Tight-binding approximation for bulk and edge electronic states in graphene

2015 ◽  
Vol 93 (5) ◽  
pp. 580-584 ◽  
Author(s):  
Oana-Ancuta Dobrescu ◽  
M. Apostol
Keyword(s):  
Finite Differences ◽  
Tight Binding ◽  
Electronic States ◽  
Surface States ◽  
Graphene Sheets ◽  
Graphene Monolayer ◽  
Edge States ◽  
Tight Binding Approximation ◽  
Edge Elements ◽  
Edge Surface

The tight-binding approximation is employed here to investigate electronic bulk and edge (“surface”) states in semi-infinite graphene sheets and graphene monolayer ribbons with various edge terminations (zigzag, horseshoe, and armchair edges). It is shown that edge states do not exist for a uniform hopping (transfer) matrix. The problem is generalized to include edge elements of the hopping matrix distinct from the infinite-sheet (“bulk”) ones. In this case, semi-infinite graphene sheets with zigzag or horseshoe edges exhibit edge states, while semi-infinite graphene sheets with armchair edges do not. The energy of the edge states lies above the (zero) Fermi level. Similarly, symmetric graphene ribbons with zigzag or horseshoe edges exhibit edge states, while ribbons with asymmetric edges (zigzag and horseshoe) do not. It is also shown how to construct the “reflected” solutions (bulk states) for the intervening equations with finite differences both for semi-infinite sheets and ribbons.

Surface-state energies and wave functions in layered organic conductors

2020 ◽  
Vol 75 (11) ◽  
pp. 987-998
Author(s):  
Danica Krstovska ◽  
Aleksandar Skeparovski
Keyword(s):  
Magnetic Field ◽  
Surface State ◽  
Tight Binding ◽  
Surface States ◽  
Organic Conductors ◽  
Wave Functions ◽  
Energy Dispersion ◽  
Quantization Rule ◽  
Tight Binding Approximation ◽  
Dispersion Law

AbstractWe have calculated and analyzed the surface-state energies and wave functions in quasi-two dimensional (Q2D) organic conductors in a magnetic field parallel to the surface. Two different forms for the electron energy spectrum are used in order to obtain more information on the elementary properties of surface states in these conductors. In addition, two mathematical approaches are implemented that include the eigenvalue and eigenstate problem as well as the quantization rule. We find significant differences in calculations of the surface-state energies arising from the specific form of the energy dispersion law. This is correlated with the different conditions needed to calculate the surface-state energies, magnetic field resonant values and the surface wave functions. The calculations reveal that the value of the coordinate of the electron orbit must be different for each state in order to numerically calculate the surface energies for one energy dispersion law, but it has the same value for each state for the other energy dispersion law. This allows to determine more accurately the geometric characteristics of the electron skipping trajectories in Q2D organic conductors. The possible reasons for differences associated with implementation of two distinct energy spectra are discussed. By comparing and analyzing the results we find that, when the energy dispersion law obtained within the tight-binding approximation is used the results are more relevant and reflect the Q2D nature of the organic conductors. This might be very important for studying the unique properties of these conductors and their wider application in organic electronics.

Electronic states at the surfaces of crystals

1939 ◽  
Vol 35 (2) ◽  
pp. 232-241 ◽  
Author(s):  
E. T. Goodwin
Keyword(s):  
Linear Chain ◽  
Tight Binding ◽  
Electronic States ◽  
Surface States ◽  
Cubic Crystal ◽  
Simple Cubic ◽  
Infinite Linear

The method of the previous paper is extended to determine the surface states of a simple cubic crystal on the approximation of tight binding. It is also applied to the case of a semi-infinite linear chain when the atomic s- and p-states are to be regarded as degenerate, the existence of surface states being again predicted.

Quantum spin Hall effect and topological insulators

2017 ◽  
Author(s):  
S. Murakami ◽  
T. Yokoyama
Keyword(s):  
Topological Insulators ◽  
Surface States ◽  
Two Dimensions ◽  
Quantum Spin ◽  
Three Dimensions ◽  
Pure Spin ◽  
Topological Order ◽  
Edge States ◽  
Edge Surface ◽  
Quantum Spin Hall

This chapter begins with a description of quantum spin Hall systems, or topological insulators, which embody a new quantum state of matter theoretically proposed in 2005 and experimentally observed later on using various methods. Topological insulators can be realized in both two dimensions (2D) and in three dimensions (3D), and are nonmagnetic insulators in the bulk that possess gapless edge states (2D) or surface states (3D). These edge/surface states carry pure spin current and are sometimes called helical. The novel property for these edge/surface states is that they originate from bulk topological order, and are robust against nonmagnetic disorder. The following sections then explain how topological insulators are related to other spin-transport phenomena.

scholarly journals Parity-time phase transition in photonic crystals with $$C_{6v}$$ symmetry

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jeng-Rung Jiang ◽  
Wei-Ting Chen ◽  
Ruey-Lin Chern
Keyword(s):  
Phase Transition ◽  
Photonic Crystals ◽  
Band Gap ◽  
Tight Binding ◽  
Symmetric System ◽  
Complex Conjugate ◽  
Edge States ◽  
Tight Binding Approximation ◽  
Common Band ◽  
Gain And Loss

Abstract We investigate the parity-time (PT) phase transition in photonic crystals with $$C_{6v}$$ C 6 v symmetry, with balanced gain and loss on dielectric rods in the triangular lattice. A two-level non-Hermitian model that incorporates the gain and loss in the tight-binding approximation was employed to describe the dispersion of the PT symmetric system. In the unbroken PT phase, the double Dirac cone feature associated with the $$C_{6v}$$ C 6 v symmetry is preserved, with a frequency shift of second order due to the presence of gain and loss. The helical edge states with real eigenfrequencies can exist in the common band gap for two topologically distinct lattices. In the broken PT phase, the non-Hermitian perturbation deforms the dispersion by merging the frequency bands into complex conjugate pairs and forming the exceptional contours that feature the PT phase transition. In this situation, the band gap closes and the edge states are mixed with the bulk states.

AN APPROACH BASED ON LANDAUER–BÜTTIKER FORMALISM TO COMPUTE THE ELECTRONIC LOCALIZED STATES AT SURFACE BOUNDARIES

2019 ◽  
Vol 27 (06) ◽  
pp. 1950164
Author(s):  
ADEL BELAYADI
Keyword(s):  
Tight Binding ◽  
Three Dimensional ◽  
Surface States ◽  
Relaxation Effect ◽  
Matching Theory ◽  
Localized States ◽  
Hamiltonian Matrix ◽  
Inhomogeneous System ◽  
Tight Binding Approximation ◽  
Characteristic Features

In this contribution, we provide a theoretical model to study the effect of different cutting edges on the appearance of localized electronic states. The system under study is a three-dimensional atomic chain that ends with an open cut forming a semi-infinite structured layer in the (1 0 0), (1 1 0) and (1 1 1) directions. We investigate the surface electronic characteristics of the monoatomic chain of a simple cube (sc), orthorhombic (orth), and tetragonal (tetr) structures. We have adopted in our approach the tight-binding approximation to build up the surface Hamiltonian matrix. Additionally, the number of secular equation, at the surface, has been determined by using phase field matching theory (FPMT). In fact, the Hamiltonian system obtained from different cutting orientations provides an inhomogeneous system. To solve the surface eigenvalue problem, we integrate the calculation of the scattering reflection probabilities as given in Landauer–Büttiker formalism. Next, based on the computed scattering probabilities, we build up the surface core states which provide the surface Hamiltonian matrix which can be solved numerically. Our model calculation has been applied to the following elements: (i) fluorite (F), manganese (Mn), polonium (Po), bromine (Br), indium (I), tin (Sn), and protactinium (Pa). The results emphasize the influence of cutting direction on the electronic characteristic of surface and on the scale of energy values. We report the appearance of new electronic curves that characterize the surface states. Those surface states are localized down, within, and above the bulk spectrum. They also provide different characteristic features, of the metals under study, in a given cutting orientation. Furthermore, we have integrated the calculation of non-structured cuts on the outer layers. The relaxation effect on the surface is a standard process which leads to stabilize the changes in the internal energy until the equilibrium. The spacing geometry caused by the relaxation on the surface could be determined by using the molecular dynamic algorithm. We account in this case the lift of degeneracy and the rise of additional localized branches within and outside the bulk range.

First-principles study of the atomic structure and edge state of zigzag carbon nanoscrolls

2015 ◽  
Vol 29 (20) ◽  
pp. 1550110
Author(s):  
Hai-Xia Dong ◽  
Bai-Hua Gong ◽  
Dang-Qi Fang ◽  
Yang Zhang ◽  
Er-Hu Zhang ◽  
...  
Keyword(s):  
Atomic Structure ◽  
First Principles ◽  
Density Functional ◽  
Graphene Nanoribbons ◽  
Tight Binding ◽  
Edge State ◽  
Van Der Waals Interactions ◽  
Edge States ◽  
Tight Binding Approximation ◽  
Carbon Nanoscrolls

The atomic structure and edge state of zigzag carbon nanoscrolls (ZCNSs) are investigated using first-principles calculations based on density functional theory. The results show a non-monotonic dependence of the total energy of ZCNS on the surface curvature due to a competition between the elasticity and the van der Waals interactions in a scroll. The edge states can be tuned by using different forms of edge hydrogenation and inner radius. It is found that the edge state range of monohydrogenated ZCNSs is smaller than that of monohydrogenated zigzag-edged graphene nanoribbons (ZGNRs), which is also verified using the tight-binding approximation. With the different edge hydrogenations, ZCNSs prefer the [Formula: see text] hybridization more than the [Formula: see text] one. Our present study could suggest the possibility of adjusting the electronic properties of ZCNSs and may provide potential applications in the electronic devices.

scholarly journals Lazy electrons in graphene

2019 ◽  
Vol 116 (37) ◽  
pp. 18316-18321
Author(s):  
Vaibhav Mohanty ◽  
Eric J. Heller
Keyword(s):  
Time Evolution ◽  
Tight Binding ◽  
Electronic States ◽  
Basis Set ◽  
Many Body ◽  
Tight Binding Approximation ◽  
Accurate Picture ◽  
Oppenheimer Approximation

Within a tight-binding approximation, we numerically determine the time evolution of graphene electronic states in the presence of classically vibrating nuclei. There is no reliance on the Born–Oppenheimer approximation within the p-orbital tight-binding basis, although our approximation is “atomically adiabatic”: the basis p-orbitals are taken to follow nuclear positions. Our calculations show that the strict adiabatic Born–Oppenheimer approximation fails badly. We find that a diabatic (lazy electrons responding weakly to nuclear distortions) Born–Oppenheimer model provides a much more accurate picture and suggests a generalized many-body Bloch orbital-nuclear basis set for describing electron–phonon interactions in graphene.

Electronic states at the surfaces of crystals

1939 ◽  
Vol 35 (2) ◽  
pp. 221-231 ◽  
Author(s):  
E. T. Goodwin
Keyword(s):  
Linear Chain ◽  
Tight Binding ◽  
Electronic States ◽  
Surface States ◽  
Wave Functions ◽  
Overlap Integrals ◽  
Finite Linear

The wave functions and energies of both the normal and surface electronic states of a finite linear chain are determined in terms of the overlap integrals by the approximation of tight binding, it being assumed that the interaction of atomic s-states with p-states can be neglected. It is shown that the existence of surface states depends on the ratio of two overlap integrals being greater than unity, and reasons are given for expecting this to be so.

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