  The mid-infrared (MIR), far-infrared (FIR) to terahertz (THz) frequencies are
the least developed parts of the electromagnetic spectrum for applications.
Traditional semiconductor technologies like laser diodes and photodetectors are
successful in the visible light range, but are still confronted with great
challenges when extended into the MIR/FIR/THz range. In this paper, we
demonstrate that topological insulators (TIs), especially those with
Mexican-hat band structure (MHBS), provide a route to overcome these
challenges. The optical responses of MHBS TIs can be one to two orders of
magnitude larger than that of normal semiconductors at the optical-transition
edge. We explore the databases of topological materials and discover a number
of MHBS TIs whose bandgaps lie between $0.05\sim 0.5~\rm eV$ and possess giant
gains (absorption coefficients) on the order of $10^4 \sim 10^5~\rm cm^{-1}$ at
the transition edge. These findings may significantly boost potential
MIR/FIR/THz applications such as photon sources, detectors, ultrafast
electro-optical devices, and quantum information technologies.
