We present the detailed characteristics of small area (33 x 33 mum(2)) InSb photodiodes grown on GaAs coated Si substrates by molecular beam epitaxy. In spite of very large lattice mismatch, 80 K peak detectivity of similar to1 x 10(10) cm Hz(1/2) W-1 has been measured under backside illumination without anti-reflection coating. Differential resistance at 80 K is limited by Ohmic leakage under small reverse bias and trap assisted tunnelling (TAT) under moderately large reverse bias. In the temperature range of 80-140 K, the zero-bias resistance is considerably degraded by Ohmic leakage which has a small activation energy of 25 meV While the excess leakage seems to be related with the dislocations, the peak quantum efficiency is not considerably affected by the presence of dislocations. 80 K 1/f noise is dominated by TAT processes, and the noise current at 1 Hz follows the empirical relation i(n) = alpha(TAT)(I-TAT) with alpha(TAT) = 7.5 x 10(-7) and beta = 0.55. Bias dependence of the generation-recombination noise is discussed and compared with the predictions of Kleinpenning's mobility fluctuation model.