Elements of Modern Physics
272
Therefore its mass may be defined as
m =
2
2
2
(
/
)
∂
∂
E k
(8.30)
This definition may be extended to apply to a particle in a periodic potential
so that the effective mass of an electron in a one-dimensional crystal is
m* =
2
2
2
(
/
)
∂
∂
E k
(8.31)
Here, however, m* is a function of k. As can be seen from Fig. 8.6(b), m* is
positive near the bottom of each zone, negative near the top of each zone and is
infinite at the point of inflection. The large effective mass can be interpreted as
being due to the strong binding force between the electron and the lattice for
some k values, which makes it difficult to move the electron. The negative
mass may be interpreted in terms of the Bragg reflection when k is close to π/a,
2π/a, etc. on account of which a force in one direction, because of reflection,
leads to a gain of momentum in the opposite direction.
A detailed analysis shows that when an external electric field E is applied,
the acceleration of the electron is given by
a =
2
2
2
1
| |
∂
−
∂
E
e
k
E
(8.32)
which again simulates a free particle motion with the effective mass m* given
in Eq. (8.31). For a three dimensional crystal, the anisotropy is taken into account
in the relation
*
∑ ij j
j
m a = – e E i
(8.33)
m* ij =
2
2
(
/
)
∂
∂ ∂
i
j
E k k
The concept of effective mass provides a satisfactory description of the
charge carriers in crystals. In normal circumstances, the conduction of current
is by the electrons, particularly in the case of crystals in which an energy band
is only partially filled (e.g., alkali metals for which the band is only half-filled).
On the other hand, consider a band which is nearly full except for a few vacancies
near the top of the band. This situation of a full band with the vacancies in the
negative charge, negative mass states may be regarded as corresponding to the
presence of positive charge, positive mass particles. These hole states with
positive charge, also act as charge carriers. In elements like Be, Zn, Cd, etc. It is
272
Therefore its mass may be defined as
m =
2
2
2
(
/
)
∂
∂
E k
(8.30)
This definition may be extended to apply to a particle in a periodic potential
so that the effective mass of an electron in a one-dimensional crystal is
m* =
2
2
2
(
/
)
∂
∂
E k
(8.31)
Here, however, m* is a function of k. As can be seen from Fig. 8.6(b), m* is
positive near the bottom of each zone, negative near the top of each zone and is
infinite at the point of inflection. The large effective mass can be interpreted as
being due to the strong binding force between the electron and the lattice for
some k values, which makes it difficult to move the electron. The negative
mass may be interpreted in terms of the Bragg reflection when k is close to π/a,
2π/a, etc. on account of which a force in one direction, because of reflection,
leads to a gain of momentum in the opposite direction.
A detailed analysis shows that when an external electric field E is applied,
the acceleration of the electron is given by
a =
2
2
2
1
| |
∂
−
∂
E
e
k
E
(8.32)
which again simulates a free particle motion with the effective mass m* given
in Eq. (8.31). For a three dimensional crystal, the anisotropy is taken into account
in the relation
*
∑ ij j
j
m a = – e E i
(8.33)
m* ij =
2
2
(
/
)
∂
∂ ∂
i
j
E k k
The concept of effective mass provides a satisfactory description of the
charge carriers in crystals. In normal circumstances, the conduction of current
is by the electrons, particularly in the case of crystals in which an energy band
is only partially filled (e.g., alkali metals for which the band is only half-filled).
On the other hand, consider a band which is nearly full except for a few vacancies
near the top of the band. This situation of a full band with the vacancies in the
negative charge, negative mass states may be regarded as corresponding to the
presence of positive charge, positive mass particles. These hole states with
positive charge, also act as charge carriers. In elements like Be, Zn, Cd, etc. It is