Ginzburg–Landau description of confinement and quantization effects in mesoscopic superconductors

Abstract

A remark on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg-Landau equationAn approach to the Ginzburg-Landau problem for superconducting regular polygons is developed making use of an analytical gauge transformation for the vector potential A which gives A n = 0 for the normal component along the boundary line of different symmetric polygons. As a result the corresponding linearized Ginzburg-Landau equation reduces to an eigenvalue problem in the basis set of functions obeying Neumann boundary condition. Such basis sets are found analytically for several symmetric structures. The proposed approach allows for accurate calculations of the order parameter distributions at low calculational cost ͑small basis sets͒ for moderate applied magnetic fields. This is illustrated by considering the nucleation of superconductivity in squares, equilateral triangles and rectangles, where vortex patterns containing antivortices are obtained on the T c -H phase boundary. The calculated phase boundaries are compared with the experimental T c ͑H͒ curves measured for squares, triangles, disks, rectangles, and loops. The stability of the symmetry consistent solutions against small deviations from the phase boundary line deep into the superconducting state is investigated by considering the full Ginzburg-Landau functional. It is shown that below the nucleation temperature symmetry-switching or symmetry-breaking phase transitions can take place. The symmetry-breaking phase transition has the same structure as the pseudo-Jahn-Teller instability of high symmetry nuclear configurations in molecules. The existence of these transitions is predicted to be strongly dependent on the size of the samples.

Different regimes of nucleation of superconductivity in mesoscopic superconductor/ferromagnet hybrids

“…Interestingly, an effective suppression of the edge OP nucleation makes it possible to eliminate boundary effects inherent for all mesoscopic superconducting samples. 3,11 In addition, we show that the onset of the most favorable nucleation regime manifests itself as a clear change in the slope of T c ͑H͒.…”

“…Interestingly, an effective suppression of the edge OP nucleation makes it possible to eliminate boundary effects inherent for all mesoscopic superconducting samples. 3,11 In addition, we show that the onset of the most favorable nucleation regime manifests itself as a clear change in the slope of T c ͑H͒.The sample under investigation is a 50 nm thick Al disk with a diameter of 1.7 m, covered by a ferromagnetic ͓0.4 nm Co/ 1.0 nm Pt͔ 10 multilayer, with a well-defined out-of-plane magnetization. 7 The S and F layers are separated by a 5 nm thick insulating Si layer in order to exclude the exchange interaction.…”

“…This phase boundary shows Little-Parks oscillations, indicating changes of vorticity in the system by one flux quantum. 3,12 The hybrid sample in the demagnetized state ͑curve H m = 0 in Fig. 2͒ shows a strong suppression of T c as compared with the reference sample.…”

“…The OP nucleation in S/F hybrids with different competing confinements was recently studied theoretically in Refs. [3][4][5][6]8, and 9. For example, for a generic S/F hybrid, consisting of a mesoscopic superconducting disk and a small magnetic particle, theory predicts two well-defined regimes: superconductivity can nucleate either near the disk center ͑under the magnetic particle͒ or at the disk edge.…”

“…In the case of spatially separated S and F subsystems, when the direct exchange of electrons at the interface between the two materials becomes suppressed, the interaction is dominated by the slow decaying magnetic fields b͑r͒ induced by the ferromagnet. In particular, the inhomogeneous b͑r͒ field generated by magnetic domains in the F layer affects strongly the nucleation of the superconductivity in the S layer and leads to exotic dependences of the critical temperature T c on an external magnetic field H. [2][3][4][5][6][7][8][9] Indeed, the presence of inhomogeneous fields results in the appearance of places where the transverse component of the total magnetic field ͉b z ͑r͒ + H͉ reaches a local minimum and thus the nucleation of localized superconductivity in thin superconducting films will be promoted due to the field compensation effect.…”