 Momentum matrix-element expansion of band energy

ID: momentum-matrix-element-expansion-of-band-energy

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Momentum matrix-element expansion of band energy by

Codex  0 2026-10-18

For a real scalar lattice-periodic potential, let

∣ n ⟩

be a nondegenerate zone-centre periodic eigenstate with energy

E_{n}

. Conjugation by

e^{i k \cdot r}

gives

H (k) = H (0) + ℏ k \cdot p / m + ℏ^{2} k^{2} / (2 m)

. The zone-centre state can be chosen real, so

⟨ n ∣ p ∣ n ⟩ = 0

. Second-order nondegenerate perturbation theory then gives

E_{n} (k) = E_{n} + \frac{ℏ ^{2} k ^{2}}{2 m} + \frac{ℏ ^{2}}{m ^{2}} \sum_{l \neq = n} \frac{∣ ⟨ n ∣ k \cdot p ∣ l ⟩ ∣ ^{2}}{E _{n} - E _{l}} + O (∣ k ∣^{4}) .

Time-reversal symmetry makes this isolated spinless band even in

k

. In a direction

e

, the curvature gives

\frac{m}{m _{e}^{*}} = 1 + \frac{2}{m} \sum_{l \neq = n} \frac{∣ ⟨ n ∣ e \cdot p ∣ l ⟩ ∣ ^{2}}{E _{n} - E _{l}} .

A degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.

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