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Nearly-free electron dispersion near a one-dimensional band gap

For

V (x) = \sum_{n \in Z} V_{n} e^{2 πin x / a},

write

k = nπ / a + κ

. Near the

n

th Bragg crossing,

E_{\pm} (k) = V_{0} + \frac{ℏ ^{2}}{2 m} [(\frac{nπ}{a})^{2} + κ^{2}] \pm (\frac{ℏ ^{2} nπκ}{ma})^{2} + ∣ V_{n} ∣^{2} .

At

κ = 0

, the two branches differ by

2∣ V_{n} ∣

.

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Solution