Two simple models of response of the Micropixel avalanche photodiode (MAPD) are proposed. The first simple discrete model describes an avalanche development and quenching in time in a single pixel. The time step is chosen as a ratio of depletion layer depth to the thermal velocity of charge carriers. Electric field and ionization coefficients for electrons and holes are recalculated before each step taking into account internal avalanche and external recharge currents and voltage drop between the pixel electrodes. The numbers of secondary carriers (electrons and holes) created within the depletion layer and collected at electrodes after each step are evaluated. One of important results is behavior of the avalanche after potential difference between cathode and anode of the pixel reaches a value of the breakdown voltage. The voltage keeps decreasing and new electron-hole pairs are being created with decreasing rate. The number of the charge carriers produced before and after the voltage reaches the breakdown voltage is approximately the same. As a result an effective pixel capacitance obtained from a slope of linear dependence of the pulse charge on bias voltage turns to be overestimated approximately twice.

In the second model the initial number of triggered pixels (or photoelectrons that initiated an avalanche process) is distributed according to Poisson law. This distribution then was distorted by optical crosstalk and dark noise counts. The crosstalk probability is assumed to follow the Poisson law. The dark noise was considered as a Poisson process. Simulation of resulting distribution of a number of triggered pixels was performed using Monte Carlo technique. Fitting results of the simulation gave simple formulae for extraction of parameters of initial distributions from the resulting distribution. The results of simulation are in good agreement with experimental data. Known fractions of events in the zeroth and first peaks of the experimental distribution and dark count rate allows one to extract the number of primary fired pixels and the crosstalk fraction.

Joint Institute for Nuclear Research, Dubna, Russia