Self-doping in boron sheets from first principles: A route to structural design of metal boride nanostructures

Based on first-principles methods, we present a self-doping picture in atomically thin boron sheets: this shows that for two-dimensional boron nanostructures, adding or removing boron atoms is essentially equivalent to simply adding or removing electrons from a fixed electronic structure. This picture allows us to propose a general design rule for pure boron nanostructures and explains the occurrence of known stable nanostructures. In addition, self-doping provides a powerful tool for finding stable metal boride nanostructures. We illustrate this last point by showing an unexpectedly stable ${\text{MgB}}_2$ sheet structure which is likely the precursor of ${\text{MgB}}_2$ nanotubes. Our results are easily generalized to other stoichiometries and other choices of metals.

Figure 1

(Color online) $N_{\sigma }/M$ and $N_{\pi }/M$ versus $\eta$. All data are extracted from ab initio plane-wave calculations, red ◻ for $\sigma$ and blue ◇ for $\pi$ states. The horizontal black dashed line shows $N/M=1/3$.

Figure 2

(Color online) (a) Evolution of boron sheets from H(1/3) to D(2/9), A(1/9), and finally T(0): green “◻” mark the centers of $\sigma$ MLWFs. (b) Isosurface contour plots of representative $\sigma$ MLWFs for H(1/3), D(2/9), A(1/9), and T(0), respectively: red for positive, blue for negative values; other $\sigma$ MLWFs are obtained by symmetry. Red solid lines show unit cells.

Figure 3

(Color online) Centers (marked by green ◻) and isosurface contour plot (red for positive and blue for negative values) of two sets of $\sigma$ MLWFs for T(0) boron sheet: (a) triangular shaped, (b) rectangular shaped. These $\sigma$ MLWFs are even with respect to reflection in the plane of the boron sheets. Red solid lines show the unit cells.

Figure 4

(Color online) (a) Isosurface contour plots of representative $\pi$ MLWFs for H(1/3), D(2/9), A(1/9), and T(0), respectively: red for positive, blue for negative values; other $\pi$ MLWFs are obtained by symmetry. These $\pi$ MLWFs are odd with respect to reflection in the plane of the boron sheets. (b) Centers of $\pi$ MLWFs shown by green ◻. Red solid lines show the unit cells.

Figure 5

(Color online) Isosurface contour plots of MLWFs associated with Mg (top, side and best-angle view, red for positive and blue for negative values), total DOS (red solid lines) and partial DOS on Mg (blue dashed lines) for (a) ${\text{MgB}}_2$ bulk, (b) ${\text{MgB}}_2$ sheet derived from bulk with Mg on the H(1/3) sheet, (c) ${\text{MgB}}_2$ sheet from a G(3/10) boron sheet, and (d) ${\text{MgB}}_2$ sheet based on an E(1/5) boron sheet. The charge transfers from Mg to B are (a) 1.82, (b) 1.37, (c) 1.05, and (d) 0.62 e/Mg. For the ${\text{MgB}}_2$ sheet in (c), there are two types of MLWFs associated with Mg: we only show one of them while the other is very similar to the one in (d). Small gray balls are boron and large blue green balls are Mg. Black solid lines are the Fermi levels.

Figure 6

(Color online) Red squares show the energies (measured relative to bulk ${\text{MgB}}_2$) per formula unit of the most stable ${\text{MgB}}_2$ sheets at each $\eta$ versus $\eta$. The optimal ${\text{MgB}}_2$ sheet structure occurs at $\eta =1/4$ [whose image is shown in Fig. 7a]. The point at $\eta =1/3$ corresponds to the bulk-derived sheet structure.

Figure 7

(Color online) The most stable ${\text{MgB}}_2$ sheets for (a) $\eta =1/4$, (b) $\eta =1/13$, (c) $\eta =1/9$, (d) $\eta =1/7$, (e) $\eta =1/5$, (f) $\eta =3/10$, and (g) $\eta =1/3$. The structure in (a) is the best ${\text{MgB}}_2$ sheet in our library. We display top views that are rotated slightly around the horizontal $\left(x\right)$ axis. Small gray balls are B, large light yellow balls are Mg lying above the boron plane, and large dark blue balls are Mg lying below the boron plane. Red solid lines show the primitive cells.

Figure 8

(Color online) ${\text{MgB}}_2$ sheet structures derived from the same F(1/4) boron sheet sublattice but with different Mg distributions from the optimal ${\text{MgB}}_2$ sheet shown in Fig. 7a. These sheets are all less stable than the optimal structure. The energy differences, in meV per formula unit, are shown below each structure. We display top views that are rotated slightly around the horizontal $\left(x\right)$ axis. Small gray balls are B, large light yellow balls are Mg lying above the boron plane, and large dark blue balls are Mg lying below the boron plane. Red solid lines show the primitive cells.