I am puzzled with impurity ionization in Semi-conductors.
Suppose $N_d$ is the density of donor impurities and $n_d$ the density of electrons bound to the single impurity orbital with energy level $\varepsilon_d$.
Defining $\mu$ as the chemical potential of the semi-conductor electron gas (with $\varepsilon_d\geq\mu$), the density of bound electrons $n_d$ can be found as :
\begin{equation} n_d = \frac{N_d}{\frac{1}{2}e^{\frac{\varepsilon_d-\mu}{k_BT}} + 1} \end{equation}
The full ionization condition is then written as $\varepsilon_d-\mu\gg k_BT$ for which almost no electrons are bound to the impurity orbital. Note that even though $\varepsilon_d-\mu\gg k_BT$, impurity ionization is still physically achieved through thermal excitation from the impurity orbital $\varepsilon_d$ to the conduction band with energy $\varepsilon_c$ where $\varepsilon_c\geq\varepsilon_d$.
What I do not understand is :
Looking at the expression of $n_d$, low temperature seems to foster impurity ionization. However, at very low temperature, we know that incomplete ionization should arise. What is the correct picture?