In Elecronic Devices and Circuits, by J Millman and CC Halkias, it is written at one place:
.....coefficient of diffusion $D$ and mobility $\mu$ aren't independent. The relationship between them is given by the Einstein equation $$ D_i=\mu_iV_T $$ where $V_T=\bar kT/e = T/11600$ ..........
Whereas later on in the same text:
For germanium the diffusion constants $D_p$, and $D_n$ vary approximately inversely proportional to $T$.
Does this mean that the mobility varies as inverse squares of temperature: $$\mu_i \propto T^{-2}$$ If yes then how, since lattice scattering mobility $\mu_L\propto T^{-3/2}$ whereas ionized impurity scattering at small $T$ values, $\mu_I\propto T^{-5/2}$ ( derived from equations given in The Dopant Density and Temperature Dependence of Electron Mobility and Resistivity in N-Type Silicon, by Sheng S. Li). If no, then how else? Does it have to do something with the low band gap of Ge? Please give a quantitative explanation.
Edit
I have come to know that an inverse square variation of mobility with temperature is resulted due to polar optical phonon scattering. However the exact derivation was missing.