# First Theoretical Realization of a Stable Two-Dimensional Boron Fullerene Network
## 3 Results and Discussion  
We first investigate the structural, bonding, and energetic characteristics of the predicted full boron nanosheet. Figure 1a shows the molecular geometry of the experimentally observed B\({}_{40}\) cage and its corresponding energy, which was predicted to be \(-6.079\) eV/atom. In order to find the local minimum energy 2D lattice, after the energy minimization of the B\({}_{40}\) cage, we applied three random rotations along the three Cartesian directions with respect to the center of the atomic mass. After randomly orienting the B\({}_{40}\) cage, the simulation box size was altered to form primary bonds with periodic images, following by the DFT energy minimization step. By conducting the calculations for around 300 randomly fabricated lattices, we could predict the first minimum energy B\({}_{40}\)-based 2D network, which is shown in Figure 1b. This novel 2D network presents an isotropic lattice with a length of 7.638 A, and corresponding energy of \(-6.230\) eV/atom. As the first important finding, the distinctly lower energy of the herein predicted B\({}_{40}\) fullerene 2D lattice, lower by \(-0.151\) eV/atom compared to the native B\({}_{40}\) cage, confirms its favorable energetic stability. The lengths of various B-B bonds were found to be close to each other and around 1.75 A. It should be noted that according to the spin-polarized calculations, it was confirmed that the herein predicted B\({}_{40}\) fullerene network is not magnetic. In order to examine the bonding mechanism, in Figure 1c the electron localization function (ELF) [30] of the B\({}_{40}\) fullerene network with an isosurface value of 0.75 is illustrated. ELF is a topological function and varies from 0 to 1. Rather large ELF values over 0.75 around the center of B-B bonds interestingly indicate the covalent nature of interactions in this novel full boron nanosheet. To facilitate the oncoming studies, atomic structures of the energy minimized 2D and 0D B\({}_{40}\) lattices are included in the Supporting Information document. We next elaborately examined the dynamical, thermal, and mechanical stability of the herein predicted B\({}_{40}\) fullerene nanosheet [32; 33; 34; 35; 36]. The phonon dispersion of the single-layer B\({}_{40}\) network is depicted in Figure 2a, along with the corresponding phonon group velocities in Figure 2b. It is clearly observable that the three acoustic (find Figure 2a inset) and all optical modes are free of imaginary frequencies, confirming the remarkable dynamical stability of the predicted lattice. Moreover, it can be seen that for the frequencies over 10 THz, the phonon branches appear, generally, with flat dispersions, confirming low group velocities, in agreement with results presented in Figure 3b. Consistent with the phonon dispersion of full carbon fullerene nanosheets [6; 10], significant band crossing is visible for the both in-plane acoustic and entire optical phonon modes, revealing short phonon lifetimes for the mainstream heat carriers in the predicted single-layer B\({}_{40}\) network. The in-plane acoustic modes show the highest group velocity of around 9.7 km/s, which is lower than that of the 11.6 km/s predicted for the 2D C\({}_{36}\) fullerene [6]. The suppressed group velocity for the phonon modes in the predicted structure suggest its lower lattice thermal conductivity compared to the C\({}_{36}\) counterpart. The mechanical properties and the corresponding stress-strain curve is shown in Figure 2c. The predicted stress-strain relation is uniaxial, meaning that during the entire deformation process, the B\({}_{40}\) monolayer is under tension only along the loading and remains acceptably stress-free along the perpendicular direction to the loading. The elastic modulus and tensile strength are predicted to be 125 and 7.8 N/m, respectively, revealing rather strong bonding in the predicted nanoporous B\({}_{40}\) network, consistent with the previously observed covalent interactions.