Epitaxy of Prestrained Graphene on a Si-Terminated SiC(0001

May 28, 2012 - When a graphene sheet is transferred onto a Si-terminated SiC(0001) substrate, covalent Si–C bonds form at the interface. These inter...
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Epitaxy of Prestrained Graphene on a Si-Terminated SiC(0001) Surface V. Sorkin and Y. W. Zhang* Institute of High Performance Computing, Singapore 138632 ABSTRACT: When a graphene sheet is transferred onto a Siterminated SiC(0001) substrate, covalent Si−C bonds form at the interface. These interfacial bonds may form domains with a regular lattice pattern. The domain size is dictated by the lattice mismatch between graphene and its substrate. In the present work, a strategy of strain engineering is employed to eliminate the lattice mismatch to achieve a perfect epitaxy between the graphene and the substrate. Both molecular mechanics simulations and density function calculations confirm the perfect epitaxial pattern formed between a prestrained graphene sheet and its underlying silicon-carbide substrate. We further examine the edge effect on the disruption of the epitaxial pattern of prestrained graphene nanoribbons. It is found that that the perfect epitaxial pattern is disrupted only along the narrow regions near the nanoribbon edges, and the size of the affected region of the arm-chair ribbon is about 3 times larger than that of the zigzag ribbon.

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INTRODUCTION Since its discovery in 2004,1 graphene has been at the center of much experimental and theoretical research.2,3 Initially, the research was primarily focused on its electronic and thermomechanical properties,4 but gradually the focus has been shifted to explore its possible applications. A wide range of applications has been envisioned for graphene, in photonics,5 spintronics,6,7 and electronics.8 Today graphene has been used for transparent electrodes in solar cells,9,10 in ultrafast high frequency transistors,11−13 in hypersensitive nanosensors,14−16 in nanomechanical transducers and actuators,17 and as precise sequencers for individual DNA strands.18,19 In many of these applications, graphene must be supported by a substrate. This can be achieved in a number of different ways: graphene can be formed on a substrate by using chemical vapor deposition13,20,21 or it can be grown epitaxially on siliconcarbide via thermal decomposition of the top substrate layers.22,23 Moreover, mechanically exfoliated graphene can be directly transferred onto a substrate via suitable manipulation of single graphene layers.21,24,25 Today, graphene can be deposited on conductors (Ni,24 Fe,26 and Cu25), semiconductors, and insulators (SiO213 and SiC22,27−29). The insulators and semiconductors are more preferable in electronic applications since the short circuits created by the metallic substrate limit the usage of graphene. Among all the semiconducting substrates, silicon-carbide has attracted a great deal of attention. The underlying reason is that silicon-carbide, a widely used wide band gap semiconductor, can serve as a bridge between modern electronics and future graphene-based electronics. Lithographic patterning of epitaxial graphene on SiC substrate may provide a direct connection of graphene-based devices to conventional electronics.22 However, © 2012 American Chemical Society

prior to any possible applications, one needs to understand how silicon-carbide substrate affects the graphene morphology and its electronic and thermal properties. These questions have recently become a subject of intensive research. It was found that covalent bonds are formed at the interface between graphene and the Si-terminated SiC substrate.27−31 As a result, ripples appear in the epitaxial graphene, and its shape and electronic properties are significantly modified.22,27,32 For example, the thermal conductivity of epitaxial graphene is considerably reduced due to damping of the phonons by the underlying substrate.33,34 Previously, we investigated35,36 epitaxial patterns formed by the interfacial bonds between a graphene nanoflake and a Siterminated SiC(0001) substrate. These bonds form intricate patterns at the interface, and each pattern type depends on the relative orientation between the graphene nanoflake and the underlying silicon-carbide substrate. It was found that the graphene nanoflake can be oriented in such a way that a set of the nanoflake atoms overlapping with the substrate atoms form finite-size domains with a perfect lattice pattern. The size of these domains is controlled by the lattice mismatch between the hexagonal lattices of the graphene nanoflake and the top substrate layer.36 A couple of questions arise immediately: Could this lattice mismatch be eliminated by prestraining graphene to achieve a perfect epitaxy between the graphene layer and the silicon-carbide substrate? If yes, what will be the perfect epitaxial pattern formed by the interfacial bonds? Received: February 8, 2012 Revised: May 15, 2012 Published: May 28, 2012 13928

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The graphene was oriented in such a way that a set of its atoms overlap with the atoms of the top substrate layer, forming a regular lattice pattern. The overlap between these atoms is not perfect due to the lattice mismatch. The lattice mismatch between the overlapping atoms36 is 4asi−si−5√3ac−c = 0.024 Å, where ac−c = 1.42 Å is the carbon−carbon bond length in graphene, and asi−si = 3.08 Å is the silicon−silicon bond length in the top surface layer. In order to eliminate this lattice mismatch, the graphene sheet was initially prestrained by 100(4asi−si/5√3ac−c − 1) ≃ 0.2%. This prestrained graphene sheet was placed on top of the substrate as shown in Figure 1b). The size of our unit computational box was set to 82.5 Å × 82.5 Å × 200 Å for the infinitely large graphene. In addition, we also investigated the epitaxy of prestrained graphene nanoribbons with arm-chair and zigzag edges, respectively. For an infinitely long graphene nanoribbon along one direction, the width was 60 Å for the nanoribbon with arm-chair edges, and 71.5 Å for the nanoribbon with zigzag edges. The total number of atoms in our simulations was about 8000. The initial distance between the graphene sheet and the substrate was 2.0 Å, which is about the Si−C bond length in the SiC bulk crystal. In our atomistic simulations, we applied a semiempirical many-body interatomic potential introduced by Tersoff39 to describe the interactions between C−C, C−Si and Si−Si atoms. Recently, this potential has been successfully employed to study silicon carbide,40 carbon and silicon nanotubes,41−43 fullerenes,44 nanodiamonds45 and graphene on silicon carbide.16,35,36 In our simulations, the total energy of the system was minimized using a conjugate gradient technique, which was implemented as part of the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code.46 The conjugate-gradient energy minimization was performed by iteratively adjusting the atom coordinates of the systems. The minimization is deemed convergent if one of the following criteria is met: (1) the difference in the total energy between two consecutive iterations is less than the specified threshold 10−6 eV, and (2) the norm of the global force vector, taken as the sum over all atoms, is less than the specified tolerance 10−6 eV/Å. Total energy minimization was performed at zero temperature. Subsequently, we investigated the morphology of the epitaxial graphene and the bond formation pattern formed at the interface. The Visual Molecular Dynamics (VMD) package47 was used for visualization. In addition, we performed DFT calculations with the CASTEP package48 to examine the epitaxial relation of the prestrained graphene accommodated on the Si-terminated SiC(0001) substrate. In our calculations, a local density approximation (LDA; CA-PZ)49,50 functional was applied, and an ultrafine energy mesh and an energy cutoff of 940 eV were used. During the optimization, the electronic ground state and the corresponding electronic charge density were found self-consistently using norm-conserving pseudopotentials.

To answer these questions, we performed atomistic simulations to examine the epitaxial pattern formed by the bonds at the interface between the prestrained graphene and the underlying Si-terminated SiC(0001) substrate. In addition, density functional theory (DFT) was also used to study the formation and properties of these interfacial bonds. Both atomistic simulation and DFT calculation confirm the existence of perfect epitaxy. Furthermore, we also investigated the epitaxial pattern formed by graphene nanoribbons with armchair and zigzag edges. It was found that the edge effect is only confined with a narrow region near the nanoribbon edge. The present work indicates that the use of strain engineering37,38 to control the epitaxy of graphene may offer a new route to fabricate novel graphene-based electronic devices.

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COMPUTATIONAL MODEL In our simulations, a prestrained graphene sheet was accommodated on a silicon−carbide substrate. The silicon− carbide substrate is a 6H-SiC allotrope with a Si-terminated SiC(0001) surface. This silicon carbide substrate consists of six silicon−carbon bilayers, two of which are fixed at the bottom to represent a semi-infinite sample (see Figure 1a). In order to

Figure 1. (a) A prestrained graphene sheet accommodated on the Siterminated 6H-SiC(0001) substrate. The carbon atoms are marked by cyan, and the silicon atoms are marked by yellow. (b) A unit cell formed by the overlapping carbon atoms of the prestrained graphene sheet and the silicon atoms of the top substrate layer.

check whether six carbon−silicon bilayers is sufficient to represent a silicon−carbide substrate, we calculated the variation in the vertical position of the silicon atoms within each silicon layer caused by the interaction of the substrate with the accommodated graphene. The variation was calculated as the difference between the highest and the lowest positions of the silicon atoms along the Z axis within each layer. We found that the variation in the first (top) substrate layer is 0.27 Å, in the second one is 0.11 Å, and in the third layer is only 0.001 Å. Thus, six silicon−carbon bilayers are sufficient to represent a semi-infinite substrate. Periodic boundary conditions were applied along the X and Y directions, while the free boundary condition was imposed for the top layers.

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RESULTS AND DISCUSSION Our atomistic simulations show that the silicon−carbon bonds were formed at the interface between the prestrained graphene and the underlying silicon-carbide substrate. As a result, the initial planar morphology of the prestrained graphene was altered, as shown in Figure 2a. The alteration of the graphene morphology is due to the ripple formation, and it is clearly visible in Figure 2b, where the graphene bonds and the 13929

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Figure 2. (a) Ripples in the prestrained graphene due to bond formation between the C-atoms of graphene (cyan) and the Si-atoms (yellow) of the top substrate layer. (b) The carbon−silicon bonds (cyan−yellow) formed at the interface and the carbon−carbon bonds (cyan) of the graphene sheet. (c) Only the carbon atoms of the graphene that form the interfacial bonds with the silicon atoms of the substrate are shown. Different colors illustrate the length of the interfacial bonds: the green-colored atoms form bonds with a length of 1.83 Å, the red-colored atoms form bonds with a length of 1.91 Å, and the gray-colored atoms form bonds with a length of 1.87 Å.

interfacial Si−C bonds are visualized. The amplitude of the ripples, calculated as the difference between the highest and the lowest position of the graphene atoms along the Z-axis, is 1.7 Å. The average gap between the graphene sheet and the top substrate layer is 2.4 Å. The interfacial Si−C bonds constitute a regular lattice with a repetitive unit cell. To illustrate this lattice unit cell, we plotted only the graphene atoms that formed interfacial bonds with the substrate atoms (see Figure 2c). The bonded atoms are colored according to the length of their bonds. This epitaxial pattern is arranged in an interesting way: the repetitive unit of the pattern consists of a green-colored atom, surrounded by a hexagon of red-colored atoms. Alongside the red-colored hexagon, there is a triangle formed by three gray-colored atoms. The greencolored carbon atom forms the shortest interfacial bonds with a length of 1.83 Å. They constitute the smallest fraction (10%) of all interfacial bonds. The red-colored carbon atoms, which form the longest Si−C bonds with a length of 1.9 Å, constitute 60% of all interfacial bonds. The remaining fraction of interfacial bonds (30%) is from the gray-colored atoms, with an intermediate bond length of 1.87 Å. The bond length distribution is plotted in Figure 3, where the percentage of the bonds with a given bond length is shown. The obtained interfacial bond pattern can be explained by geometrical arguments. An interfacial bond can be formed only if a carbon atom of graphene and a silicon atom of the top surface layer are located sufficiently close to each other. When the hexagonal lattice of the prestrained graphene overlaps with the hexagonal lattice of the top substrate layer, we can identify all interfacial pairs of atoms, which are sufficiently close to each other to form interfacial bonds. Using this information, we can predict the geometrical pattern formed by these bonds. In order to identify the atoms that can form interfacial bonds, we examined every silicon atom of the top surface layer, and obtained its nearest graphene atom. For each pair of silicon−

Figure 3. Length distribution of the interfacial bonds formed between the prestrained graphene layer and the silicon−carbide substrate.

carbon atoms, we calculated the corresponding lateral deviation, i.e., the in-plane distance (Δx2+Δy2)1/2 between the pair of atoms. All pairs were grouped according to the magnitude of the lateral deviation, as shown in Figure 4. The lateral deviation takes a discrete set of values: 0.36 Å, 0.62 Å, 0.71 Å, 1.07 Å, and so on. Apparently the pairs of atoms that completely overlap (with zero lateral deviation) form the interfacial bonds. In addition, atoms that are close to each other also form interfacial bonds, for example, the pairs of atoms with lateral displacements of 0.36 Å, 0.62 Å, or 0.71 Å. Evidently, a pair of atoms can form an interfacial bond as long as the initial lateral deviation between these atoms is smaller than a critical value. In order to find the critical value, we examined all the pairs of atoms that formed bonds at the interface (the total energy of the system was minimized). In 13930

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can be considered as an estimation for the critical value of the initial lateral deviation for the bond formation. The above results clearly show that the interfacial bond length depends on the initial lateral deviation. The atoms with zero lateral deviation formed the shortest bonds, while the atoms with the larger initial lateral deviation formed the longer bonds. So far we have discussed the epitaxy of an infinite prestrained graphene sheet on the silicon-carbide substrate using periodic boundary conditions. However, if the width of the graphene sheet is finite along one direction, then the edge of the graphene sheet may modify the perfect epitaxial pattern. In order to show this scenario, we examined two distinct cases: the epitaxial patterns formed by prestrained graphene nanoribbon with the arm-chair and the zigzag edges, respectively. For the prestrained graphene nanoribbon with arm-chair edges as shown in Figure 5a, its epitaxial pattern formed by the interfacial bonds is shown in Figure 5b. The sample geometry and the corresponding epitaxial pattern for a prestrained graphene nanoribbon with the zigzag edges are shown in Figure 5c,d. In both cases, we observe a clear manifestation of edge effect: the repetitive epitaxial pattern formed by the interfacial bonds in the case of the infinite (unbounded) graphene sheet is notably changed near the edges, but remains intact far away from the edges. The bond length distribution is slightly changed: the additional peaks corresponding to a variety of new interfacial bonds appearing near the arm-chair edge of the

Figure 4. Number of the overlapping pairs of atoms (red squares) and the interfacial bonds formed by these pairs (black circles) versus the initial lateral deviation between the atoms of the pair.

Figure 4, we plotted both the number of the paired atoms (red squares) and the number of interfacial bonds formed (black circles) versus the initial lateral deviation between the paired atoms. It is seen that only the pairs with an initial lateral deviation less than about 0.70 Å formed interfacial bonds. However, the interfacial bonds could not be formed if the initial lateral deviation exceeded this limit. Thus the value of 0.70 Å

Figure 5. Sample geometry: (a) Prestrained graphene nanoribbon with arm-chair edges accommodated on the Si-terminated SiC(0001) substrate. (b) The carbon atoms of the graphene nanoribbon forming interfacial bonds with the underlying silicon atoms of the top substrate layer. Colors are chosen to illustrate the length of the interfacial bonds: the atoms forming the shortest bonds are colored green, and atoms forming the longest bonds are marked by red. The arrows outline the regions with the extra interfacial bonds formed near the arm-chair edges. (c) Prestrained graphene nanoribbon with the zigzag edges accommodated on the Si-terminated SiC(0001) substrate. (d) The carbon atoms of the graphene nanoribbon forming interfacial bonds with the silicon atoms of the underlying substrate. The arrows outline the regions with the extra interfacial bonds formed near the zigzag edges. 13931

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pronounced for the arm-chair edge than for the zigzag one. For the arm-chair edge, this length is about 16 Å, while for the graphene nanoribbon with the zigzag edge, this length is only about 5 Å. Next, we applied the first-principle method to examine interfacial bonds formed between the prestrained graphene and the underlying silicon−carbide substrate. Using the density function method, we obtained electronic charge density and examined partial charge transfer to confirm bond formation at the interface, observed in atomistic simulations. Since the epitaxial pattern formed by the interfacial bonds has a regular repetitive structure, we selected a conventional repetitive block (“unit cell”) containing both the graphene atoms and the substrate atoms (see Figure 7a). This unit cell consists of 92 atoms and has the following dimensions: 9.95 Å × 11.44 Å × 20.0 Å. We obtained the electronic charge density (see Figure 7b), which clearly demonstrates the existence of the covalent Si−C bonds at the interface. Alternatively, to display bond formation between the carbon atoms of the graphene layer and the silicon atoms of the top surface layer, we calculated the charge density difference (see Figure 7b):

graphene nanoribbon are shown in Figure 6, where the percentage of the interfacial bonds with a given bond length is plotted.

Figure 6. Length distribution of the interfacial bonds formed between the prestrained graphene nanoribbon with arm-chair edge and the underlying silicon−carbide substrate.

Δρ(r ) = ρSiC + Gr − ρSiC − ρGr

(1)

where ρSiC+Gr is the electronic charge density for the graphene sheet supported by the silicon−carbide substrate, and ρSiC and ρGr are the separate charge densities for the substrate and graphene, respectively. The density difference makes evident the charge redistribution in the graphene layer caused by the underlying substrate. This charge redistribution is shown in Figure 7c, where we plot the regions of the enriched electron

A similar distribution is also observed for the graphene nanoribbon with the zigzag edges (not shown here). As can be seen in Figure 5, the edge sets a certain characteristic length of the affected region, far from that in which the epitaxial pattern formed by the interfacial bonds is the same as in the case of the infinitely large graphene sheet. This length characterizes the extent of the edge effect. The edge effect is much more

Figure 7. (a) A conventional “unit cell” extracted from the sample contains 92 atoms. The carbon atoms are marked by cyan, and the silicon atoms are marked by yellow. (b) Electronic charge density distribution of the sample (an isosurface with value q = 0.03). (c) Electronic charge density redistribution. The regions with the enriched electron density (red) around the interfacial Si−C bonds and the regions with depleted electron density (blue) are shown. (d) Partial atomic charges of the graphene atoms estimated by Mulliken population analysis. The graphene atoms forming interfacial bonds are marked by different colors: green specifies the shortest bonds, the red specifies the longest bonds, and the gray corresponds to the bonds with intermediate length. 13932

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zigzag edges, respectively. The perfect epitaxial pattern, typical for an unbounded prestrained graphene accommodated on the silicon-carbide, was disrupted near the edges of the graphene nanoribbons. The additional interfacial bonds formed alongside the nanoribbon edges disturb the perfect epitaxial pattern. The size of the affected region depends on the edge type. More specifically, the size of the affected region for an arm-chair edge is about 3 times larger than that of a zigzag edge. The present work shows that strain engineering37,38 can lead to a great expansion of the potential for applications of epitaxial graphene.22 The combination of strain engineering with graphene epitaxy offers new interesting possibilities for graphene-based electronics.

density (red) around the interfacial bonds and the regions of depleted electron density (blue). The charge density redistributed when the bonds were formed at the interface between the graphene and the underlying substrate. When an electrically neutral silicon atom on the top of the substrate formed a chemical bond with a neutral carbon atom of the graphene layer, the electrons of the silicon atom were partially drawn away. This left the region around the silicon nucleus with a partial positive charge, and created a partial negative charge on the carbon atom to which it was bonded. We calculated the partial atomic charges, created by the asymmetric distribution of electrons in chemical bonds, using Mulliken population analysis.51 The Mulliken population analysis implemented in the CASTEP package is extremely efficient.48 The computational cost of the Mulliken population analysis was less than 1 h when a workstation with 16 CPUs was used. It is commonly accepted that absolute magnitudes of atomic charges obtained by using population analysis have limited physical meaning, since they display a significant degree of sensitivity to the atomic basis set selected for density functional calculations.52 However, the relative values can yield useful information53,54 when a consistent basis set is used systematically. The distribution of the partial atomic charges extracted with the Mulliken population analysis is shown in Figure 7d. In Figure 7d, the red, gray, and green denote the graphene atoms that form interfacial bonds of specific length (as described above). The partial atomic charges on these carbon atoms are also indicated. As shown in Figure 7d, the partial charge on each carbon atom is closely related to the length of the interfacial bond formed between this carbon atom and its corresponding silicon atom of the substrate. The larger is the partial charge transferred to a carbon atom, the shorter is the length of the corresponding interfacial Si−C bond. For example, the partial charge of the green atoms, which form the shortest interfacial bonds with a length of 1.83 Å, is negative and the largest by its absolute value (−0.39). The red-colored atoms with the negative partial atomic charge of (−0.36) form the longest bonds with a length of 1.91 Å. The partial atomic charge of the gray atoms, forming the interfacial bonds with an intermediate length of 1.87 Å, has an intermediate value (−0.37).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the Agency for Science, Research and Technology (A*STAR), Singapore. Graphic images were made with the Visual Molecular Dynamics (VMD) visualization package.47 The Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)46 code used in our simulations was distributed by Sandia National Laboratories. We used the CASTEP package48 for density function calculations.

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CONCLUSIONS In this work, we investigated a perfect epitaxial pattern made by interfacial bonds formed between a prestrained graphene sheet and the Si-terminated SiC(0001) substrate. The graphene sheet was initially prestrained and oriented in such a way that a subset of its carbon atoms overlapping with the silicon atoms of the substrate form a regular triangular lattice. After minimization of the total energy of the system, we found out the regular repetitive epitaxial pattern formed by the Si−C bonds at the interface. We also performed DFT calculations to study the interfacial bonds formed between the prestrained graphene and the supporting SiC substrate. In particular, we examined the charge density distribution and confirmed covalent bond formation at the interface. We applied Mulliken population analysis to estimate the partial atomic charges of the graphene atoms that formed the interfacial bonds, and examine their charge transfer. We found that when the interfacial bond length increases, the magnitude of the transferred charge decreases. In addition, we studied the epitaxial patterns made by prestrained graphene nanoribbons with the arm-chair and 13933

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