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 I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. 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 This paper is available on arxiv under CC BY 4.0 DEED license. 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. Authors: 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 Abstract and 1 Introduction II. Mathematical Formalism II. Mathematical Formalism III. Work Function Distribution III. Work Function Distribution IV. Results A. Gaussian work function distribution A. Gaussian work function distribution B. Log-normal work function distribution B. Log-normal work function distribution V. Conclusion, Acknowledgements, and References V. Conclusion, Acknowledgements, and References I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. 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 This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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