266
9 Optical Properties
A is the vector potential for the electromagnetic field, i.e. E = − ˙
A, μH = ∇ × A, ∇ · A = 0
(Coulomb gauge). The Hamiltonian of an electron in the electromagnetic field is
H =
1
2m
(k − qA)
2
.
(9.32)
When terms in A
2 are neglected (i.e. two-photon processes), the perturbation Hamiltonian is thus
H em = −
q
m
A p =
i q
m
A · ∇ ≈ q r · E .
(9.33)
The latter approximation is valid for small wavevectors of the electromagnetic wave and is termed the
electric dipole approximation.
In order to calculate the dielectric function of the semiconductor from its band structure we assume
that A is weak and we can apply (9.30). The transition probability R for the photon absorption rate at
photon energy ω is then given by
3
R(ω) =
2π
k c
k v
|c|H em |v|
2
δ (E c (k c ) − E v (k v ) − ω) d
3 k c d
3 k v ,
(9.34)
with the Bloch functions |c and |v of the conduction and valence band, respectively, as given in
(6.40b).
The vector potential is written as A = Aˆ e with a unit vector ˆ
e parallel to A. The amplitude is
connected to the electric-field amplitude E via
A = −
E
2 ω
exp (i(qr − ωt)) + exp (−i(qr − ωt))
.
(9.35)
In the electric-dipole approximation the momentum conservation q + k v = k c , q being the momentum
of the light wave is approximated by k v = k c . The matrix element is then given by
|c|H em |v|
2
=
e
2
|A|
2
m 2
c|ˆ e · p|v
2 ,
(9.36)
with
c
ˆ
e · p|v
2 =
1
3
|p cv |
2
= M
2
b ,
(9.37)
and the momentum matrix element p cv given in (6.39). A k-independent matrix element |p cv |
2 is often
used as an approximation. In Fig. 9.7 the matrix elements for valence to conduction band transitions
in GaN are shown as a function of k.
In terms of the electric-field amplitude E(ω) the transition probability is
R(ω) =
2π
e
m ω
2
E(ω)
2
2
|p cv |
2
k
δ (E c (k) − E v (k) − ω) d
3 k .
(9.38)
If the integration over k is restricted to those values allowed in unit volume, the power that is lost from
the field in unit volume is given by R ω, leaving a 1/E factor. The dielectric function =
+ i
is
3 Here we assume that the valence-band states are filled and the conduction-band states are empty. If the conduction-band
states are filled and the valence-band states are empty, the rate is that of stimulated emission.
9 Optical Properties
A is the vector potential for the electromagnetic field, i.e. E = − ˙
A, μH = ∇ × A, ∇ · A = 0
(Coulomb gauge). The Hamiltonian of an electron in the electromagnetic field is
H =
1
2m
(k − qA)
2
.
(9.32)
When terms in A
2 are neglected (i.e. two-photon processes), the perturbation Hamiltonian is thus
H em = −
q
m
A p =
i q
m
A · ∇ ≈ q r · E .
(9.33)
The latter approximation is valid for small wavevectors of the electromagnetic wave and is termed the
electric dipole approximation.
In order to calculate the dielectric function of the semiconductor from its band structure we assume
that A is weak and we can apply (9.30). The transition probability R for the photon absorption rate at
photon energy ω is then given by
3
R(ω) =
2π
k c
k v
|c|H em |v|
2
δ (E c (k c ) − E v (k v ) − ω) d
3 k c d
3 k v ,
(9.34)
with the Bloch functions |c and |v of the conduction and valence band, respectively, as given in
(6.40b).
The vector potential is written as A = Aˆ e with a unit vector ˆ
e parallel to A. The amplitude is
connected to the electric-field amplitude E via
A = −
E
2 ω
exp (i(qr − ωt)) + exp (−i(qr − ωt))
.
(9.35)
In the electric-dipole approximation the momentum conservation q + k v = k c , q being the momentum
of the light wave is approximated by k v = k c . The matrix element is then given by
|c|H em |v|
2
=
e
2
|A|
2
m 2
c|ˆ e · p|v
2 ,
(9.36)
with
c
ˆ
e · p|v
2 =
1
3
|p cv |
2
= M
2
b ,
(9.37)
and the momentum matrix element p cv given in (6.39). A k-independent matrix element |p cv |
2 is often
used as an approximation. In Fig. 9.7 the matrix elements for valence to conduction band transitions
in GaN are shown as a function of k.
In terms of the electric-field amplitude E(ω) the transition probability is
R(ω) =
2π
e
m ω
2
E(ω)
2
2
|p cv |
2
k
δ (E c (k) − E v (k) − ω) d
3 k .
(9.38)
If the integration over k is restricted to those values allowed in unit volume, the power that is lost from
the field in unit volume is given by R ω, leaving a 1/E factor. The dielectric function =
+ i
is
3 Here we assume that the valence-band states are filled and the conduction-band states are empty. If the conduction-band
states are filled and the valence-band states are empty, the rate is that of stimulated emission.