Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 4 / d / Solution

For $|k_{i}a|\ll 1$, expand each cosine to second order:To this order a fixed energy fixes $k_x^2+k_y^2+k_z^2$, so the constant-energy surfaces are spheres. The band effective mass tensor satisfiesFor bonding hopping $t<0$ this is a positive electron mass at the band bottom. If $t>0$, $k=0$ is a maximum with negative electronic mass, or positive electron hole mass ${\hslash }^2/(2t{a}^2)$. The spherical shape is a quadratic approximation; cubic anisotropy appears in the fourth-order terms.