1. The kp Method
Figure Intensity as a function of viewing angle θ (or position on the screen) for (a) two slits, (b) six slits. For a diffraction grating, the number of slits is very large (≈104) and the peaks are narrower still.

2. [-(ħ22)/(2mo)+V(r)]ψnk(r) = Enkψnk(r)
The kp Method YC, Ch. 2, Sect. 6 & problems; S, Ch. 2, Sect. 1 & problems. A Brief summary only here
A Very empirical bandstructure method.
Input experimental values for the BZ center gap EG & some “optical matrix elements” (later in the course). Fit the resulting Ek using these experimental parameters.
Start with the 1e- Schrödinger Equation.
[-(ħ22)/(2mo)+V(r)]ψnk(r) = Enkψnk(r)
V(r) = Actual potential or pseudopotential (it doesn’t matter, since it’s empirical). n = Band Index

3. The 1e- Schrödinger Equation.
[-(ħ22)/(2mo)+V(r)]ψnk(r) = Enkψnk(r) (1)
Of course, ψnk(r) has the Bloch function form
ψnk(r) = eikrunk(r) (2)
unk(r) = unk(r + R), (periodic part)
Put (2) into (1) & manipulate.
This gives an Effective Schrödinger Equation for the periodic part of the Bloch function unk(r). This has the form:
[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) +
V(r)]unk(r) = Enk unk(r)
Of course, p = - iħ

4. Quantum Mechanical Perturbation Theory
Effective Schrödinger Equation for unk(r):
[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) +
V(r)]unk(r) = Enk unk(r)
Of course, p = - iħ
PHYSICS
These are NOT free electrons! p  ħk !
This should drive that point home because k & p are not simply related. If they were, the above equation would make no sense!
Normally, (ħkp)/mo & (ħ2k2)/(2mo) are “small”
Treat them using
Quantum Mechanical Perturbation Theory

5. [(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) + V(r)]unk(r)
Effective Schrödinger Equation for unk(r):
[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) + V(r)]unk(r)
= Enk unk(r) (p = - iħ)
Treat (ħkp)/mo & (ħ2k2)/(2mo) with
QM perturbation theory
First solve:
[(p)2/(2mo) + V(r)]unk(r) = Enk unk(r) (p = - iħ)
Then treat (ħkp)/mo & (ħ2k2)/(2mo) using
perturbation theory
Fit the bands using parameters for the upper valence & lower conduction bands. This gets good bands near high symmetry points in the BZ, where bands are ALMOST parabolas.

6. The Upper 3 Valence Bands: The Lowest Conduction Band:
Near the BZ center Γ = (0,0,0), in a direct gap material, results are:
The Upper 3 Valence Bands:
(P, EG, &  are fitting parameters):
Heavy Hole: Ehh= - (ħ2k2)(2mo)-1
Light Hole: Elh= - (ħ2k2)(2mo)-1 2(P2k2)(3EG)-1
Split Off: Eso= - - (ħ2k2)(2mo)-1 - (P2k2)[3(EG+ )]-1
The Lowest Conduction Band:
EC = EG+(ħ2k2)(2mo)-1
+ (⅓)(P2k2)[2(EG )-1 + (EG+ )-1]

7. (m*)-1  (mo)-1 + 2(mok)-2∑n'[|un0|kp|un'0|2][En0 -En'0]-1
The importance & usefulness of this method?
A. It gets reasonable bands near symmetry points
in the BZ using simple parameterization & computation (with a hand calculator!)
B. It gets Reasonably accurate effective masses:
YC show, near the BZ center, Γ = (0,0,0), for band n,
Enk  En0 + (ħ2k2)/(2m*),
where En0 = the zone center energy (n'  n)
(m*)-1  (mo)-1 + 2(mok)-2∑n'[|un0|kp|un'0|2][En0 -En'0]-1
This is a 2nd order perturbation theory result!
PHYSICS
The bands nearest to band n affect the effective mass of band n!