Orbital enhanced intrinsic nonlinear planar Hall effect for probing topological phase transition in CuTlSe$_{2}$

TL;DR

The paper investigates intrinsic nonlinear planar Hall effect (NPHE) as a probe of band geometry and topological transitions. Using a Weyl-point model and first-principles calculations on CuTlSe$_2$, it shows that Weyl points dramatically enhance NPHE, with the orbital magnetic moment as the main driver and a resonance-like lineshape near the Weyl energy. It reveals a strain-driven topological phase transition in CuTlSe$_2$ between a Weyl semimetal and a topological insulator, with NPHE strength tracking the topology and markedly larger in the Weyl state. This work positions CuTlSe$_2$ as a practical platform for exploring intrinsic NPHE and for using NPHE as a tool to probe topological phases and transitions.

Abstract

The intrinsic nonlinear planar Hall effect proposed in recent studies offers a new way to probe intrinsic band geometric properties in a large class of materials. However, the search of material platforms with a large response remains a problem. Here, we suggest that topological Weyl semimetals can host enhanced intrinsic nonlinear planar Hall effect. From a model study, we show that the enhancement is mainly from the orbital contribution, and the response coefficient exhibits a characteristic resonance-like lineshape around the Weyl-point energy. Using first-principles calculations, we confirm these features for the concrete material CuTlSe$_{2}$. Previous studies have reported two different topological states of CuTlSe$_{2}$. We find this difference originates from two slightly different structures with different lattice parameters. We show that the nonlinear planar Hall response is much stronger in the Weyl semimetal state than in the topological insulator state, and the large response is indeed dominated by orbital contribution amplified by Weyl points. Our work reveals a close connection between nonlinear orbital responses and topological band features, and suggests CuTlSe$_{2}$ as a suitable platform for realizing enhanced nonlinear planar Hall effect.

Figures (7)

• Figure 1: (a) Schematic of linear band dispersion around a Weyl point (Eq. (\ref{['Hw']})). Here, we take the $k_z=0$ plane. (b) The distribution of the orbital magnetic moment $\mathcal{M}_{O,x}^{v}$ for valence band in the $k_z=0$ plane. The Weyl point is indicated with an arrow. (c-d) Calculated intrinsic NPHE tensor element $\chi_{xyyx}$ for the Weyl model Eq. (\ref{['Hw']}). (c) is for chirality $C=\text{sgn}(v_F)=+1$, and (d) is for chirality $C=-1$. The red (blue) curve is the orbital (spin) contribution. $\chi$ is in unit of $10^{-5}$AT$^{-1}$V$^{-2}$. In the calculation, we take $v_F=1$ eV$\cdot$Å and a temperature of 100 K.
• Figure 2: (a) Schematic figure showing two anisotropic Weyl points, WP1 and WP2, connected by a mirror $M_{xy}$. Due to the symmetry operation, the two points have opposite chirality. (b) Intrinsic NPHE tensor element $\chi_{xyyx}$ due to WP1, WP2, and their sum. $\chi$ is in unit of $10^{-5}$AT$^{-1}$V$^{-2}$. In the calculation, we take $v_x=1.4$ eV$\cdot$Å, $v_y=0.6$ eV$\cdot$Å, $v_z=1$ eV$\cdot$Å in model (\ref{['An']}), and a temperature of 100 K.
• Figure 3: (a) Crystal structure of CuTlSe$_{2}$. The solid black lines mark the conventional cell. Some of the point-group symmetries are indicated in the figure. (b) Schematic of the measurement setup for NPHE. We take the crystal $a$ axis as $x$ direction and $b$ axis as $y$ direction. The in-plane $E$ and $B$ fields are specified by angles $\theta$ and $\varphi$, respectively. And the Hall current is in the direction normal to $E$ field. (c) Schematic of the angular dependence of intrinsic NPHE. The two curves correspond to cases with $E$ field aligned with the crystal $a$ axis ($\theta=0$) and $b$ axis ($\theta=\pi/2$), respectively.
• Figure 4: The two topological states of CuTlSe$_{2}$. (a) Band structure computed using the crystal structure of Ref. hahn1953untersuchungen, which is a topological insulator. (b) Band structure computed using the crystal structure of Ref. wang2024weyl, which is a Weyl semimetal. The Weyl points are off the high symmetry paths (see next figure).
• Figure 5: (a) Schematic illustration of positions of the eight Weyl points of CuTlSe$_{2}$ in Weyl semimetal state. They are located in the $k_x$-$k_z$ and $k_y$-$k_z$ planes of the Brillouin zone. The Weyl points with chirality $C=\pm1$ are marked with red and blue colors, respectively. (b) Phase diagram of CuTlSe$_{2}$ with respect to the lattice constants $a$ and $c$. The two experimentally reported structures are marked.