Elsevier

Ultramicroscopy

Volume 102, Issue 3, February 2005, Pages 181-187
Ultramicroscopy

Modeling and calculation of field emission enhancement factor for carbon nanotubes array

https://doi.org/10.1016/j.ultramic.2004.08.009Get rights and content

Abstract

To estimate the apex field enhancement factor associated with carbon nanotubes (CNTs) array on a planar cathode surface, the image model of floated sphere between parallel anode and cathode plates was proposed. Firstly, the field enhancement factor of individual CNT was given as the following expression, β0=h/ρ+3.5, where h is the height and ρ is the radius of CNTs. Then the field enhancement factor of CNTs array was discussed and the above expression was modified to be β=h/ρ+3.5-W, in which W is the function of the intertube distance R and represents the coulomb field interaction between the CNTs. All results show that the intertube distance of CNTs array critically affects the field emission. When the intertube distance is less than the height of tube, the field enhancement factor will decrease rapidly with decreasing the intertube distance. According to the calculated results and considering the field emission current density, the filed emission is optimal theoretically when the intertube distance is comparable with the height of CNTs.

Introduction

Owing to the unique mechanical and electrical properties, the excellent structure and the chemical stability, CNTs are certainly one of the most exciting discoveries in nanoscale physics [1], [2], [3]. The CNTs are just rolled graphite sheets and capped with half a fullerene at each end with diameter of the order of nanometer and up to a few micrometers length. The nanoscale tip and the higher aspect ratio would lead to the greater local field at the tip of CNTs, and the emission electrons could penetrate the potential barrier into the vacuum with tunnel effect [4], [5], then the higher current density could obtain with a lower voltage. Therefore, it is not surprising that CNTs are considered to be ideal candidates for the next generation field emitters for flat panel displays, field emission electron source, microwave power amplifiers and so on [6], [7], [8].

Studies have shown that many parameters, including the aspect ratio of CNT, the anode–cathode distance, the intertube distance, the CNT with open/close cap and so on, can influence the field emission properties of the CNTs array [9], [10], [11], [12], [13], [14], [15], [16]. An important factor in field emission is the relationship between the applied and the local electric field where the electron tunneling occurs. The field enhancement factor β is defined as β=Eloc/Em, where Eloc is the actual electric field at the apex of the CNT and Em is the macroscopic electric field. Kokkorakis et al. [17], [18] calculated the field enhancement factor of the open and closed CNTs with the computer simulation, the results show that the factor is the function of (h/ρ), where h is the height of the CNT and ρ is the radius of its cap. Filip et al. [19] also got the same conclusion that the enhancement factor was fitted by formula β=1+s((d-h)/ρ-1), in which d is the anode–cathode distance and s is the parameter of the screening effect.

But for the CNTs array, some experiments proved that the intertube distance critically affects the field enhancement factor [12], [13], [14], [15], [16]. Many scientists adjusted the density of CNTs array with different method, then directly tested the field emission properties of the as-prepared CNTs array and the results showed that the filed emission could be optimal when the intertube distance was about 2 times the tube height. But Suh et al. [16] produced CNTs array with porous anodic aluminum oxide (AAO) templates and was ion milled by Ar+ bombardment. The height from the surface and the intertube distance of the CNTs array are very homogeneous. The maximum emission current density of the CNTs array brought about when the intertube distance was close to the tube height. In the aspect of the theoretical research, Glukhovaet et al. [20] numerically calculated the field emission from CNTs array, and the obtained results showed that the emission enhancement factor are well fitted by expression β=1+[2+B(h/2ρ)D][1-eC(1-R/ρ)], where B, C and D are constants. But the reasons for this behavior are still unclear. Thus, field emission from CNTs array requires further modeling in order to understand their field emission mechanism.

With the simple image method of floated sphere between parallel anode and cathode plates, the apex field enhancement factor associated with CNTs arrays on a planar cathode surface was estimated. The apex field enhancement of individual CNT was given as the following expression β0=h/ρ+3.5, which is a little bigger than others’ results [21]. When the CNTs array was taken into account, the above expression was modified to be β=h/ρ+3.5-W, and the above result shows that the intertube distance critically affects the field emission properties of the CNTs array. When the intertube distance is less than the height of tube, the enhancement factor will decrease rapidly with decreasing the intertube distance [16]. Considering the field emission current density, the field emission from CNTs array is optimal theoretically when the intertube distance of CNTs array is comparable with the height of CNTs.

Section snippets

Modeling of the field emission for the CNT

To calculate the field enhancement factor for the single CNT on the cathode we used the model system shown in Fig. 1, in which one CNT stands perpendicularly on the cathode plane, having a cylindrical shape of height h and closed with a hemispherical cap with radius ρ [18], [19], and the anode–cathode distance is d. We assumed all the considered CNTs are conductive, and the potential is maintained zero over the whole surface of CNTs. For the CNTs array, in order to simplify the calculation the

Results and discussion

When the anode–cathode distance is much larger than the height of CNTs, the expression of the field enhancement factor of individual CNT is similar to Forbes, β0=h/ρ+3, which was obtained using more realistic shape for CNT [21]. Our result β0=h/ρ+3.5 is a little larger than that of Forbes about 0.5. Thus, we can conclude that the floated sphere model is reasonable for calculating the enhancement factor of CNT, especially for the case of very large aspect ratio (102–103). In the model we assumed

Conclusions

The image floated sphere model between the parallel anode and cathode plates was proposed, the apex field enhancement factor associated with individual CNT and CNTs array on a planar cathode surface have been calculated. The field enhancement of individual CNT was given as the following expression, β0=h/ρ+3.5 which is similar to the result of Forbes [21]. And the field enhancement factor of CNTs array is expressed by the formula β=h/ρ+3.5-W, which W represents the coulomb field interaction

Acknowledgements

The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (60271009).

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