Nodal adsorbate bound states in armchair graphene nanoribbons: Fano resonances and adsorbate recognition in weak disorder

We consider adsorbates on metallic armchair-graphene nanoribbons (AGNRs). An adsorbed atom on a nodal site of the gapless subband in the AGNR is found to induce a bound state. The nodal-adsorbate bound state alone does not cause reflection in the AGNR quantum transport in the one propagating-channel regime. Yet its manifestation, as a Fano resonance in $G$, is brought forth by the presence of a non-nodal adsorbate (or a scatterer) located on a longitudinally separated site. Both the adsorbate-dependent nature and the significantly enhanced energy resolution of the Fano structure in $G$ invite adsorbate recognition. The Fano peak is shown to be robust against weak disorder such as from edge vacancies, due to the localized nature of the Fano resonant state. Adsorbates F and OH, described within an extended Hückel model, are considered as examples.

Figure 1

Fano resonances in conductance $G$ of an AGNR with an adsorbate pair A1 and A2. A1 (A2) is a nodal (non-nodal) adsorbate. $G$'s are shown for a pristine AGNR (dotted curve), individual A1 (red curve), A2 (grey dashed curve), and a pair (black curve). Inset is the AGNR (width $W=20$, along $\widehat{x}$, and length along $\widehat{y}$) with the scatterers separated by $L=5$ unit cells along $\widehat{y}$. Also indicated are Bravais lattice vectors ${\mathbf{A}}_1$ (blue arrows), ${\mathbf{A}}_2$ (red arrows), transverse site-position $M$, and $M$-spacing $a$. The adsorbates are F atoms.

Figure 2

Nodal-adsorbate bound-state wave functions $\Psi$ (upper figures) and ${\left|\Psi \right|}^2$ (lower figures). Adsorbates at nodal sites $M=17$, 5, and 8 are presented in (a)–(c), respectively. The adsorbate is F. The sites in (b) marked by grey (red) open symbols are B sites to be attached upon by non-nodal adsorbates in Fig. 3. For display purpose, $s$-like functions are used in place of the site orbitals.

Figure 3

A-, B-site symmetry (asymmetry) in Fano structures for adsorbate pairs. Adsorbate pair in Fig. 2 with the non-nodal adsorbate located on a site of the A-B unit cell denoted by (a) grey (red) open square, and (b) grey (red) open circle. Solid (black) curves are for non-nodal adsorbate located on the A site, and open (red) symbols are for non-nodal adsorbate located on the B site.

Figure 4

Fano structures and Fano widths vs separation $L$ in an adsorbate pair. (a) Conductance $G$ for $L=10$, 5, and 2, in units of $2\sqrt{3}a$. (b) Exponential-type dependence of the Fano width $\Delta {\varepsilon }_{\mathrm{Fano}}$, the peak-dip energy difference, on $L$ for non-nodal adsorbate at $M=4$, 6, and 9. Inset (solid curve) is for non-nodal adsorbate at $M=3$.

Figure 5

$G$ for three nodal adsorbates and one non-nodal adsorbate. Lattice sites $(M,N)$ are (8,6), (5,$-3$), and (17,$-6$) for the nodal adsorbates, and (4,0) for the non-nodal adsorbate. All adsorbates are F in (a), and the non-nodal adsorbate is changed to OH in (b).

Figure 6

Look-up figures for adsorbate recognition. (a)–(c) are look-up figures for AGNR of $W=17$, 20, and 23, respectively. $V_{\mathrm{a}}$ curves for adsorbate OH (F) are denoted by solid left- (right-)pointing triangles with $V_{\mathrm{a}}$ referring to the right-ordinate scale. $1/\mathrm{Re}\left[G_{11}\right]$ curves for $W$-AGNR at nodal sites $MI$ are denoted by open symbols. $G$ curves in (d) and (e) denote adsorbate pairs ($L$, $M_1$, and $M_2$) for F and OH, respectively. Vertical dotted lines in (a)–(c) are guides to adsorbate recognition for (d).

Figure 7

Effects of disorder due to edge vacancies on the Fano resonance of an adsorbate pair from F (a)–(c) and OH (d)–(f). The adsorbates are at $M_1$ and $M_2$ with separation $L=5$ along a 20-AGNR. Conductance curves $G_{\mathrm{p}}$ (dashed curves) are for an adsorbate pair only, $\langle G_{\mathrm{BG}}\rangle$ (grey curves) are for edge-vacancy disorder only, and $\langle G\rangle$ (solid curves) are for both the adsorbate pair and edge-vacancy disorder. The edge vacancies distribute over $L_0\sim 1\mu \text{m}$ with concentration $P_{\mathrm{v}}=0.125\%$. Average conductance $\langle G\rangle$ is obtained for 100 disorder samples.

Figure 8

Same plot as in Fig. 7 except that the number of edge-vacancy configurations is 3. The adsorbates are at $M_1$ and $M_2$ with separation $L=5$ along a 20-AGNR. Adsorbate pair is from F (a)–(c) and OH (d)–(f). Conductance curves $G_{\mathrm{p}}$ (dashed curves) are for an adsorbate pair only, and $\langle G_{\mathrm{BG}}\rangle$ (grey curves) are for edge-vacancy disorder only (100 configurations average, $P_{\mathrm{v}}=0.125\%$ over $1\mu \text{m}$). Conductance ${\langle G\rangle }_{\Delta E}$ is averaged over both an energy interval $\Delta E$ and over three edge-vacancy configurations, where $\Delta E=2.5\times {10}^{-3}t$.

Figure 9

Effects of an additional random adsorbate A3 on the Fano structures of an adsorbate pair in the presence of edge-vacancy disorder. The adsorbate pair, located at $M_1$ and $M_2$, are separated by $L=3$. Adsorbates are F in (a)–(c), and OH in (d)–(f). A3 locates randomly over $M$ and A3-pair $L$ separation is within a $0.5\text{−}\mu \mathrm{m}$ range. ${\langle G\rangle }_{\Delta E}$ is defined as in Fig. 8, but for five configurations of edge vacancies and A3.