Comment on slope of nonlinear FN plot and field
enhancement factor
Weiliang Wang and Zhibing Li*
State Key Laboratory of Optoelectronic Materials and Technologies
School of Physics and Engineering, Sun Yat-sen University, 510275 Guangzhou, P.R. China
Email: stslzb@mail.sysu.edu.cn
Abstract:
It is common practice to extract field enhancement factor from the slope of FN plot. Many
experimentalists working on field electron emission had reported multi-(linear segment) FN
plots, which can be divided into several (usually two) linear segments. Then multi-(field
enhancement factor) were extracted from the FN plot. They claimed that the field
enhancement factor increases with applied field if the FN plot bends downward (vice versus if
the FN plot bends upward). We show that this is contrary to fact.
Keywords: field emission, FN plot, field enhancement factor
PACS: 79.70.+q
1. Introduction
Many experimentalists working on field electron emission had reported multi-(linear segment)
FN plots, which can be divided into several (usually two) linear segments [1-10]. Then
multi-(field enhancement factor  ) were extracted from the FN plot. They claimed
that  increases with applied field if the FN plot bends downward (vice versus if the FN plot
bends upward). The present paper aims to illustrate that this is contrary to fact.
2. Field enhancement factor
According to the FN equation (For simplicity, we use the elementary FN-type equation.
Experimentalists are recommended to use the technically complete FN-type equation [11])
which gives the local emission current density (LECD) JL in terms of the local work function
 and the applied macroscopic field FM

J L  a 1  2 FM2 exp  b 3 / FM

(1)
where a (  1.541434 A  eV  V 2 ) and b (  6.830890eV 3 / 2  V  nm 2 ) are universal
constants, the field enhancement factor
  b 3 / S L
(2)
where SL is the slope of an FN plot ( ln J L / FM2 versus 1/FM). Some experimentalists applied
this to each segment of a multi-(linear segment) FN plots. Therefore they found that  is
different in different FM.
Figure 1. Illustration of a two linear segment FN plot.
Figure 2. Illustration of a series of FN plots with field enhancement factor
1   2   3   4   5 (dotted) and a bending downward FN plot (solid).
Let’s take a two linear segment FN plot as an example (figure 1). This type of FN plots is
reported in many experimental papers (for example [1, 4]). We call this bending downward
FN plot. |SL| is greater (less) in low (high) field region. One would found that  in high
field region is greater than that in low field region. This is contrast to a simple reasoning:
greater  should lead to higher emission current, and then the FN plot in high field region
should be above the dashed line (figure 1). The reason is that Eq. (2) is valid only
if  and  are independent of FM. Figure 2 illustrates intuitively how  varies with FM in
bending downward FN plots. The field enhancement factor is 1 in low field region; therefore
the FN plot (solid) coincides with the upper most dotted line. The field enhancement factor
decreases to  2 ,  3 ,  4 and  5 successively in high field region, therefore the FN plot (solid)
cross the lower dotted line corresponding to  2 ,  3 ,  4 and  5 in sequence. Therefore a
bending downward (upward) FN plot means  decrease (increase) with FM (if all other
parameters are constant).
The present paper does not aim to discuses the reason of the variation of  . Because it had
been extensively discussed in many literatures, the reasons might be resistance [12, 13], space
charge effect [14], gas absorption [15], structure change of emission site [16, 17],
non-uniformity of emission sites [18, 19], localized states [20], non-Schottky-Nordheim
barrier [21, 22] or interaction between emitters [23].
3. Work function
The above discussion can be applied to work function 
straightforwardly. It is
straightforward to see that a bending downward (upward) FN plot means  increase (decrease)
with FM (if all other parameters are constant).
The potential barrier can be lowered by the applied field [12, 24-26], thus  may decrease
with FM. And  can keep constant if the Fermi level is pinned in conduction band or large
amount of surface states. What would the FN plot look like if    2 in low field region and
  1 in high field region ( 1   2 )? It should coincide with the FN plot with    2 in low
field region and jump to the FN plot with   1 in high field region. It would be a zigzag FN
plot (figure 3). These zigzag FN plots were reported in many experimental papers (for
example [27]). The present paper provides a possible explanation.
Figure 3. Illustration of a series of FN plots with local work function 1   2 (dotted) and a
zigzag FN plot (solid).
4. Conclusion
It is inappropriate to extract field enhancement factor or work function from multi-(linear
segment) FN plot. We show intuitively that a bending downward (upward) FN plot
means  decrease (increase) with FM (if all other parameters are constant), and a bending
downward (upward) FN plot means  increase (decrease) with FM (if all other parameters are
constant). Zigzag FN plot may be due to a step function like work function (versus the applied
field).
Acknowledgments
The project was supported by the National Basic Research Program of China (Grant No.
2007CB935500 and 2008AA03A314), the National Natural Science Foundation of China
(Grant No. 11274393 and 11104358). The authors thank Prof. Juncong She for inspiring
discussions.
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