Nonlinear optical conductivity of a two-band crystal I.

The structure of the electronic nonlinear optical conductivity is elucidated in a detailed study of the time-reversal symmetric two-band model. The nonlinear conductivity is decomposed as a sum of contributions related with different regions of the first Brillouin zone, defined by single or multiphoton resonances. All contributions are written in terms of the same integrals, which contain all information specific to the particular model under study. In this way, ready-to-use formulas are provided that reduce the often tedious calculations of the second and third order optical conductivity to the evaluation of a small set of similar integrals. In the scenario where charge carriers are present prior to optical excitation, Fermi surface contributions must also be considered and are shown to have an universal frequency dependence, tunable by doping. General characteristics are made evident in this type of resonance-based analysis: the existence of step functions that determine the chemical potential dependence of electron-hole symmetric insulators; the determination of the imaginary part by Hilbert transforms, simpler than those of the nonlinear Kramers-Krönig relations; the absence of Drude peaks in the diagonal elements of the second order conductivity, among others. As examples, analytical expressions are derived for the nonlinear conductivities of some simple systems: a very basic model of direct gap semiconductors and the Dirac fermions of monolayer graphene.

A polynomial approach to the spectrum of Dirac-Weyl polygonal Billiards.

The Schrödinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lamé in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac-Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method's convergence in cases where the spectrum is known exactly, and apply it to cases where no exact solution exists.

A study of the nonlinear optical response of the plain graphene and gapped graphene monolayers beyond the Dirac approximation.

In this work, we present numerical results for the second and third order conductivities of the plain graphene and gapped graphene monolayers associated with the second and third harmonic generation, the optical rectification and the optical Kerr effect. The frequencies considered here range from the microwave to the ultraviolet portion of the spectrum, the latter end of which had not yet been studied. These calculations are performed in the velocity gauge and directly address the components of the conductivity tensor. In the velocity gauge, the radiation field is represented by a power series in the vector potential, and we discuss a very efficient way of calculating its coefficients in the context of tight-binding models.

Spectral functions of one-dimensional systems with correlated disorder.

We investigate the spectral function of Bloch states in a one-dimensional tight-binding non-interacting chain with two different models of static correlated disorder, at zero temperature. We report numerical calculations of the single-particle spectral function based on the Kernel polynomial method, which has an [Formula: see text] computational complexity. These results are then confirmed by analytical calculations, where precise conditions were obtained for the appearance of a classical limit in a single-band lattice system. Spatial correlations in the disordered potential give rise to non-perturbative spectral functions shaped as the probability distribution of the random on-site energies, even at low disorder strengths. In the case of disordered potentials with an algebraic power-spectrum, [Formula: see text] [Formula: see text], we show that the spectral function is not self-averaging for [Formula: see text].

Scattering by linear defects in graphene: a tight-binding approach.

We develop an analytical scattering formalism for computing the transmittance through periodic defect lines within the tight-binding model of graphene. We first illustrate the method with a relatively simple case, the pentagon-only defect line. Afterwards, more complex defect lines are treated, namely the zz(558) and the zz(5757) ones. The formalism developed uses only simple tight-binding concepts, reducing the problem to matrix manipulations which can be easily worked out by any computational algebraic calculator.

Electronic properties of a biased graphene bilayer.

We study, within the tight-binding approximation, the electronic properties of a graphene bilayer in the presence of an external electric field applied perpendicular to the system-a biased bilayer. The effect of the perpendicular electric field is included through a parallel plate capacitor model, with screening correction at the Hartree level. The full tight-binding description is compared with its four-band and two-band continuum approximations, and the four-band model is shown to always be a suitable approximation for the conditions realized in experiments. The model is applied to real biased bilayer devices, made out of either SiC or exfoliated graphene, and good agreement with experimental results is found, indicating that the model is capturing the key ingredients, and that a finite gap is effectively being controlled externally. Analysis of experimental results regarding the electrical noise and cyclotron resonance further suggests that the model can be seen as a good starting point for understanding the electronic properties of graphene bilayer. Also, we study the effect of electron-hole asymmetry terms, such as the second-nearest-neighbour hopping energies t' (in-plane) and γ(4) (inter-layer), and the on-site energy Δ.

Dirac electrons in graphene-based quantum wires and quantum dots.

In this paper we analyse the electronic properties of Dirac electrons in finite-size ribbons and in circular and hexagonal quantum dots. We show that due to the formation of sub-bands in the ribbons it is possible to spatially localize some of the electronic modes using a p-n-p junction. We also show that scattering of confined Dirac electrons in a narrow channel by an infinitely massive wall induces mode mixing, giving a qualitative reason for the fact that an analytical solution to the spectrum of Dirac electrons confined in a square box has not yet been found. A first attempt to solve this problem is presented. We find that only the trivial case k = 0 has a solution that does not require the existence of evanescent modes. We also study the spectrum of quantum dots of graphene in a perpendicular magnetic field. This problem is studied in the Dirac approximation, and its solution requires a numerical method whose details are given. The formation of Landau levels in the dot is discussed. The inclusion of the Coulomb interaction among the electrons is considered at the self-consistent Hartree level, taking into account the interaction with an image charge density necessary to keep the back-gate electrode at zero potential. The effect of a radial confining potential is discussed. The density of states of circular and hexagonal quantum dots, described by the full tight-binding model, is studied using the Lanczos algorithm. This is necessary to access the detailed shape of the density of states close to the Dirac point when one studies large systems. Our study reveals that zero-energy edge states are also present in graphene quantum dots. Our results are relevant for experimental research in graphene nanostructures. The style of writing is pedagogical, in the hope that newcomers to the subject will find this paper a good starting point for their research.

Localized states at zigzag edges of bilayer graphene.

We report the existence of zero-energy surface states localized at zigzag edges of bilayer graphene. Working within the tight-binding approximation we derive the analytic solution for the wave functions of these peculiar surface states. It is shown that zero-energy edge states in bilayer graphene can be divided into two families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene and (ii) states with a finite amplitude over the two layers, with an enhanced penetration into the bulk. The bulk and surface (edge) electronic structure of bilayer graphene nanoribbons is also studied, both in the absence and in the presence of a bias voltage between planes.

Graphene bilayer with a twist: electronic structure.

We consider a graphene bilayer with a relative small angle rotation between the layers--a stacking defect often seen in the surface of graphite--and calculate the electronic structure near zero energy in a continuum approximation. Contrary to what happens in an AB stacked bilayer and in accord with observations in epitaxial graphene, we find: (a) the low energy dispersion is linear, as in a single layer, but the Fermi velocity can be significantly smaller than the single-layer value; (b) an external electric field, perpendicular to the layers, does not open an electronic gap.

Optimized multicanonical simulations: a proposal based on classical fluctuation theory.

We propose a recursive procedure to estimate the microcanonical density of states in multicanonical Monte Carlo simulations which relies only on measurements of moments of the energy distribution, avoiding entirely the need for energy histograms. This method yields directly a piecewise analytical approximation to the microcanonical inverse temperature beta(E) and allows improved control over the statistics and efficiency of the simulations. We demonstrate its utility in connection with recently proposed schemes for improving the efficiency of multicanonical sampling, either with adjustment of the asymptotic energy distribution or with the replacement of single spin flip dynamics with collective updates.

Disorder induced localized States in graphene.

We consider the electronic structure near vacancies in the half-filled honeycomb lattice. It is shown that vacancies induce the formation of localized states. When particle-hole symmetry is broken, localized states become resonances close to the Fermi level. We also study the problem of a finite density of vacancies, obtaining the electronic density of states, and discussing the issue of electronic localization in these systems. Our results also have relevance for the problem of disorder in d-wave superconductors.

Double exchange model for magnetic hexaborides.

A microscopic theory for rare-earth ferromagnetic hexaborides, such as Eu1-xCaxB6, is proposed on the basis of the double-exchange Hamiltonian. In these systems, the reduced carrier concentrations place the Fermi level near the mobility edge, introduced in the spectral density by the disordered spin background. We show that the transport properties such as the Hall effect, magnetoresistance, frequency dependent conductivity, and dc resistivity can be quantitatively described within the model. We also make specific predictions for the behavior of the Curie temperature T(C) as a function of the plasma frequency omega(p).

Exact solution of Ising model on a small-world network.

We present an exact solution of a one-dimensional Ising chain with both nearest-neighbor and random long-range interactions. Not surprisingly, the solution confirms the mean-field character of the transition. This solution also predicts the finite-size scaling that we observe in numerical simulations.