The kp Method

The kp 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.

[-(ħ22)/(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.[-(ħ22)/(2mo)+V(r)]ψnk(r) = Enkψnk(r) (1)

• Of course, ψnk(r) has the Bloch function form

ψnk(r) = eikrunk(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) + (ħkp)/mo+ (ħ2k2)/(2mo) + V(r)]unk(r) = Enk unk(r)• Of course, p = - iħ

• Effective Schrödinger Equation for unk(r):[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) + V(r)]unk(r) = Enk unk(r)• Of course, p = - iħ

PHYSICSThese 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, (ħkp)/mo & (ħ2k2)/(2mo) are “small”

Treat them usingQuantum Mechanical Perturbation Theory

Effective Schrödinger Equation for unk(r):[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) + V(r)]unk(r)

= Enk unk(r) (p = - iħ)

Treat (ħkp)/mo & (ħ2k2)/(2mo) withQM perturbation theory

• First solve:[(p)2/(2mo) + V(r)]unk(r) = Enk unk(r) (p = - iħ)

Then treat (ħkp)/mo & (ħ2k2)/(2mo) usingperturbation 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.

• 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]

• 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|kp|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!