- 2.7.2 Reversal of the diode effect. - 2.9 Distribution of the DOS in a disc with multiquanta vortices. - 2.27 Phase diagram of the ratchet effect in a triangle. - 2.28 Reversal of the diode effect in a triangle. - 2.30 Reversal of the diode effect in a disk. - 2.32 Compensation of the stray field of a magnetic dot. - 4.39 Time evolution of the vortex velocity. - 4.40 H –T diagram of the dynamical regimes of the vortex lattice. - and on the behavior of the vortex matter. - αψ s (1.6) is the analogue of the Schr¨odinger equation (Eq. - plays the role of the eigenvalue E in the Schr¨odinger equation. - (1.6) and (1.7)) plays the role of the energy E in Eq. - 1.8: A superconducting film (S) is placed on top of a ferromagnetic substrate (F) with the easy axis of the magnetization in the z-direction. - 2) conditions are applied to the sub- components of the order parameter ψ 1 and ψ 2 , respectively 1. - 1.6(a), while the rest of the material is in the normal state.. - The magnetization M can be written in terms of the applied field H and the flux density B in the sample:. - The superconducting order parameter vanishes in the center of the vortex core. - For the case of the type-1.5 superconductor MgB 2 , the BCS expression ξ(0. - is approximately 1% of the average value. - in the direction of the yellow arrows in Fig. - This can drive, still within the limits of the Ginzburg–. - H 2 : the expansion of the energy E(H ) in powers of H. - flowing in the different branches of the loop gain opposite phases ± πΦ/Φ 0. - (2.4), thus causing oscillations of the critical temperature T c versus magnetic field H.. - Due to the divergence of the coherence length ξ(T ) at T = T c0 (Eq. - The solution of the Hamiltonian in Eq. - (2.12) The representation of the order parameter ψ = P. - The corresponding values of the Abrikosov parameter β A. - of the problem is L. - The phase boundary of the superconducting disk (Fig. - It should be emphasized that the presence of the oscillations in the H c3 (T ) curve is crucially dependent on the im- posed Neumann boundary conditions [45]. - to the penetration of the first flux line [45]:. - Even deeper in the superconducting area, the recovery of the normal Φ 0. - The precise shape of the T c (H ) curve crucially depends on the area fraction for which T c (H ) 6 = 0.. - The superconducting properties of the different considered structures (see Tab. - The inset shows a schematics of the geometry.. - The experimentally determined T c (H) phase boundary of the disc is shown in Fig. - Similarly, the phase boundary of the ring with r i = 0.1 r o. - ψ | 2 pattern of the disc. - The experimental data have been corrected for the presence of the measuring leads. - (1.6) are characterized by irreducible representations of the cyclic group C 3 . - The unnormalized eigenfunctions of the zero field problem, i.e. - 4 a is the height and a the edge of the triangle. - The density distribution of the order parameter | ψ | 2 , shown in Fig. - The contribution of the two kinds of. - vortices (central + three corner) to the total winding number of the states, shown in Fig. - In analogy to the case of the square (Sec. - [90] to the case of a mesoscopic rectangle in the framework of the linearized Ginzburg–. - The T c (H) curves of the rectangles with ζ = 4 / 3 and ζ = 2 (Fig. - These curves are remarkably similar to the phase boundary of the square (Fig. - showing only very small changes in the positions of the cusps: the magnetic. - Even a large deformation of the square (ζ = 2) results in negligible changes in the phase boundary. - 2.17 (a – c) The density of the order parameter | ψ | 2 in the central region ( a / 10 × b / 10 ) of a superconducting rectangle with edge lengths a and b (b <. - work of the Ginzburg–Landau equations, magnetization measurements can be used. - Once that the positions of the vortices are. - Therefore, a change in the shell structure will clearly affect the height of the jumps ∆M . - 2.20), since superconductivity is enhanced in the corners of the triangle [222]. - The active area of the used Hall cross was 3.7 × 3.7µm 2 (after [334]).. - The magnetization of the square at T >. - In the case of current leads placed above the geometrical center of the triangle (Fig. - (a) Contacts along a median, (b) above the geometrical center (c) and at the base of the triangle.. - (a) Current flow through the center, (b) via the upper part of the disk (c) via the lower part of the disk. - (a) above the geometrical center (b) at the base of the triangle Fig. - θ(r n−1 ) with θ(r n ) the phase of the superconducting island centered at. - with the radius R of the ring. - Most of the multiquanta transitions observed in Fig. - Schematic sketches of the structures are shown as insets (after [193]).. - The T c (H ) phase boundaries of the three structures are shown together in Fig. - The latter takes the presence of the leads into account (after [345]).. - including the presence of the leads. - 0) in the center of the common strip, the phase ϕ having a discontinuity of π at this point. - Therefore, the additional features observed in the T c (H) phase boundary of the antidot. - 3.7) can be directly attributed to the presence of the antidots.. - (c) First period of the measured phase boundary shown in Fig. - 3.7 after sub- traction of the parabolic background (af- ter [390]).. - inside the schematic drawings of the antidot cluster. - Figure 3.8(c) shows the first period of the measured phase boundary T c (Φ) after subtraction of the parabolic background. - Within each period of the oscillations, i.e. - Insets show atomic force micrographs of the studied structures (after [334]).. - 0.95, M I i can even exceed 20% of the applied flux. - The modification of the T c (Φ) oscillations of the outer loop, due to the coupling between the outer and the inner loops, is still seen in the Fourier spectrum of the T c (Φ) line, which is shown in Fig. - 3.13 Fourier transform of the phase boundaries shown in Fig. - It is interesting to consider the evolution of the T c (Φ) phase boundary when we increase the size of a two-dimensional network starting from a single nanoscopic cell (a loop) and ending with a huge array composed of such cells (an infinite network), see Fig. - 4.4 The evolution of the T c (H ) phase boundary (London limit) when increasing the number of nanoscopic cells in a two-dimensional network from one single loop to an infinite array (after [411]).. - (1.11)) of the reference film (see the dashed lines in Fig. - up to more than 200% of the expected maximum enhancement. - with the number of the flux quanta n up to the saturation number n S [332].. - The onset of the vortex for- mation at interstices at H >. - (b) Magnifica- tion of the low resistance part of the R(H ) curve (logarithmic scale) of the nanostructured film at T /T c = 0.997. - At H = H 2 , all of the interstitial. - In the present case, the direction of the two vortices accidentally horizontal.. - did not change the configuration of the vortices. - 15, no overall order of the vortex lattice is found. - interstices now plays the role of the pinning centers in the first field pe- riod. - (b) The slopes ∆M/∆H of the dashed lines as a function of [1 − (T /T c. - (4.9) provides a better linearity of the slope ∆M/∆H versus T , see Fig. - the slopes of the M versus ln(H − H n ) lines (Fig. - In these experiments the use of the. - Indeed, the position ‘I 2 ’ in the center of the parallelogram A 1 A 2 A 3 A 4 (see the sketch of Fig. - white, antidots) of the corresponding states