The wrinkle formation in graphene on transition metal substrate: a molecular dynamics study

ABSTRACT To explore the mechanism of the wrinkle formation in graphene on transition metal substrate, a molecular dynamics (MD) simulation package that allows us to simulate systems of millions of atoms was developed. Via the MD simulation, we reveal the detailed kinetics of wrinkles formation on a Cu substrate under compressive strain, from nucleation to one-dimensional propagation and then the splitting of a large wrinkle to a few smaller ones, which is in good conformity with experimental observation. Further study reveals that both friction and the adhesion between graphene and Cu substrate are critical for the wrinkle formation and wrinkles can be easily formed with a lower frictional force and/or a smaller adhesion. Finally, we have shown that impurities in graphene or substrates can greatly facilitate the nucleation of wrinkles. The systematic exploration of the wrinkle formation in graphene on a substrate is expected to facilitate the experimental designs for the controllable synthesis of high-quality graphene.


Introduction
Graphene as a two-dimensional material has many unique physical and mechanical properties, including highest strength and elastic stiffness, in-plane electrical conductivity, thermal conductivity, and optical absorption which results in many potential applications in mechanical, electronic, and photonic devices [1][2][3][4][5][6][7][8]. Currently, chemical vapor deposition (CVD) is the widely employed method to produce graphene in large areas and high quality at a reasonable lower price [9][10][11][12]. As we cool down the system from the temperature of graphene CVD growth (~ 1000°C) to room temperature, wrinkles are formed due to the compressive strain in graphene induced by the larger thermal expansion of the substrate than that of graphene [13][14][15][16][17]. Graphene wrinkles may greatly degrade the carrier mobility [18], mechanical strength [19], thermal conductivity [20], and strain sensitivity of the CVD graphene in applications [21]. Hence, understanding the formation mechanisms of wrinkling induced by the compressive strain is highly desired for controllable graphene synthesis.
In 2015, de Lima, Amauri Libério, et al. [22] developed an analytical model with considering the binding and bending energies to analyze the criterion of wrinkle formation in graphene and have found that ~2.8% compressive strain is required to initiate a wrinkle in a perfect graphene layer on the top of another graphene layer. Baowen Li et al. [23] proposed a wrinkle nucleation model by considering the bending stiffness of graphene, adhesion between graphene and Cu substrate, and the friction force of graphene sliding on Cu surface and successfully explained that the reference of wrinkle formation in the non-epitaxial graphene region is mainly induced by smaller friction force than that of the epitaxial graphene. Wen Wang et al. [1] adopted MD simulations to explain the enlarged wrinkles on the copper substrate and Zhenqian Pang et al. [14] reported how the Stone-Wales defects and grain boundaries influence the wrinkle formation in graphene on different copper substrates using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [24]. Moreover, Kuan Zhang et al. [25,26] developed a continuum model to understand how strain anisotropy, adhesion, and friction govern spontaneous wrinkling of graphene on a substrate.
So far, the analytical models, density functional theory calculations, classical MD simulations, and continuum models are all used to study the wrinkling formation in a thin film on a substrate. But the wrinkle formation on transition metal substrates has never been systematically explored, such as how the friction, adhesion, and defects affect the formation of wrinkles and, especially, the comparison with experimental observations was rarely explored. In this paper, we firstly introduce the growth procedures and characterization of wrinkles in graphene grown on Cu surface in a CVD process. Then, details of the computational methods and parameterizations are presented. As an example, we presented a detailed wrinkle formation process during the increase of the compressive strain with atomic details. Finally, the influences of the friction, adhesion, and the defects in graphene and substrate on wrinkle formation are explored.

Graphene growth on the polycrystalline copper foil
Graphene grows on polycrystalline copper foil by the Chemical Vapor Deposition method. The copper foil was first heated to 1050°C in 60 minutes under H 2 gas flow at 2 Torr pressure. Then, 30 sccm diluted CH 4 (1% CH 4 in Ar gas) and 300 sccm H 2 gas was introduced into the chamber for 1 h. After growth, the furnace was removed to force copper foil to cool down under the same gas flow and pressure as the growth period.

Characterization
SEM was performed with FEI Verios 460 SEM, and AFM data was acquired with the Bruker Dimension Icon system.

Description of the force field for the carbon-metal interactions
So far, the C-C interaction was successfully described by the second-generation reactive empirical bond order (REBO) potential [27], which is based on the Tersoff [28] and Tersoff-Brenner potential [29]. But to a better description of some particular C-C interactions, a parameter α ij was introduced to weaken the C-C interaction in the following form [30,31]: in which the V R r ij À � is the repulsive term and V A r ij À � is the attractive interaction term. The Cu-Cu interaction in the copper substrate was described by the embedded atom method (EAM) potential, which has been successfully explored the structural properties of Cu, Ag, Au, Pd, Ni, and Pt as well as alloys containing these metals [32]. The C-Cu interaction was characterized by the Lennard-Jones (L-J) '12-6' potential [33] to study the adhesion and friction effects by adjusting the parameters ε and σ which are presented in the following equation: where ε and σ, respectively, characterize the depth of the potential well and the equilibrium position of atomic distance, and r is the distance between atomic pairs.

Parameters fitting for the C-Cu interaction
In the atomistic simulation, two types of systems were considered, monolayer graphene on single-layer rigid Cu substrate and monolayer graphene on multi-layers Cu slab with the atoms of the bottom layer fixed, which are presented in Figure 1(a,b). For different equilibrium positions of atomic distance (σ), the L-J potential parameter (the potential well depth (ε)), are fitted with the calculated C-Cu adhesion energy of ~35 meV/C atom [34]. Figure 1(c) shows the L-J potential well depth (ε) as functions of graphene-copper distance (σ). The graphene-copper interactions for single-layer and multi-layers Cu substrate are defined as Where E Gr , E Cu , E Gr=Cu represent the energies of free-standing graphene, the Cu substrate, and the graphene at the Cu surface, respectively. The parameter N is the number of C atoms. We found that the L-J potential well depth (ε) decreases with the graphenecopper distance (σ) increasing for both single-layer and multi-layers Cu systems.

Frictional force of graphene sliding on the Cu substrate
To study the effect of friction force on wrinkle formation, the friction of the graphene sliding on the substrate must be calculated. Herein, we proposed three approaches to calculate the frictional force between the graphene and the Cu substrate and the schematic illustrations of them are shown in Figure 2(a-c), in which the red and gray color represents the Cu and graphene, respectively. The first approach is to slide a graphene sheet on the Cu substrate with a constant velocity (Figure 2(a)). The dimensions of the graphene and Cu substrate are 80 � 3.6 nm 2 and 100 � 3.6 nm 2 , respectively. The second model sliding the graphene on the multi-layer Cu substrate (Figure 2(b)). The last approach starts with a small graphene sheet with initial velocity on a multi-layer Cu substrate and the friction force can be calculated based on the sliding distance ( Figure  2(c)). Finally, the relationship between the frictional force per area and the graphenecopper distance (σ) is established, as shown in Figure 2(d). The friction forces calculated by the three approaches are very close to each other, which validates the computational methods. With the same adhesion energy (35 meV), the friction can be changed drastically by tuning the distance between the graphene layer and the substrate, which allows us to explore the effect of friction force on graphene wrinkle formation. In this MD simulation, a canonical ensemble (NVT) was employed with the temperature fixed via a Nose-Hoover thermostat (T = 300 K) [35]. Each MD simulation ran for 100 ps at a time step of 1 fs. Furthermore, the periodical boundary condition was utilized in these three models.

MD simulation details for wrinkle formation
The wrinkle formation mainly depends on the interactions between the graphene sheet and the first layer of Cu substrates, including the friction force and the adhesion energy. As shown in Figure 2(b), the friction force difference of the graphene sliding on the singlelayer and multi-layers Cu substrates, respectively, is very small if the same adhesion energy (35 meV/C atom) is considered. So, both models can be used to explore the formation of wrinkles in graphene on Cu surface properly. In order to save computational time, we perform MD simulations with a graphene sheet on a single-layer rigid Cu substrate, the size of the atomic model is about 50 � 100 nm 2 with the periodical boundary condition applied and the total number of atoms is 278,600. This system was coupled to a Nose-Hoover thermostat to maintain at a temperature of 1 K. The uniaxial strain was applied gradually, the compressive strain increases by 0.1% every 10 ps during the simulation.

Wrinkles in monolayer graphene grown on Cu surface: experimental observations and atomic simulation
The graphene was grown on the polycrystalline copper foil by the CVD method, the detailed description was discussed in Section 2. As we cool down the system from the CVD temperature to room temperature, the wrinkles or folds were observed in graphene. Figure 3(a) presents the SEM image of a graphene island on a copper foil, in which the white and black arrows represent the wrinkles and folds, respectively, and the top-right enlarged SEM image represents the splitting of a graphene wrinkle. The splitting phenomena were found at the ends of both folds and wrinkles. As shown in Figure 3(b), a graphene wrinkle was visualized using AFM morphology imaging. The height profiles of the wrinkles are shown in Figure 3(c-e). Because the Cu substrate is not flat, we roughly got the height values of these wrinkles being ~3.68 nm, 0.7 nm, and 0.6 nm, respectively. To explain the experimental observations, the MD simulation for wrinkle formation was performed. In this simulation, we took σ = 2.1 Å and ε = 0.012 eV. The C-Cu distance (σ = 2.1 Å) and the potential well depth were fitted by the estimated friction force between graphene and Cu(111) surface (0.096 ~ 0.22 nN/nm 2 ) reported in Ref [23] and the C-Cu adhesion energy (35 meV/C atom) reported in Ref [34]. Based on the MD simulation, we found that a wrinkle starts to form at the compressive strain 3.2%, which is larger than that of ~2.8% in graphene due to the thermal mismatch [36]. That was because the parameters (ε, σ) we adopted induces a high friction force of graphene sliding on the Cu substrate and the ignorance of the defects in both graphene and substrate. Figure 4 shows the variation of the wrinkle amplitude and energy per C atom as a function of MD step. As we can clearly see that the energy of the system keeps decreasing as the wrinkle height becomes larger and larger. Finally, a constant wrinkle height, ~1.6 nm, is reached after the full relaxation of the compressive strain. For a better understanding of the wrinkle formation, the morphologies of graphene at serval typical MD steps are shown in Figure 4(a-h), in which the color represents the bond length distribution of graphene. Figure 4(a) shows the C-C bond length distribution in a regular  moiré pattern caused by the lattice mismatch between the graphene and the Cu substrate. Firstly, as a critical step of forming a wrinkle, the wrinkle nucleation was observed (Figure 4(b)). Then, the small nucleus of the wrinkle propagates quickly along the direction perpendicular to the compressive strain (Figure 4(c-h)). Finally, the splitting of the wrinkles was observed and there is no further propagation of the existing wrinkle, which is in good agreement with the experimental observation (Figure 3(a)). The wrinkle splitting mainly caused by the strain concentration. As a wrinkle becomes higher and higher, the local strain concentration at the ends of wrinkles will lead to the wrinkle splitting for faster strain relaxation. Moreover, during the nucleation stage, the C-C bond length near the wrinkle position is the largest, implying the relaxation of the compressive strain is the driving force of wrinkle formation.

Friction and adhesion effect on the wrinkle formation
Here we consider the effect of friction force and the adhesion energy on the formation of graphene wrinkles. To explore the effect of friction force, we consider the system of graphene on the Cu surface, where a constant adhesion energy (35 meV/C atom) is used and the friction force is tuned by varying the C-Cu bond length (σ). The detailed parameters for models are: σ = 2.1 Å, E a = 35 meV/C atom (f = 0.119 nN/nm 2 ); σ = 2.7 Å, E a = 35 meV/C atom (f = 0.039 nN/nm 2 ); σ = 3.4 Å, E a = 35 meV/C atom (f = 0.004 nN/nm 2 ), where f and E a represent the friction per unit area and adhesion energy/C atom, respectively. The simulation results are shown in Figure 5(a), from which we can see that the critical strain that initiates the wrinkle formation decreases from ~3.2% to ~2.6% with the increase C-Cu bond length or the decreased friction force. As discussed in the previous study, such a change is mainly due to the energy required to slide the graphene on the Cu surface in order to form wrinkles at a specific site. Besides, we found that the initial wrinkle height increases slightly as C-Cu distance decreases. This is due to the release of more compressive strain from the highly compressed graphene. With a brief estimation, we can see that the wrinkle height is determined by the released compressive strain and it is because that the relatively small friction force is not sufficient to main a large compressive strain in the graphene layer after a wrinkle is formed. Figure 5(b) shows the change of the critical strain by varying the adhesion energy with given the same C-Cu distances. Three types of models are considered: σ = 2.1 Å, E a = 15 meV/C atom (ε = 0.005 eV); σ = 2.1 Å, E a = 35 meV/C atom (ε = 0.012 eV); σ = 2.1 Å, E a = 50 meV/C atom (ε = 0.017 eV). As expected, the critical strain becomes larger when the adhesion energy was increased, which is in good agreement with the previous analysis [37].

The effects of defects in graphene and/or substrate on the wrinkle formation
Precious discussions presented a very large critical strain for initiating the wrinkles. Experimentally, the compressive strain due to the cooling of a Cu substrate from the graphene growth temperature (~ 1000°C) to room temperature is ~2.8% and the wrinkles in graphene were observed at 700°C [38]. So, the critical strain must be much smaller than what is shown above. Considering that a large CVD graphene always has many wrinkles and the substrate is hardly to be perfect, we believe the defects in graphene and/or substrate may facilitate the formation of a wrinkle by reducing the nucleation barrier [14,39]. To preliminarily explore the effect of defects in graphene and substrate, we consider an ad-dimer (7|5/5|7) in graphene and an adatom on Cu surface. Figure 6(a) shows the schematic and atomic illustration of addimer defects in graphene and an adatom on Cu substrate beneath the graphene layer. The Figure 6. a) The models of an ad-dimer in graphene and an adatom on Cu substrate. b) The effects of an ad-dimer in graphene and an adatom on Cu surface on the wrinkle formation in graphene on Cu substrate with given the parameters σ = 2.1 Å, ε = 0.012 eV, which ensures E a = 35 meV/C atom. c, d) The graphene wrinkle formation processes on Cu substrate with an adatom on a substrate and an addimer in graphene, respectively. position of adatom defect in Cu and that of an ad-dimer defect in graphene are presented in blue and green areas on the left side of Figure 6(a), respectively. The enlarged atomic illustrations of the adatom defect and ad-dimer defect are shown in the blue and green boxes on the right side of Figure 6(a,b) presents the comparison of wrinkle height vs. compressive strain curves obtained by simulating the perfect graphene on a flat Cu surface, defect-free graphene on Cu surface with an adatom, and graphene with an ad-dimer on a flat Cu surface, respectively. It is found that both the ad-dimer in graphene and the adatom of the Cu surface greatly reduced the critical strain by 0.6% and 1.1%, respectively. Due to the defects of ad-dimer in graphene and adatom in Cu substrate, part of the graphene was lifted off from the substrate, which prompts the wrinkle nucleation and, therefore, decreases the critical compressive strain of wrinkle nucleation. Figure 6(c,d) demonstrate the wrinkle formation processes in graphene on Cu substrate with an adatom and graphene with ad-dimer at different strains, marked by blue and green dots in Figure 6(b), respectively. From both simulations, we found that wrinkles always start to nucleate at the positions of defective sites, then propagate along the direction that is perpendicular to the direction of strain.

Conclusions
We present an in-house-developed MD simulation package that allows us to study the formation mechanism of graphene wrinkles on a transition metal substrate and some preliminary results of wrinkle formation in graphene grown by the CVD method. As observed in our simulation, a complete wrinkle formation process includes wrinkle nucleation, wrinkle propagation, and wrinkle splitting. Then, the relationship between the friction force and the graphene-copper distance (σ) was built to explore tune both the friction force and adhesion energy simultaneously during the atomic simulations. In agreement with previous estimations, a wrinkle is easily formed under a lower frictional force and smaller adhesion. Finally, the influences of the ad-dimer in graphene and adatom on Cu substrate during the wrinkle formation process were discussed and it is found that a wrinkle always likes to be formed at the position of defect either in graphene or on the substrate and the critical strain can be greatly lowered by either the defects in graphene or those on the substrate. The wrinkle formation in single-layer graphene was studied in this paper. In the future, more complicated wrinkle formation in multi-layers graphene will be explored.