Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 2 / b

For a nondegenerate band $E(\mathbf{k})$, define the reciprocal effective mass tensor byIt is the curvature tensor of the energy band. Near an extremum, $E\simeq E_0+{\hslash }^2\delta {\mathbf{k}}^T{M}^{-1}\delta \mathbf{k}/2$, so positive principal curvatures give positive electron effective masses. Together with the group velocity $\mathbf{v}={\hslash }^{-1}{\nabla }_{\mathbf{k}}E$ and $\hslash \dot{\mathbf{k}}=\mathbf{F}$, it yields ${\dot{v}}_i=(M^{-1}{)}_{ij}{F}_j$. Negative electron curvature at a valence band maximum is conventionally recast as a positive band effective mass for missing electrons.