 Solution

A hydrogenic donor state has an exponentially decaying envelope $\psi (r)\propto e^{-r/a_b}$. Neighbouring donors at typical separation $r_d\sim n^{-1/3}$ therefore have exponentially small orbital overlap. The corresponding hopping integral is of order $E_b{e}^{-r_d/a_b}$, apart from algebraic factors and factors from the number of neighbours. Hybridizing many such localized states produces an impurity band with a bandwidth of the same scale:The order-one constant $\alpha$ absorbs details of the donor spacing and envelope. The exponential dependence expresses quantum tunnelling between weakly overlapping donor orbitals, rather than thermal excitation over the donor binding energy.