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We develop an analytical theory of polarization of a vertically aligned arrayof carbon nanotubes (NTs) in external electric field. Such arrays are commonlyutilized in field-emission devices, due to the known electrostatic effect ofstrong field enhancement near the tip of an individual NT. A small ratio of theNT radius to the separation between neighboring NTs allows us to obtainasymptotically exact solution for the distribution of induced charge densityalong the NT axes. For a regular array, this solution allows us to trace thesuppression of the field penetration with increasing the density of NTs in thearray. We demonstrate that for a random array, fluctuations in the NT densityterminate the applicability of our result at distances from the NT tips muchlarger than the field penetration depth, where the induced charge density isalready exponentially small. Our prime conclusion is that, due to collectivescreening of the external field by the array, the field-emission currentdecreases drastically for dense arrays compared to an individual NT. We arguethat the reason why the strong field emission, described by the Fowler-Nordheimlaw and observed in realistic arrays, is the strong dispersion in heights ofthe constituting NTs.Comment: 10 pages, 2 figures; published version, minor change

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