On the need for a tunneling pre-factor in Fowler-Nordheim tunneling theory

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission (CFE) and in Fowler–Nordheim (FN) type equations. Except in the case of the “ideally smooth” parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrodinger equation. A review of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approach to solving the Schrodinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy–Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.

[1]  R. Fowler,et al.  Electron Emission in Intense Electric Fields , 1928 .

[2]  H. A. Kramers,et al.  Wellenmechanik und halbzahlige Quantisierung , 1926 .

[3]  R. Fowler,et al.  The Aerodynamics of a Spinning Shell , 1920 .

[4]  C. Eckart The Penetration of a Potential Barrier by Electrons , 1930 .

[5]  A. Sommerfeld,et al.  Zur Elektronentheorie der Metalle , 1927, Naturwissenschaften.

[6]  E. C. Kemble The Fundamental Principles Of Quantum Mechanics , 1937 .

[7]  Michael V Berry,et al.  Semiclassical approximations in wave mechanics , 1972 .

[8]  Theoretical analysis of field emission data , 2001 .

[9]  H. Gray,et al.  Vacuum microelectronics , 1989, International Technical Digest on Electron Devices Meeting.

[10]  Gregor Wentzel,et al.  Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik , 1926 .

[11]  L. Nordheim The Effect of the Image Force on the Emission and Reflexion of Electrons by Metals , 1928 .

[12]  R. H. Good,et al.  A WKB-Type Approximation to the Schrödinger Equation , 1953 .

[13]  N. Frank,et al.  Transmission of Electrons through Potential Barriers , 1931 .

[14]  W. Kohn,et al.  Theory of Metal Surfaces: Induced Surface Charge and Image Potential. , 1973 .

[15]  L. Felsen JWKB approximation: Contributions to the theory: by N. Froman and P. O. Fröman. 6 × 9 in. Amsterdam, North-Holland Pub., Co., 1965. Price, $4.50 , 1966 .

[16]  V. E. Chaika,et al.  Observation of the resonance-tunneling in field emission structures , 1999, Eleventh International Vacuum Microelectronics Conference. IVMC'98 (Cat. No.98TH8382).

[17]  R. Forbes,et al.  New theory for the derivation of emission area from a Fowler-Nordheim plot , 1998, Eleventh International Vacuum Microelectronics Conference. IVMC'98 (Cat. No.98TH8382).

[18]  A. Jason Field-Induced Resonance States at a Surface , 1967 .

[19]  R. Forbes The physics of generalized Fowler-Nordheim-type equations , 2007, 2007 IEEE 20th International Vacuum Nanoelectronics Conference.

[20]  K. Scheel,et al.  Handbuch der Physik , 1928 .

[21]  E. C. Kemble A Contribution to the Theory of the B. W. K. Method , 1935 .

[22]  E. Guth,et al.  Electron Emission of Metals in Electric Fields. I. Explanation of the Periodic Deviations from the Schottky Line , 1941 .

[23]  A. Messiah Quantum Mechanics , 1961 .

[24]  A. Sommerfeld,et al.  Elektronentheorie der Metalle , 1967 .

[25]  E. W. Plummer,et al.  Field Emission Energy Distribution (FEED) , 1973 .

[26]  R. Needs,et al.  Theory of field ionization , 1994 .

[27]  J. Heading,et al.  An introduction to phase-integral methods , 1962 .

[28]  R. H. Good,et al.  Thermionic Emission, Field Emission, and the Transition Region , 1956 .

[29]  Kevin L. Jensen,et al.  Electron emission theory and its application: Fowler–Nordheim equation and beyond , 2003 .

[30]  A. Modinos,et al.  Field, Thermionic and Secondary Electron Emission Spectroscopy , 1984 .

[31]  J. Deane,et al.  Reformulation of the standard theory of Fowler–Nordheim tunnelling and cold field electron emission , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[32]  J. Horn Ueber lineare Differentialgleichungen mit einem veränderlichen Parameter , 1899 .

[33]  R. Forbes Use of energy‐space diagrams in free‐electron models of field electron emission , 2004 .

[34]  H. Jeffreys On Certain Approximate Solutions of Lineae Differential Equations of the Second Order , 1925 .

[35]  R. Forbes The electrical surface as centroid of the surface-induced charge , 1999 .

[36]  K. Jensen Theory of Field Emission , 2002 .

[37]  K. Jensen Electron Emission Physics , 2007 .