306
10 Recombination
the direct transition is energetically fairly close to the fundamental, indirect L– band edge transition
(Fig. 9.15). The energy difference can be reduced from its bulk value of 136 meV by tensile strain.
Additionally, the direct transition can be favored by heavily n-doping and filling the L conduction
band minimum states (see Sect. 9.9.2). In this case, direct recombination from the conduction band
-minimum can be observed [939] and the effective energy difference has been lowered to about
100 meV.
10.2.2 Absorption
A similar consideration is made for the absorption process (Fig. 10.1b). An electron is transferred upon
light absorption from a valence-band state (occupied) to a conduction-band state that must be empty.
The coefficient is B 1 . Also, the process is proportional to the light intensity, represented by the density
of occupied photon states N ph (E),
r abs (E) =
E+E V
E C
dE e B 1 (E e , E e − E) ×
(10.4)
D e (E e ) (1 − f e (E e )) D h (E e − E) (1 − f h (E e − E)) N ph (E) .
10.2.3 Stimulated Emission
In this case, an incoming photon ‘triggers’ the transition of an electron in the conduction band into an
empty state in the valence band. The emitted photon is in phase with the initial photon (Fig. 10.1c).
The rate is (with coefficient B 2 ):
r st (E) =
E+E V
E C
dE e B 2 (E e , E e − E) ×
(10.5)
D e (E e ) f n (E e ) D h (E e − E) f h (E e − E) N ph (E) .
The photon density N ph at a given energy is given by Planck’s law and the Bose–Einstein distribution
(Appendix E)
N ph (E) = N 0
1
exp (E/kT ) − 1
.
(10.6)
The pre-factor is the density of states of the electromagnetic field
3 N 0 (E) = 8 π E
2
(n r / hc)
3 .
10.2.4 Net Recombination Rate
In thermodynamical equilibrium the rates fulfill
3 The total number of photon states in vacuum between the frequencies zero and ν is N (ν) = 8πν 3 /(3c 3 ). With ν = E/ h
and N 0 = dN (E)/dE and considering c → c/n r we obtain the given value for N 0 .
10 Recombination
the direct transition is energetically fairly close to the fundamental, indirect L– band edge transition
(Fig. 9.15). The energy difference can be reduced from its bulk value of 136 meV by tensile strain.
Additionally, the direct transition can be favored by heavily n-doping and filling the L conduction
band minimum states (see Sect. 9.9.2). In this case, direct recombination from the conduction band
-minimum can be observed [939] and the effective energy difference has been lowered to about
100 meV.
10.2.2 Absorption
A similar consideration is made for the absorption process (Fig. 10.1b). An electron is transferred upon
light absorption from a valence-band state (occupied) to a conduction-band state that must be empty.
The coefficient is B 1 . Also, the process is proportional to the light intensity, represented by the density
of occupied photon states N ph (E),
r abs (E) =
E+E V
E C
dE e B 1 (E e , E e − E) ×
(10.4)
D e (E e ) (1 − f e (E e )) D h (E e − E) (1 − f h (E e − E)) N ph (E) .
10.2.3 Stimulated Emission
In this case, an incoming photon ‘triggers’ the transition of an electron in the conduction band into an
empty state in the valence band. The emitted photon is in phase with the initial photon (Fig. 10.1c).
The rate is (with coefficient B 2 ):
r st (E) =
E+E V
E C
dE e B 2 (E e , E e − E) ×
(10.5)
D e (E e ) f n (E e ) D h (E e − E) f h (E e − E) N ph (E) .
The photon density N ph at a given energy is given by Planck’s law and the Bose–Einstein distribution
(Appendix E)
N ph (E) = N 0
1
exp (E/kT ) − 1
.
(10.6)
The pre-factor is the density of states of the electromagnetic field
3 N 0 (E) = 8 π E
2
(n r / hc)
3 .
10.2.4 Net Recombination Rate
In thermodynamical equilibrium the rates fulfill
3 The total number of photon states in vacuum between the frequencies zero and ν is N (ν) = 8πν 3 /(3c 3 ). With ν = E/ h
and N 0 = dN (E)/dE and considering c → c/n r we obtain the given value for N 0 .