ArticlePDF Available

Abstract and Figures

The Ginzburg-Landau theory, applied to superconducting materials is based on thermo-magneto-electro-dynamic concepts as phase transitions that enrich the class on this subject. Thus, in this contribution we expose the Ginzburg-Landau time-dependent equations, show the mathematical form for two nano-scale superconducting systems, one bi-dimensional homogeneous al sample with applied external current at zero magnetic field, and one three-dimensional cube in presence of a tilted magnetic fiel at zero applied current. This analysis shows the applicability of the three and two-dimensional model to superconductors. The conveniently Ginzburg-Landau theory show that the magnetic response behavior of the sample is very useful for applications in fluxtronica, SQUIDS design, magnetic resonance, among others.
Content may be subject to copyright.
Journal of Physics: Conference Series
PAPER • OPEN ACCESS
Mesoscopic superconductivty in application
To cite this article: J Barba-Ortega et al 2018 J. Phys.: Conf. Ser. 1126 012003
View the article online for updates and enhancements.
This content was downloaded from IP address 191.101.140.222 on 07/12/2018 at 13:12
1
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1234567890 ‘’“”
International Meeting on Applied Sciences and Engineering IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1126 (2018) 012003 doi :10.1088/1742-6596/1126/1/012003
Mesoscopic superconductivty in application
J Barba-Ortega1, J Bautista-Ruiz2and E Sardella3
1Departamento de F´ısica, Universidad Nacional de Colombia, Bogot´a, Colombia
2Departamento de F´ısica, Universidad Francisco de Paula Santader, San Jos´e de C´ucuta,
Colombia
3Departamento de F´ısica, Universidade Estadual Paulista, Baur´u, Brazil
E-mail: jjbarbao@unal.edu.co
Abstract. The Ginzburg-Landau theory, applied to superconducting materials is based on
thermo-magneto-electro-dynamic concepts as phase transitions that enrich the class on this
subject. Thus, in this contribution we expose the Ginzburg-Landau time-dependent equations,
show the mathematical form for two nano-scale superconducting systems, one bi-dimensional
homogeneous al sample with applied external current at zero magnetic field, and one three-
dimensional cube in presence of a tilted magnetic fiel at zero applied current. This analysis shows
the applicability of the three and two-dimensional model to superconductors. The conveniently
Ginzburg-Landau theory show that the magnetic response behavior of the sample is very useful
for applications in fluxtronica, SQUIDS design, magnetic resonance, among others.
1. Introduction
The Ginzburg-Landau theory (GLT) is one convenient tool in studying magnetic response of
superconducting. The vortex state and the confinement effects in superconducting samples
are very important when its size is of the order the coherence length ξ(T) or to the London
penetration depth λ(T) (mesoscopic sample). The size of the sample is usually considered to
be large enough such that the influence of finite dimensions on their properties are negligible
along the external applied magnetic field [1–5]. It was shown that increase of temperature
could be measured by nanoSQUIDs devices, Magnetic resonance and levitation, optical traps
of vortices in Bose-Einstein condensates, as realized by an arrangement of laser wave. T extra
degrees of freedom could open new phenomena in the studied superconducting systems due
to different possible optical patterns. GLT published in 1950 satisfactorily described several
phenomena related to superconductivity, with a phenomenological and simple aspect, due its
the researchers of the area received such work with skepticism, but in 1959, when Gorkov
showed that the Ginzburg-Landau equations (GLE) were a particular case of the superconductive
theory of BCS first principles, this was popularized in the middle. The interaction of light with
superconductivity in most cases leads to local heating of the condensate [6–10]. In 1966, Schimid
inserted a temporal dependence on GLE and, thus, the dynamics of systems out of equilibrium
could be studied [11–18]. By another handt, there are many experimental and theoretical studies
for three-dimensional-systems using the Ginzburg-Landau model has been proven to give a good
account of the superconducting properties in samples of several geometries [19–22]. GLT is also
used for study the magnetic response of mesoscopic samples in presence of a external dc current,
in this systems is possible to visualized the formation of vortex chains by scanning Hall probe
2
1234567890 ‘’“”
International Meeting on Applied Sciences and Engineering IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1126 (2018) 012003 doi :10.1088/1742-6596/1126/1/012003
microscopy after freezing the dynamical state through a field cooling procedure at a constant
bias current, also, the presence of vortex-anti-vortex (VAv) pair can nuleates in the sample
in a non-symetrical (symetrical) way due to the assymtry (simetry) of the supercurrent in the
sample, so, the VAv pairs can be controlled experimentally by including defects of other
superconducting materials at different critical temperature.
2. Theoretical formalism
In GLT, the superconducting order parameter is a complex function ψ(r) which is interpreted
as the pseudo-wave function of the superelectrons |ψ(r)|2=ns, where ψ(r) = nseis the
density of the super-electrons. The Ginzburg-Landau free energy is given by:
F=Z 1
2mi~e
cAψ
2
+α|ψ|2+β
2|ψ|4+1
8πB2!dV =ZFdV (1)
The constants in the equations are: m= 2m: the mass of a Cooper pair, e= 2e: the charge
of a Cooper pair,(α, β ): phenomenological constants. Variables ψ: the order parameter, A: the
vector potential, B=×A,B: the local magnetic field.
The equations which describe the fundamental properties of the superconducting state
described by the complex order parameter ψ, and the vector potential Acan be obtained by
using the Euler-Lagrange general equations [19, 20]:
F
¯
ψ·"F
(¯
ψ)#= 0 (2)
F
A×F
(×A)= 0 (3)
This produces the non time dependent Ginzburg-Landau equations:
1
2mi~e
cA2
ψ+αψ +βψ|ψ|2= 0 (4)
××A=4π
c
e
mRe ¯
ψi~e
cAψ(5)
The starting point to study the superconductivity of non-equilibrium is the time dependent
Ginzburg-Landau equations (TDGL). The TDGL equations are given by:
~
2mD
∂t +ie
~Φψ=1
2mi~e
cA2
ψ+αψ +βψ|ψ|2(6)
4πσ
c1
c
A
∂t +Φ=4π
c
e
mRe ¯
ψi~e
cAψ××A,(7)
The constants in the equations are: D: the diffusion coeficient, σ: the electrical conductivity,
Φ: the scalar potencial. In dimensionless units and in the gauge of zero electric potenctial Φ = 0,
the TDGL equations are given by:
∂ψ
∂t =(i+A)2ψ+ (1 T)ψ(|ψ|21) (8)
3
1234567890 ‘’“”
International Meeting on Applied Sciences and Engineering IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1126 (2018) 012003 doi :10.1088/1742-6596/1126/1/012003
A
∂t = (1 T)Re[ψ(i∇ − A)ψ]κ2∇ × ∇ × A(9)
Where ψis presented in units of (α/β)1/2, distance in units of coherence length ξ, time in
units of π~/96KBTc,Ain units of the upper critical field Hc2ξ, the Gibbs free energy Gin
units of (αTc)2and temperature Tin units of the critical temperature Tc.κis the Ginzburg-
Landau parameter, Thus, for κ < 1/2, the surface energy is positive, and we have the type I
superconductors; on the other hand, for κ > 1/2, the surface energy is negative, which defines
type II superconductors, Is becomes energetically favorable the penetration of quantized flow
(vortices) inside. We use the general boundary conditions for the order parameter and magnetic
field, they are given by the deGennes boundary conditions ˆn·(i+A)ψ|n=iψ/b, and
∇ × A=Bat the surface of the sample. ˆnis the normal vector to the surface and bis the
deGennes parameter [22–24].
3. Results and discussion
The GLT has been widely used for the study of mesoscopic superconductors. In general, in the
steady state the vortices, which in bulk materials are arranged in the network of Abrikosov,
in mesoscopic samples, the influence of the surface is so considerable that the arrangement
of the vortices follows the symmetry of the system, another exotic behavior is the formation
of vortices that have more than one quantum of magnetic flux in their nucleus (giant vortex)
[25,26]. We simulated a mesoscopic superconducting prism with lateral transversal section of size
L= 15ξ, immersed in a magnetic field H(t), the Ginzburg-Landau parameter is κ=λ/ξ = 1.3
(typical value for the Al). The sample presents an bi-dimensional thermal gradient with
Tleft =Tdown = 0.25 and Tright =Tup = 0.75 in the superior view of the lonfg prism.
In Figure 1, the square modulus of the order parameter |ψ|2and its phase ∆φfor H=
0.88,1.16 in the up and downbranch of the magnetic field are depicted. Values of the phase close
to zero are given by blue regions and close to 2πby red regions. As is well know, ∆T= 0.0,
and the votices entry always occrus by the central region, we have multi-vortex states with
N= 2,4(6,8) for H= 0.88,1.16 in the upbranch (downbranch) of H. An increased in Hcause
the vortex moves towards the interior of the sample.
Figure 1. (Color online) Square modulus of the order parameter |ψ|2at indicated values of H,
for Tleft =Tdown =Tright =Tup = 0.0 and a NN+ 2 vortex transition. Blue and yellow
regions represent values of the phase ∆φfrom 0 to 2πand ψ0 to ψ1. Arrows indicates
the upbranch and downbranch of the magnetic field.
In Figure 2 , the square modulus of the order parameter |ψ|2, its phase ∆φand magnetic
induction Bat H= 0.396,0.412,0.456 are depicted. Values of the phase close to zero are given
by blue regions and close to 2πby yellow regions. As we can appreciate the votices entry always
4
1234567890 ‘’“”
International Meeting on Applied Sciences and Engineering IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1126 (2018) 012003 doi :10.1088/1742-6596/1126/1/012003
occrus by the region a higher Tand multi-vortex states with Nto N+ 1 vortex transition are
presented. An increased in the temperature cause the vortex entry at lower magnetic field and
then, the vortices tend they moves towards the interior of the sample.
Figure 2. (Color online) Square modulus of the order parameter |ψ|2, its phase ∆φand magnetic
induction Bat H= 0.396,0.412,0.456, γ= 1.0 indicate a superconductin-normal boundary and
N= 1,2,3 respectively. Tlef t =Tdown = 0.25 and Tright =Tup = 0.75. As is well know, the
phase of the order parameter determine the vorticity in a given region, by counting its variation
in a closed path around this region. If the vorticity in this region is N, then the phase changes
by 2πN.
4. Conclusions
In summary, we solve the Ginzburg-Landau equations for studied the Abrikosov state of a Al
superconducting prims. The thermal variation into the sample allows vortices to entry the
sample interior by the regions at higher temperature. We also found that the behavior of vortex
state present Nto N+ 2 vortex transitions at low magnetic field for a sample submmersed in
a homogeneus thermal bath ∆T= 0, and a Nto N+ 1 vortex transition for ∆T6= 0.
5
1234567890 ‘’“”
International Meeting on Applied Sciences and Engineering IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1126 (2018) 012003 doi :10.1088/1742-6596/1126/1/012003
References
[1] Barba-Ortega J, Sardella E and Aguiar J 2001 Supercond. Sci. Technol. 24 015001
[2] Barba-Ortega J, Sardella E and Zadorosny R 2018 Phys. Lett. A. 382 215
[3] Moshchalkov V, Woerdenweber R and Lang W 2010 Nanoscience and engineering in superconductivity (New
York: Springer)
[4] Charles Poole Jr 2010 Handbook of superconductivity (Amsterdam: Elsevier)
[5] Barba-Ortega J, Sardella E, and Aguiar J 2015 Phys. Lett. A. 379 732
[6] Le´on J, Joya M and Barba-Ortega J 2018 Optik 172 311
[7] Jeli´c Z, Miloˇsevi´c M, and Silhanek A 2016 Sci. Rep. 635687
[8] Jeli´c Z, Miloˇsevi´c M, Van de Vodel J , and Silhanek A 2015 Sci. Rep. 514604
[9] Veshchunov I, Magrini W, Mironov S, Godin A, Trebbia J, Buzdin A, Tamarat Ph and Lounis B 2016 Nat.
Commun. 712801
[10] Ge J, Gutierrez J, Gladilin V, Devreese J and Moshchalkov V 2015 Nat. Comm. 65573
[11] Landau L and Lifshitz 1980 E Statistical physics (Oxford: Pergamon Press)
[12] Poole C, Farach H, Creswick R 1995 Superconductivity (Cambridge: Academic Press)
[13] Schmidt V 1997 The Physics of superconductors (Berlin: Spring-Verlag)
[14] Schimid A 1966 Phys. Kondens. Materie 5302
[15] He A, Xue C, and Yong H 2016 Supercond. Sci. Technol. 29 065014
[16] Silhanek A, Miloˇsevi´c, Kramer R, Berdiyorov G, Van de Vondel J, Luccas R, Puig T, Peeters F, and
Moshchalkov V (2010) Phys. Rev. Lett. 104 017001
[17] Kramer L and Watts-Tobin R 1978 Phys. Rev. Lett. 40 1041
[18] Watts-Tobin J, Kr¨ahenb¨uhl Y, and Kramer L 1981 J. Low. Temp. Phys. 42 459
[19] Gropp D, Kaper H, Leaf G, Levine D, Palumbo M, Vinokur V 1966 J. Comput. Phys. 123 54
[20] Buscaglia G, Bolech C and Lopez A 2000 Connectivity and superconductivity ed Berger J and Rubinstein J
(Heidelberg: Springer) p 200
[21] Piacente G, Peeters F 2005 Phys. Rev. B 72 205208
[22] deGennes P 1994 Superconductivity of metals and alloys (New York: Addison-Wesley)
[23] da Silva R, Milosevic M, Shanenko A, Peeters F and Aguiar J 2015 Sci. Reports 512695
[24] Shanenko A, Aguiar J, Vagov A, Croitoru M and Milosevic M 2015 Supercond. Sci. Technol. 28 054001
[25] Sardella E, Malvezzi A, Lisboa-Filho P, Ortiz W 2006 Phys. Rev. B 74 014512
[26] Chibotaru L, Ceulemans A, Bruyndoncx V, Moshchalkov V 2000 Nature 408 833
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
Article
An experimental determination of the mean vortex velocity in superconductors mostly relies on the measurement of flux-flow resistance with magnetic field, temperature, or driving current. In the present work we introduce a method combining conventional transport measurements and a frequency-tuned flashing pinning potential to obtain reliable estimates of the vortex velocity. The proposed device is characterized using the time-dependent Ginzburg-Landau formalism, where the velocimetry method exploits the resonances in mean vortex dissipation when temporal commensuration occurs between the vortex crossings and the flashing potential. We discuss the sensitivity of the proposed technique on applied current, temperature and heat diffusion, as well as the vortex core deformations during fast motion.
Full-text available
Article
Magnetic field can penetrate into type-II superconductors in the form of Abrikosov vortices, which are magnetic flux tubes surrounded by circulating supercurrents often trapped at defects referred to as pinning sites. Although the average properties of the vortex matter can be tuned with magnetic fields, temperature or electric currents, handling of individual vortices remains challenging and has been demonstrated only with sophisticated magnetic force, superconducting quantum interference device or strain-induced scanning local probe microscopies. Here, we introduce a far-field optical method based on local heating of the superconductor with a focused laser beam to realize a fast, precise and non-invasive manipulation of individual Abrikosov vortices, in the same way as with optical tweezers. This simple approach provides the perfect basis for sculpting the magnetic flux profile in superconducting devices like a vortex lens or a vortex cleaner, without resorting to static pinning or ratchet effects. Since a single vortex can induce a Josephson phase shift, our method also paves the way to fast optical drive of Josephson junctions, with potential massive parallelization of operations.
Full-text available
Article
Introducing artificial pinning centers is a well established strategy to trap quantum vortices and increase the maximal magnetic field and applied electric current that a superconductor can sustain without dissipation. In case of spatially periodic pinning, a clear enhancement of the superconducting critical current arises when commensurability between the vortex configurations and the pinning landscape occurs. With recent achievements in (ultrafast) optics and nanoengineered plasmonics it has become possible to exploit the interaction of light with superconductivity, and create not only spatially periodic imprints on the superconducting condensate, but also temporally periodic ones. Here we show that in the latter case, temporal matching phenomena develop, caused by stroboscopic commensurability between the characteristic frequency of the vortex motion under applied current and the frequency of the dynamic pinning. The matching resonances persist in a broad parameter space, including magnetic field, driving current, or material purity, giving rise to unusual features such as externally variable resistance/impedance and Shapiro steps in current-voltage characteristics. All features are tunable by the frequency of the dynamic pinning landscape. These findings open further exploration avenues for using flashing, spatially engineered, and/or mobile excitations on superconductors, permitting us to achieve advanced functionalities.
Full-text available
Article
Superconductors, ideally diamagnetic when in the Meissner state, can also exhibit paramagnetic behavior due to trapped magnetic flux. In the absence of pinning such paramagnetic response is weak, and ceases with increasing sample thickness. Here we show that in multiband superconductors paramagnetic response can be observed even in slab geometries, and can be far larger than any previous estimate - even multiply larger than the diamagnetic Meissner response for the same applied magnetic field. We link the appearance of this giant paramagnetic response to the broad crossover between conventional Type-I and Type-II superconductors, where Abrikosov vortices interact non-monotonically and multibody effects become important, causing unique flux configurations and their locking in the presence of surfaces.
Full-text available
Article
Recent progress in materials synthesis enabled fabrication of superconducting atomically flat single-crystalline metallic nanofilms with thicknesses down to a few monolayers. Interest in such nano-thin systems is attracted by the dimensional 3D–2D crossover in their coherent properties which occurs with decreasing the film thickness. The first fundamental aspect of this crossover is dictated by the Mermin–Wagner–Hohenberg theorem and concerns frustration of the long-range order due to superconductive fluctuations and the possibility to track its impact with an unprecedented level of control. The second important aspect is related to the Fabri–Pérot modes of the electronic motion strongly bound in the direction perpendicular to the nanofilm. The formation of such modes results in a pronounced multiband structure that changes with the nanofilm thickness and affects both the mean-field behavior and superconductive fluctuations. Though the subject is very rich in physics, it is scarcely investigated to date. The main obstacle is that there are no manageable models to study a complex magnetic response in this case. Full microscopic consideration is rather time consuming, if practicable at all, while the standard Ginzburg–Landau theory is not applicable. In the present work we review the main achievements in the subject to date, and construct and justify an efficient multiband mean-field formalism which allows for numerical and even analytical treatment of nano-thin superconductors in applied magnetic fields.
Full-text available
Article
One of the phenomena that make superconductors unique materials is the Meissner-Ochsenfeld effect. This effect results in a state in which an applied magnetic field is expelled from the bulk of the material because of the circulation near its surface of resistance-free currents, also known as Meissner currents. Notwithstanding the intense research on the Meissner state, local fields due to the interaction of Meissner currents with pinning centres have not received much attention. Here we report that the Meissner currents, when flowing through an area containing a pinning centre, generate in its vicinity two opposite sense current half-loops producing a bound vortex–antivortex pair, which eventually may transform into a fully developed vortex–antivortex pair ultimately separated in space. The generation of such vortex dipoles by Meissner currents is not restricted to superconductors; similar topological excitations may be present in other systems with Meissner-like phases.
Article
We studied the resistive state of a mesoscopic superconducting strip (bridge) at zero external applied magnetic field under a transport electric current, Ja, subjected to different types of boundary conditions. The current is applied through a metallic contact (electrode) and the boundary conditions are simulated via the deGennes extrapolation length b. It will be shown that the characteristic current-voltage curve follows a scaling law for different values of b. We also show that the value of Ja at which the first vortex-antivortex (V-Av) pair penetrates the sample, as well as their average velocities and dynamics, strongly depend on the b values. Our investigation was carried out by solving the two-dimensional generalized time dependent Ginzburg-Landau (GTDGL) equation.