Penetration of external field into regular and random arrays of nanotubes: Implications for field emission

TL;DR

This paper presents an analytical theory for how external electric fields penetrate regular and random arrays of carbon nanotubes, revealing how array density and fluctuations affect field emission performance.

Contribution

It introduces an asymptotically exact analytical model for charge distribution in nanotube arrays, accounting for regular and random arrangements and their impact on field penetration.

Findings

Field penetration decreases with increasing nanotube density in regular arrays.

Fluctuations in random arrays limit the applicability of the model at larger distances.

Dense arrays significantly reduce field-emission current due to collective screening.

Abstract

We develop an analytical theory of polarization of a vertically aligned array of carbon nanotubes (NTs) in external electric field. Such arrays are commonly utilized in field-emission devices, due to the known electrostatic effect of strong field enhancement near the tip of an individual NT. A small ratio of the NT radius to the separation between neighboring NTs allows us to obtain asymptotically exact solution for the distribution of induced charge density along the NT axes. For a regular array, this solution allows us to trace the suppression of the field penetration with increasing the density of NTs in the array. We demonstrate that for a random array, fluctuations in the NT density terminate the applicability of our result at distances from the NT tips much larger than the field penetration depth, where the induced charge density is already exponentially small. Our prime conclusion is that, due to collective screening of the external field by the array, the field-emission current decreases drastically for dense arrays compared to an individual NT. We argue that the reason why the strong field emission, described by the Fowler-Nordheim law and observed in realistic arrays, is the strong dispersion in heights of the constituting NTs.