Soliton states in mesoscopic two-band-superconducting cylinders

In the framework of the Ginzburg-Landau approach, we present a self-consistent theory of specific soliton states in mesoscopic (thin-walled) two-band-superconducting cylinders in external parallel magnetic fields. Such states arise in the presence of "Josephson-type" interband coupling, when phase winding numbers are different for each component of the superconducting order parameter. We evaluate the Gibbs free energy of the sysyem up to second-order terms in a certain dimensionless parameter $\epsilon\approx\frac{\mathcal{L}_{m}}{\mathcal{L}_{k}}\ll1$, where $\mathcal{L}_{m}$ and $\mathcal{L}_{k}$ are the magnetic and kinetic inductance, respectively. We derive the complete set of exact soliton solutions. These solutions are thoroughly analyzed from the viewpoint of both local and global (thermodynamic) stability. In particular, we show that rotational-symmetry-breaking caused by the formation of solitons gives rise to a zero-frequency rotational mode. Although soliton states prove to be thermodynamically metastable, the minimal energy gap between the lowest-lying single-soliton states and thermodynamically stable zero-soliton states can be much smaller than the magnetic Gibbs free energy of the latter states, provided that intraband "penetration depths" differ substantially and interband coupling is weak. The results of our investigation may apply to a wide class of mesoscopic doubly-connected structures exhibiting two-band superconductivity.Comment: 15 pages, 3 figure

Exact analytical solution of the problem of current-carrying states of the Josephson junction in external magnetic fields

The classical problem of the Josephson junction of arbitrary length W in the presence of externally applied magnetic fields (H) and transport currents (J) is reconsidered from the point of view of stability theory. In particular, we derive the complete infinite set of exact analytical solutions for the phase difference that describe the current-carrying states of the junction with arbitrary W and an arbitrary mode of the injection of J. These solutions are parameterized by two natural parameters: the constants of integration. The boundaries of their stability regions in the parametric plane are determined by a corresponding infinite set of exact functional equations. Being mapped to the physical plane (H,J), these boundaries yield the dependence of the critical transport current Jc on H. Contrary to a wide-spread belief, the exact analytical dependence Jc=Jc(H) proves to be multivalued even for arbitrarily small W. What is more, the exact solution reveals the existence of unquantized Josephson vortices carrying fractional flux and located near one of the junction edges, provided that J is sufficiently close to Jc for certain finite values of H. This conclusion (as well as other exact analytical results) is illustrated by a graphical analysis of typical cases.Comment: 21 pages, 9 figures, to be published in Phys. Rev.

Static Solitons of the Sine-Gordon Equation and Equilibrium Vortex Structure in Josephson Junctions

The problem of vortex structure in a single Josephson junction in an external magnetic field, in the absence of transport currents, is reconsidered from a new mathematical point of view. In particular, we derive a complete set of exact analytical solutions representing all the stationary points (minima and saddle-points) of the relevant Gibbs free-energy functional. The type of these solutions is determined by explicit evaluation of the second variation of the Gibbs free-energy functional. The stable (physical) solutions minimizing the Gibbs free-energy functional form an infinite set and are labelled by a topological number Nv=0,1,2,... Mathematically, they can be interpreted as nontrivial ''vacuum'' (Nv=0) and static topological solitons (Nv=1,2,...) of the sine-Gordon equation for the phase difference in a finite spatial interval: solutions of this kind were not considered in previous literature. Physically, they represent the Meissner state (Nv=0) and Josephson vortices (Nv=1,2,...). Major properties of the new physical solutions are thoroughly discussed. An exact, closed-form analytical expression for the Gibbs free energy is derived and analyzed numerically. Unstable (saddle-point) solutions are also classified and discussed.Comment: 17 pages, 4 Postscript figure

Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizes of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, non-soliton configurations (interpreted in some previous publications as «isolated fluxons» and «fluxon lattices») are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and establish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are derived: They are nothing but the «vacuum state» and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0 < H < ∞ and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H to be sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics