SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

4.9. LIGHT EMITTING DIODE (LED) 201

10 –5

10 –6

10 –7

10 –8

10 –9

10 –10

1014 1015 1016 1017 1018

Nd (for holes injected into an n-type semiconductor)

n = p (for excess electron-hole pairs injected into a region)

R

ADIATIVE LIFETIME

(

τr

)(s)

Typical carrier

densities for laser

operation

Low

injection

regime

τo

Carrier

occupation is

degenerate

fe = fh = 1

Semiconductor GaAs

Temperature is 300 K

1019 cm–3

Figure 4.35: Radiative lifetimes of electrons or holes in a direct gap semiconductor as a function
of doping or excess charge. The figure gives the lifetimes of a minority charge (a hole) injected
into ann-type material. The figure also gives the lifetime behavior of electron-hole recombina-
tion when excess electrons and holes are injected into a material as a function of excess carrier
concentration.

iv.Aregimethatisquiteimportantforlaseroperationisonewheresufficientelectronsand

holesareinjectedintothesemiconductortocause“inversion.” As will be discussed later,

this occurs iffe+fh≥ 1. If we make the approximationfe∼fh=1/ 2 for all the

electrons and holes at inversion, we get the relation

Rspon∼

n

4 τo

(4.9.12)

or the radiative lifetime at inversion is

τ∼

τo

4

(4.9.13)

This value is a reasonable estimate for the spontaneous emission rate in lasers near thresh-

old.