Toroidal Fermi-surface geometry and phonon-limited transport in nodal-line semimetals

Technical Summary: Toroidal Fermi-surface geometry and phonon-limited transport in nodal-line semimetals

Problem Statement
Nodal-line semimetals (NLSs) exhibit unconventional quasiparticle dynamics due to their extended band degeneracy and unique Fermi surface topology. While electron-phonon scattering and Bloch-Grüneisen (BG) physics have been extensively studied in conventional metals, Weyl semimetals, and Dirac semimetals, a systematic analysis for NLSs is lacking. Specifically, the toroidal geometry of the Fermi surface in doped NLSs introduces two distinct momentum scales, yet the resulting impact on quasiparticle decay rates and dc conductivity has not been rigorously derived. The authors aim to determine how this toroidal geometry modifies the temperature dependence of transport properties in the clean limit, where inelastic electron-phonon scattering dominates over disorder.

Methodology
The authors employ a minimal theoretical model of a doped circular nodal-line semimetal (NLS) characterized by a thin toroidal Fermi surface.

• Electronic Model: They utilize a two-band effective Hamiltonian for a circular nodal ring in the $k_z=0$ plane. In the low-energy, highly doped regime ($\mu \gg k_B T$), the constant-energy surfaces form a thin torus with a major radius $k_0$ and a minor radius $\kappa_F = \mu/v_0$.
• Phonon Model: Acoustic phonons are treated within an isotropic elastic continuum theory. Only longitudinal phonons are considered, coupled to electrons via a deformation potential, as piezoelectric effects are neglected for centrosymmetric materials like ZrSiS and Ca$_3$P$_2$.
• Transport Formalism: The linearized semiclassical Boltzmann transport equation (BTE) is solved exactly in the "thin-torus" limit ($\kappa_F \ll k_0$). The authors exploit the fact that the leading-order scattering kernel depends only on angular differences on the torus, allowing the collision integral to be treated as a convolution. This enables the derivation of exact integral expressions for the single-particle decay rate ($\Gamma$) and the transport lifetimes ($\tau_{tr}^i$) for different spatial directions.

Key Contributions and Results
The central finding is that the toroidal geometry generates two parametrically distinct Bloch-Grüneisen temperatures:

1. $T_{BG}^{(pol)} \propto c_l \kappa_F$: Associated with momentum transfer along the poloidal direction (minor radius).
2. $T_{BG}^{(tor)} \propto c_l k_0$: Associated with momentum transfer along the toroidal direction (major radius).

Due to the hierarchy $T_{BG}^{(pol)} \ll T_{BG}^{(tor)}$, the system exhibits three distinct temperature regimes for both the quasiparticle decay rate ($\Gamma$) and the conductivity ($\sigma$):

1. Low-Temperature Regime ($T \ll T_{BG}^{(pol)}$):

   • Both angular separations are thermally restricted.
   • Results: $\Gamma \propto T^3$ and $\sigma \propto T^{-5}$ (resistivity $\rho \propto T^5$). This recovers the standard Bloch-Grüneisen behavior found in conventional metals.

2. Intermediate Regime ($T_{BG}^{(pol)} \ll T \ll T_{BG}^{(tor)}$):

   • Poloidal momentum transfers are saturated, but toroidal transfers remain restricted.
   • Results: $\Gamma \propto T^2$ and $\sigma \propto T^{-2}$ (resistivity $\rho \propto T^2$).
   • Significance: The authors note that a $T^2$ resistivity is frequently interpreted as a hallmark of electron-electron scattering. However, this work demonstrates that in circular NLS models, this scaling arises purely from intraband electron-phonon scattering without invoking electron-electron interactions.

3. High-Temperature Regime ($T \gg T_{BG}^{(tor)}$):

   • The angular phase space is fully saturated.
   • Results: $\Gamma \propto T$ and $\sigma \propto T^{-1}$ (resistivity $\rho \propto T$), consistent with standard high-temperature behavior.

Additionally, the study reveals a temperature-dependent conductivity anisotropy. The ratio of axial to in-plane conductivity ($\sigma_{zz}/\sigma_{xx}$) changes from 2 in the low-temperature limit to 4 in the high-temperature limit, a direct consequence of the toroidal Fermi-surface geometry and the directional weighting of the transport factors.

Significance and Claims
The paper claims that the toroidal geometry of the Fermi surface in NLSs fundamentally alters the phase space for electron-phonon scattering, leading to a unique intermediate temperature window with $T^2$ scaling for resistivity. This provides a potential alternative explanation for $T^2$ resistivity in candidate NLS materials (such as ZrSiS, PbTaSe$_2$, or Ca$_3$P$_2$) that does not require electron-electron interactions.

The authors emphasize that the onset of this intermediate regime is controlled by $T_{BG}^{(pol)}$, which scales linearly with the chemical potential ($\mu$). Consequently, tuning the chemical potential (via doping or strain) should shift the lower crossover temperature, offering a specific experimental signature to distinguish this phonon mechanism from other sources of $T^2$ resistivity. The results are presented as predictions for Angle-Resolved Photoemission Spectroscopy (ARPES) linewidths and standard four-probe transport measurements in clean NLS samples.

The study acknowledges limitations, noting that the ideal toroidal geometry assumes a flat nodal line and neglects energy dispersion along the ring. In materials where the nodal line is not perfectly flat or where multiple Fermi pockets exist, the strict separation of scales may be modified, though the authors suggest the two-scale structure and intermediate regime may remain robust features if a dominant toroidal pocket exists.