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Mesoscopic superconductivty in application

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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

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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 (V−Av) 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 V−Av 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) = √nseiφ is the

density of the super-electrons. The Ginzburg-Landau free energy is given by:

F=Z 1

2m∗−i~∇−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

2m∗−i~∇−e∗

cA2

ψ+αψ +βψ|ψ|2= 0 (4)

∇×∇×A=4π

c

e∗

m∗Re ¯

ψ−i~∇−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:

~

2m∗D∂

∂t +ie∗

~Φψ=−1

2m∗−i~∇−e∗

cA2

ψ+αψ +βψ|ψ|2(6)

4πσ

c1

c

∂A

∂t +∇Φ=4π

c

e∗

m∗Re ¯

ψ−i~∇−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|2−1) (8)

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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)2/β and 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 N→N+ 2 vortex transition. Blue and yellow

regions represent values of the phase ∆φfrom 0 to 2πand ψ∼0 to ψ∼1. 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

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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.

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