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5.4. Field emission
5.4 Field emission
Electrons can be extracted from the surface of a bulk metal by
applying an electric field. This process is known as field emission
or cold emission. It relies on the quantum-mechanical process of a
single electron tunneling through a potential barrier. This process
takes place even at very low temperature. Our goal is to obtain
an explicit expression for the emission current density j in terms
of the applied field F , and the work function φ, in the limit of low
temperature T . This was first understood theoretically in 1927 by
Fowler and Nordheim [31]. This source is now widely used in a
variety of practical instruments. It is distinguished by its exceptionally high brightness. In the following sections we proceed to
derive the current density from first principles of quantum mechanics.
For the present purpose we can consider a conduction electron
to be free inside the bulk metal at x ≤ 0. The potential energy
is thus given here by U (x) = 0. Outside the metal at x ≥ 0, the
potential energy is given by
2
e
U (x) = C − F x −
,
(5.38)
16πf 0 x
where the first term on the right is a constant, associated with the
energy in the vacuum. The second term on the right is the potential energy associated with the applied electric field, where F
is given by the product of the electron charge e times the electric
field in volts per meter. In this notation F has units of force, which
is equivalent to energy per unit distance. The third term on the
right in (5.38) is the potential energy associated with the image
force. This form for the potential energy was first understood by
Nordheim [66].
For now we will ignore the third term, since it is relatively weak
in the limit of high electric field. This approximation was used by
Fowler and Nordheim [31]. The potential energy is plotted in this
approximation as a function of x in Figure 5.2. In a later section
5.4. Field emission
5.4 Field emission
Electrons can be extracted from the surface of a bulk metal by
applying an electric field. This process is known as field emission
or cold emission. It relies on the quantum-mechanical process of a
single electron tunneling through a potential barrier. This process
takes place even at very low temperature. Our goal is to obtain
an explicit expression for the emission current density j in terms
of the applied field F , and the work function φ, in the limit of low
temperature T . This was first understood theoretically in 1927 by
Fowler and Nordheim [31]. This source is now widely used in a
variety of practical instruments. It is distinguished by its exceptionally high brightness. In the following sections we proceed to
derive the current density from first principles of quantum mechanics.
For the present purpose we can consider a conduction electron
to be free inside the bulk metal at x ≤ 0. The potential energy
is thus given here by U (x) = 0. Outside the metal at x ≥ 0, the
potential energy is given by
2
e
U (x) = C − F x −
,
(5.38)
16πf 0 x
where the first term on the right is a constant, associated with the
energy in the vacuum. The second term on the right is the potential energy associated with the applied electric field, where F
is given by the product of the electron charge e times the electric
field in volts per meter. In this notation F has units of force, which
is equivalent to energy per unit distance. The third term on the
right in (5.38) is the potential energy associated with the image
force. This form for the potential energy was first understood by
Nordheim [66].
For now we will ignore the third term, since it is relatively weak
in the limit of high electric field. This approximation was used by
Fowler and Nordheim [31]. The potential energy is plotted in this
approximation as a function of x in Figure 5.2. In a later section