A hydrogenic donor state has an exponentially decaying envelope

ψ (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:

W \sim E_{b} exp (- α \frac{n ^{- 1/3}}{a _{b}}) .

(1)

The order-one constant

α

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.