Why is the Einstein relation not valid for germanium?

Question

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.

Answer 1

Quantitative explanations seems hard to come by in textbooks.

In Andy Grove's (yes, the Intel guy) Physics and Technology of Semiconductor Devices (1967) one finds

Experimentally, mobilities have been found to follow a $T^{-2.5}$ dependence rather than the theoretically predicted $T^{-1.5}$ dependence in the lattice scattering range.

(Underneath that passage I wrote, many years ago, "Explanation?" - you are not alone!)

Moving to Sze's Physics of Semiconductor Devices (I've got the second edition, 1981), one finds on pp 28-29:

For lower impurity concentrations the mobility decreases with temperature as predicted... The measured slopes, however, are different from (-$3 \over 2$) because of other scattering mechanisms. For pure materials near room temperature the mobility varies as $T^{-1.66}$ and $T^{-2.33}$ for $n$- and $p$-type Ge, respectively; as $T^{-2.42}$ and $T^{-2.20}$ for $n$- and $p$-type Si, respectively; and as $T^{-1.0}$ and $T^{-2.1}$ for $n$- and $p$-type GaAs, respectively.

So, one is left with the rather vague reason of "other scattering mechanisms" which is not very helpful, to be sure. Without digging deeply into the literature, my feeling (even way back when) was that the combination of band structure (where carriers are and where they can scatter to, including things like intervalley scattering) combined with the phonon structure (phonon populations and $E$ vs $k$ available to scatter carriers) combine to make the actual scattering calculations really hard. In other words, the fairly glib and simple relationships for mobility that are presented just don't work very well.