1. Deep Level Theory Y.-T. Shen, PhD, TTU, 1986
Extended Hjalmarson theory to treat COMPLEXES of various kinds (still no relaxation!)
Vacancy-impurity complexes in many materials.
Vacancy-impurity pairs.
Triplets: vacancy + two impurities.
Pairing: Can cause shallow levels to move deeper & deep levels to move shallower!

2. Shen & Myles Y.-T. Shen, PhD, TTU, 1986
Hjalmarson diagram for pairs (Ga vacancy, 2 impurities on nn P sites)

3. Myles, Bylander, Williams Application to understanding data in Hg1-xCdxTe (1985) & GaAs1-xPx (1990)
Vacancy-impurity complexes in semiconductor alloys!
Motivations:
Hg1-xCdxTe:
Alloy of HgTe (semimetal) &
CdTe (semiconductor; Eg ~ 1.6 eV)
Vary x & make materials with
Eg ~ 1.6 eV to 0.0 eV (near x ~0.16)
For x ~ 0.2, Eg ~ infrared  IR detector (several billion $$/year!)
GaAs1-xPx:
Alloy of GaAs (Eg ~ 1.55 eV, direct)
& GaP (Eg ~ 1.6 eV).
Light emitting diodes (red for
calculators) if contains N impurity
(deep level). (Several billion $$/year!)

4. Myles, Bylander, Williams Understanding data in Hg1-xCdxTe & GaAs1-xPx
To treat bandstructures of (random) alloy hosts (AxB1-xC):
Virtual Crystal Approximation:
Ho(alloy) = x Ho(AC) + (1-x) Ho(BC)
Not correct! Correct treatment of this (disordered alloys) would be a completely new set of lectures!

5. Myles, Bylander, Williams Understanding data in Hg1-xCdxTe Journal of Applied Physics (1985)
Vacancy-Impurity pair levels in
Hg1-xCdxTe

6. Myles, Bylander, Williams Understanding data in Hg1-xCdxTe Journal of Applied Physics (1985)
Vacancy-Impurity complex levels (Shen  triplets) in Hg1-xCdxTe
Many similar calculations found that slope of a deep level as a function of x in Hg1-xCdxTe is characteristic of:
 Impurity site in the vacancy- impurity complex
“Vacancy-like” levels have very different slopes than “impurity- like” levels (or isolated impurities)
Level symmetry (A1, T2, etc.)
 Possible use as a tool in data analysis!

7. Myles, Bylander, Williams Understanding data in Hg1-xCdxTe Journal of Applied Physics (1985)

8. Myles, Bylander, Williams Understanding data in Hg1-xCdxTe Journal of Applied Physics (1985)

9. Myles, Bylander, Shen Understanding data in GaAs1-xPx Journal of Applied Physics (1990)

10. Summary Complexes in Hg1-xCdxTe & GaAs1-xPx
Major findings:
Slopes of levels (dE/dx) can be correlated with impurity site. For both complexes & isolated impurities.
dE/dx is different for cation site & anion site impurities & complexes.
Site information can be obtained from dE/dx (as shown in theory-experiment comparison).
Vacancy-like, complex-produced levels have very different dE/dx than isolated impurities.

11. Summary Complexes in Hg1-xCdxTe & GaAs1-xPx
Major findings (continued):
dE/dx of impurity-like, complex produced levels is similar to that of isolated impurities.
dE/dx also depends on level symmetry.
dE/dx ~ same, for all levels of the same type, independent of impurity.

12. Summary Complexes in Hg1-xCdxTe & GaAs1-xPx
Important results! A potential means of distinguishing between isolated impurities & vacancy impurity complexes based on dE/dx !
An aid to defect identification (at least for some levels!).
dE/dx  “Signature” of deep level for site identification purposes. (In the spirit of the Hjalmarson theory’s chemical trends idea)

13. Summary Hjalmarson Theory & generalizations: Alloys & Vacancy-impurity complexes
dE/dx can be different for complexes & impurities
 Possible means of distinguishing them in data
dE/dx does depend on site
 Site information
dE/dx  Level “signature”
Theory can be used, with data, to help to identify defects producing deep levels in semiconductor alloys.
More generally: Hjalmarson theory is good at predicting & explaining “Chemical Trends”

14. Hjalmarson Theory Li & Myles: generalization to include lattice relaxation
Hjalmarson theory does a good job at explaining chemical trends, but suffers from a lack of accuracy.
One effect left out: Lattice Relaxation.
W.-G. Li, PhD, TTU, 1991 generalized Hjalmarson theory to include lattice relaxation.
Tightbinding based. Molecular dynamics to compute relaxation.
Ability to easily & accurately predict chemical trends is preserved!
Accuracy considerably improved!

15. Hjalmarson Theory Li & Myles: inclusion of lattice relaxation
Tested on neutral, sp3 bonded, substitutional impurities .
Could be generalized for complexes & charge state effects.
Includes effects of both first and second neighbor relaxation.
Uses a molecular dynamics (MD, F=ma) technique to calculate relaxation around an impurity.
Forces for MD calculation:
Attractive part: Obtained directly from electronic structure (including impurity) using the Hellmann-Feynman theorem.
Repulsive part: obtained from a pair potential (Harrison).

16. Li & Myles Theory Lattice Relaxation
Formally, similar to Hjalmarson theory: Deep level E in bandgap is solution to Sch rdinger Eq. in the form:
det [1 - Go(E)V]= (*)
Go(E)  (E- Ho )-1
= Host Green’s Function Operator
Ho = Host Hamiltonian (same tightbinding bands as before)
Defect potential V :
Due to lattice relaxation, is no longer a diagonal matrix.

17. Li & Myles Theory Lattice Relaxation
Defect potential
V V(U,,)
U  diagonal part. Assume
 atomic energy difference, as before
 contains chemical trends, as in Hjalmarson theory
Ul  l [(I)l - (H)l]
,   off-diagonal parts. Couple impurity to host atoms (tightbinding, LCAO type coupling). Like Vss, etc., but impurity to host matrix elements
  coupling to first neighbors
  coupling to second neighbors

18. Li & Myles Theory Lattice Relaxation
,  depend on interatomic distances
Essentially differences in impurity & host atom tightbinding overlap matrix element interactions
 CONTAIN EFFECTS OF LATTICE RELAXATION!
det [1 - Go(E)V]=0 (*)
Using Hjalmarson theory ideas:
Interesting to treat (*) as implicit equation:
E = E(U,,)

19. Li & Myles Theory Lattice Relaxation
det [1 - Go(E)V]=0 (*)
Becomes implicit equation:
E = E(U,,)
By treating ,  as parameters & computing E = E(U,,), trends in deep levels can be explored:
Chemical trends (as in Hjalmarson theory)
Ul  l [(I)l - (H)l]
Trends with varying amounts of lattice relaxation: Varying  & .

20. Li & Myles Theory Lattice Relaxation
“Breathing mode” relaxations only!
To determine ,  , use Harrison’s universal bond-length squared scaling law (see Ch. 3 of YC): For l= s,p,d, ….
l  - Cl [(1/dI)2 - (1/dH)2]
l  - Cl [(1/dII)2 - (1/dH)2]
dI  impurity - nn bond length
dII  impurity - 2nd nn bond length
dH  host - nn bond length
Cl  empirical constant, determined by host bandstructures.

21. Li & Myles Theory Lattice Relaxation
l  - Cl [(1/dI)2 - (1/dH)2]
l  - Cl [(1/dII)2 - (1/dH)2]
Often, set l = 0 in what follows:
After calculations, found it has a very small effect on results!
Can compute deep levels
E = E(U,,)
By treating , as parameters OR
By treating dI , dII , dH as parameters (or use experimental values) OR
Determine dI , dII , by molecular dynamics scheme.

22. Li & Myles Theory Lattice Relaxation
l  - Cl [(1/dI)2 - (1/dH)2]
l  - Cl [(1/dII)2 - (1/dH)2]
NOTE: For Cl0 :
l  0  dI  dH (outward relaxation)
l  0  dI  dH (inward relaxation)
Similar for l

23. Li & Myles Results “Hjalmarson diagram” (A1, s-like levels) P site impurity in GaP Chemical trends & trends with  (=0)

24. Li & Myles Results “Hjalmarson diagram” (A1, s-like levels) Ga site impurity in GaP Chemical trends & trends with  (=0)

25. Li & Myles Results D(E) =det [1 - Go(E)V]=0 Gives trends in deep level E as function of  & . A1, s-like levels (Si:Se)

26. Li & Myles Results D(E) =det [1 - Go(E)V]=0 Gives trends in deep level E as function of  & . A1, s-like levels (GaP:O)

27. Li & Myles Results D(E) =det [1 - Go(E)V]=0 Deep level E as function of dI/dH (=0). A1, s-like levels (Si:S, Si:Se, Si:Te)

28. Li & Myles Theory
To get specific dI & dII, use “Molecular Dynamics” (MD).
MD  Solve F = ma for atomic motion (allowed by Hellmann-Feynman theorem & Born-Oppenheimer Approximation)!
MD outline:
N atoms
For each atom (say atom i): Hellmann-Feynman theorem states, total force on it is:
Fi = -iEtot (r1,r2,r3,…., rN)
Etot (r1,r2,r3,…., rN) = Total energy
Difficulties: 1) classical many body problem, 2) What is Fi ? (What is Etot?)

29. Li & Myles Theory MD calculations to determine bond lengths
Replace host atom by impurity. Initially, surrounding host atoms are in original equilibrium positions.
 They experience a force. Each component:
Fx = (Fx)r + (Fx)a
(Fx)r  Repulsive part. Many electron effect. Compute from pair potential (Harrison).
(Fx)a  Attractive part. From occupied one electron levels. Compute as follows (Hellmann- Feynman theorem).

30. Li & Myles Theory MD calculations to determine bond lengths
Compute attractive part (Fx)a from electronic structure (including impurity) using Hellmann-Feynman theorem (R. Feynman, Phys. Rev., BS thesis, MIT!):
(Fx)a = -(Etot/x) = - [E(E)dE]/x
(E)= density of states, including impurity
Integral is over valence bands up to Fermi energy.

31. Li & Myles Theory MD calculations to determine bond lengths
Begin with host atoms in perfect crystal positions.
Use forces computed as just mentioned to calculate position motion of each atom over small time interval t.
Fi = mai (MD, Newton’s 2nd law)
Integration of 2nd law forward in time: standard numerical schemes.
First & second neighbor atoms relaxing  17 atoms in motion.

32. Li & Myles Theory MD calculations to determine bond lengths & deep levels
After t (~ s), calculate new positions, new bond lengths dI & dII, new l , l , new V, new forces.
Repeat for successive t until total force acting on each atom is zero (new equilibrium).
Calculate final bond lengths dI & dII, final l , l , final V
Put these into deep level problem & solve
det [1 - Go(E)V]=0
17 atoms, 4 orbitals per atom
 68 X 68 matrix

33. t ~ 10-14 s. Units of force: eV/Å
Li & Myles Results N on P site in GaP Force vs. time for one nn Ga atom
t ~ s. Units of force: eV/Å

34. Li & Myles Results N and O on P site in GaP Force vs
Li & Myles Results N and O on P site in GaP Force vs. nn distance for one Ga atom

35. Li & Myles Results MD results for bond lengths

36. Li & Myles Results D(E) =det [1 - Go(E)V]=0  &  fixed by MD A1, s-like levels (Si:Te, Si:Se, Si:S)

37. Li & Myles Results Deep level results, including lattice relaxation
Compare MD results with experiment!

38. Deep Level Theory Summary
Hjalmarson deep level theory, including lattice relaxation:
Retains its ability to predict trends. Chemical trends ~ unchanged.
Has its accuracy considerably improved in comparison with experiment.
2nd neighbor relaxation effects on deep levels are small (1st neighbor relaxation is primary effect)

39. Deep Level Theory Summary
Hjalmarson deep level theory, including lattice relaxation:
A1 (s-like) levels move deeper into bandgap with outward nn relaxation & towards conduction band with inward relaxation.
T2 (p-like) levels: opposite occurs!
Lattice relaxation effects on deep levels are “small”, but can be important in determining deep levels!

40. Deep Level Theory Conclusion
Simple deep level theory outlined here is useful for:
Predicting trends in deep levels
Helping to interpret data on unknown deep levels
Predicting level depths with reasonable accuracy (with lattice relaxation included.)