Germanium (Ge) detectors with ability of measuring a single electron-hole (e-h) pair are needed in searching for light dark matter (LDM) down to the MeV scale. We investigate the feasibility of Ge detectors with amorphous-Ge (a-Ge) contacts to achieve the sensitivity of measuring a single e-h pair, which requires extremely low leakage current. Three Ge detectors with a-Ge contacts are used to study the charge barrier height for blocking electrons and holes. Using the measured bulk leakage current and the D$ddot{o}$hler-Brodsky model, we obtain the values for charge barrier height and the density of localized energy states near the Fermi energy level for the top and bottom contacts, respectively. We predict that the bulk leakage current is extremely small and can be neglected at helium temperature ($sim$4 K). Thus, Ge detectors with a-Ge contacts possess the potential to measure a single e-h pair for detecting LDM particles.
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