Deep Level Theory
(Hjalmarson, et al.)
Generalizations, &
Applications
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
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!
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- 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]
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).
• 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).
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
Photoluminescence of the N Deep Level in GaAs:N Under 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.
Hjalmarson Theory-Chemical
Trends with hydrostatic pressure.
N Deep
Level


Phonon
Side Bands
• 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.
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.
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!
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!
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.
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
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
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
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.