Two-direction nearest-neighbour tight-binding dispersion

For hopping amplitude

- λ

from each lattice site to its neighbours at

\pm a_{1}

and

\pm a_{2}

, a Bloch state has energy

E (k) = E_{0} - 2 λ (cos (k \cdot a_{1}) + cos (k \cdot a_{2})) .

The band ranges from

E_{0} - 4∣ λ ∣

to

E_{0} + 4∣ λ ∣

and therefore has width

8∣ λ ∣

.

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 34B / Solution

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