Calculate the energy band arising from a single orbital in a two-dimensional simple hexagonal lattice (see Table 1.1), using the tight-binding approximation, and plot the energy as a function of k_x and k_y. Assume that there is one coupling energy U₁₂ for all nearest neighbors and set all other coupling energies to zero.

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Structure Table 1.1 Common crystal structures Simple cubic (sc) Body-centered cubic (bcc) Face-centered cubic (fcc) Graphite Standard primitive vectors and basis ax, ay, az ax, aŷ, a(x +ŷ + 2) or sc lattice with the basis (0,1a(x + y +2)) {a(ỹ+2), ⁄a(2+î), {a(x + ŷ) or sc lattice with the basis (0, ¼ a(x +ŷ), ¼ a(ŷ + 2), ⁄ a(2 + x)) fcc lattice with the basis (0, za(x + y +2)) Diamond Simple hexagonal (sh) aî, (fax + ay), cz Hexagonal close-packed (hcp) sh lattice with the basis (0, ½ax + 2√zaŷ + ½cê), 8 √√şa where c = Bax + ay, - Bax + ½ aŷ, c² with the basis (0, ½cz, zał + ⁄ay, −2+√x + {aỹ + {c²) -ax -ax 2√3 3 -