Higher-order Oscillatory Planar Hall Effect in Topological Kagome Metal

Exploration of exotic transport behavior for quantum materials is of great interest and importance for revealing exotic orders to bring new physics. In this Letter, we report the observation of exotic prominent planar Hall effect (PHE) and planar anisotropic magnetoresistivity (PAMR) in strange kagome metal KV$_3$Sb$_5$. The PHE and PAMR, which are driven by an in-plane magnetic field and display sharp difference from other Hall effects driven by an out-of-plane magnetic field or magnetization, exhibit exotic higher-order oscillations in sharp contrast to those following empirical rule only allowing twofold symmetrical oscillations. These higher-order oscillations exhibit strong field and temperature dependence and vanish around charge density wave (CDW) transition. The unique transport properties suggest a significant interplay of the lattice, magnetic and electronic structure in KV$_3$Sb$_5$. This interplay can couple the hidden anisotropy and transport electrons leading to the novel PHE and PAMR in contrast to other materials.


INTRODUCTION
Planar Hall effect (PHE) is a unique transport phenomena driven by an in-plane magnetic-field-induced rotation of the principal axes of the resistivity tensor 1, 2 . Because of different origins, the PHE is in many aspects quite different from those Hall effects driven by an out-of-plane magnetic field or magnetization as shown in Fig.1 (a). Obvious PHE has been observed in a few ferromagnetic metals 1, 2 and nonmagnetic semimetals with strong orbital anisotropy of electronic structure 3,4 . It is widely used for designing and fabricating commercial magnetic sensors, especially for threedimensional (3D) highly compacted, and ultra-sensitive 'lab-on-a-chip (LOC)' devices for the next generation chips. Another great interest for PHE lies in its angular dependence of the direction of magnetic field, which can be applied to infer the information of the underlying magnetic order or electronic structure. For example, to investigate this effect and pursue its origin in a quantum system may reveal nontrivial topological physics or reveal the exotic states and orders to advance the understanding of fundamental physics [5][6][7][8][9][10] .
Recently, superconductivity was observed in a new family of layered kagome metals AV 3 Sb 5 (A = K, Rb, Cs) [11][12][13][14][15][16] . Their normal states are identified as Z 2 topological metals with multiple topologically nontrivial band structures such as flat band, van Hove singularity and Dirac-fermion dispersion in close proximity to the Fermi level [17][18][19][20][21][22] . In the absence of magnetism, it is surprising to observe a giant anomalous Hall effect (AHE) in these materials. To reconcile the observations, the AHE is considered to strongly correlate to novel orders such as chiral charge-density-wave (CDW) or nematic order accompanying with symmetry breaking [23][24][25][26][27][28][29][30][31] . However, because the correlations among spin, charge, lattice, and other orders are expected to be important, it remains great challenge to get a full understanding of the origin of the various novel transport properties observed in this family of materials. In this work, we observe prominent PHE and planar anisotropic magnetoresistivity (PAMR) in KV 3 Sb 5 . More interestingly, below the CDW transition the in-plane applied magnetic field drives exotic higher-order oscillations for PHE and PAMR violating the empirical law in former materials (even in recent topological materials). The unique behaviors exhibit strong field response accompanying with non-monotonous anisotropic field dependent 3 in-plane resistivity. These novel planar transport properties suggest a strong coupling between transport electrons and anionotropies from the system with various orders in contrast to other materials. By scrutinizing various possible mechanisms, we suggest a significant interplay of the lattice, magnetism and electron associated with the fluctuations as the media leading to those novel transport behavior. Fig.1 (a), the PHE and PAMR can be characterized by measuring the transverse resistivity (ρ xy ) and longitudinal resistivity (ρ xx ) with applying an in-plane magnetic field (rotating within the ab plane of KV 3 Sb 5 shown in Fig.1 (b)) which fails to drive the ordinary Hall effect (OHE) measured with an out-of-plane magnetic field.

As shown in
Usually, obvious PHE and PAMR are only observed in a few kinds of materials and follow the angular dependence as: where ρ xy represents the in-plane Hall resistivity that directly shows the PHE, ρ xx is the PAMR, and ∆ρ = ρ ⊥ − ρ is the resistivity anisotropy (called chiral resistivity in topological materials) with ρ ⊥ and ρ representing the resistivity with the applied field µ 0 H perpendicular (90 • ) and parallel (0 • ) to the electric current respectively 1 . According to these formulas, angular dependent ρ xy (ρ xy (θ)) and ρ xx (ρ xx (θ)) exhibit twofold oscillations and a relative 45 • -angle shift for ρ xy (θ) and ρ xx (θ) as shown in Figs.1(d) and (e) [32][33][34][35][36][37] . It is observed the anisotropy from lattice, magnetism, or Fermi surface etc usually hardly affect the in-plane transport behavior such as PHE and PAMR. For KV 3 Sb 5 , obvious PHE and PAMR are observed at 2 K even with a small applied field of µ 0 H = 2 T after subtracting the OHE and out-of-plane magneto resistivity 9,13 . ρ xy and ρ xx  (1) and (2) and fail to result in high-order oscillatory components.

DISCUSSION
Usually the additional anisotropy from a system is difficult to behave in conventional PHE and PAMR even the system hosts high anisotropic Fermi surface, magnetism, 7 or lattice such as in kagome magnet Co 3 Sn 2 S 2 , tetragonal ZrSiSe, and quasi-one di- no long-range or short-range magnetic orders were observed, thus the anisotropy from conventional static magnetic order also seems not applicable. In semi-class model, the resistivity comes from the electron scattering with the lattice. Thus the anisotropy of the lattice can naturally host anisotropic carrier scattering with constrictions of the lattice's symmetry. It is noticed that even in presence of some distortion due to the CDW the lattice for KV 3 Sb 5 still keeps hexagonal structure. In the two dimensional case with C 6 rotation invariance 38 , ρ xy (θ) and ρ xx (θ) can be expressed as: where S xy2 , C xy2 , S xy4 , C xy4 , and C xy6 , are the coefficients for two, four, and sixfold symmetrical oscillations for PHE and PAMR related to lattice symmetry 38,41 . φ is the angle between the current I and a axis and here φ=0 with I//a. By using these formulas, the our data can be well fitted in consisted with former FFT analysis. Above CDW transition S xy4 , C xy4 , and C xy6 become to zero, the formula 3 and 4 become to formula 1 and 2 respectively indicating the PHE and PAMR obeying the conventional empirical law. Below the CDW transition, S xy4 , C xy4 , and C xy6 increase with the decreasing temperature and exhibit a strong field response with more contributions for oscillatory ρ xy and ρ xx at high fields. The temperature and field dependence of highorder oscillatory components (S xy4 , C xy4 , and C xy6 ) suggests the strong enhancement of a unique coupling between anisotropy from lattice and electrons scattering for planar transport behavior.
Accompanying with CDW transition, it is observed that obvious multiple fluctuations 9 emerge simultaneously. From lattice side, strong phononic fluctuations were revealed accompanying with lattice distortions 42 . From the electronic side, the electronic instability can host electronic fluctuations (including the nematic fluctuations) and result in electronic nematicity at lower temperatures with symmetry reduction 43 . Moreover, from magnetism side, for KV 3 Sb 5 though in absence of long-range or short range magnetic order, the magnetic fluctuations due to observed orbital ordering would inherit the crystalline anisotropy of the kagome lattice which plays a crucial role in the magnetic properties especially for the appearance of time-reversal symmetry breaking. We find that the observed ρ < ρ ⊥ (see Fig.2) is consistent with this picture since a magnetic field will suppress magnetic fluctuations in the field direction. In fact these fluctuations are correlated and can origin from same instability due to CDW transition indicating presentence of multiple orders in the system. These correlated fluctuations can be attributed to mediate the crystalline anisotropy and electron scatterings for planar resistivity tensors to behave higher-order oscillatory components (as shown in Fig.4 (f)) revealing a unique interplay of lattice, electron and magnetism in KV3Sb5.
In summary, prominent PHE and PAMR are observed in KV 3 Sb 5 . The applied field would drive higher-order oscillations for both ρ xy and ρ xx below the CDW transition, resulting in the violation of the empirical law. This exotic phenomena suggest a strong and complicated coupling between lattice symmetry and electron scatterings for pla-nar transport properties. We expect that such unconventional PHE and PAMR are not only potential for next generation 3D chips, but also reveal a unique interplay between various degrees of freedom for materials with frustrated crystal structures, lattice distortions and anisotropic magnetic and fluctuations.

DATA AVAILABILITY
The data supporting the findings of this study are available within the paper and in the Supplementary Information, and also are available from the corresponding authors upon reasonable request.   The field dependence of high-order oscillatory coefficients for PHE and PAMR.
S 2 , S 4 , S 6 , C 2 , C 4 , and C 6 are acquired from the amplitude of peaks of FFT spectrum of (a) and (b) . S xy2 , S xy4 , C xx2 , C xx4 , and C xx6 are acquired from the data fitting for the formulas (3) and (4).  shown with the left axis. The temperature dependent resistance along c axis R c xx is shown with the right axis with a prominent CDW anomaly around 78 K .
(f) The schematic for the coupling between distorted lattice and Fermi surface leading to high symmetrical oscillatory PHE driven by an in-plane field.