1298 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 55, NO. 6, JUNE 2008 Screening Effects Between Field-Enhancing Patterned Carbon Nanotubes: A Numerical Study Martin Dionne, Sylvain Coulombe, and Jean-Luc Meunier Abstract—A numerical investigation of the topographic field- enhancement factor β for structures including individual verti- cally aligned carbon nanotubes (VACNTs) and arrays of VACNT is presented. Some previously reported results for simple struc- tures are reviewed first. Then, the extent of the zones of field enhancement and significant screening effects surrounding a given structure is discussed. The investigation with combined VACNT confirms the criterion that the spacing Δx between identical CNT should be about twice their height to minimize screening effects. This statement is generalized to structures having different height ratios. The possibility of combining patterns of different height VACNT to minimize screening effects while allowing a larger sur- face density of such emitters is then investigated. The results show that height anisotropies in VACNT arrays can significantly reduce the field-emission current for a given applied field. A subsequent study that takes into account Joule heating and radiation losses during field emission demonstrates that, for height anisotropies larger than 5%, the VACNT tips reach temperatures above the onset temperature for selective field-assisted evaporation. This phenomenon occurs before the field-emission current from the nonideal films matches the targeted current value deduced from ideal VACNT arrays. Index Terms—Carbon nanotubes (CNTs), electric-field enhancement, field emission. I. INTRODUCTION S INCE the day they were first identified [1], carbon nano- tubes (CNTs) have received considerable interest due to their unique physical and chemical properties. In the context of application as electron emitter, their high aspect ratio and elevated electrical conductivity enabled the realization of flat- panel displays [2]. In such devices, highly localized electron emitters are sought, and CNTs, with their high length-to- diameter aspect ratios, sharp ends, mechanical and chemical stability, and relative ease of synthesis as dense mats, are ideal candidates for the localized amplification of the electric field at their tip and associated enhanced field emission of electrons. Other applications, such as cathode-ray lamps [3] and X-ray tube sources [4], have contributed to the rapid growth of the research-and-development efforts aimed at the scaling up of the synthesis processes for single- and multiwalled nanotubes with desirable and uniform properties. Among these processes, one finds several versions of plasma-enhanced chemical vapor deposition [5]–[7], where the size and surface density of the Manuscript received September 12, 2007; revised February 26, 2008. The review of this paper was arranged by Editor M. Reed. The authors are with the Chemical Engineering Department, McGill Uni- versity, Montreal, QC H3A 2B2, Canada (e-mail: mdionnemcgill@gmail.com; sylvain.coulombe@mcgill.ca; jean-luc.meunier@mcgill.ca). Digital Object Identifier 10.1109/TED.2008.920995 catalyst sites deposited onto the growth substrate control the diameter of the CNT and mat density, while changing the deposition time determines the height of the produced CNT. Some experimental [8], [9] and numerical studies [10] re- porting on the electric-field enhancement at the tip of CNT standing straight on a surface relied on the CNT height/radius ratio (h CNT /r CNT ) to estimate values of the field enhancement factor β. In the numerical studies, the decrease of β was estimated visually from the decrease of the field probed by the CNT assuming an initial value of β equal to h CNT /r CNT . However, β = h CNT /r CNT is an approximation derived from local field calculation on the top of a small conducting sphere with radius r CNT connected to the electrode surface by an infinitesimally thin wire (see [9]). Although the approximate nature of this result is usually mentioned, its origin is not always provided. The h CNT /r CNT ratios are of little assistance when structures are standing close to each other (similar or not) or are complex by themselves, such as randomly oriented or bundles of CNT. For complex cases, numerical solutions of the Laplace equation without the β = h CNT /r CNT assumption are needed. It has been reported that CNT-covered electrodes have optimal total emission when the spacing between adjacent CNT is on the order of twice their height [10], [11]. The interpretation of experimental results with the help of the Fowler–Nordheim (F–N) equation [12] requires the knowl- edge of the local work function φ 0 if the β factor is to be deduced [13]. This parameter might not be easily assessed experimentally, particularly with structures showing nanoscale geometrical features and composition variations. It was shown [14], [15] that, on the CNT cap, the strong local electric field can reduce the effective work function φ 0 significantly below the assumed 4.5 eV value. As a result, if the F–N equation is used to estimate the minimal value of the applied field E o required to produce a detectable current, this value will be lower. On the other hand, such experimental artifact can be corrected for if either φ 0 or β is corrected for the nanoscale geometrical features effect. Therefore, theoretical calculations of the β factor for combined structures could be used to guide the design optimization of patterned field-emitting cathodes. In this paper, such investigation is performed for several stand- alone structures on flat surfaces, and the results are compared to the previously reported values. Results for cases involving similar structures that are standing close to each other are obtained. Critical values for the distance between nanotubes and height ratio are deduced and used to estimate whether a given structure experiences a significant screening effect from a smaller one. A parametric equation describing the reduction of the β factor from the tips of CNT-forming specific regular 0018-9383/$25.00 © 2008 IEEE