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Armchair graphene nanoribbons (A-GNRs), an alternative material for Infrared (IR) photodetectors, attract more attention because of those tunable energy gaps by changing the width of nanoribbons and the height of interband transition. In this paper, we calculate the dark current limited detectivity, D*, of the multi layer A-GNR based IR photodetector. For this purpose, we find the band structure of A-GNRs by tight-binding model and by considering the edge deformation, the absorption coefficient using the single electron approximation, the quantum efficiency, and the optical responsivity of photodetector. Then, the dark current of photodetector has been calculated by considering two contributions: (i) The interband tunneling generation and (ii) the thermogeneration due to the optical and acoustic phonon and line edge roughness (LER) scattering in the A-GNRs. Finally, we optimize the dark current limited detectivity of the photodetector for different structural parameters. The obtained results show that for the single layer A-GNR based photodetector with W = 5 nm, L = 20 μm, Vb = 2 V, Vg = 2 V, maximum value of dark current limited detectivity, D*, at T = 300 K is ∼2.2 × 108 (cm Hz1/2/W) and at T = 77 K is ∼2.1 × 1011 (cm Hz1/2/W). Also, for narrow A-GNRs, D* increases with increasing the gate voltage, while for wider A-GNRs decreases with increasing the gate voltage. Moreover, the dark current limited detectivity increases with increasing the number of the A-GNR layers.