^{1,a)}

### Abstract

This paper reports (a) a simple dimensionless equation relating to field-emitted vacuum space charge (FEVSC) in parallel-plane geometry, namely , where is the FEVSC “strength” and is the reduction in emitter surface field (-with/field-without FEVSC), and (b) the formula , where is the ratio of emitted current density to that predicted by Child’s law. These equations apply to any charged particle, positive or negative, emitted with near-zero kinetic energy. They yield existing and additional basic formulas in planar FEVSC theory. The first equation also yields the well-known cubic equation describing the relationship between and applied voltage; a method of analytical solution is described. Illustrative FEVSC effects in a liquid metal ion source and in field electron emission are discussed. For Fowler–Nordheim plots, a “turn-over” effect is predicted in the high FEVSC limit. The higher the voltage-to-local-field conversion factor for the emitter concerned, then the higher is the field at which turn over occurs. Past experiments have not found complete turn over; possible reasons are noted. For real field emitters, planar theory is a worst-case limit; however, adjusting on the basis of Monte Carlo calculations might yield formulae adequate for real situations.

I. INTRODUCTION

II. THEORETICAL DERIVATIONS

A. Definitions and conventions

B. Planar space-charge equation

C. Derivation of a dimensionless equation

D. General solutions

III. MATHEMATICAL SPACE-CHARGE REGIMES

A. Negligible-space-charge regime

B. Small-space-charge regime

C. Branch-point neighborhood

D. Child’s law regime

E. Partial space-charge equations

F. Discussion

IV. RELATIONSHIP BETWEEN and

V. ILLUSTRATIVE APPLICATIONS

A. Field reduction at the LMIS apex

B. Field-stress reduction at the LMIS apex

C. Effect of FEVSC on Fowler–Nordheim plot shape

D. Criterion for onset of space-charge effects

E. Criterion for approach to the Child’s law regime

VI. A MODIFIED PLANAR-GEOMETRY MODEL

VII. RELATIVISTIC EFFECTS

VIII. SUMMARY

### Key Topics

- Space charge effects
- 32.0
- Current density
- 20.0
- Field emission
- 16.0
- Surface charge
- 12.0
- Charged currents
- 7.0

## Figures

To show how the field reduction factor and the emission current ratio vary with the space-charge strength .

To show how the field reduction factor and the emission current ratio vary with the space-charge strength .

To show how the factor varies with .

To show how the factor varies with .

Plots, in FN coordinates, showing how the Laplace , Poisson , and Child’s law current densities , , and vary with the Laplace field , for the parameter values and . This value corresponds to a tungsten emitter of moderate to large apex radius, in a conventional field electron microscope configuration. Curve is marked with dots and is terminated at the point where the Poisson field becomes equal to the value at which the top of the tunneling barrier goes below the emitter Fermi level. The numbers 0.90, 0.95, and 0.98 are -values corresponding to different possible criteria for the onset of FEVSC effects, and label points on the Poisson curve where approximately has these values.

Plots, in FN coordinates, showing how the Laplace , Poisson , and Child’s law current densities , , and vary with the Laplace field , for the parameter values and . This value corresponds to a tungsten emitter of moderate to large apex radius, in a conventional field electron microscope configuration. Curve is marked with dots and is terminated at the point where the Poisson field becomes equal to the value at which the top of the tunneling barrier goes below the emitter Fermi level. The numbers 0.90, 0.95, and 0.98 are -values corresponding to different possible criteria for the onset of FEVSC effects, and label points on the Poisson curve where approximately has these values.

Enlarged version of the turn-over region in Fig. 3, for values (1) and (2) . The latter value corresponds to a sharp emitter, as might be used in a Spindt array. The Laplace and Poisson curves are terminated when the top of the relevant tunneling barrier goes below the emitter Fermi level. For the high- Poisson curve this occurs before turn over. Theoretically, the Poisson curves continue into the ballistic emission regime and eventually merge with the relevant Child’s law curve, but this part of the Poisson curve may not normally be experimentally accessible. Figure 4 suggests that, for sharp emitters, FEVSC effects become significant at higher current densities than for blunter emitters, if other things are equal.

Enlarged version of the turn-over region in Fig. 3, for values (1) and (2) . The latter value corresponds to a sharp emitter, as might be used in a Spindt array. The Laplace and Poisson curves are terminated when the top of the relevant tunneling barrier goes below the emitter Fermi level. For the high- Poisson curve this occurs before turn over. Theoretically, the Poisson curves continue into the ballistic emission regime and eventually merge with the relevant Child’s law curve, but this part of the Poisson curve may not normally be experimentally accessible. Figure 4 suggests that, for sharp emitters, FEVSC effects become significant at higher current densities than for blunter emitters, if other things are equal.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content