Non-linear transport in multifold semimetals
Transport measurements are a powerful way to probe the electronic structure of quantum materials, but the information they contain is often convoluted. Yet, in particular for simple low-energy fermiologies, and by combining linear and non-linear responses, definite conclusions can be drawn -- such as, for instance, in the case of the circular photogalvanic effect in Weyl semimetals. Here, we derive the complete DC transport response functions up to third order in the applied electric field within Boltzmann theory that encode combined information about quantum geometry and band dispersion. We discuss the responses for multifold fermions at high-symmetry momenta in time-reversal symmetric crystals as well as their reduction by symmetry constraints. We exemplify in detail the cases of space group $P2_13$ and $I4_132$, which realize different multifold fermions, and show under which conditions these low-energy excitations can be differentially addressed through their bulk non-linear responses, enabling non-linear valley-tronics. We show that in the latter space group, a purely transverse third order valley current can be produced, leading to an accumulation of electrons of different valleys on opposite surfaces of the material.