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In this paper, we report the results of our theoretical investigation on the interplay of superconductivity and disorder in two-dimensional (2D) systems. The effect of disorder on superconductivity of 2D systems was found analytically using Green’s function formalism. The results of our calculation revealed that disorder induced due to randomly distributed superconducting islands enhances decoherence of Cooper pairs and suppresses superconductivity. We have also determined the critical value of disorder at which the 2D system completely loses its superconducting properties. Below this critical value of disorder, the system acts as a superconductor, a system with zero electrical resistance. Above the critical value, it acts as an insulator, a system with infinite electric resistance. This is a fascinating result because a direct transition from the state of the infinite conductivity to the opposite extreme of infinite resistivity is unexpected in the theory of condensed matter physics.

Superconductivity is a resistanceless state of matter first discovered in mercury by Onnes [

The study of the effect of disorder on superconductivity began in the late 1930s with the work of Shalnikov [

The problem of dirty superconductors gives a unique opportunity to study the competition between superconductivity which results from pairing of electrons and localization which results from scattering effects of nonmagnetic and magnetic impurities [

Investigations in this field also revealed that the pair-breaking and decoherence effects of disorder on superconductivity of materials depend on their physical dimension and superconducting pairing symmetry. According to Anderson, weak nonmagnetic disorders (impurities, dislocations, etc.) which could not affect the time-reversal symmetry have no significant effect on thermodynamic properties of three-dimensional (3D) s-wave superconductors. In the literature, this is well known by Anderson’s theorem. Until the late 1970s, most of the theoretical investigations in this field had been acting according to this theorem. However, the scaling theory of localization developed in 1979 by Abrahams et al. [

In the present paper, we have studied the effect of disorder on superconductivity of 2D systems. Based on the bosonic scenario of Mathew Fisher [

In two-dimensional superconductors, the effect of disorder can be either pair breaking or decoherence [

Superconducting islands separated by insulating regions (a). For very thin films, the islands are assumed to be on lattice sites (b) [

Based on this scenario, we developed a Hamiltonian of the form

In the case of infinite on-site repulsive interaction, our bosonic model becomes a hard-core boson model, and double occupancy of bosons is completely restricted. Therefore, the hard-core boson system is a bosonic spin-half system. In order to transform bosonic Hamiltonian (

Employing this transformation rule, we have mapped our bosonic Hamiltonian into Heisenberg-type of the form

In this section, we have calculated the expression for spin order which relates superconducting order parameter and disorder strength employing the equation of motion method of Green’s function. The Fourier transformed equation of motion for retarded Green’s function is

Replacing the Heisenberg operators

In order to reduce the higher-order Green functions which appeared in our calculations, we have employed the following decoupling approximations:

The spin operators were also transformed into

The correlation function for operators

The spectral density

Introducing normalized variables

Equation (

The plot in Figure

The behavior of superconducting order

The plot in Figure

The behavior of superconducting order

In this paper, we have studied the superconductor-insulator quantum phase transition in two-dimensional systems. We mainly considered the effect of disorder-enhanced randomness in an on-site chemical potential on the superconductivity of 2D thin films. To this end, we have calculated the relationship between the superconducting order parameter and strength of disorder in an on-site chemical potential. We have also calculated the relationship between superconducting order parameter and strength of repulsive interaction between Cooper pairs on neighbouring islands. Our calculation results revealed that disorder and repulsive Cooper pair interactions suppress superconductivity of 2D thin films. This result is very important in the study of strongly correlated electronic systems specifically in high-Tc copper-based superconductors. Here, we note that our calculations are based only on quantum mechanical mean-field approximations and can only explain qualitatively the more complex phenomenon of SIT. The quantitative explanation of this complex phenomenon requires to employ the more advanced quantum mechanical methods such as dynamical mean-field approximations and dynamical quantum Monte Carlo simulations [

The manuscript is theoretical investigation. It does not have experimental/simulation data to avail for readers. The plots on the figures are generated from equations using Matlab. Therefore, all the data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

_{c}superconductivity in the Ba–La–Cu–O system

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_{c}= 26 K