74
4 Measurement of Work Function
Fig. 4.4 Current density
versus temperature for ϕ =
4.5 eV during thermal
emission
500
1000
1500
2000
0
5
10
15
20
Js [mA/m
2
]
Temperature [K]
log
J S
T 2 − logA = −
φ
k B
×
1
T
J X
AT 2 = exp
−
φ
k B T
(4.3)
Therefore, from a plot such as that shown in Fig. 4.3, the work function is obtained.
In order to apply this thermal emission method, a material should be able to withstand
a relatively high temperature because the electron current obtained at a temperature
of, for example, 1000 K is insufficient. Because of this fact, materials measured by
this method are high-melting-point metals such as W, Mo, and Ta or compounds such
as LaB 6 and BaO. In Fig. 4.4, current density versus temperature for φ = 4.5 eV
is plotted as an example (A = 1.2 × 10
6 [A/m
2 K
2 ]). For the measurement, the
temperature dependence of the work function discussed in Chap. 2 and the effect of
the electric field applied in order to guide emitted electrons to the anode should be
taken into account.
4.1.2 Field Emission
In this measurement, electrons are emitted not by being excited to overcome the
potential barrier, but by tunneling through the potential barrier, whose thickness
becomes 1 nm upon the application of a strong electric field, as shown in Fig. 4.1.
The electron current density J 0 is described using Eq. (4.4), which is called the
Fowler–Nordheim equation [2], where F is the strength of the electric field at the
emitter surface, y is the Schottky lowering of the work function barrier, and t(y) and
ν(y) are correction terms for the image force barrier.
J 0 =
AF
2
φ
exp
−
Bφ
3
2
F
A =
e
2
8π ht 2 (y)
,
B =
4
√
2mν(y)
3e
,
y =
√
e 3 F
φ
(4.4)
4 Measurement of Work Function
Fig. 4.4 Current density
versus temperature for ϕ =
4.5 eV during thermal
emission
500
1000
1500
2000
0
5
10
15
20
Js [mA/m
2
]
Temperature [K]
log
J S
T 2 − logA = −
φ
k B
×
1
T
J X
AT 2 = exp
−
φ
k B T
(4.3)
Therefore, from a plot such as that shown in Fig. 4.3, the work function is obtained.
In order to apply this thermal emission method, a material should be able to withstand
a relatively high temperature because the electron current obtained at a temperature
of, for example, 1000 K is insufficient. Because of this fact, materials measured by
this method are high-melting-point metals such as W, Mo, and Ta or compounds such
as LaB 6 and BaO. In Fig. 4.4, current density versus temperature for φ = 4.5 eV
is plotted as an example (A = 1.2 × 10
6 [A/m
2 K
2 ]). For the measurement, the
temperature dependence of the work function discussed in Chap. 2 and the effect of
the electric field applied in order to guide emitted electrons to the anode should be
taken into account.
4.1.2 Field Emission
In this measurement, electrons are emitted not by being excited to overcome the
potential barrier, but by tunneling through the potential barrier, whose thickness
becomes 1 nm upon the application of a strong electric field, as shown in Fig. 4.1.
The electron current density J 0 is described using Eq. (4.4), which is called the
Fowler–Nordheim equation [2], where F is the strength of the electric field at the
emitter surface, y is the Schottky lowering of the work function barrier, and t(y) and
ν(y) are correction terms for the image force barrier.
J 0 =
AF
2
φ
exp
−
Bφ
3
2
F
A =
e
2
8π ht 2 (y)
,
B =
4
√
2mν(y)
3e
,
y =
√
e 3 F
φ
(4.4)