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Band nonparabolicity is the departure of an energy band's dispersion relation from a quadratic function of crystal momentum near an extremum. A parabolic isotropic approximation writes $E={\hslash }^2{k}^2/(2m_b)$, measuring $E$ from the band edge. Band coupling can make the apparent effective mass depend on energy, especially when the separation from another band is small.^[ref]

A useful isotropic model iswhere $\alpha$ has units of inverse energy. A simple two-level illustration explains the extra factor: the eigenvalue above zero of a matrix with diagonal entries $0,-E_g$ and real off-diagonal entry $Pk$ satisfies $E(E+E_g)=P^2{k}^2$. Dividing by $E_g$ gives $\alpha =1/E_g$ and ${\hslash }^2/(2m_b)=P^2/E_g$. Real materials require a more detailed multiband model, including the appropriate symmetries and spin-orbit coupling.

Differentiating the model gives the group velocity and a momentum-to-velocity mass:This mass is distinct from the mass obtained from the local second derivative of $E(k)$. For $\alpha >0$, a state with parabolic estimate ${\hslash }^2{k}^2/(2m_b)=1/\alpha$ instead has $E=(\sqrt{5}-1)/(2\alpha )$. The energy grows more slowly than the band-edge parabola predicts. Evan O. Kane's 1957 InSb calculation connected such nonparabolic band coupling with measured optical absorption and cyclotron resonance data.^[ref]

Evan O. Kane developed influential descriptions of electronic band structure in semiconductors. His 1957 calculation of indium antimonide used k·p perturbation theory to treat the interaction of conduction and valence bands near their extrema. The small band gap made that interaction important rather than a negligible correction.^[ref]

A central consequence is band nonparabolicity: electron energy need not remain proportional to the square of wave number over the relevant range. A constant band-edge effective mass can therefore fail to describe electrons farther from the minimum. The calculation connects band coupling with a measurable change in the relation between energy and motion.

Kane also calculated the fundamental optical absorption using information from the cyclotron resonance mass and compared the prediction with experimental absorption and its dependence on impurity concentration. The agreement supplied an experimental test of the adopted band picture, rather than merely fitting an arbitrary energy curve.^[ref] This combination of a tractable coupled-band model and measurable consequences explains the continuing usefulness of his work in semiconductor theory.