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A model is constructed to describe the thermal-field emission of electrons from a three-dimensional (3D) topological semimetal hosting Dirac/Weyl node(s). The traditional thermal-field electron emission model is generalised to accommodate the 3D non-parabolic energy band structures in the topological Dirac/Weyl semimetals, such as cadmium arsenide (Cd 3 As 2 ), sodium bismuthide (Na 3 Bi), tantalum arsenide (TaAs) and tantalum phosphide (TaP). Due to the unique Dirac cone band structure, an unusual dual-peak feature is observed in the total energy distribution (TED) spectrum. This non-trivial dual-peak feature, absent from traditional materials, plays a critical role in manipulating the TED spectrum and the magnitude of the emission current. At zero temperature limit, a new scaling law for pure field emission is derived and it is different from the well-known Fowler-Nordheim (FN) law. This model expands the recent understandings of electron emission studied for the Dirac 2D materials into the 3D regime, and thus offers a theoretical foundation for the exploration in using topological semimetals as novel electrodes.

Two-dimensional (2D) semi-Dirac systems, such as 2D black phosphorus and arsenene, can exhibit a rich topological phase transition between insulating, semi-Dirac, and band inversion phases when subjected to an external modulation. How these phase transitions manifest within the quantum transport and shot noise signatures remains an open question thus far. Here, we show that the Fano factor converges to the universal [Formula: see text] at the semi-Dirac phase and transits between the sub-Poissonian ([Formula: see text]) and the Poissonian shot noise ([Formula: see text]) limit at the band inversion and the insulating phase, respectively. Furthermore, the conductance of a 2D semi-Dirac system converges to the contrasting limit of [Formula: see text] and [Formula: see text] at the band inversion and the insulating phases, respectively. The quantum tunneling spectra exhibits a peculiar coexistence of massless and massive Dirac quasiparticles in the band inversion regime, thus providing a versatile sandbox to study the tunneling behavior of various Dirac quasiparticles. These findings reveal the rich interplay between band topology and quantum transport signatures, which may serve as smoking gun signatures for the experimental studies of semi-Dirac systems near the topological phase transition.

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