Electromigration Forces on Atoms on Graphene Nanoribbons: The Role of Adsorbate-Surface Bonding

Abstract

The synthesis of the two-dimensional (2D) material graphene and nanostructures derived from graphene has opened up an interdisciplinary field at the intersection of chemistry, physics, and materials science. In this field, it is an open question whether intuition derived from molecular or extended solid-state systems governs the physical properties of these materials. In this work, we study the electromigration force on atoms on 2D armchair graphene nanoribbons in an electric field using ab initio simulation techniques. Our findings show that the forces are related to the induced charges in the adsorbate-surface bonds rather than only to the induced atomic charges, and the left and right effective bond order can be used to predict the force direction. Focusing in particular on 3d transition metal atoms, we show how a simple model of a metal atom on benzene can explain the forces in an inorganic chemistry picture. This study demonstrates that atom migration on 2D surfaces in electric fields is governed by a picture that is different from the commonly used electrostatic description of a charged particle in an electric field as the underlying bonding and molecular orbital structure become relevant for the definition of electromigration forces. Accordingly extended models including the ligand field of the atoms might provide a better understanding of adsorbate diffusion on surfaces under nonequilibrium conditions.

Figure 1

Single atom sitting on top of a 2D carbon nanostructure in an electric field, E, applied along the surface direction (“transverse”) and under a current flow, I. The atom experiences electromigration forces (red arrow), which consist of a contribution, F_dir, due to the electric field and a contribution, F_wind, due to scattering from electrons from the current. The inset shows the adatom and a benzene ring, representing a simplified model of the extended structure.

Figure 2

Single atoms adsorbed on a 7-aGNR in a transverse electric field/under finite bias experiencing electromigration forces. (a) Co atom on 7-aGNR with semi-infinite aGNR electrodes. A bias voltage is applied between the left and right electrode, which leads to an electric field and current flow. (b) Co atom on finite 7-aGNR in the transverse electric field E_x. (c) Adsorption sites of single atoms (Co, Al, and Ag) on 7-aGNR: Co and Al prefer the hollow site, while Ag prefers the top site. (d) Top: Current- and field-induced force F_x on the Co atom (black line) from the transport setup shown in (a) and purely field-induced force (red line) from the setup shown in (b). Bottom: Current through system from (a) over an electric field. (e) Forces on Co, Al, and Ag atoms on 7-aGNR in an electric field. (f) Field-induced charge density on Ag and Co for E_x = 0.15 V/Å. (g) Top: Electrostatic forces F_el(0) due to induced net charge Q(0) on Co, Al, and Ag upon adsorption on the GNR, obtained by Hirshfeld (solid) and Mulliken (dotted line) analysis. Bottom: Electrostatic forces ΔF_el(E) due to field-induced charges ΔQ(E) over electric field E_x.

Figure 3

Analysis of the change in bond order (ΔBO) and the charge redistribution in the left and right bonds of Co, Al, and Ag on a finite 7-aGNR in a transverse electric field E_x. (a) Left- and right field-dependent ΔBO of Co on 7-aGNR over electric field E_x. The inset shows the partitioning into left and right bonds. (b) Visualization of induced bond charges (top and side view) for Co at E_x = 0.15 eV/Å, where blue depicts electron depletion and red accumulation. (c,d) Same analysis for Al on the hollow site and (e,f) for Ag on the top site. The charge redistribution in the bonds between atom and surface described by ΔBO predicts the direction of the field-induced force on the atom.

Figure 4

Comparison of electromigration forces, bonding and antibonding electrons, and energy states for Co and Sc on benzene (“C₆H₆ + M” model) in an electric field E_x. (a) Field-induced force F_x on Co and Sc on benzene. (b) Field-induced bonding and antibonding electrons, n_B and n_A, on the left- and right-hand side of the adatoms. While for C₆H₆ + Sc, only bonding orbitals are occupied (n_A = 0), C₆H₆ + Co exhibits electron density from antibonding orbitals 2e₁. (c) Density of states (DOS) of C₆H₆ + M and crystal orbital overlap population (COOP) between the adatom and C₆H₆ for Co (top) and Sc (bottom). In the COOP, the energetic positions of the C₆H₆ + M states is shown. The COOP indicates the nature of the states (positive = bonding and negative = antibonding). (d) C₆H₆ + M molecular orbitals denoted according to their symmetry.

Figure 5

Correlation between the electromigration force and energy states close to E_F = 0 eV. (a) Force F_x at field E_x = 0.15 V/Å on adatoms (gray bars) and energetic position of state 2e₁(x) (blue), 1e₂(x) (red), and 1e₂(xy) (blue dashed) for atoms on benzene (C₆H₆ + M) and (b) on 7-aGNRs (GNR + M). (c) Molecular orbitals 2e₁(x), 1e₂(x), and 1e₂(xy) of Co on 7-aGNR. The forces on the early TMs are related to the populations of the 1e₂ states, while the late TMs are dominated by state 2e₁(x), being closer to E_F for both C₆H₆ + M and GNR + M.

Figure 6

Electromigration forces on a Co atom on 7-aGNR in a transverse electric field, E_x, depending on the doping level and atom site. (a) Force F_x over electric field E_x on a Co atom on a n-, p-, and undoped 7-aGNR. (b) Comparison of force F_x over electric field E_x on Co on top and hollow site of an undoped 7-aGNR.