Image charge force of keV ions through insulating nanocapillaries

In nanocapillaries of large aspect ratio, the attractive image charge force is strong enough to affect the trajectory of ions passing through capillaries and consequently to diminish the fraction of transmitted beam ions. We calculated the theoretically transmitted fraction, using an expression of the image charge force valid in the case of a static ion and an infinite cylindrical dielectric interface. When comparing the theoretically transmitted fraction to the available experimental data for nanocapillaries with an inner diameter of less than 200 nm, we found a surprisingly large disagreement, i.e., the theoretically transmitted fractions were easily an order of magnitude lower than the experimental ones. Noting that the image charge force depends on the velocity of the ion via the frequency-dependent relative permittivity of the insulator, we investigated whether the disagreement could be lifted using a velocity depend image charge force. We give the exact expressions of the dynamical image charge force for a plane and cylindrical dielectric interface as a function of the ion velocity. We then reevaluated the theoretically transmitted fractions in the case of ${\mathrm{SiO}}_2$ and polyethylene terephthalate dielectric interfaces. Our findings are discussed in light of the available experimental data.

Figure 1

Simulated transmitted beam fractions (black curve) through an insulating nanocapillary. The gray curve, for which the electric filed due to deposited charges was removed, is added for comparison.

Figure 2

Ratio $F_{\text{im}}/F_{\text{im}}^0$ as a function of $\rho /R$ for various ${\varepsilon }_r$. The label “metal” stands for a perfectly conducting interface. The full curves are the function $h_{{\varepsilon }_r}(\rho /R)$ for ${\varepsilon }_r=2,4,10,100$ and $|{\varepsilon }_r|\to \infty$.

Figure 3

Trajectories for three different initial values of ${\varrho }_0$ in the case of $\mathrm{\Gamma }=1$. Only for ${\varrho }_0^{\text{max}}<0.435$ are the ions transmitted, so that, for $\mathrm{\Gamma }=1$, the transmitted fraction $f_{\text{th}}$ equals ${\varrho }^2=18.6$%.

Figure 4

Theoretical transmitted fraction $f_{\text{th}}$ (in percentage) as a function of the dimensionless parameter $\mathrm{\Gamma }$ defined by Eq. (9) in the case of a uniform, divergence-less ion beam aligned with the capillary axis. Full black line is computed using $F_{\text{im}}^{\circ }$ while the red circles were obtained with $F_{\text{im}}^{\text{cyl}}=F_{\text{im}}^{\circ }\times h_{{\varepsilon }_r}$ and ${\varepsilon }_r=2$.

Figure 5

Schematic representation of the presence (left) or absence (right) of a cold collar of thickness $d$ at the inlet and outlet of a capillary after a gold deposition occurred on the front and back side of the insulating sheet containing the capillaries.

Figure 6

Comparison between experimentally transmitted fractions (red triangles) extracted from [15, 16] and our theoretically transmitted fractions as a function of the capillary diameter $2R_i$. Experimental data are normalized such that fractions coincides for capillaries with 400-nm diameter. Filled blue circles give the theoretically transmitted fractions in the case where the inlet and outlet diameters include a 20-nm collar, filled black squares are evaluated without collar.

Figure 7

Arbitrary dielectric permittivity spectrum over a wide range of frequencies. The real ${\varepsilon }_r^{\prime }$ and imaginary ${\varepsilon }_r^{″}$ parts of permittivity are shown and various processes are labeled: Interface polarization, dipolar relaxation, atomic, and electronic resonances at higher frequencies.

Figure 8

Real part of the dielectric response function of ${\mathrm{SiO}}_2$ (red dashed line) and PET (black full line) extracted from [27] and [22, 28, 29], respectively.

Figure 9

Normalized image charge force $F_z(z_p,v)/F_z(z_p,0)$ as a function of the ion velocity $v$, for three different distances $z_p$ from the ${\mathrm{SiO}}_2$ interface, $z_p$ = 5, 10, and 20 nm. The arrow indicates the velocity of 7 keV ${\mathrm{Ne}}^{7+}$ ions as used by Sahana et al. [4].

Figure 10

Normalized image charge force, $F_{\text{im}}(\rho ,v)/F_{\text{im}}^0\left(\rho \right)$ of a moving charge inside a cylindrical dielectric interface of radius $R=50$ nm as a function of the dimensionless radial distance $\rho /R$ for three different velocities, namely, $v=2\times {10}^4$, $v=1.8\times {10}^5$, and $v=2.8\times {10}^5$ m/s. Upper panel: dielectric response function of ${\mathrm{SiO}}_2$ as given in Fig. 8 is used. Bottom panel: dielectric response function of PET as given in Fig. 8 is used.

Figure 11

Simulated ion trajectories (gray curves) through a 25-$\mu \mathrm{m}$-long nanocapillary of inner radius of 60 nm, as indicated by the black dashed line. The red curves indicate the accumulated charge in arbitrary units at the inner interface.