## 1. Introduction

To present day overwhelming majority works on theory of superconductivity were devoted to single gap superconductors. More than 50 years ago the possibility of superconductors with two superconducting order parameters were considered by V. Moskalenko

(Moskalenko, 1959) and H. Suhl, B.Matthias and L.Walker (Suhl et al., 1959). In the model of superconductor with the overlapping energy bands on Fermi surface V.Moskalenko has theoretically investigated the thermodynamic and electromagnetic properties of two-band superconductors. The real boom in investigation of multi-gap superconductivity started after the discovery of two gaps in *e.g.* in heavy fermion compounds (Jourdan et al., 2004; Seyfarth et al., 2005), high-T_{c} cuprates (Kresin & Wolf, 1990), borocarbides (Shulga et al., 1998), liquid metallic hydrogen (Ashcroft, 2000; Babaev, 2002; Babaev et. al, 2004). Recent discovery of high-temperature superconductivity in iron-based compounds (Kamihara et al., 2008) have expanded a range of multiband superconductors. Various thermodynamic and transport properties of

Two-band superconductivity proposes new interesting physics. The coexistence of two distinctive order parameters

In this chapter we are focusing on the implication of the _{1}-I-S_{2} junctions (I - dielectric) between two- and one- band superconductors have been studied recently in a number of articles (Agterberg et al., 2002; Ota et al., 2009; Ng & Nagaosa, 2009). Another basic type of Josephson junctions are the junctions with direct conductivity, S-C-S contacts (C – constriction). As was shown in (Kulik & Omelyanchouk, 1975; Kulik & Omelyanchouk, 1978; Artemenko et al., 1979) the Josephson behavior of S-C-S structures qualitatively differs from the properties of tunnel junctions. A simple theory (analog of Aslamazov-Larkin theory( Aslamazov & Larkin, 1968)) of stationary Josephson effect in S-C-S point contacts for the case of two-band superconductors is described in Sec.4).

## 2. Ginzburg-Landau equations for two-band superconductivity.

The phenomenological Ginzburg-Landau (GL) free energy density functional for two coupled superconducting order parameters

Where

and

The terms

By minimization the free energy F=

and expression for the supercurrent

In the absence of currents and gradients the equilibrium values of order parameters

For the case of two order parameters the question arises about the phase difference _{12} (3), one can obtain that for

If the interband interaction is ignored, the equations (7) are decoupled into two ordinary GL equations with two different critical temperatures

Let the first order parameter is stronger then second one, i.e.

Phenomenological constants

For arbitrary value of the interband coupling

Expanding expressions (11) over

Considered above case (expressions (10)-(12)) corresponds to different critical temperatures

Consider now another situation, which we will use in the following as the model case. Suppose for simplicity that two condensates in current zero state are identical. In this case for arbitrary value of

## 3. Homogeneous current states and GL depairing current

In this section we will consider the homogeneous current states in thin wire or film with transverse dimensions

The current density *j* and modules of the order parameters do not depend on the longitudinal direction *x*. Writing

for the free energy density (2)-(4) we obtain

Where

The current density j in terms of phases

Total current j includes the partial inputs

In contrast to the case of single order parameter (De Gennes, 1966), the condition

Note, that now the value of

We will parameterize the current states by the value of superfluid momentum

The system of equations (21-23) describes the depairing curve *T*. It can be solved numerically for given superconductor with concrete values of phenomenological parameters.

In order to study the specific effects produced by the interband coupling and dragging consider now the model case when order parameters coincide at

Here we normalize

which for

For *q* stable branches, which corresponds to possibility of bistable state. In Fig. 4 the numerically calculated from equations (24,25) curve

The interband scattering (

If dragging effect (

## 4. Little-Parks effect for two-band superconductors

In the present section we briefly consider the Little–Parks effect for two-band superconductors. The detailed rigorous theory can be found in the article (Yerin et al., 2008). It is pertinent to recall that the classical Little–Parks effect for single-band superconductors is well-known as one of the most striking demonstrations of the macroscopic phase coherence of the superconducting order parameter (De Gennes, 1966; Tinkham, 1996). It is observed in open thin-wall superconducting cylinders in the presence of a constant external magnetic field oriented along the axis of the cylinder. Under conditions where the field is essentially unscreened the superconducting transition temperature

where

How the Little–Parks oscillations (28) will be modified in the case of two order parameters with taking into account the proximity (

We proceed with free energy density (20), but now the momentum

At fixed flux

## 5. Josephson effect in two-band superconducting microconstriction

In the Sec.3 GL-theory of two-band superconductors was applied for filament’s length

For

In the case

with the boundary conditions:

Calculating the current density

The current density

Introducing the phase difference on the contact

The critical current *e.g.* (Golubov et. al, 2004)) (see illustration at Fig.11).

This phenomenological theory, which is valid for temperatures near critical temperature

## 6. Conclusion

In this chapter the current carrying states in two-band superconductors are described in the frame of phenomenological Ginzburg-Landau theory. The qualitative new feature, as compared with conventional superconductors, consists in coexistence of two distinct complex order parameters

## Acknowledgments

The author highly appreciates S. Kuplevakhskii and Y.Yerin for fruitful collaborations and discussions. The research is partially supported by the Grant 04/10-N of NAS of Ukraine.