Screened field enhancement factor for the floating sphere model of a carbon nanotube array

The screened field enhancement factor for a carbon nanotube (CNT) placed in a CNT array (which is reduced due to the screening effect) is derived based on the "floating sphere" model. We obtain an expression for the field enhancement factor for a CNT in the array as γ = 3 + 2 ( 1 + η ) / { ( 2 + η ) [ 2 π α ( 2 + η ) δ 2 + η ] } , where ρ is the radius of sphere, h is the distance from cathode to the center of sphere, and D is the distance between the nearest spheres, η = ρ / h , δ = ρ / D , and α = 1 for square or 2 / 3 for hexagonal lattice made of CNTs. Explicit algebraic formulas for optimizing the distance between tubes, areal density of emitters, and the anode current are also obtained.