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

2. As we’ve discussed, the Hjalmarson et al. theory was designed to predict & 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 A 1-x B x C. As we’ll see, it was QUITE SUCCESSFUL in this in comparison with large amounts of data!

4. Hjalmarson Theory “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- H o ) -1 V] = 0. Also, the Central Cell Potential V is diagonal (no lattice relaxation) & the diagonal matrix elements have the form V ℓ = β ℓ [(ε I ) ℓ - (ε H ) ℓ ]

5. 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). N in GaAs 1-x P x An Example of a “Good” Deep Center

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. 13 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 GaAs x P 1-x :N for various alloy compositions x. Obviously, the theoretical depth is wrong, but the slope as a function of x is ~ correct. GaAs 1-x P x :N Interesting, beautiful data! A very useful aspect of Hjalmarson Theory: Chemical Trends as a function of alloy composition.

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

12. 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! Deep Levels Due to Impurity Pairs Sankey, Hjalmarson & Dow, 1982

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 Schrödinger Equation solutions: det[1 - (E- H o ) -1 V] = 0 for Vacancy-Impurity (V,X) Pairs in Si A 1 or s-like Levels

15. Impurity Pairs in Si: Sankey, Hjalmarson & Dow A 1 & T 2 (s-like) Levels Schrödinger Equation solutions: det[1 - (E- H o ) -1 V] = 0 for Impurity (X,X) Pairs in Si

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

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