3.1. BASIC CONCEPTS 79
In thermal equilibrium,ρemshould be identical with the spectral density of blackbody
radiation given byPlanck’s formula[18]
ρem=8 πhν^3 /c^3
exp(hν/kBT)− 1. (3.1.5)
A comparison of Eqs. (3.1.4) and (3.1.5) provides the relations
A=( 8 πhν^3 /c^3 )B; B′=B. (3.1.6)These relations were first obtained by Einstein [17]. For this reason,AandBare called
Einstein’s coefficients.
Two important conclusions can be drawn from Eqs. (3.1.1)–(3.1.6). First,Rsponcan
exceed bothRstimandRabsconsiderably ifkBT>hν. Thermal sources operate in this
regime. Second, for radiation in the visible or near-infrared region (hν∼1 eV), spon-
taneous emission always dominates over stimulated emission in thermal equilibrium at
room temperature (kBT≈25 meV) because
Rstim/Rspon=[exp(hν/kBT)− 1 ]−^1 1. (3.1.7)Thus, all lasers must operate away from thermal equilibrium. This is achieved by
pumping lasers with an external energy source.
Even for an atomic system pumped externally, stimulated emission may not be
the dominant process since it has to compete with the absorption process. Rstimcan
exceedRabsonly whenN 2 >N 1. This condition is referred to aspopulation inversion
and is never realized for systems in thermal equilibrium [see Eq. (3.1.2)]. Population
inversion is a prerequisite for laser operation. In atomic systems, it is achieved by using
three- and four-level pumping schemes [18] such that an external energy source raises
the atomic population from the ground state to an excited state lying above the energy
stateE 2 in Fig. 3.1.
The emission and absorption rates in semiconductors should take into account the
energy bands associated with a semiconductor [5]. Figure 3.2 shows the emission pro-
cess schematically using the simplest band structure, consisting of parabolic conduc-
tion and valence bands in the energy–wave-vector space (E–kdiagram). Spontaneous
emission can occur only if the energy stateE 2 is occupied by an electron and the energy
stateE 1 is empty (i.e., occupied by a hole). The occupation probability for electrons in
the conduction and valence bands is given by theFermi–Dirac distributions[5]
fc(E 2 )={ 1 +exp[(E 2 −Efc)/kBT]}−^1 , (3.1.8)
fv(E 1 )={ 1 +exp[(E 1 −Efv)/kBT]}−^1 , (3.1.9)whereEfcandEfvare the Fermi levels. The total spontaneous emission rate at a
frequencyωis obtained by summing over all possible transitions between the two
bands such thatE 2 −E 1 =Eem=h ̄ω, whereω= 2 πν, ̄h=h/ 2 π, andEemis the
energy of the emitted photon. The result is
Rspon(ω)=∫∞EcA(E 1 ,E 2 )fc(E 2 )[ 1 −fv(E 1 )]ρcvdE 2 , (3.1.10)