Graphene: Free electron scattering within an inverted honeycomb lattice

Abstract

Theoretical progress in graphene physics has largely relied on the application of a simple nearestneighbor tight-binding model capable of predicting many of the electronic properties of this material. However, important features that include electron-hole asymmetry and the detailed electronic bands of basic graphene nanostructures (e.g., nanoribbons with different edge terminations) are beyond the capability of such simple model. Here we show that a similarly simple plane-wave solution for the one-electron states of an atom-based two-dimensional potential landscape, defined by a single fitting parameter (the scattering potential), performs better than the standard tight-binding model, and levels to density-functional theory in correctly reproducing the detailed band structure of a variety of graphene nanostructures. In particular, our approach identifies the three hierarchies of nonmetallic armchair nanoribbons, as well as the doubly-degenerate flat bands of free-standing zigzag nanoribbons with their energy splitting produced by symmetry breaking. The present simple planewave approach holds great potential for gaining insight into the electronic states and the electrooptical properties of graphene nanostructures and other two-dimensional materials with intact or gapped Dirac-like dispersions.The two-dimensional (2D) honeycomb carbon-atom lattice known as graphene 1 is a promising material for applications in optical and electronic devices 2-4 . In particular, its peculiar conical electronic dispersion 5,6 and 2D character enable a uniquely large optical tunability 7,8 and a suitable playground for quantum electrodynamics phenomena, such as the relativistic Klein tunneling 9 , as well as a customizable zoo of exotic band structures when decorated with defects 10 , arranged in twisted bilayers 11 , or laterally patterned into ribbons 12,13 . Energy-gap engineering in graphene, an essential prerequisite for nanoelectronics applications, demands controlled and selective sub-lattice perturbations at the atomic scale, such as chemical doping 14,15 or gating 16 , lateral strain 17,18 , and substrate-induced sublattice asymmetry [19][20][21][22] .Graphene nanoribbons (GNRs) have been extensively studied as simple, appealing nanostructures that lead to electronic band features, such as gap opening, due to quantum confinement, and peculiar edge states that can readily be tuned through their width, shape, and edgeterminations 12,13 . The rapidly progressing on-surface chemistry, which allows controlled-synthesis of novel graphene-based nanostructures, such as GNRs with complex architectures 23-28 , combined with the precise mapping of their electronic structures using angle-resolved photomission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS) 29-32 , make GNRs promising candidates for the realization of exotic graphene-based nanodevices [33][34][35] .Theoretical understanding and prediction of extended graphene and GNRs properties has been instrumental in the development of the field. Density-functional theo...

Synthesis of Graphene Nanoribbons on a Kinked Au Surface: Revealing the Frontier Valence Band at the Brillouin Zone Center

“…3d-f, where we display the theoretical simulation of the photoemission intensity from free standing 7-AGNRs, as determined from our electron plane wave expansion (EPWE) method. 36 The simulation proves that, at k x = Au (161415) Au ( 788 In all cases a downward dispersing GNR-related band is detected around k x = 0, below the Shockley state. We attribute this band to the first occupied GNR band (VB 1 ), which agrees with our EPWE simulations.…”

“…We obtained the band structure and photoemission intensity by solving Schrödinger equation for this defined potential landscape following the procedures detailed in Ref. 36…”

Electronic Structure Tunability by Periodic meta-Ligand Spacing in One-Dimensional Organic Semiconductors

“…Within the EPWE approach, the photoemission intensity for a given binding energy and photoelectron wave vector is obtained from Fermi's golden rule applied to the in-plane wave function (an initial state) and a normalized plane wave (a final state) for the parallel component of the photoelectron wave function, as detailed in ref. (47). In this semiempirical method, zigzag chains are considered free-standing and planar, which implies that the simulated bands are substrate independent and free of Br interactions.…”

Atomically‐Precise Texturing of Hexagonal Boron Nitride Nanostripes

“…The pore/wire nature of the 𝜋 𝛼 /𝜋 𝛽 split-bands in the hBN/Rh(111) interface is well captured in our electron-planewave-expansion (EPWE) and photoemission model. [26] Using 4c (see also Figure S4, Supporting Information for more details). We indeed find that a 𝜋 band splitting arises for ΔV < 0.…”

“…The pore/wire nature of the π α / π β split‐bands in the hBN/Rh(111) interface is well captured in our electron‐plane‐wave‐expansion (EPWE) and photoemission model. [ 26 ] Using only four parameters, namely, the different scattering potentials at N and B atoms, a rigid potential shift Δ V from wires to pores (first Fourier component of the nanoscale potential), and an effective mass m *, we fit the experimentally observed energies at , , and points, and obtain the theoretical bands of Figure 4c (see also Figure , Supporting Information for more details). We indeed find that a π band splitting arises for Δ V < 0.…”