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Recent technology and experiments have fabricated high-quality superconducting MgB_{2} nanoparticles.
We investigate properties of two-gap superconductivity in nanosized systems by using a
two-sublevel model. In the present work, we analyze the results obtained for superconducting granules
in the case of multiband superconductivity. We discuss the finite size effect in multiband
superconductors. A definition of the critical level spacing of two-gap superconductivity is also presented,
and we discuss the condensation energy and the parity gap of two-gap superconductivity
in relation to the size dependence of those properties with two bulk gaps and the effective pair
scattering process between two sublevels.

Recent advances in nanoscience have demonstrated that fundamentally new physical phenomena are found when systems are reduced in size to dimensions that become comparable to the fundamental microscopic length scales of a material under study. Superconductivity is a macroscopic quantum phenomenon, and therefore, it is especially interesting to see how this quantum state is influenced when the samples are reduced to nanometer sizes. In such systems, new states of matter can be engineered that do not occur in bulk materials.

The properties of superconducting materials with ultra-small sizes differ from those in bulk [

After the discovery of multiband superconductors such as MgB_{2} with

Recent discovery of superconductivity of MgB_{2} has much attracted great interest with multigap superconductivity [

MgB_{2} is the first material whose effects are so dominant and implications are so thoroughly explored. Magnesium diboride is reported to be an anisotropic superconductor with conventional Bardeen-Cooper-Schrieffer electron-phonon coupling [_{2} calculated in several works since the discovery of the superconductivity is similar to that of graphite and is formed by by the

In the recent years, great efforts have been devoted to the fabrication of MgB_{2} nanostructures that could play a crucial role in the field of applied superconductivity [_{2}, nanoparticles of approximately 20–100 nm in size are available (see Figure

The micrograph of 20–100 nm nanoparticles of MgB_{2}.

For ultra-small superconducting grains, the experiments [

In the case of a two-gap superconductor, we can consider a model with two sublevels corresponding to two independent bands. We consider a pairing Hamiltonian with two sublevels

In this study, we assume that the Debye energies for two sublevels are the same

We now adapt the developed technique of path integration in the case of two-gap superconductivity. In what follows, we will use this method to study the influence of fluctuations on the two-gap superconductivity in ultra-small superconducting grains. This approach gives an exact expression for the canonical partition function of a superconductor [

In the case of stronger interaction,

In nanosized single-band superconductivity, the condensation energy can be defined as

To discuss the critical level spacing for a two-gap system, we start from the coupled gap equation of (

In the case of two sublevel spacings, the chemical potential lies halfway between the highest occupied and the lowest unoccupied levels with smaller level spacing in the half-filled case as shown in Figure

Positioning of the chemical potential relative to the electronic energy levels in a two-gap superconducting grain. Solid and dotted lines mean two sublevels. (a) Half-filled system with

Thus, we have investigated the properties of nanosized two-gap superconductivity by using a two-sublevel model in the framework of the mean-field approximation. In view of the discussion for the condensation energy in nanosized two-gap superconductivity, the phases of the gaps are very important to stabilize the superconductivity. At the same phases, the two-gap superconductivity is instable by the coupling constant

The parity effect can be observed only at sufficiently low temperatures. It becomes especially significant in ultra-small granules about 10 Å in size. The experimental investigation of superconductivity in such granules becomes possible now due to the developed elegant experimental technique [_{2}.

In summary, a model corresponding to nanosized two-gap superconductivity has been presented, and the expression of the partition function of a nanosized system has been analytically derived by using the path integral approach. A definition of the critical level spacing of the two-gap superconductivity has been also presented, and we discuss the condensation energy and the parity gap of the two-gap superconductivity in relation to the size dependence of those properties with two bulk gaps and the effective pair scattering process between two sublevels. The results of this work can be tested in the tunneling experiments with MgB_{2} nanoparticles [

The authors thank Professors Jun Akimitsu and Karl Bennemann for their continued encouragement and helpful discussions. S. P. Kruchinin thanks the Research Center for Structural Thermodynamics, Osaka University, for hosting the author as a guest professor to accomplish this work.

_{2}nanowires

_{2}: structural characterization and in situ study of synthesis kinetics

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