Range of Nordheim parameter y

[pafcu]

Hi,

The current density due to field emission on a surface is described by the Fowler-Nordheim equation
$$J = t(y)^{-2}a\phi^{-1}F^2\exp(-v(y)b\phi^{3/2}/F)$$

where $$v(y)=1 - y^2+ (1/3)y^2ln y$$ and $$y=cF^{1/2}\phi^{-1}$$
$$F$$ is the applied electric field above the surface.

(See for example "Simple good approximations for the special elliptic functions in standard
Fowler-Nordheim tunneling theory for a Schottky-Nordheim barrier" by Forbes)

Anyway, my specific problem is that y should be in the range [0,1] as stated in the paper. However, y is clearly proportional to the field strength. Is there some limiting factor that forces the applied field to be small enough so that y is never larger than 1? Or is it simply the case that once y=1 it doesn't grow anymore, no matter how powerful the electric field F gets? This doesn't seem to be explained anywhere, and many papers don't mention the range of y at all.

 

[eljose79]

I would like to address your concerns regarding the range of y in the Fowler-Nordheim equation. First, it is important to note that the Fowler-Nordheim equation is used to describe the current density due to field emission on a surface, which is a phenomenon that occurs at very high electric fields. Therefore, it is unlikely that the electric field F would reach such high levels where y would exceed 1.

Furthermore, the range of y is limited by the physical properties of the material and the geometry of the surface. For example, the material may have a maximum electric field that it can withstand before breaking down, or the geometry of the surface may prevent the electric field from reaching high levels.

It is also worth mentioning that the Fowler-Nordheim equation is derived from a simplified model and may not accurately describe the behavior of the current density at very high electric fields. In such cases, experimental data and more sophisticated models may be needed to accurately describe the behavior of the current density.

In conclusion, the range of y in the Fowler-Nordheim equation is limited by the physical properties of the material and the geometry of the surface, and it is unlikely that the electric field would reach levels high enough for y to exceed 1. However, it is important for researchers to carefully consider the limitations of the equation and to use experimental data and more sophisticated models when necessary.

 

[charng]

Hello,

Thank you for bringing up this issue with the range of Nordheim parameter y. The range of y is indeed an important factor in the Fowler-Nordheim equation and it is crucial to understand its limitations.

Firstly, it is important to note that the range of y being stated as [0,1] is a general guideline and may vary depending on the specific system or material being studied. As you mentioned, y is proportional to the field strength and therefore, as the field strength increases, y will also increase. However, there are limiting factors that can prevent y from exceeding 1.

One such limiting factor is the breakdown of the material at high electric fields. When the electric field becomes too strong, it can cause the material to break down and lose its ability to sustain the field emission process. This can occur at different values of y depending on the material properties and structure. Therefore, it is important to consider the breakdown limit when determining the range of y for a specific system.

Additionally, as you mentioned, once y reaches 1, it does not continue to increase with increasing field strength. This is because the term (1/3)y^2lny in the expression for v(y) becomes dominant and causes the value of v(y) to decrease. This, in turn, limits the growth of y and prevents it from exceeding 1.

In summary, the range of Nordheim parameter y is not a fixed value and can vary depending on the system being studied. However, there are limiting factors such as material breakdown and the behavior of the v(y) term that prevent y from exceeding 1. It is important to consider these factors when interpreting the results of the Fowler-Nordheim equation. I hope this helps to clarify your question. Thank you for your interest in this topic.