Quantum oscillations and nonsaturating magnetoresistivity in nodal-line semimetals

Technical Summary: Quantum Oscillations and Nonsaturating Magnetoresistivity in Nodal-Line Semimetals

Problem Statement
While topological semimetals have been extensively studied, understanding their magnetotransport behaviors remains a critical avenue for extracting intrinsic information such as band structures, Berry phases, and Fermi surface topologies. Specifically, recent experimental observations in the magnetic nodal-line semimetal EuGa4 revealed a "giant" nonsaturating magnetoresistivity (MR) ratio ($R \simeq 2 \times 10^5\%$) persisting up to 40 Tesla. Previous theoretical analyses of this phenomenon relied on semiclassical Boltzmann transport equations. However, a comprehensive quantum mechanical study within the framework of Landau quantization was lacking. Furthermore, while quantum oscillations in nodal-line semimetals have been studied regarding Berry phase accumulation, it remained unclear whether unambiguous signatures of the nodal-line state (specifically the torus-shaped Fermi surface) could be extracted from these oscillations, distinct from the ambiguity often found in intercept analysis.

Methodology
The authors employ a simplified torus model to describe the nodal-line semimetal state, defined by a Hamiltonian involving a radial wave vector, Fermi velocity in the $z$-direction, and a Dirac mass term. The study proceeds through the following steps:

1. Landau Level (LL) Calculation: Using the Peierls substitution and ladder operators in the Landau gauge, the authors derive the analytical expressions for the energy spectra and eigenfunctions of the Landau levels under a perpendicular magnetic field.
2. Chemical Potential Evolution: Assuming a fixed carrier density, the chemical potential ($\mu$) is calculated as a function of the magnetic field by solving the carrier density equation involving the density of states (DOS) under quantization.
3. Transport Coefficients: The magnetoconductivity (MC) tensors ($\sigma_{xx}$ and $\sigma_{xy}$) are calculated using the Kubo-Bastin formula, which incorporates the effects of impurity scatterings via a linewidth broadening parameter ($\eta$).
4. Magnetoresistivity (MR): The longitudinal resistivity ($\rho_{xx}$) is derived from the tensor inversion of the conductivity. The authors compare two calculation methods: one neglecting the Hall conductivity ($\sigma_{xy} \approx 0$) and one including the full tensor inversion.
5. Analysis: Fast Fourier Transform (FFT) analysis is performed on the oscillations of the chemical potential and conductivity to identify frequencies. The results are compared across "low-energy" and "high-energy" regions relative to a Lifshitz transition point.

Key Contributions and Results

• Dual Oscillation Frequencies as a Signature: The study reveals that in the low-energy region (where the Fermi energy lies within the torus structure), both the chemical potential and magnetoconductivity exhibit two distinct oscillation frequencies. These frequencies correspond to the two extremal circular orbits ($\alpha$ and $\beta$) of the characteristic torus Fermi surface. In contrast, the high-energy region (where the Fermi surface is a single closed loop) exhibits only one frequency. The authors identify this dual-frequency signature as a robust experimental indicator of the nodal-line semimetal state, supported by existing de Haas-van Alphen (dHvA) data in materials like CaAgAs and ZrSiTe.
• Nonsaturating MR in the Low-Energy Region: The calculations confirm that nonsaturating MR behavior occurs in the low-energy region, driven by the magnetic field-driven expansion of Landau level spacings which reduces the number of participating levels. However, this behavior is absent in the high-energy region, where the MR tends to saturate.
• Discrepancy in MR Magnitude: When calculating the MR ratio ($R$) using the full Kubo-Bastin formalism (including the Hall conductivity), the authors find that while nonsaturating behavior exists, the magnitude of the MR ratio is significantly smaller than experimental reports. For the parameters chosen to mimic EuGa4, the calculated $R$ ranges between 50% and 400%. This is orders of magnitude smaller than the experimentally reported $2 \times 10^5\%$.
• Role of Hall Conductivity: The study demonstrates that the Hall conductivity ($\sigma_{xy}$) is comparable in magnitude to the longitudinal conductivity ($\sigma_{xx}$) and cannot be neglected. Neglecting $\sigma_{xy}$ (as done in some previous semiclassical analyses) artificially enhances the nonsaturating MR prediction, but even with this simplification, the theoretical values do not reach the "giant" experimental scale.

Significance and Claims
The paper claims to provide a deeper quantum mechanical understanding of magnetotransport in nodal-line semimetals. Its primary significance lies in:

1. Identifying a Topological Signature: Establishing the presence of two distinct oscillation frequencies in the low-energy regime as a definitive experimental signature of the torus Fermi surface inherent to nodal-line semimetals.
2. Re-evaluating the Origin of Giant MR: The authors modestly conclude that the "giant" MR observed in EuGa4 likely does not arise solely from the nodal-line semimetal state. Instead, they suggest the discrepancy between their theoretical results (hundreds of percent) and experimental results (hundreds of thousands of percent) may stem from missing physical mechanisms in the model, specifically the coupling between localized Eu magnetic moments and itinerant carriers (spin-dependent scatterings), which were not included in the current calculation.

The work serves as a theoretical baseline, suggesting that while nodal-line topology explains the nonsaturating nature of the MR and the oscillation frequencies, it is insufficient to explain the extreme magnitude of the MR ratio without considering additional magnetic interactions specific to the material system.