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The multiple-quantum-well (MQW) phototransistor is considered theoretically. The MQW phototransistor utilizes intersub-band optical absorption and exhibits giant photocurrent gain which can lead to very high responsivity and detectivity. This effect is due to the thermionic injection of hot electrons across the emitter barrier and fast electron transit through the MQW base. An analytic theory of the MQW phototransistor utilizing parameters evaluated by Monte Carlo simulation is proposed. Transition from near ballistic hot electron transport to diffusive transport decreases the responsivity but its value can be significant in this case as well.NOTATIONe absolute value of electron chargem effective mass of electron Planck's constantL W quantum well widthL B width of energy barrier between quantum wellsW E emitter barrier widthW C collector barrier widthW B multiple quantum well base widthΔ E emitter barrier heightε 0 energy of injected electronsε W quantum well depthε i ionization energyε F Fermi energyT temperature multiplied by Boltzmann's constantτ electron momentum relaxation timeτ c electron capture time base potentialV bias voltageβ efficiency of hot electron transport through the MQW baseη + probability of the photoexcited electron escape into the collectorη - probability of the photoexcited electron escape into the emitterσ photoionization cross-sections quantum well capture velocityI optical radiation intensityN emitter and collector donor densityN B base donor densityν 0 velocity of injected electronsν T thermal electron velocityD diffusion coefficientj H hot electron current densityj T current density of thermally excited electronsj P current density of photoexcited electronsj current densityj 0 dark current densityΔ j photocurrent densityg P photocurrent gainR responsivityD * detectivity