Solid State Physics
271
| A | ≈ | B |, which is associated with the π/a < k ≤ 2π/a branch of the solutions.
The most important feature of these branches is that there is an energy gap of V 1
at k = π/a between the two Brillouin zones described by these branches, and
there are 2N allowed states (including the negative k states) in each zone. This
is qualitatively similar to the energy bands. It also follows from Eq. (8.28) that
since k – k′ = 2π/a, ∂E/∂k = 0 at k = π/a. The energy bands are illustrated in
Fig. 8.6(b). The gap at k = 2π/a comes from the V 2 term etc.
In the case of the three dimensional problem the condition in Eq. (8.27) is
modified to read k – k′ = q where q is a reciprocal lattice vector defined in
Eq. (8.16). The corresponding Brillouin zones are obtained form the requirement
that | k | = | k′ | at the boundaries. This leads to the condition 2k . q – q
2
= 0
which defines the boundaries as some plane perpendicular to the reciprocal
lattice vectors q. The perturbations of energies at the edges of the Brillouin
zones lead to distorted equal-energy surfaces in the k-space. In particular, the
electrons in a crystal occupy the lowest energy levels at T = 0 K, subject to the
Pauli principle. The surface of the region of all occupied states in the k-space is
called the Fermi surface. Since the energy distortions are prominent mainly
near the surfaces of the Brillouin zones, the nearness of the Fermi surface to the
surfaces of the Brillouin zones, and its shape are of importance for the
understanding of the properties of electrons in crystals in general and metals in
particular.
Another property of the energy bands worth noting is the density of states.
The free particle energy density is proportional to E
1/2
[see Eq. (7.29)]. However,
the periodic potential distorts the energy levels in such a way that there are no
energy levels in the energy gaps and the density of states goes to zero at the
bottom and the top of the energy band.
For obtaining information about the energy bands and the density of states,
x-rays are used to knock out electrons in the energy bands. An analysis of the
x-rays emitted when electrons for higher energy bands undergo transitions to
these vacant levels, provides information about the energy bands and the density
of states.
Effective Mass
A useful idea in the band theory of solids is that of the effective mass of an
electron in a solid. One is led to this idea in an effort to simulate an electron in
a periodic potential by a free electron but with an effective mass.
The energy of a free electron is given by
E =
2 2
2
k
m
(8.29)
271
| A | ≈ | B |, which is associated with the π/a < k ≤ 2π/a branch of the solutions.
The most important feature of these branches is that there is an energy gap of V 1
at k = π/a between the two Brillouin zones described by these branches, and
there are 2N allowed states (including the negative k states) in each zone. This
is qualitatively similar to the energy bands. It also follows from Eq. (8.28) that
since k – k′ = 2π/a, ∂E/∂k = 0 at k = π/a. The energy bands are illustrated in
Fig. 8.6(b). The gap at k = 2π/a comes from the V 2 term etc.
In the case of the three dimensional problem the condition in Eq. (8.27) is
modified to read k – k′ = q where q is a reciprocal lattice vector defined in
Eq. (8.16). The corresponding Brillouin zones are obtained form the requirement
that | k | = | k′ | at the boundaries. This leads to the condition 2k . q – q
2
= 0
which defines the boundaries as some plane perpendicular to the reciprocal
lattice vectors q. The perturbations of energies at the edges of the Brillouin
zones lead to distorted equal-energy surfaces in the k-space. In particular, the
electrons in a crystal occupy the lowest energy levels at T = 0 K, subject to the
Pauli principle. The surface of the region of all occupied states in the k-space is
called the Fermi surface. Since the energy distortions are prominent mainly
near the surfaces of the Brillouin zones, the nearness of the Fermi surface to the
surfaces of the Brillouin zones, and its shape are of importance for the
understanding of the properties of electrons in crystals in general and metals in
particular.
Another property of the energy bands worth noting is the density of states.
The free particle energy density is proportional to E
1/2
[see Eq. (7.29)]. However,
the periodic potential distorts the energy levels in such a way that there are no
energy levels in the energy gaps and the density of states goes to zero at the
bottom and the top of the energy band.
For obtaining information about the energy bands and the density of states,
x-rays are used to knock out electrons in the energy bands. An analysis of the
x-rays emitted when electrons for higher energy bands undergo transitions to
these vacant levels, provides information about the energy bands and the density
of states.
Effective Mass
A useful idea in the band theory of solids is that of the effective mass of an
electron in a solid. One is led to this idea in an effort to simulate an electron in
a periodic potential by a free electron but with an effective mass.
The energy of a free electron is given by
E =
2 2
2
k
m
(8.29)