The extension of DFPT to two dimensional crystals, in particular gated two-dimensional heterostructure, s described in the following paper:
T. Sohier, M. Calandra, and F. Mauri, Phys. Rev. B 96, 075448 (2017), https://doi.org/10.1103/PhysRevB.96.075448
See example PHonon/example/example17/.