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We report large-area, 250-mum-diameter, 4H-SiC avalanche photodiodes with low dark current and high gain. At room temperature, the dark current density is 59.5 nA/cm2 at a gain of 1000. An external quantum efficiency of 48% at 280 nm is achieved at unity gain with a recessed-window structure. The differential resistance of the recessed-window device at zero bias is estimated to be 6times1014 Omega. As a result of high external quantum efficiency, large area, and large differential resistance, a record high specific detectivity of 4.1times1014 cmHz1/2 W-1, has been achieved. In Geiger mode operation, high single photon detection efficiency of 30% with dark count probability of 8times10-4 is reported.