Length dependence of the number of phase slip lines in a superconducting strip

Journal Pre-proof Length dependence of the number of phase slip lines in a superconducting strip C.A. Aguirre, E. Sardella, J. Barba-Ortega

PII: DOI: Reference:

S0038-1098(19)30723-9 https://doi.org/10.1016/j.ssc.2019.113799 SSC 113799

To appear in:

Solid State Communications

Received date : 16 August 2019 Revised date : 14 November 2019 Accepted date : 19 November 2019 Please cite this article as: C.A. Aguirre, E. Sardella and J. Barba-Ortega, Length dependence of the number of phase slip lines in a superconducting strip, Solid State Communications (2019), doi: https://doi.org/10.1016/j.ssc.2019.113799. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Journal Pre-proof

of

Length dependence of the number of phase slip lines in a superconducting strip C. Aguirre 1, E. Sardella2, J.Barba-Ortega1C. A. Aguirre

pro

Departamento de Física, Universidade Federal de Mato Grosso, Cuiabá - Brasil E-mail: [email protected] 3

2

re-

E. Sardella Departamento de Física, Universidade Estadual Paulista, Baurú - Brasil E-mail: [email protected] 3

J. Barba - Ortega Departamento de Física, Universidad Nacional de Colombia, Bogotá - Colombia E-mail: [email protected]

Jo

urn al P

Corresponding Author: Cristhian Andres Aguirre. Universidade Federal de Mato-Grosso Av. Fernando Corrêa da Costa, nº 2367 - Bairro Boa Esperança. Cuiabá - MT - 78060-900 Fone/PABX: +55 (65) 3615-8000 / FAX: +55 (65) 3628-1219

Noname manuscript No. (will be inserted by the editor)

Journal Pre-proof

C. A. Aguirre · E. Sardella · J. Barba-Ortega

of

Length dependence of the number of phase slip lines in a superconducting strip

pro

Received: date / Accepted: date

re-

Abstract We study the superconducting state of a strip under an transport current Ja and an external magnetic field H. We show that the value of the critical currents at which the vortex-antivortex (V-Av ) pair penetrates the sample, the number of phase-slip lines, and their average velocities and dynamics strongly depend on the size of the sample L. Our investigation was carried out by numerically solving the two-dimensional generalized timedependent Ginzburg-Landau equations (GTDGL). Keywords Ginzburg-Landau · Anti-vortex · Mesoscopics · Superconductor · Phase slip lines

1 Introduction

urn al P

PACS 74.20.De · 74.25.Bt · 74.25.Uv · 74.25.Wx

One mechanism that can explain the resistive state of a superconducting sample in presence of an externally applied current is the so-called kinematic vortices 1 . This was first proposed by Andronov et al. and later observed experimentally by Sivakov et al. [1,2]. They consist of V-Av pairs that move at a C. A. Aguirre Departamento de F´ısica, Universidade Federal de Mato Grosso, Cuiab´ a - Brasil E-mail: [email protected] E. Sardella Departamento de F´ısica, Universidade Estadual Paulista, Baur´ u - Brasil E-mail: [email protected] J. Barba - Ortega Departamento de F´ısica, Universidad Nacional de Colombia, Bogot´ a - Colombia E-mail: [email protected] 1

Jo

It is already well accepted that the kinematic vortices are constituted of a V-Av pair which moves at a much larger velocity than a normal Abrikosov vortex. The

Journal Pre-proof 2

re-

pro

of

much larger velocity than a normal Abrikosov vortex [1–9]. Several theoretical and experimental investigations have been performed on superconductors in the presence of external applied transport currents. For instance, A. G. Sivakov et al., using low-temperature scanning laser microscopy, observed the phase-slip lines and also observed Shapiro steps under microwave radiation, which showed that the frequency of the order parameter oscillation is equal to the Josephson frequency [10]. G. R. Berdiyorov et al. reported variations in voltage for superconducting systems with an external current, which accounts for Andreev-type states exhibiting non-monotonic behavior in the magnetoresistance, which is common in granular superconductors. They also analyzed the Little-Parks-like oscillations stemming from vortex crossings through the weak links of normal metals in the presence of both magnetic and electrical fields by using the time-dependent Ginzburg-Landau (TDGL) theory and found that the vortices penetrate the weak superconducting region with a smaller critical temperature and that large magnetoresistance oscillations are due to current-excited moving vortices, [11–13]. J. Barba-Ortega et al., using TDGL theory, analyzed the resistive state of a mesoscopic superconducting strip at zero external applied magnetic field under an electric transport current subjected to different types of boundary conditions and roughness of the sample and showed that the value of the current at which the first vortex/anti-vortex pair penetrates the sample, as well as their average velocities and dynamics, strongly depend on the boundary conditions and roughness percentage [4, 14]. also Arutyunov et al. experimentally, show that superconducting samples close Tc exhibits thermal fluctuations, presenting states with finite resistivity in a nanowires [15] it has also been reported measurements of slip phase in indium oxide sample [16], with possible applications in a quantum metrology.

urn al P

P. Sanchez-Lotero et al. studied the flux-flow phenomena in a superconducting mesoscopic stripe submitted to an applied current and external magnetic field, and showed that the I − V curves, present a universal behaviour [17]. In the present contribution, we study the superconducting state of a strip under an external applied current and magnetic field (see Figure 1). The sides of the bridge are attached to two electrodes positioned symmetrically (indicated by the blue color). We found that the value of the critical current density at which the pairs of kinematic vortices enter the sample and the number of phase-slip lines strongly depend on the size of the sample L. This paper is organized as follows: In Section 2 we describe the theoretical formalism used to study a mesoscopic thin film in the presence of an applied current at zero magnetic field. Then in Section 3 we present the results of the numerical solution of the time-dependent Ginzburg Landau equations for a sample with a variable surface. In Section 4, we present our conclusions.

Jo

order parameter nearly vanishes along the line where the kinematic vortices move, although it has two minima which carry the singularities of its phase.

Journal Pre-proof 3

2 Theoretical Formalism

  ∂A ¯ = Re ψ(−i∇ − A)ψ − κ2 [∇ × ∇ × A] ∂t

of

We consider a very thin bridge of thickness d  ξ and intermediate width (ξ  W  λ). Within this approximation, we can neglect the magnetic field produced by the transport current itself. Therefore, this can be treated as a two-dimensional problem. The general form of the generalized timedependent Ginzburg Landau (GTDGL) equation in dimensionless units is given by [18–21]:   ∂ψ Γ 2 ψ ∂|ψ|2 µ p + + iΦψ = −(i∇ + A)2 ψ + (1 − |ψ|2 )ψ(1) 2 ∂t 1 + Γ 2 |ψ|2 ∂t (2)

urn al P

re-

pro

where ψ represents the order parameter, A the potential vector, and κ = 1.0 is the Ginzburg-Landau parameter. The constants are taken with the values Γ = 10 and µ = 5.75, the range 10 ≤ Γ ≤ 20 is suitable for most metals like the N b compound [7, 22, 23]. The equations p are presented in an adimensional form, as follows: |ψ| in units of ψ∞ = −α/β; lengths in units of the coherence length ξ; A in units of Hc2 ξ, where Hc2 is the second critical field; time in units of Ginzburg-Landau time tGL = π¯h/8KB Tc η; the scalar potential Φ in Φ0 = h ¯ /2etGL units; and the external applied current J in J0 = cσ¯h/2etGL , where σ is the conductivity in the normal state. δ, x = δ, andy = 0.1ξ(0) are the dimensions of mesh sample. The usual superconducting-normal boundary conditions (∇ − iA)ψ · n = 0 are taken in non-contact sections, and in the contact sections we use the Dirichlet boundary condition ψs = 0. The TDGL equations must comply with the continuity equation ∇ · J + ∂t ρ(r, t) = 0 and the external current ∇2 Φ = ∇ · Js and iteratively solve the GTDGL and the Poisson equations for the sample [24–27]. The main idea of the link variable method is to use a new variable U that depends on A, to transform the equations and then proceed with a finite element scheme (see[28]). About the wellposedness, unicity and existence of the solution for the problem, there is no simple answer in nonlinear cases, but for the link variable method the wellposedness is guaranteed by the convergence for the time   ∆x2 ∆x2 , (3) ∆t = min 4 4κ

with κ the Ginzburg-Landau parameter and ∆x the grid for the spatial size of the sample. 3 Results

3.1 Kinematic Vortex

Jo

In Figure 1, we illustrate the studied sample. Here, L = 12ξ, 24ξ, 36ξ, 48ξ, w = 8ξ, and a = 2ξ. A dc current density Ja is uniformly applied through

Journal Pre-proof

of

4

pro

Fig. 1: (Color online) Schematic view of the studied system: a strip of length L and width w; the width of the electrodes is a, through which a uniform dc current density Ja is injected in presence of an external magnetic field H.

Jo

urn al P

re-

the electrodes. In Figure 2, we show (a) the time-averaged voltage as a function of the applied current-density Ja for sizes L = 12ξ, 24ξ, 36ξ, and48ξ of the sample, and (b) resistivity ∂V /∂Ja as a function of the applied current Ja for L = 12ξ, 36ξ, and48ξ. Note that Jc1 is practically constant for all L used, while a new Jc appears when L increases, finding a Jc4 independent of L. In addition, it is shown that the local maxima in the resistivity increase as L increases. This is the result of the inclusion of kinematic vortices in the sample. This increase in resistivity can be explained from two different points of view. The first and most common is that since the condensate is in the presence of an external electric field, the speed increases and in general the condensate becomes unstable, accounting for variations of the order parameter |ψ|, bringing it to zero, and after a certain time, the maximum value of that parameter is recovered but is offset by 2π. Initially, this loss of the superconducting state is generated, followed by its restoration. The other point of view is that given the electroctrostatic interaction between the system, the superposition of the supercurrent density, and the external current overlap, since the superconducting condensate is subject to periodic conditions, i.e. superposition causes the two current densities to be out of phase (breaking) at certain points, leading to metastable states, when locally the superconductivity state is lost, accounting for an increase in voltage (resistive states) and in the subsequent movement of kinematic vortices in the sample. For a greater applied current, the state is stabilized, approaching the normal state, and therefore negative magnetoresistence has been reported, which accounts for these systems, explaining these variations in voltage [29]. In Figure 3, the average velocity of a vortex V during the annihilation with an anti/vortex (Av) (V −Av interaction) is plotted for L/ξ = 24, 36, and48. It can be seen that the local maxima coincide with the values of Jci (i =1,2,3,4), i.e. the movement of the V − Av is drastically increased, since when generating higher values of current, this increases the Lorentz force on them by increasing their local speed. Additionally, it shows how the maxima exhibit

Journal Pre-proof

urn al P

re-

pro

of

5

Fig. 2: (Color online) (a) Time-averaged voltage V as a function of the applied current-density Ja for L = 24ξ, 36ξ, 48ξ, (b) Resistivity ∂V /∂Ja as a function of the applied current Ja for L = 24ξ, 36ξ, 48ξ, at H = 0.

Jo

a decreasing asymptotic behavior for L/ξ = 36, 48, in particular the case L/ξ = 24 only exhibits one maximum. This gives rise to dependence on the size of the sample, since for samples of smaller sizes the behavior after Jc1 is

Journal Pre-proof

pro

of

6

Fig. 3: (Color online) Average velocity of the V-Av pairs, for L/ξ = 24, 35, 48, at zero magnetic field. J1 1.70 1.64 1.62 1.62 1.80 1.62

J2 None None 1.74 1.66 1.90 1.76

PLines 1 1 1-3 1- 3 3-5 3-5

re-

L 18 21 24 36 42 48

urn al P

Table 1: Current at wich occurs the first (J1 ) and second (J2 ) apparition of the phase slip line vortex PLines , for a sample os size L at H = 0.

Jo

linear, not generating jumps in the voltage (only the resistive state), added to the fact that for the sizes L/ξ = 18, 21 there is no second critical current Jc2 (see Tab 1). In Figure 4, the behavior of ln|ψ| and the behavior of the different fringes where the V − Av pairs are moved are presented, which are expected to be perpendicular to the direction of the application of the external current. As the size increases, larger lines are generated, for which their coordinated movement is generated and the saturation of these paths is greater, accounting for resistive states in the system even in the absence of an external field, i.e. the movement of the kinematic vortices in the sample is at high speed, which is why the macro behavior of the pairs V − Av is seen, since the oscillations in the order parameter account for the periodic creation of these pairs, but so quickly that upon being annihilated (expelled from the sample), it accounts for the creation of a new pair in the central position of the sample, a process that is repeated several times. As it is, it softens the condensate; this explains why we cannot distinguish the individual movement of each pair. In Figure 5, the time-averaged voltage V as a function of the time for a) L = 24ξ, b) L = 36ξ, and c) L = 48ξ is plotted. As can

Journal Pre-proof 7

of

Fig. 4: (Color online) Snapshots of the logarithm of the order parameter, ln|ψ| for L = 24ξ, 36ξ, 48ξ at Ja = 1.74 with a number of PLine = 3, 3, 5 phase split lines respectively at zero magnetic field.

urn al P

3.2 Abrikosov Vortex State: Lorentz Force

re-

pro

be seen, the voltage oscillates periodically in time, with a global minimum corresponding to the Meissner state. Over time, a V − Av pair penetrates the sample, leading to a local maximum in the voltage curve. This V − Av pair resides inside the sample for a short time, during which the voltage reaches a local minimum. Later, it leaves the sample through a maximum in the V (t), and the system relaxes into its initial state. Immediately afterward, a new V − Av pair penetrates the sample, and the entire V − Av vortex entry and exit sequence repeats. Thus the finite voltage is due to the periodic entrance of V − Av pairs [3].

Jo

In Figure 6, we show the behavior of |ψ| for the same external magnetic field in the presence of the different external currents. We show that as the current increases, the Lorentz force also increases (F = −J × B), and hence the velocity of the vortice in the sample. This linear increase is described in Figure 2, since linear monotonical behavior is expected for values less than Jc1 , given the Abrikosov vortices entry. The voltage in the sample is modified and generates an increase in resistivity as it is altered due to the entry of more vortices and interaction in the systems different densities of currents between the vortices. With this, the macroscopic effect is to generate a type of interference in the accelerated movement that perceives the vortex, similar to the viscous effect in a fluid. In Figure 7, the order parameter and the phase difference ∆Φ are shown for constant values of the external field H = 0.25 and the applied current Ja = 0.50 at different times. The movement of the Abrikosov vortice can be seen, according to the direction of the Lorentz force line. Since the force is constant (H and J constant), a constant acceleration of the vortex in direction −y can be seen. In Fig 7, the difference of phase (2π) due to the reestablishment of the superconducting state can be seen.

Journal Pre-proof

urn al P

re-

pro

of

8

Fig. 5: Time-averaged voltage V as a function of the time for a) L = 24ξ, b) L = 36ξ, and c) L = 48ξ. Right figure, zoom from a section of the left figure.

4 Conclusions

Jo

By solving the generalized time-dependent Ginzburg-Landau equation, we analyzed the resistive state of superconducting bridges under an applied dc electrical current and an applied magnetic field H. We found that the critical current for the transition to the resistive state, Jc1 , exhibits a weak dependence on the size of the sample, while the number of jumps in the I −V curve, corresponding to different critical currents, exhibits a strong dependence on L. We showed the process of phase-shifting of the system in the presence of an external current, due to the loss of the superconducting state, because of the momentary instability (high velocities) of the condensate.

Journal Pre-proof

of

9

re-

pro

Fig. 6: (Color online) Snapshots of the order parameter, |ψ| (yellow/blue corresponds to largest/zero |ψ|) for H = 0.50 at indicates values of Ja . FL is the Lorentz force

urn al P

Fig. 7: (Color online) Snapshots of the time evolution of the order parameter, |ψ| for H = 0.25 and Ja = 0.50. FL indicates the Lorentz force. 5 Acknowledgements

C. A. Aguirre would like to thank the Brazilian agency CAPES, for financial support and the Ph.D fellowship, grand number (089.229.701-89). References

Jo

1. A. Andronov, I. Gordion, V. Kurin, I. Nefedov, and I. Shereshevsky, Physica C, 213, 193 (1993). 2. A. G. Sivakov, A.M. Glukhov, A. N. Omelyanchouk, Y. Koval, P. Muller, A. V. Ustinov, Phys. Rev. Lett. , 91, 267001 (2003). 3. W. J. Skocpol, M. R. Beasley, M. Tinkham, J. Appl. Phys. 45, 4054 (1974) 4. J. Barba-Ortega, E. Sardella, R. Zadorosny, Phys. Lett. A, 382, 215 (2018) 5. G. Berdiyorov, M. V. Miloˇsevi´c, F. M. Peeters, Phys. Rev. B, 79, 184506 (2009). 6. G. Berdiyorov, K. Harrabi, F. Oktasendra, K. Gasmi, A. I. Mansour, J. P. Maneval, F.M. Peeters, Phys. Rev. B, 90, 054506 (2014) 7. G. Berdiyorov, K. Harrabi, J.P. Maneval, F.M. Peeters, Supercond. Sci. Technol. 28, 025004 (2015).

Journal Pre-proof 10

Jo

urn al P

re-

pro

of

8. J. Barba-Ortega, M. R. Joya, E. Sardella, Eur. Phys. J. B, 92, 143 (2019). 9. R. B. Laibowitz, A. N. Broers, J. T. C. Yeh, J. M. Viggiano, App. Phys. Lett. 35, 891 (1979). 10. A. G. Sivakov, A. M. Glukhov, A. N. Omelyanchouk, Y. Koval. P. Muller. and A. V. Ustinov, Phys. Rev. Lett. 91, 267001 (2003). 11. G. R. Berdiyorov, X. H. Chao, F. M. Peeters, H. B. Wang, V. V. Moshchalkov, and B. Y. Zhu, Phys. Rev. B, 86 224504 (2012). 12. G. Berdiyorov, A. R. de C. Romaguera, M. V. Miloˇsevi´c, M. M. Doria, L. Covaci, and F. M. Peeters, Eur. Phys. J. B 85, 130 (2012). 13. G. R. Berdiyorov, M. V. Miloˇsevi´c, M. L. Latimer, Z. L. Xiao, W. K. Kwok, and F. M. Peeters, Phys. Rev. Lett. 109, 057004 (2012). 14. J. Barba-Ortega, H. Blas and M. R. Joya, Physica C, 569, 9 (2019). 15. K. Yu. Arutyunov, T. Hongisto, J. Lehtinen, L. Leino, and A. Vasiliev, Sci. Rep. 2, (293), 1-7 (2012). 16. O. V. Stafiev , L. B. Ioffe, S. Kafanov, Yu. A. Pashkin , K. Yu. Arutyunov , D. Shahar, O. Cohen , and J. S. Tsai, Nature, 484, 355-357, (2012). 17. P. S´ anchez-Lotero, D. Dom´ınguez and J. A. Aguiar, Eur. Phys. J. B, 89, 141 (2016). 18. Q. Du, M.D. Gunzburger, Physica D 69, 215 (1993). 19. S. J. Chapman, Q. Du, M. S. Gunzburger, Z. Angnew. Math. Phys. 47, 410 (1996). 20. Q. Du. M. D. Gunzburger, J. S. Peterson, Phys. Rev. B, 51, 16194 (1999). 21. C. A. Aguirre, Q. Martins and J. Barba-Ortega, Rev. UIS Ing. , 18, (2), 213 (2019). 22. J. Watts-Tobin, Y. Kr¨ ahenb¨ uhl, and L. Kramer, J. Low. Temp. Phys. 42, 459 (1981). 23. L. Kramer, R. J. Watts-Tobin, Phys. Rev. Lett. 40, 1041 (1978). 24. A. van Blaanderen, R. Ruel, and P. Wiltzius, Nature, 385, 321 (1997) 25. L. Van Look, B. Y. Zhu, R. Jonckheere, B. R. Zhao, Z. X. Zhao, and V. V. Moshchalkov, Phys. Rev. B, 66, 214511 (2002). 26. C. Aguirre, Q. Martins, A. de Arruda, J. Barba, Heliyon, 5, e01570 (2019). 27. D. Y. Vodolazova, G. Berdiyorov, F. M. Peeters, Physica C, 552, 64 (2018). 28. G. C. Buscaglia, C. Bolech and A. Lopez, Connectivity and Superconductivity, J. Berger, and J. Rubinstein (Eds.), Springer, (2000). 29. A. K. Elmurodov, F. M. Peeters, D. Y. Vodolazov, S. Michotte, S. Adam, F. de Menten de Horne, L. Piraux, D. Lucot and D. Mailly, Phys. Rev. B. 78 214519 (2008).

Journal Pre-proof

Conflict of Interest and Authorship Conformation Form Please check the following as appropriate: All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

o

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

o

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

o

The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:

Jo

urn al P

Author’s name Cristhian Andres Aguirre Edson Sardella Jose Barba

re-

pro

of

o

Affiliation Universidade Federal de Mato Grosso Universidade Estadual Paulista Universidad Nacional de Colombia

Journal Pre-proof

Dependence of the Slip phases with the size of the sample.

·

Kinematic vortex.

·

Creation/Destruction Vortex/anti-Vortex.

Jo

urn al P

re-

pro

of

·