It was authored by Prof. Peter Y. Yu, Dept. of The following is excerpted from a lecture found on-line. It was authored by Prof. Peter Y. Yu, Dept. of Physics, U. of California at Berkeley

*Peter Y. Yu, U.C.-Berkeley Impurities & Defects Effects on Semiconductor Devices Outline: Point Defects 1. “Shallow” & “Deep Impurities” (“Centers”) 2. Examples of “Deep Centers” a. Isoelectronic centers (“good” defects!) b. Fe in Si (“bad” defects for solar cells). 3. Conclusions “Human beings and semiconductors are interesting because of their defects”* *Peter Y. Yu, U.C.-Berkeley

Defect Classification by their Electronic Properties 1. Donors: Examples: PSi in Si, SiGa in GaAs 2. Acceptors: Examples: BSi in Si, SiAs in GaAs 3. Isovalent or Isoelectronic: Example: NP in GaP 4. Amphoteric: Example: Si in GaAs SiGa is a donor, SiAs is an acceptor. 5. Vacancies & Interstitials

Hydrogenic or “Shallow” Impurities Screened Coulomb Potential: Consider a single donor impurity, such as P substituting for Si. There is one extra valence electron e- in P in comparison to Si. This e- is very weakly bound to P+ by the Screened Coulomb Potential: V = -[(e2)/(εr)] ε = Material dielectric constant. This accounts for the screening of the impurity potential by the valence electrons.

Screened Coulomb Potential: V = -[(e2)/(εr)] Hydrogenic or “Shallow” Impurities Single Donor Impurity The donor e- is very weakly bound to the donor atom Screened Coulomb Potential: V = -[(e2)/(εr)] ε = Dielectric Constant. Accounts for the screening of the impurity potential by the valence electrons. Schrödinger Eqtn. for that e- is the same as that for an Effective “Hydrogen Atom” {(p2)/(2m*) - (e2)/(εr)]}ψ(r) = Eψ(r) m* = Effective mass of e- in the conduction band.

Effective “Hydrogen Atom” Single Donor Impurity: The donor e- is very weakly bound to the donor atom . The Schrödinger Equation for that e- is equivalent to that for an Effective “Hydrogen Atom” {(p2)/(2m*) - (e2)/(εr)]}ψ(r) = Eψ(r) m* = Effective mass of electron in the conduction band. This approach is therefore known as the Effective Mass Approximation It is also known as the Effective Hydrogen Atom Approximation

“Hydgrogenic” Impurity Levels Just as in the Hydrogen (H) Atom, the energy eigenvalues E for the donor energy levels form a Rydberg series. Because of this, they are called “Hydgrogenic” Levels They have the form: En = E – (R*)/(n2) (n = 1,2,3,4,.) where the Effective Rydberg constant is defined as: R* [(m*/m)/(ε)2]R R = Rydberg Constant from the H Atom Problem R  13.6 eV E  Energy when the e- becomes free (ionized in H). This is the conduction band edge in semiconductors. R*  Energy to ionize the electron in the ground state.

“Hydgrogenic” Impurity Levels for a single donor electron have the form: En = E – (R*)/(n2) (n = 1,2,3,4,.) The effective Rydberg constant is R*  [(m*/m)/(ε)2]R E  Energy of the conduction band edge in semiconductors. R*  Energy to ionize the electron in the ground state. Using the above equation in semiconductors. It is, the binding energy of the e- in the 1s level of the donor ion is ~ 10-100 meV that small because, generally m* << m and ε >> 1.

Effective “Bohr radius”: a*  ε (m/m*)a0 “Hydgrogenic” Impurity Levels for a single donor: En = E – (R*)/(n2) (n = 1,2,3,4,.) R*  [(m*/m)/(ε)2]R E  Energy of the conduction band edge in semiconductors. R*  Energy to ionize the electron in the ground state. For this Effective “Hydrogen Atom”, there is also an Effective “Bohr radius”: a*  ε (m/m*)a0 H atom Bohr Radius = a0 = 0.5Å a* is a measure of the average distance that the 1s donor electron can move in the lattice away from the donor atom. Typical numbers give: a* > 10Å. This is a large distance (several lattice spacings)!

“Hydrogenic” Donor Levels hydrogenic donor levels Schematic Diagram of hydrogenic donor levels in the bandgap region of a direct gap material Note that, since the donor levels are a few meV & the band gap Eg is in the eV range, this diagram is obviously NOT to scale!! 8

Comparison: Measured Shallow Donor & Acceptor Levels & Effective Mass Approximation Predictions in Some Semiconductors 9

Lattice Relaxation. 1. The Chemical Nature of the Defect. Two “Local Factors” which help to determine impurity/defect electronic properties are 1. The Chemical Nature of the Defect. 2. The Defect “Size” or Spatial Extent. Impurities with d & f valence electrons tend to retain their atomic nature in the material. That is, the electron is localized & doesn’t travel very far from it’s donor atom. Impurities with a large size difference from the host atoms tend to induce (sometimes very large!) Lattice Relaxation. Defects involving dangling bonds (e.g. vacancies) also tend to induce (sometimes very large!) 10

Deep Levels or “Deep Centers” The binding energies E of hydrogenic donor or acceptor impurities are typically < 100 meV & therefore E << Eg. (Eg is the host material bandgap). So these impurities are also labeled as Shallow Impurities or Shallow Levels. The earliest understanding was that defects which produce energy levels where the Effective Mass (hydrogenic) Approximation is not valid were known as Deep Centers or Deep Levels. 11

The more modern (35+ year old!) Deep Levels or “Deep Centers” It was assumed that these defects always produced levels E in the host band gap of the order of ~ (½)Eg from a band edge. The more modern (35+ year old!) understanding: Deep centers may have energy levels 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 effects can be important. 11

Qualitative Reasons for Lattice Relaxation Host atoms may have to change their equilibrium positions (displace from equilibrium) in order to accommodate defects. 12

Qualitative Reasons for Lattice Relaxation Host atoms may have to change their equilibrium positions (displace from equilibrium) in order to accommodate defects. For example, this happens if the size of the defect is either >> larger than the size of the host atom it displaces or if the impurity atom size is << the size of the host atom. 12

Qualitative Reasons for Lattice Relaxation Host atoms may have to change their equilibrium positions (displace from equilibrium) in order to accommodate defects. For example, this happens if the size of the defect is either >> larger than the size of the host atom it displaces or if the impurity atom size is << the size of the host atom. The most severe case of this happens if there is a vacancy. In that case, the host lattice tends to form new bonds so that there are no “dangling bond” defects left. The figure (next slide) shows a schematic of what happens when there is a vacancy in Si. 12

Qualitative Reasons for Lattice Relaxation The most severe case of this happens if there is a vacancy. In that case, the host lattice tends to form new bonds so that there are no “dangling bond” defects left. The figure shows qualitatively a schematic of what happens when there is a vacancy in Si. 12

Energy Considerations in Lattice Relaxation It costs a small amount in energy to displace atoms (~phonon energy for small displacements) but this more than balanced out by the gain in lowering the electronic energy. The details of all of this are complex & beyond the scope of this course. They involve so-called “Negative & Centers” and Jahn-Teller distortions. There also can be strong electron-phonon coupling as in molecules 13

Energy Considerations in Lattice Relaxation The configuration and the size of the atomic displacements from equilibrium are determined by a complex balance between electronic and lattice energies and therefore are difficult to predict. These displacements also depend strongly on the charge state of the defect. 13

Isoelectronic “Traps” Deep Centers with Small Binding Energies & Lattice Relaxation: Isoelectronic “Traps” Consider a substitutional NAs in GaAs and GaAsP (alloy): N has the same valence as As or P so there is No Coulomb Potential & thus No shallow donor or acceptor Hydrogenic levels! There also are no dangling bonds. Can there be bound states? Yes!! That is, can there be levels in the band gap? Yes!! 14

The impurity potential is short ranged & weak Isoelectronic “Traps” NAs in GaAs and GaAsP (alloy): No Coulomb Potential  No Hydrogenic levels! Can there be levels in the band gap? Yes!!!! Chemically, N is much more electronegative than P & As. The electronegativity of N = 3 and for P = 1.64, respectively. This means that electrons are more attracted to N than to P or As. So, The impurity potential is short ranged & weak  A small binding energy is expected That is, the level in the band gap is expected to be near a band edge. 14

The impurity potential is short ranged & weak Isoelectronic “Traps” NAs in GaAs and GaAsP (alloy): The “Effective H Atom” Model does not apply! No shallow Hydrogenic levels! The impurity potential is short ranged & weak  A small binding energy is expected A simple approximation to this potential would be a δ-function. The potential is weak & highly localized in nature  The EMA does not apply! 14

An Example of a “Good” Deep Center  The electron wavefunction is N in GaP & GaAs: 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. 14

(A historically important problem!) An Example of a “Good” Deep Center N in GaP (A historically important problem!) An Example of a “Good” Deep Center 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 (just as 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. 13

(A historically important problem!) An Example of a “Good” Deep Center N in GaP (A historically important problem!) An Example of a “Good” Deep Center So, GaP:N was one of the earliest materials used for red LED’s. More recently, GaP:N has been replaced by the more efficient emitter: GaInP (alloy). 13

GaAsP Alloy with N Impurities: Interesting, beautiful data! The N impurity level is a deep level in the bandgap in GaP but it is a level resonant in the conduction band in GaAs. The figure is photoluminescence data in the alloy GaAsxP1-x:N for various alloy compositions x. 13