20
INTRODUCTION TO PHYSICS OF THE SOLID STATE
Brillouin zone ha,,, = n/a, where a is the distance between atoms m and M at
equilibrium. The Brillouin zone is a unit cell in wavenumber or reciprocal space, as
will be explained later in this chapter. The optical branch vibrational frequencies are
in the infrared region of the spectrum, generally with frequencies in the range from
lo’, to 3 x lOI4 Hz, and the acoustic branch frequencies are much lower. In three
dimensions the situation is more complicated, and there are longitudinal acoustic
(LA), transverse acoustic (TA), longitudinal optical (LO), and transverse optical
(TO) modes.
The atoms in molecules also undergo vibratory motion, and a molecule containing N atoms has 3N - 6 normal modes of vibration. Particular molecular groups
such as hydroxyl -OH, amino -NH, and nitro -NO, have characteristic normal
modes that can be used to detect their presence in molecules and solids.
The atomic vibrations that we have been discussing correspond to standing-wave
types. This vibrational motion can also produce traveling waves in which localized
regions of vibratory atomic motion travel through the lattice. Examples of such
traveling waves are sound moving through the air, or seismic waves that start at the
epicenter of an earthquake, and travel thousands of miles to reach a seismograph
detector that records the earthquake event many minutes later. Localized traveling
waves of atomic vibrations in solids, called phonons, are quantized with the energy
fio = hv, where v = w/2n is the frequency of vibration of the wave. Phonons play
an important role in the physics of the solid state.
2.2. ENERGY BANDS
2.2.1. Insulators, Semiconductors, and Conductors
When a solid is formed the energy levels of the atoms broaden and form bands with
forbidden gaps between them. The electrons can have energy values that exist within
one of the bands, but cannot have energies corresponding to values in the gaps
between the bands. The lower energy bands due to the inner atomic levels are
narrower and are all full of electrons, so they do not contribute to the electronic
properties of a material. They are not shown in the figures. The outer or valence
electrons that bond the crystal together occupy what is called a valence band. For an
insulating material the valence band is 1 1 1 of electrons that cannot move since they
are fixed in position in chemical bonds. There are no delocalized electrons to carry
current, so the material is an insulator. The conduction band is far above the valence
band in energy, as shown in Fig. 2.1 la, so it is not thermally accessible, and remains
essentially empty. In other words, the heat content of the insulating material at room
temperature T = 300 K is not sufficient to raise an appreciable number of electrons
from the valence band to the conduction band, so the number in the conduction band
is negligible. Another way to express this is to say that the value of the gap energy Eg
far exceeds the value kBT of the thermal energy, where kB is Boltzmann’s constant.
In the case of a semiconductor the gap between the valence and conduction bands
is much less, as shown in Fig. 2.1 1 b, so Eg is closer to the thermal energy kB T, and
INTRODUCTION TO PHYSICS OF THE SOLID STATE
Brillouin zone ha,,, = n/a, where a is the distance between atoms m and M at
equilibrium. The Brillouin zone is a unit cell in wavenumber or reciprocal space, as
will be explained later in this chapter. The optical branch vibrational frequencies are
in the infrared region of the spectrum, generally with frequencies in the range from
lo’, to 3 x lOI4 Hz, and the acoustic branch frequencies are much lower. In three
dimensions the situation is more complicated, and there are longitudinal acoustic
(LA), transverse acoustic (TA), longitudinal optical (LO), and transverse optical
(TO) modes.
The atoms in molecules also undergo vibratory motion, and a molecule containing N atoms has 3N - 6 normal modes of vibration. Particular molecular groups
such as hydroxyl -OH, amino -NH, and nitro -NO, have characteristic normal
modes that can be used to detect their presence in molecules and solids.
The atomic vibrations that we have been discussing correspond to standing-wave
types. This vibrational motion can also produce traveling waves in which localized
regions of vibratory atomic motion travel through the lattice. Examples of such
traveling waves are sound moving through the air, or seismic waves that start at the
epicenter of an earthquake, and travel thousands of miles to reach a seismograph
detector that records the earthquake event many minutes later. Localized traveling
waves of atomic vibrations in solids, called phonons, are quantized with the energy
fio = hv, where v = w/2n is the frequency of vibration of the wave. Phonons play
an important role in the physics of the solid state.
2.2. ENERGY BANDS
2.2.1. Insulators, Semiconductors, and Conductors
When a solid is formed the energy levels of the atoms broaden and form bands with
forbidden gaps between them. The electrons can have energy values that exist within
one of the bands, but cannot have energies corresponding to values in the gaps
between the bands. The lower energy bands due to the inner atomic levels are
narrower and are all full of electrons, so they do not contribute to the electronic
properties of a material. They are not shown in the figures. The outer or valence
electrons that bond the crystal together occupy what is called a valence band. For an
insulating material the valence band is 1 1 1 of electrons that cannot move since they
are fixed in position in chemical bonds. There are no delocalized electrons to carry
current, so the material is an insulator. The conduction band is far above the valence
band in energy, as shown in Fig. 2.1 la, so it is not thermally accessible, and remains
essentially empty. In other words, the heat content of the insulating material at room
temperature T = 300 K is not sufficient to raise an appreciable number of electrons
from the valence band to the conduction band, so the number in the conduction band
is negligible. Another way to express this is to say that the value of the gap energy Eg
far exceeds the value kBT of the thermal energy, where kB is Boltzmann’s constant.
In the case of a semiconductor the gap between the valence and conduction bands
is much less, as shown in Fig. 2.1 1 b, so Eg is closer to the thermal energy kB T, and