Linear optical properties of zigzag single-walled BN nanotube ensembles from a model calculation

We present a simple theoretical description of the linear optical properties of zigzag single-walled BN nanotube (BN-SWNT) ensembles within a single-particle approach that uses the Král-Mele-Tománek effective Hamiltonian for modeling the electronic structure of such tubes. The perturbation-theory method of Genkin and Mednis is applied to derive analytical expressions for both the real and imaginary parts of the linear optical susceptibility ${\chi }^{\left(1\right)}\left(\omega \right)$, and these are used to calculate numerically the optical functions (dielectric function, refractive index, reflectance and optical absorption coefficient) for several representative BN-SWNT ensembles. The results of our calculations are discussed in the light of the recent experimental observation by Lauret et al. [Phys. Rev. Lett. 94, 037405 (2005)] of the optical absorption spectrum of an assemble of large-diameter BN-SWNT’s. It is shown that the developed theory is capable of explaining the dominant features of the experimental data, thus suggesting that the characteristic peaks observed in the optical absorption spectrum of BN-SWNT’s are due to direct interband electron transitions between pairs of van Hove singularities in the electronic density of states of these tubes.

Figure 1

Fragment of a two-dimensional BN sheet with B and N occupying alternate sites, which is rolled up in order to construct a BN-SWNT specified by a chiral vector $\mathbf{C}$. Also shown the Cartesian coordinate axes $x$ and $y$ with the $x$ direction along the circumference of the tube and the $y$ direction along the axis of the tube. The vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ with length $a_0$ are primitive translation vectors. The shaded hexagon represents a unit cell of the BN sheet.

Figure 2

(a) Static dielectric constant ${\epsilon }_0$ calculated for uniform ensembles of BN-SWNT’s $(l,0)$ with $l$ ranging from 7 to 25 and (b) refractive index $n_0$ at zero frequency for the same ensembles. Solid dots, square, and open triangles show the results obtained for the three types of BN-SWNT ensembles specified by Eqs. (36, 37, 38), respectively.

Figure 3

Theoretical spectra of the real ${\epsilon }_1\left(\omega \right)$ (left panels) and imaginary ${\epsilon }_2\left(\omega \right)$ (right panels) parts of the complex dielectric function $\epsilon \left(\omega \right)$ calculated for five uniform BN-SWNT ensembles consisting, respectively, of the tubes with chirality indices (15,0), (17,0), (18,0), (19,0), and (21,0). The latter ones are indicated in the upper part of the panels. The numbers $0,\pm 1,\pm 2,\dots$ on the top of the peaks correspond to our labeling convention for the interband electron transitions (for details, see the text).

Figure 4

Calculated spectra of the refractive index $n\left(\omega \right)$ for the same BN-SWNT ensembles, as in Fig. 3. Resonant interband electron transitions are indicated by the numbers on the top of the peaks, as in Fig. 3.

Figure 5

Calculated spectra of the reflectance $\mathcal{R}\left(\omega \right)$ for the same BN-SWNT ensembles, as in Fig. 3. Resonant interband electron transitions are indicated by the numbers on the top of the peaks, as in Fig. 3.

Figure 6

Calculated optical absorption spectra for the same BN-SWNT ensembles, as in Fig. 3. Resonant interband electron transitions are indicated by the numbers on the top of the peaks, as in Fig. 3.