= Effective mass for exponentially decaying tight-binding hopping
For $t(a)=-2E_1e^{-a/a_1}$ with $E_1,a_1>0$, a nearest-neighbour cubic <tight-binding model> has zone-centre <band effective mass>
$$
m^*(a)=-\frac{\hbar^2}{2t(a)a^2}=\frac{\hbar^2 e^{a/a_1}}{4E_1a^2}.
$$
Its logarithmic derivative is $1/a_1-2/a$. The unique minimum occurs at $a=2a_1$ and has value $\hbar^2\exp(2)/(16E_1a_1^2)$. The explicit $a^2$ in band curvature competes with the exponentially weakening <hopping integral>. Extrapolation to vanishing atomic separation lies outside the localized-orbital approximation.
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