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

TL;DR

This paper develops a physics-informed model to predict the critical electric field for charge amplification in cryogenic germanium detectors, enabling better design and operation at 77K and 4K.

Contribution

It introduces a unified framework combining a mobility-based bound with impact-ionization physics, providing closed-form relations and calibration methods for device design.

Findings

Realistic transport models lower the predicted critical field compared to single-free-flight bounds.

The model captures the temperature dependence of the critical field between 77K and 4K.

Portable formulas link material properties to device geometry and noise characteristics.

Abstract

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.