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 which 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_eff² · 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_eff² · 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_eff² · 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_eff² ·	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

R_se + R_sp	=	R_fa

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_eff² · u · Dn · A_fa · f(E) · (1 – f(E + hn) A_se · N_eff² · 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_ref³(hn)² h³ · c³ · 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

A_fa	=	A_se

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_ref³ · (hn)² · A_se 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/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^e₀ + Dn^e

n^h	=	n^h₀ + Dn^h

n^e₀ · n^h₀	=	n_i²

We then have the cases

Dn^e	=	Dn << n^e₀, n^h₀

i.e almost equilibrium, and

Dn	>>	n^e₀, n^h

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^e₀ · n^h₀)

for D» 0, or

R^eq_sp	=	B_sp · n_i²

For non-equilibrium, which is the condition we are ususally considering, so we drop the index on R_sp, we have generally

R_sp	=	B_sp(n^e₀ + Dn) · (n^h₀ + Dn)

	=	B_sp[n_i² + Dn · (n^e₀ + n^h₀ + Dn

	=	R^eq_sp + B_sp · Dn · (n^e₀ + n^h₀ + 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^e₀ + n^h₀ + 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^e₀ + n^h) we have

t^li_sp	=	1 B_sp · (n^e₀ + 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 Shockley-Read-Hall formula where we had

t	=	1 v · s · n^maj

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!

To index

6.1.1 LASER conditions

Radiation Equilibrium in Direct Semiconductors

Detailed Derivation of the Inversion Condition

Exercise 6.1-1