Released power in a vortex-antivortex pairs annihilation process

1. Introduction

The magnetic properties of type I and type II are given by the Ginzburg - Landau parameter κ, the superconductor being of type II when κ > 1/ √ 2 and of type I if κ < 1/ √ 2. Tinkham [1] first showed, that even of material with κ < 1/ √ 2 can behave like type II if the film is sufficiently thin. The manipulation of single, multi, and giant vortex states in mesoscopic superconductors are very important for applications of spins, fluxtronic, medicine, etc [2, 3, 4, 5, 6, 7]. It is possible that a vortex anti vortex pairs becomes stable, this stability correspond to the symmetry of the system with pinning centers [8, 9], magnetic dots [10, 11], arrays of small current loops, hot spot. As is well know, in this kind of process the heat is always present, the heat is diffused and depend on the velocity of such diffusion [12, 13, 14].

Gulevich and Kusmartsev, proposed a experiment based on a sample with a Josephson coupling term, where the dynamics of vortex and antivortex take place [15]. Sardella et al, analyzed the annihilation of a vortex- antivortex pair in a square superconducting system with a central pinning center. Show that when the vortex enters the superconducting sample, its average velocity is ∼ 103 ms ⁻¹ and during the annihilation process, its average velocity is ∼ 105 ms⁻¹ [8].

Zadorosny et al studied the dynamics of a vortex- antivortex pair in several superconducting systems with different areas and with an antidot inserted in the center, they found that in the annihilation process the vortexantivortex pair acquire an elongated format [9].

E. Duarte et al, studied the vortex antivortex annihilation in a mesoscopic superconducting system with a central pinning center, they found a dissipative effect, generating appreciable changes in the order parameter, and the power released in the annihilation of a vortex-antivortex pair is detectable in measurements of the total magnetic moment as a function of time [16].

J. Barba-Ortega et al, investigated the process of annihilation of a vortex-antivortex pair in a superconducting square with a central anti-pinning center. They found that when a metallic anti-pining center is used the vortex-antivortex collision occurs close to the edge of the defect, and the antivortex speed decrease when the anti-dot becomes more metallic [17].

In this work we studied the dissipative process (total energy released in the collision) between a vortex- antivortex collision in a superconducting square with a concentric hole. We calculated the magnetization and density of superconducting electrons as a function of the time for several values of the Ginzburg - Landau parameter. This work is organized as follows. In section 2 we present the theoretical formalism used to study the mesoscopic systems. In section 3 we analyze the results obtained from the simulations and, subsequently, discuss them. In section 4, we present our conclusions.

2. Theoretical Formalism

We solve the time dependent Ginzburg Landau equations (TDGL), proposed by Schmid [18], provides a temporal evolution of the order parameter ψ and the potential vector A for a superconducting sample immersed in an external applied magnetic field H. For our purposes, we solved the equations for the energy dissipated due to both the applied magnetic field and the order parameter (ψ) during the vortex-antivortex process [19, 20, 21, 22, 23, 24]:

(1)

(2)

In the equations 1 and 2, J_s = 𝑅𝑒 [𝜓̅(−𝑖∇ − A)𝜓], is the super-current density, 𝜓 represents the order parameter, A the vector potential, T temperature. How is usual we presented, in dimensionless form with |𝜓| in units of ψ ∞ = − α β with β = 1, lengths in units of the coherence length ξ, T = 0; A in units of H_c2ξ, where H_c2 is the second critical field, temperature in units of the critical temperature T_c and time in units of time Ginzburg-Landau t_GL = πh/8K_BT_cη. τ(x,y) is the deformed parameter α explain the variation of the Cooper pairs density locally in the sample, even this modified model have been added in superconducting samples for different specific geometries. In the Ginzburg-Landau model, α represents regions at different T_c, and κ(x,y) is the Ginzburg-Landau Parameter for an anysotropic sample. We simulated a superconducting square of size L = 12ξ with a central defect with size L_d = 2ξ, respectively, immersed in an oscillatory magnetic field as (See Figure 1):

(3)

In the equation 3, H goes to 0 → 1, and then the magnetic field is reversed to 1 → −1, and finally the magnetic loop is completed to −1 → 0. We take H₀= 1,0. The equation for the dissipated power energy was obtained by using the Helmholtz free energy for a superconductor in presence of external magnetic field. Such equation, in dimensionless form, is given by:

(4)

he first term in the equation 4 represents the dissipation due to the induced electrical field E, and the second one is due to the dissipation related to the relaxation of ψ. The dissipated power energy is given in units of H c 2 2 (0)/[8πκ² t_GL] As W_total diffuses through the system, we couple the thermal diffusion equation to the Ginzburg-Landau ones [12, 16]. We take u = 5,79 and γ = 10 (for more details see Ref. [25]). We studied two scenarios with κ = 1,0 and τ = 1,0 in the whole sample, except in a) the cases where κ_d = 0,4,0,7,0,9,1,1 within the defect, b) the cases where τ = 0,4,0,7,0,9,1,1 within the defect. Note that, given the thin film geometry, this is an effective κ value, which depends on the thickness of the stripe.

3. Results and discussion

In the Figure 2 we show the magnetization loops as a function of the applied magnetic field, −4πM(H), for κ = 1,0 in whole sample and a) τ = 0,4 b) τ = 0,7, c) τ = 0,9, and d) τ = 1,1. We calculate the vortex number into the sample, we found that, due to the size of the defect and the sample, just one vortex nucleate into the sample at H ≃ 0,9 being trapped by the defect, as the magnetic field is decreased, this vortex leaves the sample and when H is inverted, an antivortex enter the sample while a vortex remains trapped in the defect.

Thus, a vortex-antivortex pair is formed and each specimen moves toward each other until their mutual annihilation. A similar process occurs in taking τ = 0,9 and τ = 1,1, respectively, but in these cases two and six vortices penetrates the sample. As we can see, while τ increases into the defect, the number of jumps in the magnetization curve increases, and as is well known, the number of jumps in the magnetization curve is related to the number of vortices entering or leaving the sample.

A similar behaviour we can appreciate in the Figure 3, where the loop for the magnetization is plotted for τ = 1,0 in whole sample and a) κd = 0,9, b) κd = 1,1.

In this case the effect of κd on the vortex number that penetrates the sample is negligible, a paramagnetic effect in the down branch of the magnetization curve, evidence the pinning vortex effect of the nature of the defect. In the Figure 4 we plot the average of the modulus square of the order parameter | ψ | ² as a function of time for κ = 1,0 in whole sample and a) τ = 0,4 b) τ = 0,7, c) τ = 0,9, and d) τ = 1,1.

As is well know, | ψ | ² depends on the vorticity N of the system and presents a sequence of periodic pulses and the number of maximus increases with τ. We calculated the power spectrum (2W_Total), Figure 5, as a function of time t, for κ = 1,0 in whole sample and a) τ = 0,4 b) τ = 0,7, c) τ = 0,9, and d) τ = 1,1

For a defect filled with another superconductor at higher Tc (τ > 1,0), the time averaged power spectrum presents an harmonic behaviour as a function of time t for 4 maximus, whereas for a defect filled with a metallic material 0 < τ < 1,0, one and two maximus or the first and second harmonic can be distinguished.

A similar behaviour is appreciated in the Figure 6 where the average of the modulus square of the order parameter | ψ |² as a function of time for τ = 1,0 in whole sample and κ_d = 0,9, b) κ_d = 1,1. A similar behaviour occurs when κ_d varies into the sample, but the number of maximum in the order parameter independent on κ_d < 1,0. We observed an anomalous behaviour in this oscillations when κ_d= 1,1 is used. When κ_d changes into the sample, remains τ invariant, the power spectrum (2wtotal), Figure 7, as a function of time t, is very similar in all the studied cases for 0 < τ < 1,0, and present a non-conventional behaviour for τ > 1,0.

4. Conclusions

In the thin film limit we studied the released power energy in a vortex-antivortex process in superconducting square with a central defect at several temperatures and Ginzburg-landau parameter. We showed that when the temperature inside the defects increases, the number of maximum in the dissipated power increases with τ, while the Ginzburg-Landau parameter varies into the defects, remains the temperature invariant, the number of maximum is independent of κ_d . We obtained profiles of the average order parameter as a function of time for several κ and τ and calculated their power spectrum.

We found that, three harmonic frequency are obtained for τ < 1,0, the power spectrum is affected by the entry and exit of vortices and anti-vortices in the defect. Since that the released power in these process are not desirable for several applications of superconducting systems, we claim that the detailed knowledge about vortex- antivortex dynamics is an important and relevant element to avoid them. Therefore, our results could contribute to important applications of technological devices.

Acknowledgments

The Colombian Geological Survey, Laboratory Direction Cali – Colombia, and the Gualconda’s gold mine Manager Mrs. Rolberto Álvarez.

References

Additional information

How to cite: C. A. Aguirre, M. R. Joya, J. Barba-Ortega, “Released power in a vortex-antivortex pairs annihilation process,” Rev. UIS Ing., vol. 20, no. 1, pp. 153-160, 2021, doi: 10.18273/revuin.v20n1-2021014

Revista UIS Ingenierías
ISSN: 1657-4583
Vol. 20
Num. 1
Año. 2021

Released power in a vortex-antivortex pairs annihilation process

CristianMiryamJosé José Aguirre-TellezRincón-JoyaBarba-Ortega
Universidad Federal de Mato GrossoUniversidad Nacional de ColombiaUniversidad Nacional de Colombia,BrasilColombiaColombia