4.1 Utilizing Electron Emission Current Measurement
75
When J 0 [A/cm
2 ], F [V/cm], and φ [eV] are used, A = 1.54 × 10
−6 (for t
2
(y) = 1),
B = 6.83 × 10
7 (for ν(y) = 1), and y = 3.78 × 10
−4
×
√
F
φ
. Because y represents
the effect of the field on the barrier height, when F → 0 (zero-field limit), t
2
(y) and
ν(y) are equal to 1. Approximations of t
2
(y) = 1.1 and ν(y) = 0.95 − y
2 have been
proposed [3].
In practice, the field emission current I and the applied voltage V are experimentally obtained. Therefore, J 0 and F in Eq. (4.4) are respectively replaced by I and V
using the emission area α and the local field conversion factor β as follows.
I = J 0 × α, F = β × V
Then Eq. (4.4) can be rewritten as Eq. (4.5) using I and V instead of J 0 and F,
respectively [4].
I = 1.4 × 10
−6
α
(βV )
2
φ
exp
−6.49 × 10
7 φ
3
2
βV
+
9.8
√ φ
(4.5)
Here, it should be noted that β has a unit of cm
−1 . When both the emitter and
the collector of emitted electrons are flat parallel plates, F = E (the strength of a
macroscopic electric field) =
V
d
, where d is the distance between the emitter and the
collector. In this case, β =
1
d
. However, in some studies the local field conversion
factor has been defined as F = β −
V
d
for nonflat emitters, and one should be careful
when using a local field conversion factor. Equation (4.5) can be transformed into
Eq. (4.6).
log
I
V 2
= a +
b
V
(4.6)
Upon plotting the relationship in Eq. (4.6) with
I
V
as the abscissa and the logarithm
of
I
V 2 as the ordinate, a graph such as Fig. 4.5 is obtained. If the work function,
emission area, and local field conversion factor are constant for the entire range of
the applied voltage, then the slope is constant, meaning that the graph is a straight
line. Here, base of a logarithm is taken as 10 instead of e (Napier’s constant) for
convenience in later discussion. Then, the slope b is represented by Eq. (4.7) and the
intercept a is represented by Eq. (4.8).
b = −2.82 × 10
7 φ
3
2
β
(4.7)
a = log
1.4 × 10
−6
α
β
2
φ
+
4.26
√ φ
(4.8)
75
When J 0 [A/cm
2 ], F [V/cm], and φ [eV] are used, A = 1.54 × 10
−6 (for t
2
(y) = 1),
B = 6.83 × 10
7 (for ν(y) = 1), and y = 3.78 × 10
−4
×
√
F
φ
. Because y represents
the effect of the field on the barrier height, when F → 0 (zero-field limit), t
2
(y) and
ν(y) are equal to 1. Approximations of t
2
(y) = 1.1 and ν(y) = 0.95 − y
2 have been
proposed [3].
In practice, the field emission current I and the applied voltage V are experimentally obtained. Therefore, J 0 and F in Eq. (4.4) are respectively replaced by I and V
using the emission area α and the local field conversion factor β as follows.
I = J 0 × α, F = β × V
Then Eq. (4.4) can be rewritten as Eq. (4.5) using I and V instead of J 0 and F,
respectively [4].
I = 1.4 × 10
−6
α
(βV )
2
φ
exp
−6.49 × 10
7 φ
3
2
βV
+
9.8
√ φ
(4.5)
Here, it should be noted that β has a unit of cm
−1 . When both the emitter and
the collector of emitted electrons are flat parallel plates, F = E (the strength of a
macroscopic electric field) =
V
d
, where d is the distance between the emitter and the
collector. In this case, β =
1
d
. However, in some studies the local field conversion
factor has been defined as F = β −
V
d
for nonflat emitters, and one should be careful
when using a local field conversion factor. Equation (4.5) can be transformed into
Eq. (4.6).
log
I
V 2
= a +
b
V
(4.6)
Upon plotting the relationship in Eq. (4.6) with
I
V
as the abscissa and the logarithm
of
I
V 2 as the ordinate, a graph such as Fig. 4.5 is obtained. If the work function,
emission area, and local field conversion factor are constant for the entire range of
the applied voltage, then the slope is constant, meaning that the graph is a straight
line. Here, base of a logarithm is taken as 10 instead of e (Napier’s constant) for
convenience in later discussion. Then, the slope b is represented by Eq. (4.7) and the
intercept a is represented by Eq. (4.8).
b = −2.82 × 10
7 φ
3
2
β
(4.7)
a = log
1.4 × 10
−6
α
β
2
φ
+
4.26
√ φ
(4.8)