1. “Deep” Levels

2. “Deep Levels”  “Deep Centers”
 “Deep Traps”
An old research area for me.
My discussion is partially from old research notes, published papers, & research talks I’ve given.

3. Vlr  long-ranged, screened Vsr  short-ranged, “central
Just considering properties of substitutional impurities, the treatment of impurities we’ve seen so far has left out a lot!
A general impurity (defect) potential V
has the form: V  Vlr + Vsr
Vlr  long-ranged, screened
Coulomb potential
This causes the “shallow” donor & acceptor levels just discussed.
Vsr  short-ranged, “central
cell” potential (includes chemical differences of impurity & host atoms + lattice distortion).

4. A general impurity (defect) potential V has the form: V  Vlr + Vsr
It’s now understood that it’s primarily the short-ranged, central cell potential Vsr which is
Responsible for the existence of “deep levels”.

5. “Deep Levels” or “Deep Centers”
Reminder of our recent discussion of
“shallow” donors & acceptors:
The binding energies E of these donor &
acceptor impurities are typically < 100 meV
(away from one of the band edges).
So, E << Eg, (Eg is the host bandgap).
So these impurities are often labeled
Shallow Impurities or
Shallow Levels

6. MANY defects in MANY materials.
Shallow Impurities or Shallow Levels
Also, as discussed earlier, an
Effective H Atom Model
can be used to obtain an understanding of the
basic physics of these defect energies E. In
addition, a more sophisticated & accurate theory,
Effective Mass Theory (EMT)
(the Effective Mass Approximation)
has been very successful at predicting the defect
energy levels E. It obtains results which are in
excellent agreement with experiment on
MANY defects in MANY materials.

7. Deep Centers or Deep Levels.
The earliest understanding was that
defects which produce energy levels E
where EMT is not valid & is not able to
quantitatively explain the defect energy
levels E were known as
Deep Centers or Deep Levels.
Earlier, it was assumed that these
defects always produced levels E in the
host bandgap of the order of ~ (½)Eg
from a band edge.

8. The more recent, modern understanding,
which we’ll now discuss, is that energy
levels E produced by Deep Centers may
have energies E in the bandgap which can
be close to either the conduction band edge
or the valence band edge.
It turns out that, for such defects, lattice
relaxation (or distortion) effects are often
important, but are most often still not the
dominant effect which explains the observed
defect energy levels E.

9. So, EMT & its generalizations work well for shallow levels, but FAIL for “deep” levels.
In the previous statement, the conventional (experimental) definition of “deep” was used.
“Deep”  E  ~ 150 meV
from the conduction band or valence band edges
(somewhere in the middle of the bandgap)
In a few minutes, another (theoretical) definition of “deep” will be used which actually contradicts the usual meaning of the English word “deep”.

10. We’ve already discussed many motivations for needing a theoretical understanding of defects & impurities in semiconductors.
Many of these are also motivations to understand deep levels theoretically: Semiconductor properties are strongly influenced by defects with which produce deep levels in the bandgap.
So, a theoretical understanding of deep levels is important technologically.
But, it also contains very interesting basic physics.
There are many observed deep levels in many materials. Even now, little is known about the origin of many of these.

11. First Principles  Semi-empirical
There are also many deep level theories
First Principles  Semi-empirical
We’ll outline a relatively simple, semi-empirical theory that gets the basic physics correct, while not necessarily getting all quantitative predictions correctly. Then, we’ll discuss various applications of it in attempts to understand a variety of data in a variety of materials.

12. Importance of Deep Levels Contrast of Them with Shallow Levels
Shallow Donor & Acceptor Levels
(discussed earlier):
1. Mainly control the conductivity, as we will see. For electron number density n, velocity v in external electric field E, the current density j (Ohm’s Law) is given in terms of the conductivity σ as: j = nev = σE

13. Importance of Deep Levels Contrast of Them with Shallow Levels
Shallow Donor & Acceptor Levels
(discussed earlier):
2. Can be introduced in concentrations up to
n ~ 1020 cm-3  The conductivity can be made to vary over many orders of magnitude: (Ω-cm)-1 ~  σ ~  103 (Ω-cm)-1
3. Can be introduced non-uniformly the into the material.
 p-n junctions are possible!
 Other device applications are possible!

14. Deep level defects play a very different role than shallow level impurities.
Concentrations of defects producing them are usually much smaller than for shallow levels:
~ 1012 cm-3 to (max) ~ 1018 cm-3
So, Deep Levels usually have a negligible effect on the electrical conductivity. Their concentrations are usually too small to affect the electron number density n.
So, Ohm’s Law j = nev = σE is usually not
affected very much by them. However, deep levels in the bandgap can either help or hinder device performance, depending on the energy level & on the device!

15. Deep Level Defects: Can act as centers for e--e+ recombination
“Deep Traps”
 They can have a very Strong effect on the material optical & electronic properties!
Fig. 8, from Ch. 8 of the book
Deep Levels in Semiconductors
by Milan Jaros (Adam Hilger, 1982).

16. Some Effects of Deep Levels on Devices
They can shorten e- & e+ lifetimes.
This can be good or bad, depending on the application!
This is Bad, for example, for a photocell.
This is Good, for example, for a fast switch!
They are used in some GaAs fast switches!

17.  Deep Levels can degrade device performance!
Some Effects of Deep Levels on Devices
They can enhance radiative recombination.
This can be good or bad, depending on the application!
This is Good, for example, if they are purposely introduced in a LED to produce specific color. This is Bad, for example, if they are of unknown origin and/or uncontrollable!
 Deep Levels can degrade device performance!

18. “Radiative Killer Centers”
Due to their potential bad effects on optoelectronic devices,
Deep Level Defects
are sometimes known as
“Radiative Killer Centers”
 Deep Levels need to be controlled for many device applications.
A first step in controlling them is understanding them!

19. Vlr  Long-ranged, screened Vsr  Short-ranged, “Central
The Modern Understanding of the Basic Picture of the Physics of Deep Levels
only emerged starting in the early 1980’s.
As we’ve said, a general impurity or defect potential has form:
V  Vlr + Vsr
Vlr  Long-ranged, screened
Coulomb Potential
responsible for the “shallow” donor & acceptor levels.
Vsr  Short-ranged, “Central
Cell” Potential
includes the Chemical Differences in the Impurity & Host Atoms + Lattice Relaxation or Distortion.

20. The general defect potential:
V  Vlr + Vsr
It is now well-understood & accepted (see BW & YC):
Deep Levels are Produced by the short
ranged, Central Cell Part of the Defect
Potential Vsr!
 Deep levels & shallow levels are in
opposite physical regimes from each other.
Not surprisingly,
The physics of deep levels is very different from that of shallow levels!

21. it emphasizes the part of the defect
Since the early 1980’s (H.P. Hjalmarson, PhD dissertation, U. of Illinois, a HUGE number of papers using his theory), & thanks to J.D. Dow, in the literature on deep levels people have often used theorist’s definitions of the terms “deep” & “shallow”, which may be contrary to the ordinary English usage of these words!
This theory jargon ignores the energy level depth in the bandgap. Instead,
it emphasizes the part of the defect
potential V which produces the level.

22.  “Deep” levels it emphasizes the part of the defect
This theory jargon ignores the energy level depth in the bandgap. Instead,
it emphasizes the part of the defect
potential V which produces the level.
Vlr = The long ranged, screened Coulomb potential. The resulting electron wavefunctions Ψ are hydrogen-like & (relatively) spread out in r space.  “Shallow” levels
Vsr = The short ranged, central cell potential.
The resulting electron wavefunctions Ψ are (relatively) localized in r space.
 “Deep” levels

23. SO using this Jargon
“Shallow Levels” are produced by the long ranged potential Vlr & have wavefunctions  which are spread out in the direct lattice.
“Deep Levels” are produced by the short-ranged potential Vsr & have wavefunctions  which are localized in the direct lattice.

24. SO using this Jargon
But, this Theory Jargon goes even further! It says that
If the short-ranged, central cell potential alone produces a level, it is, by definition, a “DEEP” level, whether or not it is energetically
deep in the bandgap!
It could, be energetically shallow, or even resonant with one of the bands!

25.  “Shallow deep levels are possible”!#
But, this Theory Jargon goes even further! It says that
 “Shallow deep levels are possible”!#
#J.D. Dow ~ 1979 or 1980

26. Schematically, it helps to think about a one-dimensional model lattice, such as the Krönig-Penney Model from our bandstructure discussions:
Defect Site
Vsr
Vsr
 Vlr
Consider a defect at one site.
The Defect Potential is V  Vlr + Vsr l

27. The simple deep level theory discussed
Schematically, it helps to think about a one-dimensional model lattice, such as the Krönig-Penney Model from our bandstructure discussions:
Defect Site
Vsr
Vsr
 Vlr
Consider a defect at one site.
The Defect Potential is V  Vlr + Vsr
The simple deep level theory discussed
next makes the assumption: V  Vsr l

28.  There are no shallow (H-atom-like) levels in this theory!
Consider a defect at one site. The Defect Potential is V  Vlr + Vsr
The simple deep level theory discussed
next makes the assumption: V  Vsr
That is, in this theory, the long- ranged Vlr (screened Coulomb) is ignored.
 There are no shallow (H-atom-like) levels in this theory!
(They could be added back later, if we
wanted, using EMT) l

29. Molecular (LCAO) Model of Deep Levels
The model discussed next is not a quantitative model & not the actual theory to be described.
It is qualitative, in order to help gain an understanding of the physics of deep levels.
It gets the qualitative physics of deep level defects correctly!!!
It is analogous (in this way only!) to the simple “H-atom” model for shallow levels!

30. Molecular (LCAO) Deep Level Model
This model, as well as the full detailed accurate theory was due originally to H.P. Hjalmarson, PhD Dissertation, U. of Illinois, (+ PhD advisor, Jack Dow & collaborator, Peter Vogl).
The quantitative theory discussed later takes the “Chemist’s Viewpoint” of a solid & treats the bands in the
Tightbinding (LCAO) Approximation.
However, to obtain a first understanding of deep level defects, this LCAO model first considers the bonding & antibonding levels of a “host molecule” & a “defect molecule”.

31. Molecular (LCAO) Deep Level Model
From the book
Deep Levels in
Semiconductors
by M. Jaros
(Adam Hilger,
1982).
Originally discussed by H.P. Hjalmarson,
PhD Dissertation U. of Illinois 1980

32. Molecular (LCAO) Deep Level Model
From Deep Levels in
Semiconductors
by M. Jaros (Adam
Hilger, 1982).

33. This qualitative model can obtain features of deep level physics which are present in the actual calculations discussed later.
In this figure, the illustration is for substitutional impurities for A1, or s-like defect levels.
However, it can easily also treat T2 or p-like levels & more complex defects.

34. A primary result from this
qualitative model:
If the impurity is more electronegative than the host atom it replaced (in which case there will be an antibonding level below the host antibonding levels), it will produce a level in the host bandgap.
This Completely Explains “Chemical Trends” (discussed later)

35. HOST-LIKE WAVEFUNCTION
More results from this
qualitative model:
There are two energy levels due to the defect:
1. The DEEP LEVEL
This is the observed impurity level. It is in or near the host bandgap & is derived mostly from host conduction band states.
It is an impurity-host molecule anti-bonding level with a
HOST-LIKE WAVEFUNCTION

36. 2. The HYPER-DEEP LEVEL# IMPURITY-LIKE WAVEFUNCTION
More results from this
qualitative model:
There are two energy levels due to the defect
2. The HYPER-DEEP LEVEL#
#J.D. Dow ~ 1979 or 1980
This is resonant with the valence band.
It is an impurity-host molecule bonding level with an
IMPURITY-LIKE
WAVEFUNCTION