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The kp Method

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

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  1. The kp Method

  2. 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

  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. 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

  5. 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.

  6. 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. 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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