Theoretical study of scattering in graphene ribbons in the presence of structural and atomistic edge roughness

We investigate the diffusive electron-transport properties of charge-doped graphene ribbons and nanoribbons with imperfect edges. We consider different regimes of edge scattering, ranging from wide graphene ribbons with (partially) diffusive edge scattering to ribbons with large width variations and nanoribbons with atomistic edge roughness. For the latter, we introduce an approach based on pseudopotentials, allowing for an atomistic treatment of the band structure and the scattering potential, on the self-consistent solution of the Boltzmann transport equation within the relaxation-time approximation and taking into account the edge-roughness properties and statistics. The resulting resistivity depends strongly on the ribbon orientation, with zigzag (armchair) ribbons showing the smallest (largest) resistivity and intermediate ribbon orientations exhibiting intermediate resistivity values. The results also show clear resistivity peaks, corresponding to peaks in the density of states due to the confinement-induced subband quantization, except for armchair-edge ribbons that show a very strong width dependence because of their claromatic behavior. Furthermore, we identify a strong interplay between the relative position of the two valleys of graphene along the transport direction, the correlation profile of the atomistic edge roughness, and the chiral valley modes, leading to a peculiar strongly suppressed resistivity regime, most pronounced for the zigzag orientation.

Figure 1

(a) A schematic representation of three possible GNR orientations with supercell indicated by a rectangle: zigzag (Z), armchair (A), and armchair zigzag (AZ). (b) The $\pi$-bond bands of graphene are shown as a function of the wave numbers $k_x$ and $k_z$ in the hexagonal first Brillouin zone, as obtained from a nearest-neighbor tight binding model, leading to Dirac cones at special points $K$ and $K^{\prime }$. (c) A visualization of the Dirac cone projection procedure, explained in Appendix pp1-s1, for the ribbon orientations depicted in (a). (d) The energy spectrum for a $\sim 10\text{−}\mathrm{nm}$-wide armchair-zigzag GNR, consisting of bulk conduction (orange) and valence (blue) subbands, as well as edge states (black), according to the simplified GNR model of Appendix pp1-s1.

Figure 2

The Kurokawa pseudopotentials of a single carbon and hydrogen atom as defined in Eq. (6) are shown in (a) reciprocal (normalized to the atomic volume of C in diamond, approximately equal to $5.7\AA {}^3$) and (b) real space, using the Kurokawa parameters provided in Table 1. The wave vector cutoff (discretization) is indicated in reciprocal (real) space by the red dashed line.

Figure 3

(a)–(c) The (sub)band structures of 10-nm-wide GNRs with (a) armchair (A), (b) zigzag (Z), and (c) armchair-zigzag (AZ) edge configuration [see Fig. 1], which are obtained with the pseudopotential approach of Sec. 2b, are presented as a function of the transport wave vector $k_{\parallel }$. (d) The charge density is shown as a function of the Fermi level. A gray dashed line intersects at a doping level of $-0.4\mathrm{eV}$ with respect to the charge-neutrality point, in all subfigures.

Figure 4

The three different types of edge scattering for a graphene (nano)ribbon with (average) width $W$ are represented schematically: DE scattering (left) with edge scattering parameter $P$; SER scattering (center) with, in addition, the SER standard deviation ${\sigma }_{\mathrm{SER}}$ and correlation length ${\mathrm{\Lambda }}_{\mathrm{SER}}$; AER scattering (right), which is characterized by the AER correlation length ${\mathrm{\Lambda }}_{\mathrm{AER}}$ and the type of edge modifications (only the removal or addition of a single row of hydrogen-terminated carbon atoms here).

Figure 5

(a) Top view of a $\sim 1\text{−}\mathrm{nm}$-wide armchair GNR (C atoms in gray) with standard hydrogen (blue circles) termination. The supercell is delineated by the black dashed lines. (b) The four different AER potentials [see Eq. (11)] of an approximately 1-nm-wide GNR are plotted in the transversal plane, with the depicted values being the average along the transport direction. The atom positions of the edge modifications are superimposed on the figure, the large gray (small blue) circles representing C (H) atoms and + ($-$) denoting an addition (removal) with respect to the unmodified edge.

Figure 6

(a),(b) The 2D resistivity $\rho$ due to AER of 10-nm-wide armchair (A), zigzag (Z), and armchair-zigzag (AZ) GNRs as a function of the roughness correlation length with (a) pseudopotential based (see Sec. 3c), (b) simplified scattering rates (see Appendix pp1-s2). (a) The gray vertical lines in (a) indicate the lengths of the supercell (for each orientation, with matched line type) along the transport direction. (c)–(e) The 2D resistivity as a function of the GNR width $W$ and roughness correlation length ${\mathrm{\Lambda }}_{\mathrm{AER}}$ for (c) armchair, (d) zigzag, and (e) armchair-zigzag GNRs. (b)–(e) The simplified AER scattering model of Appendix pp1-s2 is considered with $E_{\mathrm{AER}}=5\mathrm{eV}$ and ${\sigma }_{\mathrm{AER}}=\sqrt{3}{a}_0/cos{\gamma }_{\mathrm{GNR}}$, corresponding to the width of a single row of carbon atoms.

Figure 7

(a) A comparison of the 2D resistivity $\rho$ of GNRs due to DE, SER, and AER scattering [legend in (b) and details on the parameters can be found in the text] as a function of the ribbon width $W$. For DE and SER scattering, two values for the probability of DE scattering $P$ are considered. For AER scattering, the maximum value of the resistivity, for any value of the correlation length ${\mathrm{\Lambda }}_{\mathrm{AER}}$, is presented. The value of ${\mathrm{\Lambda }}_{\mathrm{AER}}$, for which the resistivity is maximal, is shown in (b) as a function of the GNR width for the three GNR orientations under consideration: armchair (A), zigzag (Z), and armchair-zigzag (AZ).