This cartoon mixes 2 legends:

1. Legend of Newton, the

apple & gravity which

led to the

Universal Law of

Gravitation. 2. Legend of William Tell &

the apple.

Outline of Hjalmarson, Vogl, Wolford, & Dow

Deep Level Theory

A brief outline of this theory, then some results.

• The Schrödinger Equation including the defect or impurity is (Dirac notation, for convenience):

(Ho +V)|Ψ> = E|Ψ> (1)Ho= Hamiltonian for a perfect, periodic, crystal.

It produces the bandstructure.V = The defect potential (to be discussed).

It produces defect levels. • Solve (1) in r space (the direct lattice) to take advantage of the localized nature of V for

deep levels. Use a tightbinding (LCAO) approach to the bandstructure & a Green’s function formalism. See YC, Ch. 4, which discusses this in detail.

• Manipulate with (1) (using an operator or matrix formalism):

(E- Ho)|Ψ> = V|Ψ> or |Ψ> = (E- Ho )-1 V|Ψ> or [1 - (E- Ho )-1V]|Ψ> = 0

Finally det[1 - (E- Ho )-1V] = 0 (2)Emphasize: (2) the Schrödinger Equation (1)

in different notation!

• So, the Schrödinger Equation including the defect or impurity is

det[1 - (E- Ho )-1V] = 0 (2)• It is convenient to define the

Host Green’s Function (Matrix) Operator

Go(E) (E- Ho)-1

• The Schrödinger Equation (2) then has the form:

det[1 - Go(E)V] = 0 (3)

The GOAL is then:Given Ho, & V, find the energy E which makes the determinant in (3) vanish! (3) is an equation for the deep level E, which is the eigenvalue of the Hamiltonian with the defect, H = Ho+ V & the solution to the Schrödinger Equation we seek!

det[1 - Go(E)V] = 0 (3)• To solve (3), models for Ho (bandstructures) & V (defect potential) are

needed. • The calculations use a tightbinding (LCAO) representation for Ho & V,

and for the Green’s function Go(E).

• For the host bandstructures Ho use the formalism in the paper:“A Semi-Empirical Tightbinding Theory

of the Electronic Structure of Semiconductors”P. Vogl, H. Hjalmarson, & J. Dow,

Journal of the Physics and Chemistry of Solids, 44, 365-378 (1983)

(directly linked from the course lecture page).• For the defect potential V, use the formalism in the paper:

“Theory of Substitutional Deep Traps in Covalent Semiconductors”, H. Hjalmarson, P. Vogl, D. Wolford, & J. Dow

Physical Review Letters 44, 810 (1980). (directly linked from the course lecture page).

See also, H.P. Hjalmarson, PhD dissertation, U. of Ill., 1980

det[1 - Go(E)V] = 0 (3)

Hjalmarson Deep Level Theory:• Rather than a quantitative theory, it is a theory designed for & best suited for

predictions of chemical trends in deep levels (discussed next) & explanations of such trends.

• It & generalizations have been successfully used to predict chemical trends in a variety of problems. Its simplicity allows for qualitative & semi-quantitative predictions of a number of defect properties. It’s quantitative accuracy is limited.

Chemical Trends• Given the host, how does the deep level change as the impurity is changed

or as one type of defect is changed to another.• Given the impurity or defect, how does the deep level change as the host

changes (especially, the alloy composition dependence in alloy semiconductors).

• Explaining chemical trends will help to explain a lot of data!

det[1 - Go(E)V] = 0

• Consider substitutional impurities only at first.

• The host Ho is described by Vogl, Hjalmarson, Dow semi-empirical tightbinding

bandstructures.

• Model for the Defect Potential V:

– Considers the central cell part of V only.

Neglects the long ranged Coulomb potential.

There are no shallow (hydrogenic) levels in this theory (these could be accounted for later using EMT!)

– Considers ideal defects only, neglects lattice relaxation.

(Generalized to include this by W.G. Li & C.W. Myles in the late 1980’s)

– Considers nearest-neighbor interactions only.

– V is a diagonal matrix in the LCAO representation.

– Considers neutral impurities only: No charge state effects (added later by Lee & Dow).

det[1 - Go(E)V] = 0

• Model for the Defect Potential V. Assume that:

– V is diagonal & proportional a difference in “atomic energies” between impurity & host atom it replaces.

– The matrix V = H - Ho.

In the LCAO representation, the diagonal matrix elements are “atomic energies, so the diagonal elements of V have the form:

Vℓ (εI)ℓ - (εH)ℓ

Here (εI)ℓ & (εH)ℓ are the impurity & the host atomic energies for the orbital of symmetry type ℓ (ℓ = s, p, d,…) or (ℓ = A1, T2, ….) That is

Vℓ βℓ[(εI)ℓ - (εH)ℓ]

βℓ is an empirical parameter.

– This form explicitly accounts for chemical shifts & their effects on the defect potential.

• So, finally:

det[1 - Go(E)V] = 0 (1)

• V is a diagonal matrix with diagonal elements

Vℓ = βℓ[(εI)ℓ - (εH)ℓ] (2)

• Given V, the E which solves (1) is the deep level of interest.• Independently of (2), for given V given, (1) can be viewed as

An IMPLICIT EQUATION (with a numerical solution)

for the deep level E as a function of V.

• That is, (1) can be thought of as a function:

E = E(V) (3)• Now, using (2), (3) becomes

E = E(εI) (4)

• Numerical solution to (4) gives predictions of Chemical Trends!

det[1 - Go(E)V] = 0 (1) Vℓ = βℓ[(εI)ℓ - (εH)ℓ] (2)• A plot of the numerical results for E vs. the diagonal part of V or, equivalently E versus (εI)ℓ looks

schematically like:

• For a specific impurity (fixed V or fixed atomic energies), drop a vertical line from that V. Where this crosses the curve (the solution to (1)), is the predicted deep level. A1 (s-like) levels are shown. Other symmetries are similar.

E

From this graph, we obtain the implicit function

E = E(εI)That is, it predicts how the

deep level E depends on the impurity. Or, it predicts a

chemical trend!

Vℓ or (εI)ℓ

specific impurity

14

Consider N in GaP & GaAsAn 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 opticaltransition 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 introducedinto GaP but only small amount of N can be introducedinto GaAs because of a larger difference in atomic sizes.

13

N in GaPA “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 goodmaterial for LED’s (just as Si & Ge also aren’t for the same reason).

• It turns out that the presence of N actually enhances theoptical 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).

13

The GaAsP Alloy with N Impurities:

Interesting, beautiful data!

The N impurity level is a deep level in the bandgapin GaP but is a level resonant in the conduction band in GaAs. The figure is photoluminescence datain the alloy GaAsxP1-x:N under large hydrostatic pressure for various alloy compositions x.