“Deep” Levels

“Deep Levels”  “Deep Centers”  “Deep Traps” An old research area for me. My treatment is similar to, but different then YC, Ch. 4. BW mention Deep Levels only briefly in Sect. 5.6, pp 97 & 98. My discussion is partially from my old research notes, published papers, & research talks I’ve given.

Just considering properties of substitutional impurities, the treatment of impurities we’ve seen so far has left out a lot! A general impurity or defect potential V has the form: V  V lr + V sr V lr  the long-ranged, screened Coulomb potential responsible for the “shallow” donor & acceptor levels just discussed. V sr  the short-ranged, “central cell” potential (includes chemical differences of impurity & host atoms + lattice distortion). It is now understood that it is primarily the shortranged, central cell potential V sr which is Responsible for the existence of “deep levels”, as we will see.

11 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 << E g, where E g is the host bandgap. So these impurities are often labeled 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. ––––

11 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)E g from a band edge. 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. ––––

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”.

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. –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.

The Importance of Deep Levels A Contrast of Them with Shallow Levels Shallow Donor & Acceptor Levels (discussed last time): 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 2. Can be introduced in concentrations up to n ~ cm -3  The conductivity can be made to vary over many orders of magnitude: (Ω-cm) -1 ~  σ ~  10 3 (Ω-cm) Can be introduced non-uniformly the into the material.  p-n junctions are possible!  Other device applications are possible!

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: ~ cm -3 to (max) ~ cm -3  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!

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

Some Effects That Deep Level Defects Can Have on Devices They can shorten e - & e + lifetimes. –Good or bad, depending on the application! Bad, for example, for a photocell. Good, for example, for a fast switch! Needed in GaAs fast switches! They can enhance radiative recombination. –Good or bad, depending on the application! Good, for example, if they are purposely introduced in a LED to produce specific color. Bad, for example, if they are of unknown origin and/or uncontrollable! They can degrade device performance! So, they 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!

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  V lr + V sr V lr  Long-ranged, Screened Coulomb Potential which is responsible for the “shallow” donor & acceptor levels. V sr  the Short-ranged, “Central Cell” Potential which includes the Chemical Differences in the Impurity & Host Atoms (which, as we’ll see, is often the dominant effect) + Lattice Relaxation or Distortion.

The general defect potential: V  V lr + V sr It is now well-understood & accepted (see BW & YC) that Deep Levels are Produced by the Chort- ranged,Central Cell Part of the Defect Potential V sr !  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!

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. V lr = The long ranged, screened Coulomb potential. The resulting electron wavefunctions Ψ are hydrogen-like & (relatively) spread out in r space.  “Shallow” levels V sr = The short ranged, central cell potential. The resulting electron wavefunctions Ψ are (relatively) localized in r space.  “Deep” levels

SO using this Jargon “Shallow Levels” are produced by the long ranged potential V lr & have wavefunctions  which are spread out in the direct lattice. “Deep Levels” are produced by the short-ranged potential V sr & have wavefunctions  which are localized in the direct lattice. 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! But, this Theory Jargon goes even further! It says that  “Shallow deep levels are possible”! # # J.D. Dow ~ 1979 or 1980

Schematically, it helps to think about a one-dimensional model lattice, such as the Krönig-Penney Model from our bandstructure discussions: l Consider a defect at one site. The Defect Potential is V  V lr + V sr The simple deep level theory discussed next makes assumption: V  V sr –That is, in this theory, the long- ranged V lr (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) Defect Site  V lr V sr   V sr

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! It was due originally to H.P. Hjalmarson, PhD Dissertation, U. of Illinois, 1980 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”.

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

This qualitative model can obtain features of deep level physics which are present in the actual calculations discussed later. In the figure shown, the illustration was for substitutional impurities for A 1, or s-like defect levels. However, it can easily also treat T 2 or p-like levels & more complex defects. A main result from this qualitative model is 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)

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 2. The HYPER-DEEP LEVEL # This is resonant with the valence band. It is an impurity-host molecule bonding level with an IMPURITY-LIKE WAVEFUNCTION # J.D. Dow ~ 1979 or 1980