Figure 4
(Color online) Panel (1) is the normal-superconducting phase diagram of the mesoscopic square with a magnetic dot on top shown in Fig. 1. The field interval has been chosen to cover the regime where the vortex patterns presented in Fig. 3 nucleate so that in the
curve (thick solid line) we have identified the nucleation region of each of these vortex states by using the same lower case letter than in the corresponding panel of that figure. The thin solid lines are associated with the different symmetry-breaking and symmetry-switching transitions that the order parameter undergoes in the superconducting phase. Therefore, they separate different vortex patterns that have been labeled by using upper case letters followed or not by a number, the latter corresponding to those found at the lowest studied temperature. The configurations with the same letter have the same vorticity, such as
and
,
and
,
and
,
and
,
and
,
,
, and
,
and
, and
,
,
, and
. Note that the lines tagged with 1 have a negative (positive) slope for
and, thus, they correspond to transitions involving the spontaneous nucleation of vortices (antivortices) with decreasing temperature. This phenomenon may also happen in the transition regions
and
, characterized by a high instability associated to small energy differences between the different vortex states. A first indication of the origin of this instability is shown in panel (2), where we present the
dependence of the energy of the first three Landau levels of each irrep involved in the order-parameter expansion. As it can be seen, these levels are closer in energy to each other in the field regime where both transition regions appear than in other
intervals. Panels (a)–(t) are the vortex patterns found at
in the magnetic-field region covered by the phase diagram presented in panel (1) [the vortex pattern corresponding to the phase U at
is separately shown in Fig. 5d]. Note that the usual increment of the vorticity by one is recovered at such a low temperature so that the vortex patterns in (a)–(t) cover
values between
and 9 in steps of 1. For negative vorticities, the antivortices nucleate in a well defined sequence: they first progressively fill the corners of the square in the region out of the dot and, once the corners have the same number of antivortices, the next one nucleates in the center of the square and the cycle restarts. In contrast, for positive vorticities [panels (l)–(t)] all the vortices are located in the region below the dot and they behave similarly to the case of an individual superconducting microdisk, i.e., they arrange themselves in a multishell structure trying to form a triangular lattice.
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