Systematic ab initio study of the optical properties of BN nanotubes

A systematic ab initio study of the optical and electronic properties of BN nanotubes within density functional theory in the local density approximation is performed. Specifically, the optical dielectric function $\epsilon$ and the band structure of the single-walled zigzag [(5,0),(6,0),(9,0),(12,0),(15,0),(20,0),(27,0)], armchair [(3,3),(4,4),(6,6),(8,8),(12,12),(15,15)], and chiral [(4,2),(6,4),(8,4),(10,5)] as well as the double-walled zigzag (12,0)@(20,0) BN nanotubes are calculated. The underlying atomic structure of the BN nanotubes is determined theoretically. It is found that though the band gap of all the single-walled nanotubes with a diameter larger than 15 Å is independent of diameter and chirality, the band gap of the zigzag nanotubes with smaller diameters decreases strongly as the tube diameters decrease and that of the armchair nanotubes has only a weak diameter dependence, while the band gap of the chiral nanotubes falls in between. It is also found that for the electric field parallel to the tube axis $(E\Vert \widehat{z})$, the absorptive part ${\epsilon }^{″}$ of the dielectric function for all the nanotubes except a few with very small diameters, is very similar to that of bulk hexagonal (h) BN with the electric field parallel to the BN layers $(E\perp c)$. In other words, in the low-energy region $(4–9\mathrm{eV})$ the ${\epsilon }^{″}$ consists of a single distinct peak at $\sim 5.5\mathrm{eV}$, and in the high-energy region $(9–25\mathrm{eV})$ it exhibits a broad peak centered near 14.0 eV. For the electric field perpendicular to the tube axis $(E\perp \widehat{z})$, the ${\epsilon }^{″}$ spectrum of all the nanotubes (except the ultrasmall-diameter nanotubes) in the low-energy region also consists of a pronounced peak at $\sim 6.0\mathrm{eV}$, while in the high-energy region it is roughly made up of a broad hump starting from 10.0 eV. The magnitude of the peaks is in general less than half of the magnitude of the corresponding ones for $E\Vert \widehat{z}$, showing a moderate optical anisotropy in the nanotubes that is smaller than in h-BN. Interestingly, the static dielectric constant $\epsilon \left(0\right)$ for all the nanotubes is almost independent of diameter and chirality with $\epsilon \left(0\right)$ for $E\Vert \widehat{z}$ being only about 30% larger than for $E\perp \widehat{z}$. For both electric-field polarizations, the static polarizability $\alpha \left(0\right)$ is roughly proportional to the tube diameter, suggesting that, unlike carbon nanotubes, the valence electrons on the BN nanotubes are tightly bound. The calculated electron energy-loss spectra of all the nanotubes studied here for both electric field polarizations are similar to those of $E\perp c$ of h-BN, being dominated by a broad $\pi +\sigma$-electron plasmon peak at $\sim 26\mathrm{eV}$ and a small $\pi$-electron plasmon peak at $\sim 7\mathrm{eV}$. Interwall interaction is found to reduce the band gap slightly and to have only minor effects on the dielectric functions and energy-loss spectra. The calculated dielectric functions and energy-loss spectra are in reasonable agreement with the available experimental data.

Figure 1

(Color online) Calculated curvature energies [total energy relative to that of a single BN (graphene) sheet] of BN (carbon) nanotubes. The results for carbon nanotubes are taken from Ref. 4 and the lines are a fitting of $\alpha ∕R^2$ to the data sets with $\alpha =2.004$ and $1.248\mathrm{eV}{\mathrm{\AA }}^2∕\mathrm{atom}$ for BN and carbon nanotubes, respectively.

Figure 2

Energy bands and density of states of h-BN and the single BN sheet. The mid-gap energy level is at 0 eV.

Figure 3

(Color online) Theoretical dielectric function (a)–(d) and energy-loss function (e)–(f) of the single BN sheet and h-BN. For comparison, the experimental (exp.) (from Ref. 28 and references therein) dielectric function and energy loss function of h-BN are also shown. The solid and dashed lines represent the theoretical results for the electric field parallel to the BN layers $(E\Vert a)$ and perpendicular to the BN layers $(E\Vert c)$, respectively. The dot-dashed and double-dot-dashed lines denote the experimental spectra for $E\Vert a$ and $E\Vert c$, respectively. The dotted lines denote the zero value of the real part $\left({\epsilon }^{\prime }\right)$ of the dielectric function.

Figure 4

Energy bands of the representative zigzag, armchair, and chiral BN nanotubes. The midgap energy level is at 0 eV. Note that the length of the Brillouin zone (line $\overline{\Gamma Z}$), which is $\pi ∕T$ where $T$ is the lattice constant (Table ), is different for the different nanotubes.

Figure 5

(Color online) Calculated band gaps of the BN nanotubes vs their tube diameters. For comparison, the band gaps of h-BN and also the single BN sheet are also shown as dash-dotted and dash-double-dotted horizontal lines, respectively.

Figure 6

Density of states of the representative zigzag BN nanotubes. The midgap energy level is at 0 eV.

Figure 7

Density of states of the representative armchair and chiral BN nanotubes. The midgap energy level is at 0 eV.

Figure 8

(Color online) Calculated dielectric function of the zigzag BN nanotubes. “Parallel” and “perpendicular” denote electric fields polarized parallel and perpendicular to the nanotube axis, respectively. The solid, dashed, dot-dashed, and dotted lines represent the dielectric functions for the (6,0), (9,0), (15,0), and (27,0) BN nanotubes, respectively.

Figure 9

(Color online) Calculated dielectric function of the armchair BN nanotubes. “Parallel” and “perpendicular” denote electric fields polarized parallel and perpendicular to the nanotube axis, respectively. The solid, dashed, dot-dashed, and dotted lines represent the dielectric functions for the (4,4), (6,6), (8,8), and (15,15) BN nanotubes, respectively.

Figure 10

(Color online) Calculated dielectric function of the chiral BN nanotubes. “Parallel” and “perpendicular” denote electric fields polarized parallel and perpendicular to the nanotube axis, respectively. The solid, dashed, dot-dashed, and dotted lines represent the dielectric functions for the (4,2), (6,2), (8,4), and (10,5) BN nanotubes, respectively.

Figure 11

(a) ${\alpha }_{\mathit{xx}}\left(0\right)$ and (b) ${\alpha }_{\mathit{zz}}\left(0\right)$ vs $D$ for the BN nanotubes studied. The solid line is a linear least-squares fit.

Figure 12

(Color online) Calculated energy-loss function of the representative BN nanotubes studied. “Parallel” and “perpendicular” denote electric fields polarized parallel and perpendicular to the nanotube axis, respectively. The measured energy-loss function of multiwalled BN nanotubes (Ref. 36) is also plotted in (a) and (b) for comparison.

Figure 13

Calculated density of states (DOS) of the double-walled (12,0)@(20,0) as well as single-walled (12,0) and (20,0) BN nanotubes. The DOS of the (20,0) and (12,0) have been shifted upwards by 0.8 and 0.4 states/eV/atom, respectively. The midgap energy level is at 0 eV.

Figure 14

(Color online) Calculated dielectric function of the double-walled (12,0)@(20,0) (solid lines) as well as single-walled (12,0) (dashed lines) and (20,0) (dot-dashed lines) BN nanotubes. “Parallel” and “perpendicular” denote electric fields polarized parallel and perpendicular to the nanotube axis, respectively. The experimental dielectric function of multiwalled BN nanotubes (Ref. 36) (dotted lines) is also plotted in (a) and (c) for comparison.