1. Deep Level Theory (Hjalmarson, et al.) Generalizations, & Applications

2. Trends in the ordering of deep levels
As we’ve discussed, the Hjalmarson et al.
theory was designed to predict & explain
Chemical Trends in deep levels.
Trends in the ordering of deep levels
1. Due to different defects in the same host.
2. Due to the same defect in different hosts
(e.g. as the alloy composition changes in alloys).
 A GLOBAL THEORY
of Deep Level Defects

3. Hjalmarson et al. Theory & Chemical Trends
It is somewhat crude quantitatively, but it is now understood that it contains the
Correct Qualitative Physics of Deep Levels.
Further, it is a GLOBAL THEORY OF Chemical
Trends in Deep Levels: Ordering of Deep Levels
It was designed to be useful in
A. Predicting, for a given host, which impurities will
produce deep levels & which will not.
B. Sorting out data on deep levels of unknown origin.
C. Understanding the dependence of deep levels on the
composition x in semiconductor alloys like A1-xBxC.
As we’ll see, it was QUITE SUCCESSFUL in this in
comparison with large amounts of data!

4. Hjalmarson Theory Hundreds of Predictions Trends!
“Hjalmarson Diagram” From Y.C. Ch. 4 . Originally from H. Hjalmarson PhD Dissertation, U. of Illinois, 1980
Hundreds of
Predictions
of Chemical
Trends!
Recall theory details
discussed previously.
Look for solutions to the
Schrödinger
Equation in the form:
det[1 - (E- Ho )-1V] = 0. Also, the Central Cell Potential V is diagonal
(no lattice relaxation) & the diagonal matrix elements have the form
Vℓ = βℓ[(εI)ℓ - (εH)ℓ]

5. N in GaAs1-xPx An Example of a “Good” Deep Center
The short-ranged potential means that the wavefunction in r space will be highly localized around the N.
 The electron wavefunction is spread out in k-space.
Although GaP is an indirect bandgap material, the optical transition is very strong in GaP:N
 Red LED’s used to be made from GaP:N
It turns out that a large amount of N can be introduced into GaP but only small amount of N can be introduced into GaAs because of a larger difference in atomic sizes.
The N impurity in GaP is a “good” deep center because it makes GaP:N into a material which is useful for light-emitting diodes (LED).

6. GaP has an indirect band gap so, pure GaP is not a good material for LED’s (Si & Ge also aren’t for the same reason).
It turns out that the presence of N actually enhances the optical transition from the conduction band to the N level which makes GaP:N an efficient emitter.
So, GaP:N was one of the earliest materials for red LED’s.
More recently, GaP:N has been replaced by the more efficient emitter: GaInP (alloy).

7. GaAs1-xPx:N Interesting, beautiful data!
A very useful aspect of Hjalmarson Theory: Chemical Trends as a function of alloy composition.
The N impurity level is a deep level in the bandgap in GaP but is a level resonant in the conduction band in GaAs.
The figure is photoluminescence
data (Wolford, Streetman, et al.) in GaAsxP1-x:N for various alloy
compositions x.
Obviously, the theoretical depth is wrong, but the slope as a function of x is ~ correct. 13

8. Photoluminescence of the N Deep Level in GaAs:N Under Hydrostatic Pressure
Hjalmarson Theory-Chemical Trends with hydrostatic pressure.
Data (Wolford, et al.) in GaAs:N.
At atmospheric pressure, the N level is resonant in the conduction band in GaAs. As the pressure increases, the conduction band minimum at the Γ- point moves up, while the minimum at the X-point moves down. Direct to indirect bandgap crossover at P ~ 40 kbar.
Also, the N deep level comes out of the conduction band at P ~ 30 kbar!!
Obviously, the theoretical slope as a function of P is ~ correct.
N Deep
Level  
Phonon
Side Bands

9. This theory is crude, but it is now known that it gets the essential physics of deep levels correctly.
The predicted level depths are often in disagreement with experiment by ~ eV.
It’s ability to predict Chemical Trends means that it could be used to help to sort out data!
Over the years, various refinements, corrections, generalizations have been made. Some of these will be discussed next. Most of these move the levels by
~  0.1 to 0.2 eV.

10. Charge State Effects: Ren, Hu, Sankey, Dow, 1982
Hjalmarson Theory neglects “Charge State Effects”:
Deep levels depend on the charge state of the defect. The original theory assumption was neutral defects.
 The defect potential V had no Coulomb effects in it.
Ren et al. added e- - e- coupling. This is straightforward, but tedious. The results are that:
1. The predicted Chemical Trends are unchanged.
2. Shifts in the level depths due to charge state effects are
ΔE ~ 0.1 eV per electron charge.

11. Charge State Effects: Ren, Hu, Sankey, Dow, 1982
ENDOR Data on S in Si
A measurement of the spatial extent of the
impurity charge density: ρ  |Ψ|2
Deep Level Theory fails at large R.
Consistent with the assumption of spatial
localization. EMT is valid at large R!

12. Deep Levels Due to Impurity Pairs Sankey, Hjalmarson & Dow, 1982
Hjalmarson Theory, but for nearest-neighbor impurity pairs.
Same ideas, but a larger defect potential matrix V!
Use group theory to classify the defect states.
Included vacancy-impurity pairs.
A beginning to the treatment of complexes!

13. Sankey, Hjalmarson & Dow, 1982 Qualitative Physics: Vacancy Impurity Pairs
The simplest Vacancy-Impurity Complex: The vacancy-impurity pair.
Figure: The P-Vacancy pair in Si.
Pairing can cause shallow levels to move deeper & deep levels to become shallower.

14. Vacancy-Impurity Pairs in Si: Sankey, Hjalmarson & Dow
A1 or s-like
Levels
Schrödinger Equation solutions:
det[1 - (E- Ho)-1V] = 0
for Vacancy-Impurity (V,X) Pairs in Si

15. Impurity Pairs in Si: Sankey, Hjalmarson & Dow
A1 & T2 (s-like) Levels
Schrödinger Equation solutions:
det[1 - (E- Ho)-1V] = 0
for Impurity (X,X) Pairs in Si

16. Impurity Pairs in GaP: Sankey, Hjalmarson & Dow
A1 (s-like)
Levels
Solid Dots () are experimental data.
Schrödinger Equation
solutions:
det[1 - (E- Ho)-1V] = 0
for Impurity (X,O) Pairs in GaP

17. Impurity Pairs in GaAsxP1-x: Sankey, Hjalmarson & Dow
A1 (s-like) Levels
Schrödinger Equation
solutions:
det[1 - (E- Ho)-1V] = 0
for (Zn,O) & (V,O)
Pairs in GaAsxP1-x
Solid Dots () are experimental data.