Structure of porous two-dimensional boron crystals

In this work, we foresee the structure of a new class of borophenes with smaller two-dimensional (2D) densities of atoms than those explored so far for 2D boron crystals. Boron atoms in the porous borophenes tend to be five-coordinated in contrast with commonly investigated structures with hexagonal holes for which the number of nearest neighbors of each atom varies from three to six. High metallic character is the usual feature of borophenes, however, we have also identified a semimetallic borophene that turns into a semiconductor upon unit-cell expansion. A $10\%$ increase in lattice constant of this structure gives rise to an energy gap of $0.3\text{eV}$. This extends to semiconductor industry the possible application of 2D boron crystals.

Figure 1

Results of calculations based on our model. The binding energy is calculated using Eq. (2) and is plotted as function of hole density. Three distinct regions can be identified in the figure corresponding to structures with different dimensionality. The largest one (highlighted in gray) corresponds to 2D structures. Representatives of each region are shown in Fig. 2.

Figure 2

Examples of structures found in our search using the cluster-expansion-like model. For each structure are shown four unit cells each containing $4\times 5$ or $6\times 6$ lattice points. The top-left and top-right 2D structures correspond to $\eta =1/20$ and $\eta =1/5$, respectively. The bottom-left structure corresponds to $\eta =1/2$ and is a boron-triple-chain (BTC) stripe. The bottom-right structure corresponds to $\eta =13/18$ and is a cluster consisting of ten boron atoms.

Figure 3

Results of calculations based on DFT. Binding energy versus hole density for several 2D boron structures described in the text. The red solid circles correspond to structures with hexagonal vacancies that have been obtained experimentally. The structures with larger holes are shown in Fig. 4.

Figure 4

Two-dimensional boron structures with larger holes. In parentheses the plane symmetry groups of the structures are included. The structure struc-$1/4$ coincides with that reported in Ref. [7], whereas structures struc-$7/25$ and struc-$19/49$ concur with those reported in Ref. [5].

Figure 5

Average number of NNs as a function of hole density. Structures with hexagonal holes exhibit much wider range of average number of NNs than structures with larger holes.

Figure 6

Electronic band structure for a $10\%$ elongated lattice constant of struc-$7/25$. At the K point, VBM (CBM) is singly degenerate and corresponds to the irreducible representation $A_1^{″}$ ($A_2^{″}$) of the $D_{3h}$ point group. The $E_g$ dependence on the lattice constant elongation $\mathrm{\Delta }a/a_0$ is shown in the inset.

Figure 7

Binding-energy difference $\mathrm{\Delta }{E}_b$ between a structure with out-of-plane distortions and a structure confined to planarity as a function of hole density $\eta$. Inset shows buckling height $\mathrm{\Delta }z$ as a function of hole density.