1. Semiconductors and Light
A non-standard way to deal with it
Assumes confidence with
Basic Solid State Physics:
Crystals
Electrons, Holes, direct-indirect Band Gap
Their foundation on the Bloch’s Theorem
pn junction DC characteristics
Basic light-matter interaction theory
Quantum-mechanic perturbation theory
Spontaneous and stimulated emission, absorption

2. Semiconductors and Light
Does NOT aim to be complete
Aims to build up a self-consistent view of:
Electron, Hole and Photon densities
Their link with measurable quantities: I, V, P
Their call for technological solutions

3. Semiconductors and Light
It gives a new insight on:
Threshold current
Gain and loss coefficients
The theoretical foundation of known empirical formulas
It should be compared with literature.
Coldren LA, Corzine SW, Mašanović ML. Diode lasers and photonic integrated circuits, Wiley series in microwave and optical engineering. Jhon Wiley & Sons, Inc., Hoboken, New Jersey; Second Edition 2012.
J.T. Verdeyen. Laser Electronics. Third Edition, Prentice Hall, 1995.

4. Semiconductors and Light
Absorption
Emission
Partial reflection
Transparency
Refraction Eg n
Easily explained by Eg:
Absorbption
Transparency
Refraction
Partial Reflection

6. Diamond has the closest lattice spacing
It is
Mechanically Hard
Electrically Insulating
Optically Transparent
Optically Refractive

7. … but Light Emission is another tale
Fermi Golden Rule
(holds for any system)
Momentum Selection Rule
(specific for crystals)
Si excluded from light emitting devices
Recombination rate is no more
It must obey the selection rules
High probability
Low probability

8. Something may be anticipated about light emission in direct gap semiconductors
Few electrons
Few holes
Few photons
Many electrons
Many holes
Many photons
No electrons
No holes
No photons h𝜈 Eg 𝜙𝜈
= photon density
Typical ≈kT
Typical 40kT

9. How to get many electrons and many holes together?
By injecting a forward current into a pn junction
But in an ideal diode, they meet inside the depletion layer without recombining.
Then recombine, separately, entering the neutral regions

10. This is a Light Emitting Diode = LED
The practical solution is to force e-h recombination inside the depletion layer introducing a thin layer at smaller bandap between the p and n regions.
In a well designed device, a forward bias V will cause a current I to flow and an optical power POUT to leave the structure.
This is a Light Emitting Diode = LED

11. Possibility for population inversion
But something more may be anticipated about light emission in direct gap semiconductors
More empty than filled states
More filled than empty states
More filled than empty states
More empty than filled states 𝜙n
Possibility for population inversion 𝜙p
More top-down (emitting) than bottom-up (absorbing) transitions stimulated by light.
GAIN = Light Amplification
The LED achieves Light Amplification by Stimulated Emission of Radiation
This is a LASER DIODE

12. Form qualitative to quantitative:
we should:
List all mechanisms involving photons and electrons and holes
Balance them
Bring optical properties properties inside the world of diodes:
Spectrum 𝜙𝜈, gain g, loss α, optical power POUT
Bring diode properties properties inside the world of lasers:
Current I, Voltage V
Include concurring phenomena non involving photons
Find lumped equations for V, I and POUT
The starting point will be a Rate Equation for photons

13. Our program: Consider a Double Heterostructure ideal diode
made of direct gap semiconductor
where all recombinations are radiative
and happens only inside the central active layer qV p n
depletion layer
active layer
Inside that layer all electron, holes and photon densities are uniform
We will look first for the electro-optical characteristics of such ideal diode
Non-radiative recombination inside the layer
and later on, we will include
Other recombinations and currents outside the layer

14. A quick recall of the Einstein’s treatment of Black Body radiation (1905)
Search for the spectral power density u𝜈 at equilibrium
Fermi’s Golden rule not yet discovered
No Fermi-Dirac or Bose-Einstein statistics available: only Boltzmann. No exclusion principle
No quanta. Planck (1901) not sure of their existence. Einstein going to explain the phototelectric effect on the same year (1905)
Classical results from Thermodynamics
Stefan’s Law
Wien’s (Displacement) Law
Rayleigh and Jeans’ Ultraviolet Catastrophe . But good for low 𝜈.

15. 2-level system with N0 particles
Energy
Density of states
Population E2 g2
Level 2 E1 g1
Level 1
Rate of spontaneous 2→1 transitions
Rate of stimulated 2→1 transitions
Rate of stimulated 1→2 transitions
Coefficients A and B can depend on 𝜈 but not on T
At equilibrium 2→1 = 1 →2

16. At high temperature:
Increase with T4
Do not change with T
Go to unity
Wien’s (Displacement) Law
Do not include T

17. Must be (Rayleigh and Jeans):
Planck’s Law

20. Top-down optical transitions proportional to
Bottom-up optical transitions proportional to

23. The diode at equilibrium must give back the Black Body formula
Conduction-to-valence spontaneous transitions (e-h recombination, photon emission)
Conduction-to-valence stimulated transitions(e-h recombination, photon emission)
Valence-to-conduction stimulated transitions (e-h generation, photon absorption)
At equilibrium V=0
As for Einstein’s treatment:

24. Out of equilibrium, rates can be not constant, V≠0 and an escape term must be introduced, that must vanish at equilibrium
In the steady state:
Using the previous forms
we can solve for

25. B may be and 𝜙0𝜈 is a slow function of h𝜈.
We can safely assume for both:
B and 𝜙0𝜈 are constant
where h𝜈 Eg 𝜙𝜈
= photon density
𝜙0𝜈
Is the joint density of states for electrons and holes. It is null for
and is linear in for thick layers
and constant for thin layers (the normal case)
𝜏C is the average permanence time of radiation inside the active layer. It is a function of 𝜈 because of refraction and resonances. But we start keeping it constant.

26. Let us suppose Eg= 1 eV and qV=0.5 eV
Thick layer case. Linear vertical scale. Pay attention to the values in the abscissa
calculated
Linear scale h𝜈 Eg 𝜙𝜈
= photon density
Expected (qualitative)

27. Let us now change qV:
Thick layer case. Log vertical scale. Pay attention to the values of qV
Logarithmic scale

28. Thin layer case. Log vertical scale. Pay attention to the values of qV
Logarithmic scale
At qV=1.02 eV
Something happens when qV approaches Eg

29. For possible emission, we have
exponentials at the denominator are extremely small

30. But the denominator vanishes as
The total optical power
increases unlimited
Infinite energy is required to further increase qV.
Voltage clamps at a threshold value
that is the minimum among the ones allowed
For a thin layer this is

31. What is it the happening?
Resc
Rsp
Rst
Rabs
qVth Eg
Spontaneous emission and absorption dominate
over stimulated emission. It is the LED regime.
Stimulated emission balances absorption.
It the the transparency condition.
Stimulated emission overcomes absorption.
Light starts to be amplified (super-radiance)
Spontaneous emission is blocked. Voltage is calmped.
Stimulated emission dominates. It is the LASER regime.

32. The more the bathtube is filled, the higher is the output fluxqV
PTOT
But when the edge is reached, the flux can increase without increasing the water level

33. Current in the ideal laser diode
The current is q times the net recombination rate.
For the ideal LED/Laser diode,
where recombination is always radiative
Integration factor. h𝜈 Eg 𝜙𝜈
= photon density

34. This is the DC transfer function I(V) of an ideal laser diode.
Equation of an ideal diode with saturation current =
V<Vth: LED range
Laser region
V=Vth: Laser range
Shockley region (LED)
This is the DC transfer function I(V) of an ideal laser diode.

35. Optical Power in the ideal laser diode
Optical Power=Photon Density x Photon Energy x Volume / Lifetime
But:
Iph
POUT
Collection efficiency (coupling+conversion+…)

36. Optical Power in the real laser diode
Iph is not the only current
measured
For V<Vth
non-radiative recombination dominates
A non-radiative current Inr exists
qVth
Ith
The total current is made of I= Inr + Iph
As qV clamps at qVth,
Inr stops, and Iph grows alone
Quantum Efficiency:
A threshold current Ith defines the transiton
In practise:

37. Threshold current

38. What do textbooks tell? for any V,I for V> Vth (that is I> Ith )
Collected escaping power
Total escaping power
What do textbooks tell?
Conversion efficiency (often omitted)
My opinion: a big mistake. It must go to unity for I> Ith

39. Lumped equations for the DC regime
I(V), POUT(V) → direct substitution
POUT(I) → eliminate
is analytic under the form I(POUT)
calculates and embeds the threshold condition
deals with the sub-thresholdregime
It comes out a huge formula, but it