Concept of the Hole in ELECTRICAL CONDUCTION IN SOLIDS
In considering the two-dimensional representation of the covalent bonding shown in
Figure 3.13a, a positively charged "empty state" was created when a valence electron
was elevated into the conduction band. For T > 0 K, all valence electrons may gain
thermal energy; if a valence electron gains a small amount of thermal energy, it may hop
into the empty state. The movement of a valence electron into the empty state is equivalent
to the movement of the positively charged empty state itself. Figure 3.17 shows the
movement of valence electrons in the crystal alternately filling one empty state and creating a new empty state, a motion equivalent to a positive charge moving in the valence
band. The crystal now has a second equally important charge carrier that can give rise to
a current. This charge carrier is called a hole and, as we will see, can also be thought of
as a classical particle whose motion can be modeled using Newtonian mechanics.
The drift current density due to electrons in the valence band, such as shown in
Figure 3.14b, can be written as
where the summation extends over all filled states. This summation is inconvenient
since it extends over a nearly full valence band and takes into account a very large
number of states. We may rewrite Equation (3.49) in the form
If we consider a band that is totally full, all available states are occupied by electrons.
The individual electrons can be thought of as moving with a velocity as given
by Equation (3.39):
The band is symmetric ink and each state is occupied so that, for every electron with
a velocity v l , there is a corresponding electron with a velocity -|v|. Since the band is
full, the distribution of electrons with respect to k cannot he changed with an
externally applied force. The net drift current density generated from a completely full
band, then, is zero, or
We can now write the drift current density from Equation (3.50) for an almost
full band as
where the u, in the summation is the
associated with the empty state. Equation (3.52) is entirely equivalent to placing a
positively charged particle in the empty states and assuming all other states in the hand
are empty, or neutrally charged. This concept is shown in Figure 3.18. Figure 3.18a
shows the valence band with the conventional electron-filled states and empty states,
while Figure 3.18b shows the new concept of positive charges occupying the original
empty states. This concept is consistent with the discussion of the positively charged
"empty state" in the valence band as shown in Figure 3.17.
The vi in the summation of Equation (3.52) is related to how well this positively
charged particle moves in the semiconductor. Now consider an electron near the top of
the allowed energy band shown in Figure 3.16b. The energy near the top of the allowed
energy hand may again he approximated by a parabola so that we may write
The energy E,, is the energy at the top of the energy hand. Since E i E,, for electrons
in this band, then the parameter C? must be a positive quantity.
Taking the second derivative of E with respect to k from Equation (3.53). we
obtain
We may rearrange this equation so that
Comparing Equation (3.55) with Equation (3.41), we may write
where m* is again an effective mass. We have argued that C2 is a positive quantity,
which now implies that m *is a negative quantity. An electron moving near the topo
an allowed energy hand behaves as if it has a negative mass.
We must keep in mind that the effective mass parameter is used to relate quantum
mechanics and classical mechanics. The attempt to relate these two theories
leads to this strange result of a negative effective mass. However, we must recall that
solutions to Schrodinger's wave equation also led to results that contradicted classical
mechanics. The negative effective mass is another such example.
In discussing the concept of effective mass in the last section, we used an analogy
of marbles moving through two liquids. Now consider placing an ice cube in the center
of a container filled with water: the ice cube will move upward toward the surface
in a direction opposite to the gravitational force. The ice cube appears to have a negative
effective mass since its acceleration is opposite to the external force. The effective
mass parameter takes into account all internal forces acting on the particle.
If we again consider an electron near the top of an allowed energy band and use
Newton's force equation for an applied electric field, we will have
However, m* is now a negative quantity, so we may write
An electron moving near the top of an allowed energy band moves in the same direction
as the applied electric field.
The net motion of electrons in a nearly full hand can he described by considering
just the empty states, provided that a positive electronic charge is associated with
each state and that the negative of m* from Equation (3.56) is associated with each
state. We now can model this band as having particles with a positive electronic
charge and a positive effective mass. The density of these particles in the valence
band is the same as the density of empty electronic energy states. This new particle
is the hole. The hole, then, has a positive effective mass denoted by mg and a positive
electronic charge, so it will move in the same direction as an applied field.
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Figure 3.13a, a positively charged "empty state" was created when a valence electron
was elevated into the conduction band. For T > 0 K, all valence electrons may gain
thermal energy; if a valence electron gains a small amount of thermal energy, it may hop
into the empty state. The movement of a valence electron into the empty state is equivalent
to the movement of the positively charged empty state itself. Figure 3.17 shows the
movement of valence electrons in the crystal alternately filling one empty state and creating a new empty state, a motion equivalent to a positive charge moving in the valence
band. The crystal now has a second equally important charge carrier that can give rise to
a current. This charge carrier is called a hole and, as we will see, can also be thought of
as a classical particle whose motion can be modeled using Newtonian mechanics.
The drift current density due to electrons in the valence band, such as shown in
Figure 3.14b, can be written as
where the summation extends over all filled states. This summation is inconvenient
since it extends over a nearly full valence band and takes into account a very large
number of states. We may rewrite Equation (3.49) in the form
If we consider a band that is totally full, all available states are occupied by electrons.
The individual electrons can be thought of as moving with a velocity as given
by Equation (3.39):
The band is symmetric ink and each state is occupied so that, for every electron with
a velocity v l , there is a corresponding electron with a velocity -|v|. Since the band is
full, the distribution of electrons with respect to k cannot he changed with an
externally applied force. The net drift current density generated from a completely full
band, then, is zero, or
We can now write the drift current density from Equation (3.50) for an almost
full band as
where the u, in the summation is the
associated with the empty state. Equation (3.52) is entirely equivalent to placing a
positively charged particle in the empty states and assuming all other states in the hand
are empty, or neutrally charged. This concept is shown in Figure 3.18. Figure 3.18a
shows the valence band with the conventional electron-filled states and empty states,
while Figure 3.18b shows the new concept of positive charges occupying the original
empty states. This concept is consistent with the discussion of the positively charged
"empty state" in the valence band as shown in Figure 3.17.
The vi in the summation of Equation (3.52) is related to how well this positively
charged particle moves in the semiconductor. Now consider an electron near the top of
the allowed energy band shown in Figure 3.16b. The energy near the top of the allowed
energy hand may again he approximated by a parabola so that we may write
The energy E,, is the energy at the top of the energy hand. Since E i E,, for electrons
in this band, then the parameter C? must be a positive quantity.
Taking the second derivative of E with respect to k from Equation (3.53). we
obtain
We may rearrange this equation so that
Comparing Equation (3.55) with Equation (3.41), we may write
where m* is again an effective mass. We have argued that C2 is a positive quantity,
which now implies that m *is a negative quantity. An electron moving near the topo
an allowed energy hand behaves as if it has a negative mass.
We must keep in mind that the effective mass parameter is used to relate quantum
mechanics and classical mechanics. The attempt to relate these two theories
leads to this strange result of a negative effective mass. However, we must recall that
solutions to Schrodinger's wave equation also led to results that contradicted classical
mechanics. The negative effective mass is another such example.
In discussing the concept of effective mass in the last section, we used an analogy
of marbles moving through two liquids. Now consider placing an ice cube in the center
of a container filled with water: the ice cube will move upward toward the surface
in a direction opposite to the gravitational force. The ice cube appears to have a negative
effective mass since its acceleration is opposite to the external force. The effective
mass parameter takes into account all internal forces acting on the particle.
If we again consider an electron near the top of an allowed energy band and use
Newton's force equation for an applied electric field, we will have
However, m* is now a negative quantity, so we may write
An electron moving near the top of an allowed energy band moves in the same direction
as the applied electric field.
The net motion of electrons in a nearly full hand can he described by considering
just the empty states, provided that a positive electronic charge is associated with
each state and that the negative of m* from Equation (3.56) is associated with each
state. We now can model this band as having particles with a positive electronic
charge and a positive effective mass. The density of these particles in the valence
band is the same as the density of empty electronic energy states. This new particle
is the hole. The hole, then, has a positive effective mass denoted by mg and a positive
electronic charge, so it will move in the same direction as an applied field.
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