### Article

## Quantum elasticity of graphene: Thermal expansion coefficient and specific heat

We explore thermodynamics of a quantum membrane, with a particular application to suspended graphene membrane and with a particular focus on the thermal expansion coefficient. We show that an interplay between quantum and classical anharmonicity-controlled fluctuations leads to unusual elastic properties of the membrane.

The effect of quantum fluctuations is governed by the dimensionless coupling constant, g0 1, which vanishes

in the classical limit ( → 0) and is equal to 0.05 for graphene. We demonstrate that the thermal expansion

coefficient αT of the membrane is negative and remains nearly constant down to extremely low temperatures,

T0 ∝ exp(−2/g0).We also find that αT diverges in the classical limit: αT ∝ −ln(1/g0) for g0 → 0. For graphene

parameters, we estimate the value of the thermal expansion coefficient as αT −0.23 eV−1, which applies below

the temperature Tuv ∼ g00 ∼ 500K(where 0 ∼ 1 eVis the bending rigidity) down to T0 ∼ 10−14 K. ForT <T0,

the thermal expansion coefficient slowly (logarithmically) approaches zero with decreasing temperature. This

behavior is surprising since typically the thermal expansion coefficient goes to zero as a power-law function.We

discuss possible experimental consequences of this anomaly.We also evaluate classical and quantum contributions

to the specific heat of the membrane and investigate the behavior of the Gr¨uneisen parameter.

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.

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 f-,d+*id*-, and p+*ip*-wave symmetries of the order parameter.

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.