Image-potential states of single- and multiwalled carbon nanotubes

The existence of low angular momentum image-potential states is predicted both for single- and multiwalled nanotubes. The states are confined between the self-induced potential on the vacuum side and the surface barrier, created by the central ascent in the transverse nanotube potential. Effective interactions near the surface of the nanotube are modeled with a cylindrical jelliumlike surface barrier, parameterized to ensure the correct transition into the long-range image potential. Binding energies and wave functions are calculated for (12,0), (10,10), (9,0) single-walled and $\left(d=9.48\mathrm{nm}\right)$ multiwalled nanotubes for different values of electron angular momenta. In addition, the expected relative lifetimes were calculated for the case of zero–angular momentum states of a (10,10) SWNT. The possible formation of image-potential states in nanotube bundles is briefly discussed.

Figure 1

The total effective potential for an $l=1$ image electron in the vicinity of a SWNT (a) and MWNT (b). The inset in (a) displays the effective potential for different values of electron angular momenta.

Figure 2

Binding energies of image-potential states calculated for the (10,10) SWNT $\left(a=0.68\mathrm{nm}\right)$. Series of states with angular momentum in the range of 0–10 are considered. Numbers next to the energy bars indicate principal quantum numbers, $n$.

Figure 3

Squares of the wave functions for the first four image-potential states of a (10, 10) SWNT $\left(l=2\right)$. The lower panel displays the associated potential in the radial direction.

Figure 4

Binding energies of $n=1,l=0–10$ image-potential states calculated for the (10,10), (12,0), and (9,0) SWNTs with associated diameters of 1.36, 0.94, and $0.70\mathrm{nm}$, respectively. The shaded area represents the location of the positive centrifugal barrier.

Figure 5

Wave functions for the lowest three image-potential states calculated for 19-walled $\left(d=14.2\mathrm{nm}\right)$ MWNT $\left(l=1\right)$. Wave function for the highest bound state forming within the jellium shell is also shown. The lower panel displays the total effective potential in radial direction.

Figure 6

Binding energies of image-potential states are calculated for the case of a MWNT that has an outer diameter of $9.48\mathrm{nm}$. Series of states with angular momenta in the range of 0–2 are considered. Numbers next to the energy bars indicate a state’s principle quantum number, $n$.

Figure 7

The height of the positive centrifugal barrier as a function of the electron angular momentum, $l$, and the nanotube diameter. The results are valid both for SWNTs and MWNTs (see text). The curve, indicated as $l=l_{\text{barrier}}$, is formed by the crossing of the potential surface with the $XY$ plane, and displays the marginal combination of the nanotube parameters ($a$ and $l$) needed for the formation of the positive barrier.

Figure 8

Effective potential, $V(x,y)$, for the two adjacent (12,0) SWNTs. The $XY$ plane is perpendicular to the axis of both nanotubes.