Excess energy and deformation along free edges of graphene nanoribbons

Change in the bonding environment at the free edges of graphene monolayer leads to excess edge energy and edge force, depending on the edge morphology (zigzag or armchair). By using a reactive empirical bond-order potential and atomistic simulations, we show that the excess edge energy in free-standing graphene nanoribbons can be partially relaxed by both in-plane and out-of-plane deformation. The excess edge energy and edge force are calculated for graphene nanoribbons with parallel zigzag or armchair edges. Depending on the longitudinal constraint, the compressive edge force leads to either in-plane elongation of the ribbon or out-of-plane buckling deformation. In the former case, the longitudinal strain is inversely proportional to the ribbon width. In the latter case, energy minimization predicts an intrinsic wavelength for edge buckling to be 6.2 nm along the zigzag edge and 8.0 nm along the armchair edge. For graphene nanoribbons of width less than the intrinsic wavelength, interaction between the two free edges becomes significant, leading to antiphase correlation of the buckling waves.

Figure 1

(Color online) Illustration of the bond structures, bond energies, and unbalanced interatomic forces at the unrelaxed (a) zigzag and (b) armchair edges. $V_0$ is the bond energy at the reference state; $V_{Z1}$, $V_{A1}$, and $V_{A2}$ are the bond energies near the free edges. Dark arrows indicate directions of unbalanced forces acting on the edge atoms.

Figure 2

(Color online) Average energy per atom of graphene nanoribbons with (a) zigzag edges and (b) armchair edges, as a function of the ribbon width $\left(W\right)$, for unrelaxed edges and after 1D edge relaxation. $\gamma$ is the excess edge energy per length.

Figure 3

(Color online) Excess energy per unit length vs ribbon width $\left(W\right)$: (a) zigzag edge and (b) armchair edge. The dashed lines are analytical predictions [Eq. (7)] for 2D relaxation based on the edge energy and edge force from 1D relaxation.

Figure 4

(Color online) The bond structures (with bond lengths in angstrom) at (a) zigzag and (b) armchair edges after 1D relaxation. The dashed rectangular boxes represent effective edge layers that are subjected to compressive edge forces ($f_Z$ and $f_A$).

Figure 5

(Color online) Average energy per atom and longitudinal force $\left(F\right)$ versus imposed longitudinal strain for a graphene ribbon with zigzag edges. The ribbon width $W=3.8\text{nm}$. The dashed lines are predicted by Eqs. (7, 8) for the energy and force, respectively.

Figure 6

(Color online) Longitudinal strain of graphene nanoribbons after 2D relaxation: comparing MM calculations (open symbols) and the predictions by Eq. (10). $f_Z$ and $f_A$ are the edge forces for the zigzag and armchair edges, respectively.

Figure 7

(Color online) Edge buckling of graphene nanoribbons with (a) zigzag edges $\left(W=7.7\text{nm},L=23.6\text{nm}\right)$ and (b) armchair edges $\left(W=7.9\text{nm},L=23.4\text{nm}\right)$. $W$ and $L$ are the width and length of the ribbons, respectively. Color bar indicates the out-of-plane displacement (in angstrom).

Figure 8

(Color online) Excess energy of graphene nanoribbons with edge buckling for different wave numbers $\left(n\right)$. (a) Zigzag edge $\left(W=7.7\text{nm}\right)$ and (b) armchair edge $\left(W=7.9\text{nm}\right)$. $W$ is the ribbon width.

Figure 9

(Color online) Excess energy of graphene nanoribbons with edge buckling, replotted against the buckle wavelength using the same results in Fig. 8. (a) Zigzag edge $\left(W=7.7\text{nm}\right)$ and (b) armchair edge $\left(W=7.9\text{nm}\right)$. $W$ is the ribbon width and $n$ is the wave number. The MM results are fitted by fourth-order polynomial functions (plotted as solid lines): $y=0.00002x^4-0.00071x^3+0.0093x^2-0.054x+10$ in (a) and $y=0.000002x^4-0.00011x^3+0.0019x^2-0.014x+11$ in (b).

Figure 10

(Color online) Antiphase correlation of edge buckling in a narrow graphene ribbon $\left(W=3.4\text{nm},L=14.8\text{nm}\right)$. $W$ and $L$ are the width and length of the ribbon, respectively.