Electron knock-on cross section of carbon and boron nitride nanotubes

We present a theoretical description of electron irradiation of single-walled carbon and boron nitride nanotubes. In a first step, the anisotropy of the atomic emission energy threshold is obtained within extended molecular-dynamics simulations based on the density-functional tight-binding method. In a second step, we numerically derive the total Mott cross section for different emission sites as a function of the incident electron energy. Two regimes are then described: at low irradiation energies (below $300\mathrm{keV}$), the atoms are preferentially ejected from the upper and lower parts of the tube, while at high energies (above $300\mathrm{keV}$), the atoms are preferentially ejected from the side walls. Typical values from a fraction of barn (at side wall for $150\mathrm{keV}$ electron) up to around $20\text{barn}$ (for $1\mathrm{MeV}$ electrons) are obtained for the total cross section of knock-on processes for both C and BN nanotubes. These values are smaller than those previously reported using isotropic models and the main reasons for the discrepancies are discussed. Finally, in boron nitride nanotubes, we report that the emission energy threshold maps show boron sputtering to be more favorable for low irradiation energies, while nitrogen sputtering is more favorable at high energies. These calculations of the total knock-on cross section for various nanotubes can be used as a guideline for transmission electron microscopy experimentalists using high energy focused beams to shape nanotubes, and also more generally if electron irradiation is to be used to change nanotube properties such as their optical behavior or conductivity.

Figure 1

(Color online) (a) Schematic representation of the irradiation geometry for a layered structure lying in the $XY$ plane. The target atom sits at the center of the referential. $\vec{v}$ is atom emission direction and $\vec{e}$ is the incidence direction of the electron lying in the $XZ$ plane. (b) Schematic representation of the irradiation geometry for a carbon nanotube, radius $R$, projected onto the $XZ$ plane. The nanotube has its axis along $Y$ perpendicular to the incident electron beam. The projection of $\vec{v}$ is represented as a dashed vector in this figure, although we note that in the general geometry assumed in our calculation the direction of the emitted atom $\vec{v}$ is not confined to the $XZ$ plane.

Figure 2

(Color online) Three-dimensional representation of the map of the emission threshold function $E_d(\delta ,\gamma )$ for a carbon atom in a graphitic layer as a function of the spherical coordinates $\delta$ and $\gamma$. The color scale indicates the emission energy values from $20\mathrm{eV}$ to more than $100\mathrm{eV}$. The sphere indicates the emission direction for the ejected carbon atom. The sphere is centered on the initial position of the targeted C atom.

Figure 3

Total knock-on cross section for carbon atoms in a single-walled carbon nanotube as a function of their position $\alpha$ around the tube circumference. Angles $\alpha =0°$ and $\alpha =90°$ refer, respectively, to carbon atoms in the tube base and in the tube side. The electron beam is entering vertically from the top of the figure. Total cross section for the full tube $(0°<\alpha <360°)$ can be obtained by symmetrization of the plot. The curves are plotted for incident electron energies between $130\mathrm{keV}$ and $1\mathrm{MeV}$ representative of the voltages used in TEM.

Figure 4

Total knock-on cross section for boron atoms in a single-walled BN nanotube as function of their position $\alpha$ around the tube circumference.

Figure 5

Total knock-on cross section for nitrogen atoms in a single-walled BN nanotube as a function of their position $\alpha$ around the tube circumference.

Figure 6

(Color online) Transmitted energy $T$, emission energy threshold $E_d$, and cross section $\sigma \left(T\right)$ as a function of the emission angle $\Omega$. The energy of the electron beam is $500\mathrm{keV}$ for both orientations $\alpha =0°$ (tube section perpendicular to the electron beam) and $\alpha =90°$ (tube section parallel to the electron beam).