The kp Method Figure 35-18. 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.

[-(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

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

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

[(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.

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]

(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!