  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.
