Using the square-lattice band from the previous problem

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch Tight binding, band filling and the three kinds of solid

Problem

Using the square-lattice band

E(k_x,k_y) = \varepsilon_{\rm at} - 2t(\cos k_xa + \cos k_ya)

from the previous problem, expand the energy near the top of the band,

k_x = \pi/a + \delta k_x

and

k_y = \pi/a + \delta k_y

with

\delta k_x, \delta k_y

small, to second order. Show that the effective mass there is isotropic (the same along

\delta k_x

and

\delta k_y

) and equal to

-\hbar^2/(2ta^2)

, the same magnitude as at the band bottom but with the opposite sign.

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