Weyl and Dirac semimetals represent a class of topological materials characterized by unique electronic band structures that host quasiparticles analogous to the relativistic Weyl and Dirac fermions predicted in high-energy physics. These materials exhibit extraordinary electronic properties, including high carrier mobility, topological surface states, and the chiral anomaly, making them promising candidates for next-generation high-speed electronics and quantum technologies.

The defining feature of Weyl semimetals is the presence of non-degenerate band crossings near the Fermi level, known as Weyl nodes. These nodes occur in pairs of opposite chirality and are protected by crystal symmetry or time-reversal symmetry breaking. In momentum space, the dispersion around a Weyl node is linear in all three directions, forming a conical energy-momentum relationship. The separation of Weyl nodes in momentum space gives rise to Fermi arc surface states, which connect the projections of the bulk Weyl points on the surface Brillouin zone. Dirac semimetals, on the other hand, host four-fold degenerate band crossings, effectively behaving as two overlapping Weyl nodes of opposite chirality. These crossings are stabilized by additional crystalline symmetries, such as rotational symmetry.

The chiral anomaly is a hallmark phenomenon in Weyl semimetals, arising when parallel electric and magnetic fields induce a charge imbalance between Weyl nodes of opposite chirality. This leads to a negative magnetoresistance, where the resistivity decreases with increasing magnetic field. The chiral anomaly has been experimentally observed in materials like TaAs and Na3Bi through transport measurements. The effect is quantifiable via the Adler-Bell-Jackiw anomaly equation, which relates the charge pumping rate between Weyl nodes to the applied electromagnetic fields.

Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool for probing the band structure of Weyl and Dirac semimetals. ARPES measurements on Cd3As2 and Na3Bi have directly visualized the Dirac cones and Fermi arcs, confirming their topological nature. For instance, Cd3As2 exhibits two Dirac points along the Γ-Z direction in the Brillouin zone, protected by C4 rotational symmetry. Similarly, TaAs displays multiple pairs of Weyl nodes, as verified by ARPES and scanning tunneling microscopy.

Quantum oscillations, such as Shubnikov-de Haas and de Haas-van Alphen effects, provide additional insights into the Fermi surface topology and carrier dynamics. In TaAs, quantum oscillation studies reveal non-trivial Berry phases and high carrier mobilities exceeding 10,000 cm²/Vs, attributed to the linear dispersion and low effective mass of Weyl fermions. These measurements also help distinguish between trivial and topological carriers, as the latter exhibit π Berry phase shifts in oscillation patterns.

Several materials have emerged as prototypical Weyl and Dirac semimetals. TaAs is a well-studied Weyl semimetal with 12 pairs of Weyl nodes and strong spin-orbit coupling. Cd3As2 is a Dirac semimetal with ultrahigh mobility and robust stability against environmental degradation. Na3Bi, another Dirac semimetal, exhibits a single pair of Dirac points and has been extensively investigated for its quantum transport properties. These materials often require precise stoichiometry and crystal growth conditions, with molecular beam epitaxy and chemical vapor deposition being common synthesis techniques.

The exceptional electronic properties of Weyl and Dirac semimetals open avenues for high-speed electronic applications. Their linear dispersion and high carrier mobility make them suitable for low-power, high-frequency transistors and terahertz detectors. The chiral anomaly and non-linear optical responses also enable novel device concepts, such as chiral plasmons and quantum sensors. Furthermore, the topological protection of surface states may enhance device reliability by mitigating scattering from defects and impurities.

Challenges remain in the practical deployment of these materials, including controlled doping, contact engineering, and integration with conventional semiconductor platforms. However, ongoing advances in material synthesis and nanofabrication techniques continue to bridge these gaps. The exploration of new compounds, such as transition metal phosphides and rare-earth monopnictides, further expands the scope of potential applications.

In summary, Weyl and Dirac semimetals offer a rich playground for fundamental physics and technological innovation. Their unique band structures, chiral anomaly, and topological surface states distinguish them from conventional semiconductors and insulators. Experimental probes like ARPES and quantum oscillations have been instrumental in uncovering their electronic properties, while high-speed electronics and quantum technologies stand to benefit from their extraordinary characteristics. Future research will likely focus on optimizing material quality, discovering new compounds, and translating laboratory findings into practical devices.