Comparisons between apex-radius values extracted from Fowler-Nordheim plots and from SEM measurements, for carbon-based emitters Richard G. Forbes Advanced Technology Institute & Department of Electrical and Electronic Engineering, Faculty of Engineering & Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, UK Permanent e-mail alias:
[email protected]
Ala’a A. Al-Qudah, Shadi Alnawasreh, Mazen A. Madanat, and Marwan S. Mousa Department of Physics, Mu'tah University, Al-Karak 6170, Jordan
Abstract—This conference paper discusses different ways of defining the emission area of a field emitter, and the relationship between this and apex radius. Methods of extracting apex-radius estimates from current-voltage (i-V) measurements are indicated. Comparisons are made between apex-radius estimates derived from i-V measurements and from scanning-electron-microscope (SEM) images. The difficulty in extracting meaningful radius estimates when the emission appears to be non-orthodox is illustrated, showing that further studies are needed.
the emitter tip. An approach that is workable in principle (e.g., [1]) is to assume some specific model-emitter shape, and define the emission area either as the area over which the local barrier field is greater than some appropriate "minimum" value, or greater than some percentage of its maximum value. Ideally, one might prefer to use local emission current density (ECD) JL rather than barrier field, but the "minimum" approach will not work well, because (at present) JL cannot be reliably calculated.
Keywords—field electron emission; emission area; field emitter apex radius; Fowler-Nordheim plots; scanning electron microcope.
A simpler geometrical approach is to assume that the emission area varies only weakly with voltage and applied field and that a (constant) effective geometrical area Ag is given by
I. INTRODUCTION: APEX RADIUS VERSUS EMISSION AREA
Ag = εg (πra)2 ,
Despite the long history of field electron emission (FE), there have been relatively few attempts to compare different methods of estimating the emission area, or––more-or-less equivalently––emitter apex radius. This paper discusses the background issues and reports some comparisons for tungstenbased and for carbon-based forms of single-tip field emitter (STFE). The comparisons are based on apex radius, rather than emission area, for the following reason. Although there may be practical difficulties, it seems likely that (at least for some emitters) the concept of "apex radius" ra is well defined, and that one can get a "reasonable" estimate rSEM of apex radius from a scanning electron microscope (SEM) image. But in a FE context (as shown below), the concept of "emission area" is not uniquely defined, and there are also problems in deciding how to relate emission area to apex radius. Thus, the apex radius seems a parameter of better scientific standing. II. DIFFERENT TYPES OF AREA-PARAMETER To a large extent, the "area" of a field emitter depends on how you attempt to define it and how you attempt to measure it. Several categories of area-parameter exist. (A) Geometrical parameters. A pointed field electron emitter emits from its "tip", but the current density is not uniform, because the local field and work-function vary across
(1)
where εg is an appropriately chosen constant. (B) Fowler-Nordheim-theory parameters. As set out in detail in another poster [2], the total emission current ie from a STFE can be written via the three linked equations: JkC = aφ–1(ζC–1Ve)2 exp[–νFbφ3/2ζC/Ve] ,
(2)
JC = λC JkC ,
(3)
ie = An JC = AnλCJkC ≡ Af JkC ,
(4)
where all symbols have their usual meanings, as in [2]. The notional emission area An is obtained [3] by integrating JL over the whole emitting area A of the STFE and writing the result as ∫ JLdA = An JC ,
(5)
where JC is the characteristic local ECD at some chosen characteristic point "C" on the emitter surface (in modeling usually the emitter apex). An is roughly equivalent to a "percentage" geometrical area, as discussed above. The value of the characteristic local pre-exponential correction factor λC is not reliably known (an informed guess [4] is that, for a metal, λC may lie in the range 0.005
