1. 10.1.1 Degenerate Semiconductors
the semiconductor materials, we have discussed so far have been fairly pure
doping represented a small fraction of the total atomic density
1018 impurities/cm3 compared with ~51022 atoms/cm3 for Si
since the impurity atoms were so widely spaced we could assume that there was no charge transport within the impurity levels themselves
as we increase the doping levels, eventually we must begin to consider the interactions between the impurity levels
the impurity levels are no longer discrete, but rather begin to form a band
these impurity bands may even overlap the conduction or valence bands
Example: donors present in high concentrations > 1020 cm-3 are so close together that the donor states form a band which may overlap the bottom of the conduction band

2. if the concentration of electrons in the conduction band, n, exceeds the effective density of states, Nc
an equivalent situation exists for the case where the acceptor impurity concentration is large, in this case, the material is called degenerate p-type, and the Fermi level lies within the valence band
energy states below EF are mostly filled, and states above EF are mostly empty
in a degenerate n-type material, the region between Ec and EF is mostly filled with electrons
in a degenerate p-type material, the region between Ev and EF is mostly filled with holes
EFn
EFp Ec Ev Ei Eg
then the Fermi level is no longer within the band gap
it lies somewhere in the conduction band
when this occurs, the material is called degenerate n-type

3. Equilibrium of a p-n junction formed between degenerate materials
Transition region
- At equilibrium, within the transition region, there are few electrons in the conduction band, and few holes in the valence band

4. 8.4.1 Population Inversion at a Junction
- Let us now apply a “large enough” forward bias to the junction
the quantity (Fn – Fp) is a measure of the departure from equilibrium at a particular location

5. if the bias voltage (and thus the current) is large enough, then electrons and holes are injected into and across the transition region
this was also the case for an ordinary p-n junction
only now the injected carrier concentrations are quite considerable
the region around the transition region is far from being depleted of charge carriers
in fact, the transition region now has a large concentration of electrons in the conduction band, and a large concentration of holes in the valence band
if these population densities are high enough, then a condition of Population Inversion exists
the region around the junction over which a population inversion exists is called an Inversion Region
the steady-state carrier concentrations are given by:
(4-15)

6. - Let us take a closer look at the inversion regionFn Fp Ec Ev Ei Eg
(Fn – Fp)
The condition for population inversion must take into account the distribution of energies available for transitions between the bands
For dominance of stimulated emission between two energy levels separated by an energy hν, the population of the upper level must be larger than that of the lower level
unlike the two-level case, in semiconductors, there are bands of levels available for transitions

7. Transitions between levels from up to Fn in the conduction band down to as low as Fp in the valence band may occur
For any transition energy hν in a semiconductor, a population inversion exists when:
(Fn – Fp) > hν
- For band-to-band transitions, the minimum requirement for population inversion occurs for photons with hν = Eg
(Fn – Fp) > Eg
When Fn and Fp lie within their respective bands, stimulated emission can dominate over a range of transitions from hν = (Fn – Fp) to hν = Eg
The dominant transitions for LASER action are primarily determined by the resonant cavity and the strong recombination radiation occurring near hν = Eg

8. - Recall that in order to obtain LASER action we require:
A large photon field energy density ρ(ν) in order to make stimulated emission dominant over spontaneous emission
an optical resonant cavity is used to allow the photon density to build up to a large value
A population inversion in order to make stimulated emission dominant over absorption
A population inversion is accomplished by applying a bias current through the junction
The material used for junction LASER fabrication must:
be a direct semiconductor (ie: no dominant trapping processes)
be able to be doped both n-type and p-type
We must also be able to construct a resonant cavity in the junction region

9. A population inversion is accomplished by applying a bias current through the junction

10. 8.4.2 Emission Spectra for p-n Junction LASERs
Under forward bias, an inversion layer (large electron population at same location as a large hole population) can be established along the plane of the junction
an electron having an energy between Fn and Ec can directly recombine with a hole having energy between Ev and Fp
the resulting photons can have any energy Fn – Fp ≥ hν ≥ Eg
the photon wavelengths that participate in stimulated emission are determined by the length of the resonant optical cavity
(a) incoherent emission below threshold
(b) LASER modes at threshold
(c) dominant LASER mode above threshold
Note: the intensity scale is greatly compressed from (a) to (b) to (c)

11. 8.4.3 The Basic Semiconductor LASER