A two-dimensional square lattice of spacing has one atomic level per site

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

A two-dimensional square lattice of spacing aa has one atomic level εat\varepsilon_{\rm at} per site, and an electron can hop with the same amplitude t>0t > 0 to each of its four nearest neighbours (one step in +x+x, −x-x, +y+y and −y-y). Following the same argument as the text's one-dimensional chain, write down the tight-binding energy E(kx,ky)E(k_x, k_y), and show that the band runs from εat−4t\varepsilon_{\rm at} - 4t at kx=ky=0k_x = k_y = 0 to εat+4t\varepsilon_{\rm at} + 4t at the zone corner kx=ky=π/ak_x = k_y = \pi/a, a width of 8t8t.

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