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4.3 Work function
Energy required to extract an electron from a
solid
The work function
 is the energy required to remove an electron from the highest
filled level in the Fermi distribution of a solid so that it is stationary at a
point in a field-free zone just outside the solid, at absolute zero. An
estimate of the work function can be obtained thermionically from
Richardson's equation
|
I = AT 2 exp(−/kT) |
where I is the thermionic current, T the
absolute temperature, k is Boltzmann's constant and A is a
constant having the theoretical value 120 amp cm−2
deg−2. Since A contains the reflection coefficient,
which varies with temperature, observed A values usually differ from
the theoretical value.
too varies very slightly with temperature, but for most metals
the difference between
at absolute zero and at room temperature is less than
experimental errors.
may also be estimated photoelectrically for metals.
Einstein’s expression for the photoelectric effect is hv =
e + E, where E is the kinetic
energy of the ejected photoelectron. The photoelectric current J
released when light of energy hv falls on the surface of a metal, for
which the threshold frequency is given by hv0 = e (for then E = 0), is given by the Fowler equation
|
J = B(kT2).f{(hv
− hv0)/kT} |
where f is a universal function of (hv
− hv0)/kT and B is constant provided that
hv is near to hv0.
The third common method of measuring
is by the contact potential difference (c.p.d.)
VAB that exists between the surfaces of two solids A and B of
work functions
A and
B, when connected electrically, since
|
B −
A = eVAB |
for the two solids at the same temperature. The method
involves a prior knowledge of the work function of one of the solids if that of
the other is to be measured absolutely.
A
fourth method involves the field emission of electrons when an external
electric field F is applied. The field emission current density,
J, is given by
|
|
J = 1.54 x 10−6 |
F2 |
t2(y)exp[−6.83 ×
107
3/2f(y)/F] |
|
|
where f(y) and t(y) are
slowly varying elliptic functions of F and
.
Typical errors of the
tabulated quantities for the metals are 0.02 eV. Note that adsorbates or
contaminants will usually reduce the measured
and that different crystal faces of the same material have
differing values of
.
Work
functions
|
Metal |
Work function φ/eV |
Metal |
Work function φ/eV |
|
|
Photoelectric |
C.P.D. |
|
Thermionic |
Photoelectric |
C.P.D. |
| |
|
|
|
|
|
|
|
Li . .
. . . |
— |
2.32 |
Nb
. . . .
. |
4.30 |
— |
4.37 |
|
Na . .
. . . |
2.36 |
2.46 |
Mo
. . . .
. |
4.33 |
4.49 |
4.21 |
|
K . .
. . . |
2.30 |
2.01 |
Ta
. . . .
. |
4.33 |
4.30 |
4.22 |
|
Rb . .
. . . |
2.05 |
— |
W
. . . .
. |
4.55 |
4.55 |
4.55 |
|
Cs . .
. . . |
1.95 |
1.82 |
Re
. . . .
. |
4.72 |
— |
— |
| |
|
|
|
|
|
|
|
Be . .
. . . |
— |
3.91 |
Ti
. . . .
. |
4.10 |
4.33 |
4.20 |
|
Mg . .
. . . |
— |
3.61 |
Cr
. . . .
. |
4.60 |
4.44 |
— |
|
Ca . .
. . . |
2.87 |
— |
Mn
. . . .
. |
— |
4.08 |
— |
|
Ba . .
. . . |
2.52 |
2.35 |
Fe
. . . .
. |
— |
4.60 |
4.16 |
|
|
|
|
Co
. . . .
. |
— |
4.97 |
— |
|
Zn . .
. . . |
3.63 |
4.11 |
Ni
. . . .
. |
5.24 |
5.15 |
5.25 |
|
Cd . .
. . . |
— |
4.22 |
|
|
|
|
|
|
|
|
Zr
. . . .
. |
4.00 |
— |
— |
|
Al . .
. . . |
4.28 |
4.19 |
Hf
. . . .
. |
3.65 |
— |
— |
|
Ga . .
. . . |
4.35 |
— |
|
|
|
|
|
In . .
. . . |
4.08 |
— |
Ru
. . . .
. |
— |
4.71 |
4.73 |
|
|
|
|
Rh
. . . .
. |
4.72 |
— |
— |
|
Sn . .
. . . |
4.28 |
4.43 |
Pd
. . . .
. |
— |
5.40 |
— |
|
Pb . .
. . . |
4.25 |
3.83 |
Ir
. . . .
. |
4.57 |
— |
— |
|
|
|
|
Pt
. . . .
. |
5.36 |
5.63 |
— |
|
Cu . .
. . . |
4.65 |
4.51 |
|
|
|
|
|
Ag . .
. . . |
4.26 |
4.29 |
Th
. . . .
. |
— |
— |
3.71 |
|
Au . .
. . . |
5.10 |
5.28 |
U
. . . .
. |
3.47 |
3.47 |
3.63 |
| |
|
|
|
|
|
|
|
As . .
. . . |
4.79 |
— |
C (dag)
. . . . |
— |
— |
4.65–5.0 |
|
Sb . .
. . . |
4.56 |
— |
Si
. . . .
. |
— |
4.95 |
4.75 |
|
Bi . .
. . . |
4.34 |
— |
Ge
. . . .
. |
— |
5.15 |
4.83 |
|
|
|
|
|
|
|
|
|
Note: Values selected from the articles by J. C.
Rivière, Solid State Surface Science (ed. Mino Green), Vol. 1,
1969 (Marcel Dekker, New York), and by J. Hölzl and F. K. Schulte,
Springer Tracts in Modern Physics, 85, 1
(1979). |
M.P. Seah
|
|
