Modeling field emitter arrays using nonlinear line charge distribution

Abstract

Modeling high aspect ratio field emitter arrays is a computational challenge due to the enormity of the resources involved. The line charge model (LCM) provides an alternate semi-analytical tool that has been used to model both infinite as well as finite sized arrays. It is shown that the linearly varying charge density used in the LCM generically mimics ellipsoidal emitters rather than a Cylindrical-Post-with-an-Ellipsoidal-Tip (CPET) that is typical of nanowires. Furthermore, generalizing the charge density beyond the linear regime allows for modeling shapes that are closer to a CPET. Emitters with a fixed base radius and a fixed apex radius are studied with a view to understanding the effect of nonlinearity on the tip enhancement factor and the emitter current in each case. Furthermore, an infinite square array of the CPET emitters is studied using the nonlinear line charge model, each having a height h=1500 μm and a base radius b=1.5 μm. It is found that for moderate external field strengths (0.3−0.4 V/μm), the array current density falls sharply for lattice spacings smaller than 43h. Beyond this value, the maximal array current density can be observed over a range of lattice spacings and falls gradually thereafter.

Variation of field enhancement factor near the emitter tip

“…Note that a surface obtained by fixing η = η 0 in this coordinate system is an ellipsoid. For a prolate hemiellipsoid on a grounded plane in the presence of an external electrostatic field −E 0ẑ , the solution of Laplace equation may be written as 8,22 ,…”

The tunneling potential for field emission from nanotips

“…Common mechanisms for producing an electron beam are thermionic, field and photo emission. A topic of current research centres around large area arrays of pointed field emitters [7][8][9][10][11][12] that offer high brightness, high current density beams having a small spread in energy at low operational temperatures.…”

Scaling in large area field emitters and the emission dimension

“…In the above, λ is the slope of the line charge density Λ(z) (i.e. Λ(z) = λz), obtained by projecting the surface charge density along the emitter axis 34 . The approximation λ i /λ j ≈ 1 is found to be reasonable so long as the pair of emitters are not too close compared to their height.…”