1. Summary, Qualitative Hydrogenic Model
Typical numbers:
 ~ 10, 0.03mo  m  mo
 10 meV  |En=1- Eb|  150 meV
 “Shallow Levels” (compared to bandgaps of ~ eV)

2. Hydrogenic Model
Simple hydrogenic model qualitatively explains shallow donor & acceptor levels.
TOO SIMPLE! Does not work well quantitatively.
Unphysical results, contrary to intuition:
Levels En do not depend on impurity (absurd)!
 All shallow donors in a given host have the same energies! En depend on host only through m and 
A more accurate theory is clearly needed!

3. Effective Mass Theory Simple hydrogenic model is wrong!
Need a more accurate theory which
Uses true impurity potential, which is more complicated than simple screened Coulomb! Need ~ pseudopotential for impurity.
Need also to account for k dependence of effective mass m.
“Anisotropic H atom”
Such a theory has been developed & works well quantitatively
 “Effective Mass Theory”

4. Effective Mass Theory (EMT)
EMT (& generalizations) quantitatively explains most shallow donor/acceptor level data.
Detailed discussion in YC, Ch. 4.
Here, brief outline.
Need to REALLY solve Schrdinger Eq., including accurate impurity potential:
(Ho +V)= E
Ho = p2/(2mo) + Vo : Gives perfect crystal bandstructure
V  Impurity potential. More complicated than simple screened Coulomb -e2/(r)

5. Effective Mass Theory (EMT)
Schrdinger Eq. Including impurity:
(Ho +V)= E (1)
Presence of impurity (V)
 Translational symmetry is destroyed
 Bloch’s Theorem no longer valid.  is NOT of Bloch form.
Goal: Solve Eq. (1) for impurity levels E in bandgap using accurate V

6. Effective Mass Theory (EMT)
(Ho +V)= E (1)
Two possible approaches:
Expand  in complete set of Bloch functions nk (complete, orthonormal set of eigenfunctions for perfect crystal). Eigenfunctions in k space.
Effective mass theory (EMT). Expand  in another complete basis set, to be described later, which are eigenfunctions in r space.

7. Effective Mass Theory (EMT)
(Ho +V)= E (1)
Briefly first method.
Expand  in complete set of Bloch functions nk.
Expansion  = nk ank nk (2) Requires HUGE number of terms (a huge number of ank are not small). Bloch functions nk are states extended over entire crystal.
Impurity wavefunction  is (relatively) spatially localized.
Eq. (2) tries to express a spatially localized quantity  in terms of a number of spatially extended quantities nk.  Requires a huge number in sum! Difficult, but has been done.

8. Effective Mass Theory (EMT)
(Ho +V)= E (1)
Better method is EMT.
Bloch functions have form:
nk(r) = eikr unk(r)
Extended or delocalized in r space. (Localized in k space)
Define Wannier functions:
an(r,Ri)k exp(-ikRi) nk(r)/(N)½
Ri direct lattice point, k BZ sum
an(r,Ri) ~ discrete spatial Fourier transform of Bloch function
nk(r)  Extended in r space
 an(r,Ri)  Localized in r space!

9. Wannier Functions
an(r,Ri)k exp(-ikRi) nk(r)/(N)½
Ri direct lattice point, k BZ sum
an(r,Ri) ~ discrete spatial Fourier transform of Bloch function
Bloch functions obey inverse Fourier transform relation
nk(r)i exp(ikRi) an(r,Ri) /(N)½
This form: Very similar to LCAO/ tightbinding approx. for nk(r). However, in LCAO, an’s were atomic functions (from “bare” atom). Here, an’s are localized (in r space) states constructed from true eigenfunctions of e- in periodic solid. (an’s are r space eigenfunctions for perfect solid!)

10. Wannier Functions Properties: Translationally invariant
an(r,Ri)  an(r - Ri)
Complete, orthonormal set of (r space) electronic eigenfunctions of perfect crystal.
 Any function defined in space of crystal lattice can be expanded in terms of them.
Alternate (to Bloch functions) representation of electronic eigenfunctions of perfect crystal.  Could develop bandstructure theory using these instead of Bloch functions.

11. Effective Mass Theory (EMT)
(Ho +V)= E (1)
Expand impurity wavefunction in a complete set of Wannier functions.
(r) = i,n Cn(Ri)an(r - Ri)
an(r - Ri)  Wannier function for band n & lattice site Ri
Cn(Ri)  expansion coefficient for band n & lattice site Ri

12. Effective Mass Theory (EMT)
(Ho +V)= E (1)
(r) = i,n Cn(Ri)an(r - Ri) (2)
Insert (2) into (1). Many math details follow. These make use of
Expansion of Wannier functions an(r- Ri) in terms of Bloch functions nk(r),
The fact that nk(r), are eigenfunctions of Ho with eigenvalues being the bands En(k)
Ho nk(r) = En(k) nk(r)
Quasi-momentum p= k. Momentum operator p = -i.
 Wavevector k becomes an operator everywhere it appears in Schrdinger Eq.: k -i /R

13. Effective Mass Theory (EMT)
(Ho +V)= E (1)
Everywhere it appears in (1) k  -i /R
Specifically, in bands of perfect crystal En(k), make replacement
En(k)  En(-i /R)
MUCH manipulation (pp in YC) results in new, effective Schrdinger Eq.
Specialize to one band only. Assume that only one band
 n dominates problem.
Normally lowest CB (donors) or highest VB (acceptors).
Assume impurity is localized on one particular lattice site  R

14. Effective Mass Theory (EMT)
Under these assumptions, (band n dominates, impurity at R) effective Shrdinger Eq. is (see YC):
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
En(-i /R)  energy band n with k  -i /R
Cn(R)  coefficient for band n & impurity site R in expansion of impurity wavefunction in terms of Wannier functions:
(r) = i,n Cn(Ri)an(r - Ri)
U(R) impurity potential

15. Effective Mass Theory (EMT)
Effective Shrdinger Eq. for impurity is
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
Accurate, quantitative solution, using U(R) = pseudopotential for impurity, requires extensive computation.
True impurity wavefunction is not Cn(R)! It is
(r) = i,n Cn(Ri)an(r - Ri)
With band n dominating & impurity at site R, this becomes
(r)  Cn(R)an(r - R)
with an(r - R) = Wannier function
for band n (perfect crystal eigenfunction;
get from bandstructure problem!)

16. Effective Mass Theory (EMT)
Band n dominating, impurity at R, impurity wavefunction is
(r)  Cn(R)an(r - R)
YC show that this is also
(r)  Cn(R)unk=0(r)
unk=0(r) = periodic part of Bloch function at BZ center
Cn(R) solutions to
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
In general, must be done numerically. However, consider special case where band n is a non-degenerate, isotropic, parabolic CB, with a minimum at .

17. Effective Mass Theory (EMT)
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
Special case: band n is a non-degenerate, isotropic, parabolic CB, with a minimum at . Expand band for k near its minimum. (Can show linear term = 0)
En(k)  En(0) + 2k2/(2me)
En(0) = band minimum
Using this in effective Schrdinger Eq. (k  -i /R)
En(-i/R)  En(0)-[2/(2me)](2/R2)
gives
[-{2/(2me)}(2/R2) + U(R)]Cn(R) =[E- En(0)]Cn(R)

18. Effective Mass Theory (EMT)
Non-degenerate, isotropic, parabolic CB, min. at .
[-{2/(2me)}(2/R2) + U(R)]C(R) =[E- E(0)]C(R)
U(R) = pseudopotential for impurity
Simplest approx. for potential is U(R) = -e2/(R) (screened Coulomb). Get effective H atom problem back again!
In this case, solutions C(R) are H atom wavefunctions. But
Impurity wavefunction (from before) is (r)  C(R)u(r)
= H atom function times periodic part of Bloch function for CB at 

19. Effective Mass Theory (EMT)
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
Have been many modifications & approximations to this between the simplest “effective H atom” case & the true numerical problem!
Hydrogenic, shallow donors.
R treated as continuous, when actually it is discrete.
H atom-like solutions, with some modifications.
Ok as long as effective Bohr radius a >> ao

20. Effective Mass Theory (EMT) Effective H atom approximation
Focus on discrete, bound states of impurity.
Classify states according to hydrogenic notation:
n = principle quantum number
l = orbital angular momentum quantum number
In simplest case, get Rydberg series as before:
Energies:
En = EC -(13.6 eV)(me/mo)/(2n2)
Effective Bohr radius
a = (0.53 A)  (mo/me)
1s “wavefunction” C1s(R)  exp(-R/a)

21. Effective Mass Theory (EMT) Effective H atom approximation
Can show that energy errors using simplest, “Effective H atom” approximation
 (ao/a)2
Note: We’ve assumed CB minimum is at  & that band is parabolic there.
This is ~ valid for direct gap materials like GaAs, but is NOT valid for Si & other indirect gap materials.!

22. Effective Mass Theory (EMT)
Simple, “Effective H atom” model cannot be correct!
Rydberg energies En & “wavefunctions” Cn(R) are independent of impurity & depend on host only by me & 
Detailed, correct results require numerical solutions of effective Schrdinger Eq. for impurity
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
with U(R) = pseudopotential for impurity.

23. Effective Mass Theory (EMT)
Effective Schrdinger Eq. for impurity
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
For some impurities  “Deep Impurities” even the numerical solution fails because the approximations break down. (Discussed next).
Shallow level theory (EMT) works best if core electron shell of impurity resembles that os replaced host atom.

24. Effective Mass Theory (EMT)
Donors in indirect gap materials (Si, etc.)
CB minimum is at (or near) BZ edge. For Si this is at X point. BUT 6 equivalent X points!
 CB at its minimum is 6-fold degenerate!
Also, effective mass of CB is highly anisotropic.
 me depends on k direction away from X point!
Need MODIFIED Effective Mass Theory to account for this!

25. Effective Mass Theory (EMT)
Si, at CB minimum, effective mass has tensor or matrix form: ml
me = mt 0
0 0 mt
ml  “longitudinal effective mass” (k || [100])
mt  “transverse effective mass” (k  [100])

26. Effective Mass Theory (EMT)
In effective Hamiltonian, where impurity problem  Effective H atom problem, replace KE by “anisotropic KE”:
p2/(2me)  (p||)2/(2ml) + (p)2/(mt)
In EMT, Effective Schrdinger Eq. for impurity
[En(-i /R)+U(R)]Cn(R) =E Cn(R)
becomes, on expansion of En about X point & on using screened Coulomb potential for U(R):
[-{2/(2ml)}(2/R||2) -{2/(mt)}(2/R2)
-e2/(R)]CX(R) = [E- EX(0)]CX(R)
“Elliptically deformed H atom”:
Still label eigenstates by quantum numbers n, l. m. Degeneracy of quantum number m is (partially) lifted.

27. Effective Mass Theory (EMT)
“Elliptically deformed H atom”
Impose lattice symmetry:
Use group theory notation for Td point group.
Get “A1 levels”  “s-like”
Get “T2 levels”  “p-like”
Get “E levels”  “p-like”
Wavefunction assumed to be H atom-like, but anisotropic to correspond to “deformed H atom”. Contains parameters which are either fit to data or determined by a variational procedure. (See YC)

28. Effective Mass Theory (EMT)
Further improvements to “deformed H atom”  “Valley-orbit coupling” model:
6 equivalent s-like states from 6 equivalent X valleys with CB minima.
Use degenerate perturbation theory & group theory, to see how degeneracies are lifted. Must diagonalize a 6X6 matrix.

29. Effective Mass Theory (EMT)
Treatment of acceptor levels is similar, but there are added complications!
Valence bands are always (4 fold) degenerate at BZ center ( point)!
Hole effective masses are highly anisotropic at .
 Need to represent effective mass as a tensor or 3X3 matrix. This is even more complicated than for donor levels in indirect materials because effective mass tensor is not diagonal!

30. Effective Mass Theory (EMT)
Generalizations of EMT for acceptor levels: easy to see how to do. Tedious, highly computational.
YC show: Due to tensor mass, effective Schrdinger Eq. for impurity becomes set of 4 coupled tensor-like eqtns (4X4 matrix eqtn):
[j  (Dij)(/R)(/R)
-e2/(R)]Ci(R) = E Ci(R)
(i, j = 1, 2, 3, 4), (, = x, y, z)
(Dij) contains anisotropic mass tensor. Represent with empirical parameters (Eq. 4.38)
Need to include spin-orbit coupling
DOESN’T LOOK MUCH LIKE H ATOM!
A mess to deal with. Results: see tables.