Past exam of the condensed matter physics course of the University of Oxford / 2014 / 3 / b / i

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- Solution i

Solution

An electron hole is a missing electron in an otherwise almost filled valence band. A completely filled band has no net transport current: its velocities cancel over the Brillouin zone. Describing a few missing states as mobile excitations is therefore simpler than following almost every occupied electron state.

Near an isotropic valence maximum, write the electron dispersion as

E_{v} (k) = E_{v} (0) - ℏ^{2} k^{2} / (2 m_{h})

with

m_{h} > 0

. The electron curvature mass is negative there. Removing that electron creates an excitation with momentum

ℏ k_{h} = - ℏ k

and energy measured upward from the filled reference,

E_{h} (k_{h}) = E_{v} (0) - E_{v} (- k_{h}) = \frac{ℏ ^{2} k _{h}^{2}}{2 m _{h}} .

Thus

m_{h}^{*} = - m_{e, valence}^{*} > 0.

The sign refers to the positive excitation energy of a missing electron near a maximum, not to negative vacuum mass. A general maximum gives the corresponding positive hole-curvature tensor.^[ref]

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