### Article

## Influence of Landau level mixing on the properties of elementary excitations in graphene in strong magnetic field

Massless Dirac electrons in graphene fill Landau levels with energies scaled as square roots of their numbers. Coulomb interaction between electrons leads to mixing of different Landau levels. The relative strength of this interaction depends only on dielectric susceptibility of surrounding medium and can be large in suspended graphene. We consider influence of Landau level mixing on the properties of magnetoexcitons and magnetoplasmons—elementary electron-hole excitations in graphene in quantizing magnetic field. We show that, at small enough background dielectric screening, the mixing leads to very essential change of magnetoexciton and magnetoplasmon dispersion laws in comparison with the lowest Landau level approximation.

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.

The behavior of the TE and TM electromagnetic waves in graphene at the interface between two semi-infinite dielectric media is studied. The dramatic influence on the TE waves propagation even at very small changes in the optical contrast between the two dielectric media is predicted. Frequencies of the TE waves are found to lie only in the window determined by the contrast. We consider this effect in connection with the design of graphene-based optical gas sensor. Near the frequency, where the imaginary part of the conductivity of graphene becomes zero, ultrahigh refractive index sensitivity and very low detection limit are revealed. The considered graphene-based optical gas sensor outperforms characteristics of modern volume refractive index sensors by several orders of magnitude.

Magneto-fermionic condensate under study is a Bose-Einstein condensate of cyclotron spin-flip magnetoexcitons in a quantum Hall insulator. This condensate features unique properties such as millisecond range lifetime and hundreds of micrometers of propagation length. In this study, utilizing the photo-induced resonant reflection technique, we measured the exciton escape time. Finally, we estimated the exciton condensate propagation velocity as 25 m/s, which is much higher than a single particle propagation velocity. We also proposed a mechanism of exciton condensation.

A novel type of spaser with the net amplification of surface plasmons (SPs) in a doped graphene nanoribbon is proposed. The plasmons in the THz region can be generated in a doped graphene nanoribbon due to nonradiative excitation by emitters like two level quantum dots located along a graphene nanoribbon. The minimal population inversion per unit area, needed for the net amplification of SPs in a doped graphene nanoribbon, is obtained. The dependence of the minimal population inversion on the surface plasmon wave vector, graphene nanoribbon width, doping, and damping parameters necessary for the amplification of surface plasmons in the armchair graphene nanoribbon is studied.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.