2.3 Factors Determining the Work Function
17
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
EN (JCP_1956)
work function (WF, eV)
EN (renewal)
WF(eV) = 2.27 * EN + 0.34
Pauling’s electronegaavity (EN)
[7]
[9,10]
Fig. 2.10 Correlation between Pauling’s electronegativity [7, 9, 10] and work function of elemental
(polycrystalline) metal [8]
to extend it to multielement systems, we need more general parameters. As shown
in Fig. 2.7, the bulk term of the work function, B , is B = E V AC(bulk) − E F =
V XC −
2
2m
k
2
F . In DFT calculations for the jellium model, both V XC and k
2
F are functions
of the electron density in the bulk ρ, which is related to the Wigner–Seitz radius r s
by the following equation.
4
3
π r
3
s =
1
ρ
(2.9)
It was shown that both the bulk term and the surface term of the work function can
be calculated as functions of the Wigner–Seitz radius in the jellium model (uniform
background) [11], which is shown in Fig. 2.11. When the atomistic structure is
neglected and the system is treated as a uniform “sea” of valence electrons, the
surface term is also determined by the bulk quantity, the electron density in the bulk
ρ, or the Wigner–Seitz radius r s , because the surface electron distribution is generated
in order to achieve an equilibrium with the bulk. By using this relationship between
the Wigner–Seitz radius and the work function, the work function of multielement
systems can be estimated in the framework of the jellium model, which will be
discussed in Chap. 3. In the framework of the jellium model, if r s increases, the bulk
term increases but the surface term decreases, resulting in the work function, which
is the sum of the bulk term and the surface term, decreasing with increasing r s . The
figure gives an idea of the size of the surface term relative to the total work function.
According to the above discussion, bulk properties such as the electronegativity
and Wigner–Seitz radius exhibit a correlation with the work function when the
atomistic structure can be neglected (the work function for polycrystals).
The surface term originates from the surface electrostatic potential generated
by the deviation of the electron distribution from the positive charge distribution.
17
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
EN (JCP_1956)
work function (WF, eV)
EN (renewal)
WF(eV) = 2.27 * EN + 0.34
Pauling’s electronegaavity (EN)
[7]
[9,10]
Fig. 2.10 Correlation between Pauling’s electronegativity [7, 9, 10] and work function of elemental
(polycrystalline) metal [8]
to extend it to multielement systems, we need more general parameters. As shown
in Fig. 2.7, the bulk term of the work function, B , is B = E V AC(bulk) − E F =
V XC −
2
2m
k
2
F . In DFT calculations for the jellium model, both V XC and k
2
F are functions
of the electron density in the bulk ρ, which is related to the Wigner–Seitz radius r s
by the following equation.
4
3
π r
3
s =
1
ρ
(2.9)
It was shown that both the bulk term and the surface term of the work function can
be calculated as functions of the Wigner–Seitz radius in the jellium model (uniform
background) [11], which is shown in Fig. 2.11. When the atomistic structure is
neglected and the system is treated as a uniform “sea” of valence electrons, the
surface term is also determined by the bulk quantity, the electron density in the bulk
ρ, or the Wigner–Seitz radius r s , because the surface electron distribution is generated
in order to achieve an equilibrium with the bulk. By using this relationship between
the Wigner–Seitz radius and the work function, the work function of multielement
systems can be estimated in the framework of the jellium model, which will be
discussed in Chap. 3. In the framework of the jellium model, if r s increases, the bulk
term increases but the surface term decreases, resulting in the work function, which
is the sum of the bulk term and the surface term, decreasing with increasing r s . The
figure gives an idea of the size of the surface term relative to the total work function.
According to the above discussion, bulk properties such as the electronegativity
and Wigner–Seitz radius exhibit a correlation with the work function when the
atomistic structure can be neglected (the work function for polycrystals).
The surface term originates from the surface electrostatic potential generated
by the deviation of the electron distribution from the positive charge distribution.