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The k·p Method in Physics: Effective Band Structure Calculation

Learn the empirical k·p method for band structure calculations in condensed matter physics. Understand the Schrödinger Equation, perturbation theory, and fitting parameters for upper and lower bands.

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The k·p Method in Physics: Effective Band Structure Calculation

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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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