Complex Vortex Configuration in a Disk with a Higher ...

Complex Vortex Configuration in a Disk with a Higher/Lower Critical Temperature Superconducting Geometrical Central Defect J. D. González, Jairo Barón-Jaimez & J. Barba-Ortega

Journal of Low Temperature Physics ISSN 0022-2291 Volume 179 Combined 3-4 J Low Temp Phys (2015) 179:264-275 DOI 10.1007/s10909-015-1279-9

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Author's personal copy J Low Temp Phys (2015) 179:264–275 DOI 10.1007/s10909-015-1279-9

Complex Vortex Configuration in a Disk with a Higher/Lower Critical Temperature Superconducting Geometrical Central Defect J. D. González · Jairo Barón-Jaimez · J. Barba-Ortega

Received: 21 November 2014 / Accepted: 14 January 2015 / Published online: 5 February 2015 © Springer Science+Business Media New York 2015

Abstract The influence of a pentagonal/hexagonal defect at higher/lower critical temperature in a thin superconducting disk on the critical fields and vortex configuration is investigated numerically. The coupled nonlinear Ginzburg–Landau equations are solved self-consistently to calculate and compare quantities such as the magnetization, super-current, free Gibbs energy, Cooper pair density, magnetic induction, and phase of order parameter. For certain magnetic field ranges, we found a coexistence of a giant vortex state, placed in the center of the disk and a complex multi-vortex state, toward the corners of the defect. Keywords

Ginzburg–Landau · Geometrical defects · Mesoscopics

1 Introduction Vortices (tubes of quantized magnetic flux) allow the superconductor state to survive at high applied magnetic field in type-II superconductors, making them technologically relevant. The vortices have a non-superconducting core with a radius on the scale of the coherence length ξ and a circulating super-current that generates one quantum of magnetic flux, Φ0 = h/2e, over the scale of the London penetration depth. Although

J. D. González Grupo de Teoría de la Materia Condensada, Universidad del Magdalena, Santa Marta, Colombia e-mail: [email protected] J. Barón-Jaimez Universidad Nacional Autónoma de México (UNAM), Mexico, Mexico J. Barba-Ortega (B) Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia e-mail: [email protected]

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the paired electrons in a superconductor carry charge without resistance, a current will exert a Magnus force on all magnetic flux, which results in dissipation, if any of them move. The vortex pinning is an opportunity to understand and reduce uncontrolled vortex motion. A vortex may be pinned by collocating its energetically costly nonsuperconducting core with a defect that locally suppresses superconductivity [1–5]. Continued reduction in uncontrolled vortex motion will open up applications both for quiet circuits in sensing and communication and for large currents in high-field magnets and power distribution. Collectively controlled vortex motion can serve as a rectifier [6], a vortex ratchet mechanism can perform clocked logic [7], and vortices can control spins in an adjacent diluted magnetic semiconductor [8], while vortices adjacent to an electron gas in a quantum-Hall state may allow the creation of exotic quantum states [9]. A proposed test of the long-standing idea that vortices may entangle like polymers requires controlled local manipulation of single vortices [10,11]. Vortices are of theoretical interest for their own sake [12,13], as clues to the underlying superconductivity [14,15], as analogs for interacting bosons [16], and as model systems for soft condensed matter [12]. In mesoscopic samples, there is a competition between the Abrikosov lattice, being the lowest-energy configuration in bulk material and films, and the boundary, which tries to impose its geometry on the vortex lattice. Experimentally, the magnetization of superconducting disks and rings has been measured as a function of the externally applied magnetic field [17–19]. In small systems, vortices may overlap so strongly that it is more favorable to form one giant vortex [20–23]. Experimental reports show that when the order parameter is enhanced, as a consequence, the surface contact energy has an enhancing effect upon the transition temperature Tc [24]. This could be accomplished, for example, via deposition of a superconductor with a higher intrinsic Tc [25–27] or surface cold working [28,29], or, perhaps, via disordering a thin surface film [30,31], or depositing a thin film of a normal metal [32–34]. The possibility of reversible surface enhancement of superconductivity has been examined experimentally [29]. It is shown that single crystal tin samples with cold-worked surfaces represent a superconductor with a surfaceenhanced order parameter or negative surface extrapolation length, whose magnitude can be controlled. Other experimental results discuss the enhancement of the order parameter in two principally different ways: either as an effect of stresses induced by surface treatment [28] or as the effect of stress-free defects [35]. On the other hand, Kozhevnikov et al. found that a surface modification provided a controllable enhancement of the order parameter [29]. Associated with this is the prediction that surface enhancement can yield a significant increase of the critical temperature for samples, in consequence the practical applications are expanded. In the present paper, we consider a superconducting thin disk in presence of an external magnetic field, and we study the influence of a pentagonal/hexagonal defect formed by a superconductor at higher/lower critical temperature (Fig. 1) on the vortex configuration, vorticity, and magnetization. 2 Theoretical Formalism In the present paper, we consider a defect in a thin superconducting disk immersed in an insulating medium in the presence of a uniform magnetic field H0 . To solve this

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problem we use the thin film limit, d  ξ, λ, (d is the thickness of the disk and Re its radius). Using dimensionless variables, the general form of the TDGL equations in the zero electric potential gage are given by [36–38]:   ∂ψ = − (i∇ + A)2 ψ + ψ 1 − |ψ|2 ∂t

(1)

  ∂A = Re ψ¯ (−i∇ − A) ψ − κ 2 ∇ × ∇ × A. ∂t

(2)

In Eqs. (1) and (2), dimensionless √ units were introduced as follows: |ψ| is the order parameter in units of ψ∞ (0) = −α(0)/β, where α(0) and β are two phenomenological constants; lengths in units of the coherence length ξ(0); time in units of t0 = π h¯ /8K B Tc ≈ 10−12 s ; A in units of Hc2 (0)ξ(0), where Hc2 (0) is the second critical thermodynamic field, T = 0 temperature in units of the critical temperature ¯ − A)ψ] in units of J0 = h¯ c2 /8π eξ , λ(0) Tc [25–27], super-current J = Re[ψ(−i∇ is the penetration depth. For a thin disk with regions on its surface at different critical temperature, Eq. (1) can be rewritten as  2 ∂ψ(r, θ ) = − i∇r,θ + A(r, θ ) ψ(r, θ ) + ψ(r, θ )( f (r, θ ) − |ψ(r, θ )|2 ), ∂t

(3)

where f (r, θ ) is a function which describes points (r, θ ) at different critical temperature. We choose d = ξ(0) and Re = 13ξ(0). Largest unit cells lengths were ar = Re /Nr and aθ = 2π/Nθ , taking Nr = Nθ = 64 as the number of cells in the r and θ directions, respectively. In a microscopic analysis, it is possible to establish a link between the microscopic theory of deGennes and the Ginzburg–Landau theory but this is not an objective of this paper [39,40]. 3 Results and Discussion Let us now turn to the presentation of the results that arise from the numerical solution of TDGL equations. Here, we are interested only in the vortex configurations and the choice of geometrical defects that is adequate for this purpose. The disk and defects areas are ADisk = 132 ξ(0)2 π  531ξ(0)2 , APentagon  116ξ(0)2 , and AHexagon  127ξ(0)2 , respectively. We consider f (r, θ ) = 1 everywhere, except at defects position in the disk which are simulated by using f (r, θ ) = 0.8 for the superconductor at lower critical temperature and f (r, θ ) = 1.2 for the superconductor with higher critical temperature, see Fig. 1. 3.1 Magnetization and Vorticity In the first step, we will compare the magnetization Fig. 2 (up) and vorticity Fig. 2 (down) as a function of the applied magnetic field for the two geometries considered. The disk with a hexagonal defect formed by a superconductor with a higher critical temperature (Sup > Tc ) is analyzed by using the green curves in both magnetization and vorticity, and the blue one for a pentagonal defect formed by a superconductor with lower critical temperature (Sup < Tc ). It is clear to see in Fig. 2 (up) that the

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Fig. 1 Layout of the studied sample: superconducting disk with a central superconductor defect with higher critical temperature (green) or at lower critical temperature (blue), for pentagonal (left) or hexagonal (right) case (Color figure online)

superconducting state is not always the Meissner state, further, for the chosen boundary condition superconductor/vacuum b → ∞, the first transition field H1 /Hc2 = 1.0556 is independent of the defect geometry and the kind of the superconductor used. For 1.0556 < H0 < 1.3975 (in the mixed state), we observe significant differences in the magnetization in the samples with a Sup < Tc and a Sup > Tc . While for 1.3975 < H0 < 1.38, the magnetization curves behave similarly, increasing the magnetic field, the superconductivity is destroyed at H0 /Hc2 = 1.8. Finally, in the down branch of the magnetic field, at H0 = 0, we find that the magnetization is zero for the hexagon case, while for a pentagonal defect, the magnetization is −4π M = −0.2501/Hc2 , with obvious paramagnetism in the sample. This behavior is clearly reflected in the vorticity Fig. 2 (down), which confirms that the value H0 for the first entry of vortices occurs in H0 /Hc2 = 1.0556 for each defect, and additionally that the vortices are pinned in the case of a pentagon defect at H0 = 0. Now, in the interval 1.0556 > H0 /Hc2 > 0.9542, we can ensure that the number of vortices is similar (pentagon and hexagon cases, both in the down branch of the magnetic field) and from H0 /Hc2 = 0.9542 H0 = 0, the vorticity is different, as we can see in Fig. 2 (down), where the vorticity for the hexagon case is N = 1, while for the pentagon case N = 22 at H0 = 0. 3.2 Cooper Pair Density and Magnetic Field Distribution Figure 2 is in a good agreement with Figs. 3, 4, 5, and 6. Therefore it is easier to find the corresponding values in which the different transitions occur. Additionally, it is possible to count the number of vortices in those figures, and compare them with Fig. 2 (down). In this subsection, we analyze the vortex configuration using the results obtained from the numerical calculations for the Cooper pair density and the

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Fig. 2 Magnetization curve (up) and vorticity N (down) as a function of the applied magnetic field H0 (Sup superconductor) (Color figure online)

magnetic field distribution as a function of the applied magnetic field (increasing ↑ H0 to H2 and then decreasing ↓ H0 to zero, Figs. 3 and 4). In the two upper panels of Fig. 3, we can appreciate a multi-vortex state with N = 24 and Nd = 4 (N implies the number of vortices in the superconductor area, whereas Nd inside the defect) for H0 /Hc2 = 1.0619, 1.0623 respectively. High (low) Cooper pair density is given by red (blue) regions. Following the panels corresponding from left to right, the vortices adopt the form of a hexagon, remaining outside of the defect, but increasing H0 the vortices can go into it. It can be explained due to the enhancement of the order parameter at the interface by choosing the suitable superconductor with higher critical temperature to form the defect. Nd take values of 8, 10, and 14 for H0 /Hc2 = 1.1074,

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Fig. 3 Square modulus of the order parameter |ψ|2 at the indicated applied magnetic field for a thin disk with a hexagonal defect at higher critical temperature (two superior lines). Oblique view magnetic induction h (two inferior lines). (↑ H0 (↓ H0 ) the applied magnetic field is increasing (decreasing) (Color figure online)

1.1265, and 1.2771, respectively; it is precisely in this last one, where we can observe a normal region outside of the defect, and when the force of vortex–vortex interaction overcomes the repulsive force exerted by the defect, the vortices can penetrate into the defect. It is interesting to analyze the behavior of the vortices when the magnetic field is decreasing: the vortices start to exit, as we can see in the second panel in Fig. 3, from H0 /Hc2 = 0.6111 to H0 /Hc2 = 0.0349. Figure 3 indicates that the superconductor/defect interface can be easily tunneled, due to the order parameter suppression, from the defect to outside of it. The two panels in the bottom in Fig. 3 show the plot of the magnetic field distribution in the sample, high (low) magnetic field is given by red (blue) regions. The red spots in the sample represent the nucleus of the vortices and the red regions at the surface of the sample represent the compression of the magnetic field. Also it is possible to find that the magnetic field distribution showed a correspondence with the Cooper pair density in the same values of applied magnetic field H0 . In fact, even when only a few quanta of magnetic flux are nucleated into a mesoscopic sample, the vortex interactions with other vortices and with the screening currents circulating around the boundary give rise to a variety of configurations such as multi-vortex states. These configurations are strongly influenced by both the geometry and size of the sample and defect. In Fig. 4, choosing the following superior panel from left to right at H0 /Hc2 = 1.0617 the vortices “feel” the boundary of the pentagonal defect (at lower critical temperature), then the vortices go inside the defect at lower

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Fig. 4 Square modulus of the order parameter |ψ|2 at H0 for a thin disk with a pentagonal defect at lower critical temperature (two superiors lines). Oblique view of the magnetic induction h (two inferior lines) (Color figure online)

values of H0 , from which the order parameter in the defect frontier is suppressed as long as Figs. 3 and 4 are compared. As a consequence, this region reaches the normal state before the rest of the disk (contrary to what we note in Fig. 3). So, the pinning effect in the down branch of the magnetic field is enhanced, see the second panel in Fig. 4. At H0 /Hc2 = 0.1419 it is possible to see the vortices caged in the defect despite the fact that H0 = 0, it can be noted also in Fig. 2 (blue curve, for pentagonal defect at lower Tc ) where at H0 = 0 a negative value of the magnetization remains (Fig. 2, up) as well as a positive value of vorticity (Fig. 2, down). Notice that in the magnetic field distribution (two lower panels in Fig. 4) the vortices move towards the center when the magnetic field is increasing. At higher magnetic field, individual vortices are no longer visible in the contour plots of the magnetic field distribution at H0 /Hc2 = 1.5777. Also, at H0 /Hc2 = 1.5777, the vortices are too close to each other and the corresponding spots are overlapping in the picture. How does the geometry of the defect influence the vortex states? Figures 3 and 6 show in both cases pentagonal and hexagonal defects which are formed by a superconductor at higher critical temperature than the rest of the disk. We found that the number of vortices on the defect depends on its geometry. In Fig. 3 (hexagonal defect) at H0 = 1.1256 we see Nd = 8, and in Fig. 6 (pentagonal defect) at H0 = 1.1577, Nd = 1, just one vortex can penetrate the defect and this vortex remains from H0 = 1.1000 to H0 = 1.1577, which shows the influence of the geometry in the anti-pinning effect. The opposite situation occurs in Figs. 4 and 5 due

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Fig. 5 Square modulus of the order parameter |ψ|2 at H0 for a thin disk with an hexagonal defect at lower critical temperature (two superiors lines). Phase of the order parameter (two inferior lines) (Color figure online)

to a defect at lower critical temperature regarding to the rest of the disk. If we compare the panels at H0 = 1.0626 and H0 = 1.0642 the vortices entrance might occur first in the pentagonal case then the hexagonal case itself. It shows that the geometry of the defect has a clear effect over the force pinning in our system.

3.3 Phase of Order Parameter In Fig. 5, we can appreciate that the vortices are caged in hexagonal disposition at H0 = 1.0606 and the superficial superconductivity is reached at H0 = 1.3710. Now, if we compare snapshot of two superior panels in Figs. 5 and 6 for pentagonal case respectively, a multi-vortex configuration can be maintained in the defect with f (r, θ ) = 1.2, regarding to the defect where f (r, θ ) = 0.8, because the vortices entrance is more gradual in the upper branch of the magnetic field which can be more highly perceived in the second case (Fig. 6) than in the first one (Fig. 5). It is possible to note, comparing the snapshots at H0 /Hc2 = 1.3710 in Fig. 5 that the normal state is reached in the center of the disk, whereas in Fig. 6 at H0 /Hc2 = 1.6078 remain a multi-vortex state. This behavior can be confirmed observing the lower panels in Figs. 5 and 6, owing to this the phase of order parameter allows to determine the number of vortices in a given region and the fact that the phase of ψ varies by 2π by making a complete

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Fig. 6 Square modulus of the order parameter |ψ|2 at H0 for a thin disk with a pentagonal at higher critical temperature (two superiors lines). Phase of the order parameter (two inferior lines) (Color figure online)

circuit, corresponding to the existence of a single quantum flux associated with the vortex. If the vorticity in this region is N , then the phase changes φ/2π . The color code for values of the phase close to one are given by red regions and close to 2π by red regions. The discrete configuration of the vortices into the defect are found in the pentagonal case (Fig. 6) at higher values of H0 than the hexagonal case (Fig. 5) before the defect to reach the normal state.

3.4 Superconducting Current Density When a superconducting sample is placed in an external magnetic field, the magnetic field is expelled from the superconductor owing to screening currents near the sample boundary. The direction of the screening currents is such that the corresponding magnetic field is opposite to the external one, which leads to a lower total field in the superconductor. The magnetic field penetrating the superconductor creates currents flowing in a direction opposite to the screening currents. The competition between these currents and the screening currents results in the existence of vortices. Figure 7 shows vector plots of the super-current in the superconducting disk with the conditions described in the caption, for selected vortex configurations (see in the panels of Fig. 7). In this figure, it is clear that the screening currents near to the boundary of the sample

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Fig. 7 (Panel up) Square modulus of the order parameter |ψ|2 at H0 for a thin disk with and hexagonal defect at higher critical temperature with the corresponding simulation of the behavior of the super-current. (Panel down) Similar simulation using a pentagonal defect formed by a superconductor of lower critical temperature (Color figure online)

flow clockwise and the currents around the vortex in the center counter-clockwise. The magnitude of the current is indicated by the length of the arrows; as we can see, it is not the same everywhere in the sample. We see in the upper panel of Fig. 7 at H0 /Hc2 = 1.0691 and in the lower panel at H0 /Hc2 = 0.2239 in the same figure, that when the vortices have entered in the respective defects, in the region between the two vortices the currents around these two vortices cancel each other out. Additional possible information that could be obtained is about the presence of anti-vortices in the sample, looking for some spots in the figure, where the currents flow in clockwise direction. In order to understand the process of nucleation, we calculated the corresponding free energy for the whole system as a function of time (Fig. 8). The results are a series of cusps as a function of time, each of which corresponding to a change

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Fig. 8 Free Gibbs energy loop as a function of the time t for a disk with hexagonal and pentagonal defect (Color figure online)

in the vorticity in the system. Furthermore, the magnetization curve is in agreement with the vorticities of the superconducting system, which represents the diamagnetic variation in the sample. It is seen from Fig. 8 for the green line that the critical fields H1 /Hc2 = 1.0104 (a), H2 /Hc2 = 1.8000 (b), and decreasing to H0 = 0 (c) at the time t1 /t0 ≈ 815, t2 /t0 ≈ 1,436, and t3 /t0 ≈ 32,544, respectively, whereas for the red line we obtain H1 /Hc2 = 1.0709 (d), H2 /Hc2 = 1.8000 (e), and H0 = 0 (f) at the time t1 /t0 ≈ 907, t2 /t0 = 15,564, and t3 /t0 ≈ 43,776, respectively. Additionally, the points (b) and (e) mark the peaks of the curves in which the free Gibbs energy is zero, this helps us to find the time used for the system to reach the normal state. 4 Conclusions We investigated theoretically the influences of the centered geometrical defects formed by higher/lower critical temperature superconducting sample on the vortex configuration in a thin superconducting disk. We calculated the magnetization, Cooper pair density, magnetic field, super-current density, and free energy as a function of time for hexagonal and pentagonal defects. The vortex lattice was different in the two geometries used due to the fact that it tries to adapt to the geometry of the sample. This influences considerably the stability range of the different vortex states. We also want to emphasize that the surface barrier generated by given f (r, θ ) values plays a crucial role for the pinning/anti-pinning effect, which not only accelerates the vortices to rush into the central area but also decelerates or even stops the vortex to escape from the defects due to the surface barrier. Acknowledgments The authors would like to give thanks to Edson Sardella for his useful discussions. This work was partially financed by the Universidad del Magdalena (Fonciencias) and the Colombian Agency Colciencias through doctoral scholarships 567.

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