  We present the design of mid-infrared and THz quantum cascade laser cavities
formed from planar photonic crystals with a complete in-plane photonic bandgap.
The design is based on a honeycomb lattice, and achieves a full in-plane
photonic gap for transverse-magnetic polarized light while preserving a
connected pattern for efficient electrical injection. Candidate defects modes
for lasing are identified. This lattice is then used as a model system to
demonstrate a novel effect: under certain conditions - that are typically
satisfied in the THz range - a complete photonic gap can be obtained by the
sole patterning of the top metal contact. This possibility greatly reduces the
required fabrication complexity and avoids potential damage of the
semiconductor active region.
