5.2 Lattice Vibrations
111
An exponential decrease of the LVM amplitude occurs for negative K that is ensured for M + 2 C < 0
(and M > −1 and C > −1). The defect frequency is given by
ω d = ω m
(1 + C ) (2 + C (3 + M ))
2(1 + M ) (2 C + 1 − M )
.
(5.45)
We note that for C = 0 (5.41) and (5.42) are recovered.
For a given mass defect, the change of frequency with is (in linear order, i.e. for C 1)
∂ω d (( M , , C )
∂∂ C
=
1 − 4 M −
2
M
4 (1 − M )
1 −
2
M
C .
(5.46)
The linear coefficient diverges for M → −1. For M between −0.968 and 0 the linear coefficient
varies between 2 and 1/4. Therefore, a larger force constant ( C > 0) increases the LVM frequency of
the defect, as expected for a stiffer spring.
In a binary compound the situation is more complicated. We assume here that the force constants
remain the same and only the mass of the substitution atom M d is different from the host. The host
has the atom masses M 1 and M 2 with M 1 < M 2 . Substitution of the heavy atom with a lighter one
creates a LVM above the optical branch for M d < M 2 . Additionally, a level in the gap between the
optical and acoustic branch is induced. Such LVM is called a gap mode. Substitution of the lighter
atom of the binary compound induces a LVM above the optical branch for M d < M 1 . A gap mode
is induced for M d > M 1 . The situation for GaP is depicted in Fig. 5.18. LVM in GaAs have been
reviewed in [382].
The energy position of a local vibrational mode is sensitive to the isotope mass of the surrounding atoms. In Fig. 5.19, a high-resolution (0.03 cm
−1 ) spectrum of the
12 C As LVM in GaAs
is shown together with a theoretical simulation. The various theoretical peak positions are given
as vertical bars, their height indicating the oscillator strength. Five experimental peaks are obvious that are due to a total of nine different transitions. The C atom can experience five different
Fig. 5.17 Energy of local
vibrational modes in Si and
Ge. Experimental values at
T = 300 K (B in Ge:
T = 80 K) taken from
[381] and references
therein and from [385]
(C in Ge). The dashed lines
are the mass dependence
according to (5.42) scaled
to the experimental
frequency of the 10 B LVM
111
An exponential decrease of the LVM amplitude occurs for negative K that is ensured for M + 2 C < 0
(and M > −1 and C > −1). The defect frequency is given by
ω d = ω m
(1 + C ) (2 + C (3 + M ))
2(1 + M ) (2 C + 1 − M )
.
(5.45)
We note that for C = 0 (5.41) and (5.42) are recovered.
For a given mass defect, the change of frequency with is (in linear order, i.e. for C 1)
∂ω d (( M , , C )
∂∂ C
=
1 − 4 M −
2
M
4 (1 − M )
1 −
2
M
C .
(5.46)
The linear coefficient diverges for M → −1. For M between −0.968 and 0 the linear coefficient
varies between 2 and 1/4. Therefore, a larger force constant ( C > 0) increases the LVM frequency of
the defect, as expected for a stiffer spring.
In a binary compound the situation is more complicated. We assume here that the force constants
remain the same and only the mass of the substitution atom M d is different from the host. The host
has the atom masses M 1 and M 2 with M 1 < M 2 . Substitution of the heavy atom with a lighter one
creates a LVM above the optical branch for M d < M 2 . Additionally, a level in the gap between the
optical and acoustic branch is induced. Such LVM is called a gap mode. Substitution of the lighter
atom of the binary compound induces a LVM above the optical branch for M d < M 1 . A gap mode
is induced for M d > M 1 . The situation for GaP is depicted in Fig. 5.18. LVM in GaAs have been
reviewed in [382].
The energy position of a local vibrational mode is sensitive to the isotope mass of the surrounding atoms. In Fig. 5.19, a high-resolution (0.03 cm
−1 ) spectrum of the
12 C As LVM in GaAs
is shown together with a theoretical simulation. The various theoretical peak positions are given
as vertical bars, their height indicating the oscillator strength. Five experimental peaks are obvious that are due to a total of nine different transitions. The C atom can experience five different
Fig. 5.17 Energy of local
vibrational modes in Si and
Ge. Experimental values at
T = 300 K (B in Ge:
T = 80 K) taken from
[381] and references
therein and from [385]
(C in Ge). The dashed lines
are the mass dependence
according to (5.42) scaled
to the experimental
frequency of the 10 B LVM