  Frequency multiplication of terahertz signals on a solid state platform is
highly sought-after for the next generation of high-speed electronics and the
creation of frequency combs. Solutions to efficiently generate extreme
harmonics (up to the $31^{\rm{st}}$ harmonic and beyond) of a terahertz signal
with modest input intensities, however, remain elusive. Using fully
nonperturbative simulations and complementary analytical theory, we show that
3D Dirac semimetals (DSMs) have enormous potential as compact sources of
extreme terahertz harmonics, achieving energy conversion efficiencies beyond
$10^{-5}$ at the $31^{\rm{st}}$ harmonic with input intensities on the order of
$10$ MW/cm$^2$, over $10^5$ times lower than in conventional THz high harmonic
generation systems. Our theory also reveals a fundamental feature in the
nonlinear optics of 3D DSMs: a distinctive regime where higher-order optical
nonlinearity vanishes, arising as a direct result of the extra dimensionality
in 3D DSMs compared to 2D DSMs. Our findings should pave the way to the
development of efficient platforms for high-frequency terahertz light sources
and optoelectronics based on 3D DSMs.
