Localized electron states near the armchair edge of graphene

It is known that zigzag graphene edge is able to support edge states: There is a nondispersive single-electron band localized near the zigzag edge. However, it is generally believed that no edge states exist near the unmodified armchair edge, while they do appear if the edge is subjected to suitable modifications (e.g., chemical functionalization). We explicitly present two types of edge modification which support the localized states. Unlike zigzag edge states, which have zero energy and show no dispersion, properties of the armchair localized states depend sensitively on the type of edge modification. Under suitable conditions they demonstrate pronounced dispersion. While the zigzag edge state wave function decays monotonously when we move away from the edge, the armchair edge state wave function shows nonmonotonous decay. Such states may be observed in scanning tunneling spectroscopy experimentally.

Figure 1

Honeycomb graphene lattice and armchair edge. Sublattice $A$ is blue (dark-gray); sublattice $B$ is yellow (light-gray). Primitive lattice vectors ${\mathit{a}}_1,{\mathit{a}}_2$ have coordinates $\frac{a_0}{2}(3,\pm \sqrt{3})$. Nearest-neighbor vectors: ${\mathit{\delta }}_1=a_0(-1,0)\text{and}{\mathit{\delta }}_{2,3}=\frac{a_0}{2}(1,\pm \sqrt{3})$. The armchair edge is located at the $y=0$ line. The sample is located in the $y\le 0$ half-plane.

Figure 2

The model studied in Sec. 3. Graphene occupies the $y\le 0$ half-plane. The nearest-neighbor hopping integral $t^{\prime }$ between atoms at the edge is different from the hopping integral $t$ in the bulk. Localized states propagate along the $x$ axis and decay for $y\to -\infty$.

Figure 3

Numerical solution of Eq. (28) gives $\overline{\varkappa }$ as a function of ${\overline{k}}_x$ for different $t^{\prime }$: (1) $\delta \tau =0.7$, (2) $\delta \tau =0.3$, and (3) $\delta \tau =0.2$. One can observe that the closer $t^{\prime }$ to $t$, the closer $\overline{\varkappa }$ to zero. If $t^{\prime }=t$, then $\overline{\varkappa }=0$. This means that on the unmodified armchair edge the localized states do not exist.

Figure 4

Armchair edge states and the bulk states of graphene. For given $k_x$ there is a continuum of the graphene bulk states, corresponding to different $k_y$. This continuum is shown as a red shaded area. The curves $1–4$ and $1^{\prime }–4^{\prime }$ represent the bands of edge states, calculated numerically for different $t^{\prime }$. For $t^{\prime }<t$ electron-hole symmetrical pairs of edge bands exist in the gap between the continuum of bulk states: (1–1${}^{\prime }$) $t^{\prime }=0.5t$, (2–2${}^{\prime }$) $t^{\prime }=0.1t$. These are type A solutions. When $t^{\prime }>2t$ localized states emerge above and below the bulk states: (3–3${}^{\prime }$) $t^{\prime }=2.01t$, (4–4${}^{\prime }$) $t^{\prime }=2.5t$. These are type B solutions.

Figure 5

Eigenenergy $\varepsilon$ as a function of ${\overline{k}}_x$ for $\delta \tau =0.3$. The solid (red) curve shows the numerical solution of Eqs. (15) and (28). The dotted (blue) curve corresponds to the approximate formula (34), which is valid for small ${\overline{k}}_x$. We can see that Eq. (34) is in good agreement with the numerical result for ${\overline{k}}_x\lesssim \pi /4$.

Figure 6

The function $\varepsilon \left({\overline{k}}_x\right)$ calculated numerically (solid curves) and using approximation (37) in the limit of $\delta \tau \ll 1$: (1) $\delta \tau =0.2$, (2) $\delta \tau =0.3$, (3) $\delta \tau =0.5$. Even for $\delta \tau$ as large as 0.3 Eq. (37) works very well.

Figure 7

Functions ${\overline{\varkappa }}_1\left({\overline{k}}_x\right)$ (solid curve) and ${\overline{\varkappa }}_2\left({\overline{k}}_x\right)$ (dotted curve) found numerically by solving Eqs. (19), (21), and (40) (type B solution). The parameter $t^{\prime }=3t$. One can see that ${\overline{\varkappa }}_1\left({\overline{k}}_x\right)={\overline{\varkappa }}_2({\overline{k}}_x+\pi )$.

Figure 8

Graphene armchair edge with radical atoms attached. Here $R_{\alpha ,\beta }$ are wave functions of electrons at the radical sites, ${\varepsilon }^{\prime }$ is the on-site potential, $t_R$ is a hopping integral between radicals and nearest carbon atoms, which may be different from $t$, the hopping integral in graphene sheets.

Figure 9

Energy of localized electron states near the functionalized armchair edge calculated for large $t_R\gg t$. There are two edge bands ${\varepsilon }_{1,2}\left(k_x\right)$ for type A solution (blue, dashed) found numerically for ${\varepsilon }^{\prime }=0.1t$ and $t_R=2t$. For such a choice of the parameters type B solutions are absent. For same ${\varepsilon }^{\prime }=0.1t$, but larger $t_R=4t$ there are four nondispersive edge bands of type B (black solid lines). In addition, two type A solutions exist. However, since their eigenenergies lie very close to the edge of the continuum, these branches are not shown. The shaded area (red) corresponds to the bulk graphene states.

Figure 10

The length of the C-C bond in the benzene-like molecule C${}_6$Ha${}_6$, where Ha denotes halogen atoms, as a function of the atomic weight of a halogen atom. The weight is given in atomic units (a.u.) and the bond length is in angströms. Halogen substitution leads to elongation of the C-C bond length as compared to the bond length in an ordinary benzene molecule. The dependence illustrates the fact that suitably chosen functionalization may, in principle, induce sufficient elongation of the edge bond to stabilize the edge states described in Sec. 3. Based on the data provided by NIST Chemistry WebBook.

Figure 11

Energy of localized electron states near the functionalized armchair edge calculated for four different sets of parameters ${\varepsilon }^{\prime }$ and $t_R$ in the $t_R\ll t$ case. For each set of parameters, there are two edge bands ${\varepsilon }_{1,2}\left(k_x\right)$ (solid and dashed curves). The parameters are the following: (1) ${\varepsilon }^{\prime }=0.5t$, $t_R=0.1t$ (type A); (2) ${\varepsilon }^{\prime }=0.2t$, $t_R=0.1t$ (type A); (3) ${\varepsilon }^{\prime }=2.7t$, $t_R=0.1t$ (type B); and (4) ${\varepsilon }^{\prime }=3.7t$, $t_R=0.2t$ (type B). The (red) shaded area corresponds to the bulk graphene states with positive energy.