Internal Charge Amplification in Germanium at 77K and 4K: From Single-Free-Flight Bounds to a Physics-Informed Ionization Model

Internal charge amplification (ICA) in cryogenic high-purity germanium (HPGe) can lower detection thresholds by providing gain inside the detector crystal, but reliable operation requires a predictive estimate of the avalanche-onset \emph{critical electric field} $E_{\mathrm{crit}}$. We present a compact framework for $E_{\mathrm{crit}}$ at 77~K and 4~K (typical HPGe operating temperatures) that bridges (i) a mobility-based single-free-flight (SFF) upper bound with (ii) a physics-informed impact-ionization model incorporating energy-dependent scattering, nonparabolic (Kane) dispersion, intervalley transfer, and the high-energy ``lucky-drift'' tail. This unified treatment yields closed-form, design-useful relations, including $E_{\mathrm{crit}}^{(\mathrm{PI})}=B(T)/\ln[A(T)d]$, and a practical calibration workflow that maps measured low-field mobility $\mu(T)$ and gain curves $M(V)$ (Chynoweth analysis) to device-level bias targets with propagated uncertainty bands. Example electron and hole estimates indicate that realistic transport typically lowers $E_{\mathrm{crit}}$ relative to SFF and increases the predicted change in $E_{\mathrm{crit}}$ between 77~K and 4~K. The resulting portable formulas connect materials/transport inputs to geometry, excess noise, and field shaping, providing design-ready guidance for stable, unipolar-favored ICA with controlled quenching in Ge and other cryogenic semiconductors.