Droplets of the Order Parameter in a Low Density Attracting Electron System in the Presence of a Strong Random Potential


The properties of a two-dimensional low density (n ≪ 1) electron system with strong onsite Hubbard attraction U > W (W is the bandwidth) in the presence of a strong random potential V uniformly distributed in the range from –V to +V are considered. Electronic hops only at neighboring sites on the square lattice are taken into account, thus W = 8t. The calculations were carried out for a lattice of 24 × 24 sites with periodic boundary conditions. In the framework of the Bogoliubov–de Gennes approach we observed an appearance of inhomogeneous states of spatially separated Fermi–Bose mixture of Cooper pairs and unpaired electrons with the formation of bosonic droplets of different size in the matrix of the unpaired normal states We observed a decrease in the droplet size (from larger droplets to individual bielectronic pairs) when we decrease the electron density at fixed values of the Hubbard attraction and random potential. The obtained results are important for the construction of the gross phase diagram and understanding of the nature of the phase transition between superconducting, normal metallic and localized states in quasi-2D (thin) film of a dirty metal. In a more practical sense it is interesting also for the experimental implementation of superconducting qubits on quantum circuits with high impedances in granular superconductors.

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We are grateful to R.Sh. Ihsanov, E.A. Burovskii, K.I. Kugel, A.Ya. Tzalenchuk, A.S. Vasenko, N.N. Degtyarenko, A.V. Krasavin, A.A. Golubov for useful discussions of this work.


This work was supported by the Competitiveness Enhancement Project of NRNU MEPhI (contract no. 02.a03.21.0005, 08.27.2013) using the equipment of the collective use center “Complex for modeling and processing data from research facilities of the mega-class” SIC “Kurchatov Institute” (subsidy of the Ministry of Education and Science, work identifier RFMEFI62117X0016), http://ckp.nrcki.ru/.

M.Yu.K. thanks for the support the HSE Program of Basic Research and expresses his gratitude to the RFBR fund (grant no. 20-02-00015).

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Correspondence to M. Yu. Kagan or E. A. Mazur.

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Contribution for the JETP special issue in honor of I.E. Dzyaloshinskii’s 90th birthday

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Kagan, M.Y., Mazur, E.A. Droplets of the Order Parameter in a Low Density Attracting Electron System in the Presence of a Strong Random Potential. J. Exp. Theor. Phys. 132, 596–605 (2021). https://doi.org/10.1134/S1063776121040233

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