Letter pubs.acs.org/NanoLett
Tunable Generation of Correlated Vortices in Open Superconductor Tubes Vladimir M. Fomin,*,† Roman O. Rezaev,†,‡ and Oliver G. Schmidt*,†,§ †
Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstraße 20, D-01069 Dresden, Germany Laboratory of Mathematical Physics, Tomsk Polytechnic University, Lenin Ave. 30, 634050 Tomsk, Russia § Material Systems for Nanoelectronics, Chemnitz University of Technology, Reichenhainer Straße 70, D-09107 Chemnitz, Germany ‡
S Supporting Information *
ABSTRACT: As shown theoretically, geometry determines the dynamics of vortices in the presence of transport currents in open superconductor micro- and nanotubes subject to a magnetic field orthogonal to the axis. In low magnetic fields, vortices nucleate periodically at one edge of the tube, subsequently move along the tube under the action of the Lorentz force and denucleate at the opposite edge of the tube. In high magnetic fields, vortices pass along rows closest to the slit. Intervortex correlations lead to an attraction between vortices moving at opposite sides of a tube. Open superconductor nanotubes provide a tunable generator of superconducting vortices for fluxon-based quantum computing. KEYWORDS: Superconducting vortices, Ginzburg−Landau theory, open superconductor nanotubes, tunable generator of superconducting vortices
H
ybridization of reduced dimensionality in a superconductor with curved geometry and nontrivial topology has been a rich source of novel physics. For instance, measurements of the magnetoresistance in superconducting cylinders1 and loops2 have revealed fluxoid quantization and a huge impact of nanostructuring on the normal-superconducting phase boundary. Curvature was found to be the key factor to control the depression of the critical temperature in a thin superconductor cylindrical shell as a function of the angle between the axis of the cylinder and the magnetic field.3,4 The coexistence of the Meissner state with various vortex patterns in thin superconducting spherical shells5 was shown to drive the phase transition to higher magnetic fields.6 The multipleperiodicity of magnetoresistance recently observed in YBa2Cu3O6+x (YBCO) nanoscale rings7 implies the existence of a concentric vortex structure with nonuniform vorticity.8 In high-temperature superconducting films patterned into a network of nanoloops, periodic changes in the interaction between thermally excited moving vortices and the oscillating persistent current induced in the nanoloops give rise to the observed large oscillations of the magnetoresistance.9 In the present Letter, we investigate theoretically effects of curvature on the dynamics of vortices in open superconductor tubes, which are placed in a magnetic field perpendicular to the axis. Such a configuration is interesting because the normal to the surface component of the magnetic field is inhomogeneous. Consequently, the Lorentz (Magnus) force due to the interaction of vortices with the transport current flowing in the angular direction occurs to be maximal by amplitude and oppositely directed at the top and bottom parts of a © 2012 American Chemical Society
superconductor tube. Therefore, it can be expected that the vortices channel in the regions close to the top and bottom of a tube and that they move there in the opposite directions with the characteristic times of vortex dynamics in a tube being dependent on radius and significantly different from those in a planar film. Advancements in fabrication of rolled-up micro- and nanotubes10 including superconductor layers (e.g., InGaAs/ GaAs/Nb)11,12 open new possibilities for experimental investigation of the vortex matter in superconductors with curved geometries. Following the experimental choice, we consider a membrane of Nb, rolled-up into an open cylindrical tube of radius R and length L with its axis parallel to the x-axis (Figure 1a). The tube sizes must be much larger than the coherence length in order to embed superconducting vortices. The coherence length ξ and penetration depth λ for a Nb membrane (eqs 4.24b and 4.26b of ref 13) are determined by the parameters ξ0 = 38 nm, λL(0) = 39 nm (ref 14) and the mean free electron path (ref 15) l = 10 nm/(1 + 40 nm/d), which equals 5.6 nm for the thickness of membrane d = 50 nm. The Ginzburg−Landau (GL) parameter is κ = λ/ξ = 5. With the Fermi velocity vF = 6 × 105 m/s (ref 15), the diffusion coefficient is D = lvF/3 = 11.2 × 10−4 m2/s. A narrow paraxial slit is present throughout the whole cylinder. The slit width is taken Δ =1/30 × 2πR for all radii. The applied uniform magnetic field is parallel to the z-axis, B = Bez. Received: October 25, 2011 Revised: January 30, 2012 Published: February 14, 2012 1282
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Without a transport current, the superconductor tube governed by the GL eq 1 evolves during the time of about 1 ns from any initial state to the steady equilibrium state, which contains fragments of a deformed hexagonal vortex lattice in the top and bottom areas of the tube, where the magnetic field is normal (at φ = π/2, 3π/2) or close to normal to the surface. There are no vortices in the side areas of the tube, where the magnetic field is tangential (at φ = 0, π) or close to tangential to the surface. Curvature-induced inhomogeneity of the normal component of the magnetic field Bn(φ) = −B sin(φ) leads to an inhomogeneous deformation of the hexagonal vortex lattice. An increase of the magnetic flux produces more prominent Meissner currents jM as shown in Figure 1b. The existence of supercurrent circulation in a thin-wall cylinder due to a magnetic field component perpendicular to the axis was described in ref 3. Furthermore, for cylindrical thin-film shells in a tilted magnetic field, a superposition of superconducting vortices and supercurrent circulation was anticipated for large enough magnetic field components normal to the axis.4 Our equilibrium patterns of the order parameter (Figure 2) clearly reveal a coexistence of superconductor vortices and Meissner currents in tubes. In a superconductor tube of radius R = 280 nm (which is of the same order of magnitude as the penetration depth and 1 order of magnitude larger than the coherence length for the temperature under consideration) at low magnetic fluxes, Meissner currents flow in the regions free of vortices. At Φ/Φ0 = 54, Meissner currents begin to penetrate the region with vortices. The occurring Lorentz (or Magnus) force pushes the vortices away from the regions where the magnetic field is close to tangential. With further increase of the relative magnetic flux, the sample becomes strongly inhomogeneous; the superconducting state is expelled from the side areas of the tube, while at the top and bottom, the vortices form chains. The number of vortex rows in a chain at Φ/Φ0 = 56 increases from 2 to 3 with doubling the radius from 280 to 560 nm. In tubes, the shape of a vortex configuration is determined by a competition between the triangular vortex lattice and the boundary region on the cylindrical surface, where vortices occur. This boundary region in tubes, as distinct from planar films, is not a geometric boundary, but rather results from interplay between the magnetic field and the curvature of the cylindrical surface. Now, we switch on a transport current with density j = jeφ, which is imposed through the electrodes attached to both sides of the slit as shown in Figure 1a. Each of the vortices moves under the influence of the Lorentz force density13 F = jtot × B due to the total current jtot = j + jM+jvortices, which includes the transport current j, a Meissner current jM, and a current jvortices of other vortices (if present). At low magnetic fields, individual vortices periodically nucleate at edges of the tube (at x = L at around φ = π/2 and x = 0 at around φ = 3π/2) (see Supporting Information). Subsequently, they move along the tube under the action of the
Figure 1. (a) Scheme of the structure. An external current is represented with heavy (red) arrows. The electrodes, which are connected to both sides of a slit, are shown semitransparent. The x-axis is selected as the polar axis of cylindrical coordinates (ρ,φ,x). The angle φ is counted from the direction of the y-axis passing through the middle of the slit. (b) Scheme of Meissner currents in a superconductor tube. Two systems of Meissner currents jM develop flowing along both sides of the cut (represented in red and pink online). The magnetic field BM induced by Meissner current tends to partly compensate the applied magnetic field B. As distinct from a superconductor tube, only one Meissner current flows in a planar film along its perimeter.
The first GL equation16 for the superconductor order parameter ψ is represented in dimensionless form17 ∂ψ = (∇ − iA)2 ψ + 2κ 2ψ(1 − |ψ|2 ) ∂t
(1)
with the GL parameter κ. At the relative temperature T/Tc = 0.95, λ = 279 nm and ξ = 56 nm. Because the membrane’s thickness d ∼ 50 nm is substantially smaller than the effective penetration depth for the Nb membrane, the 2D approximation is applicable. The magnetic field in the sample is approximated by the applied magnetic field B. According to Figure 1a, the normal to the cylindrical surface component of the magnetic field is Bn(φ) = −B sin(φ) and an effective magnetic flux is Φ = 2RLB, which can be expressed in units of the magnetic flux quantum Φ0 = h/(2e) with Planck’s constant h and electron charge e. The vector potential A of the magnetic field is taken in the Landau gauge: A = Aex, A = −By. The transport current density j = jeφ is imposed via the modified boundary conditions at the sides of the slit to which electrodes are attached (∇ − i A)ψ|n ,boundary = 0, [∇ − i(A + j/|ψ|2 )]ψ|n ,electrode = 0
(2)
The calculations are performed within an adaptive finitedifference scheme: after a steady distribution of the order parameter is reached in a sample exposed to a given magnetic flux Φ, a transport current j is switched on, and a stationary regime of the vortex dynamics is determined.17
Figure 2. Equilibrium distribution of the superconducting order parameter vortices at Φ/Φ0 = 8.5, 17.0, and 25.5 in an open tube of radius R = 280 nm and length L = 3.5 μm with a slit of width Δ = 1/30 × 2πR = 59 nm. 1283
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Figure 3. Snapshots of the superconducting order parameter distribution (first row) and accumulated trajectories (second row) of vortices at Φ/Φ0 = 8.5, 17.0, and 25.5 in the presence of a transport current with dimensionless density j = 1.65 (the current density unit is 8.57 × 109 Am−2) in an open tube of radius R = 280 nm and length L = 3.5 μm with a slit of width Δ = 1/30 × 2πR = 59 nm. Our calculations show that there is no appreciable effect of the slit width Δ on vortex dynamics when the ratio Δ/2πR < 0.055.
Lorentz force until they denucleate at the opposite edge of the tube (case Φ/Φ0 = 8.5 in Figure 3). In higher magnetic fields (cases Φ/Φ0 = 17.0 and 25.5 in Figure 3), the steady vortex pattern persists in the presence of a transport current, but is subject to deformations (see Supporting Information). These deformations are more prominent in tubes of larger radius and are visualized by the accumulated vortex trajectories (increments of the local values of the order parameter integrated over time). After a nucleation of a new vortex at one tube edge, its motion governed by the Lorentz force leads to a shift of all the other vortices at the sites in a row closest to the slit, toward the opposite edge, until the last vortex of the row denucleates at the latter edge. The accumulated vortex trajectory represents just this passing of vortices through the sites in the row reminiscent of conduction on a percolating lattice.18 Vortices remote from the slit remain almost immobile, because there the transport current is significantly perturbed by the currents of the vortices, which are closer to the slit. Figure 4 describes quantitatively what is represented in Figure 3 about the dynamics of vortices in tubes subject to magnetic fields of varying magnitude. For a magnetic field that is lower than a “first critical” value Bcr, indicated in Figure 4c, there are no vortices in tubes. At B ≃ Bcr, there appear vortices moving in one and the same direction at the bottom (or top) of a tube. At B > Bcr, two sets of paths correspond to vortices moving in opposite directions in the top and bottom areas of the tube, because they meet opposite transport currents. The opposite directions of the Lorentz force at φ = π/2 and φ = 3π/2 are illustrated in Figure 1a. This behavior is peculiar for tubes. With increasing magnetic field beyond a “second critical” value, there are no vortices because of the strong Meissner currents. In Figure 4a,b, x-coordinates of vortices in tubes with L = 3.5 μm, R = 280 nm for Φ/Φ0 = 8.5 and 11.0, correspondingly, are shown as a function of time. In the sparse-vortex regime, represented in Figure 4a, the time needed for a vortex to move from one to another edge of the tube Δt1 is larger than the time between two consecutive vortex nucleation events at one edge of the tube Δt2. In this regime, a new vortex (at the bottom or at the top part) does not nucleate before an “old” vortex (at the same part) has denucleated, and hence, the time average of the number of vortices at each part of the tube is Btr, the multivortex regime occurs, represented in Figure 4b, where Δt1 > Δt2. Consequently, a new vortex (in the bottom or top area) 1284
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Figure 5. Dynamics of the squared modulus of the order parameter |ψ|2 distribution in the tube with R = 280 nm for Φ/Φ0 = 8.5. In this and subsequent figures, the time (in ns) is indicated on the panels. The external current is applied through the top and bottom boundaries in the panels. The cut corresponds to the upper and lower sides in the sweep (black lines).
Figure 6. Dynamics of the current density jφ distribution in the vicinity of the edge x = L at different time just before nucleation of a vortex in the tube with R = 280 nm for Φ/Φ0 = 8.5.
mT, which corresponds to the relative magnetic flux Φ/Φ0 = 8.5. In the state shown in Figure 5 for t = 55.65 ns, the penetration of a vortex in a tube is impeded by a potential barrier at the edge, which occurs due to the current jφ. As illustrated in Figure 6, a steady redistribution of currents leads to a local decrease of the current density, especially near the point φ = π/2, accompanied with a widening of the current profile and a developing depression of the superconductor order parameter. It is accompanied with a slight increase of the free energy of the system (calculated here for illustrative purposes). When the total current at the edge jφ is close to zero, a barrier effectively disappears, and a vortex nucleates at the edge x = L (panels at t = 55.86 ns in Figures 5 and 6). This is accompanied by a lowering of the free energy of the system. Importantly, simultaneously with a nucleation of a vortex at one edge x = L in the top area of the tube another vortex nucleates at the other edge x = 0 in the bottom area of the tube. This is a
nucleates before the previously nucleated (in the same area) vortex has denucleated, and the time average of the number of vortices in each area of the tube is >1. In the interval 200 nm < R < 600 nm, both Bcr and Btr increase almost linearly with the inverse radius of the tube (lower inset of Figure 4c). As our calculations show, for a moderate number of pinning centers, the vortex dynamics changes only quantitatively. Analysis of the effect of specific pinning on the vortex dynamics in tubes will be performed elsewhere. The vortex penetration in superconductor structures is known to be delayed due to the occurrence of surface potential barriers.19−21 In the case under consideration, as shown below, a barrier significantly changes in time, providing a temporary local depression, which allows for nucleation of a vortex. The calculated dynamics of the squared modulus of the order parameter |ψ|2 distribution in the tube with R = 280 nm is represented in Figure 5 for the applied magnetic field B = 8.8 1285
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Figure 7. Patterns of the superconductor currents in the tube with R = 280 nm for Φ/Φ0 = 8.5, when two vortices are at the shortest distance from each other at t = 56.28 ns. On the unrolled cylindrical surface, the superconductor currents are circulating in the opposite directions. This makes two vortices on the cylindrical surface attract each other.
not occur. This state “signals” the system to start a new cycle of preparing a pattern of the superconducting current, which allows for a synchronous vortex nucleation at the top and bottom parts of the tube as shown in Figure 5. The correlation effect is manifested when the tube sizes are smaller that the decoherence radius. The experiment12 implies that the decoherence radius in the rolled-up structures under analysis is larger than 100 μm. Any effects of decoherence are beyond the scope of the present Letter. Comparison between the characteristic times of vortex dynamics for a planar film17 with Lx = 3.5 μm and Ly = 2π × 420 nm ≈ 2.6 μm and for a tube with radius R = 420 nm (upper inset of Figure 4c) reveals a significant (by a factor of 3 in the analyzed case) increase of both critical magnetic field Bcr and the transition magnetic field Btr by virtue of rolling up that planar film into a tube. This increase is due to a remarkable difference between superconducting vortex dynamics in a tube and in a planar film. The vortices in the latter repel each other. Two correlated vortices on a cylindrical surface effectively attract each other in the analyzed region of radii and magnetic fields. Hence, their motion under the action of the Lorentz force leads to a simultaneous denucleation of both vortices on the opposite edges of the tube as illustrated in Figure 4a. Qualitatively similar behavior of vortex dynamics is revealed for open superconductor tubes with the materials parameters of YBCO. Consequently, the experimentally achievable superconductor tubes of ∼1 μm radius act as a periodic generator of correlated vortex pairs in the tunable frequency range ∼0.1 to 10 GHz. The frequency is tuned by changing the magnetic field. Thus, open superconductor nanotubes provide a controllable vortex transmission line for the superconducting-vortex-based platform of quantum computing.22 For experimental validation of our theory, we suggest to use the technique, which has been already proved to be efficient to analyze the vortex dynamics in patterned superconductors.23 The main idea of the method is to measure the voltage induced by moving vortices. A vortex moving with a velocity v generates13 an electric field E = B × v. The resulting voltage can be measured through the dedicated contacts on the two sides of the slit, which can be realized lithographically. Being of the order of several microvolts by magnitude, the voltage generated by moving vortices in open tubes at the micrometer scale is quite detectable by modern devices.12 In conclusion, characteristic times of nonequilibrium vortex dynamics in an open tube are significantly different to the characteristic times in a planar film under the same magnetic field. This difference is caused not only by a spatial dependence
clear manifestation of a correlation between the dynamics of vortices in both top and bottom areas of the tube. At this stage, there are two vortices moving in the tube under the influence of the Lorentz force: one moving to the left at the top and another one moving to the right at the bottom (t = 55.93 ns to t = 56.91 ns). Although the circulations of superconductor currents in both vortices are of the same sense in 3D, they are opposite when plotted on the unrolled 2D cylindrical surface of the tube in Figure 7. As a consequence, the vortices in the analyzed region of radii and magnetic fields appear to effectively attract each other like a vortex−antivortex pair in the bulk. This scenario is confirmed by the fact that the free energy is minimal when the vortices moving in the top and bottom areas of the tube are just on one diameter. At the moment t = 56.28 ns, when the vortices are at the shortest distance from each other, the free energy reaches its minimum. Further on, under the influence of the Lorentz force the vortices continue their motion over the tube, overcoming the mutual attraction (t = 56.56 ns). Accordingly, the free energy starts to rise. At t ≈ 57.00 ns, the vortex at around φ = π/2 arrives at the edge of the tube x = 0, where there is no barrier because the current is vanishingly small, and denucleates, while the other vortex denucleates at around φ = 3π/2 at the edge x = L. Denucleation of vortices is accompanied with an increase in the free energy of the system. Finally, at t = 57.19 ns the same picture is restored, which existed at the beginning of the above-considered period. Interestingly, our numerical data for Re ψ at t = 57.19 ns (Figure 8) imply that the denucleation of vortices in a tube
Figure 8. Pattern of the Re ψ in the tube with R = 280 nm for Φ/Φ0 = 8.5 at t = 57.19 ns.
results in a state with a strong correlation between the order parameter at the points of future nucleation of both vortices, which are separated from each other by the length of the tube L in the x-direction and by the half of the circumference πR in the φ-direction. In this respect, cylindrical structures are unique compared to planar structures,17 where such a correlation does 1286
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(19) Schuster, Th.; Indenbom, M. V.; Kuhn, H.; Brandt, E. H.; Konczykowski, M. Phys. Rev. Lett. 1994, 73, 1424−1427. (20) Zeldov, E.; Larkin, A. I.; Geshkenbein, V. B.; Konczykowski, M.; Majer, D.; Khaykovich, B.; Vinokur, V. M.; Shtrikman, H. Phys. Rev. Lett. 1994, 73, 1428−1431. (21) Burlachkov, L. Physica A 1993, 200, 403−412. (22) Matsuo, S.; Furuta, K.; Fujii, T.; Nagai, K.; Hatakenaka, N. Appl. Phys. Lett. 2007, 91, 093103. (23) Wördenweber, R.; Hollmann, E.; Schubert, J.; Kutzner, R.; Ghosh, A. K. Appl. Phys. Lett. 2009, 94, 202501.
of the magnetic field component normal to the cylindrical surface but also by correlations between the states of the superconducting order parameter in the opposite areas of the cylindrical surface. Our results demonstrate new perspectives of tailoring nonequilibrium properties of vortices in curved superconductor nanomembranes and of their application as tunable superconducting flux generators for fluxon-based information technologies.
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ASSOCIATED CONTENT
S Supporting Information *
Videos demonstrating the dynamics of vortices in open superconductor tubes. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: (V.M.F.)
[email protected]; (O.G.S.)
[email protected]. Phone: 0049-351-4659-780. Fax: 0049-351-4659-782. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge discussions with C. C. Bof Bufon, E. H. Brandt, A. V. Chaplik, Ch. Deneke, V. N. Gladilin, E. Hollmann, A. Lorke, U. Markwardt, A. S. Melnikov, R. Müller, A. Rosch, J. A. Salus, D. J. Thurmer, A. V. Ustinov, A. Wixforth, and R. Wördenweber. The present work was partly supported by the collaborative project with ZIH of Dresden University of Technology.
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