We analyze the features of the graphene mono-and multilayer reflectance in the far-infrared region as a function of frequency, temperature, and carrier density taking the intraband conductance and the interband electron absorbtion into account. The dispersion of plasmon mode of the multilayers is calculated using Maxwell's equations with the influence of retardation included. At low temperatures and high electron densities, the reflectance of multilayers as a function of frequency has the sharp downfall and the subsequent deep well due to the threshold of electron interband absorbtion.PACS numbers: 81.05. Uw, 78.67.Ch, Monolayer and bilayer graphenes 1,2,3 are gapless twodimensional (2D) semiconductors 4,5,6 whereas its 3D predecessor-graphite is a semimetal 7,8,9 . Hence the dimensionality effects for the unique substance can be studied 10 . Monolayer graphene has a very simple electron band structure. Near the energy ε = 0, the energy bands are cones ε 1,2 (p) = ±vp at the K points in the 2D Brillouin zone with the constant velocity parameter v = 10 8 cm/s. Such a degeneration is conditioned by symmetry because the small group C 3v of the K points has two-dimensional representation.While the carrier concentration is decreasing in the field gate experiment, the graphene conductivity at low temperatures goes to the finite minimal values 1,2 . Much theoretical efforts 11,12,13,14 have been devoted to evaluate the minimal conductivity in different approaches. Theoretical 15,16,17 and experimental researches show that the main mechanism of the carrier relaxation is provided by the charged impurities and gives the collision rate τ −1 ∼ 2π 2 e 4 n imp /hǫ 2 g ε, where ǫ g is the dielectric constant of graphene, ε is the characteristic electron energy (of the order of the Fermi energy or temperature), and n imp is the density of charged impurities per the unit surface. Plasmons in graphene are considered in Refs.18,19 . The optical visibility of both monolayer and bilayer graphene is studied in Ref.20 focusing on the role of the underlying substrate.In the present paper, we analyze the spectroscopy of the graphene monolayer and multilayers in the infra-red region. In order to calculate the reflection coefficient for the multilayers, we follow the method used in Ref.21 and determine the spectrum of electromagnetic excitationsplasmons. We use the appropriate boundary conditions at interfaces and the complex conductivity σ as a function of frequency ω, temperature T , and chemical potential µ. The chemical potential of ideal pure graphene equals to zero at any temperature. With the help of the gate voltage, one can control the density and type (n or p) of carriers varying their chemical potential.The general expression for the conductivity used here is obtained in our previous paper 18 and is valid under a restriction that the collision rate of carriers is less than the frequency and spatial dispersion of the electric ac field, τ −1 ≪ ω, kv. In limiting cases, our result coincides with the formulas of Ref.22,23 . For hig...

The discovery of the Dirac electron dispersion in graphene [1] led to the question of the Dirac cone stability with respect to interactions. Coulomb interactions between electrons were shown to induce a logarithmic renormalization of the Dirac dispersion. With a rapid expansion of the list of compounds and quasiparticle bands with linear band touching [2], the concept of bosonic Dirac materials has emerged. We consider a specific case of ferromagnets consisting of the Van der Waals-bonded stacks of honeycomb layers, e.g chromium trihalides CrX3 (X = F, Cl, Br and I), that display two spin wave modes with energy dispersion similar to that for the electrons in graphene. At the single particle level, these materials resemble their fermionic counterparts. However, how different particle statistics and interactions affect the stability of Dirac cones has yet to be determined. To address the role of interacting Dirac magnons, we expand the theory of ferromagnets beyond the standard Dyson theory [3, 4] to a case of non-Bravais honeycomb layers. We demonstrate that magnon-magnon interactions lead to a significant momentumdependent renormalization of the bare band structure in addition to strongly momentumdependent magnon lifetimes. We show that our theory qualitatively accounts for hitherto unexplained anomalies in a nearly half century old magnetic neutron scattering data for CrBr3 [5,6]. We also show that honeycomb ferromagnets display dispersive surface and edge states, unlike their electronic analogs.

We show that the surface states in topological insulators can be understood based on a well-known Shockley model, a one-dimensional tight-binding model with two atoms per elementary cell, connected via alternating tunneling amplitudes. We generalize the one-dimensional model to the three-dimensional case corresponding to the sequence of layers connected via the amplitudes, which depend on the in-plane momentum p = (p_x,p_y). The Hamiltonian of the model is described a (2 x 2) Hamiltonian with the off-diagonal element t(k,p) depending also on the out-of-plane momentum k. We show that the complex function t(k,p) defines the properties of the surface states. The surface states exist for the in-plane momenta p, where the winding number of the function t(k,p) is non-zero as k is changed from 0 to 2pi. The sign of the winding number defines the sublattice on which the surface states are localized. The equation t(k,p)=0 defines a vortex line in the three-dimensional momentum space. The projection of the vortex line on the two-dimensional momentum p space encircles the domain where the surface states exist. We illustrate how our approach works for a well-known TI model on a diamond lattice. We find that different configurations of the vortex lines are responsible for the "weak" and "strong" topological insulator phases. The phase transition occurs when the vortex lines reconnect from spiral to circular form. We discuss the Shockley model description of Bi_2Se_3 and the applicability of the continuous approximation for the description of the topological edge states. We conclude that the tight-binding model gives a better description of the surface states.Comment: 18 pages, 17 figures; version 3: Sections I-IV revised, Section VII added, Refs. [33]-[35] added; Corresponds to the published versio