Fractional Fowler–Nordheim Law for Field Emission From Rough Surface With Nonparabolic Energy Dispersion

Abstract

The theories of field electron emission from perfectly planar and smooth canonical surfaces are well understood, but they are not suitable for describing emission from rough, irregular surfaces arising in modern nanoscale electron sources. Moreover, the existing models rely on Sommerfeld's free-electron theory for the description of electronic distribution which is not a valid assumption for modern materials with nonparabolic energy dispersion. In this paper, we derive analytically a generalized Fowler-Nordheim (FN) type equation that takes into account the reduced space-dimensionality seen by the quantum mechanically tunneling electron at a rough, irregular emission surface. We also consider the effects of non-parabolic energy dispersion on field-emission from narrow-gap semiconductors and few-layer graphene using Kane's band model. The traditional FN equation is shown to be a limiting case of our model in the limit of a perfectly flat surface of a material with parabolic dispersion. The fractional-dimension parameter used in this model can be experimentally calculated from appropriate current-voltage data plot. By applying this model to experimental data, the standard field-emission parameters can be deduced with better accuracy than by using the conventional FN equation.

Fowler–Nordheim Law Correlated with Improved Field Emission in Self‐Assembled NiCr2O4 Nanosheets

“…[43] In the previous literature, the value of α was chosen as 1 for the ideal planner emitter surface which was not realistic accurately. [42] In our case, the experimental result of Figure 6a fit better with Equation (3) for the value of α % 0.986, and it prescribed a realistic nearly planner emitter surface. As the emitter surface is slightly rough due to the deposition of nanosheets on the planner cathode, the local electric field conflicts lightly with the applied electric field at an emission site.…”

“…Fowler–Nordheim ( F – N ) law is one of the important tools for explaining the electric field‐induced electron tunneling from a perfectly flat nanometric field emitter deposited on a planner substrate through a nearly triangular potential‐energy barrier. Our experiment leads to F – N type equation, where the generalized fractional FE density can be expressed aswhere J FNα represents the local FE current density in terms of the applied surface electric field ( E ) and local work function (Φ); λ m and β are the macroscopic pre‐exponential correction factor and local field enhancement factor, respectively. The first ( a FNα ) and second ( b FNα ) F – N constants were set as and for fitting our experimental result with the barrier shape correction factor ν = 0.7 for the most realistic “Schottky–Nordheim (SN)” barrier .…”

“…The first ( a FNα ) and second ( b FNα ) F – N constants were set as and for fitting our experimental result with the barrier shape correction factor ν = 0.7 for the most realistic “Schottky–Nordheim (SN)” barrier . In the previous literature, the value of α was chosen as 1 for the ideal planner emitter surface which was not realistic accurately . In our case, the experimental result of Figure a fit better with Equation for the value of α ≈ 0.986, and it prescribed a realistic nearly planner emitter surface.…”

“…Fowler-Nordheim (F-N) law is one of the important tools for explaining the electric field-induced electron tunneling from a perfectly flat nanometric field emitter deposited on a planner substrate through a nearly triangular potential-energy barrier. Our experiment leads to F-N type equation, where the generalized fractional FE density ðJ FNα Þ can be expressed as [42]…”

Thickness Dependence of Space-Charge-Limited Current in Spatially Disordered Organic Semiconductors

“…where c(α, x) = π α/2 Γ(α/2) |x| α−1 [21], ρ is the carrier charge density, ν is the drift velocity, E is the electric field, and is the dielectric permittivity of the material, and V is the electric potential. Using ν = −µE, Eq.…”