5.2 Lattice Vibrations
99
We are used to such linear relations for sound (and also light) waves. The phase and group velocity are
the same and do not depend on k. Thus, such solutions are called acoustic. The sound velocity of the
medium (v g (k = 0) in (5.9)) is given by v s = a
√
C/M.
It is characteristic of the non-continuous medium that for k approaching the boundary of the Brillouin
zone, the behavior of the wave is altered. For k = π/a the wavelength is just λ = 2π/k = 2a, and
thus samples the granularity of the medium. The maximum phonon frequency ω m is
ω m =
4C
M
.
(5.11)
The group velocity is zero at the zone boundary, thus a standing wave is present.
Since the force constants of the longitudinal and transverse waves can be different, the dispersion
relations are different. The transverse branch of the dispersion relation is two-fold degenerate when
the two directions perpendicular to k are equivalent.
5.2.2 Diatomic Linear Chain
Now we consider the case that the system is made up from two different kinds of atoms (Fig. 5.5).
This will be a model for semiconductors with a diatomic base, such as zincblende. We note that the
diamond structure also needs to be modeled in this way, although both atoms in the base are the same.
The lattice will be the same and the lattice constant will be a. Alternating atoms of sort 1 and 2 with
a relative distance of a/2 are on the chain. The displacements of the two atoms are labeled u
1
n and u
2
n ,
both belonging to the lattice point n. The atoms have the masses M 1 and M 2 . The force constants are
C 1 (for the 1–2 bond within the base) and C 2 (for the 2–1 bond between different bases) (Fig. 5.1d).
The total energy of the system is then given as
U =
1
2
C 1
n
u
1
n − u
2
n
2 +
1
2
C 2
n
u
2
n − u
1
n+1
2 .
(5.12)
The equations of motion are
Fig. 5.5 Visualization of
acoustic and optical waves
in a diatomic linear chain
99
We are used to such linear relations for sound (and also light) waves. The phase and group velocity are
the same and do not depend on k. Thus, such solutions are called acoustic. The sound velocity of the
medium (v g (k = 0) in (5.9)) is given by v s = a
√
C/M.
It is characteristic of the non-continuous medium that for k approaching the boundary of the Brillouin
zone, the behavior of the wave is altered. For k = π/a the wavelength is just λ = 2π/k = 2a, and
thus samples the granularity of the medium. The maximum phonon frequency ω m is
ω m =
4C
M
.
(5.11)
The group velocity is zero at the zone boundary, thus a standing wave is present.
Since the force constants of the longitudinal and transverse waves can be different, the dispersion
relations are different. The transverse branch of the dispersion relation is two-fold degenerate when
the two directions perpendicular to k are equivalent.
5.2.2 Diatomic Linear Chain
Now we consider the case that the system is made up from two different kinds of atoms (Fig. 5.5).
This will be a model for semiconductors with a diatomic base, such as zincblende. We note that the
diamond structure also needs to be modeled in this way, although both atoms in the base are the same.
The lattice will be the same and the lattice constant will be a. Alternating atoms of sort 1 and 2 with
a relative distance of a/2 are on the chain. The displacements of the two atoms are labeled u
1
n and u
2
n ,
both belonging to the lattice point n. The atoms have the masses M 1 and M 2 . The force constants are
C 1 (for the 1–2 bond within the base) and C 2 (for the 2–1 bond between different bases) (Fig. 5.1d).
The total energy of the system is then given as
U =
1
2
C 1
n
u
1
n − u
2
n
2 +
1
2
C 2
n
u
2
n − u
1
n+1
2 .
(5.12)
The equations of motion are
Fig. 5.5 Visualization of
acoustic and optical waves
in a diatomic linear chain