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The authors report on the current-voltage (I-V) characteristic under forward biases obtained in low leakage, small size p-(In,Ga)N-GaN-n multiquantum well diodes. Under barrier illumination, the devices present a high optical response with capabilities to detect optical powers in the pW range without further amplification. This response is attributed to the screening of the internal electric fields. Recombination times of a few seconds are found to be associated to this mechanism. Moreover, a step-like feature is found in the I-V characteristic before the diode turn-on voltage. Our model proposes tunneling current through the multiquantum-well structure as responsible of this feature. Fast modulation of the tunneling effect under barrier illumination is used to evaluate the detection of low photon fluxes.