We present the analytical solution of the wave function and energy dispersion of armchair graphene nanoribbons (GNRs) based on the tight-binding approximation. By imposing the hard-wall boundary condition, we find that the wave vector in the confined direction is discretized. This discrete wave vector serves as the index of different subbands. Our analytical solutions of wave function and associated energy dispersion reproduce the results of numerical tight-binding and the solutions based on the $\mathbf{k}\bullet \mathbf{p}$ approximation. In addition, we also find that all armchair GNRs with edge deformation have energy gaps, which agrees with recently reported first-principles calculations.

In this paper, the average density of states (ADOS) in graphene with binary alloy disorders is calculated by the recursion method. We observed an obvious resonant peak and a dip in ADOS curves near the Dirac point, which result from interactions with surrounding impurities. We also found that the resonance energy $\left(E_r\right)$ and the dip position $\left({\epsilon }_{\text{dip}}\right)$ are strongly dependent on the concentration of disorders $\left(x\right)$ and their on-site potentials $\left(v\right)$. A linear relation, ${\epsilon }_{\text{dip}}=xv$, holds when the impurity concentration is low. This relation can also be extended when the impurity concentration is high, but with certain constraints. We also compute ADOS with a finite density of vacancies.

In this paper, a real-space analytical expression for the free Green’s function (propagator) of bilayer graphene is derived based on the effective-mass approximation. Green’s function displays highly spatial anisotropy with threefold rotational symmetry. The calculated local density of states (LDOS) of a perfect bilayer graphene produces the main features of the observed scanning tunneling microscopy (STM) images of graphite at low bias voltage. Some predicted features of the LDOS can be verified by STM measurements. In addition, we also calculate the LDOS of bilayer graphene with vacancies by using the multiple-scattering theory (scatterings are localized around the vacancy of bilayer graphene). We observe that the interference patterns are determined mainly by the intrinsic properties of the propagator and the symmetry of the vacancies.

In this Letter, we calculate the density of states of graphene under a highly uniform magnetic field and white-noise random potential. We discover that the disorder-broadened zero-energy Landau band has a Gaussian shape and its width is proportional to the random potential variance and the square root of magnetic field. We also use the Wegner-type calculation to justify our simulation results.

In this paper, we numerically study bound electron states induced by long-range Coulomb impurities in gapped graphene and quasibound states in the supercritical region based on the lattice model. We present a detailed comparison between our numerical calculations and the prediction of quantum electrodynamics $2+1\left(\text{QED}\right)$ continuum model. Furthermore, the supercritical charge has been determined by both the lattice model and the continuum model. The numerical results show that the behavior of quasibound state is consistent with the prediction using Fano’s formalism.

Exploiting the enabling power of the Lanczos method in momentum space, we determine accurately the quasiparticle and scaling properties of disordered three-dimensional Dirac semimetals surrounding the quantum critical point separating the semimetal and diffusive metal regimes. We unveil that the imaginary part of the quasiparticle self-energy obeys a common power law before, at, and after the quantum phase transition, but the power law is nonuniversal, whose exponent is dependent on the disorder strength. More intriguingly, whereas a common power law is also found for the real part of the self-energy before and after the phase transition, a distinctly different behavior is identified at the critical point, characterized by the existence of a nonanalytic logarithmic singularity. This nonanalytical correction serves as the very basis for the unusual power-law behaviors of the quasiparticles and many other physical properties surrounding the quantum critical point. Our approach also allows the ready and reliable determination of the scaling properties of the correlation length and dynamical exponents. We further show that the central findings are valid for both uncorrelated and correlated disorder distributions and should be directly comparable with future experimental observations.