Collective $\pi$-electronic excitations in BN double-walled nanotubes

We report a systematic theoretical study of the collective $\pi$-electronic excitations in boron nitride double-walled nanotubes (BN-DWNTs). For simplicity, it is assumed that both shells (inner and outer) of such tubes have a zigzag achiral structure. Taking into account intershell Coulomb coupling and neglecting intershell electron tunneling, we introduce the effective dynamic-dielectric-response function of the BN-DWNTs, which depends on frequency $\omega$, wave number $q$, and angular momentum $L$. An explicit expression for this function is derived within the random-phase approximation using standard many-body techniques based on the Green’s function method. Numerical results are presented for the wave-number dispersion and damping of the $\pi$-plasmon modes with different $L$’s, demonstrating a unified picture of the $\pi$-plasmon-energy variation with $q$ for the BN-DWNTs of different diameters. According to this picture, the spectrum of the $\pi$ plasmons, which are shown to be long lived and hence well-defined collective electronic excitations in the BN-DWNTs, consists of a set of nonintersecting upward-dispersed branches, which are well separated in their energies at small values of $q$, but which tend to merge with increasing $q$. Each of the branches corresponds to one and only one value of the angular momentum $L=0,1,2,\dots$ and none of the branches starts from $q=0$. The present calculations also show that the $\pi$ plasmons in the BN-DWNTs can exist even at those $q$ values at which the $\pi$-plasmon modes are not supported by either of the nanotube shells alone. It is found that the threshold value of the wavelength, at which the $L=0$ $\pi$-plasmon dispersion curve in the BN-DWNTs makes its start, is redshifted as compared to that in the inner and outer nanotube shells if they are considered separately. The most important features of our calculated results seem to be consistent, more or less reasonable, with those derived from the recent electron-energy-loss-spectroscopy experiment of Fuentes et al. [Phys. Rev. B 67, 035429 (2003)] on a macroscopic assembly of BN multiwalled nanotubes of different diameters and chiralities, thus suggesting a universality of the properties of $\pi$ plasmons in double-walled and multiwalled BN nanotubes.

Figure 1

Schematic illustration of the BN-DWNT with length $A$, the innermost radius $R_1$, and the outermost radius $R_2$. The vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2^{\prime }$ specify the positions of two $\pi$ electrons (marked with heavy dots) localized on shells 1 and 2 of the nanotube, respectively. The angles $\phi$ and ${\phi }^{\prime }$ of the cylindrical coordinate system are also shown.

Figure 2

(a)–(h) define the basic graphical vocabulary, i.e., the Feynman diagrams representing the basic quantities ${\mathcal{V}}_{ij}$, $V_{ij}$, $G_{jmk\pm }$, ${\Lambda }_{j-+\left(+-\right)}$, ${\Lambda }_{j-+\left(+-\right)}^{\ast }$, and ${\Pi }_j$ of the problem. The indices $i$ and $j$ denote the shells of the nanotube, ${\mathcal{V}}_{ij}$ is the effective screened interaction, $V_{ij}$ is the “bare” Coulomb interaction, $G_{jmk\pm }$ is the Green’s function of the noninteracting $\pi$ electrons, ${\Lambda }_{j-+\left(+-\right)}$ and ${\Lambda }_{j-+\left(+-\right)}^{\ast }$ are the vertices at which Coulomb lines join fermion lines, and ${\Pi }_j$ is a generalized irreducible interband polarization function. (i) Diagrammatic representation of a pair of coupled equations for the effective interactions ${\mathcal{V}}_{11}$ and ${\mathcal{V}}_{21}$ in the RPA.

Figure 3

The real and imaginary parts of the effective dynamic-dielectric-response function ${\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$ are plotted versus $\hslash \omega /t_0$ for the BN-DWNT (17,0)@(25,0) (solid lines). For comparison the real and imaginary parts of the true dynamic-dielectric functions ${\epsilon }_{1,2}\left(\omega ,q,L\right)$ of the two constituent BN-SWNTs (17,0) (dotted lines) and (25,0) (dashed lines) are also shown. The results for the BN-DWNT and the BN-SWNTs are obtained using Eqs. (23, 24) and Eq. (22), respectively. The value of the wave number $q$ is fixed at $0.3k_{\text{BZ}}$; the angular momentum $L$ is chosen to be 0 (left panels) and 1 (right panels). The other parameters used to generate this figure are given in the text. The sign $\times 3$ above the solid lines means that the values of the functions $\text{Re}{\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$ and $\text{Im}{\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$ are obtained from those in the figure by multiplying them by factor 3. The open and solid circles mark the zeros of $\text{Re}{\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$. The arrow on the horizontal axis indicates the location of the $\pi$-plasmon energy.

Figure 4

Plot of the real and imaginary parts of the effective dynamic-dielectric-response function ${\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$ versus $\hslash \omega /t_0$ for the three BN-DWNTs (17,0)@(25,0), (25,0)@(33,0), and (37,0)@(45,0). The wave number $q$ and the angular momentum $L$ are the same as in Fig. 3. The parameters $\Delta$, $t_0$, and $\Gamma$, used to generate this figure, are given in the text. The open and solid circles mark the zeros of $\text{Re}{\epsilon }_{\text{eff}}\left(\omega ,q,L\right)$. The arrow on the horizontal axis indicates the location of the $\pi$-plasmon energy. The figure demonstrates a universality of the $\pi$-plasmon energy in the zigzag BN-DWNTs of different diameters.

Figure 5

The calculated dispersion curves for the $L=0$ $\pi$-plasmon modes in the BN-DWNT (17,0)@(25,0) (triangles) and in the two constituent BN-SWNTs (17,0) (squares) and (25,0) (circles). The arrows on the horizontal axis indicate the threshold values of the wave number $q$, at which the dispersion curves in the former and in the latter make their start. The parameters used to generate this figure are given in the text.

Figure 6

The calculated energies $\hslash {\omega }_L$ of $L=0,1,2$ $\pi$-plasmon modes in the BN-DWNT (17,0)@(25,0) at several values of the wave number $q$ are shown by solid squares, triangles, and circles, respectively. The open circles show the $\pi$-plasmon energies derived from the EELS experiment of Fuentes et al. (Ref. 5) on a sample of BN-MWNTs and presented in Fig. 6 in Ref. 5. The transfer integral $t_0$ between $\pi$ orbitals of nearest-neighboring B and N atoms is chosen to be equal to 2 eV in order to give the best fit of our calculated results to those obtained experimentally in the above-mentioned paper. The solid lines are intended as a guide to the eye.