Analysis of Lateral Orientation of Single-Walled Carbon Nanotube on Graphite

We investigate the stable lateral orientation of the single-walled carbon nanotube (SW-CNT) physically adsorbed onto the graphite substrate surface using molecular mechanics simulation. The system of the (3,3) armchair-type SW-CNT comprised of 198 carbon atoms with a length of 40.3 Å interacting with the rigid graphene sheet is considered. Effect of the initial lateral orientation on the final lateral orientation is discussed. The stability of the initial and the final stable orientations can be explained by analyzing the interaction energy between the SW-CNT and the substrate, as a function of the rotational angle θ and the center position rc = (xc, yc) of the SW-CNT, within the lateral (0001) plane of the graphite substrate. Molecular mechanics simulations for the perfect substrate surface under the condition T = 0 give the final stable minima near the initial states, instead of the atomic-scale locking around the global minima. [DOI: 10.1380/ejssnt.2009.48]


I. INTRODUCTION
To use the carbon nanotube (CNT) as electronic, optical and mechanical devices on the substrate, it is important to control the lateral position and the orientation of the CNT on the substrate.Recently the lattice oriented growth [1,2] and step-templated growth [3] of the single-walled CNT (SW-CNT) have been reported by several groups.It has been clarified that both the surface atomic arrangement and the geometrical structure contribute to the mechanism of growth [4].The atomic-force microscopy (AFM) measurement of the multi-walled CNT (MW-CNT) physically adsorbed on the graphite substrate by our group [5] has showed that the lattice of the outermost shell of the MW-CNT takes the AB stacking registry with that of the graphite substrate surface.Thus it can be expected that the interaction or lattice matching between the CNT and the substrate contributes to the lateral alignment of the CNT on the substrate.
However the mechanism of the lateral orientation of the CNT on the substrate has yet to be clarified.Therefore in this work we have paid attention to the lateral alignment of the CNT on the substrate, particularly the effect of the surface atom arrangement and its interaction with the CNT lattice.Considering the importance of the SW-CNT as elements of various types of devices because of its unique properties of the electronic and the thermal transports, the stable lateral orientation of the armchair-type SW-CNT physically adsorbed onto the graphene substrate surface is studied.The stable orientation of the SW-CNT is simulated using molecular mechanics simulation and it is analyzed by using the interaction energy between the SW-CNT and the graphite substrate.
Effect of the initial orientation θ in on the final stable orientation θ fin of the SW-CNT is investigated by using the following procedure used in our previous works [6,7].First a SW-CNT of the (3,3) armchair type with a length of 40.3 Å and a diameter of 4.2 Å, is considered.This SW-CNT comprised of 198 carbon atoms, has the left-and the right-most open edges and consists of repeated structures of two-different kinds of carbon-atom rings [6,7].Next the rigid equilateral hexagonal-shaped graphene sheet with a length of 65.8 Å of each side, comprised of 4056 carbon atoms, is adopted as a substrate surface.Both the SW-CNT and graphene structures are separately optimized beforehand by minimizing the total energy described by the covalent Tersoff potential energy [8], using the Polak-Rebiere-type conjugate gradient (CG) method [9].The convergence criterion is set so that the maximum of absolute value of all the forces acting on the movable atoms, becomes smaller than 10 −4 eV/ Å.
Then the SW-CNT is placed on the rigid graphene surface.The initial rotational angle of the CNT, θ in = 0 • , is defined so that the AB stacking registry between the bottom part of the CNT lattice and the graphene lattice is achieved (Fig. 1).The initial center position of the CNT, r in c , is also defined as the lateral position of the carbon atom on the bottom part of the CNT (Fig. 1).In the present simulation, r in c is fixed as the origin such as r in c = (0, 0).The CNT is rotated within 0 • ≤ θ in ≤ 60 • around r in c = (0, 0).For each fixed rotational angle θ in , the structure of the CNT interacting with the rigid graphene sheet is obtained by minimizing the total energy V total = V cov + V vdW , using the CG method [9].It should be noted again that the graphene sheet is assumed to be rigid.Here, as the covalent bonding interaction potential of the CNT, V cov , and as the nonbonding van der Waals interaction potential between the CNT and the graphene, V vdW , the Tersoff [8] and Lennard-Jones (LJ) [10] potentials are also used, respectively.The initial vertical distances between the bottom part of the carbon atom of the CNT and the graphene surface are set as d z = 2.7 Å for all the initial rotational angles θ in .Thus, if the CNT for the initial state P (r in c = (0, 0), θ in ) is structurally optimized, the CNT for the final stable state Q (r fin c , θ fin ) is obtained.

A. Final stable states
Simulated results of the final stable states Q (r fin c , θ fin ) as a function of the initial states P (r in c = (0, 0), θ in ) are shown in Figs. 2 and 3. Figure 2 shows the calculated final stable rotational angle θ fin plotted as a function of the initial rotational angle θ in , for the fixed center position r in c = (0, 0).It is clarified that θ fin deviates a little from θ in around the straight line, θ fin = θ in , which means the CNT finds the local minimum state near the initial orientation θ = θ in .The relation θ fin ≡ θ fin (θ in ) is symmetric with respect to the point (θ in , θ fin ) = (30 • , 30 • ) because the structure of the CNT-graphene system is symmetric with respect to the rotational angle θ = 30 • .Figure 3 shows the calculated final stable center positions r fin c = (x fin c , y fin c ) corresponding to the initial center positions r in c = (0, 0) for the initial rotational angles 0 • ≤ θ in ≤ 60 • .It is clarified that r fin c are distributed around the initial position, r in c = (0, 0).It is noted again θ fin is basically different from θ in due to the relaxation by the CG method.In the following, to explain the final state Q (r fin c , θ fin ), three different kinds of simulated results corresponding to the initial states P (r in c = (0, 0), First the initial state P (r in c = (0, 0), θ in = 0 • ) is considered.Since this initial state corresponds to the stable AB stacking registry between the bottom part of the armchair type SW-CNT and the graphene lattice, the final stable state Q(r fin c , θ fin ) = P(r in c = (0, 0), θ in = 0 • ), is obtained.Since the deformation of the CNT is negligiable, we pay attention to only the interaction energy between the CNT and the graphene substrate surfaces, V vdW .Therefore both the initial state P and the final stable state Q are analyzed using the interaction potential energy surface, E(r c , θ) ≡ V vdW , hereafter.sition r in c = (0, 0).Next Fig. 5 shows E(r c , θ in = 0 • ), a two-dimensional map of the interaction potential energy surface, as a function of the center atom position r c for the fixed rotational angle θ in = 0 • .When E(r c , θ in = 0 • ) is calculated, the initial distance between the bottom part of the CNT and the graphene surface is fixed as d z = 2.7 Å, because the energy surface for the initial center atom position of CNT, (r in c , d z ), is considered here.Since the initial state P (r in c = (0, 0), θ in = 0 • ) is located on the local minimum positions of both E(r in c = (0, 0), θ) [red circle in Fig. 4] and E(r c , θ in = 0 • ) (red circle in Fig. 5), it can be easily understood that this is an energetically stable state.Now the sharp minima appeared in every 60 • and the small variation except for the sharp-minima regions on E(r in c = (0, 0), θ) (Fig. 4) are in very good agreement with previous static simulated results by Buldum and Lu [11].These periodic sharp minima can easily induce the atomic-scale locking of the CNT.In our simulation, the energy barrier required to jump from Q to the neighboring local miminum is ∆E QR = 0.41 eV as illustrated in Fig. 4.

Initial rotational angle [deg.]
Next the initial state P (r in c = (0, 0), θ in = 5 • ) is studied (Fig. 6 tial state P is clearly unstable on both the potential surfaces, E(r in c = (0, 0), θ) and E(r c , θ in = 5 • ), and the CNT tends to move toward the local minimum position along the steepest descent direction indicated by arrows (Fig. 6(a)).Therefore it can be expected that θ changes toward θ = 0 • on E(r in c = (0, 0), θ), and r c changes toward the negative-y region on E(r c , θ in = 5 • ) as illustrated in Fig. 6(a).As shown in Fig. 6(b), the new stable state Q (r fin c =(0, −0.7), θ fin =3 • ) is located in the direction of arrows of Fig. 6(a).Thus, in this case, the final stable state Q can be easily expected to be located in the steepest descent direction starting from the initial state P. Furthermore, the energy barrier required to jump from Q to the neighboring local miminum is ∆E QR = 0.42 eV as illustrated in Fig. 6.
Finally the initial state P (r in c = (0, 0), θ in = 10 • ) is discussed (Fig. 7(a)).In this case the CNT takes no rotational motion such as θ in = θ fin = 10 • (Figs.7(a (Fig. 7(a)).Therefore it can be expected that θ moves toward θ = 0 • on E(r in c = (0, 0), θ) and r c moves toward the positive-y region on E(r c , θ in ).However, after r c changes, the shape of the potential energy surface E(r fin c , θ), markedly changes compared to E(r in c , θ).As a result, as shown in Fig. 7(b), the new stable minimum position Q (r fin =(1.5, −0.8), θ fin =10 • ), is not located in the direction indicated by arrows of Fig. 7(a).Thus, unlike for the cases of θ in = 0 • (Sec.IIIB) and 5 • (Sec.IIIC), the final stable state Q cannot be simply expected from the steepest descent directions from the initial state P. The other cases of θ in can be explained similarly to the cases of θ in = 0 • , 5 • and 10 • mentioned above.

IV. CONCLUSIONS AND DISCUSSIONS
In this paper the interaction-energy analysis of the lateral stable orientation of the (3,3) armchair-type SW-CNT physically adsorbed onto the rigid graphene surface is demonstrated.Using two degrees of freedom, the center position r in c and the rotational angle θ in , the stability and the unstability of the initial state P(θ in , r in c ) and the final stable state Q(θ fin , r fin c ) can be successfully explained.The global stable structure of the CNT physically adsorbed onto the graphene surface can be expected from the present simulation as follows: First, on the energy surface E(r in c = (0, 0), θ) as shown in Fig. 4, the steep minimum peak with a barrier height of 0.4 eV appears at every 60 • except for the other relatively flat regions.Next, on the energy surface E(r c , θ in = 0 • ) as shown in Fig. 5, the atomic sites equivalent to the minimum points, r in c = (0, 0), periodically appear at the hollow sites (center positions of the six-membered ring of the graphene http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) lattice) on the relatively flat potential energy surface corresponding to the blue-colored regions, where the energy variations are less than 100 meV.Therefore the CNT can easily move between equivalent hollow sites located periodically.Considering the features of the above two kinds of energy-surface shapes, the global stable structure can be expected as 'the structure where the center of the CNT is located on the hollow site (ex.r in c = (0, 0)) of the sixmembered ring of the graphene surface with an orientation of θ = nπ/3(n : integer)', that's to say, 'the structure which satisfies the AB stacking registry between the bottom part of the CNT and the graphene lattices.' In Figs. 6 and 7, the shapes of E(r in c = (0, 0), θ) and E(r c , θ in = 0 • ) markedly change after the relaxation at first sight.However, since the blue regions on E(r c , θ in = 0 • ) of Figs.6(b) and 7(b) are flat, where the energy variations are only several meV, the CNT can easily take a translational motion toward the hollow sites located periodically within x−y plane.Therefore it can be said that the CNT tends to move onto the hollow site and rotate toward the stable orientation near θ = 0 • (Fig. 6(b)) and θ = 60 • (Fig. 7(b)) to make the AB stacking registry.
Since the present molecular mechanics simulation has been performed under the condition of T = 0, the SW-CNT finds the local minima near the initial state.On the other hand, under the room temperature condition, the CNT can easily jump over the local energy barrier of about several tens of meV, which can easily induce the atomic-scale locking of the CNT around the position of the periodic sharp minima θ = 0 • and 60 • as shown in Fig. 4, due to the thermal fluctuation and the disordered structure on the surface.To check the effect of the disordered structure such as defects introduced on the perfect lattice of graphite, we have performed a preliminary simulation for the graphite surface including the row of the point defects in line.It is clarified that, as the length of the row increases, an energy path to the lower stable minimum on the energy surface E(r c , θ) can be more easily opened, and the atomic-scale locking around θ = 0 • and 60 • can more easily occur, instead of the locking around the other local minima which are energetically higher.In the present simulation, the local stable structure is obtained by the CG method.To obtain the energetically most stable structure, the molecular dynamics simulation including the frictional terms or the modified CG method exploring the global minimum is required.The molecular dynamics simulation under the finite termperature is useful for not only taking into consideration the effect of the finite temperature but also giving clue to understanding the lateral growth mechanism of the SW-CNT on the substrate.
Unlinke the macroscopic rigid continuous body on the floor, the nanoscale objects do not necessarily move to an intial steepest descent direction of the force due to the effect of the atomic-scale interaction acting between the nanoscale objects and the substrate.Therefore analysis of the interaction energy performed in this work is useful for controlling not only the lattice oriented growth of the SW-CNT [1,2,4] but also the molecular manipulation of the SW-CNT by using AFM [5].For example, the present analysis can be used to expect and control in which direction the growth mode occurs.Furthermore, analysis of the stable lateral orientation gives us information in which direction the force should be given to move the nanoscale objects to a desired position, and how the nanoscale peeling process from the substrate occurs theoretically and experimentally [6,7,12].

FIG. 1 :
FIG. 1: Model used in the simulation.The (3,3) armchair-type SW-CNT (red-colored) is placed on the rigid graphene surface (green-colored).The rotational angle θ in of the SW-CNT is defined around the center atom position r in c = (0, 0)(blue colored).

FIG. 2 :
FIG.2:The final stable rotational angle θ fin plotted as a function of the initial rotational angle θ in , for the fixed center position r in c = (0, 0).

FIG. 3 :FIG. 4 :
FIG.3:The final stable center positions r fin c = (x fin c , y fin c ) corresponding to the initial center positions r in c = (0, 0) for the initial rotational angle 0 • ≤ θ in ≤ 60 • .The carbon network of the graphene lattice is also indicated.θ fin different from θ in is obtained using the relaxation by the CG method as shown in Fig.2.

FIG. 5 :
FIG. 5: The interaction potential energy surface, E(rc, θ in = 0 • ), between the CNT and the graphene substrate surface, as a function of the center position rc = (xc, yc) for the fixed rotational angle θ in = 0 • .The initial height of CNT, dz = 2.7 Å, is used.The energy is plotted by the mesh of 0.1 Å × 0.1 Å.The carbon network of the graphene lattice is also indicated.Here the initial state located at the local minimum [red circle] is clearly a final stable state such as r in c = r fin c = (0, 0).

FIG. 6 :
FIG. 6: The initial orientation θ in = 5 • corresponds to the final orientation θ fin = 3 • .(a) E(r in c = (0, 0), θ) as a function of the rotational angle θ, and E(r c , θ in = 5 • ) as a function of the center position r c = (x c , y c ).The initial height of CNT, d z = 2.7 Å, is used.(b) E(r fin c = (0, −0.7), θ) as a function of the rotational angle θ, and E(r c , θ fin = 3 • ) as a function of the center position r c = (x c , y c ).The final height of CNT obtained by the CG method, d z = 3.2 Å, is used.In this case, the final stable state Q can be easily expected to be located in the steepest descent direction starting from the initial state P indicated by arrows in (a).R (θ = 8 • ) [blue circle] corresponds to the local maximum.∆E QR = 0.42 eV between Q and R means the sharp energy minimum compared to the other relatively flat regions.

-4. 16 FIG. 7 :
FIG. 7: The initial orientation θ in = 10 • corresponds to the final orientation θ fin = 10 • .(a) E(r in c = (0, 0), θ) as a function of the rotational angle θ, and E(r c , θ in = 10 • ) as a function of the center position r c = (x c , y c ).The initial height of CNT, d z = 2.7 Å, is used.(b) E(r fin c = (1.5, −0.8), θ) as a function of the rotational angle θ, and E(r c , θ fin = 10 • ) as a function of the center position r c = (x c , y c ).The final height of CNT obtained by the CG method, d z = 3.2 Å, is used.In this case, the final stable state Q cannot be simply expected from the steepest descent directions from the initial state P indicated by arrows in (a).