  We propose a scheme for coupling 2D materials to an engineered cavity based
on a defective rod type photonic crystal lattice. We show results from
numerical modelling of the suggested cavity design, and propose using the
height profile of a 2D material transferred on top of the cavity to maximise
coupling between exciton recombination and the cavity mode. The photonic
structure plays a key role in enhancing the launch efficiency, by improving the
directionality of the emitted light to better couple it into an external
optical system. When using the photonic structure, we measured an increase in
the extraction ratio by a factor of 3.4. We investigated the variations in the
flux spectrum when the radius of the rods is modified, and when the 2D material
droops to a range of different heights within the cavity. We found an optimum
enhancement when the rods have a radius equal to 0.165 times the lattice
constant, this enhancement reduces when the radius is reduced or increased.
Finally, we discuss the possible use of solid immersion lenses, in combination
with our photonic structure, to further enhance the launch efficiency and to
improve vertical confinement of the cavity mode.
