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D(W ) ≈ 1, corresponding to perfect transmission for energies
W ≥ C. We define a new variable ξ = W − C. We further define a quantity φ called the work function by
φ = C − ζ.
(5.33)
Physically, φ represents the energy which must be supplied to an
electron at the Fermi level in order for emission to occur. This
quantity is a unique property of the bulk material. In practice, the
work function differs from the electron affinity defined above for
an actual emitter.
The integral for the emission current density j is approximated
as
4πmekT ∞
−φ − ξ
j =
dξ ln exp
+ 1 .
(5.34)
h 3
0
kT
We further approximate
−φ − ξ
exp
« 1.
(5.35)
kT
Expanding the logarithm, the integral for j becomes
∞
4πmekT
φ
ξ
j =
exp −
dξ exp −
.
(5.36)
h 3
kT
0
kT
The integral over ξ is readily performed. The emission current
density j is given by
4πme
φ
j =
(kT )
2 exp −
.
(5.37)
h 3
kT
This represents the main result of this section. It is known as the
Richardson–Dushman law for thermionic emission [60].
308
Chapter 5. Electron emission from solids