Anisotropy included in a nanoscale superconductor: Theoretical development

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1 Journal of Physics: Conference Series PPER OPEN CCESS nisotropy included in a nanoscale superconductor: Theoretical development To cite this article: O J Gaona et al 207 J. Phys.: Conf. Ser View the article online for updates and enhancements. This content was downloaded from IP address on /0/208 at 0:39

2 IOP Conf. Series: Journal of Physics: Conf. Series doi :0.088/ /935//02070 nisotropy included in a nanoscale superconductor: Theoretical development O J Gaona, J D González and J R Beltrán Universidad del Magdalena, Santa Marta, Colombia jgonzaleza@unimagdalena.edu.co bstract. theoretical procedure to obtain the Ginzburg-Landau GL equations with the inclusion of localized anisotropy in the superconducting sample is shown. Using this theory, it is possible to study the vortex structures in two-dimensional mesoscopic superconductors with defects in the presence of a uniform magnetic field. The defects would be included through variation of parameter γx, y, z considering different shapes into the sample. This theory would allow to find unconventional vortex states which can show vortex clusters or exhibiting asymmetry. In samples with defects the vortex strings are formed owing to the interactions of vortices with variable Meissner currents through the sample due to spatial change of the anisotropy, but also the thermodynamics parameters of the superconducting sample could be determined as a function of the degree of the anisotropy generated in the defect.. Introduction Whenever a scientific advances is made, the reseachers make a effort to explain it theoretically, in the particular case of superconductivity the experimental discoveries occurs before of the theoretical study in the most of cases. For instance, the London theory [] was the first theory which explain the results obtained in the laboratory but it took more than two decades, but also in this theory vortices are treated as point-like objects but valuable information regarding to the vortex structure cannot be demonstrated. Over the past 50 years, Landau and Ginzburg [2] developed a phenomenological theory which had great success in explaining macroscopic properties of superconductors but it was also immensely useful for description of mesoscopic superconducting samples [3, 4]. The geometry of the system is important as well as the defects included in the sample, not only to determine equilibrium flux structures, but also to determine the critical parameters of the mesoscopic superconductor [5, 6] and physics becomes richer when a magnetic field is applied to this systems and vortices also show very rich behavior when different kind of defects are included in the superconducting sample [7, 5]. For this reason, this paper hopes to contribute theoretically to the study of superconducting systems that include defects generated by anisotropies in the superconducting sample. 2. Theoretical formalism The GL theory was implement by solving the equations obtained by minimizing the functional energy that can be expanded in powers of ψ as G s = G n + α ψ 2 + β 2 ψ 4 + 2m iħ 2e c ψ 2 + h2 8π h h 0 h 0 Content from this work may be used under the terms of the Creative Commons ttribution 3.0 licence. ny further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd

3 IOP Conf. Series: Journal of Physics: Conf. Series doi :0.088/ /935//02070 Where G n is the free energy density of a superconductor in the normal state, whereas h denotes the applied magnetic field, and h 0 is the total local field, including the response of the superconductor. It will be scale used as for the order parameter ψ 0 2 = α 0 β which is the order parameter of the Mesissner state. Now in order to find the equations for ψ and such that their solutions give the minimum value of the functional, we solve the variational problem with respect of ψ : αψψ + β 2 ψψ ψψ + 2m iħ 2e c ψ iħ 2e c ψ + h2 8π h h 0 h 0 dv = 0 2 αψδψ + 2 β 2 ψψψ δψ + 2m iħ 2e c ψ iħ 2e c δψ dv = 0 3 it is obtained: and its boundary condition αψ + βψ ψ 2 + 2m iħ 2e 2 c ψ = 0 4 n iħ 2e 2 c ψ s = 0 5 where n is the unit vector normal to the surface of the superconductor. In order to obtain the equation for, we shall minimize the functional with respect to : αψψ + β 2 ψψ ψψ + 2m iħ 2e c ψ iħ 2e c ψ + 2 h 0 h 0 dv = 0 6 8π 2m iħ ψ 2e ψ c 2e c δ ψ 2e c δ ψ iħ ψ 2e ψ c dv + 2 δ δ h 0 h 0 dv = 0 7 8π Finally, it is obtained the second equation of GL theory, for the vector potential iħ 2m ψ ψ ψ ψ e 2 m c ψ 2 = J 8 The anisotropy is included in the equations trough in the GL coefficient α, as α = α 0 /γx, y, z. In the presence of such anisotropy, by minimization of equation, we obtain the equations for order parameter and vector potential, following the same procedure shown before. Now, normalizing the first GL equation αψ + βψ ψ 2 2m iħ 2e 2 c ψ = 0 9 αψ + βψ ψ 2 = 2m iħ 2e 2 c ψ 0 2

4 IOP Conf. Series: Journal of Physics: Conf. Series doi :0.088/ /935//02070 αψ 0 ψ + βψ0 ψ ψ0 2 ψ 2 = ψ 0 2m α 0 γx, y, z ψ β α 0 β Now, the equation is divided by α 0 ψ γx, y, z ψ 2 Considering that 2e cħ ξ0 = iħ 2e 2 ψ ψ = 2m iħ 2e c = ħ2 2m α 0 ξ 2 0 H c2 ξ0 = and ħ 2 2m α 0 γx, y, z ψ 2 Following the same procedure for the second GL equation: = 2 c ψ 2 ψ 2 i 2e 2 cħ ξ0 ψ 3 = ξ 2 0, we obtain i 2 ψ. 4 iħ 2m ψ ψ ψ ψ e 2 m c ψ 2 = J 5 It is normalized the right term as follow: Hence c = c H c2 ξ0 6 iħe 2m ψ ψ ψ ψ e 2 m c ψ 2 H c2 ξ0 7 ξ0 iħe c H c2 2m ψ ψ ψ ψ e 2 m c ψ 2 Hc2 ξ0 8 i ξ λ 2 0 ψ 0 2 ψ ψ ψ ψ ξ2 0 λ 2 0 ψ 2 ψ 2 9 i 2 κ ψ 0 2 ψ ψ κ ψ 0 2 ψ 2 20 i 2 ψ 0 2 ψ ψ ψ 0 2 ψ 2 = κ 2 Finally, the GL equations it is written as follow: γx, y, z ψ 2 ψ = i 2 ψ. 22 i 2 ψ ψ ψ ψ ψ 2 = κ Conclusions In summary, we have explained in detail the the theoretical procedure to obtain the Ginzburg- Landau GL equations to make numerical experiments to simulate anisotropic samples and is hopes to have presented a brief information which can be of use for researchers in the years to come. 3

5 IOP Conf. Series: Journal of Physics: Conf. Series doi :0.088/ /935//02070 cknowledgments This work was partially financed by the Universidad del Magdalena Fonciencias and the Colombian agency Colciencias through doctoral scholarships 567. References [] London F, London H 935 Proc. Roy. Soc [2] Moshchalkov V V 999 Handbook of nanostructured materials and nanotechnology New York: cademic Press p 45 [3] Schweigert V, Peeters F M 998 Phys. Rev. B [4] Schweigert V, Peeters F M 998 Phys. Rev. Lett [5] Berdiyorov G R, Yu S H, Xiao Z L, Peeters F M, Hua J, Imre and Kwok W K 2009 Phys. Rev. B [6] Romaguera R, Doria M M, Peeters F M 2007 Phys. Rev. B 76 R [7] Harada K, Kamimura O, Kasai H, Matsuda T, Tonomura, Moshchalkov V V 996 Science [8] Nori F 996 Science [9] Misko V R, SavelâĂŹev S, Rakhmanov L, Nori F 2006 Phys. Rev. Lett [0] González J D, Rincon M, Barba-Ortega J 204 Int. Journal of Mod. Phys. B [] Barba-Ortega J, González J D, Sardella E 204 Journal of low Temp. Phys [2] Berdiyorov G R, Milošević M V, Peeters F M 2006 Phys. Rev. Lett [3] Berdiyorov G R, Milošević M V, Peeters F M 2007 Phys. Rev. B [4] Berdiyorov G R, Milošević M V, Peeters F M 2006 Phys. Rev. B [5] Berdiyorov G R, Milošević M V, Peeters F M 2006 Europhys. Lett