Einstein Coefficients

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 » Eg. We then have the three basic processes between electrons (and holes) and radiation:
Fundamental absorption
The rate Rfa with which fundamental absorption takes place was given by (we use the simple version)
  Rfa  =  Afa · Neff2 · u(n) · Dn · æ
è
1  –  fh in V (E v, EFh, T) ö
ø
 ·  æ
è
1  –  fe in C (E c, EFe, T) ö
ø
Since we now consider thermal equilibrium, we have EFh = EFe = EF. We also can replace 1  –  fh in V(Ev, EFh, T) by f(E, EF, T) because the probability of not finding a hole at E v = E is equal to the probability of finding an electron; and fh in V(E v, EFh, T) can be written as by 1  –  f(E, EF, T). Moreover, wherever we have fe in C, we simply substitute by f(E + hn, EF,T). This yields
  Rfa  =  Afa · Neff2 · u(n) · Dn · æ
è
f (E, EF, T) ö
ø
 ·  æ
è
1  – f (E  +  hn, EF, T) ö
ø
Stimulated emission.
The rate Rse for stimulated emission (in the form rewritten for equilibrium exactly as above) was
Rse  =  Ase · Neff2 · u(n) · Dn · æ
è
f (E  +  hn , EF, T) ö
ø
 ·  æ
è
1  –  f (E , EF, T) ö
ø
Spontaneous emission.
We have not yet considered the rate Rsp for spontaneous emission in the same formalism as the other two, but that is easy now. We have
Rsp  =  Bsp·Neff2 · æ
è
f(E + hn, EF, T) ö
ø
 ·   æ
è
1 – f(E, EF, T) ö
ø
Combining everything gives a surprisingly simple equation for Rsp:
Rsp  =  Rse · Bsp
Ase · 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
Rse + Rsp  =  Rfa
Inserting the equation for Rsp yields
Rfa  –  Rse  =  Rse · Bsp
Ase · u(n) · Dn
     
      Rfa
Rse
  = Bsp
Ase · u(n) · Dn
   
From this we obtain
u(n) · Dn  =   æ
ç
è
Ase · Rfa
Bsp · Rse
  –  Ase
Bsp
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 Rfa and Rse, and obtain
u(n) · Dn  =  Ase · Neff2 · u · Dn · Afa · f(E) · (1 – f(E + hn)
Ase · Neff2 · u · Dn · Bsp · f(E + hn) · [1 – f(E)]
  –  Ase
Bsp
Now insert the Fermi distribution and shuffle once more - good exercise! - , and you get
u(n) · Dn  =   Bsp
Afa · exp (hn/kT)  –  Ase
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 · nref3(hn)2
h3 · c3 · exp (hn/kT)  –  1
· d(hn)  =   Bsp
Afa · exp (hn/kT)  –   Ase
With this equation we have reached our goal and proved that

Afa  =  Ase

Can you see why? Well - the equation thus must be valid at all temperatures. This is only possible if Afa = Ase! Think about it!
Using this equality we finally obtain

Bsp  =   8p · nref3 · (hn)2 · Ase
h3 · c3

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 X-ray 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/tsp.
On the other hand, the definition of the spontaneous emission rated from above can be rewritten as
Rsp  =  Bsp · ne · nh
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
ne  =  ne0  +  Dne
     
nh  =  nh0  +  Dnh
     
ne0 · nh0  =  ni2
We then have the cases
Dne  =  Dn  <<  ne0, nh0
i.e almost equilibrium, and
Dn  >>  ne0,  nh
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 >> nmin and express this in rates of spontaneous recombination. For Dn = 0 we would have equilibrium with a recombination rate for the spontaneous recombination of
Reqsp  =  Bsp(ne0 · nh0)
for D» 0, or
Reqsp  =  Bsp · ni2
For non-equilibrium, which is the condition we are ususally considering, so we drop the index on Rsp, we have generally
Rsp  =  Bsp(ne0  +  Dn) · (nh0  +  Dn)
     
   =  Bsp[ni2  +  Dn · (ne0  +  nh0  +  Dn
     
   =  Reqsp  +  Bsp · Dn · (ne0  +  nh0  +  Dn)
Reqsp becomes negligible as soon as Dn >> nmin which is not yet high injection and which we will have in all interesting cases. We thus finally approximately
Rsp  »  Bsp · Dn · (ne0  +  nh0  +  Dn)
Equating these expression with the simple formula Rsp = Dn/tsp 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 Rlisp» Bsp · Dn · (ne0 + nh) we have
tlisp  =  1
Bsp · (ne0  +  nh)
For high injection, i.e Dn >> nmaj, meaning Rhisp » Bsp · Dn·(Dn), we have
thisp  =  1
Bsp · Dn
This compares favorably with our old Shockley-Read-Hall formula where we had
t  =  1
v · s · nmaj
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!
 

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go to 6.1.1 LASER conditions

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go to Exercise 6.1-1

© H. Föll (Semiconductor - Script)