Field Enhancement of Full-3D Carbon Nanotube Arrays Evaluated in an Axisymmetric 2D Model

In field emission studies of parallel-oriented carbon nanotube (CNT) arrays, there is the necessity to model experimental data using simulations in a large variety of cases, of which only a small number are covered in the literature. Arrays of CNTs are, fundamentally 3D systems, but simulation in full 3D-geometry is time and memory consuming and is also less accurate than simulating in 2D-geometry. Here we show an alternative that overcomes these hurdles. We describe the field enhancement factor (β) of CNTs in a 2D-rotationally-symmetric coordinate system and compare the results with square and hexagonal 3D arrays. We find that the deviation between the 2D rotational model and the 3D-models is at most 25%. Moreover, values of β calculated in the 2D-geometry can be corrected with a factor that is a function of the aspect ratio and spacing in the array. We apply the axis symmetric model to open tip CNT arrays, which are notoriously difficult to simulate in 3D.