Dynamic dielectric response and electron-energy-loss spectra of individual single-walled BN nanotubes

Motivated by the recent electron-energy-loss-spectroscopy (EELS) experiment of Arenal et al. [Phys. Rev. Lett. 95, 127601 (2005)], we investigate the collective $\pi$-electronic excitations in individual boron nitride single-walled nanotubes (BN-SWNTs) with a zigzag wrapping. The dynamic dielectric response of such tubes is treated within a self-consistent-field approach using the one-particle states of the $\pi$-electron system of the BN-SWNTs derived from a simple two-band tight-binding model. Based on this approach, we obtain explicit analytic expressions for the real and imaginary parts of the frequency- and wave-number-dependent dielectric function of the system, whereby the electron-energy-loss function can easily be calculated. Numerical results for this function are presented for several zigzag BN-SWNTs in order to illustrate the possible electron-energy-loss spectra, and each tube is found to support only one branch of the wave-number-dispersed collective $\pi$-plasmon mode for each transferred quantum angular momentum $L$. In this respect, our results are in good agreement with the experiment, where only one pronounced peak, which can presumably be attributed to the $\pi$-plasmon excitations with a small $L$ and a small transferred momentum $q$, dominates the electron-energy-loss spectra in the range up to $8\mathrm{eV}$. The wave-number dispersion of the plasmon modes with different $L$’s is extracted from the calculated spectra of the real part of the dielectric function, and it is shown that the plasmon dispersion curves exhibit a number of unusual characteristics that could be observable in momentum-resolved EELS experiments.

Figure 1

Lattice structure of a two-dimensional BN sheet (a), geometrical structure of a BN-SWNT (b), and the first Brillouin zone of the BN sheet (c). The shaded hexagon represents a unit cell of the BN sheet. Two primitive translation vectors of lattice structure are denoted by ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. Three vectors directed from a B site to nearest-neighboring N sites are given by ${\mathbf{d}}_i(i=1,2,3)$. A BN-SWNT is specified by a chiral vector $\mathbf{C}$ indicating the direction in which the BN sheet is rolled up to form a nanotube. For the zigzag BN-SWNTs under consideration we choose the coordinate system $xyz$ with the $x$ axis along the circumference direction and the $y$ axis along the tube axis. For other notations, see the text.

Figure 2

Plot of the $\pi$-electronic energy subbands $E_{smk}$ for the BN-SWNT (17,0), as obtained within the tight-binding model. One pair of arrows indicates the highest valence and lowest conduction subbands corresponding to the azimuthal quantum number $m=m_g$. The other pair of arrows points out the pair of subbands, specified by the quantum number $m=m_p$, which play a key role in the formation of the interband $L=0$ plasmon mode in the nanotube under consideration. The above-mentioned quantum numbers $m_g$ and $m_p$ are equal to 9 and 8, respectively, in the case of the BN-SWNT (17,0). For details, see the text.

Figure 3

The real and imaginary parts of the dielectric function ${\epsilon }_L(\omega ,q)$ are plotted versus $\hslash \omega ∕t_0$ for the BN-SWNT (17,0) at the value of the longitudinal momentum transfer $q$ equal to $0.2k_{\mathrm{BZ}}$. The broadening parameter $\gamma$ is chosen to be 0.001. The left (right) panels correspond to the angular momentum transfer $L=0\left(1\right)$. In panels (a) and (b), the numbers 5,6,…,16 on the top of the resonant and antiresonant peaks indicate the value of the azimuthal quantum number $m$ corresponding to the single-particle interband excitation $v_m\to c_m$. In panels (c) and (d), the above-mentioned peaks are labeled by the capital alphabetic letters A,B,…,K. The correspondence between those letters and the interband transitions involved is as follows: $\mathrm{A}-v_9\to c_{10}$, $v_{10}\to c_9$; $\mathrm{B}-v_{10}\to c_{11}$, $v_{11}\to c_{10}$; $\mathrm{C}-v_{11}\to c_{12}$, $v_{12}\to c_{11}$; $\mathrm{D}-v_{12}\to c_{13}$, $v_{13}\to c_{12}$; $\mathrm{E}-v_{13}\to c_{14}$, $v_{14}\to c_{13}$; $\mathrm{F}-v_{14}\to c_{15}$, $v_{15}\to c_{14}$; $\mathrm{G}-v_{15}\to c_{16}$, $v_{16}\to c_{15}$; $\mathrm{H}-v_8\to c_9$, $v_9\to c_8$; $\mathrm{I}-v_8\to c_7$, $v_7\to c_8$; $\mathrm{J}-v_7\to c_6$, $v_6\to c_7$; $\mathrm{K}-v_6\to c_5$, $v_5\to c_6$.

Figure 4

Plot of the real and the imaginary parts of the dielectric function ${\epsilon }_L(\omega ,q)$ versus $\hslash \omega ∕t_0$ for the BN-SWNT (17,0) in the region where the $\mathrm{Re}{\epsilon }_L(\omega ,q)$ vanishes. The momentum transfer $q$ and the parameter $\gamma$ are the same as in Fig. 3. The angular momentum transfer $L=0$. The open and solid circles mark the zeros of $\mathrm{Re}{\epsilon }_L(\omega ,q)$. The arrow on the horizontal axis indicates the location of the $\pi$ plasmon energy. Number 8 on the top of the resonant and antiresonant peaks indicates the value of the azimuthal quantum number $m_p$ (see the captions in Figs. 2, 3).

Figure 5

The calculated electron-energy-loss function $S_L(\omega ,q)$ is plotted versus $\hslash \omega ∕t_0$ at zero angular momentum transfer $(L=0)$ for the BN-SWNTs (17,0) (a), (21,0) (b), and (25,0) (c) at the three different values of the broadening parameter $\gamma$ given in the figure.

Figure 6

The calculated electron-energy-loss spectra $S_L(\omega ,q)$ for the BN-SWNTs (17,0) (a), (21,0) (b), and (25,0) (c) at the transferred momentum $q=0.3k_{\mathrm{BZ}}$ and the four different values of the angular transferred momentum $L(=0,1,2,3)$. The broadening parameter $\gamma$ is equal to 0.08.

Figure 7

The calculated electron-energy-loss spectra $S_L(\omega ,q)$ for the BN-SWNTs (17,0) [(a) and (d)], (21,0) [(b) and (e)], and (25,0) [(c) and (f)] at $L=0$ (left panels) and $L=1$ (right panels). The solid, dotted, dashed, and dot-dashed curves on each panel correspond to the four values of the transferred momentum $q=0.3k_{\mathrm{BZ}}$, $0.4k_{\mathrm{BZ}}$, $0.5k_{\mathrm{BZ}}$, and $0.6k_{\mathrm{BZ}}$, respectively. The broadening parameter $\gamma$ is equal to 0.08.

Figure 8

The calculated dispersion curves for the $L=0,1,2,3$ $\pi$-plasmon modes in the BN-SWNTs (17,0) (a), (21,0) (b), and (25,0) (c) are shown by open squares, circles, triangles, and diamonds, respectively.