 Solution

Taking $E_1>0$ so that the zone centre is a minimum, the effective mass for exponentially decaying tight-binding hopping isIts logarithmic derivative isThe curve decreases for $a<2a_1$, has a unique minimum at $a=2a_1$, and increases for $a>2a_1$:The formal limits are an $a^{-2}$ divergence as $a\to 0$ and an exponential divergence as $a\to \infty$. The small-separation extrapolation eventually invalidates the localized-orbital tight-binding model; the sketch describes the specified formula.