Decomposition into propagating and evanescent modes of graphene ribbons

Bulk modes (BM) are basis solutions to the Schrödinger equation, and they are useful in a number of physical problems. In the present paper, we establish a complete set of BMs for graphene ribbons at arbitrary energy. We derive analytical expressions for these modes and systematically classify them into propagating or evanescent mode. We also demonstrate their uses in efficient electronic transport simulations of graphene-based electronic devices within both the mode-matching method and the Green's function framework. Explicit constructions of Green's functions for infinite and semi-infinite graphene ribbons are presented.

Figure 1

(a) Schematic of graphene lattice. In the $x$-direction, the lattice is viewed as repetition of the supercell (numbered by $m$) indicated by the shaded rectangle. In the $y$-direction, the lattice is a collection of zigzag chains, which are labeled by $n$. A supercell extends from $n=-\infty$ to $n=\infty$. Each lattice site is specified as $(m,n,\nu )$, where $\nu =A,B$ refers to the sublattices. (b) By a transformation expressed in Eq. (6), the supercell is transformed into a diatomic chain. The effective hopping energies between adjacent sites are either $\Lambda$ or 1, depending on whether the sites sit in the same zigzag chain or not.

Figure 2

(a) Schematic of ZGR lattice. The symbol “x” means that the wave function should be zero at this site. (b) Energy band structure of ZGR for $N_z=9$. Shaded region represents the spectrum of graphene. (c) Distribution of roots of Eq. (24) in the complex $\sigma$ plane for the energy region $0<\left|\varepsilon \right|\le {\varepsilon }_c$. In this plot, the energies are taken as $\varepsilon =0.01,0.04,0.07,0.10$. The arrows indicate the direction of increasing energy. None of the roots [except when $\left|\varepsilon \right|={\varepsilon }_c$; note ${\varepsilon }_c=1/(N_z+1)=0.1$ for $N_z=9$] sit on the unit circle. (d) Same as (c) for the energy region of ${\varepsilon }_c\ge \left|\varepsilon \right|\le 1$. In this plot, the energies are taken as $\varepsilon =0.1,0.4,0.7,1.0$. (e) Same as (c) for the energy region of $1\le \left|\varepsilon \right|$. In this plot, the energies are taken as $\varepsilon =1.0,1.5,2.0,2.5,3.0$. Some of the circles are overlapping because of the tiny shift. Here all roots lie on the unit circle (dashed curve), i.e., $\left|\sigma \right|=1$ and the resulting $p$ are real.

Figure 3

(a) The lattice structure of singly bearded ZGR. (b) Energy band structure for $N_z=10$. The central flat subband extends throughout the 1BZ.

Figure 4

(a) Structure of a graphene aperture connecting two graphene nanoribbons (width $N_z$), where the aperture consists of $N_c$ connecting bonds. We use $n_L$ and $n_R$ to label the zigzag chains for the left and right nanoribbons, respectively. The local wave functions vanish on crossed sites while they match on the sites on connecting bonds marked by black and white circles. (b) The energy dependence of dimensionless conductance $g$ for $N_z=115$ and various $N_c$. (c) For $N_c\ll N_z$, the $g$ shows a resonance of width $\Gamma$ that scales as $N_z^{-3}$. The $\Delta$ is indicated in Fig. 2.

Figure 5

A generic setup in two-probe transport measurements. The conducting channel, represented by the Hamiltonian $H_s$, is connected to the left electrode and the right electrode via couplings $V_L$ and $V_R$, respectively. Both electrodes are represented by semi-infinite ideal ZGRs each consisting of $N_z$ zigzag chains.

Figure 6

Lattice structure of armchair graphene ribbons.