Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP

TL;DR

This paper investigates the electronic, spin, and topological properties of four Weyl semimetals, revealing how anisotropy, tilt, and pairing of Weyl nodes influence their physical characteristics and topological invariants.

Contribution

It provides a detailed analysis of how anisotropy, tilt, and pairing of Weyl nodes affect the electronic structure and topological properties of TaAs, TaP, NbAs, and NbP.

Findings

Weyl bands exhibit strong anisotropy and tilt, especially for W2 nodes.

Node tilts and positions significantly impact the density of states.

Pairing of nodes alters spin textures and Berry curvature, indicating topological changes.

Abstract

By means of $ab \ initio$ band structure methods and model Hamiltonians we investigate the electronic, spin and topological properties of four monopnictides crystallizing in body centered tetragonal structure. We show that the Weyl bands around a Weyl point W1 or W2 possess a strong anisotropy and tilt of the accompanying Dirac cones. These effects are larger for W2 nodes than for W1 ones. The node tilts and positions in energy space significantly influence the density of states of single-particle Weyl excitations. The node anisotropies destroy the conventional picture of (anti)parallel spin and wave vector of a Weyl fermion. This also holds for the Berry curvature around a node, while the monopole charges are independent as integrated quantities. The pairing of the nodes strongly modify the spin texture and the Berry curvature for wave vectors in between the two nodes. Spin components may change their orientation. Integrals over planes perpendicular to the connection line yield finite Zak phases and winding numbers for planes between the two nodes, thereby indicating the topological character.

Tables

Table 1. Table SM1: Weyl node positions in the Brillouin zone.

	TaAs			TaP
	$k_x\left(2\pi /a\right)$	$k_y\left(2\pi /a\right)$	$k_z\left(2\pi /c\right)$	$k_x\left(2\pi /a\right)$	$k_y\left(2\pi /a\right)$	$k_z\left(2\pi /c\right)$
W1	0.0078	0.5103	0.0000	0.0077	0.5174	0.0000
	-0.0078	0.5103	0.0000	-0.0077	0.5174	0.0000
	-0.5103	0.0078	0.0000	-0.5174	0.0077	0.0000
	-0.5103	-0.0078	0.0000	-0.5174	-0.0077	0.0000
	0.0078	-0.5103	0.0000	0.0077	-0.5174	0.0000
	-0.0078	-0.5103	0.0000	-0.0077	-0.5174	0.0000
	0.5103	0.0078	0.0000	0.5174	0.0077	0.0000
	0.5103	-0.0078	0.0000	0.5174	-0.0077	0.0000
W2	0.0198	0.2818	0.5905	0.0161	-0.2741	0.5885
	-0.0198	0.2818	0.5905	-0.0161	-0.2741	0.5885
	-0.2818	0.0198	0.5905	0.2741	0.0161	0.5885
	-0.2818	-0.0198	0.5905	0.2741	-0.0161	0.5885
	0.0198	-0.2818	0.5905	0.0161	0.2741	0.5885
	-0.0198	-0.2818	0.5905	-0.0161	0.2741	0.5885
	0.2818	0.0198	0.5905	-0.2741	0.0161	0.5885
	0.2818	-0.0198	0.5905	-0.2741	-0.0161	0.5885
	0.0198	0.2818	-0.5905	0.0161	-0.2741	-0.5885
	-0.0198	0.2818	-0.5905	-0.0161	-0.2741	-0.5885
	-0.2818	0.0198	-0.5905	0.2741	0.0161	-0.5885
	-0.2818	-0.0198	-0.5905	0.2741	-0.0161	-0.5885
	0.0198	-0.2818	-0.5905	0.0161	0.2741	-0.5885
	-0.0198	-0.2818	-0.5905	-0.0161	0.2741	-0.5885
	0.2818	0.0198	-0.5905	-0.2741	0.0161	-0.5885
	0.2818	-0.0198	-0.5905	-0.2741	-0.0161	-0.5885
	NbAs			NbP
	$k_x\left(2\pi /a\right)$	$k_y\left(2\pi /a\right)$	$k_z\left(2\pi /c\right)$	$k_x\left(2\pi /a\right)$	$k_y\left(2\pi /a\right)$	$k_z\left(2\pi /c\right)$
W1	0.0026	0.4859	0.0000	0.0029	0.4921	0.0000
	-0.0026	0.4859	0.0000	-0.0029	0.4921	0.0000
	-0.4859	0.0026	0.0000	-0.4921	0.0029	0.0000
	-0.4859	-0.0026	0.0000	-0.4921	-0.0029	0.0000
	0.0026	-0.4859	0.0000	0.0029	-0.4921	0.0000
	-0.0026	-0.4859	0.0000	-0.0029	-0.4921	0.0000
	0.4859	0.0026	0.0000	0.4921	0.0029	0.0000
	0.4859	-0.0026	0.0000	0.4921	-0.0029	0.0000
W2	0.0064	0.2790	0.5736	0.0047	0.2710	0.5770
	-0.0064	0.2790	0.5736	-0.0047	0.2710	0.5770
	-0.2790	0.0064	0.5736	-0.2710	0.0047	0.5770
	-0.2790	-0.0064	0.5736	-0.2710	-0.0047	0.5770
	0.0064	-0.2790	0.5736	0.0047	-0.2710	0.5770
	-0.0064	-0.2790	0.5736	-0.0047	-0.2710	0.5770
	0.2790	0.0064	0.5736	0.2710	0.0047	0.5770
	0.2790	-0.0064	0.5736	0.2710	-0.0047	0.5770
	0.0064	0.2790	-0.5736	0.0047	0.2710	-0.5770
	-0.0064	0.2790	-0.5736	-0.0047	0.2710	-0.5770
	-0.2790	0.0064	-0.5736	-0.2710	0.0047	-0.5770
	-0.2790	-0.0064	-0.5736	-0.2710	-0.0047	-0.5770
	0.0064	-0.2790	-0.5736	0.0047	-0.2710	-0.5770
	-0.0064	-0.2790	-0.5736	-0.0047	-0.2710	-0.5770
	0.2790	0.0064	-0.5736	0.2710	0.0047	-0.5770
	0.2790	-0.0064	-0.5736	0.2710	-0.0047	-0.5770

Full text

\DeclareAcronym

WP short=WP, long=Weyl point,

\DeclareAcronymDOS short=DOS, long=density of states,

\DeclareAcronymbct short=bct, long=body-centered tetragonal,

\DeclareAcronymARPES short=ARPES, long=angle-resolved photoemission spectroscopy,

\DeclareAcronymSOC short=SOC, long=spin-orbit coupling,

\DeclareAcronym3D short=3D, long=three-dimensional,

\DeclareAcronym2D short=2D, long=two-dimensional,

\DeclareAcronymDFT short=DFT, long=density functional theory,

\DeclareAcronymXC short=XC, long=exchange and correlation,

\DeclareAcronymBZ short=BZ, long=Brillouin zone,

\DeclareAcronymTWS short=TWS, long=topological Weyl semimetal

\DeclareAcronymTDS short=TDS, long=topological Dirac semimetal

\DeclareAcronymPBE short=PBE, long=Perdew, Burke and Ernzerhof,

\DeclareAcronymGGA short=GGA, long=generalized gradient approximation,

\DeclareAcronymPW short=PW, long=plane wave,

\DeclareAcronymEELS short=EELS, long=electron energy loss spectra,

\DeclareAcronymQE short=QE, long=QUANTUM ESPRESSO,

\DeclareAcronymIR short=IR, long=infrared,

Influence of anisotropy, tilt and pairing of Weyl nodes:

The Weyl semimetals TaAs, TaP, NbAs, and NbP

Davide Grassano and Olivia Pulci and Elena Cannuccia

Dipartimento di Fisica, Università ”Tor Vergata” Roma and INFN, via della Ricerca Scientifica, I-00133 Rome, Italy

  

Friedhelm Bechstedt

Institut für Festkörpertheorie und -optik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

I Weyl vs Trivial points

The crystals of TaAs, TaP, NbAs, NbP contain two mirror planes, $M_{x}$ and $M_{y}$ and two glide mirror planes, $M_{xy}$ and $M_{-xy}$ (Fig.SM4,SM5). Thereby, the Cartesian axes $x$ and $y$ are perpendicular to the tetragonal axis, which defines the $z$ axis. The mirror symmetries protect four nodal rings (Fig.SM3 in the mirror planes, which are represented by gapless points when \acSOC is not included. With \acSOC a single nodal ring evolves into three pairs of (gapless) band touching points, one pair is in the $k_{z}=0$ plane of the \acBZ (labeled W1) and two pairs off the $k_{z}=0$ plane (labeled W2).

For one WP at ${\bf k}_{W}$, the time-reversal symmetry requires another one at $-{\bf k}_{W}$. Because of their pairing, there must exist two more Weyl points at ${\bf k}^{\prime}_{W}$ and $-{\bf k}_{W}^{\prime}$. Pairs of \acpWP are formed at ${\bf k}_{W}$ and ${\bf k}_{W}^{\prime}$ as well as $-{\bf k}_{W}$ and $-{\bf k}_{W}^{\prime}$. In the W1 case the other four \acpWP follow because of the $M_{x}$ and $M_{y}$ mirror operations. In the W2 case the total number of ${\bf k}_{W}$ points is doubled because of their occurrence in planes at $k_{z}=k_{Wz}$ and $k_{z}=-k_{Wz}$. The two \acpWP in one pair are close to each other in $k_{x}$- or $k_{y}$-direction around $k_{x}=0$ and $k_{y}=0$, respectively. The ${\bf k}$-space splittings of such a pair along these two Cartesian directions amount to 0.0156 (TaAs), 0.0154 (TaP), 0.0052 (NbAs), and 0.0058$\frac{2\pi}{a}$ (NbP) in the W1 case or 0.0396 (TaAs), 0.0322 (TaP), 0.0128 (NbAs), and 0.0094$\frac{2\pi}{a}$ (NbP) in the W2 one. The positions given in Table SM1 fully obey the discussed $M_{x}$- and $M_{y}$-mirror symmetries up to four digits. This fact underlines the accuracy of our numerical procedures to determine the Weyl node positions.

For TaAs the absolute positions of W1 and W2 are in very good agreement with similar \acDFT calculations Huang et al. (2015); Xu et al. (2015); Lee et al. (2015); Buckeridge et al. (2016). This also holds for the calculations for the other three monopnictides, if the same denotations of W1 and W2 as well as the same $k_{z}$-zero are applied Lee et al. (2015); Xu et al. (2017). Also the measured ${\bf k}$-space separations between two nodes of a pair, e.g. for W2 in TaAs Xu et al. (2015); Souma et al. (2016), or the absolute position of the \acpWP in NbP Xu et al. (2017) agree well with the computed results.

II Nodal lines

The nodal lines can be observed by running the calculation without SOC. Each point of the nodal line is selected, from a band structure calculation over a fine mesh of k-points on the Y=0 plane, using the condition $E_{gap}<1meV$

III Symmetry planes

IV Electronic properties