  We study the operation of infrared photodetectors based on van der Waals
heterostructures with the multiple graphene layers (GLs) and n-type emitter and
collector contacts. The operation of such GL infrared photodetectors (GLIPs) is
associated with the photoassisted escape of electrons from the GLs into the
continuum states in the conduction band of the barrier layers due to the
interband photon absorption, the propagation of these electrons and the
electrons injected from the emitter across the heterostructure and their
collection by the collector contact. The space charge of the holes trapped in
the GLs provides a relatively strong injection and large photoelectric gain. We
calculate the GLIP responsivity and dark current detectivity as functions of
the energy of incident infrared photons and the structural parameters. It is
shown that both the periodic selective doping of the inter-GL barrier layers
and the GL doping lead to a pronounced variation of the GLIP spectral
characteristics, particularly near the interband absorption threshold, while
the doping of GLs solely results in a substantial increase in the GLIP
detectivity. The doping "engineering" opens wide opportunities for the
optimization of GLIPs for operation in different parts of radiation spectrum
from near infrared to terahertz.
