.11ε0.IV. CONCLUSIONSThe edges of the samples do produce a geometrical con-finement effect in mesoscopic vortex matter that is put inevidence by the orientation of the outer shells of vorticeswith compact planes parallel to the edges what producesa concomitant increase of the density of topological de-fects. By means of molecular dynamics simulations weshow that the density of defects found experimentally isnot an out-of-equilibrium feature but rather the effect in-troduced by geometrical confinement. By means of differ-ences in the mean value of the inter-vortex interaction ofthe mesoscopic and macroscopic vortex structures we areable to quantify the confinement energy per unit length.We find that is just 0.11− 0.13 ε0, the larger value in thecase of disks than cuboid geometries.41 V. V. Moshchalkov, et al., Nature 373, 319 (1995).2 A. K. Geim, et al., Nature 390, 256 (1997).3 V. A. Schweigert, F. M. Peeters, and P. S. Deo, Phys. Rev.Lett. 81, 2783 (1998).4 J. J. Palacios, Phys. Rev. B 58, R5948 (1998).5 V. Bruyndoncx, et al., Phys. Rev. B 60, 10468 (1999).6 L. R. E. Cabral, B. J. Baelus, and F. M. Peeters, Phys.Rev. B 70, 144523 (2004).7 Y. Fasano, J. Herbsommer, and F. de la Cruz, Phys. Stat.Sol. (b) 215, 563 (1999).8 M. Menghini, et al., Phys. Rev. Lett. 90, 147001 (2003).9 A. P. Petrovic, et al., Phys. Rev. Lett. 103, 257001 (2009).10 S. Demirdis, et al., Phys. Rev. B 84, 094517 (2011).11 Y. M. Wang, et al., Phys. Rev. B 65, 184506, (2002)12 H. Pastoriza, et al., Phys. Rev. Lett. 72, 2951 (1994).13 E. Zeldov, et al., Nature 375, 373 (1995).14 D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).15 L. I. Glazman, and A. E. Koshelev, Phys. Rev. B 43, 2835(1991).16 H. Pastoriza and P. H. Kes, Phys. Rev. Lett. 75, 3525(1995).17 Y. Fasano,et al., Proc. Nat. Acad. Sci. 102, 3898 (2005).18 M. I. Dolz, A. B. Kolton and H. Pastoriza, Phy. Rev. B81, 092502 (2010).19 Y. Fasano, et al., Solid State Commun. 128, 51 (2003).20 M.I. Dolz, et al., submitted to Phys. Rev. B, cond-mat:1212.4564. (2014).21 M. Konczykowski, et al., cond-mat: arXiv:1212.4564.22 A. Kolton, D. Domı́nguez, and N. Grønbech-Jensen, Phys.Rev. Lett. 83, 3061 (1999).

English

Geometrical Confinement Effects in Layered Mesoscopic Vortex Matter

CONICET Digital

Geometrical confinement effects in layered mesoscopic vortex-matterN. R. Cejas Bolecek,1 M. I. Dolz,2 A. Kolton,3 H. Pastoriza,1 C. J. vander Beek,4 M. Konczykowski,4 M. Menghini,5 G. Nieva,1 and Y. Fasano11Low Temperature Division, Centro Ato´mico Bariloche, CNEA, Argentina2Universidad Nacional de San Luis, San Luis, Argentina3Solid State Theory group, Centro Ato´mico Bariloche, CNEA, Argentina4Laboratoire des Solides Irradie´s, Ecole Polytechnique, Palaiseau, France5Katholieke Universitat Leuven, Leuven, Belgium(Dated: September 26, 2014)We study geometrical confinement effects in Bi2Sr2CaCu2O8+δ mesoscopic vortex-matter withedge-to-surface ratio of 7 − 12 %. Samples have in-plane square and circular edges, 30µm widths,and ∼ 2µm thickness. Direct vortex imaging reveals the compact planes of the structure align withthe sample edge by introducing topological defects. The defects density is larger for circular thanfor square edges. Molecular dynamics simulations suggest this density is not an out-of-equilibriumproperty but rather determined by the geometrical confinement.PACS numbers:I. INTRODUCTIONUnderstanding the confinement effects introduced bysample geometry is crucial for characterizing the staticand dynamic properties of mesoscopic vortex mat-ter. This subject was actively investigated for low-temperature superconductors with dimensions compa-rable or smaller than coherence length or penetrationdepth, λ1–6. Mesoscopic vortex matter in these materi-als have structural properties strongly influenced by thegeometry of the specimens3, in contrast with results inmacroscopic samples for several compounds7–10. Con-finement effects are in competition with inter-vortex in-teraction that increases with field and temperature. Ma-terials with an important electronic anisotropy such aslayered high-Tc’s have quite a large value of λ and theninter-vortex interactions become more relevant.Due to the technical difficulties for fabricating micron-sized samples of layered high-Tc’s complex oxides, thereare few works in the literature investigating the ef-fect of confinement in vortex matter nucleated in thesematerials11. In this work we study this issue in theparadigmatic Bi2Sr2CaCu2O8+δ compound that presentsa rich vortex phase diagram governed by thermal fluctu-ations and extremely anisotropic magnetic properties. Inthis compound, the phase diagram of macroscopic as wellas mesoscopic21 vortex matter is dominated by a first-order transition12,13 between a solid phase at low tem-peratures and a liquid14 or decoupled gas15,16 of pancakevortices at high temperatures. The vortex solid phase ofmacroscopic samples presents quasi long-range positionalorder17.Here we report on the structural properties of themesoscopic vortex solid nucleated in Bi2Sr2CaCu2O8+δat low fields and with single-vortex resolution. We studyboth, experimentally and with simulations, the effectof confinement and inter-vortex interactions for sampleswith square and circular edges experimental.II. METHODSWe engineered micron-sized superconducting samplesfrom bulk Bi2Sr2CaCu2O8+δ crystals (Tc = 89 K). Wefabricated circular and square samples with typical di-mensions of 30µm by means of optical lithography andsubsequent physical ion-milling of the negative of thesamples18. Freestanding 2µm thick disks and cuboids areobtained after cleaving the towers resulting from milling.We directly imaged the solid vortex phase with single-vortex resolution by means of magnetic decoration exper-iments performed at 4.2 K after field-cooling19. In theseexperiments the evaporated magnetic nanoparticles landin the sample surface at the places where the gradient oflocal inductance is maximum, therefore decorating thevortex positions. The imaged structure corresponds tothe vortex solid frozen at the temperature at which pin-ning sets in, Tfreez ∼ Tirr7, of the order of 90 − 87 K forthe low-fields studied here20. Decreasing the sample sizedown to microns does not significantly affect the value ofTirr21.We also performed molecular dynamics simulationsof two-dimensional vortex matter in order to emulatethe experimentally-observed vortex matter structuralproperties22. We studied the case of 30µm diameterdisks and focused on the density of topological defectswhen varying the simulation cooling-rate.III. RESULTS AND DISCUSSIONFigure 1 shows snapshots of the mesoscopic vor-tex structure nucleated in the disk and cuboidBi2Sr2CaCu2O8+δ samples after field-cooling down to4.2 K at applied fields of 20 and 40 Oe. The local in-duction calculated as the number of vortices times theflux quantum yields B ∼ 0.75H, a reduction due to de-magnetizing effects (aspect ratio of the disks and cuboidsof ∼ 15). The direction of the compact planes of thearXiv:1409.7305v1  [cond-mat.supr-con]  25 Sep 20142vortex structure in micron-sized specimens is affected bythe confinement effect introduced by the edges of thesamples, particularly in the case of the outer shells ofvortices. This finding is in contrast to observations inmacroscopic samples7. The alignment is more evident inthe Delaunay triangulations of Fig. 2, a well known ge-ometrical algorithm that determines the first-neighborsfor every vortex in the structure17. First-neighbors vor-tices are bounded with lines and non-sixfold coordinatedones are highlighted in grey. In the case of the cuboidsamples, irrespective of the vortex density (447 vs. 1092),one of the compact planes of the structure is parallel tothe sample edge. For the disks, only a few outer shells ofvortices mimic the sample edges, the number dependinginversely with the vortex density. Towards the center ofthe sample, a rather ordered vortex crystallite is formedwith the compact planes having no register with the sam-ple edges. The transition between the orientation of theouter and inner shells is done via the plastic deformationsentailed by topological defects.For the vortex structure studied here these topologicaldefects are generally disclinations, namely vortices withfive or seven first-neighbors, and pairs of them or screw-dislocations associated to an extra plane of vortices. Forexample, isolated dislocations are observed in the mid-dle of the vortex structure nucleated in the cuboid at anFigure 1: Magnetic decoration images of the mesoscopicvortex matter nucleated in field-cooling processes in micron-sized Bi2Sr2CaCu2O8+δ samples. (a) Disk with 30µm di-ameter and (b) cuboid with 30µm sides length, all sampleswith 2µm thickness. Magnetic decorations were performed at4.2 K and at applied fields of 20 (left panel) and 40 Oe (rightpanel). The scale-bar corresponds to 10µm.applied field of 40 Oe. The density of non-sixfold coordi-nated vortices, ρdef , strongly depends on the local induc-tion. In the case of macroscopic Bi2Sr2CaCu2O8+δ vor-tex matter, ρdef decreases exponentially up to 20 Gaussand then saturates around 2% as shown in Fig. 3 (a).This is due to the enhancement of inter-vortex interac-tion on increasing field. This magnitude follows the sameB-evolution for mesoscopic vortex matter but is at least50% larger than for bulk samples. In addition, ρdef is al-ways larger in disks than in cuboids for roughly the samevortex density. This can be explained by considering thataligning a compact plane of vortices with the edges of acuboid does not imply to change the orientational orderof the structure whereas in order to do so in a disk thevortex planes have to bend.In the case of macroscopic samples, it has been provedthat the structure observed by means of field-cooling dec-orations at 4.2 K, and therefore its ρdef , is quite closeto the equilibrium17. The possibility of the increase onthe ρdef on decreasing the system size being an out-of-equilibrium phenomena can not be discarded. There-fore we performed molecular dynamics simulations of themesoscopic vortex matter nucleated in a 30µm disk witha density of 15 Gauss in order to test this possibility. Inparticular, we performed tests on the dependence of ρdefwith the cooling rate, inversely proportional to the timeallowed to the system to relax. First, we performed sim-ulations in a macroscopic sample in order to find thepinning magnitude that has to be considered in order toreproduce the observed ρdef . Then we used this magni-Figure 2: Delaunay triangulations of the mesoscopic vor-tex matter shown in Fig. 1. Disclinations are highlighted ingray; sixfold(non-sixfold)-coordinated vortices are indicatedin blue(red). The scale-bar corresponds to 10µm.3tude of pinning to perform simulations in micron-sizedsamples. The results of ρdef as a function of the coolingrate, see Fig. 3 (b), indicate that the observed experimen-tal values correspond to the case of large relaxation times.Therefore, we can ascertain that the amount of topolog-ical defects observed in the experiments in micron-sizedsamples is not an out-of-equilibrium feature and that theobserved structure is quite close to the equilibrium.Having this certainty in mind, we tried to estimatethe geometrical confinement energy induced by the sam-ple edges. In the case of bulk samples, the mean value ofthe inter-vortex interaction energy distribution is slightlyshifted upward with respect to the value for a perfectAbrikosov lattice with the same vortex density10. Sinceafter a field-cooling process vortices are close to equilib-rium, this shifting can only be accounted by the effect ofbulk pinning10. This can be expressed, by unit length,as < int >b −Abr = bp, where b stands for the bulksample and < int > is the mean value of a distributionof inter-vortex interaction energies, Abr the value of theinter-vortex interaction energy in a perfect Abrikosov lat-tice (a delta-function), and p the pinning energy. In thecase of mesoscopic vortex matter, an extra term entersinto the energy-balance, namely the confinement energyεconf and therefore < int >meso −Abr = mesop + εconf ,where meso means for the case of mesoscopic vortex mat-ter and the energies are noted similarly as in the pre-vious case. Therefore, one can have access to an es-timation of the confinement energy in mesoscopic vor-tex matter just by assuming that the pinning magni-tude is the same irrespective of the sample size, and thenεconf =< int >meso − < int >b.The inter-vortex interaction energy per unit length de-pends on the inter-vortex distances rij , and for a vor-tex i has a value iint =∑j 20K0(rij/λ), with the sumover neighbor-vortices j, 0 ∝ λ2 the vortex line ten-Figure 3: Density of topological defects (non-sixfold coordinated vortices) in mesoscopic and macroscopicBi2Sr2CaCu2O8+δ vortex matter. (a) Experimental data as afunction of applied field for the mesoscopic vortex structuresnucleated in the 30µm disk and cuboid, and macroscopic vor-tex matter. (b) Results from molecular dynamics simulationsas a function of the simulations relaxation time in the case ofa 30µm disk and B = 15 Gauss.sion, and K0 the zeroth-order modified Bessel function.In real cases this magnitude is spatially inhomogeneousdue to the elastic and plastic deformations of the struc-ture, and therefore there is a distribution of iint withan almost-Gaussian shape10. Only in the case of anideal Abrikosov lattice this magnitude is space invari-ant and its distribution is a delta function. We haveperformed inter-vortex energy-distribution calculationsin vortex structures observed by magnetic decoration inthe macroscopic samples from which were engineered thedisks and cuboids. We also performed the same calcu-lations in the mesoscopic vortex matter nucleated in thedisks and cuboids at both applied fields. Irrespective ofthe field, the mean values of < int >meso are alwayslarger than in the case of macroscopic vortex matter,what can be reasonably ascribed to the extra deforma-tions introduced by the larger amount of topological de-fects nucleated in the micron-sized samples. In accor-dance with this reasoning, the < int >meso value is al-ways larger for the structure nucleated in the disks thanin the cuboids by a 6-9% on increasing field. Thereforethe estimated confinement energy is in the case of disksequal to 1 ± 0.1 × 10−8 erg/cm∼ 0.130, larger than inthe case of cuboids, 0.8± 0.1× 10−8 erg/cm∼ 0.110.IV. CONCLUSIONSThe edges of the samples do produce a geometrical con-finement effect in mesoscopic vortex matter that is put inevidence by the orientation of the outer shells of vorticeswith compact planes parallel to the edges what producesa concomitant increase of the density of topological de-fects. By means of molecular dynamics simulations weshow that the density of defects found experimentally isnot an out-of-equilibrium feature but rather the effect in-troduced by geometrical confinement. By means of differ-ences in the mean value of the inter-vortex interaction ofthe mesoscopic and macroscopic vortex structures we areable to quantify the confinement energy per unit length.We find that is just 0.11− 0.13 0, the larger value in thecase of disks than cuboid geometries.41 V. V. Moshchalkov, et al., Nature 373, 319 (1995).2 A. K. Geim, et al., Nature 390, 256 (1997).3 V. A. Schweigert, F. M. Peeters, and P. S. Deo, Phys. Rev.Lett. 81, 2783 (1998).4 J. J. Palacios, Phys. Rev. B 58, R5948 (1998).5 V. Bruyndoncx, et al., Phys. Rev. B 60, 10468 (1999).6 L. R. E. Cabral, B. J. Baelus, and F. M. Peeters, Phys.Rev. B 70, 144523 (2004).7 Y. Fasano, J. Herbsommer, and F. de la Cruz, Phys. Stat.Sol. (b) 215, 563 (1999).8 M. Menghini, et al., Phys. Rev. Lett. 90, 147001 (2003).9 A. P. Petrovic, et al., Phys. Rev. Lett. 103, 257001 (2009).10 S. Demirdis, et al., Phys. Rev. B 84, 094517 (2011).11 Y. M. Wang, et al., Phys. Rev. B 65, 184506, (2002)12 H. Pastoriza, et al., Phys. Rev. Lett. 72, 2951 (1994).13 E. Zeldov, et al., Nature 375, 373 (1995).14 D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).15 L. I. Glazman, and A. E. Koshelev, Phys. Rev. B 43, 2835(1991).16 H. Pastoriza and P. H. Kes, Phys. Rev. Lett. 75, 3525(1995).17 Y. Fasano,et al., Proc. Nat. Acad. Sci. 102, 3898 (2005).18 M. I. Dolz, A. B. Kolton and H. Pastoriza, Phy. Rev. B81, 092502 (2010).19 Y. Fasano, et al., Solid State Commun. 128, 51 (2003).20 M.I. Dolz, et al., submitted to Phys. Rev. B, cond-mat:1212.4564. (2014).21 M. Konczykowski, et al., cond-mat: arXiv:1212.4564.22 A. Kolton, D. Domı´nguez, and N. Grønbech-Jensen, Phys.Rev. Lett. 83, 3061 (1999).

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Geometrical Confinement Effects in Layered Mesoscopic Vortex Matter

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