304
10 Recombination
Fig. 10.1 Processes of
band–band recombination:
a spontaneous emission, b
absorption and c stimulated
emission. A full (empty)
circle represents an
occupied (unoccupied)
electron state
E=h
E=h
E=h
E=h
E=h
v
E
g
E
c
E
(c)
(b)
(a)
The other time constant is τ D = the dielectric relaxation time; it describes the transport of carriers
due to mobility (and diffusion). Large dielectric relaxation times are present in semiconductors with
high mobility (low defect density, small carrier mass), small τ D typically for hopping conduction.
The relaxation case is given for τ 0 τ D ; carriers will recombine quickly and it is hard to build up
non-equilibrium carriers and separate them with an applied electric field. In the recombination case
τ D τ 0 , non-equilibrium carriers can assume non-uniform distributions and an applied electrical field
generates separate quasi-Fermi levels for electrons and holes.
1 (cmp. Sect. 7.6).
10.2 Band–Band Recombination
The band–band recombination is the relaxation from an electron in the conduction band into the valence
(the empty state there is the hole). In a direct semiconductor, electrons can make an optical transition
between the bottom of the conduction band to the top of the valence band. In an indirect semiconductor,
this process is only possible with the assistance of a phonon and is thus much less probable.
10.2.1 Spontaneous Emission
We consider the spontaneous recombination of an electron of energy E e and a hole of energy E h
(Fig. 10.1a). C(E e , E h ) is a constant proportional to the matrix element of the optical transition (cf.
Sect. 9.6). The spontaneous recombination rate r sp at photon energy E ≥ E C − E V = E g is (assuming
energy conservation, i.e. E = E e − E h , but without k-conservation in a dense plasma [938]),
r sp (E) =
∞
E C
dE e
E V
−∞
dE h C(E e , E h ) ×
(10.2)
D e (E e ) f e (E e ) D h (E h ) f h (E h ) δ(E − E e + E h )
=
E+E V
E C
dE e C(E e , E e − E) ×
D e (E e ) f e (E e ) D h (E e − E) f h (E e − E) ,
where f h denotes the hole occupation f h = 1 − f e .
1 In the relaxation case, the separation of quasi-Fermi levels is kT .
10 Recombination
Fig. 10.1 Processes of
band–band recombination:
a spontaneous emission, b
absorption and c stimulated
emission. A full (empty)
circle represents an
occupied (unoccupied)
electron state
E=h
E=h
E=h
E=h
E=h
v
E
g
E
c
E
(c)
(b)
(a)
The other time constant is τ D = the dielectric relaxation time; it describes the transport of carriers
due to mobility (and diffusion). Large dielectric relaxation times are present in semiconductors with
high mobility (low defect density, small carrier mass), small τ D typically for hopping conduction.
The relaxation case is given for τ 0 τ D ; carriers will recombine quickly and it is hard to build up
non-equilibrium carriers and separate them with an applied electric field. In the recombination case
τ D τ 0 , non-equilibrium carriers can assume non-uniform distributions and an applied electrical field
generates separate quasi-Fermi levels for electrons and holes.
1 (cmp. Sect. 7.6).
10.2 Band–Band Recombination
The band–band recombination is the relaxation from an electron in the conduction band into the valence
(the empty state there is the hole). In a direct semiconductor, electrons can make an optical transition
between the bottom of the conduction band to the top of the valence band. In an indirect semiconductor,
this process is only possible with the assistance of a phonon and is thus much less probable.
10.2.1 Spontaneous Emission
We consider the spontaneous recombination of an electron of energy E e and a hole of energy E h
(Fig. 10.1a). C(E e , E h ) is a constant proportional to the matrix element of the optical transition (cf.
Sect. 9.6). The spontaneous recombination rate r sp at photon energy E ≥ E C − E V = E g is (assuming
energy conservation, i.e. E = E e − E h , but without k-conservation in a dense plasma [938]),
r sp (E) =
∞
E C
dE e
E V
−∞
dE h C(E e , E h ) ×
(10.2)
D e (E e ) f e (E e ) D h (E h ) f h (E h ) δ(E − E e + E h )
=
E+E V
E C
dE e C(E e , E e − E) ×
D e (E e ) f e (E e ) D h (E e − E) f h (E e − E) ,
where f h denotes the hole occupation f h = 1 − f e .
1 In the relaxation case, the separation of quasi-Fermi levels is kT .