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We fabricated AlGaN solar-blind avalanche photodiodes (APDs) that were based on separate absorption and multiplication (SAM) structures. It was determined experimentally that the dark current in these APDs is rapidly enhanced when the applied voltage exceeds 52 V. Theoretical analyses demonstrated that the breakdown voltage at 52 V is mainly related to the local trap-assisted tunneling effect. Because the dark current is mainly dependent on the trap states as a result of modification of the lifetimes of the electrons in the trap states, the tunneling processes can be modulated effectively by tuning the trap energy level, the trap density, and the tunnel mass.