A Systematic Analytical and Numerical Study of Incomplete Dopant Ionization in Germanium Over 4-400 K

Incomplete dopant ionization critically influences the electrical properties of germanium (Ge), particularly under low-temperature and low-doping conditions relevant to advanced electronic and optoelectronic devices. In this work, we present a systematic numerical investigation of temperature- and concentration-dependent dopant ionization in Ge over the temperature range 4–400 K and dopant concentrations from 1×1014 to 1×1018 cm-3. Ionization probabilities are evaluated for common acceptor dopants (Boron, Gallium, and Indium) and donor dopants (Phosphorus, Arsenic, and Antimony), with activation energies spanning 10–16 meV. The results reveal severe dopant freeze-out at cryogenic temperatures, where ionization probabilities drop below 0.1–0.2 for lightly doped Ge (N≤1015cm-3), leading to carrier density reductions exceeding 80–90% compared to full-ionization assumptions. Donor dopants with lower activation energies achieve near-complete ionization (P(T)>0.9) at 100–150 K, while higher-energy acceptors require temperatures above 200–250 K. Increasing dopant concentration to 1017- 1018 cm-3 significantly suppresses freeze-out, enabling ionization probabilities above 0.8 at temperatures as low as 50–70 K. At and above room temperature, all dopants exhibit near-unity ionization across the investigated concentration range. These findings provide quantitative guidelines for dopant selection and concentration optimization in Ge-based electronic, optoelectronic, and cryogenic devices, emphasizing the necessity of explicitly accounting for incomplete ionization in low-temperature device modeling and design.

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