
Again, we come back to the question: Do (direct)
semiconductors glow
in the dark? 


The answer was yes – but only to the extent that all
(black) bodies glow in the dark, following
Plancks famous
formula. 


Here we will look at this question in a different way that
also will allow us to obtain the Einstein coefficients. 

Instead of looking at the equilibrium distribution
of all kinds of radiation in a "black body", we now consider only the
frequencies prevalent in direct semiconductors, i.e. radiation with hn »
E_{g}. We then have the three basic processes between
electrons (and holes) and radiation: 

Fundamental absorption 


The rate R_{fa} with which fundamental
absorption takes place was given by (we use the
simple version) 



R_{fa} 
= 
A_{fa} ·
N_{eff}^{2} · u(n) · Dn · 
æ
è 
1 – f_{h in V} (E
^{v}, E_{F}^{h}, T) 
ö
ø 
· 
æ
è 
1 – f_{e in C} (E
^{c}, E_{F}^{e}, T) 
ö
ø 




Since we now consider thermal equilibrium, we have
E_{F}^{h} = E_{F}^{e} =
E_{F}. We also can replace 1 – f_{h
in V}(E^{v}, E_{F}^{h},
T) by f(E, E_{F}, T) because the
probability of not finding a hole at
E ^{v} = E is equal to the probability of finding
an electron; and f_{h in V}(E ^{v},
E_{F}^{h}, T) can be written as by 1
– f(E, E_{F}, T). Moreover,
wherever we have f_{e in C}, we simply substitute by
f(E + hn,
E_{F},T). This yields 



R_{fa} 
= 
A_{fa} ·
N_{eff}^{2} · u(n) · Dn · 
æ
è 
f (E, E_{F}, T) 
ö
ø 
· 
æ
è 
1 – f (E + hn, E_{F}, T) 
ö
ø 



Stimulated emission. 


The rate R_{se} for stimulated emission
(in the form rewritten for equilibrium exactly as above) was 


R_{se} 
= 
A_{se} ·
N_{eff}^{2} · u(n) · Dn · 
æ
è 
f (E + hn ,
E_{F}, T) 
ö
ø 
· 
æ
è 
1 – f (E , E_{F},
T) 
ö
ø 



Spontaneous emission. 


We have not yet considered the rate
R_{sp} for spontaneous emission in the
same formalism as the other two, but that is easy now. We have 


R_{sp} 
= 
B_{sp}·N_{eff}^{2}
· 
æ
è 
f(E + hn,
E_{F}, T) 
ö
ø 
· 
æ
è 
1 – f(E, E_{F}, T) 
ö
ø 



Combining everything gives a surprisingly simple
equation for R_{sp}: 


R_{sp} 
= 
R_{se} · B_{sp}
A_{se} · u(n) ·
Dn 



Thermodynamic equilibrium now demands that the
number of photons produced must be equal to the number of photons absorbed. In
other words, the sum of the emission rates must equal the absorption rate, or






Inserting the equation for R_{sp} yields



R_{fa} –
R_{se} 
= 
R_{se} ·
B_{sp}
A_{se} · u(n) ·
Dn 




R_{fa}
R_{se} 
= 
B_{sp}
A_{se} · u(n) ·
Dn 





From this we obtain 


u(n) · Dn = 
æ
ç
è 
A_{se} · R_{fa}
B_{sp} · R_{se} 
– 
A_{se}
B_{sp} 
ö
÷
ø 
–1 



All we have to do now is to insert all the lengthy
equations we derived for the rates. The math required for that is easy, but
tedious. 


For ease of writing we now drop all indices and
functionalities which are not desparately needed, insert the equations for
R_{fa} and R_{se}, and obtain 


u(n) ·
Dn = 
A_{se} ·
N_{eff}^{2} · u · Dn · A_{fa}
· f(E) · (1 – f(E + hn)
A_{se} · N_{eff}^{2} · u
· Dn ·
B_{sp} · f(E + hn) ·
[1 – f(E)] 
– 
A_{se}
B_{sp} 




Now insert the Fermi distribution and shuffle once more 
good
exercise!  , and you get 


u(n) · Dn = 
B_{sp}
A_{fa} · exp (hn/kT)
– A_{se} 



We now have an equation for the density of photons
at some particular frequencies defined by the semiconductor. However, we have
not made any specific assumptions about this frequency except that it is in
thermodynamic equilibrium 


This requires that u(n)
· Dn obtained in
this special way must be precisely
identical to the radiation density as expressed in Plancks
fundamental formula (which was derived in
another advanced
module) and we have 


8p ·
n_{ref}^{3}(hn)^{2}
h^{3} · c^{3} · exp (hn/kT) – 1 
· d(hn) = 
B_{sp}
A_{fa} · exp (hn/kT)
– A_{se} 



With this equation we have
reached our goal and proved that 





Can you see
why? Well  the equation thus must be valid at all temperatures. This is only possible if
A_{fa} = A_{se}! Think about it! 

Using this equality we finally obtain 


B_{sp} = 
8p · n_{ref}^{3} ·
(hn)^{2} · A_{se}
h^{3} · c^{3} 



This is an important, if slightly sad equation. It
says that the Einstein coefficient of spontaneous emission is some constant
times the Einstein coefficient of
stimulated emission times the square of the frequency. 


In other words: At frequencies high enough,
spontaneous emission always wins  it will be hard to
make an Xray Laser! 


Unfortunately, the result we obtained does not change by doing
more fancy math, e.g. by using the more precise equation for the transition
rates from the advanced module. We have
to live with it. 

We could go on now. After all, spontaneous
emission is a recombination channel that we have treated before  in
chapter 2 and
chapter 5. 


In any case we simply
had for the net
recombination rate U = Dn/t and U was the net recombination
rate. For the fraction that recombines via spontaneous radiation, we simply
have to take the lifetime t for that process
and obtain 


U = Dn/t_{sp}. 

On the other hand, the definition of the
spontaneous emission rated from above can be
rewritten as 


R_{sp} 
= 
B_{sp} · n^{e} ·
n^{h} 




because the effective density of states times the relevant
Fermi distribution gives simply the density of electrons and holes in their
bands. 


The density of carriers we write, as ever so often, as 


n^{e} 
= 
n^{e}_{0} +
Dn^{e} 



n^{h} 
= 
n^{h}_{0} +
Dn^{h} 



n^{e}_{0} ·
n^{h}_{0} 
= 
n_{i}^{2} 




We then have the cases 


Dn^{e} 
= 
Dn <<
n^{e}_{0}, n^{h}_{0} 




i.e almost equilibrium, and 





i.e. the
high injection
case. 

For the rate of spontaneous recombination, we then
may distinguish the extreme cases of near equilibrium (Dn » = 0, and
Dn >> n_{min} and
express this in rates of spontaneous recombination. For Dn = 0 we would have equilibrium with a
recombination rate for the spontaneous recombination of 


R^{eq}_{sp} 
= 
B_{sp}(n^{e}_{0} ·
n^{h}_{0}) 




for D» 0, or 


R^{eq}_{sp} 
= 
B_{sp} · n_{i}^{2} 



For nonequilibrium, which is the condition we are
ususally considering, so we drop the index on R_{sp}, we
have generally 


R_{sp} 
= 
B_{sp}(n^{e}_{0} +
Dn) · (n^{h}_{0}
+ Dn) 




= 
B_{sp}[n_{i}^{2}
+ Dn ·
(n^{e}_{0} +
n^{h}_{0} + Dn 




= 
R^{eq}_{sp} +
B_{sp} · Dn ·
(n^{e}_{0} +
n^{h}_{0} + Dn) 




R^{eq}_{sp} becomes negligible
as soon as Dn >>
n_{min} which is not yet
high injection and which we will have in all interesting cases. We thus finally
approximately 


R_{sp} 
» 
B_{sp} · Dn ·
(n^{e}_{0} +
n^{h}_{0} + Dn) 



Equating these expression with the simple formula
R_{sp} = Dn/t_{sp} under all
conditions, we can now express the life time in terms of the
Einstein coefficient and the carrier concentration. 


For low injection conditions, i.e. relatively small Dn meaning
R^{li}_{sp}»
B_{sp} · Dn ·
(n^{e}_{0} + n^{h}) we have 


t^{li}_{sp} 
= 
1
B_{sp} · (n^{e}_{0} +
n^{h}) 




For high injection, i.e Dn >> n_{maj}, meaning
R^{hi}_{sp} »
B_{sp} · Dn·(Dn), we have 


t^{hi}_{sp} 
= 
1
B_{sp} · Dn 



This compares favorably with our
old
ShockleyReadHall formula where we had 





with v = thermal velocity and s = capture cross section . 

Here some circle closes. But we
will delve no more into this subject but simply remember: The Einstein
coefficients of stimulated emission and fundamental absorption are identical
for very fundamental reasons! 

