2.4 Effect of Temperature on the Work Function
23
2.4 Effect of Temperature on the Work Function
One of the most common methods of measuring the work function, as well as one
of the most common applications of electron emission, is to use thermionic electron
emission, where a metal wire is heated at a high temperature. Therefore, the temperature dependence of the work function is important and described here. We saw that
the work function is determined by the Wigner–Seitz radius in the jellium model
in the previous section. Increasing the temperature of a material usually causes the
expansion of the lattice, resulting in a decrease in the density of valence electrons
and an increase in the Wigner–Seitz radius. According to Fig. 2.11, the bulk term
increases whereas the surface term decreases with increasing Wigner–Seitz radius.
The work function, which is the sum of the two terms, decreases with increasing
Wigner–Seitz radius. Therefore, it is expected that the work function will decrease
with increasing temperature.
The temperature dependence of a metal work function based on the jellium model
has been reported [16] and is reproduced in Table 2.4. The work function is expressed
as a simple power function of the Wigner–Seitz radius in the jellium model, where
the power factor is −0.5, but the numerical fitting to experiments gives a power
factor of −0.674. The lattice expansion changes not only the average positive charge
density but also the electron distribution at the surface, and these changes have
opposite effects on the work function. However, we can see in Table 2.4 that the
thermal coefficient of the work function is negative for all metals, which agrees
with the above expectation based on the jellium model. Because the surface term is
dependent on the crystal orientation, as discussed in Sect. 2.3, the effect of the crystal
plane on the surface term (α hkl ) in the framework of the jellium model has also been
examined [17]. In this reference, the effect of the temperature dependence on the ion
core radius caused by atom vibration was taken into account in the calculation of the
effect of the crystal plane on the surface term (α
vib
hkl ). The result of this calculation is
reproduced in Table 2.5 and Fig. 2.18a–c for Na, Al, and Cu, respectively [17]. As
shown in Table 2.5 and Fig. 2.18, above 300 K, results including the vibration effect
show that a closely packed plane has a more positive temperature dependence for
both bcc and fcc metals. However, the results without the vibration effect appear to
have no general trend among the metals.
Obtaining experimental results on the temperature dependence of the work function is not easy owing to possible contamination at elevated temperatures including
the surface segregation of impurity elements and the difficulty of precise temperature
control. Therefore, only a limited amount of data has been reported. In a report on
Cu(111) and Cu(110) [18], the importance of cleaning the sample in the measurement of the temperature dependence on the work function is emphasized. Figure 2.19a
shows that the temperature dependence of the work function on insufficiently cleaned
Cu(111) is completely different from that for clean Cu(111) shown in Fig. 2.19b. The
temperature coefficients in Fig. 2.19b are −(10 ± 6) × 10
–5 eV/K for Cu(111) and
−(20 ± 10) × 10
–5 eV/K for Cu(110) [18], whereas they are −1.7 × 10
–5 eV/K
for Cu(111) and ~−17 × 10
–5 eV/K for Cu(110) in the above calculations [17].
23
2.4 Effect of Temperature on the Work Function
One of the most common methods of measuring the work function, as well as one
of the most common applications of electron emission, is to use thermionic electron
emission, where a metal wire is heated at a high temperature. Therefore, the temperature dependence of the work function is important and described here. We saw that
the work function is determined by the Wigner–Seitz radius in the jellium model
in the previous section. Increasing the temperature of a material usually causes the
expansion of the lattice, resulting in a decrease in the density of valence electrons
and an increase in the Wigner–Seitz radius. According to Fig. 2.11, the bulk term
increases whereas the surface term decreases with increasing Wigner–Seitz radius.
The work function, which is the sum of the two terms, decreases with increasing
Wigner–Seitz radius. Therefore, it is expected that the work function will decrease
with increasing temperature.
The temperature dependence of a metal work function based on the jellium model
has been reported [16] and is reproduced in Table 2.4. The work function is expressed
as a simple power function of the Wigner–Seitz radius in the jellium model, where
the power factor is −0.5, but the numerical fitting to experiments gives a power
factor of −0.674. The lattice expansion changes not only the average positive charge
density but also the electron distribution at the surface, and these changes have
opposite effects on the work function. However, we can see in Table 2.4 that the
thermal coefficient of the work function is negative for all metals, which agrees
with the above expectation based on the jellium model. Because the surface term is
dependent on the crystal orientation, as discussed in Sect. 2.3, the effect of the crystal
plane on the surface term (α hkl ) in the framework of the jellium model has also been
examined [17]. In this reference, the effect of the temperature dependence on the ion
core radius caused by atom vibration was taken into account in the calculation of the
effect of the crystal plane on the surface term (α
vib
hkl ). The result of this calculation is
reproduced in Table 2.5 and Fig. 2.18a–c for Na, Al, and Cu, respectively [17]. As
shown in Table 2.5 and Fig. 2.18, above 300 K, results including the vibration effect
show that a closely packed plane has a more positive temperature dependence for
both bcc and fcc metals. However, the results without the vibration effect appear to
have no general trend among the metals.
Obtaining experimental results on the temperature dependence of the work function is not easy owing to possible contamination at elevated temperatures including
the surface segregation of impurity elements and the difficulty of precise temperature
control. Therefore, only a limited amount of data has been reported. In a report on
Cu(111) and Cu(110) [18], the importance of cleaning the sample in the measurement of the temperature dependence on the work function is emphasized. Figure 2.19a
shows that the temperature dependence of the work function on insufficiently cleaned
Cu(111) is completely different from that for clean Cu(111) shown in Fig. 2.19b. The
temperature coefficients in Fig. 2.19b are −(10 ± 6) × 10
–5 eV/K for Cu(111) and
−(20 ± 10) × 10
–5 eV/K for Cu(110) [18], whereas they are −1.7 × 10
–5 eV/K
for Cu(111) and ~−17 × 10
–5 eV/K for Cu(110) in the above calculations [17].