Comparison of the stopping power of plasmons and single-particle excitations for nanotubes

We consider the motion of a charged particle near the surface of a single-walled cylindrical nanotube of radius $R$. We formulate the problem by considering the case when the charged particle moves parallel to the axis of the nanotube. We calculate the frictional force on the particle due to the electrostatic interaction with the electrons on the surface of the nanotube in terms of its polarization function. We compare the contributions to the energy loss from the plasmons and particle-hole modes as a function of $r_0$, the impact parameter of the charged particle on the axis of the nanotube. Our calculations show that the plasmon contribution exceeds that from particle-hole modes when the charged particle is not moving close to the surface of the cylinder. However, when $R\approx r_0$, the role of these excitations is reversed. We also calculate the image potential of the cylindrical nanotube and show that the effective potential for a charged particle has bound states.

Figure 1

The energy loss for plasmons (indicated) and particle-hole modes (other curve) as a function of the charged particle velocity parallel to the axis of the cylinder. The energy loss is expressed in units of $(e^2{k}_F^2∕4\pi {ϵ}_0)v_F$ and the velocity in units of $v_F$. In this notation, $k_F=\sqrt{2m^{*}{E}_F}∕\hslash$ and $v_F=\hslash k_F∕m^{*}$. The radius of the cylinder is $R=11.0\mathrm{\AA }$, ${ϵ}_b=2.4$, the electron effective mass $m^{*}=0.25m_e$ where $m_e$ is the bare electron mass, and the impact parameter is (a) $r_0=0$, (b) $r_0=5.0\mathrm{\AA }$, (c) $r_0=10.0\mathrm{\AA }$, and (d) $r_0=15.0\mathrm{\AA }$.

Figure 2

The contributions to the stopping power for Fig. 1d from (a) plasmons and (b) particle-hole excitations for subband transitions $L$.

Figure 3

The $L=0$ plasmon excitation energies in units of the Fermi energy (solid lines) as a function of $q∕k_F$ where $k_F$ is the Fermi wave vector of the ground subband for a nanotube. The dashed lines are $\omega =vq$, i.e., $\hslash \omega ∕E_F=(2v∕v_F)(q∕k_F)$, for various $v$. The values for ${ϵ}_b,m^{*},E_F$, and $R$ are the same as Fig. 1.

Figure 4

The effective potential for a charged particle with angular momentum quantum number $4⩽l_0⩽8$. The values for ${ϵ}_b,m^{*}$, and $R$ are the same as Fig. 1. We chose the mass of the external charge as $M=m_e$. For these parameters, the effective potential has bound states for $l_0<10$.