Polymerized $s{p}^2\text{−}s{p}^3$ hybrid metallic phase of ${\text{C}}_{60}$ as obtained via constant-pressure molecular dynamics

Using the tight-binding molecular-dynamics method, we study the pressure-induced structural phase transition of the two-dimensional “tetragonal” ${\text{C}}_{60}$ polymer. It is found that the structural transition to a three-dimensionally (3D) polymerized ${\text{C}}_{60}$ phase takes place for a wide range of pressure from 12 to 28 GPa. Although the lattice constants of this 3D polymer phase are close to those of experimentally observed values [S. Yamanaka et al., Phys. Rev. Lett. 96, 076602 (2006)], the present phase has much more $s{p}^3$ C sites than the experimentally proposed phase and each ${\text{C}}_{60}$ has interfullerene chemical bonds with all the 12 neighbor ${\text{C}}_{60}$ fullerenes. From the subsequent total-energy electronic-structure analysis in the framework of the density-functional theory, it is found that this completely polymerized ${\text{C}}_{60}$ phase has much lower total energy than the previously proposed 3D polymer phases and possesses the metallic electronic structure.

Figure 1

Geometry of the “tetragonal” ${\text{C}}_{60}$ polymer, which actually possesses the body-centered-orthorhombic unit cell. In this phase, ${\text{C}}_{60}$ fullerenes form 2D covalent-bond network shown in (a), and these 2D network planes are stacked with the “$AB$” sequence shown in (b). Within a polymerized ${\text{C}}_{60}$ plane, each ${\text{C}}_{60}$ is connected to adjacent ${\text{C}}_{60}$ fullerenes via a four-membered ring (the $\left[2+2\right]$ cycloaddition-type bonding pattern) not only in the direction of the $x$ axis but also in the direction of the $y$ axis as is evident in (b).

Figure 2

(Color online) Trajectory of $\left(T,p\right)$ in the MD simulation process for $P_{\text{ext}}=20\text{GPa}$. The final $T$ and $p$ values after 2000 steps, $\left(T_2,p_2\right)$, as well as their values at 1200 steps, $\left(T_1,p_1\right)$, are indicated by open circles. It is evident that $p_1$ is already almost equal to $P_{\text{ext}}$, and $T_1$ is also very close to the final value, $T_2$ (941 K in this case). In the two different ways ($A$ and $B$, see text) of studying the stability of the final structure under ambient pressure and low temperature, $T$ and $p$ values change along the trajectories $A_1+A_2$ and $B_1+B_2$ for processes A and B, respectively. In process B after switching the target pressure from $P_{\text{ext}}$ to zero, potential energy is partly converted to the kinetic energy and the temperature of the system gradually increases (trajectory $B_1$).

Figure 3

Schematic picture of three-dimensionally polymerized ${\text{C}}_{60}$ phase at $p=0$ and $T=0$ viewed from (a) $y$ and (b) $x$ directions, respectively. The geometry is obtained via process A after the constant-pressure MD simulation of 20 GPa. (c) ${\text{C}}_{60}$ units on the initial two-dimensional polymer plane. Although there are five interfullerene bonds along the $y\left(a_2\right)$ direction, two of them are overlapping with other bonds in this figure and do not show up. (d) Similarly, the side view of this complex interfullerene bonding along the $y$ direction, where only three bonds show up again. It is apparent from (d) that there are two kinds of this complex interfullerene bonding along the $y$ direction although they are mirror symmetric to each other with respect to the $xy$ plane. Therefore, the material possesses not the body-centered-orthorhombic unit cell but the orthorhombic primitive unit cell.

Figure 4

A variety of interfullerene bonding patterns that appear in the ${\mathit{a}}_2$ direction. In each panel, the horizontal direction corresponds to the direction of ${\mathit{a}}_2$. In order to avoid the overlapping of bonds and to show the number of interfullerene bonds clearly, the line of sight is set to be slightly above the ${\mathit{a}}_1$ axis in each panel except in panel (a) where the line of sight is slightly below the ${\mathit{a}}_1$ axis. Arrows indicate the interfullerene bonds, and decorated spheres represent the atoms on the four-membered ring at the initial two-dimensional phase. Patterns (a) and (b) are essentially the same but they should be distinguished if both appear simultaneously as in the case of $P_{\text{ext}}=20\text{GPa}$ and process A [Fig. 3d]. Patterns (e) and (f) are also the same as in the case of (a) and (b). Therefore, patterns (a) and (c)–(e) are the main different patterns obtained in the present TBMD simulation.

Figure 5

(a) X-ray powder-diffraction pattern simulated for the LDA-optimized three-dimensional polymer geometry in using $\text{Cu}K\alpha$ spectral line $\left(\lambda =1.54060\text{Å}\right)$, and (b) that using $\text{Mo}K\alpha$ spectral line $\left(\lambda =0.70926\text{Å}\right)$. Intensity is normalized so that the third peak’s top is equal to one in both (a) and (b). (c) Radial distribution function for the LDA-optimized three-dimensional polymer geometry. Broadening of peaks is expressed by a Gaussian function with a full width at half maximum of $0.96\text{Å}$.

Figure 6

LDA electronic density of states (DOS) of the 3D polymer phase. The energy is measured from the Fermi energy $E_F$. Interestingly, the system is metallic while a small gap of 0.34 eV opens within the valence band.