The kp Method

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# The kp Method - PowerPoint PPT Presentation

The kp Method. The kp Method YC, Ch. 2, Sect. 6 &amp; problems; S, Ch. 2, Sect. 1 &amp; problems. A Brief summary only here. A Very empirical bandstructure method.

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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 Ekusing these experimental parameters.

[-(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
Effective Schrödinger Equationfor unk(r):

[(p)2/(2mo) + (ħkp)/mo+ (ħ2k2)/(2mo) +

V(r)]unk(r) = Enk unk(r)

• Of course, p = - iħ

PHYSICS

These are NOTfree 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

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 areALMOST parabolas.

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*),

whereEn0 = 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!