Giant magneto-cubic in-plane Hall effect in a nonmagnetic material

Domain: physics
Published: 2026-03-20
Editorial outcome: major_revision

Abstract
In-plane Hall effect triggered by an external magnetic field applied in the transport plane has
attracted significant experimental attentions in recent few years. However, most experiments focus
on magnetic materials, where the existence of magnetic ordering may complicate understanding
the physics behind, and the relatively small signal magnitudes limit the application of the effect.
Here, we report a giant in-plane Hall effect in a nonmagnetic half-Heusler compound LuAuSn,
with a magnitude exceeding all the previously reported values. A 2𝜋⁄3-period of in-plane Hall
effect and the consistent cubic dependence on the magnetic field are observed, realizing the long-
sought theoretical prediction of magneto-cubic in-plane Hall effect under threefold rotational
symmetry in an unexpected material. The scaling law analysis and first-principles calculations
indicate that extrinsic side jump and skew scattering processes from both impurity and phonon
scatterings dominate the observed effect. These findings unravel a new type of magneto-nonlinear
in-plane Hall effect, and its large magnitude and wide-temperature operation may open the door
to practical applications of in-plane Hall effect.

Introduction
Hall effect is one of the most important physical phenomena in condensed matter physics 1-3.
Experimentally, the Hall effect is most usually manifested with the external magnetic field applied
along the normal direction of the transport plane defined by the driving electric current and the
generated transverse voltage (Fig. 1a). In 2018, the observation of Hall effect in ZrTe induced by
5
magnetic field applied within the transport plane unravels the in-plane Hall effect (IPHE) (Fig.1b)
4, 5, which was later systematically studied in VS-VS , showing a perfect linear dependence on the
2
in-plane magnetic field and a 2𝜋 -periodic angular dependence 6. Such studies recall earlier
theoretical investigations on the quantized anomalous Hall effect induced by in-plane
magnetization 4, 7, 8, and then inspired intensive experimental attentions on IPHE in magnetic
materials, such as Fe Sn , RuO , CuPt/CoPt heterostructure, EuCd Sb , Fe films, SrRuO ,
3 2 2 2 2 3
Co Sn S , and EuZn Sb 5, 9-16. The key issues in these studies are how the magnetic order
3 2 2 2 2
parameter respond to the magnetic field and how this interplay result in the observed angular
dependence and scaling on the field 9, 17, 18. On the other hand, theoretical studies of IPHE in recent
few years mostly focused on nonmagnetic materials 8, 19-29. This is because nonmagnetic materials
avoid the complexity introduced by the intricate interplay of magnetic field and the magnetic order
parameter. Therefore, to observe this fundamental transport effect in nonmagnetic materials is
critical for revealing the IPHE physics induced by magnetic field alone. However, compared to the
large body of IPHE experiments in magnetic materials, IPHE in nonmagnetic materials has so far
only been achieved in Cd As 30, PtBi 31, ZrTe 32-34, LaAlO /SrTiO interfaces 35, and IrO 10.
3 2 2 5 3 3 2
While the IPHE in IrO has been attributed to Lorentz force, the origins of those in ZrTe and
2 5
LaAlO /SrTiO interfaces are still elusive 32-35, partly due to their complex dependences on
3 3
magnetic field beyond simple low order power-law dependence. Recently, two important advances
concerning Cd As and PtBi have confirmed the theoretically proposed field-induced Berry
3 2 2
curvature 19, 21, 25-27. However, the extrinsic contribution to IPHE 25, 36, like side-jump and skew-
scattering, has not been detected in experiments.

In this work, we make a step forward by observing the IPHE in a nonmagnetic half-Heusler
compound LuAuSn, which shows a 2𝜋⁄3 -period angular dependence on the magnetic field
orientation and a cubic scaling on the field strength up to 𝐵 = 3 T, realizing the long-sought
theoretical prediction of magneto-cubic IPHE under 𝐶 symmetry in an unexpected material 4,
3𝑧
22, 25. The scaling law analysis and the comparison with the results of first-principles calculations
indicate that extrinsic side jump and skew scattering processes from both impurity and phonon
scatterings dominate the observed IPHE. At 𝐵 = 3 T and 𝑇 = 2 K, the Hall conductivity is 306
Ω−1cm−1, which is more than one order of magnitude larger than the value in ZrTe
5
measured at
similar temperature and field and is also larger than all the up-to-now reported in-plane Hall
conductivities in magnetic materials. The IPHE conductivity decreases rapidly with increasing
temperature but still remains sizable up to room temperature. The nonmagnetic nature of the
material, the clean 𝐵3 scaling at a wide range of magnetic field, the giant in-plane Hall
conductivity, and the broad operating temperature range jointly make LuAuSn an ideal platform
for investigating IPHE and related in-plane Hall families.

Results
Cubic-B dependent IPHE in C -symmetric systems
3z
The symmetry determines whether the IPHE can exist, as well as its angle dependent and B-
dependent behaviors. We first review the IPHE from a symmetry perspective. As a time-reversal-
odd effect, Hall effects require the presence of time-reversal-breaking fields, such as the magnetic
field in nonmagnetic materials. The Hall effect can be described by a magnetic-field-dependent
Hall vector 𝐄H = 𝛒H(𝐁)×𝐣, where 𝛒H = (𝜌H ,𝜌H ,𝜌H ), 𝐣 is the applied current density, and
𝑦𝑧 𝑧𝑥 𝑥𝑦
𝐄H is the induced Hall electric field, which leads to the experimentally measured Hall voltage.
The Hall vector is an odd function of magnetic field: 𝛒H(𝐁) = −𝛒H(−𝐁). The component of the
Hall vector perpendicular to the magnetic field applied in the transport plane corresponds to the
IPHE of interest. Importantly, the IPHE is independent of the relative direction between the
magnetic field and the applied current density 12. This distinguishes it from the planar Hall effect,
which is essentially an anisotropic magnetoresistance effect. Since inversion symmetry imposes
no constraints on the Hall vector, it is sufficient to consider the constraints imposed by proper
rotational symmetries. Consider an infinitesimal rotation described by ℜ = 1−d𝜃𝐋∙𝐧, where 𝐧
is the direction of the rotation axis, and 𝐿 𝑗𝑘 = 𝜖 are the generators of 𝔰𝔬(3) Lie group with
𝑖 𝑖𝑗𝑘
𝜖 being the Levi-Civita symbol. This infinitesimal rotation enforces ℜ𝛒H(ℜ−1𝐁) = 𝛒H(𝐁),
𝑖𝑗𝑘
which leads to a differential equation: 𝐧×𝛒H = (𝐧×𝐁)∙𝛁 𝛒H. To fulfill this condition, the
𝐁
Hall vector must be parallel to the magnetic field. This implies that the out-of-plane Hall effect is
unrestricted by rotational symmetry, while the IPHE requires reducing symmetry.

In the following, we define the 𝑥𝑦 plane as the transport plane. The relevant Hall resistance
is 𝜌H , and the magnetic field is applied in 𝑥𝑦 plane. Under 𝐶 symmetry, 𝜌H remains
𝑦𝑥 2𝑧 𝑦𝑥
unchanged, while the in-plane magnetic field reverses sign. Therefore, the IPHE is forbidden. Due
to the same reason, the IPHE is also forbidden under 𝐶 , and 𝐶 symmetries. On the other hand,
4𝑧 6𝑧
the 𝐶 -symmetric may allow IPHE. It should be noted that 𝐶 forbids IPHE in the linear
3𝑧 3𝑧
order of 𝐵, and the leading order effect should exhibit 𝐵3 dependence (Figs. 1d and 1f). Given
that the 𝐶 symmetry is possessed by wide classes of topological materials, such a magneto-
3𝑧
cubic IPHE should be the leading magneto-nonlinear Hall effect 9 in these materials, greatly
enlarging the platform of IPHE. In fact, a 𝐵3-dependent IPHE has recently been observed 15 in a
magnetic Weyl semimetal Co Sn S . However, the 𝐵3-dependent IPHE is in principle allowed in
3 2 2
nonmagnetic materials without the impact of magnetic ordering, though it has not yet been
experimentally demonstrated.

LuAuSn is a half-Heusler compound with a non-centrosymmetric crystal structure, which
belongs to 4̅3𝑚 point group (Fig. 1g) 37, 38. To investigate the IPHE, LuAuSn single crystal were
synthesized using Sn-flux method 39. The (111) plane is natural growth surfaces and can easily
obtain the flat plane that can reduce the misleading in IPHE measurement. The (111) plane of
single crystal are determined by X-ray diffraction and Laue diffraction (Fig. S1). As shown in Fig.
1g, the (111) plane is represented by the gray plane, where the Lu, Sn, Au atomic layers are stacked
in order along [111] direction. Fig. 1h shows the top view of (111) plane. It exhibits 3𝑚 point-
group symmetry, consisting of 𝐶 rotation and three mirror symmetries as highlighted by dashed
3𝑧
lines. It meets the mentioned symmetry requirement for observing the 𝐵3 IPHE. Due to the 𝐶
3𝑧
symmetry, the IPHE will show 2𝜋/3 period angular dependence by rotating the magnetic field
(Fig. 1j), which is distinguished from the 𝜋-period planar Hall response.

Observation in nonmagnetic LuAuSn
The temperature dependent resistivity 𝜌 shows typical metallic behavior at zero magnetic
𝑥𝑥
field for sample #1 (see Fig. S2a). We first investigate the conventional out-of-plane Hall effect,
which as mentioned exists even in the isotropic systems. The magnetic field was applied along the
normal direction of the (111) plane, and the current was applied along the [1̅12] direction (see
Figs. S2 and S3). The Hall resistivity was identified as the field-odd part of the measured transverse
resistivity: 𝜌H (𝐁) = [𝜌 (𝐁)−𝜌 (−𝐁)]/2. As shown in the inset of Fig. S2c, 𝜌H exhibits
𝑦𝑥 𝑦𝑥 𝑦𝑥 𝑦𝑥
linear dependence on the magnetic field across temperature range from 2 K to 300 K and 𝐵 ≤
3 T . For fixed 𝐵 = 3 T , we rotated the magnetic field in the 𝑧𝑦 plane (see Fig. 2a), and the
measured Hall resistivity is given in Fig. 2b. The Hall signal persists even when the magnetic field
lies in the plane, and it is even more pronounced than in the out-of-plane configuration. Fig. 2d
shows the Hall resistivity measured by rotating the in-plane magnetic field, where 𝜑 is defined
as the angle relative to the [11̅0] axis (Fig. 2c). The angular dependence can be well decomposed
as:
𝜌H = 𝜌i sin(3𝜑)+𝜌o sin(𝜑+𝜑 ) (1)
𝑦𝑥 𝑦𝑥 𝑦𝑥 0
and the decomposed components are plotted in Figs. 2e and 2f. Here, the first term is identified as
the in-plane Hall resistivity, and the second term arises from a small misalignment of the magnetic
field, with 𝜑 quantifying the deviation between the (111) plane and the magnetic field rotation
0
plane (see Fig. S4c). This decomposition is reasonable given the small out-of-plane component of
B field. The 2𝜋/3 period angular dependence is consistent with the IPHE in a 𝐶 -symmetric
3𝑧
system. Moreover, the in-plane component shows sin(3𝜑) dependence with no phase shift. This
can be explained by the presence of ℳ symmetry, since it imposes the condition
y
𝜌i (90°−𝜑) = −𝜌i (90°+𝜑) . The extracted in-plane Hall resistivity curves at different
𝑦𝑥 𝑦𝑥
magnetic fields are shown in Fig. 2g. All of them exhibit a sin(3𝜑) angular dependence. As
shown in Fig. 2h, 𝜌i shows a good agreement with cubic B-dependence fitting below 3 T. And
𝑦𝑥
the cubic B-dependence behavior is persisted to room temperature (see Fig. S5). This for the first
time verifies the magnetic-cubic IPHE in nonmagnetic compound.

A key distinction between the IPHE and the planar Hall effect is that the IPHE is independent
of the relative direction between the magnetic field and the applied current 12. To verify this in
LuAuSn, we designed an experiment in which the magnetic field direction is fixed relative to the
crystal, while the driving current is applied along two inequivalent crystallographic directions, the
[11̅0] axis (Fig.3a) and its perpendicular direction, the [1̅12] axis (Fig.3d). Figs. 3b and 3e
depict the longitudinal resistivity (𝜌 and 𝜌 ), both of which show a 𝜋 period oscillation. It is
𝑥𝑥 𝑦𝑦
independent of the relative orientation between the magnetic field and the crystal axis. Both
configurations show maximal resistivity when 𝐵 ⊥ 𝐼. The two resistivities 𝜌 and 𝜌 differ
𝑥𝑥 𝑦𝑦
by approximately a 𝜋/2 phase, consistent with the 3m point group symmetry. Figs. 3c and 3f
present the IPHE resistivity measured in the two configurations. They show identical results with
no phase difference. This again confirms the IPHE nature of our observation, which is
distinguished from an in-plane anisotropic magnetoresistance (AMR) effect.

Giant in-plane Hall conductivity
We next extract the in-plane Hall conductivity and investigate its temperature and magnetic
field dependence. The conductivity were obtained by 𝜎 = 𝜌 ⁄(𝜌 𝜌 −𝜌 𝜌 ) and 𝜎H =
𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑦 𝑦𝑥 𝑦𝑥
−𝜌H ⁄(𝜌 𝜌 −𝜌 𝜌 ), ,herer, 𝜌 𝜌 = 𝜌2 −(𝜌H ) 2 , ,hite, 𝜌 ,, bring,ter,firld-rvrn,
𝑦𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑦 𝑦𝑥 𝑥𝑦 𝑦𝑥 (𝑦𝑥) 𝑦𝑥 (𝑦𝑥)
paet,of,ter,mrasuerd,teansvresr,ersistivity,𝜌 (𝐁) = [𝜌 (𝐁)+𝜌 (−𝐁)]/2.,Tern ,ter,in-planr,
(𝑦𝑥) 𝑦𝑥 𝑦𝑥
Hall,conductivity,𝜎i ,is,idrntifird,as,ter, 2𝜋/3,preiod,componrnt,(srr,Fig.,S6b).,Figs.4a and 4b
𝑦𝑥
show the angular-dependence of the anisotropic longitudinal conductivity 𝜎 under an in-plane
𝑥𝑥
magnetic field, and 𝜎i at 2 K- 300 K. The magnetic field magnitude was fixed at 3 T. The
𝑦𝑥
maximum value of 𝜎i reaches up to 306 Ω−1cm−1 at 2 K. Fig. 4d presents the temperature
𝑦𝑥
dependence of 𝜎i under magnetic fields of 3 T, 6 T and 9 T, respectively. The magnetic field
𝑦𝑥
was fixed along the [1̅12] direction (𝜑 = π/2), where the IPHE is the most significant due to the
sin(3𝜑) factor. The in-plane Hall conductivity decreases with increasing temperature but remains
finite even at room temperature (17 Ω−1cm−1 at 9 T, 6 Ω−1cm−1 at 6 T, and 1 Ω−1cm−1 at 3
T) (see Fig. S7). The insert figure shows the corresponding in-plane Hall angle (IPHA). It can
reach about 0.1, comparable to conventional anomalous Hall effect in topological semimetals 40-
42. Fig. 4e also shows the magnetic field dependence of 𝜎i at 2 K. Below 3 T, the IPHE follows
𝑦𝑥
a clear cubic dependence. It gradually deviates from a cubic dependence at higher magnetic fields.
The out-of-plane Hall conductivity 𝜎o at 2 K is also presented. It shows a two-carrier type
𝑦𝑥
normal Hall curves, which is consistent with the band structure, the hole and electron pockets at X
and Γ points 43. The different behaviors of 𝜎i and 𝜎o hint fundamentally different physical
𝑦𝑥 𝑦𝑥
origins. Notably, the IPHE conductivity of LuAuSn at 2 K and 3 T is an order of magnitude larger
than that of non-magnetic ZrTe . It is even larger than all magnetic IPHE systems (Fig. 4f and
5
Table S2). Moreover, the in-plane Hall conductivity in LuAuSn can be further enhanced to
approximately 687 Ω−1cm−1 at 7 T.

Methods
Single crystal growth. LuAuSn single crystals were grown using the Sn flux method. A
mixture of pure elements with molar ratio Lu: Au: Sn = 1: 1: 10 was placed in an alumina crucible,
which was then sealed in an evacuated quartz ampoule. The quartz ampoule was placed in a pit-
type furnace and heated from room temperature to 1150 ℃, where it was held for 24 hours. It was
then slowly cooled to 650 ℃ at the rate of 2 ℃ per hour for sample #1 and #2. At this point, the
excess Sn flux was removed by centrifugation.

Single crystal characterization. Single-crystal X-ray diffraction and crystal orientation were
done at room temperature using a TD-3500 XRD system and TD-M3000 Laue system, respectively.
The chemical composition of the LuAuSn sample was examined using scanning electron
microscope (SEM) equipped with energy dispersive spectrometer (EDS). EDS measurements were
performed at multiple positions on the crystal surfaces, with results consistent within the
instrumental error margin of 1%−2%.

Physical property characterization. Magneto-transport properties were carried out using a
Physical Property Measurement System (PPMS, 9 T). The angle-dependent electrical transport
measurements were performed using the standard four-probe method with a horizontal rotator. A
universal sample holder board was used for out-of-plane transport measurements, while a parallel
magnetic field sample holder board was used for in-plane transport measurement.

First-principles calculations. First-principles calculations based on density functional
theory were performed using the Vienna ab initio Simulation Package 48-50 with the projector-
augmented wave method 51. The Perdew-Burke-Ernzerhof 52 generalized gradient approximation
was employed to treat the exchange-correlation energy. The lattice constant was taken as
6.5627 Å. The inner positions are fully relaxed with force tolerance 0.01 eV Å−1. The Γ-centered
𝑘-point mesh with size 8×8×8 were used for Brillouin zone sampling. The Liechtenstein et
al.'s approach 53 was used to treat localized f orbitals of Lu atoms with an effective 𝑈 = 6.7 eV
and 𝐽 = 0.7 eV 39, 54. The ab initio Wannier tight-binding models were constructed using the
Wannier90 package 55. A dense 𝑘-point mesh with size 300×300×300 were used for Hall
conductivity calculations.

The intrinsic and the Lorentz force contributions were evaluated using first-principles
calculations. The intrinsic response is given by 𝜎 = −(𝑒2/ℏ)∫𝑓 Ω 56, 57, where 𝑓 is the
𝑦𝑥 0 𝑧 0
Fermi-Dirac distribution function, and Ω is the Berry curvature. In calculations, both the band
𝑧
energy and Bloch states were dressed by considered the effect of magnetic field through the
Zeeman's coupling 𝐻 = 𝜇 (𝐋+2𝐬)∙𝐁/ℏ , where 𝜇 is the Bohr magneton, 𝐬 is the spin
Z B 𝐵
operator, and 𝐋 is the orbital angular moment operator. The orbital angular momentum matrix
elements in the basis of Bloch eigenstates are given by 𝐋 = 𝑚 (𝝃 ×𝐯 −𝐯 ×𝝃 )/2 58,
𝑚𝑛 e 𝑚𝑙 𝑙𝑛 𝑚𝑙 𝑙𝑛
where 𝝃 = (1−𝛿 )𝐀 , 𝐀 = 〈𝑢 |𝑖𝜕 𝑢 〉 is the interband Berry connection, and 𝐯
𝑚𝑛 𝑚𝑛 𝑚𝑛 𝑚𝑛 𝑚 𝐤 𝑛 𝑚𝑛
is the matrix element of velocity operator. The in-plane Hall effect due to the Lorentz force was
evaluated as 𝜎 = −(𝜏𝑒3/ℏ)∫𝑓′𝑣𝑥𝜏(𝐯×𝐁)∙𝜕 𝑣𝑦 59, where again, the band energy and Bloch
𝑦𝑥 0 𝐤
states are dressed to include the Zeeman's coupling. The relaxation time was estimated by
comparing the measured longitudinal conductivity with the calculated Drude weight. It depends
on the chemical potential, and 𝜏 ≈ 0.1 ps at the Fermi level.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The in-plane Hall effect (IPHE), which was referred to as the \"longitudinal Hall effect\" in the 1950s, has recently attracted\nrenewed interest because of its possible connection to band topology. This manuscript reports a systematic study of the\nIPHE in the paramagnetic metal LuAuSn, highlighting (i) a B^3-scaling IPHE and (ii) a record-high in-plane Hall conductivity\nat 2 K. By comparing the data with first-principles calculations, the authors argue that side-jump and skew-scattering\nmechanisms dominate the observed in-plane Hall conductivity. The experimental data are of good quality. However, I am\nnot convinced that the present findings have sufficient novelty or impact to merit publication in Nature Communications, for\nthe reasons below.\n(i) The authors claim that the paper reports the first experimental observation of a B^3-scaling IPHE in paramagnetic metals.\nHowever, experimental indications of such behavior were already reported as early as the 1950s (Refs. [R1, R2] below). For\ninstance, in n-type paramagnetic germanium, which shares the same Laue point group (m-3m) as LuAuSn, B^3-scaling and\neven B^5-scaling IPHEs were observed [R1]; similarly, a B^3-scaling IPHE was reported in paramagnetic bismuth, albeit not\nas the main focus of the study [R2]. Therefore, the authors' claim of \"the first observation of magneto-cubic IPHE in a three-\ndimensional nonmagnetic material\" lacks sufficient justification. I believe the authors should cite these prior works and\nclearly describe what is genuinely new in their study.\n[R1] L. Grabner, Phys. Rev. 117, 689 (1960).\n[R2] T. Okada, J. Phys. Soc. Jpn. 11, 89 (1956); J. Phys. Soc. Jpn. 12, 1327 (1957).\n(ii) The manuscript does not sufficiently discuss the physical significance of the B^3-scaling IPHE compared with the more\nconventional B-linear IPHE. What new insights—if any—does the analysis of the B^3-scaling IPHE offer that cannot be\nobtained from the linear-in-B IPHE? Without such a discussion, the physical significance of the B^3-scaling IPHE remains\nunclear.\n(iii) I have concerns regarding the symmetry analysis in lines 103–113. First, the statement that there are 14 Laue point\ngroups is incorrect; the correct number is 11. Moreover, the discussion appears to be formulated in the crystallographic\nprincipal-axis frame. Since tensor components transform under a change of orthonormal basis, a symmetry-forbidden B-\nlinear IPHE in the principal-axis frame may become allowed in a different coordinate frame. In fact, whether such an IPHE is\nsymmetry-allowed depends on the choice of the transport plane. Given that the experiments were performed in a tilted frame,\nthe symmetry argument should be reformulated in a coordinate-independent manner to avoid internal inconsistencies.\n(iv) Although the large in-plane Hall conductivity at 2 K is impressive, the conductivity at room temperature is much smaller.\nWithout a discussion of how in-plane Hall conductivity might be enhanced near room temperature or why the current values\nare technologically meaningful, the claim that this work \"opens the door to practical applications of IPHE\" seems premature.\n(v) The scaling law employed in the analysis was originally proposed for the anomalous Hall effect in ferromagnetic metals.\nIt would be helpful if the authors could clarify why this scaling approach is applicable to the in-plane Hall conductivity in a\nparamagnetic metal, especially given its B^3-dependence.\nGiven the concerns regarding the novelty and impact of the manuscript, I am unable to recommend it for publication in\nNature Communications.", "reviewer_2": "In this paper, the authors have demonstrated the in-plane Hall effect in a nonmagnetic half-Heusler compound LuAuSn. The\nobserved Hall signal with a period of 2pi/3 seems to be large, and the presented analysis using the scaling law may be\nhelpful for further examining the scattering effects in the in-plane Hall effect. On the other hand, similar observation has been\nreported for a nonmagnetic semimetal Cd3As2, which has been publicly available in this past March (arXiv:2503.04195, not\ncited) and recently published in Phys. Rev. Lett. Given that the introduction needs to be updated to reflect recent progress on\nthe “unexpected” nonmagnetic compound, I am currently uncertain about the manuscript’s impact. Further concerns are\noutlined below.\n1. While it is straightforward from symmetry considerations that the B^3 term contributes to the in-plane Hall effect on the\n(111) cubic plane, it is not entirely clear whether the presented data indeed support the proposed interpretation. The curve\nshown in Fig. 2g itself appears to be well fitted by the B^3 dependence. On the other hand, a comparable nonlinear behavior\nis evident even in the out-of-plane field sweep (theta = 90 deg), as presented in Fig. 2b, where only the field range\ndominated by B dependence appears to shift to higher fields. How does the dependence evolve for theta varying from 0 to\n90 deg? My concern is that the curve shown in Fig. 2g may be affected by the presence of two types of carriers. In particular,\nthe density and mobility of holes and electrons are found to be nearly identical (5x10^19 cm-3, 1,700 cm2V-1s-1 and\n4x10^19 cm-3, 2,000 cm2V-1s-1), as summarized in Fig. S2, which make it difficult to simply interpret that the in-plane Hall\neffect below 3 T is dominated by a single type of carrier.\n2. It may be difficult to position LuAuSn this compound as an ideal system of the in-plane Hall effect, and the authors are\nencouraged to clarify the limitations more explicitly. As already noted, this compound intrinsically contains two types of\ncarriers, which is also supported by the band structures in Fig. 5a. This inevitably\ncomplicates the field and temperature dependence of the Hall effect. Furthermore, in\nthe nonmagnetic cubic system, octupolar as well as linear couplings between the\nHall vector and the magnetic field are allowed, but they are restricted to diagonal ones.\nIn this situation, the in-plane anomalous Hall effect accidentally appears only when the\nmeasurement plane is not defined by two principal axes, such as the (111) and (103)\nplanes (see Phys. Rev. X 15, 031006 (2025)., not cited).\n3. It is noted that rho_xy of sample #1 in Fig. 2b does not change sign for the\nout-of-plane field sweep, whereas sigma_xy in\nFig. 4c changes from negative to positive. Is this from sample #2, or from yet another sample? If so, it would be helpful to\nprovide basic transport characterization of these samples, as the two-carrier behavior\nalso differs from sample to sample.\n4. Related to the point above, the manuscript would benefit from a more detailed discussion of how off-stoichiometry and\nstructural disorder in the half-Heusler compound affect presented two-carrier transport and scattering effects, which can differ\nfrom sample to sample. The EDX analysis indicates a slight deviation from the nominal stoichiometric ratio, and the degree\nof order/disorder could in principle be quantified from the relative XRD peak intensities. It remains unclear how much\ndisorder is present in the examined samples including sample #2 or the one displayed in Fig. 4c. Considering that side-jump\nand skew-scattering contributions play a large role, it is essential to present whether the first-principles calculations were\ncarried out for the ordered phase, and how disorder—if included—would alter the theoretical predictions.\n5. Considering that multiple fitting parameters are used, how accurately can each contribution be decomposed? How about\nthe fitting results for sample #2? I also have concerns regarding the direct application of the sigma_xx scaling from single-\ncarrier systems (Ref. 39) to the multi-carrier system. Does the calculation of the Lorentz-force contributions in Fig. 5c also\ntake into account multi-carrier effects? The results should be displayed over the same energy range as in Fig. 5b.\n6. I am concerned that the main results presented as rho_xy may in fact correspond to rho_yx. In addition, it seems that the\ndefinition of axis handedness is inconsistent between Figs. 2a and 2c, which may result in additional confusion.", "reviewer_3": "This manuscript reports the observation of a giant in-plane Hall effect (IPHE) in the nonmagnetic half-Heusler compound\nLuAuSn, with clear characteristics (cid:1)^3. Dependency relationship and 2(cid:2)/3 Periodicity, and (cid:3)3v Symmetry are consistent.\nThe working experiments are solid and supported by first-principles calculations, representing an important advancement in\nthe field of nonlinear Hall effects. The amplitude of the effect is very significant and can last up to room temperature, showing\ngreat application potential. I therefore recommend publication in Nature Communications, pending minor revisions to\naddress the following points.\n(1)The manuscript mentioned that high-quality single crystals were synthesized, but the residual resistivity (RRR) was only 3\n(Figure S2a). This value is relatively low, indicating that there is a considerable concentration of defects or impurities in the\nsample. This may affect the analysis of the scattering mechanism and raise questions about whether the effect can maintain\nthe same scale relationship or even be stronger in higher-quality samples. It is suggested that the author provide more\ninformation about the sample quality, such as whether samples from different batches exhibit repeatable RRR and IPHE\nsignals. Have you ever attempted to improve the growth process to obtain samples with a higher RRR? The manuscript\nmentioned that their crystals are “high-quality,” yet the residual-resistivity ratio (RRR) is only ~3 (Fig. S2a). That value is low\nenough to imply a sizeable concentration of defects or impurities, which could skew any quantitative discussion of\nscattering. Please show RRR and IPHE data for at least two additional growth batches to confirm\nreproducibility. Comment on whether growth conditions (flux ratio, cooling rate, annealing) have been varied in an attempt to\nraise the RRR. In the discussion, it should be clearly pointed out that the limited mass of the current sample may have an\nimpact on the quantitative analysis of the scattering mechanism.\n(2)In Page 12, Line 188-189; Page 13, Line 254-259, the author points out that after B > 3T,(cid:4)(cid:5)_(cid:6)(cid:7) gradually deviates (cid:1)^3. The\ndependency relationship was merely touched upon with \"perturbation theory failure\" and \"non-perturbation effect\", lacking\nin-depth physical discussion and experimental data analysis. Here, I suggested that the author conduct a more in-depth\nanalysis of the behavior in high-field Settings. Can authors try using higher-order items (such as aB^3+b(cid:1)^5, or another\nsuitable form) to perform fitting? Is this deviation related to the magnetic field correction of the band structure (such as the\nZeeman effect and the formation of Landau energy levels)? Adding some angle-dependent measurements under high field\nconditions might reveal more information. A speculative discussion on high-field behavior should be added to the discussion\nsection.\n(3)The authors candidly state that the scaling-law analysis cannot disentangle side-jump and skew-scattering contributions.\nWhile this is an intrinsic theoretical limitation, the manuscript never discusses how this limitation affects the conclusions,\nnor does it suggest concrete ways to overcome it (for example, by systematic doping experiments that tune the scattering\nchannels). I will recommend that the authors add a paragraph in “Discussion on the physical mechanisms” that explicitly\nhighlights the inability of the scaling law to separate individual scattering mechanisms. Frame this limitation as a future\nresearch direction rather than a mere analytical outcome.\n(4)Figure 4f compares the IPHE conductivity of LuAuSn with that of other material systems, a valuable comparison.\nHowever, the cited data (Refs. [1–3, 5, 17, 18, 29]) may have been measured under different temperatures and magnetic\nfields. It is needed to explicitly annotate to undermine the rigor of this comparison (i.e., same T and B).\n(5)In the first-principles estimate of the Lorentz-force contribution, the authors adopt a constant relaxation time τ = 0.1 ps,\nextracted by matching the calculated conductivity to the measured Drude conductivity. Yet τ is likely energy-dependent, and\nit is questionable whether a single value applies to all carriers and all temperatures. Please briefly discuss in the Methods\nsection why this τ was chosen, acknowledge that an energy-independent τ is an approximation, and comment on how this\napproximation might affect the order-of-magnitude estimate of the Lorentz-force contribution.\n(6)The caption of Fig. 4f reads: “The data were taken from references [1–3, 5, 17, 18, 29].” In the reference list (p. 7),\nhowever, Refs. [17] and [18] are 2025 preprints or papers marked “in press” on Fe thin films and Co3Sn3S2, respectively.\nCiting not-yet-published work—especially non-peer-reviewed preprints—in a quantitative comparison demands extra\ncaution and explicit labeling. Amend the caption to state that Refs. [17] and [18] are preprints or “in press.” Whenever\npossible, replace them with formally published data. If their inclusion is unavoidable, add a sentence in the main text or\nSupplementary Information noting the preliminary nature of these data.\n(7)On pages 9&10, both “C_3z” and “C_3v” symmetry appear. Although both are relevant for the (111) surface, the notation\nshould be uniform and precise; please define it explicitly at its first appearance.\nOverall, this work represents a significant step forward in the study of in-plane Hall effects. The combination of giant\nresponse, room-temperature operation, and clear symmetry-based interpretation makes it a strong candidate for publication\nin Nature Communications. I recommend acceptance pending minor revisions as outlined above."}, "author_response": "Response to the Report of Reviewer #1\nThe in-plane Hall effect (IPHE), which was referred to as the \"longitudinal Hall\neffect\" in the 1950s, has recently attracted renewed interest because of its possible\nconnection to band topology. This manuscript reports a systematic study of the IPHE\nin the paramagnetic metal LuAuSn, highlighting (i) a B3-scaling IPHE and (ii) a\nrecord-high in-plane Hall conductivity at 2 K. By comparing the data with\nfirst-principles calculations, the authors argue that side-jump and skew-scattering\nmechanisms dominate the observed in-plane Hall conductivity. The experimental data\nare of good quality. However, I am not convinced that the present findings have\nsufficient novelty or impact to merit publication in Nature Communications, for the\nreasons below.\nResponse: We thank the reviewer for your positive evaluation of our work, including\nrecord-high in-plane Hall conductivity, high quality of our data and systematic nature\nof our study. And we agree with your opinion that the in-plane Hall effect (IPHE) has\nrecently attracted renewed interest because of its connection to band topology.\nDuring the review process, we also noticed that two nonmagnetic topological\nmaterials Cd As [1] and PtBi [2] were reported to exhibit IPHE with the small\n3 2 2\nin-plane Hall conductivity arising from the intrinsic contribution dictated solely by\nband structures. Compared to these works and other previous works, we not only\nachieved record-high in-plane Hall conductivity and in-plane Hall angle (about 10%),\nbut provided the first experimental observation that scattering induced extrinsic\nmechanisms other than the Lorentz-force one plays an important role in IPHE. Our\nobservation uncovers a new strategy to enhance IPHE by utilizing extrinsic\nmechanism, and opens an unexplored area within the field of experimental research\non IPHE.\nIn the following, we provide point-by-point responses to your insightful\ncomments, elaborate on the novelty and significance of the findings in the present\nmanuscript.\n[1] Shinichi Nishihaya, et al, Phys. Rev. Lett. 135 (10), 106603 (2025)\n[2] Arthur Veyrat, et al, Nat. Commun. 16 (1), 6711 (2025)\nComment 1: The authors claim that the paper reports the first experimental\nobservation of a B3-scaling IPHE in paramagnetic metals. However, experimental\nindications of such behavior were already reported as early as the 1950s (Refs. [R1,\nR2] below). For instance, in n-type paramagnetic germanium, which shares the same\nLaue point group (m-3m) as LuAuSn, B3-scaling and even B5-scaling IPHEs were\nobserved [R1]; similarly, a B3-scaling IPHE was reported in paramagnetic bismuth,\nalbeit not as the main focus of the study [R2]. Therefore, the authors' claim of \"the\nfirst observation of magneto-cubic IPHE in a three-dimensional nonmagnetic\nmaterial\" lacks sufficient justification. I believe the authors should cite these prior\nworks and clearly describe what is genuinely new in their study.\n[R1] L. Grabner, Phys. Rev. 117, 689 (1960).\n[R2] T. Okada, J. Phys. Soc. Jpn. 11, 89 (1956); J. Phys. Soc. Jpn. 12, 1327 (1957).\nResponse: We thank the reviewer for bringing the two interesting references (Refs.\n[R1] and [R2]) into our attention. We have cited them in the revised manuscript\n(References 11 and 12). These prior studies explored B-dependent transport behavior\nin paramagnetic germanium and bismuth, and Ref. [R1] includes a B3 scaling signal.\nHowever, we would like to point out that the experimental data presented in\nthese articles are not likely to arise from IPHE. The main reasons are as follows:\n First, it is impossible to obtain genuine IPHE from the measurements in Ref. [R1]\non the cubic n-type germanium (according to symmetry constraint as pointed out\nbelow), making it difficult to attribute the B3 scaling therein to IPHE. In Ref. [R1],\ntwo types of measurements were performed. Type A: the magnetic field rotated in\nthe (110) plane and the current was along [110] direction. Type B: the magnetic\nfield rotated in the (100) plane and the current was along [100] direction. Ge\nshows diamond-type cubic lattice structure. According to theoretical analysis [3],\nthe IPHE in odd powers of B is forbidden for the measurement planes (100) and\n(110) hosting C symmetry. Therefore, the observed B3 scaling in Ref. [R1] is not\n2\nlikely to arise from genuine IPHE.\n Second, in the measurement of Refs. [R1] and [R2] on germanium and bismuth,\nthe transport plane changes as B rotates. Thus, the signals of two paired probes\nare a mixture of the out-of-plane and in-plane Hall signals with different transport\nplanes. Separating the out-of-plane and in-plane Hall signals using simple\ntrigonometric functions is inappropriate because IPHE is determined by the\nsymmetry of different transport planes in these measurements. Moreover, the\nout-of-plane Hall signal is nearly two orders of magnitude larger than the in-plane\nHall signal in Ref. [R1], making accurate extraction of the in-plane Hall signal\nchallenging, which may strongly influence the determination of B-dependent\nscaling.\nIn addition to the finding of genuine IPHE with B3 scaling, the novelty and\nsignificance of our work also lie in that it reports the record-high in-plane Hall\nconductivity, the extrinsic mechanism of IPHE and a new IPHE family materials\nhosting it - half-Heusler RAuSn (R= rare earth). Previous researches primarily\nfocused on IPHE induced by the intrinsic mechanism related to nontrivial topology,\nsuch as in VS-VS , Fe Sn , Cd As , EuCd Sb and PtBi . These studies demonstrate\n2 3 2 3 2 2 2 2\nthat IPHE holds significant physical importance. However, the limitation of the\nintrinsic mechanism is also evident. The intrinsic contribution usually provides a\nsmall in-plane Hall conductivity. In contrast, our findings indicate that extrinsic\nmechanisms possess unique advantages in achieving record-high in-plane Hall\nconductivity. Furthermore, by tuning the quality of single crystal, we obtained even\nlarger in-plane Hall conductivities (Table R1). Our work thus unveils a new strategy\nto enhance IPHE by utilizing the extrinsic mechanism in highly conductive materials.\nTherefore, this work, which presents the first experimental demonstration of the\ndominant extrinsic contribution to IPHE and the unprecedentedly strong in-plane Hall\nconductivity, will certainly trigger follow-up researches exploring the extrinsic IPHE.\nMoreover, half-Heusler compounds is a large family with the same crystal structure.\nThey form a good platform for further researches on IPHE. These results provide new\ninsight into the IPHE and broaden the scope of its potential applications.\nWe hope the above explanations have resolved your concern regarding the\nnovelty and impact of our manuscript.\nTable R1 The RRR value and in-plane Hall conductivity for sample #1-#5\n[3] Takashi Kurumaji, Physical Review Research 5 (2), 023138 (2023).\nComment 2: The manuscript does not sufficiently discuss the physical significance of\nthe B3-scaling IPHE compared with the more conventional B-linear IPHE. What new\ninsights—if any—does the analysis of the B3-scaling IPHE offer that cannot be\nobtained from the linear-in-B IPHE? Without such a discussion, the physical\nsignificance of the B3-scaling IPHE remains unclear.\nResponse: We appreciate your suggestion. The physical significance of B³-scaling\nIPHE includes at least the following two aspects:\n From the symmetry perspective, the -scaling IPHE is subject to stringent\n1\nsymmetry constraints. In fact, any rotation axis perpendicular to the transport\nB\nplane, such as and , forbids the -scaling IPHE. By contrast, the\n1\n-scaling IPHE\nC\n2izs not f\nC\no3rZbidden by\nB\nsymmetry. This difference largely\nextends the playground of IPHE, because\nC\n3mZany interesting topological materials\nbelong to hexagonal lattice, and exhibit symmetry. Typical examples include\nthe transition metal dichalcogenides, C3tZhe topological semimetals Cd\n2\n,\nEuCd Sb , PtBi , and the kagome materials Fe Sn , Co Sn S . In these wide\n2 2 2 3 2 3 2 2\nclasses of extensively studied materials, the magneto-cubic IPHE stands as the\nleading magneto-nonlinear Hall effect.\n Moreover, because the out-of-plane normal Hall effect always contains a\n1\ncontribution, the -scaling IPHE can easily mix with the out-of-plane Hall\nB\neffect, making it difficult to separate the IPHE contribution. For example, Ref. [4]\nreports that distinct IPHE in RuO , where symmetry forbids an inherent IPHE,\n2\nwhich originates from the superposition of inequivalent out-of-plane Hall effects.\nTherefore, the -scaling IPHE in low-symmetry materials is more uncertain in\n1\nexperiments. In C symmetry systems, three-fold in-plane Hall signal is distinct\nB 3Z\nfrom the out-of-plane signal, thus the“clean” IPHE can be easily separated,\nallowing for a sharp interpretation of the underlying physics. The results of B3\nscaling in C system provide more compelling verification of the relation\n3Z\nbetween symmetry and B-dependent behavior.\nWe have expanded the discussion of the physical significance of B3-scaling IPHE\nby adding the above points in the revised manuscript (Line 85-86, Page 3; Lines\n110-115, Page 4).\n[4] Meng Wang, Jianbing Zhang, Di Tian, Pu Yu, and Fumitaka Kagawa,\nCommunications Physics 8 (1), 28 (2025).\nComment 3: I have concerns regarding the symmetry analysis in lines 103–113. (1)\nFirst, the statement that there are 14 Laue point groups is incorrect; the correct\nnumber is 11. (2) Moreover, the discussion appears to be formulated in the\ncrystallographic principal-axis frame. Since tensor components transform under a\nchange of orthonormal basis, a symmetry-forbidden B-linear IPHE in the\nprincipal-axis frame may become allowed in a different coordinate frame. In fact,\nwhether such an IPHE is symmetry-allowed depends on the choice of the transport\nplane. Given that the experiments were performed in a tilted frame, the symmetry\nargument should be reformulated in a coordinate-independent manner to avoid\ninternal inconsistencies.\nResponse: We thank the reviewer for this valuable comment, which helps strengthen\nour manuscript. The phrase \"14 Laue classes\" was a typo, the correct number is 11.\nWe agree with the reviewer that the symmetry-allowed IPHE depends on the choice of\nthe transport plane. In the revision, we have made two modifications to avoid\nmisleading:\n(i) We explicitly define the plane as the transport plane [Lines 106-107, Page 4].\nWith this setup, the relevant xHyall resistance is , and the magnetic field is applied\n𝐻\nwithin the plane. Then, we find that a\n𝜌𝑦\n-\n𝑥\nsymmetric system forbids the\n(ii) We have removed the sentences about Laue groups, since they do not specify the\ndirection of transport plane.\nComment 4: Although the large in-plane Hall conductivity at 2 K is impressive, the\nconductivity at room temperature is much smaller. Without a discussion of how\nin-plane Hall conductivity might be enhanced near room temperature or why the\ncurrent values are technologically meaningful, the claim that this work \"opens the\ndoor to practical applications of IPHE\" seems premature.\nResponse: We greatly appreciate this valuable comment. Researches on IPHE are still\nin the early stage of exploring its physical mechanisms and in discovering new\nmaterials. Nevertheless, exploring the potential of this effect for practical applications\nrepresents an important future direction. Our findings suggest that LuAuSn is a\npromising material for potential practical applications. We address the practical\nimplications as follows:\n First, currently, a major obstacle hindering the application of this effect is that\nmany IPHE materials exhibit low in-plane Hall conductivity and small in-plane\nHall angle, combined with a narrow operating temperature range confined to the\nlow-temperature region [1,5,6]. However, LuAuSn not only shows the largest\nin-plane Hall conductivity and Hall angle (see Fig. 4d), but also exhibits wide\noperating-temperature range from 2K to 300K. These properties make LuAuSn\none of the most promising materials for practical IPHE applications, particularly\nin cryogenic environments. We think that both cryogenic and room temperature\nenvironments are technologically meaningful for this type of devices. As shown\nin Fig. 4d, both in-plane Hall conductivity and Hall angle exhibit large value\nbelow 150K. Therefore, the application of this effect in cryogenic environments\nis feasible.\n Second, I would like to emphasize that although the room-temperature in-plane\nHall conductivity in LuAuSn is much smaller than that of 2 K, it is still much\nlarger than the low-temperature in-plane Hall conductivities of previously studied\nIPHE materials (Please see the comparison in Table R2). This comparison\nunequivocally shows the prominent performance of LuAuSn even at room\ntemperature. Therefore, LuAuSn indeed holds great potential for applications at\nroom temperature.\n Third, to further resolve your concern, we also design another experiment aimed\nat enhancing in-plane Hall conductivity at 300K. Higher-quality single crystals\nwere obtained by optimizing the growth condition. Table R2 shows the in-plane\nHall conductivity at room temperature for different samples of LuAuSn. The\nin-plane Hall conductivity is enhanced by 11% for sample #5. The results indicate\nthat improving single crystal quality is an effective method for enhancing room\ntemperature performance. We plan to conduct further in-depth research on this\noptimization path as an independent research initiative.\nBased on the above discussions, we believe that IPHE of LuAuSn holds practical\napplication value. We hope these discussions and the added measurements have fully\nresolved your concern regarding the practical applicability of our findings.\nTable R2 The in-plane Hall conductivity in LuAuSn, SrRuO , EuCd Sb , EuZn Sb\n3 2 2 2 2,\nand Cd As\n3 2\n[5] A. Nakamura, et al, Phys. Rev. Lett. 133 (23), 236602 (2024).\n[6] E. Lesne, et al, Nat. Mater. 22 (5), 576 (2023).\nComment 5: The scaling law employed in the analysis was originally proposed for\nthe anomalous Hall effect in ferromagnetic metals. It would be helpful if the authors\ncould clarify why this scaling approach is applicable to the in-plane Hall conductivity\nin a paramagnetic metal, especially given its B3-dependence.\nResponse: We thank the Reviewer for this valuable suggestion.\n The IPHE also contains intrinsic, side-jump, and skew-scattering contributions, as\nin the anomalous Hall effect, and therefore the scaling law analysis can be applied.\nThe identification of intrinsic, side-jump, and skew-scattering contributions does\nnot rely on the magnetic state of the system. As an example, please note that the\nsame scaling analysis is widely applied in studies of the nonlinear Hall effect in\nnonmagnetic systems (please see, e.g. [7][8]), since the time-reversal-even\nnonlinear Hall effect also contains these three contributions. Moreover, the\nscaling law analysis applied to the anomalous Hall effect in ferromagnetic\nmaterials does not concern the power dependence on the background\nmagnetization. By the same taken, the scaling law analysis applied to the IPHE in\nnonmagnetic materials does not concern the power dependence on the magnetic\nfield.\nIn the revision, we have added the following explanation in manuscript [Lines\n214-217, Page 8]:\n\"We note that this scaling relation was originally applied to the anomalous Hall\neffect in ferromagnetic systems, and can also be applied to the IPHE in nonmagnetic\nsystems because intrinsic, side-jump, and skew-scattering contributions are likewise\npresent.\"\n[7] Z. Z. Du, et al, Nat. Commun. 10 (1) (2019).\n[8] T. Ma, et al, Nat. Commun. 13 (1) (2022).\nComment 6: Given the concerns regarding the novelty and impact of the manuscript,\nI am unable to recommend it for publication in Nature Communications.\nWe thank you a lot for the highly valuable comments and suggestions that help\nus greatly in improving our manuscript. We hope that our detailed reply and revision\nhave fully resolved your concerns.\nResponse to the Report of Reviewer #2\nIn this paper, the authors have demonstrated the in-plane Hall effect in a nonmagnetic\nhalf-Heusler compound LuAuSn. The observed Hall signal with a period of 2pi/3\nseems to be large, and the presented analysis using the scaling law may be helpful for\nfurther examining the scattering effects in the in-plane Hall effect. (1) On the other\nhand, similar observation has been reported for a nonmagnetic semimetal Cd As ,\n3 2\nwhich has been publicly available in this past March (arXiv:2503.04195, not cited)\nand recently published in Phys. Rev. Lett. Given that the introduction needs to be\nupdated to reflect recent progress on the “unexpected” nonmagnetic compound, I am\ncurrently uncertain about the manuscript’s impact. Further concerns are outlined\nbelow.\nResponse: We sincerely appreciate your valuable feedback and professional advice,\nwhich have significantly helped us to improve our manuscript. Below, we address\nyour concerns and clarify key aspects of our study.\nThe Cd As is an important advancement in IPHE. After submitting our paper to\n3 2\nNature communications, we noticed that two significant articles (Cd As [1]and PtBi\n3 2 2\n[2]) in this field were published online or arXiv available. The field-induced intrinsic\nIPHE was observed in nonmagnetic Dirac semimetal Cd As and topological nodal\n3 2\nline semimetal PtBi . In revision, we have updated our introduction accordingly and\n2\nprovided a discussion of these advances [Line 59, Page 2; Lines 63-66, Page 3;\nReference [33], [34]].\nMeanwhile, we would like to emphasize that beyond the novelty of\nnon-magnetic systems, in our work we also demonstrate: (1) in new IPHE family of\nmaterials RAuSn (R= rare earth), the record-high values of in-plane Hall conductivity\ncan be achieved; (2) extrinsic mechanism can be responsible for large IPHE; and (3)\nthe B3 scaling of IPHE in a C system.\n3Z\nPrevious researches primarily focused on the IPHE induced by intrinsic\nmechanism independent of scattering, such as VS-VS , Fe Sn , Cd As , EuCd Sb\n2 3 2 3 2 2 2\nand PtBi . These studies demonstrated that IPHE holds significant physical\n2\nimportance. However, the limitation of the intrinsic mechanism is also evident. The\nintrinsic contribution usually shows a small in-plane Hall conductivity. In contrast,\nexperimental research on IPHE related to extrinsic mechanisms remains unexplored.\nHere, our findings show that extrinsic mechanisms possess unique advantages for\nachieving large in-plane Hall conductivity, which is indispensable towards application\nof IPHE. A good platform, half-Heusler compounds, is uncovered for realizing\nrecord-high IPHE by extrinsic mechanism and exploring its further enhancement.\nTherefore, this work, which presents the first experimental demonstration of dominant\nextrinsic contribution to IPHE and the unprecedentedly strong in-plane Hall\nconductivity, will certainly trigger many follow-up researches exploring the extrinsic\nIPHE. Moreover, we experimentally realized for the first time the long-sought\ntheoretical prediction of magneto-cubic IPHE under three-fold symmetry in\nnon-magnetic systems [3-6], confirming the direct correlation between crystalline\nsymmetry and B scaling of IPHE. These results provide new insights into the IPHE\nand broaden the scope of its potential applications. We hope that our explanations\nhave fully resolved your concern on the impact of our manuscript.\n[1] Shinichi Nishihaya, et al, Phys. Rev. Lett. 135 (10), 106603 (2025).\n[2]A. Veyrat, et al, Nat. Commun. 16 (1), 6711 (2025).\n[3] X. Liu, et al, Phys Rev Lett 111, 086802 (2013).\n[4] S. Sun, et al, Phys Rev B 106, L241105 (2022).\n[5] C. M. Wang, et al, Phys. Rev. B 108 (12), L121301 (2023).\n[6] V. A. Zyuzin, Phys. Rev. B 102, 241105(R) (2020).\nComment 1: While it is straightforward from symmetry considerations that the B3\nterm contributes to the in-plane Hall effect on the (111) cubic plane, it is not entirely\nclear whether the presented data indeed support the proposed interpretation. The curve\nshown in Fig. 2g itself appears to be well fitted by the B3 dependence. On the other\nhand, a comparable nonlinear behavior is evident even in the out-of-plane field sweep\n(theta = 90 deg), as presented in Fig. 2b, where only the field range dominated by B\ndependence appears to shift to higher fields. How does the dependence evolve for\ntheta varying from 0 to 90 deg? My concern is that the curve shown in Fig. 2g may be\naffected by the presence of two types of carriers. In particular, the density and\nmobility of holes and electrons are found to be nearly identical (5x1019 cm-3, 1,700\ncm2V-1s-1 and 4x1019 cm-3, 2,000 cm2V-1s-1), as summarized in Fig. S2, which make it\ndifficult to simply interpret that the in-plane Hall effect below 3T is dominated by a\nsingle type of carrier.\nResponse: We appreciate your insightful remarks regarding the potential influence of\nmulti-carrier transport on the observed Hall response. Phenomenally, the B3\ndependent IPHE curves show similar nonlinear behavior to the out-of-plane Hall (or\nnormal Hall effect) curves. However, while the nonlinear behavior of normal Hall\neffect comes from the two-carrier nature, the origin of the IPHE differs distinctly:\n First, as shown in Fig. 2e, the total Hall signal can be decomposed into two\ncomponents: the in-plane Hall signal with three-fold symmetry and\n𝑖\nout-of-plane (normal) Hall component𝜎 𝑦𝑥sin (3𝜙) with one-fold symmetry,\n𝑜\noriginating from a slight misleading bet𝜎w𝑦e𝑥esnin c (u𝜙rr+ent𝜙 p0l)ane and B rotating plane\n(see Fig. R1). The distinct angular dependence implies that their origins are\ndistinctive.\n Second, to further demonstrate this distinction, we provide the data of separated\n, and normal Hall conductivity at 2K for sample#1 in Fig. R1. Obviously,\n𝑖 𝑜\n𝜎𝑦𝑥 s𝜎h𝑦o𝑥ws similar B dependent behavior to the normal Hall conductivity, but\no\n𝜎th 𝑦 e 𝑥 se two are very different from . These features imply that shares the\n𝑖 𝑜\n𝜎𝑦𝑥 𝜎𝑦𝑥\nsame origin as the normal Hall effect, whereas is completely different from\n𝑖\nthe two-carrier nature induced normal Hall effect𝜎. 𝑦𝑥\n Third, we also provide Hall curves for θ varying from to for sample #1\nin Fig. R2. As the B field is rotated away from the normal line, the Hall signal\n0° 105°\nincreases first and then decreases, but it remains non-zero even when B // I. It\nindicates that there is another contribution (i.e., IPHE) to Hall effect as B\napproaches the direction of I. IPHE and normal Hall effect are two different\neffects.\nFig. R1a Magnetic field dependence of the in-plane Hall conductivity and R1b\nmisalignment-induced out-of-plane Hall conductivity for sample #1 at 2K. R1c and\nR1d are normal Hall measurement schematic and normal Hall conductivity for\nsample #1 at 2K. The dashed line the guide line for sign change of normal Hall.\nFig. R2 Hall resistivity for sample #1 versus magnetic field applied at different angles\nto the [-112] direction, as schematically shown in the inset. [Manuscript, Fig. 2b]\nComment 2: It may be difficult to position LuAuSn this compound as an ideal system\nof the in-plane Hall effect, and the authors are encouraged to clarify the limitations\nmore explicitly. As already noted, this compound intrinsically contains two types of\ncarriers, which is also supported by the band structures in Fig. 5a. This inevitably\ncomplicates the field and temperature dependence of the Hall effect. Furthermore, in\nthe nonmagnetic cubic system, octupolar as well as linear couplings between the\nHall vector and the magnetic field are allowed, but they are restricted to diagonal ones.\nIn this situation, the in-plane anomalous Hall effect accidentally appears only when the\nmeasurement plane is not defined by two principal axes, such as the (111) and (103)\nplanes (see Phys. Rev. X 15, 031006 (2025)., not cited).\nResponse: We thank you for this comment regarding the influence of two-carriers\nconductivity in LuAuSn. To clarify the issue, we provide IPHE data at different\ntemperatures and the corresponding analysis.\n First, if the B-cubic behavior of IPHE comes from two-carrier model, it will\ninevitably destroy the B3-scaling and forms complex B-dependent behavior,\nespecially when the hole and electron concentrations are not identical. Second, as\nshown in Fig. S2, the concentrations of holes and electrons are nearly identical at\nlow temperatures, and the density of electrons increases as T increase, while the\ndensity of holes is almost constant over the whole temperature range from 2K to\n300K. The growing imbalance between the concentrations of hole and electron\nmakes the normal Hall increasingly linear (i.e. closer to a single-carrier model).\nBased on the above reasoning, the IPHE curves will deviate from B3 scaling at\nhigh temperatures. To identify this property, we carried out measurement of the\ntemperature dependent IPHE (see Fig. R3). We found that IPHE exhibits B3\nscaling over a broader magnetic-field range at higher temperatures. This result\ncontradicts what would be expected from the two-carrier model, and therefore\nconfirms that B3 scaling arises from the IPHE in LuAuSn under C crystal\n3Z\nsymmetry.\n Regarding the reviewer’s second point, from the perspective of magnetic\noctupoles, we do not think this is a difficulty for positioning LuAuSn as an ideal\nsystem of IPHE. In fact, cubic system is one of the suitable candidates for\nstudying B3-scaling IPHE as mentioned by the paper [Phys. Rev. X 15, 031006\n(2025)]. This is because the dipole contribution (in the language of our paper, the\nB1-scaling IPHE) is isotropic and therefore vanishes for any measurement plane.\nThis was precisely the motivation for examining the octupolar structure in cubic\nsystems in that work. We agree with the reviewer that the IPHE cannot appear in\ncubic systems when the measurement plane is defined by two principal axes.\nHowever, in the face-centered cubic structure, the (111) plane is the close-packed\nplane, the atoms preferentially form flat (111) planes during crystal growth, and\nthus do not pose any difficulty in experiment.\nNote: We have cited Phys. Rev. X 15, 031006 (2025) (reference [23]) in revised\nmanuscript.\nFig. S2a The temperature-dependent resistivity at zero magnetic field. The residual\nresistivity ratio (RRR= ) is 3 for sample #1. S2b and S2c The\nmagnetic field-dependen\n𝜌\ntx xl\n(\no\n3\n0\n)\ni\n/\nn\n𝜌\naxlx\n(\nr\nsi\n)\nresistivity and Hall resistivity respectively\nmeasured from 2K to 300K. The inset of S2c shows Hall resistivity at B < 3T. S2d\nThe two-carrier model fitting curves to the longitudinal resistivity and Hall resistivity\nat 2K. S2e and S2a The temperature dependence of extracted carrier mobilities and\ncarrier densities, respectively.\nFig.R3a Magnetic field dependence of in-plane Hall resistivity measured from 2K to\n300K for sample #1. The magnetic field is applied along direction. The inset\nshows the curves at high temperatures. R3b and R3d are B3\n[\n1̅f\n1\nit\n2\nti\n]\nng of IPHE at 2K and\n300K, respectively. R3c The magnetic field-temperature diagram indicating the range\nover which of B3 scaling is observed in LuAuSn (sample #1). [Supplementary\ninformation, note 3]\nComment 3: It is noted that of sample #1 in Fig. 2b does not change sign for the\nout-of-plane field sweep, whe\n𝜌\nr𝑥e𝑦as in Fig. 4c changes from negative to positive.\nIs this from sample #2, or from y𝜎e𝑥t 𝑦another sample? If so, it would be helpful to\nprovide basic transport characterization of these samples, as the two-carrier behavior\nalso differs from sample to sample.\nResponse: We appreciate your careful review of the data. (1) Fig.R4 shows the basic\ntransport characterization for sample #2. The results indicate that the carrier density\nand mobility of sample #1 and #2 show a small difference, but both samples clearly\nshow two-carrier behavior of out-of-plane (normal Hall effect) data (Fig. S2 and\nFig.R4). Sample #2 shows higher carrier density and smaller mobility. The observed\nsign change for sample #1 comes from the nearly perfect charge carrier compensation\nand similar mobility of electrons and holes. (2) The data in Fig. 2b were obtained for\nsample #2, while the data in Fig. 4c were obtained for sample #1. We have now\nreplaced the data in Fig. 2b with that of sample #1 (Fig. R2). [Fig. 2, page 19]\nFig.R4a and R4b Magnetic field-dependent longitudinal resistivity and Hall\nresistivity, respectively, measured from 2K to 300K for sample #2, respectively. R4c -\nR4d Temperature dependence of carrier density and mobility, respectively, extracted\nfor sample #2. [Supplementary information, note 2]\nComment 4: Related to the point above, the manuscript would benefit from a more\ndetailed discussion of how off-stoichiometry and structural disorder in the\nhalf-Heusler compound affect presented two-carrier transport and scattering effects,\nwhich can differ from sample to sample. The EDX analysis indicates a slight\ndeviation from the nominal stoichiometric ratio, and the degree of order/disorder\ncould in principle be quantified from the relative XRD peak intensities. It remains\nunclear how much\ndisorder is present in the examined samples including sample #2 or the one displayed in Fig. 4c. Considering that side-jump\nand skew-scattering contributions play a large role, it is essential to present whether the first-principles calculations were\ncarried out for the ordered phase, and how disorder—if included—would alter the theoretical predictions.\nResponse: We appreciate your insightful comment on the role of defects and disorder\nin LuAuSn. We agree that understanding the quantitative relationship between defects\n(including off-stoichiometry and structural disorder) and the IPHE is important. But to\nbe the best of our literature knowledge, establishing a fully quantitative relation\nbetween them is not an easy task, if not impossible [7-9]. Standard characterization\ntechniques as EDX and XRD may not be sufficient for determining the exact\ncontribution of these defects to the IPHE.\nNevertheless, the residual resistivity ratio (RRR) is an indicator characterizing\nthe relative defect concentration. To explore the relation between RRR and in-plane\nHall conductivity, we grow another two batches of single crystal in different growth\nconditions (temperature regimes B and C shown in Fig. R5). The single crystal quality\nis improved by optimizing the growth progress (regimes B and C). The RRR values\nfor regimes A, B and C are about 3, 5 and 8, respectively. According to the current\nunderstanding of extrinsic mechanisms, the skew scattering contribution becomes\nmore pronounced in cleaner samples, leading to higher value of Hall conductivity\n[7-9]. The results summarized in Table 1 indicate that samples with higher RRR\ndemonstrate larger in-plane Hall conductivity, confirming that extrinsic mechanisms\nplay an important role in LuAuSn.\nRegarding the question on first-principles calculations, we would like to clarify\nthat in the theory of Hall effect (and thus of IPHE) for metallic states, disorder is\nusually treated as a perturbation. Therefore, the first-principles calculations were\ncarried out for the ordered phase. In addition, we also note that the side-jump and\nskew-scattering contributions are analyzed by scaling law, and are not related to\nfirst-principles calculations.\nFig. R5 The temperature regimes used for the single crystal growth of samples #1-#5\nFig. R6 The temperature-dependent resistivity and in-plane Hall conductivity for\nsamples #3-#5. Note that the measured in-plane Hall conductivity for samples #3- #5\nincludes the contribution from misalignment-induced out-of-plane Hall conductivity.\nHowever, this contribution does not influence the observation of record-high in-plane\nHall conductivity for the high-quality sample.\nTable 1 The RRR value and in-plane Hall conductivity for sample #1-#5\n[7] Bijun Tang, et al, Nature Electronics 5 (4), 224 (2022).\n[8] Shuo-Ying Yang, et al, Science Advances 6 (31), eabb6003 (2020).\n[9] Y. Fujishiro, et al, Nat. Commun. 12 (1), 317 (2021).\nComment 5: (1) Considering that multiple fitting parameters are used, how\naccurately can each contribution be decomposed? (2) How about the fitting results for\nsample #2? (3) I also have concerns regarding the direct application of the\nscaling from single-carrier systems (Ref. 39) to the multi-carrier system. (4) Does\n𝜎\nth𝑥𝑥e\ncalculation of the Lorentz-force contributions in Fig. 5c also take into account\nmulti-carrier effects? The results should be displayed over the same energy range as in\nFig. 5b.\nResponse: We appreciate your detailed comments on the fitting procedure. Our\nresponses for the four questions are as follows.\n A: To clarify the accuracy of our fitting, we have included error bars for the\ndecomposed contributions in Fig. 5d and list the standard errors in Fig. R7. The\nfitting errors are smaller than 5%, indicating the reliability of our analysis (see Fig.\nR5, the standard errors of fitting parameters are much smaller than their values). B:\nRegarding the accurate decomposition of the different contributions, there is an\nintrinsic theoretical limitation to the scaling-law analysis: In general, it cannot\ncompletely disentangle side-jump and skew-scattering contributions if both linear\nand quadratic scaling terms coexist. C: As shown in Table R1, we can obtain\nsingle crystals of different quality (RRR) by tuning the growth conditions.\nTheoretically, by characterizing samples with different RRR at the low\ntemperature limit, we can separate C (conventional skew-scattering) and C\n1 2\n(impurity scattering induced side-jump and intrinsic skew-scattering) by using\nlinear fitting , and no further separation can be made in general.\nNevertheless,\n𝑦\nth\n=\nthe\n𝐶\nc2ur\n+\nre\n𝐶\nn1t\n𝜎\nr𝑥e𝑥s0ults of the scaling law analysis give the definite\nconclusion that both impurity and phonon scatterings play important roles in our\nsample. \n We carried out additional measurements for sample #2 at 3T, the results of which\nare shown in Fig. R8. The in-plane Hall conductivity at 2K and 3T is about\n145 . The scaling analysis yields: ,\n−1 −1 −1 −1\n𝛺 𝑐𝑚\n, and , w\n𝐶\ni2th\n+\nt\n𝐶\nh1e\n𝜎\n𝑥fi𝑥t0tin\n=\ng\n1\ne\n8\nr\n7\nΩ\nr le\n𝑐\nss\n𝑚\nthan\n𝐶\n83% \n=\n. \n−1 −1 −1 −1\n−\nTh\n3\ne\n2\ns\n1\ne\nΩ\nresu\n𝑐\nl\n𝑚\nts obtained\n𝐶\nf4o\n=\nr s\n1\na\n4\nm\n5\np\nΩ\nle #\n𝑐\n2\n𝑚\nalso confirm that that both impurity and\nphonon scattering play important roles, consistent with our findings for sample #1.\n The application of for scaling relies on the fact that it measures the\ncharacteristic time sca\n𝜎\nl𝑥e𝑥 of transport. The scaling law analysis does not depend on\nwhether the system contains single or multiple types of carriers. In fact, using\nfor scaling has been widely applied to typical multi-carrier systems, such\n𝜎\n𝑥a𝑥s\nKV Sb [8] and Co Sn S [10-11].\n3 5 3 2 2\n The calculation of the Lorentz-force contribution was based on the first-principles\nband structure, which automatically takes into account the electron and hole\ncontributions at different doping levels. In the revision, we have replotted Fig. 5c\nover an energy range of eV, the same as the plot in Fig. 5b.\n±0.1\nFig.R7 The fitting and standard error for sample #1\nFig.R8a Angular dependence of in-plane Hall conductivity at 300K and B\n= 3 T for sample #2. R8b The scaling of in-plane Hall cond2uKct𝜎iviTty≤ (3T) as a\ni\nfunction of sample #2. [Supplementary informati𝜎o 𝑦 n 𝑥 , note 6]\n[10] Enke\n𝜎\nL𝑥𝑥iu\n(\n,\n3\na\n𝑇\nt\n)\ne\n⁄\nl\n𝜎\n𝑥𝑥\n(\n2\n𝐾\n,\n3\n𝑇\n)\n,N\nat\n.Ph\nys.14 (11), 1125 (2018).\n[11] Jianlei Shen, at el, Adv. Funct. Mater., 2000830 (2020).\nComment 6: I am concerned that the main results presented as may in fact\ncorrespond to . In addition, it seems that the definition of axi\n𝜌\ns𝑥 𝑦handedness is\ninconsistent bet𝜌w𝑦e𝑥en Figs. 2a and 2c, which may result in additional confusion.\nResponse: Thank you for your kind suggestion. We apologize for any confusion\ncaused by these mistakes. We have standardized all the symbols used in the revised\nmanuscript.\nWe hope that our detailed responses have satisfactorily address your comments\nand improve the clarity of manuscript. We truly appreciate your thoughtful review and\nthank you once again for your time and consideration.\nResponse to the Report of Reviewer #3\nThis manuscript reports the observation of a giant in-plane Hall effect (IPHE) in the\nnonmagnetic half-Heusler compound LuAuSn, with clear characteristics B3.\nDependency relationship and 2/3 periodicity, and C symmetry are consistent.\nThe\n3v\nworking experiments are solid and supported by first-principles calculations,\nrepresenting an important advancement in the field of nonlinear Hall effects. The\namplitude of the effect is very significant and can last up to room temperature,\nshowing great application potential. I therefore recommend publication in Nature\nCommunications, pending minor revisions to address the following points.\nResponse: Thank you for your positive evaluation of our study, including its\ninnovative nature, impact, and potential application value. We also sincerely\nappreciate your insightful review and constructive feedback, which have significantly\nenhanced the clarity and rigor of our manuscript. We have carefully addressed each\ncomment, added necessary data, and made corresponding revisions to improve the\nwork's quality. We would also like to note that based on your review comments, we\ndiscovered an effective approach to enhance in-plane Hall performance. Below, we\nprovide detailed responses to your comments.\nComment 1: (1) The manuscript mentioned that high-quality single crystals were\nsynthesized, but the residual resistivity (RRR) was only 3 (Figure S2a). This value is\nrelatively low, indicating that there is a considerable concentration of defects or\nimpurities in the sample. This may affect the analysis of the scattering mechanism and\nraise questions about whether the effect can maintain the same scale relationship or\neven be stronger in higher-quality samples. (2) It is suggested that the author provide\nmore information about the sample quality, such as whether samples from different\nbatches exhibit repeatable RRR and IPHE signals. Have you ever attempted to\nimprove the growth process to obtain samples with a higher RRR? The manuscript\nmentioned that their crystals are “high-quality,” yet the residual-resistivity ratio (RRR)\nis only ~3 (Fig. S2a). That value is low enough to imply a sizeable concentration of\ndefects or impurities, which could skew any quantitative discussion of scattering.\nPlease show RRR and IPHE data for at least two additional growth batches to confirm\nreproducibility. Comment on whether growth conditions (flux ratio, cooling rate,\nannealing) have been varied in an attempt to raise the RRR. (3) In the discussion, it\nshould be clearly pointed out that the limited mass of the current sample may have an\nimpact on the quantitative analysis of the scattering mechanism.\nResponse: We thank the reviewer very much for the insightful comments on the\nsample quality and very helpful suggestions.\n First, we have revised the inaccurate description of single crystal quality in the\nrevised manuscript [Lines 120-121, Page 5; Line 279, Page 10].\n Second, according to your suggestions, we design another two experiments on\nsingle crystal growth. The detailed growth conditions are shown in Fig. R1. The\nsingle crystal quality was improved by optimizing the growth procedure\n(Temperature regimes B and C). The RRR values are approximately 3 for regime\nA, 5 for regime B, and 8 for regime C. These results indicate that the growth\nconditions, like flux ratio, cooling rate, growth temperature, can influence the\nquality of single crystal. Thus, the single crystal quality can be controlled through\nadjusting growth conditions.\n Third, we also performed preliminary measurement of in-plane Hall conductivity\nat 2K for different samples. The results in Fig. R2 and Table R1 indicate that the\nhigh quality of sample is a key factor to obtain high in-plane Hall conductivity.\nAccording to the current understanding of extrinsic mechanisms of IPHE, the\nskew-scattering contribution becomes more pronounced in cleaner samples,\nleading to higher value of Hall conductivity [5-7]. The results summarized in\nTable R1 indeed indicate that samples with higher RRR demonstrate larger\nin-plane Hall conductivity, confirming that extrinsic mechanisms, especially the\nskew-scattering, play an important role in LuAuSn.\nFinally, we also clarified the limitation of current scaling analysis due to the\nsmall number of samples in the discussion section. [Line 233-239, Page 9].\nFig. R1 Temperature regimes used for the single crystal growth for sample #1-#5\nFig. R2 The temperature-dependent resistivity and in-plane Hall conductivity for\nsamples #3-#5. Note that the measured in-plane Hall conductivity for samples #3-\n#5 includes the contribution from misalignment-induced out-of-plane Hall\nconductivity. However, this contribution does not influence the observation of\nrecord-high in-plane Hall conductivity for the high-quality sample.\nTable R1 The RRR and the maximum values of in-plane Hall conductivity for\nsamples #1-#5\nComment 2: (1) In Page 12, Line 188-189; Page 13, Line 254-259, the author points\nout that after B > 3T, gradually deviates B3. The dependency relationship was merely\ntouched upon with \"perturbation theory failure\" and \"non-perturbation effect\", lacking\nin-depth physical discussion and experimental data analysis. Here, I suggested that the\nauthor conduct a more in-depth analysis of the behavior in high-field Settings. Can\nauthors try using higher-order items (such as aB3+bB5, or another suitable form) to\nperform fitting? (2) Is this deviation related to the magnetic field correction of the\nband structure (such as the Zeeman effect and the formation of Landau energy levels)?\n(3) Adding some angle-dependent measurements under high field conditions might\nreveal more information. A speculative discussion on high-field behavior should be\nadded to the discussion section.\nResponse: We appreciate the reviewer's valuable suggestions to explore the high-field\nbehavior of IPHE in more detail.\n We have conducted a more comprehensive analysis of the high-field data (5T ~\n9T) at 2K using fitting with higher-order terms . The results\n3 5\nindicate that in high field range a B-linear scaling emerges (Fig. R3). Meanwhile,\naB+bB +cB\nthe behavior from 3T to 5T manifests the transition from to -linear scaling.\nB B\nLuAuSn in the perturbative regime of B field, and thus the fitted B-linear scaling\nat high fields can only be a strong-field non-perturbative behavior and an\napproximation. As the reviewer pointed out exactly, large B field can\nconsiderably change the band structure, and the magneto-transport may no longer\nconform to polynomial dependence on B. The specific scaling power on B is thus\nnot informative in this regime. Indeed, a previous theoretical work [1] discussed\nthe deviation from B3 scaling of IPHE under high magnetic field, and attributed it\nto Zeeman-field induced band structure modifications.\n We also measured the temperature dependence of the critical field for the\nB3-scaling regime in Fig.R4. The results show that the critical field shifts to\nhigher magnetic field as temperature increases. This remarkable feature is also\nconsistent with the above argument that the high-field behavior deviating from B3\nscaling originates from field-induced change of band structures, as this change is\nless effective at higher temperatures due to increasing thermal smearing.\nIn revision, we have added the above results and discussions in the revised\nmanuscript [Lines 265-274, Page 10]\nFollowing the reviewer’s suggestion, we have also conducted angle-dependent\nmeasurements under higher magnetic fields, as shown in Fig. R5.\nFig. R3 Linear fitting for high-filed in-plane Hall resistivity.\nFig.R4a The magnetic field dependence of in-plane Hall resistivity measured from\n2K to 300K for sample #1. The magnetic field is applied along . The inset\nshows the curves at high temperatures. R4b and R4d are B3 fitting of\n[\n1̅IP\n2]\nE at 2K and\n300K, respectively. R3c The magnetic field-temperature diagram indicating the range\nover which B3 scaling is observed in LuAuSn (sample #1). [Supplementary\ninformation, note 3]\nFig.R5a and R5b The in-plane anisotropic MR and in-plane Hall resistivity measured\nat B =10T, 12T and 14T. The presence of significant SdH oscillation in high magnetic\nfield make the in-plane Hall curves not smooth as the B scanning in (111) plane.\n[Supplementary information, note 7]\n[1] Vladimir A. Zyuzin, Phys. Rev. B 102, 241105(R) (2020).\nComment 3: The authors candidly state that the scaling-law analysis cannot\ndisentangle side-jump and skew-scattering contributions. While this is an intrinsic\ntheoretical limitation, the manuscript never discusses how this limitation affects the\nconclusions, nor does it suggest concrete ways to overcome it (for example, by\nsystematic doping experiments that tune the scattering channels). I will recommend\nthat the authors add a paragraph in “Discussion on the physical mechanisms” that\nexplicitly highlights the inability of the scaling law to separate individual scattering\nmechanisms. Frame this limitation as a future research direction rather than a mere\nanalytical outcome.\nResponse: Thank you for your valuable suggestions. First, the scaling analysis in\ncombination with the first-principles calculations concluded that the observed IPHE\noriginates from side-jump and skew scattering mechanisms. Although one cannot\nfurther separate the side-jump and skew-scattering contributions, our conclusion is not\naffected by this theoretical limitation. Second, additional experiments can indeed\nprovide more information. It has been proposed that the C term (conventional skew\n1\nscattering) can be separated by performing measurements on samples with different\nthicknesses at low temperatures. Once C is obtained, one can obtain C contribution\n1 2\n(impurity scattering induced side-jump and intrinsic skew-scattering) by linear fitting\n. Following the reviewer’s suggestion, we have added a dedicated\n𝑦\npa\n=\nrag\n𝐶\nr2ap\n+\nh 𝐶1in\n𝜎\n𝑥𝑥th0e \"Discussion\" section that explicitly addresses this theoretical\nlimitation and suggest possible ways to further separate the C and C contributions\n1 2\n[Lines 233-239, Page 9].\nComment 4: Figure 4f compares the IPHE conductivity of LuAuSn with that of other\nmaterial systems, a valuable comparison. However, the cited data (Refs. [1–3, 5, 17,\n18, 29]) may have been measured under different temperatures and magnetic fields. It\nis needed to explicitly annotate to undermine the rigor of this comparison (i.e., same T\nand B).\nResponse: We selected data in other samples also with 2K and 3T for comparison,\nand there are some materials in previous experiments that were not measured at 2K.\nAs such, we have chosen the lowest temperature in those measurements for\ncomparison. Following the reviewer’s suggestion, in the revised manuscript and\nSupplemental information, we have provided more detailed comparisons. [Fig.4f;\nsupplementary information table S1]\nComment 5: In the first-principles estimate of the Lorentz-force contribution, the\nauthors adopt a constant relaxation time τ = 0.1 ps,\nextracted by matching the calculated conductivity to the measured Drude conductivity.\nYet τ is likely energy-dependent, and it is questionable whether a single value applies\nto all carriers and all temperatures. Please briefly discuss in the Methods section why\nthis τ was chosen, acknowledge that an energy-independent τ is an approximation, and\ncomment on how this approximation might affect the order-of-magnitude estimate of the\nLorentz-force contribution.\nResponse: We thank the reviewer for this valuable suggestion. We agree that the\nrelaxation time can be energy dependent. Using the conductivity of S/cm,\n4\nthe estimated relaxation time as a function of chemical potential is shown in Fig. R6a.\n1.6×10\nThis result indicates it is reasonable to take ps within energy range of\nmeV. In revision, we have replotted Fig. 5c in range of eV according to the\nτ≈ 0.1 ±10\nreviewer’s suggestion [see also Fig. R6b]. Since the relaxation time deviates from\n±0.1\nps over this range, we instead use the energy dependent relaxation time and illustrate\n0.1\nthis point in Method section [lines 320-322, page 11-12]. The results show that the\nenergy dependence of the relaxation time does not affect the estimation of the\nLorentz-force contribution.\nFig.R6a Estimated relaxation time as a function of chemical potential. R6b\nCalculated Lorentz-force-induced in-plane Hall conductivity. (Manuscript,Fig. 5c)\nComment 6: The caption of Fig. 4f reads: “The data were taken from references [1–3,\n5, 17, 18, 29].” In the reference list (p. 7),\nhowever, Refs. [17] and [18] are 2025 preprints or papers marked “in press” on Fe thin films and Co Sn S , respectively.\n3 3 2\nCiting not-yet-published work—especially non-peer-reviewed preprints—in a\nquantitative comparison demands extra caution and explicit labeling. Amend the\ncaption to state that Refs. [17] and [18] are preprints or “in press.” Whenever\npossible, replace them with formally published data. If their inclusion is unavoidable,\nadd a sentence in the main text or Supplementary Information noting the preliminary nature\nof these data.\nResponse: Thank you for the very valuable suggestion. In revision, we have updated\nthe citation information of Refs. 17 and 18, which have been published on Phys. Rev.\nB in 2025.\nComment 7: On pages 9&10, both “C ” and “C ” symmetry appear. Although both\n3z 3v\nare relevant for the (111) surface, the notation should be uniform and precise; please\ndefine it explicitly at its first appearance.\nResponse: We thank the reviewer very much for the helpful suggestions. We have\ncorrected the notation for symmetry throughout the manuscript to maintain uniformity\n[Line 515, Page 18]. A precise definition of C in the context of the (111) surface\n3Z\nappears in the revised manuscript to enhance clarity [Lines 126-128, Page 5].\nOverall, this work represents a significant step forward in the study of in-plane Hall\neffects. The combination of giant response, room-temperature operation, and clear\nsymmetry-based interpretation makes it a strong candidate for publication in Nature\nCommunications. I recommend acceptance pending minor revisions as outlined\nabove.\nOnce again, we thank the reviewer very much for the thoughtful comments and\nsuggestions, which have significantly helped improve the manuscript. We look\nforward to the opportunity to publish this work in Nature Communications.\nResponse to the Report of Reviewer #1\nThe authors have made a serious and thoughtful effort to address the concerns I raised\nin the first round. In particular, they have substantially clarified the distinction\nbetween their results and earlier reports, strengthened the symmetry-based arguments,\nand improved the overall positioning of the work. The revised presentation is clearer\nand more careful, and the additional analyses help to solidify the main physical claims.\nIn view of the significant improvements made during the revision, I believe the\nmanuscript is now suitable for publication.\nResponse: We sincerely thank the reviewer for the very helpful review of our\nmanuscript, and we are grateful for the recommendation of our work.\nResponse to the Report of Reviewer #2\nWhile the authors have made a clear effort to address the reviewers' comments, the B3\ndependence remains relatively straightforward for a cubic (111) plane due to\nsymmetry constraints on the B dependence. As also pointed out by other reviewers,\nthe magneto-cubic behaviour weakens at higher magnetic fields and does not appear\nto be directly linked to a giant response. Accordingly, the strong emphasis on both\n'giant' and 'magneto-cubic' behaviour especially in the title may be somewhat\noverstated. By contrast, the investigation of Heusler compounds with in-plane\nanomalous Hall effects, and the possible enhancement associated with scattering\nmechanisms, constitutes an interesting direction that could merit publication.\nResponse: We are pleased that the reviewer finds our work constitutes an interesting\ndirection and recommends it for publication. Regarding the reviewer's question, as\nshown in Fig. 4f, the IPHE in LuAuSn at 3 T is larger than those in other nonmagnetic\ncompounds. At the same time, the magneto-cubic behavior remains well preserved at\nmagneto-cubic.\nResponse to the Report of Reviewer #3\nHaving thoroughly examined the authors’ detailed point-by-point rebuttal and the\nrevised manuscript, I am pleased to state that my initial concerns have been\ncomprehensively and convincingly addressed. The authors have gone to exceptional\nlengths to provide not only clear theoretical and experimental clarifications but also\nsubstantial new data that significantly strengthen the manuscript's core claims. The\nquality of the response and the depth of the additional analysis are commendable. My\nprimary reservations regarding the novelty and impact of the work have been fully\nresolved. The authors' symmetry-based argument distinguishing their observation in\nLuAuSn from earlier reports in materials like germanium is persuasive. They\ncorrectly point out that the experimental configurations in those historical studies\ncould not isolate a genuine symmetry-protected in-plane Hall effect, making their\nwork the first clear demonstration of a magneto-cubic IPHE in a well-defined,\nthree-dimensional nonmagnetic crystalline system. Furthermore, the newly added data\non samples with progressively higher residual resistance ratios (RRR) is a\nmasterstroke. The clear trend of dramatically enhanced IPHE conductivity with\nincreasing sample quality provides direct and compelling experimental evidence that\nextrinsic scattering mechanisms—namely side-jump and skew-scattering—are\ndominant. This successfully shifts the narrative from merely observing a large effect\nto understanding its origin and establishing a material-based strategy for its\noptimization. The authors have also done an excellent job tackling more technical\nconcerns. The additional analysis of high-field behavior, including attempts at\nhigher-order fitting and the discussion of a possible crossover regime, adds valuable\nnuance. The demonstration that the scaling law analysis holds for a second sample (#2)\nand the clarification of the multi-carrier effects reinforce the robustness of their\nconclusions. I appreciate the careful revisions to the symmetry discussion, the updated\ncomparisons in Figure 4f, and the candid discussion in the manuscript regarding the\ninherent limitations of separating individual scattering contributions within the scaling\nframework.\nIn summary, the revised manuscript presents a highly complete and robust study.\nIt makes a significant contribution by unveiling a new, high-performance material\nplatform for the in-plane Hall effect, firmly establishing the role of extrinsic\nmechanisms in generating a record-large response, and providing a clear path for\nfuture material engineering. The work undoubtedly meets the high standards for\npublication in Nature Communications.\nFor final polishing, I only suggest a few minor editorial points. (1) Please ensure\na brief note is added to the figure captions or methods to clarify which specific sample\n(#1, #2, etc.) corresponds to each main figure panel, as this was discussed in the\nrebuttal. (2) The phrasing on the high-field behavior in the discussion could be\nslightly refined to more explicitly frame the deviation from B³ scaling as an entry\npoint into the interesting physics of non-perturbative magnetic field regimes. (3)\nFinally, a final check to ensure all key data from the rebuttal (like the RRR vs.\nconductivity table and the temperature-field phase diagram) are seamlessly integrated\ninto the main text or supplementary information would be beneficial.\nI am fully satisfied with the authors' responses and the state of the manuscript. I\nrecommend acceptance in its current form.\nResponse: We sincerely thank the reviewer for the insightful review of our\nmanuscript, the highly positive evaluations and the recommendation for publication.\nWe are also grateful for the final polishing suggestions, and we have made the\nfollowing corresponding revisions:\n(1) We have clarified this point in the figure captions.\n(2) The relevant statement has been rephrased in the Discussion section. (page 10,\nlines 270-271)\n(3) We have incorporated the key data and related discussion into the Supplementary\nInformation. (Supplementary Note 8)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made a serious and thoughtful effort to address the concerns I raised in the first round. In particular, they have substantially clarified the distinction between their results and earlier reports, strengthened the symmetry-based arguments, and improved the overall positioning of the work. The revised presentation is clearer and more careful, and the additional analyses help to solidify the main physical claims.\nIn view of the significant improvements made during the revision, I believe the manuscript is now suitable for publication.", "reviewer_2": "While the authors have made a clear effort to address the reviewers' comments, the B3 dependence remains relatively straightforward for a cubic (111) plane due to symmetry constraints on the B dependence. As also pointed out by other reviewers, the magneto-cubic behaviour weakens at higher magnetic fields and does not appear to be directly linked to a giant response. Accordingly, the strong emphasis on both 'giant' and 'magneto-cubic' behaviour especially in the title may be somewhat overstated. By contrast, the investigation of Heusler compounds with in-plane anomalous Hall effects, and the possible enhancement associated with scattering mechanisms, constitutes an interesting direction that could merit publication.", "reviewer_3": "Having thoroughly examined the authors’ detailed point-by-point rebuttal and the revised manuscript, I am pleased to state that my initial concerns have been comprehensively and convincingly addressed. The authors have gone to exceptional lengths to provide not only clear theoretical and experimental clarifications but also substantial new data that significantly strengthen the manuscript's core claims. The quality of the response and the depth of the additional analysis are commendable.\nMy primary reservations regarding the novelty and impact of the work have been fully resolved. The authors' symmetry-based argument distinguishing their observation in LuAuSn from earlier reports in materials like germanium is persuasive. They correctly point out that the experimental configurations in those historical studies could not isolate a genuine symmetry-protected in-plane Hall effect, making their work the first clear demonstration of a magneto-cubic IPHE in a well-defined, three-dimensional nonmagnetic crystalline system. Furthermore, the newly added data on samples with progressively higher residual resistance ratios (RRR) is a masterstroke. The clear trend of dramatically enhanced IPHE conductivity with increasing sample quality provides direct and compelling experimental evidence that extrinsic scattering mechanisms—namely side-jump and skew-scattering—are dominant. This successfully shifts the narrative from merely observing a large effect to understanding its origin and establishing a material-based strategy for its optimization.\nThe authors have also done an excellent job tackling more technical concerns. The additional analysis of high-field behavior, including attempts at higher-order fitting and the discussion of a possible crossover regime, adds valuable nuance. The demonstration that the scaling law analysis holds for a second sample (#2) and the clarification of the multi-carrier effects reinforce the robustness of their conclusions. I appreciate the careful revisions to the symmetry discussion, the updated comparisons in Figure 4f, and the candid discussion in the manuscript regarding the inherent limitations of separating individual scattering contributions within the scaling framework.\nIn summary, the revised manuscript presents a highly complete and robust study. It makes a significant contribution by unveiling a new, high-performance material platform for the in-plane Hall effect, firmly establishing the role of extrinsic mechanisms in generating a record-large response, and providing a clear path for future material engineering. The work undoubtedly meets the high standards for publication in Nature Communications.\nFor final polishing, I only suggest a few minor editorial points. Please ensure a brief note is added to the figure captions or methods to clarify which specific sample (#1, #2, etc.) corresponds to each main figure panel, as this was discussed in the rebuttal. The phrasing on the high-field behavior in the discussion could be slightly refined to more explicitly frame the deviation from B³ scaling as an entry point into the interesting physics of non-perturbative magnetic field regimes. Finally, a final check to ensure all key data from the rebuttal (like the RRR vs. conductivity table and the temperature-field phase diagram) are seamlessly integrated into the main text or supplementary information would be beneficial.\nI am fully satisfied with the authors' responses and the state of the manuscript. I recommend acceptance in its current form."}, "author_response": "Response to the Report of Reviewer #1\nThe authors have made a serious and thoughtful effort to address the concerns I raised in the first round. In particular, they have substantially clarified the distinction between their results and earlier reports, strengthened the symmetry-based arguments, and improved the overall positioning of the work. The revised presentation is clearer and more careful, and the additional analyses help to solidify the main physical claims.\nIn view of the significant improvements made during the revision, I believe the manuscript is now suitable for publication.\nResponse: We sincerely thank the reviewer for the very helpful review of our manuscript, and we are grateful for the recommendation of our work.\nResponse to the Report of Reviewer #2\nWhile the authors have made a clear effort to address the reviewers' comments, the B3 dependence remains relatively straightforward for a cubic (111) plane due to symmetry constraints on the B dependence. As also pointed out by other reviewers, the magneto-cubic behaviour weakens at higher magnetic fields and does not appear to be directly linked to a giant response. Accordingly, the strong emphasis on both 'giant' and 'magneto-cubic' behaviour especially in the title may be somewhat overstated. By contrast, the investigation of Heusler compounds with in-plane anomalous Hall effects, and the possible enhancement associated with scattering mechanisms, constitutes an interesting direction that could merit publication.\nResponse: We are pleased that the reviewer finds our work constitutes an interesting direction and recommends it for publication. Regarding the reviewer's question, as shown in Fig. 4f, the IPHE in LuAuSn at 3 T is larger than those in other nonmagnetic compounds. At the same time, the magneto-cubic behavior remains well preserved at 3 T. Therefore, it is appropriate to describe the IPHE in LuAuSn as both giant and magneto-cubic.\nResponse to the Report of Reviewer #3\nHaving thoroughly examined the authors’ detailed point-by-point rebuttal and the revised manuscript, I am pleased to state that my initial concerns have been comprehensively and convincingly addressed. The authors have gone to exceptional lengths to provide not only clear theoretical and experimental clarifications but also substantial new data that significantly strengthen the manuscript's core claims. The quality of the response and the depth of the additional analysis are commendable. My primary reservations regarding the novelty and impact of the work have been fully resolved. The authors' symmetry-based argument distinguishing their observation in LuAuSn from earlier reports in materials like germanium is persuasive. They correctly point out that the experimental configurations in those historical studies could not isolate a genuine symmetry-protected in-plane Hall effect, making their work the first clear demonstration of a magneto-cubic IPHE in a well-defined, three-dimensional nonmagnetic crystalline system. Furthermore, the newly added data on samples with progressively higher residual resistance ratios (RRR) is a masterstroke. The clear trend of dramatically enhanced IPHE conductivity with increasing sample quality provides direct and compelling experimental evidence that extrinsic scattering mechanisms—namely side-jump and skew-scattering—are dominant. This successfully shifts the narrative from merely observing a large effect to understanding its origin and establishing a material-based strategy for its optimization. The authors have also done an excellent job tackling more technical concerns. The additional analysis of high-field behavior, including attempts at higher-order fitting and the discussion of a possible crossover regime, adds valuable nuance. The demonstration that the scaling law analysis holds for a second sample (#2) and the clarification of the multi-carrier effects reinforce the robustness of their conclusions. I appreciate the careful revisions to the symmetry discussion, the updated comparisons in Figure 4f, and the candid discussion in the manuscript regarding the inherent limitations of separating individual scattering contributions within the scaling framework.\nIn summary, the revised manuscript presents a highly complete and robust study.\nIt makes a significant contribution by unveiling a new, high-performance material platform for the in-plane Hall effect, firmly establishing the role of extrinsic mechanisms in generating a record-large response, and providing a clear path for future material engineering. The work undoubtedly meets the high standards for publication in Nature Communications.\nFor final polishing, I only suggest a few minor editorial points. (1) Please ensure a brief note is added to the figure captions or methods to clarify which specific sample (#1, #2, etc.) corresponds to each main figure panel, as this was discussed in the rebuttal. (2) The phrasing on the high-field behavior in the discussion could be slightly refined to more explicitly frame the deviation from B³ scaling as an entry point into the interesting physics of non-perturbative magnetic field regimes. (3) Finally, a final check to ensure all key data from the rebuttal (like the RRR vs. conductivity table and the temperature-field phase diagram) are seamlessly integrated into the main text or supplementary information would be beneficial.\nI am fully satisfied with the authors' responses and the state of the manuscript. I recommend acceptance in its current form.\nResponse: We sincerely thank the reviewer for the insightful review of our manuscript, the highly positive evaluations and the recommendation for publication. We are also grateful for the final polishing suggestions, and we have made the following corresponding revisions:\n(1) We have clarified this point in the figure captions.\n(2) The relevant statement has been rephrased in the Discussion section. (page 10, lines 270-271)\n(3) We have incorporated the key data and related discussion into the Supplementary Information. (Supplementary Note 8)"}]

Decision letter