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## The Kohn–Luttinger superconductivity in idealized doped graphene

Idealized graphene monolayer is considered neglecting the van der Waals potential of the substrate and the role of the nonmagnetic impurities. The effect of the long-range Coulomb repulsion in an ensemble of Dirac fermions on the formation of the superconducting pairing in a monolayer is studied in the framework of the Kohn–Luttinger mechanism. The electronic structure of graphene is described in the strong coupling Wannier representation on the hexagonal lattice. We use the Shubin–Vonsowsky model which takes into account the intra- and intersite Coulomb repulsions of electrons. The Cooper instability is established by solving the Bethe–Salpeter integral equation, in which the role of the effective interaction is played by the renormalized scattering amplitude. The renormalized amplitude contains the Kohn–Luttinger polarization contributions up to and including the second-order terms in the Coulomb repulsion. We construct the superconductive phase diagram for the idealized graphene monolayer and show that the Kohn–Luttinger renormalizations and the intersite Coulomb repulsion significantly affect the interplay between the superconducting phases with *id*-, and *ip*-wave symmetries of the order parameter.

These notes have appeared as a result of a one-term course in superfluidity and superconductivity given by the author to fourth-year undergraduate students and first-year graduate students of the Department of Physics, Moscow State University of Education. The goal was not to give a detailed picture of these two macroscopic quantum phenomena with an extensive coverage of the experimental background and all the modern developments, but rather to show how the knowledge of undergraduate quantum mechanics and statistical physics could be used to discuss the basic concepts and simple problems, and draw parallels between superconductivity and superfluidity.

Superconductivity and superfluidity are two phenomena where quantum mechanics, typically constrained to the microscopic realm, shows itself on the macroscopic level. Conceptually and mathematically, these phenomena are related very closely, and some results obtained for one can, with a few modifications, be immediately carried over to the other. However, the student of these notes should be aware of important differences between superconductivity and superfluidity that stem mainly from two facts: (1) electrons in a superconductor carry a charge, therefore one has to take into account interaction with electromagnetic radiation; (2) electrons move in a lattice, therefore phonons play a role not only a mediators of attractive interaction between pairs of electrons, but also as scatterers of charge carriers.

Although these are notes on superfluidity *and *superconductivity, and there are a few cross-references, the two subjects can be studied independently with, perhaps, a little extra work by the student to fill in the gaps resulting from such study. The material of Chapter 1 introduces the method of second quantisation that is commonly used to discuss systems with many interacting particles. It is then applied in Chaper 2 to treat the uniform weakly interacting Bose gas within the approach by N. Bogoliubov, and in Chapter 4 to formulate the theory of the uniform superconducting state put forth by J. Bardeen, L. Cooper and R. Schrieffer. Chapter 3 presents the theory proposed independently by E. Gross and L. Pitaevskii of a non-uniform weakly interacting Bose gas, with a discussion of vortices, rotation of the condensate, and the Bogoliubov equations. In Chapter 5 we discuss the Ginzburd-Landau theory of a non-uniform superconductor near the critical temperature and apply it to a few simple problems such as the surface energy of the boundary between a normal metal and a superconductor, critical current and critical magnetic field, and vortices.

The optical properties of graphene-based structures are discissed. The universal optical absorption in graphene is reviewed. The photonic band structure and transmission of graphene-based photonic crystals are considered. The spectra of plasmon and magnetoplasmon excitations in graphene layers and grapehene nanoribbons (GNR) are analyzed. The localization of the electromagnetic waves in the photonic crystals with defects, which play a role of waveguide, is studied. Properties of plasmons and magnetoplasmons in graphene layers and GNR are reviewed. The surface plasmon amplification by stimulated emission of radiation with the net amplification of surface plasmons in the doped GNR is described. The minimal population inversion per unit area needed for the net amplification of plasmons in a doped GNR is reported. The various applications of graphene for photonics and optoelectronics are reviewed. The tunability of photonic and plasmonic properties of various graphene structures by doping achieved by applying the gate voltage is discussed.

Graphene synthesis technology on substrates is promising, as is compatible with existing CMOS-technology. Knowledge about how to affect the substrate of choice for structural and electronic properties of graphene is important and opens up new opportunities in targeted influence on the properties of this unique material. Specialized measuring system was established to measure the galvanomagnetic characteristics of substrates multigraphene. Its structure and the measurement results are presented in the paper. For surface resistivity measurements we obtained samples were higher than that of natural graphite, but much lower than for samples of colloidal suspensions.

A numerical study of the thermodynamic properties of a superconducting quantum cylinder in a longitudinal magnetic field is carried out. Closed-form expressions for the critical temperature, the free energy, the heat capacity jump, and the magnetization difference between the superconducting and normal phases as functions of the nanotube parameters are obtained in limit cases.

Recently bright-light control of the SSPD has been demonstrated. This attack employed a "backdoor" in the detector biasing scheme. Under bright-light illumination, SSPD becomes resistive and remains "latched" in the resistive state even when the light is switched off. While the SSPD is latched, Eve can simulate SSPD single-photon response by sending strong light pulses, thus deceiving Bob. We developed the experimental setup for investigation of a dependence on latching threshold of SSPD on optical pulse length and peak power. By knowing latching threshold it is possible to understand essential requirements for development countermeasures against blinding attack on quantum key distribution system with SSPDs.

We demonstrate evidence of coherent magnetic flux tunneling through superconducting nanowires patterned in a thin highly disordered NbN film. The phenomenon is revealed as a superposition of flux states in a fully metallic superconducting loop with the nanowire acting as an effective tunnel barrier for the magnetic flux, and reproducibly observed in different wires. The flux superposition achieved in the fully metallic NbN rings proves the universality of the phenomenon previously reported for InOx .We perform microwave spectroscopy and study the tunneling amplitude as a function of the wire width, compare the experimental results with theories, and estimate the parameters for existing theoretical models.

The thermodynamical potential of a superconducting quantum cylinder is calculated. The dependence of the critical temperature and the heat capacity of a superconducting system of the surface concentration of electrons and on the radius of the nanotube is studied.

Overview This book concisely presents the latest trends in the physics of superconductivity and superfluidity and magnetismin novel systems, as well as the problem of BCS-BEC crossover in ultracold quantum gases and high-Tc superconductors. It further illuminates the intensive exchange of ideas between these closely related fields of condensed matter physics over the last 30 years of their dynamic development. The content is based on the author’s original findings obtained at the Kapitza Institute, as well as advanced lecture courses he held at the Moscow Engineering Physical Institute, Amsterdam University, Loughborough University and LPTMS Orsay between 1994 and 2011. In addition to the findings of his group, the author discusses the most recent concepts in these fields, obtained both in Russia and in the West. The book consists of 16 chapters which are divided into four parts. The first part describes recent developments in superfluid hydrodynamics of quantum fluids and solids, including the fashionable subject of possible supersolidity in quantum crystals of 4He, while the second describes BCS-BEC crossover in quantum Fermi-Bose gases and mixtures, as well as in the underdoped states of cuprates. The third part is devoted to non-phonon mechanisms of superconductivity in unconventional (anomalous) superconductors, including some important aspects of the theory of high-Tc superconductivity. |The last part considers the anomalous normal state of novel superconductive materials and materials with colossal magnetoresistance (CMR). The book offers a valuable guide for senior-level undergraduate students and graduate students, postdoctoral and other researchers specializing in solid-state and low-temperature physics.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.