Page 112 - Electrical Properties of Materials

94 The free electron theory of metals

calculated from the condition,

d e 2
– – eE x = 0, (6.42)
dx 16π 0 x
leading to

1/2

The energy needed to escape from V max =–e eE . (6.43)
the metal is reduced by –V max . 4π 0
The effective work function is thus reduced from φ to

eE
1/2
φ eff = φ – e . (6.44)
4π 0
Substituting this into eqn (6.37) we get the new formula for thermionic
emission
√

{φ – e (eE /4π 0 )}
2
J = A 0 (1 – r)T exp – . (6.45)
k B T
The reduction in the effective value of the work function is known as the
Schottky effect, and plotting log J against E 1/2 , we get the so-called Schottky
line. A comparison with experimental results in Fig. 6.6 shows that above a
certain value of the electric field the relationship is quite accurate. Do not be
too impressed, though; in graphs of this sort the constants are generally fiddled
to get the theoretical and experimental curves on top of each other. But it cer-
tainly follows from Fig. 6.6 that the functional relationship between J and E 1/2
is correct.

T=1566 K

log J
T=1437 K

Theoretical curve
Fig. 6.6 Experimental points
Experimental verification of the
1
Schottky formula [eqn (6.45)]. 2