Authors:
(1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India;
(2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India.
Table of Links
Abstract and 1 Introduction
II. Mathematical Formalism
III. Work Function Distribution
IV. Results
A. Gaussian work function distribution
B. Log-normal work function distribution
V. Conclusion, Acknowledgements, and References
III. WORK FUNCTION DISTRIBUTION
In the previous section we have summarized field emission current for a given local work function Φ. For a system of nano-particles or systems with inhomogeneity the work function will be different over the length scale of the size of the collector for emitted electrons. In such a situation we need to average the field emission current over the distribution of work function as follows:
where P(Φ) is the distribution of the work function, Φ. We choose two widely used distribution functions, namely (i) Gaussian and (ii) log-normal distribution. It is well known that systems with bulk disorder follows Gaussian distribution and the work function distribution function, P(Φ), is given by
where σ is the standard deviation of the distribution and Φ0 is the known bulk value for a given material.
We also use log-normal distribution for the work function :
It is important to mention that we were inspired by the experimental result of Gamez et. al.[12]. They showed that for a system of Pt nano-particles the pair distribution function (PDF) for the radius of nano-particles follows log-normal distribution
Various statistical properties of this distribution are summarized in the table I :
IV. RESULTS
In this section we show our results for current density averaged over two choice of probability distributions as have been discussed in the previous section.
A. Gaussian work function distribution
We first consider the case of Gaussian distribution for the work function. In Fig. 1 we show the histogram plot of the work function, sampled over Gaussian distribution for four choices of bulk work function Φ0 = 3.0, 3.5, 4.0 and 4.5 eV with σ = 0.05. In each case we also fit the histogram plot to Gaussian distribution. From this fit we can see that our choice of random variables for Φ are well sampled over Gaussian distribution.
1. Case with Φ ≪ EF
2. Case with Φ ≫ EF
