1. Note! The following is excerpted from a lecture found on-line. The original author is Professor Peter Y. Yu Department of Physics University of California at Berkley

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

3. 1. Donors: Examples: P Si in Si, Si Ga in GaAs 2. Acceptors: Examples: B Si in Si, Si As in GaAs 3. Isovalent or Isoelectronic: Example: N P in GaP 4. Amphoteric: Example: Si in GaAs Si Ga is a donor, Si As is an acceptor. 5. Vacancies & Interstitials Defect Classification by their Electronic Properties

4. 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 = -[(e 2 )/(εr)] ε = Material dielectric constant. This accounts for the screening of the impurity potential by the valence electrons. The Schrödinger Equation for that e - is “equivalent” to that for an Effective “Hydrogen Atom” {(p 2 )/(2m * ) - (e 2 )/(εr)]}ψ(r) = Eψ(r) m * = The effective mass of the 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 Hydrogenic or “Shallow” Defects

5. Just as in the Hydrogen 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: E n = E  – (R*)/(n 2 ) (n = 1,2,3,4,.) where the Effective Rydberg constant is R* = (m*/m)(13.6 eV)/(ε) 2 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

6. “Hydgrogenic” Impurity Levels for a single donor electron have the form: E n = E  – (R*)/(n 2 ) (n = 1,2,3,4,.) the effective Rydberg constant is R* = (m*/m)(13.6 eV)/(ε) 2 E   Energy of the conduction band edge in semiconductors. R*  Energy to ionize the electron in the ground state. Using the above equation, the binding energy of the e - in the 1s level of the donor ion is ~ 10-100 meV in semiconductors. It is that small because, generally m* > 1. For this effective “Hydrogen Atom”, there is also an effective “Bohr radius”, a * = ε(0.5Å)(m/m*) 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)! “Hydgrogenic” Impurity Levels

7. 8 “Hydrogenic” Donor Levels Note that, since the donor levels are a few meV & the bandgap E g is in the eV range, this diagram is obviously NOT to scale!! Schematic Diagram of hydrogenic donor levels in the bandgap region of a direct gap material

8. 9 Comparison: Measured Shallow Donor & Acceptor Levels & Effective Mass Approximation Predictions in some compound semiconductors

9. 10 Two “Local Factors” which help to determine defect properties are 1. The chemical nature of the defect 2. The defect “size”. –––– 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!) Lattice Relaxation.

10. 11 Deep Levels or “Deep Centers” The binding energies E of hydrogenic donor or acceptor impurities are typically < 100 meV & therefore E << E g, where E g 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. 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 is that 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. 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 impurity atom 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 shows a schematic of what happens when there is a vacancy in Si.

12. 13 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 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. 14 Deep Centers with Small Binding Energies & Lattice Relaxation Isoelectronic Traps Consider substitutional N As 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? (Can there be levels in the bandgap?) N is much more electronegative than P and 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. This impurity potential is short-ranged and weak  A small binding energy is expected (the level in the bandgap is expected to be near a band edge). A simple approximation to this potential would be a δ-function Although potential is shallow, it is highly localized nature  The EMA is not a good approximation.

14. 14 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.

15. 13 N in GaP 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. 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).

16. 13 The 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 GaAs x P 1-x :N under large hydrostatic pressure for various alloy compositions x.