Communication: Surface-to-bulk diffusion of isolated versus interacting C atoms in Ni(111) and Cu(111) substrates: A first principle investigation Free

Chemical vapor deposition (CVD) is now a widely accepted, low-cost, and scalable method of growing macroscale graphene sheets.^1,2 The CVD process starts with the decomposition of methane into active C species on catalytic metal substrates which are usually late transition metals (Ni, Pd, Pt) and coinage metals (Cu, Ag, Au).^3,4 Of these metal substrates, Cu and Ni are the most common due to their low cost, controllable microstructure, and high etchability.⁵ However, the quality and homogeneity of the CVD-grown graphene are found to depend on the type of catalytic metal substrate used.^6–9 While monolayer graphene homogeneously grows on Cu, multi-layer graphene with non-uniform layer thicknesses typically forms on Ni.¹⁰ The common understanding is that the extremely low solubility of C in Cu confines the active C species to the surface resulting in monolayer graphene, while the high solubility of C in Ni results in C segregation and precipitation to form multilayer graphene.¹¹ This argument has been used to explain why the CVD growth of graphene on Cu is self-limiting and will be independent of both the substrate thickness and the growth time, since the formation of monolayer graphene prevents further dissociation of methane.¹² It also correctly predicts that the number of graphene layers on Ni can be controlled by limiting the carbon content in bulk Ni.^13,14

The above solubility argument, however, assumes an equilibrium saturation of C atoms in the metal substrate, which may not be the case during the CVD growth of graphene. For example, Lander et al. showed that the equilibrium solubility of C in Ni was reached only after 7 h,¹⁵ while the heating and cooling stages during CVD occur over several minutes.² In addition, the average 1-3 layers of graphene grown on 300 nm thick Ni substrates during CVD¹⁶ is short of the expected ∼10 graphene layers based on solubility argument. Here, we perform density functional theory (DFT) calculations to study the kinetics of surface-to-bulk transport of C atoms in both Ni(111) and Cu(111) substrates during early-stage CVD processes. Our results demonstrate that the contrasting mechanisms of graphene growth are due to competing C–C versus C–substrate interaction effects, rather than the previous solubility argument. The preferential Ni–C bonding over C–C bonding causes the surface-to-bulk diffusion of interacting C atoms in Ni(111) to proceed rapidly in a chain-like fashion with reduced barrier energies of ∼1 eV, compared to ∼1.4-1.6 eV for isolated C atom diffusion. On the other hand, the preferential C–C bonding for Cu favors the formation of C–C dimer pairs which limits the CVD reaction to the Cu subsurface, even though the barrier energy for isolated C atom diffusion in Cu is lower than in Ni.

Our DFT calculations are performed using VASP.^17,18 We use the Vanderbilt ultra-soft pseudo potentials for calculating the interaction between the ionic core electrons and the valence electrons,¹⁹ and adopt the local density approximation for exchange and correlation.²⁰ We employ both 4 × 4 × 4 and 4 × 4 × 6 FCC Ni and Cu supercells, with the (111) plane oriented normal to the vertical (Z) axis. A 2 nm thick vacuum layer is introduced at the top of the lattice cell to model the free surface of the Cu(111) and Ni(111) substrates, while the bottom layer of metal atoms in each supercell is rigidly fixed to represent the bulk region. We adopt a plane wave basis energy cutoff of 800 eV, and use a 10 × 10 × 1 uniform Monkhorst-Pack k-point sampling scheme for Brillouin zone sampling. Geometric relaxations of the supercell are performed using the conjugate gradient method with a force residual of <0.01 eV/Å.

Fig. 1 shows the top and side views of a (111) metal lattice. A surface C atom placed at either the top (A4) or bridge (A3) site is unstable and spontaneously moves to the closest tetrahedral (A2) or octahedral (A1) site during relaxation, which correspond to hcp and fcc stacking. Defining the free energy of binding as

where E_S is the total energy of the relaxed configuration, E_M the energy of the relaxed metal substrate prior to C atom insertion, and E_C the energy of the lone C atom at infinity (∼0.154 eV from DFT), we find that ΔE = − 9.54 eV and −9.44 eV for a C atom placed at the equilibrium tetrahedral and octahedral surface sites of Ni(111), respectively. Therefore, the tetrahedral site of Ni(111) is only slightly more stable than the octahedral site by ∼0.1 eV. Active C species from the decomposition of methane can reside at either of these two surfaces sites, with a barrier energy of E_b = 0.5 eV for hopping between these sites.²¹ In comparison, ΔE = − 7.31 eV and −7.16 eV at the tetrahedral and octahedral surface sites of Cu(111), respectively.

The subsurface octahedral and tetrahedral voids are located directly beneath the octahedral (A1) and tetrahedral (A2) surface sites (Fig. 1). Surface-to-subsurface diffusion of C atoms can occur from the tetrahedral surface site, through the subsurface tetrahedral void, to the subsurface octahedral void.²² Alternatively, C atoms can also diffuse directly from the octahedral surface site to the subsurface octahedral void, through the diffusion path (P1) to (P3) in Fig. 2(a). Using a 4 × 4 × 4 FCC lattice, we show in Figs. 2(b) and 2(c) (case S1), for both Ni(111) and Cu(111), respectively, the change in ΔE of a C atom moving directly from (P1) to (P3) in increments of 0.25 Å. After each increment, we fix the C atom position and relax the FCC lattice to calculate E_S in (1). We observe that the barrier energy associated with this surface-to-subsurface diffusion is almost negligible (E_b ∼ 0.1 eV at (P2) for both Ni and Cu) due to the presence of the free surface, allowing C atoms to rapidly diffuse to the subsurface octahedral voids in both Ni(111) and Cu(111).

We next trace the evolution of ΔE as the C atom diffuses from (P3) to one of the lower diagonal octahedral voids (P7) through the diffusion path shown in Fig. 2(a). In agreement with previous nudged elastic band studies,²³ this diffusion pathway passes through a bridge site (P5) at which a peak ΔE develops, resulting in an E_b of 1.71 eV and 1.50 eV for Ni and Cu, respectively. We have also introduced a larger 4 × 4 × 6 supercell, and have found no difference in the diffusion barrier energies of an isolated C atom in Ni from (P1) to (P7). Using this larger supercell, the barrier energy for C atom diffusion from (P7) to one of the diagonal octahedral voids below is found to be 1.41 eV. This barrier energy saturates with continued diffusion of this C atom further into bulk (E_b ∼ 1.42 eV). We have also calculated E_b for the diffusion of isolated C atoms in bulk Ni and Cu lattices using a full-periodic 4 × 4 × 6 supercell, i.e., without the vacuum layer, and obtained E_b of 1.61 eV and 1.16 eV, respectively.

The transport of active C species during CVD, however, will unlikely be governed by random diffusion of isolated C atoms due to the high concentration of active C species on the metal surface during CVD. As shown for case S1 in Figs. 2(b) and 2(c), the active C atoms on the surface of Ni(111) and Cu(111) will readily diffuse to the subsurface octahedral voids due to the low E_b of ∼0.1 eV. Subsequent surface C atoms introduced by the methane dissociation process during CVD can reside at the octahedral surface sites directly above the occupied subsurface octahedral voids (case S2) or even at diagonal tetrahedral surface sites (case S3), resulting in a two C atom configuration as shown schematically in the inset in Fig. 2(c). We study both these cases using the 4 × 4 × 4 supercell, by fixing the surface C atom at its equilibrium position at the surface site, and moving the second C atom at the octahedral void from (P3) to (P7) along the same path outlined earlier; ΔE of the second C atom is calculated by taking E_M in (1) as the total relaxed energy state of the metal substrate with the surface C atom. For Ni(111) in Fig. 2(b), we show that the local minimum energy position for the second C atom at the subsurface octahedral void is now slightly perturbed due to the presence of the surface C atom. More importantly, ΔE increases from −10.43 eV for case S1 (isolated C atom diffusion) to −9.06 eV and −9.59 eV for case S2 and S3, respectively, indicating that the second C atom at the subsurface octahedral void is now less stable due to the presence of the surface C atom. This repulsive interaction between the surface and subsurface C atoms results in a dramatic reduction in the barrier energy for the diffusion of the subsurface C atom to the diagonal octahedral void below. While these results are based on a rigidly fixed surface C atom, we have also performed calculations where we remove the constraints on the surface C atom while moving the subsurface C atom, and have confirmed that the surface C atom still remains attached to the pre-designated Ni surface site during relaxation.

In contrast, the presence of a C atom at the octahedral surface site of Cu(111) in case S2 in Fig. 2(c) dramatically decreases ΔE of the subsurface C atom from −7.82 eV to −8.57 eV. The equilibrium distance between these two C atoms is now 1.2 Å, which is comparable to the bond distance between C dimers. The surface and subsurface C atoms therefore interact to form very stable C dimers. With the surface C atom fixed, we trace the diffusion path of the subsurface C atom to the diagonal octahedral void below and obtain an unrealistically high E_b of 2.29 eV. Hence, diffusion of C atoms into the bulk is no longer possible once the C dimers are formed. The presence of a C atom at the tetrahedral surface site of Cu(111) in case S3 similarly lowers ΔE. This time, however, the equilibrium site of the subsurface C atom is located near Z = 0, implying that the surface C atom at the tetrahedral site will draw the diagonal subsurface C atom out to the surface to form C–C dimers. Even for case S2 where the subsurface C atom continues to reside within the subsurface octahedral void, a slight lateral perturbation can up-float the entire C–C dimer pair to the surface during relaxation.²⁴ Previous studies show that these C–C dimer pairs are highly mobile on the Cu(111) surface and assemble into hexagonal graphene units.²⁵ 

Therefore, it is the strong bonding between surface/subsurface C atoms on Cu(111) that results in the formation of C–C dimer pairs. Once formed, the C atoms can no longer diffuse into the bulk and are effectively confined to the surface. In contrast, it is the repulsive interaction between surface/subsurface pairs of C atoms within Ni(111) that dramatically reduces E_b for diffusion of the subsurface C atom further into the bulk. In the presence of a three-atom C chain in a 4 × 4 × 6 Ni(111) supercell linking diagonal voids (Fig. 3(a)), the diffusion of the third C atom to the diagonal octahedral void below requires an E_b of 1.08 eV. The barrier energy does not change with continued extension of the chain, and the migration of the last C atom in a four-atom C chain in Fig. 3(b) has a similar E_b of 1.07 eV. The barrier energy also does not change for the subsequent diffusion of the third C atom to the diagonal octahedral void below. However, the diffusion of intermediate C atoms along the chain (e.g., Fig. 3(c)) would require an E_b of 1.54 eV, which is even higher than the barrier energy required for isolated C atom diffusion. These results suggest that the diffusion of a continuous chain of C atoms in Ni will proceed in a sequential atom-by-atom fashion. During each sequential event, however, the C atom can diffuse to any one of the three diagonal octahedral voids in the layer below with similar E_b of 1.07 eV, and need not follow the exact path dictated by the previous C atom. Branching of the atomic chain as it progresses deeper in the substrate is therefore expected. Based on the Arrhenius relation, the reduction in E_b from ∼1.42 eV for isolated C atom diffusion to ∼1.07 eV for sequential chain diffusion translates to a 23 times faster diffusion rate at the CVD temperature of 1000 °C, which explains the ultra-fast segregation and precipitation of C atoms on Ni(111) during CVD.¹⁴ This reduced barrier energy, however, applies only to a continuous chain of C atoms occupying diagonal octahedral voids. Even for a two-atom C chain, as shown in Fig. 3(d), a reduced E_b of 1.17 eV is observed. In the event that a gap exists along the chain, the diffusion process for the two unconnected sections of the chain will now proceed independently. To illustrate this point, we introduce a three C atom configuration with a gap between the second and third C atom, as shown in Figs. 3(e) and 3(f). The diffusion barrier energy for the second octahedral void is now 0.43 eV (Fig. 3(e)) which is comparable to that in the absence of the third unconnected C atom (case S2 in Fig. 2(b)), while the E_b of 1.45 eV for diffusion of the third C atom (Fig. 3(f)) is close to that for the bulk diffusion of isolated C atoms.

Fig. 4 summarizes the barrier energies associated with isolated C atom diffusion versus many-atom C–C interactions for Ni(111) and Cu(111). We conclude that the contrasting graphene growth mechanisms on Ni and Cu substrates are caused by the presence of multiple interacting C atoms. The interaction between surface/subsurface C atoms for Cu(111) favors the formation of C–C dimer pairs which prevents surface-to-bulk C atom diffusion, while the presence of proximate C atoms for Ni(111) in fact accelerates both surface-to-bulk and bulk C atom diffusion processes. These two classes of possible reactions solely depend on the C–C versus C-substrate interactions. The much stronger binding energy of C with Ni compared to C with Cu is consistent with the d-band model.²⁶ In the same vein, Chen et al. identified two distinct categories of metal substrates: one with preferential metal-C bonding (Ni, Pt, Pd), and another with preferential C–C bonding (Cu, Ag, Au).²⁷ We expect these groups of metals to exhibit mono- versus multi-layer CVD graphene growth mechanisms akin to those displayed by either Cu or Ni. Understanding these substrate-dependent mechanisms provides important insights for tailoring the microstructure and layer thickness of the CVD-grown graphene.

The authors acknowledge the support of the AE Multi disciplinary Initiative (UIUC), as well as computational time provided by TACC grant number TG-MSS130007, and Blue Waters sustained-petascale computing project which is supported by the National Science Foundation (Award Nos. OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. A.H. acknowledges the support of a CSE fellowship.