Direct Energy, 2018a

120 6.4 Crystallography Revisited
Lattice vectors in real space have units of length, m. Lattice vectors in
reciprocal space have units m −1 .
The reciprocal lattice gives information about the spatial frequency of
atoms. If the planes of atoms in a crystal are closely spaced in one direction,
| −→ a 1 | is relatively small. The corresponding reciprocal vector | −→ b 1 | is relatively
large. The reciprocal lattice represents the spatial frequency of the atom
in units m −1 . If the planes of atoms in a crystal are far apart, | −→ a 1 | is large
and | −→ b 1 | is small.
If a beam of light shines on a crystal where the wavelength of light
is close to the crystal spacing, light will be diracted, and the diraction
pattern is related to the reciprocal lattice. The Brillouin zone is a primitive
cell for a reciprocal lattice. The volume of a unit cell in reciprocal space
over a unit cell in real space is given by
vol. Brillouin zone
vol. primitive cell in real space = −→
b1 · −→ b 2 × −→ b 3
−→ a1 · −→ a 2 × −→ a 3
=(2π) 3 . (6.18)
As for the real space lattice, to understand the reciprocal space lattice,
we need to only understand one cell because the reciprocal space lattice is
periodic.
6.4.2 E versus k Diagrams
The energy level diagrams, discussed in Section 6.3, plot allowed energies
of electrons where the vertical axis represented energy. No variation is
shown on the horizontal axis. The most useful energy level diagrams for
semiconductors are zoomed in so that only the valence and conduction
band are shown. In many cases, it is useful to plot energy level diagrams
versus position in real space. For such a diagram the vertical axis represents
energy, and the horizontal axis represents position. It is also useful to plot
energy level diagrams versus position in reciprocal space.
Kinetic energy is given by
E kinetic = 1 2 m|−→ v | 2 = 1
2m |−→ M| 2 (6.19)
where −→ v represents velocity in m s and m represents mass in kg. Momentum
is given by −→ M = m −→ v in units kg·m
s = J·s
m . Electrons in crystals at
T>0 K vibrate, and certain vibrations are resonant in the crystal. The
crystal momentum −→ M crystal represents the internal momentum of due to
vibrations. It can be expressed as
−→
M crystal = −→ k (6.20)