T. Y. Tan
Tan
1
16. ELECTRONS AND HOLES IN SEMICONDUCTING CRYSTALS
Semiconducting crystals such as Si and GaAs are the base materials of manufacturing
modern electronic and optoelectronic devices. A small amount (as compared to the crystal
atomic density) of mobile or free electrons and holes in a semiconducting crystal constitutes the
electrical charge carriers. As has been briefly noted in Chapter 14, the electrons and holes are
defects resulting from some of the crystal bond electrons becoming 'free' at a nonzero
temperature. This is a thermal equilibrium property of the crystal, just as the crystal also contains
(atomic) point defects. From Chapter 15, we see that calculation of the thermal equilibrium point
defect concentrations of metals is carried out by consider the Gibbs free energy of the system
consisting of the crystal and the point defects using Eq. (15.1):
G = Cg
f
 TS
m
.
That is, the problem is treated as a classical one. From the point of view of quantum mechanics,
this can be done because point defects are Bosons for which many can occupy the same energy
level, and for a point defects species, i.e., vacancies, there exists but one energy level:
g
v
f
. The
calculation of the thermal equilibrium concentrations of electrons and holes is, however, not as
straightforward as that of point defects. This is because electrons are Fermions for which some
aspects must be treated by quantum mechanics. Fermions such as electrons obey the Pauli
exclusion principle in that each energy level can only be occupied by one particle. While
compared to atomic density the number of electrons and holes exists in a semiconducting crystal
is small, this number itself is nevertheless sizable. At room temperature, it reaches 10
13
cm
3
in
Ge, 10
10
cm
3
in Si, and 10
7
cm
3
in GaAs. When 'doped', the concentration of the appropriate
type of carrier can become larger by many orders of magnitude. This means that at least the same
number of energy levels are involved for a 1 cm
3
crystal and a calculation of the charge carrier
thermal equilibrium concentrations using expressions similar to Eq. (15.1) becomes quite
impossible, because there will be more than 10
7
g
f
values with two adjacent ones differing by a
negligibly small amount. Thus, the calculation of the thermal equilibrium electron and hole
concentrations in a semiconductor is carried out using a different method, which involves
essentially three steps. In the first, the FermiDirac distribution function of electrons is derived
(Chapter 4). This function states the probability of an energy level with energy E is occupied by
an electron. In the second, the
density of states
is calculated. A state is the synonym of an energy
level, and the density of the states is needed because the energy level density for the same
δ
E
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T. Y. Tan
Tan
2
value is different at different E values. In the third, the results obtained from the first two steps
are combined together to obtain the thermal equilibrium electron and hole concentrations in a
semiconducting crystal.
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 Fall '11
 DR.TAN
 T. Y. Tan

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