14
2 What is the Work Function?: Definition and Factors …
Here, ε is the dielectric constant of a semiconductor, φ is the potential induced by
the surface charge distribution, and N is the carrier density. Note that band bending
occurs at a clean surface of a semiconductor, although it is often introduced at the
interface between a metal and a semiconductor in semiconductor physics. Because
band bending is caused by the electron distribution near the surface, surface modification such as HF treatment can modify the band bending of a semiconductor.
The band of n-Si, which initially bends upward on a clean surface, is flattened by
HF treatment (Fig. 2.8b). For semiconductors whose Fermi level is located near the
valence band (the conduction band is close to E V AC ), if the band bends downwards
by an amount larger than the electron affinity (EA = E V AC − E C ), the band diagram
is similar to that shown in Fig. 2.8c. Then, electrons in the conduction band can be
emitted without any potential barrier. This situation is called negative electron affinity
(NEA). A hydrogen-terminated diamond (111) surface is a well-known NEA surface
[4]. One should note that an NEA does not mean a negative work function. Even
if bands bend downwards, E V AC is still above the Fermi level. This means that one
should excite electrons up to the conduction band in some way, and thus electron
emission from an NEA surface is not a spontaneous phenomenon.
Here, we discuss the image force. The concept of the image force was originally
introduced for the method of image charges in electrostatics, where the surface electrostatic potential was not taken into account. Therefore, we start by considering the
image force without a surface term, where the charge distribution induced by ion
cores and that induced by valence electrons are uniform and overlap. The potential
diagram without the image force taken into account is shown in Fig. 2.9a, where
the potential changes abruptly at the surface. We consider adding the effect of the
image potential to this. This image force in vacuum is expressed using the following
equation.
F =
e
2
4πε 0
·
1
r 2
(2.6)
Here, e is the charge of electrons and is 1.602 × 10
−19 [C], ε 0 is the permittivity of
vacuum and is 8.85 × 10
−12 [F/m], and r [m] is the distance between an electron and
the surface. By integrating Eq. (2.6), the image potential at distance z is expressed
as [5]
∞
∫
z
e
2
4πε 0
·
1
r 2 dr = 14.42 · 10
−10
·
1
z
[eV ].
(2.7)
The magnitude of the image potential at different z values is shown in Table 2.1.
From the table, the image potential is of practical importance for z ≤ 100 nm.
Therefore, when the work function is defined as in Fig. 2.1b with a d L, d
should be large enough not to be influenced by the image potential, giving one more
condition: d > 1000 nm (to avoid a minor influence, the distance should be one order
of magnitude larger than the minimum distance of practical importance). When the
image force is taken into account, the potential change near the surface becomes
2 What is the Work Function?: Definition and Factors …
Here, ε is the dielectric constant of a semiconductor, φ is the potential induced by
the surface charge distribution, and N is the carrier density. Note that band bending
occurs at a clean surface of a semiconductor, although it is often introduced at the
interface between a metal and a semiconductor in semiconductor physics. Because
band bending is caused by the electron distribution near the surface, surface modification such as HF treatment can modify the band bending of a semiconductor.
The band of n-Si, which initially bends upward on a clean surface, is flattened by
HF treatment (Fig. 2.8b). For semiconductors whose Fermi level is located near the
valence band (the conduction band is close to E V AC ), if the band bends downwards
by an amount larger than the electron affinity (EA = E V AC − E C ), the band diagram
is similar to that shown in Fig. 2.8c. Then, electrons in the conduction band can be
emitted without any potential barrier. This situation is called negative electron affinity
(NEA). A hydrogen-terminated diamond (111) surface is a well-known NEA surface
[4]. One should note that an NEA does not mean a negative work function. Even
if bands bend downwards, E V AC is still above the Fermi level. This means that one
should excite electrons up to the conduction band in some way, and thus electron
emission from an NEA surface is not a spontaneous phenomenon.
Here, we discuss the image force. The concept of the image force was originally
introduced for the method of image charges in electrostatics, where the surface electrostatic potential was not taken into account. Therefore, we start by considering the
image force without a surface term, where the charge distribution induced by ion
cores and that induced by valence electrons are uniform and overlap. The potential
diagram without the image force taken into account is shown in Fig. 2.9a, where
the potential changes abruptly at the surface. We consider adding the effect of the
image potential to this. This image force in vacuum is expressed using the following
equation.
F =
e
2
4πε 0
·
1
r 2
(2.6)
Here, e is the charge of electrons and is 1.602 × 10
−19 [C], ε 0 is the permittivity of
vacuum and is 8.85 × 10
−12 [F/m], and r [m] is the distance between an electron and
the surface. By integrating Eq. (2.6), the image potential at distance z is expressed
as [5]
∞
∫
z
e
2
4πε 0
·
1
r 2 dr = 14.42 · 10
−10
·
1
z
[eV ].
(2.7)
The magnitude of the image potential at different z values is shown in Table 2.1.
From the table, the image potential is of practical importance for z ≤ 100 nm.
Therefore, when the work function is defined as in Fig. 2.1b with a d L, d
should be large enough not to be influenced by the image potential, giving one more
condition: d > 1000 nm (to avoid a minor influence, the distance should be one order
of magnitude larger than the minimum distance of practical importance). When the
image force is taken into account, the potential change near the surface becomes