= Band nonparabolicity
<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 $E=\hbar^2k^2/(2m_b)$, 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.<Band structure of indium antimonide by Evan O. Kane>[Abstract and pages 249–261: coupled conduction and valence bands in small-gap InSb]{ref}
A useful isotropic model is
$$
E(1+\alpha E)=\frac{\hbar^2k^2}{2m_b},
$$
where $\alpha$ 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 $E(E+E_g)=P^2k^2$. Dividing by $E_g$ gives $\alpha=1/E_g$ and $\hbar^2/(2m_b)=P^2/E_g$. 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:
$$
v=\frac1\hbar\frac{dE}{dk}=\frac{\hbar k}{m_b(1+2\alpha E)},\qquad \frac{\hbar k}{v}=m_b(1+2\alpha E).
$$
This mass is distinct from the mass obtained from the local second derivative of $E(k)$. For $\alpha>0$, a state with parabolic estimate $\hbar^2k^2/(2m_b)=1/\alpha$ instead has $E=(\sqrt5-1)/(2\alpha)$. 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.<Band structure of indium antimonide by Evan O. Kane>[Publisher abstract: absorption calculated using the cyclotron-resonance mass and compared with impurity-dependent absorption]{ref}
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