^{1}

^{2}

^{1}

^{2}

Using Bogoliubov-de Gennes formalism for inhomogeneous systems, we have studied superconducting properties of a bundle of packed carbon nanotubes, making a triangular lattice in the bundle's transverse cross-section. The bundle consists of a mixture of metallic and doped semiconducting nanotubes, which have different critical transition temperatures. We investigate how a spatially averaged superconducting order parameter and the critical transition temperature depend on the fraction of the doped semiconducting carbon nanotubes in the bundle. Our simulations suggest that the superconductivity in the bundle will be suppressed when the fraction of the doped semiconducting carbon nanotubes will be less than 0.5, which is the percolation threshold for a two-dimensional triangular lattice.

Single wall carbon nanotubes (SWCNTs) represent a unique class of quasione-dimensional nanoscale systems exhibiting various interesting phenomena. Among other exciting features, it was demonstrated that individual single wall carbon nanotubes may have intrinsic superconducting properties [

It is expected that doping of SWCNTs in a bundle by, for example, boron, may significantly improve their superconducting properties [

However, synthesis of SWCNTs by currently known methods usually results in a mixture of semiconducting and metallic nanotubes. Since the nanotubes after the synthesis initially are not doped (or unintentionally slightly p-type doped, e.g., by oxygen of atmosphere), those are only metallic tubes, which may have superconducting transition, while semiconducting tubes will be “diluting” superconductivity in the bundle by the inverse proximity effect [

One may try to estimate a spatially averaged order parameter and the corresponding effective critical temperature for a bundle consisting of a mixture of these two types of SWCNTs. From an experimentalist’s point of view, it is even more important to solve a bit more complex problem: for a given fraction of doped semiconducting SWCNTs in the bundle and the experimentally determined critical temperature

Spatial variations of the superconducting order parameter are significant for nanoscale systems, including nanotubes [

For the description of the system, we utilized the Hamiltonian form:

Using the Bogoliubov transformation, which diagonalizes the Hamiltonian equation (

Here, we assumed the same constant hopping parameter

We studied how the spatially averaged superconducting order parameter

The results of simulations are shown in Figure

Spatially averaged superconducting order parameter

In Figure

Spatially averaged superconducting order parameter

Using the data plotted in Figure

Critical temperature

According to our model for an optimally doped bundle consisting of 100% semiconducting SWCNTs, the

One has to note that even for relatively small bundles 16 × 16, the physical properties have relatively small variations for different realizations of the spatial distributions of semiconducting nanotubes. To support this observation, we plot a whole set of the spatially averaged order parameters for 50 random configurations of the nanotubes for different temperatures. We choose the equal number of the metallic and semiconducting nanotubes in the bundle

A set of spatially averaged order parameters (in units of

In Figure

A particular realization of the spatial distribution of the superconducting order parameter (in units of

In conclusion, we introduced a mean field microscopic model to describe superconductivity in a bundle of a mixture of carbon nanotubes of different superconducting properties. We have studied the dependence of the spatially averaged superconducting gap

Note that our mean field BdG model is unable to predict and properly describe quantum phase fluctuations of the order parameter in isolated doped semiconducting nanotubes, where the superconductivity will be suppressed even stronger. Future research using, for example, Ginzburg-Landau inhomogeneous equations [

_{13}

_{2}superconductor: a case study on a 2D triangular lattice in the repulsive Hubbard model