Biomedical Engineering Reference
In-Depth Information
because the excitation energy required to span the forbidden gap is of the order of
8.5 eV. Generally, when the gap is greater than about 5 eV, the material is an insula-
tor. While the details are different, a similar situation describes the covalent solids.
The width of the band gap in carbon is 5.4 eV, making it an insulator. In silicon
the gap is 1.14 eV and in germanium, 0.67 eV. At absolute zero temperature the
valence bands in these two metals are completely filled and the conduction band is
empty. They are insulators. At room temperatures (
kT
∼
0.025
eV), a small fraction
of their electrons are thermally excited into the conduction band, giving them some
conductivity. Covalent solids having an energy gap
∼
1
eV are called intrinsic semi-
conductors. In conductors, the valence and conduction bands merge, as indicated
in Fig. 10.13, providing mobility to the valence electrons. Sodium, with its single
atomic ground-state 3s electron is a conductor in the solid phase.
We focus now on semiconductors and the properties that underlie their impor-
tance for radiation detection and measurement. One can treat conduction electrons
in the material as a system of free, identical spin-
2
particles (Sections 2.5, 2.6). They
can exchange energy with one another, but otherwise act independently, like mole-
cules in an ideal gas, except that they also obey the Pauli exclusion principle. Under
these conditions, it is shown in statistical mechanics that the average number
N
(
E
)
of electrons per quantum state of energy
E
is given by the Fermi distribution,
1
e
(
E
-
E
F
)/
kT
+1
.
N
(
E
)
=
(10.6)
Here
k
is the Boltzmann constant,
T
is the absolute temperature, and
E
F
is called
the
Fermi energy
. At any given time, each quantum state in the system is either
empty or occupied by a single electron. The value of
N
(
E
)
is the probability that a
state with energy
E
is occupied. To help understand the significance of the Fermi
energy, we consider the distribution at the temperature of absolute zero,
T
= 0
.For
states with energies
E
>
E
F
above the Fermi energy, the exponential term in the de-
nominator of (10.6) in infinite; and so
N
(
E
)
=
0
. For states with energies below
E
F
,
the exponential term is zero; and so
N
(
E
)
1
. Thus, all states in the system be-
low
E
F
are singly occupied, while all above
E
F
are empty. This configuration has
the lowest energy possible, as expected at absolute zero. At temperatures
T
>0
,the
Fermi energy is defined as that energy for which the average, or probable, number
of electrons is
2
.
It is instructive to diagram the relative number of free electrons as a function of
energy in various types of solids at different temperatures. Figure 10.14(a) shows
the energy distribution of electrons in the conduction band of a conductor at a
temperature above absolute zero (
T
>0). Electrons occupy states with a thermal
distribution of energies above
E
C
. The lower energy levels are filled, but unoccupied
states are available for conduction near the top of the band. A diagram for the same
conductor at
T
=
0
is shown in Fig. 10.14(b). All levels with
E
<
E
F
are occupied and
all with
E
>
E
F
are unoccupied.
Figure 10.15(a) shows the electron energy distribution in an insulator with
T
>0
.
The valence band is full and the forbidden-gap energy
E
G
(
∼
=
5
eV) is so wide that
the electrons cannot reach the conduction band at ordinary temperatures. Fig-
