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)[2πα(2+η)δ2+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.