SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

4.9. LIGHT EMITTING DIODE (LED) 199

CONDUCTION

BAND

VALENCE

BAND

Eg

Ec

Ev

k

h^2 k^2

2m*

h^2 k^2

2m*

hω

e

h

Figure 4.34: A schematic of theE-kdiagram for the conduction and valence bands. Optical
transitions are vertical; i.e., thek-vector of the electron in the valence band and in the conduction
band is the same.

and the recombination time becomes (ωis expressed in electron volts)

τo=

0. 67

ω[eV]

ns (4.9.6)

Therecombinationtimediscussedaboveistheshortestpossiblespontaneousemissiontime
sincewehaveassumedthattheelectronhasaunitprobabilityoffindingaholewiththesame
k-value.
When carriers are injected into the semiconductors, the occupation probabilities for the elec-
tron and hole states are given by the appropriate quasi-Fermi levels. The emitted photons leave
the device volume so that the photon density never becomes high in thee-hrecombination re-
gion. In a laser diode the situation is different, as we shall see later. The photon emission rate is
given by integrating the emission rateWemover all the electron-hole pairs after introducing the
appropriate Fermi functions.