Chapter 2
What is the Work Function?: Definition
and Factors that Determine the Work
Function
2.1 Definition of the Work Function
The work function (φ) is conventionally defined as the minimum energy required
to extract one electron from a metal. In this definition, the final state of the electron
needs to be specified. For the energy to be minimum at the final state, the electron
must be finally at rest. Thus, the work function depends on the final position of the
electron. Here, the magnitude relationship among the atomic distance in the metal
(a), the size of the metal (L), and the distance between the surface of the metal
and the final position of the electron (d ) is classified into three types (Fig. 2.1).
When an electron is extracted from a metal with a finite size to the final position
at a large distance (a point at infinity), the energy required does not depend on the
crystal orientation (L d , Fig. 2.1a), which is not the work function in the usual
sense. If the final distance is sufficiently large compared with the atomic distance
but sufficiently small compared with the size of the metal, the electron at the final
position is under the influence of the ambient electric fields in vacuum and the energy
required is dependent on the crystal orientation (a d L, Fig. 2.1b). This is the
definition of the work function [1]. The condition of distance d > 10
3 nm is explained
later in the text (Sect. 2.2). For an extremely small d value, an electron is extracted
near the surface (d ∼ a L, Fig. 2.1c).
At zero temperature, by considering the energy difference between the initial state
(a system containing N electrons in the ground state with energy E N ) and the final
state (a system with one electron extracted, containing (N − 1) electrons), it can be
seen that the work function is equal to the difference between E N and the energy in
the final state, with the energy of (E N −1 + φ V ), where φ V denotes the electrostatic
energy of the extracted electron at rest far from the surface (at distance d ).
φ = (E N −1 + φ V ) − E N
(2.1)
Equation (2.1) can be rewritten as
© National Institute for Materials Science, Japan 2021
M. Yoshitake, Work Function and Band Alignment of Electrode Materials,
NIMS Monographs, https://doi.org/10.1007/978-4-431-56898-8_2
7
What is the Work Function?: Definition
and Factors that Determine the Work
Function
2.1 Definition of the Work Function
The work function (φ) is conventionally defined as the minimum energy required
to extract one electron from a metal. In this definition, the final state of the electron
needs to be specified. For the energy to be minimum at the final state, the electron
must be finally at rest. Thus, the work function depends on the final position of the
electron. Here, the magnitude relationship among the atomic distance in the metal
(a), the size of the metal (L), and the distance between the surface of the metal
and the final position of the electron (d ) is classified into three types (Fig. 2.1).
When an electron is extracted from a metal with a finite size to the final position
at a large distance (a point at infinity), the energy required does not depend on the
crystal orientation (L d , Fig. 2.1a), which is not the work function in the usual
sense. If the final distance is sufficiently large compared with the atomic distance
but sufficiently small compared with the size of the metal, the electron at the final
position is under the influence of the ambient electric fields in vacuum and the energy
required is dependent on the crystal orientation (a d L, Fig. 2.1b). This is the
definition of the work function [1]. The condition of distance d > 10
3 nm is explained
later in the text (Sect. 2.2). For an extremely small d value, an electron is extracted
near the surface (d ∼ a L, Fig. 2.1c).
At zero temperature, by considering the energy difference between the initial state
(a system containing N electrons in the ground state with energy E N ) and the final
state (a system with one electron extracted, containing (N − 1) electrons), it can be
seen that the work function is equal to the difference between E N and the energy in
the final state, with the energy of (E N −1 + φ V ), where φ V denotes the electrostatic
energy of the extracted electron at rest far from the surface (at distance d ).
φ = (E N −1 + φ V ) − E N
(2.1)
Equation (2.1) can be rewritten as
© National Institute for Materials Science, Japan 2021
M. Yoshitake, Work Function and Band Alignment of Electrode Materials,
NIMS Monographs, https://doi.org/10.1007/978-4-431-56898-8_2
7