264
9 Optical Properties
The first term is the kinetic energy (p stands for the momentum of the relative motion of the atoms 1
and 2 in the base, p = M r ˙
u), the second the potential energy, the third the dipole interaction and the
fourth the electric-field energy. The equation of motion for a plane wave u = u 0 exp[−i(ωt − k · r)]
( ¨
u = −ω
2 u) yields
M r ω
2 u = b 11 u + b 12 E .
(9.20)
Thus, the electric field is
E = (ω
2
− ω
2
TO )
M r
b 12
u .
(9.21)
Here, the substitution ω
2
TO = b 11 /M r was introduced that is consistent with (9.18) and b 11 = 2C.
ω TO represents the mechanical oscillation frequency of the atoms undisturbed by any electromagnetic
effects. Already now the important point is visible. If ω approaches ω TO , the system plus electric field
oscillates with the frequency it has without an electric field. Therefore the electric field must be zero.
Since the polarization P = ( − 1) 0 E is finite, the dielectric constant thus diverges.
The polarization is
P = −∇ E ˆ
H = − (b 12 u + b 22 E) .
(9.22)
The displacement field is
D = 0 E + P = 0 E −
b 22 −
b
2
12 /M r
ω
2
TO − ω 2
E = 0 (ω) E .
(9.23)
Therefore, the dielectric function is
(ω) = (∞) +
(0) − (∞)
1 − (ω/ω TO ) 2 .
(9.24)
Here, (∞) = 1 − b 22 / 0 is the high-frequency dielectric constant and (0) = (∞) + b
2
12 /(b 11 0 ) the
static dielectric constant. The relation (9.24) is shown in Fig. 9.6.
From the Maxwell equation ∇ · D = 0 for zero free charge we obtain the relation
0 (ω) ∇ · E = 0 .
(9.25)
Thus, either (ω) = 0 or ∇ · E = 0, i.e. u is perpendicular to k. In the latter case we have a TO
phonon and, neglecting retardation effects, using ∇ × E = 0 we find E = 0 and therefore ω = ω TO ,
justifying our notation. In the case of (ω) = 0, we call the related frequency ω LO and find the so-called
Lyddane–Sachs–Teller (LST) relation [839]
ω
2
LO
ω
2
TO
=
(0)
(∞)
.
(9.26)
This relation holds reasonably well for optically isotropic, heteropolar materials with two atoms in
the basis, such as NaI and also GaAs. Since at high frequencies, i.e. ω ω TO , only the individual
atoms can be polarized, while for low frequencies the atoms can also be polarized against each other,
(0) > (∞) and therefore also ω LO > ω TO . For GaAs, the quotient of the two phonon energies is
1.07. Using the LST relation (9.26), we can write for the dielectric function
(ω) = (∞)
ω
2
LO − ω
2
ω
2
TO − ω 2
.
(9.27)
9 Optical Properties
The first term is the kinetic energy (p stands for the momentum of the relative motion of the atoms 1
and 2 in the base, p = M r ˙
u), the second the potential energy, the third the dipole interaction and the
fourth the electric-field energy. The equation of motion for a plane wave u = u 0 exp[−i(ωt − k · r)]
( ¨
u = −ω
2 u) yields
M r ω
2 u = b 11 u + b 12 E .
(9.20)
Thus, the electric field is
E = (ω
2
− ω
2
TO )
M r
b 12
u .
(9.21)
Here, the substitution ω
2
TO = b 11 /M r was introduced that is consistent with (9.18) and b 11 = 2C.
ω TO represents the mechanical oscillation frequency of the atoms undisturbed by any electromagnetic
effects. Already now the important point is visible. If ω approaches ω TO , the system plus electric field
oscillates with the frequency it has without an electric field. Therefore the electric field must be zero.
Since the polarization P = ( − 1) 0 E is finite, the dielectric constant thus diverges.
The polarization is
P = −∇ E ˆ
H = − (b 12 u + b 22 E) .
(9.22)
The displacement field is
D = 0 E + P = 0 E −
b 22 −
b
2
12 /M r
ω
2
TO − ω 2
E = 0 (ω) E .
(9.23)
Therefore, the dielectric function is
(ω) = (∞) +
(0) − (∞)
1 − (ω/ω TO ) 2 .
(9.24)
Here, (∞) = 1 − b 22 / 0 is the high-frequency dielectric constant and (0) = (∞) + b
2
12 /(b 11 0 ) the
static dielectric constant. The relation (9.24) is shown in Fig. 9.6.
From the Maxwell equation ∇ · D = 0 for zero free charge we obtain the relation
0 (ω) ∇ · E = 0 .
(9.25)
Thus, either (ω) = 0 or ∇ · E = 0, i.e. u is perpendicular to k. In the latter case we have a TO
phonon and, neglecting retardation effects, using ∇ × E = 0 we find E = 0 and therefore ω = ω TO ,
justifying our notation. In the case of (ω) = 0, we call the related frequency ω LO and find the so-called
Lyddane–Sachs–Teller (LST) relation [839]
ω
2
LO
ω
2
TO
=
(0)
(∞)
.
(9.26)
This relation holds reasonably well for optically isotropic, heteropolar materials with two atoms in
the basis, such as NaI and also GaAs. Since at high frequencies, i.e. ω ω TO , only the individual
atoms can be polarized, while for low frequencies the atoms can also be polarized against each other,
(0) > (∞) and therefore also ω LO > ω TO . For GaAs, the quotient of the two phonon energies is
1.07. Using the LST relation (9.26), we can write for the dielectric function
(ω) = (∞)
ω
2
LO − ω
2
ω
2
TO − ω 2
.
(9.27)