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
III. Work Function Distribution
IV. Results
A. Gaussian work function distribution
B. Log-normal work function distribution
V. Conclusion, Acknowledgements, and References
II. MATHEMATICAL FORMALISM
We closely follow and summarize the results obtained by Lopes et. al.[13],[14] for field-emission current density, J. In the standard FN-type MG theory the field emission current density is given by the well known expression[15–18],
K(λ) and E(λ) are the complete elliptic integral of the first and second kind, with
From the relations above it is quite evident that calculation of transmission current requires evaluation of numerical integrals for complete elliptic integrals. Due to the singularities present in complete elliptic integrals it is very hard to extract meaningful results purely numerically[19]. Under this circumstances we can consider series expansion for complete elliptic integrals[20] as follows
where
ls A detailed calculation gives the following form for the field emission current density
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