Effect of random edge-vacancy disorder in zigzag graphene nanoribbons

The magnetic and coherent transport properties of small-width zigzag graphene nanoribbons (ZGNRs) with monohydrogen edge passivation are investigated as a function of random edge-vacancy disorder and ribbon length. Results from noninteracting tight-binding models with (i) nearest and (ii) up to third nearest neighbor hopping are compared against those obtained from an extended mean-field Hubbard model for edge-defected ZGNRs (length = 48.02 $\AA$ and width = 9.24 $\AA$). Through ensemble averaging, a persistent magnetism and Hubbard-$U$ (i.e., spin-generated) conductance gap is found irrespective of the extent of random edge-vacancy disorder. At longer device lengths (up to 144.1 $\AA$) and at high disorder (42.5%), gaps open in the noninteracting model systems, whereas the gap in the Hubbard-calculated systems becomes spin dependent. In all cases, the conductance gaps increase as a function of increasing system length, although the gaps in the Hubbard systems remain smaller due to increased robustness against edge disorder. The continuance of the magnetic state and gap robustness in the ensemble-averaged Hubbard results indicates a complex interplay between the kinetics, disorder, system size, and spin interaction. Such findings may serve to reinform previous studies that have used noninteracting models to investigate disorder in ZGNRs.

Figure 1

(a) Ideal 20-5-ZGNR device used in this work that is 20 carbon atoms long (48.02 $\AA$) and 5 carbon atoms wide (9.24 $\AA$). 20-5-ZGNRs with randomly-generated (b) $7.5\%$, (c) $42.5\%$, and (d) $90\%$ edge-vacancy concentrations.

Figure 2

Schematic of an edge-vacancy-defected portion of a 20-5-ZGNR showing (a) a Klein defect (circled), and (b) a notch-defect extending into the second atomic row after the Klein defect has been removed.

Figure 3

GTB results of the local net spin polarization for the systematic edge-vacancy-defected ZGNRs studied by Huang et al. [15]. Blue circles are net spin-up and red circles are net spin-down. Green circles are net zero spin polarization. The edge-vacancy concentrations are 0.00, 0.034, 0.068, and 0.136 ${\AA }^{-1}$ from left to right (Huang et al.'s convention) with equivalent percentage edge-vacancy concentrations also shown.

Figure 4

(a) Quantitative values taken from the GTB results (Fig. 3) of the local magnetic moment as a function of the atomic-edge position superimposed over Huang et al.'s results. (b) Example of the structural and magnetic differences between the GTB and Huang et al. systems. Images from Huang et al. used with permission.

Figure 5

(a) The local net spin polarization at the lead-device interface on an edge atom (x) and internal atom ($\circ$) for a 51.3% edge-vacancy-defected 20-5-ZGNR device as a function of the number of ideal ZGNR units in the lead region. Convergence against the ideal system results (horizontal lines) occurs at a nondefected lead length of eight 5-ZGNRs. (b) Structure of the defected 20-5-ZGNR device with the required eight 5-ZGNR-unit ideal leads (black background).

Figure 6

Convergence of the maximum standard error associated with the ensemble-averaged spin-dependent GTB conductance determined in the range 1.5 $\mathrm{eV}\le E-E_F\le 1.5$ eV for a 7.5% random edge-vacancy-defected 20-5-ZGNR.

Figure 7

GTB prediction for the ensemble-averaged net spin polarization per atom in 20-5-ZGNRs showing persistent magnetism throughout the edge-vacancy concentration range. Error bars correspond to the standard errors associated with these measurements. A parabolic fit to these data is included to guide the eye.

Figure 8

STB, ETB, and GTB results for the ensemble-averaged conductance ($G$) as a function of the energy $E$, relative to the Fermi energy $E_F$, in an ideal 20-5-ZGNR, and with (a) 7.5%, (b) 42.5%, and (c) 90% random edge-vacancy concentrations. The insets are zoomed-in views of the conductance about $E_F$. A maximum uncertainty of $\pm 0.06e^2/h$ occurs in the $SE$ of these results. However, in the $42.5\%$ noninteracting solutions, a $\pm 0.008e^2/h$ local $SE$ uncertainty occurs at $E_F$, thus maintaining finite conductance at this point.

Figure 9

(a) The integrated STB, ETB, and GTB ensemble-averaged conductance ($\int G$) measured over the energy range $-0.5\mathrm{eV}<E-E_F<0.5$ eV, and (b) comparison of the ensemble-averaged conductance measured at $E_F$ for the STB-, ETB-, and GTB-calculated 20-5-ZGNRs, as a function of increasing edge-vacancy concentration. The conductance at $E_F$ in the noninteracting models does not go to zero despite accounting for the $\pm SE$ uncertainties of these measurements.

Figure 10

Effect of increasing the edge-vacancy concentration on the ensemble-averaged localization length ($\zeta$) calculated for a 20-5-ZGNR with (a) 7.5% and (b) 42.5% edge-vacancy concentrations. The Hubbard-$U$ gap for an ideal GTB system is indicated by the vertical dashed lines. In (a), a horizontal dotted line has been included to signify the device length of 48.02 Å. The maximum $SE$ uncertainty associated with the localization lengths in the ensemble average is $\pm 3$ Å.

Figure 11

Effect of the system length (Å) on the ensemble-averaged conductance (G) measured at $E_F$ for random edge-disordered 5-ZGNRs at 42.5% edge-vacancy concentration. Green symbols represent the STB results and black symbols represent the ETB results. According to the zero conductance criterion [5, 20], a value of the conductance below the green line (0.01*$e^2/h$ threshold) is equivalent to zero, thus indicating a conductance gap. Error bars correspond to the $SE$ uncertainties associated with these measurements.

Figure 12

Criteria used by Cresti and Roche (light green [Eq. (13)] and black [Eq. (14)]) as a function of increasing device length, measured in Å, for random edge-vacancy disordered 5-ZGNRs with 42.5% edge-vacancy concentration. Circles correspond to the STB model and diamonds correspond to the ETB model. The horizontal line is where these criteria are established. Straight lines joining the data points are a guide to the eye only.

Figure 13

Effect of device length on the ensemble-average conductance gap $\mathrm{\Delta }G$ for random edge-disordered STB, ETB and GTB 5-ZGNRs at 42.5% edge-vacancy concentration. The $0.01*e^2/h$ conductance threshold [5, 20] is assumed for determining zero conductance and in defining the formation of a conductance gap.

Figure 14

Effect of the standard error ($SE$) uncertainty on the coherent transport results. For the $L=96.1$ Å case shown in (a), the edge state appears above the 0.01*$e^2/h$ threshold criterion in the ensemble-averaged result (main graph) and in the $G-SE$ result (top-right inset) leading to zero gaps, whereas a finite gap occurs in the $G+SE$ solution (top-left inset). In the $L=120$ Å case shown in (b), the gap for $G-SE$ (top-inset right) is similar to the ensemble-averaged gap results (main graph), whereas the $G+SE$ case (top-inset left) exhibits a larger gap.