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

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

2. As we’ve discussed, theHjalmarson et al. theorywas designed topredict & explain Chemical Trends in deep levels. Chemical Trends 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 in deep levels. 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 Chemical Trends: Ordering of Deep Levels 1. Due to different defects in the same host. 2. Due to the same defect in different hosts 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 Diagram” From Y.C. Ch. 4 . Originally from H. Hjalmarson PhD Dissertation, U. of Illinois, 1980 Hjalmarson Theory Hundreds of Predictions of Chemical Trends! Recall theory details discussed previously. Look for solutions to the Schrödinger Equationin the form: det[1 - (E- Ho )-1V] = 0. Also, the Central Cell PotentialV is diagonal (no lattice relaxation) & the diagonal matrix elements have the form Vℓ = βℓ[(εI)ℓ- (εH)ℓ]

5. N in GaAs1-xPxAn 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:Ninto a material which is useful for light-emitting diodes (LED).

6. GaPhas an indirect band gap so, pure GaPis 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:Nan efficient emitter. • So, GaP:N was one of the earliest materials for red LED’s. • More recently, GaP:Nhas been replaced by the more efficient emitter: GaInP(alloy).

7. GaAs1-xPx:NInteresting, beautiful data! A very useful aspect of Hjalmarson Theory: Chemical Trendsas 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:Nfor various alloy • compositions x. • Obviously, the theoretical depth is wrong,but the slope as a function of xis ~ correct. 13

8. Photoluminescence of theN Deep LevelinGaAs: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- pointmoves 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 depthsare often in disagreement with experiment by ~ 0.1 - 0.3 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 Theoryneglects“Charge State Effects”: • Deep levels depend on the charge state of the defect. The original theory assumption was neutral defects.  The defect potentialV 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.Shiftsin 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 Dataon 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 PairsSankey, 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, 1982Qualitative Physics: Vacancy Impurity Pairs • The simplestVacancy-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 Equationsolutions: 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 Equationsolutions: 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 Schrödinger Equation solutions: det[1 - (E- Ho)-1V] = 0 for (Zn,O) & (V,O) Pairs in GaAsxP1-x A1 (s-like) Levels Solid Dots () are experimental data.