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The kp Method. 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.

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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
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.

  • 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


  • 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.


  • 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*),

    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!


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