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Register a new userAnalytical expressions for the Electromagnetic Dyadic Greens Function in Graphene and thin layers
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An analytical general analysis of the electromagnetic Dyadic Greens Function for two-dimensional sheet (or a very thin film) is presented, with an emphasis on on the case of graphene. A modified steepest descent treatment of the fields from a point dipole given in the form of Sommerfeld integrals is performed. We sequentially derive the expressions for both out-of-plane and in-plane fields of both polarizations. It is shown that the analytical approximation provided is very precise in a wide range of distances from a point source, down to a deep subwavelength region (1/100 of wavelength). We separate the contribution from the pole, the branch point and discuss their interference. The asymptotic expressions for the fields are composed of the plasmon, Norton wave and the components corresponding to free space.
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We study the electronic structure of graphene with a single substitutional vacancy using a combination of the density-functional, tight-binding, and impurity Greens function approaches. Density functional studies are performed with the all-electron spin-polarized linear augmented plane wave (LAPW) method. The three $sp^2 sigma$ dangling bonds adjacent to the vacancy introduce localized states (V$sigma$) in the mid-gap region, which split due to the crystal field and a Jahn-Teller distortion, while the $p_z pi$ states introduce a sharp resonance state (V$pi$) in the band structure. For a planar structure, symmetry strictly forbids hybridization between the $sigma$ and the $pi$ states, so that these bands are clearly identifiable in the calculated band structure. As for the magnetic moment of the vacancy, the Hunds-rule coupling aligns the spins of the four localized V$sigma_1 uparrow downarrow$, V$sigma_2 uparrow $, and the V$pi uparrow$ electrons resulting in a S=1 state, with a magnetic moment of $2 mu_B$, which is reduced by about $0.3 mu_B$ due to the anti-ferromagnetic spin-polarization of the $pi$ band itinerant states in the vicinity of the vacancy. This results in the net magnetic moment of $1.7 mu_B$. Using the Lippmann-Schwinger equation, we reproduce the well-known $sim 1/r$ decay of the localized V$pi$ wave function with distance and in addition find an interference term coming from the two Dirac points, previously unnoticed in the literature. The long-range nature of the V$pi$ wave function is a unique feature of the graphene vacancy and we suggest that this may be one of the reasons for the widely varying relaxed structures and magnetic moments reported from the supercell band calculations in the literature.
An analytical method for diffraction of a plane electromagnetic wave at periodically-modulated graphene sheet is presented. Both interface corrugation and periodical change in the optical conductivity are considered. Explicit expressions for reflection, transmission, absorption and transformation coefficients in arbitrary diffraction orders are presented. The dispersion relation and decay rates for graphene plasmons of the grating are found. Simple analytical expressions for the value of the band gap in the vicinity of the first Brillouin zone edge is derived. The optimal amplitude and wavelength, guaranteeing the best matching of the incident light with graphene plasmons are found for the conductivity grating. The analytical results are in a good agreement with first-principle numeric simulations.
We present a numerically efficient technique to evaluate the Greens function for extended two dimensional systems without relying on periodic boundary conditions. Different regions of interest, or `patches, are connected using self energy terms which encode the information of the extended parts of the system. The calculation scheme uses a combination of analytic expressions for the Greens function of infinite pristine systems and an adaptive recursive Greens function technique for the patches. The method allows for an efficient calculation of both local electronic and transport properties, as well as the inclusion of multiple probes in arbitrary geometries embedded in extended samples. We apply the Patched Greens function method to evaluate the local densities of states and transmission properties of graphene systems with two kinds of deviations from the pristine structure: bubbles and perforations with characteristic dimensions of the order of 10-25 nm, i.e. including hundreds of thousands of atoms. The strain field induced by a bubble is treated beyond an effective Dirac model, and we demonstrate the existence of both Friedel-type oscillations arising from the edges of the bubble, as well as pseudo-Landau levels related to the pseudomagnetic field induced by the nonuniform strain. Secondly, we compute the transport properties of a large perforation with atomic positions extracted from a TEM image, and show that current vortices may form near the zigzag segments of the perforation.
We present a further development of methods for analytical calculations of Greens functions of lattice fermions based on recurrence relations. Applying it to tight-binding systems and topological superconductors in different dimensions we obtain a number of new results. In particular we derive an explicit expression for arbitrary Greens function of an open Kitaev chain and discover non-local fermionic corner states in a 2D p-wave superconductor.
The boundary Greens function (bGF) approach has been established as a powerful theoretical technique for computing the transport properties of tunnel-coupled hybrid nanowire devices. Such nanowires may exhibit topologically nontrivial superconducting phases with Majorana bound states at their boundaries. We introduce a general method for computing the bGF of spinful multi-channel lattice models for such Majorana nanowires, where the bGF is expressed in terms of the roots of a secular polynomial evaluated in complex momentum space. In many cases, those roots, and thus the bGF, can be accurately described by simple analytical expressions, while otherwise our approach allows for the numerically efficient evaluation of bGFs. We show that from the behavior of the roots, many physical quantities of key interest can be inferred, e.g., the value of bulk topological invariants, the energy dependence of the local density of states, or the spatial decay of subgap excitations. We apply the method to single- and two-channel nanowires of symmetry class D or DIII. In addition, we study the spectral properties of multi-terminal Josephson junctions made out of such Majorana nanowires.
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