Quantum transport in topological nodal-line semimetals

Figures & data

Figure 1. (a) Sketch of the band structure of the nodal-line semimetal and the drumhead-like surface states in the surface Brillouin zone. (b) Torus-shaped Fermi surface.

Figure 2. (a) Torus-shaped Fermi surface, with major radius $k_0$, minor radius $\kappa$, toroidal angle $\theta$, and poloidal angle $\phi$. (b) The impurity potentials results in 3D and 2D diffusion behaviors, respectively. (c) A coherent backscattering process from wave vector $k$ to $-k$ around the toroidal direction, which is a 3D diffusion behavior. (d) Backscattering from wave vector $\delta k$ to $-\delta k$ around poloidal direction, which is a 2D diffusion behavior [54].

Figure 3. Relevant Feynman diagrams. (a) Three leading-order diagrams of the quantum interference corrections to the conductivity, with one bare Hikami box and two dressed Hikami boxes. The arrowed solid line represents Green functions and the arrowed dashed line represents impurity scattering. (b) Ladder diagram vertex correction to the velocity. (c) The Cooperon correction [54].

Figure 4. The magnetoconductivity $\delta {\sigma }_z(B)$ in the (a) SR limit and (b) LR limit for different phase coherence lengths ${\mathrm{\ell }}_{\varphi }$, which are represent by the different color [54].

Figure 5. (a) The maximum ($\alpha$) and minumum ($\beta$) cross sections in the nodal-line plane of the torus Fermi surface. (b) The maximum ($\gamma$) and minimum ($\delta$) cross sections out of the nodal-line plane of the torus Fermi surface [66].

Table 1. The phase shift $\phi$ of nodal-line semimetals in Figure 5 [66]

Cross section	Berry phase	Min./Max	Electron	Hole
α	0	Max.	-1/2+0-1/8=-5/8	+5/8
β	0	Min.	-1/2+0+1/8=-3/8↔5/8	-5/8
γ	0	Max.	-1/2+0-1/8=-5/8	+5/8
δ	π	Min.	-1/2+π/2π+1/8=1/8	-1/8

Figure 6. (a) Schematic diagram of the torus-shaped Fermi surface in nodal-line semimetal, the major radius is $k_0$, minor radius is $\kappa$, the toroidal angle and poloidal angle are $\theta$,$\varphi$, respectively. (b) Finite berry curvature distribution in the poloidal cross section. (c-e) The deformation process of the Fermi surface induced by the increasing external magnetic field [97].

Figure 7. (a) MR as a function of the external magnetic field with and without sign reversal. (b) Correspondence between MR’s sign reversal point and the critical magnetic field $B^{\ast }$ (open circles) [97].

Figure 8. Three types of AR, (a) RAR, (b) SAR, (d) AAR. (c) torus-shaped Fermi surface [101].

Table 2. Different features in the RAR, SAR, and AAR [101]

	$v_x$	$v_y$	$v_z$
RAR	$-$	$-$	$-$
SAR	$-$	$+$	$+$
AAR	$-$	$-\left(+\right)$	$+\left(-\right)$

Figure 9. (a)The two-terminal setup for detecting the AAR. (b) Nonlocal conductance in RAR regime. (c) (d) Nonlocal conductance in AAR regime with different interface barrier [101].

Figure 10. (a) A junction between normal metal and nodal-line semimetal, The drumhead-like surface state at the boundary are encircled by the projection of the bulk nodal loop onto the surface Brillouin zone. The inset show trivial (dashed line) and nontrivial (solid line) band dispersion. (b) Probability of spin-flipped reflection [101].

Figure 11. (a) Setup for detecting DSS with spin transport. (b) The spin (thick lines) and charge (thin lines) with different interface barrier heights. (c) Effect of finite dispersion of DSS on the spin conductance, where the interface barrier $Z=10$. (d) Setup for detecting DSS with charge transport. (e) Charge conductance with different interface barrier heights. (f) Effect of finite dispersion of DSS on the charge conductance, where the interface barrier $Z=10$ [105].

Figure 12. MR measurements of ZrSiS. (a) The curves of MR in different temperatures. (b) The MR curves with different measurement angles. (c) The MR data in the log-log frame. (d) The MR’s angular dependence, forming a butterfly shape [87].

Chen W, Lu H-Z, Zilberberg O. Weak localization and antilocalization in nodal-line semimetals: dimensionality and topological effects. Phys Rev Lett. 2019;122:196603.

 PubMed Web of Science ®Google Scholar

Li C, Wang C, Wan B, et al. Rules for phase shifts of quantum oscillations in topological nodal-line semimetals. Phys Rev Lett. 2018;120:146602.

 PubMed Web of Science ®Google Scholar

Yang M-X, Geng H, Luo W, et al. Sign reversal of magnetoresistivity in massive nodal-line semimetals due to the Lifshitz transition of the Fermi surface. Phys Rev B. 2021;104:165149.

 Web of Science ®Google Scholar

Luo W, Chen W, Xing D. Anomalous Andreev reflection on a torus-shaped Fermi surface. Sci China Phys Mech Astron. 2021;64:267262.

 Web of Science ®Google Scholar

Chen W, Luo K, Li L, et al. Proposal for detecting nodal-line semimetal surface states with resonant Spin-flipped reflection. Phys Rev Lett. 2018;121:166802.

 PubMed Web of Science ®Google Scholar

Wang X, Pan X, Gao M, et al. Evidence of both surface and bulk Dirac bands and anisotropic nonsaturating magnetoresistance in ZrSiS. Adv Electron Mater. 2016;2:1600228.

 Web of Science ®Google Scholar