Thermal expansion behavior of holes in graphene nanomeshes

The thermal expansion of a hole, in a planar system, follows the same trend as the thermal expansion of the whole system, i.e., the hole expands (contracts) if the material expands (contracts) under thermal excitation. At nanoscale, this phenomenon has not been studied so far. Here, using tools of classical molecular dynamics simulations, we show that graphene nanomeshes (GNMs) behave oppositely: While the whole structure contracts (expands), the nanoholes expand (contract) under thermal excitation. We propose and test a simple mechanism to describe this unexpected behavior in terms of out-of-plane vibrations of the atoms close to and far from the edges of the holes. This mechanism allows us to see that, contrary to usual planar systems, this behavior comes from nonuniform thermal expansion along the structure. Although the thermal expansion of holes in GNMs is contrary to the classical prediction, we verify that the thermal expansion of the whole GNM structure is the same as that of pristine graphene.

Figure 1

GNMs considered in the present study. They are named (a) GNM-46, (b) GNM-62, and (c) GNM-67, where the numbers represent the approximated values of the lateral dimensions in angstroms. The holes are passivated with hydrogen. Carbon (hydrogen) atoms are represented by gray (white) spheres. The length of the structure along zigzag, $l_x$, and armchair, $l_y$, directions and $x$ and $y$ dimensions of hole and neck sizes, $(h_x,h_y)$ and $(n_x,n_y)$, respectively, are depicted in (d).

Figure 2

Variation of (a) $l_{x0}$ and (b) $l_{y0}$ with temperature of the G-67 structure. Points are results from MD simulations and the line is the fitting of the points to a fourth order polynomial in $T$.

Figure 3

Linear TECs of pristine graphene structures G-46 (dotted line), G-60 (dashed line), and G-67 (full line).

Figure 4

Average linear TECs of the whole structure of GNMs (full line), their holes (dashed line), and necks (dot-dashed line). The TEC of the corresponding graphene (dotted line) is shown for comparison. (a)–(c) The data for the GNM-46, GNM-62, and GNM-67, respectively.

Figure 5

Area of the holes of the GNM-46 (circle), GNM-62 (square), and GNM-67 (triangle) as functions of the temperature. The full lines are fittings of the points to a fourth order polynomial in $T$.

Figure 6

Area TECs of the whole structure (full line) and its hole (dashed line) of the (a) GNM-46, (b) GNM-62, and (c) GNM-67. The area TECs of the corresponding pristine graphenes (dotted lines) are also shown for comparison.

Figure 7

Snapshots of the GNM-67 at (a) 50 K, (b) 300 K, (c) 700 K, and (d) 1000 K. Circles are drawn around some pieces of the structure close to the hole edge (highlighted in yellow) to show their amount of displacement along the out-of-plane direction.

Figure 8

Scheme proposed to understand the increase of the size of the hole with temperature. Horizontal black lines represent the plane of the GNM and the inclined black and gray lines represent the amplitudes of out-of-plane oscillations of the edges of the hole at large and low temperatures, respectively. Black and gray arrows represent the hole sizes at large and low temperatures, respectively.

Figure 9

Time and space averages of the absolute values of the $z$ coordinate of the carbon atoms of the GNM-67 (up) GNM-62 (middle), and GNM-46 (bottom) at regions I (square) and II (circle) as defined in the upper-left part of the picture. Circles and squares are values obtained from MD simulations, and full lines are guides to the eyes.

Figure 10

In-plane lattice parameter of GNM-67 as function of temperature in regions I (square) and II (circle). Circles and squares are values obtained from MD simulations, and full lines are guides to the eyes.