[1,2]Nitin Samarth

[1]Department of Physics, The Pennsylvania State University, University Park,
16802, PA, USA

2]Department of Materials Science & Engineering, The Pennsylvania State
University, University Park, 16802, PA, USA

# Off-Resonant Detection of Domain Wall Oscillations Using Deterministically
Placed Nanodiamonds

Jeffrey Rable  Jyotirmay Dwivedi<EMAIL_ADDRESS>*  [

###### Abstract

Nitrogen-vacancy (NV) centers in diamond offer a sensitive method of measuring
the spatially localized dynamics of magnetization and associated spin textures
in ferromagnetic materials. We use NV centers in a deterministically
positioned nanodiamond to demonstrate off-resonant detection of GHz-scale
microwave field driven oscillations of a single domain wall (DW). The
technique exploits the enhanced relaxation of NV center spins due to the
broadband stray fields generated by an oscillating DW pinned at an engineered
defect in a lithographically patterned ferromagnetic nanowire. Discrepancies
between the observed DW oscillation frequency and predictions from
micromagnetic simulations suggest extreme sensitivity of DW dynamics to
patterning imperfections such as edge roughness. These experiments and
simulations identify potential pathways toward quantum spintronic devices that
exploit current driven DWs as nanoscale microwave generators for qubit
control, greatly increasing the driving field at an NV center and thus
drastically reducing the $\pi$ pulse time.

###### keywords:

ferromagnetic domain wall, NV center

##  1 Introduction

GHz-scale ferromagnetic dynamics, such as spin waves, vortex gyrations, and
domain wall oscillations, offer both a fascinating target for quantum sensing
and a potential qubit driving source in quantum computing applications. One of
these qubits, the nitrogen-vacancy (NV) center in diamond, has been previously
used to detect ferromagnetic resonance[1, 2, 3, 4], spin wave scattering[5],
spin-wave frequency multiplication [6], and vortex gyrations[7]. Ferromagnetic
dynamics have also been explored as potential sources of microwave driving for
NV centers: NV-magnon coupling was first reported in 2017[8], then
electrically tuned in subsequent works [9, 10], and vortices have been shown
to enhance NV spin rotation during qubit addressibility experiments[11].

Domain walls (DWs) should offer similar advantages to vortices for
addressability since they produce large stray fields and they can be moved
electrically. Pinned DWs act like a mass in a potential well, which is
determined by the ferromagnet’s properties; they can oscillate in the GHz
range, making them useful for direct NV driving. They can be pinned to
specific sites via geometric patterning of the magnetic feature, and can be
driven via external microwave fields[12], DC current[13, 14], AC current[15],
and spin transfer torques[16]. However, the literature on DW sensing using NVs
has been largely focused on characterizing their static structures.[17, 18,
19]

Figure 1: Schematic of the measurement concept; an NV center containing
nanodiamond is placed over a geometrically-determined DW pinning site in a
ferromagnetic nanowire.

In this paper, we take the first steps towards NV-DW coupling by off-
resonantly detecting GHz-scale transverse domain wall dynamics in NiFe
(Permalloy, Py) nanowires using deterministically placed nanodiamonds at a DW
pinning site (Fig. 1). We demonstrate that, similar to previous ferromagnetic
resonance and vortex gyration measurements [7], driving of domain wall
oscillations generates magnetic fields that overlap the NV ground state
transition frequency, reducing NV polarized state lifetimes and producing an
optical contrast.

###  1.1 Domain Wall Oscillation Measurement

##  2 Results

Figure 2: (a) Simulations showing the frequency of DW oscillations over
applied fields in a 300 nm wide bowed nanowire with a 120 nm radius half-
circle defect. The dotted red lines denote the NV ground state transition for
a field perfectly aligned along the NV axis (b) Spatially resolved image of
the mode in (a) at 1 mT. (c) Spatially resolved image of a higher frequency
mode at 1 mT, showing that more uniform dynamics occur at much higher
frequencies than DW dynamics.

Initial simulations with external microwave field driving allow us to find an
ideal nanowire geometry for DW detection and NV-DW coupling, with the goal of
producing oscillations that would be easily detectable and overlap with the NV
ground state transition at an applied field below the DW nucleation field.
This would allow us to independently detect the DW away from an NV transition,
tune the oscillation frequency and NV ground state transitions into resonance
for coupling measurements, and perform control measurements for both these
experiments by denucleating the DW. These results led to the fabrication and
study of nanowires with widths in the 100’s of nm range, with pinning sites
composed of a half-circle defect with a radius of approximately 40% the wire
width, most of which should have enabled coupling. For example, the simulation
results shown in Fig. 2 suggest that, for a 300 nm wide wire with a 120 nm
defect, DW oscillations should be detectable at about 300 MHz below the NV
ground state transition in the absence of an applied field, and that the two
should come into resonance at a field below 10 mT. Simulations on the wire
initialized with no DW show that one will nucleate at 11 mT applied field.

Figure 3:  (a) Atomic force microscopy (AFM) image of nanodiamond placed at
the pinning site (bright spot) in a 300 nm wide wire with a 120 nm radius
half-circle defect. (b) MFM image of the same site, showing the presence of a
DW (dark spot) near the nanodiamond and pinning site. The DW is partly
obscured by the nanodiamond, which leaves a small bright streak on the image.
(c) AFM image of nanodiamond placed at the pinning site in a 350 nm wide wire
with a 140 nm radius half-circle defect. (d) MFM image of the same site,
showing the presence of a DW in the wire.

We begin our measurements on a 300 nm wide nanowire with a 120 nm wide defect
where a nanodiamond is placed over the pinning site (Fig. 3 (a) and (b)) using
a pick-and-place protocol reported earlier [4] and in Methods. Using the
nanodiamond located there, we begin measuring the DW nucleation field by
monitoring the frequency of the NV ground state transition over increasing
applied field; because the Zeeman splitting is approximately 28 GHz/T
$\vec{B}\cdot\boldsymbol{\hat{\textbf{n}}}$, where $\vec{B}$ is the applied
field and $\boldsymbol{\hat{\textbf{n}}}$ is the NV axis unit vector, we can
detect the nucleation and the accompanying increase in stray field at the
pinning site via the detection of a discontinuity in the ground state
transition. This is seen in Fig. 4 (a) at 13 mT, slightly off from the
predicted 11 mT from our simulations. Next, we search for DW oscillations
using a $T_{1}$ spectroscopy measurement (Fig. 4 (b)). In this measurement, we
use the 532 nm laser to polarize the NV into the $m_{s}=0$ state, then apply
microwaves for 5 µs with the laser off, and turn the laser back on to read out
the state during the first 500 ns it is on (signal measurement). A reference
measurement is then taken 15 µs later, after the NV spin is re-polarized, with
our result being the contrast between the signal and reference measurements.
Sweeping across a 400 MHz to 3.5 GHz range both with and without a DW
nucleated, this measurement, shown in Fig. 4 (c), yields a broader range of
dynamics than expected in simulations (Fig. 4 (d)). Rather than a single broad
peak centered at 2.4 GHz, we find two broad peaks at approximately 1.8 and 2.3
GHz, with an additional, sharper peak at 1.9 GHz. Earlier studies have
suggested that DW dynamics in nanowires are extremely sensitive to edge
roughness[12]. This leads us to hypothesize that the difference between
experimental and simulated results is likely due to the deviation of the shape
of the fabricated wires from the ideal design used in our simulations.
Furthermore, we find that the signal in the $T_{1}$ spectroscopy measurement
fades rapidly with increasing field, preventing us from realizing resonant
coupling (data not shown). At minimal applied field, directly driving the NV
ground state transition through Rabi oscillations with and without a DW fails
to demonstrate DW-mediated power enhancement at this site; we also do not see
any signs of DW-mediated power broadening in our ESR signals (data not shown).

Figure 4: (a) Continuous-wave electron spin resonance (ESR) measurement
performed on a nanodiamond positioned over the pinning site of a 300 nm wide
nanowire with a 120 nm defect. The circled discontinuity marks where the DW
nucleated. (b) Diagram of the $T_{1}$ spectroscopy measurement used to detect
DW dynamics; green blocks denote the 532 nm laser. (c) $T_{1}$ spectroscopy
measurements of the nanodiamond from (a) performed both with a nucleated
domain wall and without in minimal applied field (0.7 mT). (d) Normalized
linecut of the simulation results from Fig. 2 (a) at 1 mT, along with the
normalized results of a simulation in the same wire without a DW nucleated.
Figure 5: (a) $T_{1}$ spectroscopy measurement on the nanodiamond in Fig. 3
(c), located on top of a 350 nm wide wire with a 140 nm radius semicircular
defect. (b) Normalized simulations of dynamics in a 350 nm wide bowed wire
with a 140 nm radius semicircular defect with and without a DW.

We then repeat these DW oscillation measurements using $T_{1}$ spectroscopy on
a second, 350 nm wide wire with a 140 nm defect, shown in Figs. 3 (c), with
the results shown in Fig. 5 (a) and Fig. 6 (a). This wire does not appear
ideal for coupling in simulations (Fig. 6 (b)), but is the largest in our
series of wires and, as such, should produce the largest stray fields, making
it a useful system to confirm our previous results. Similar to the 300 nm wide
wire, we see the strongest signal at approximately 2 GHz. However, we also
detect some additional peaks near 1.4 GHz and 3.2 GHz; this result is somewhat
surprising considering that the 350 nm wire appears to have fewer fabrication
errors (Fig. 3 (a) and (c)). Fig. 6 (a) shows that upon sweeping the applied
magnetic field, these additional peaks rapidly fade, and the feature at 2 GHz
splits into two distinct features - one that increases in frequency, but fades
rapidly, and one that remains constant at 1.9 GHz; neither of these match our
simulated domain wall frequency dispersion. This persistent, constant signal
at 1.9 GHz, allows us to detect the nucleation field using the domain wall
dynamics, rather than the static stray field, demonstrated in 6 (c); we find
that the nucleation field is 11.25 mT, matching our simulations that show
nucleation occurs between 11 and 12 mT. Thus, while our dynamic simulations
fail to accurately predict DW oscillation frequency, our quasi-static
simulations accurately predict the nucleation field.

Figure 6: (a) $T_{1}$ spectroscopy measurement performed across different
applied fields on the nanodiamond in Fig. 3 (c), placed on top of a 350 nm
wide wire with a 140 nm radius semicircular defect. (b) Simulations of the DW
dynamics of the wire measured in (a); the dashed red line represents the NV
ground state transition frequency for an applied field oriented perfectly
along the NV axis. (c) Measurement of DW nucleation in the 350 nm wire using
the 1.9 GHz peak

To elucidate why NV centers can detect DW oscillations off-resonantly, we
perform micromagnetic simulations where the DW is driven by a constant
sinusoidal driving field and compute the stray field at the NV location. To
determine appropriate driving field amplitudes to use in these simulations, we
perform Rabi oscillation measurements on our NV at the pinning sites; these
results give us a maximum driving amplitude of 1.1 mT at 2.87 GHz, which we
take as an approximation of our 1.9 GHz domain wall driving field. For our
simulated wire, we measure the generated field at 3.3 GHz to replicate the
frequency difference between our measured DW oscillations and NV center zero
field splitting, rather than looking directly at the ZFS. We find that the
Fourier components at this frequency reach magnitudes of nearly 0.1 mT at the
driving field amplitudes we achieved in our experiment, which is sufficient to
explain the measured relaxation enhancement.

Figure 7: NV center DW oscillation peak contrast (red dots) in the 300 nm wide
wire and the corresponding simulated 3.3 GHz Fourier component magnitude
(dashed line) versus microwave driving field amplitude. The simulated DW stray
field used to perform this analysis was measured at 50 nm above the center of
the wire, where the pinning site was located.

Furthermore, comparing the 3.3 GHz Fourier component to our measured domain
wall oscillation peak contrast, a rough proxy for stray field amplitude, in
Fig. 7, we can see a similar pattern between the experimental and simulated
results, where the stray field amplitude peaks and then drops off
substantially. Two possibilities explain the difference in peak locations.
First, there are inherent discrepancies between our measured and simulated
wires, which we also believe caused the mismatch between observed and
predicted DW oscillation frequencies. Second, we measured microwave driving
power at 2.87 GHz, the NV ground state spin transition frequency, rather than
the domain wall frequency. It is likely that our microwave stripline does not
produce the same field strength at all frequencies. These results suggest
that, similar to earlier studies of vortex gyrations [7], DW driving results
in off-resonant stray fields that will overlap with the NV transition
frequency.

Additionally, extracting the Fourier component of the DW-generated microwave
field in these simulations shows a strong enhancement; 0.1 mT driving fields
result in a domain wall stray microwave field amplitude exceeding 3 mT,
suggesting that bringing a DW and NV center into resonance can drastically
reduce the $\pi$ pulse time.

##  3 Conclusion

In summary, we have shown that it is possible to detect localized GHz-scale DW
dynamics in ferromagnetic systems using NV centers by deterministically
placing them on top of a DW pinning site in Py nanowires. This detection, via
enhanced spin-relaxation from the broadband stray fields generated, further
expands the menagerie of magnetic dynamics that can be measured using NV
centers. Additionally, the prospect of resonant NV-DW coupling opens up
potential avenues for qubit control; in the microwave driven case, our
simulations suggest that the presence of a proximal DW could increase the
driving field at an NV center by over a factor of 30, drastically reducing the
$\pi$ pulse time. Furthermore, this work opens up the possibility of driving
NV centers using current driven DW oscillations, exploiting DWs as nanoscale
microwave generators.

##  4 Methods

###  4.1 Domain Wall Oscillation Measurement

We use standard electron beam lithography, electron beam thin film deposition,
and a liftoff process to prepare 10 nm thick Py semicircular nanowires with a
notch-shaped defect in the center. 100 nm diameter nanodiamonds (Adámas
Nanotechnologies, 3 ppm NV) are placed using an AFM pick-and-place technique
outlined elsewhere[4], yielding results like those shown in Fig. 3 (a), (c).
MFM measurements, carried out in a Bruker ICON AFM with a low moment CoCr
coated tip (Bruker, MESP-LM-V2), confirm nanodiamond proximity to a DW (Fig. 3
(b), (d)). DWs in the nanowires are created and destroyed using the shape
anisotropy of the curved wires; they are nucleated by applying a magnetic
field perpendicular to the wire at the defect site, and denucleated by
applying a magnetic field tangential to the wire at the defect site. The in-
plane nucleation field is applied using an N52 permanent magnet mounted on a
highly repeatable stepper motor stage (Newport, ILS200PP Stepper Motor), and
the denucleation field is applied using an N52 magnet positioned near the
sample by hand. The nucleation magnetic field is calibrated in the plane of
the sample using the NV centers in a single crystal CVD diamond film with
known orientation; due to stage limits, the smallest field we can apply is 0.7
mT.

For optical polarization and readout of the NV center fluorescence, we use
excitation via a 532 nm wavelength CW laser (Oxxius, LCX-532L-200-CSB-PP) and
detection with an avalanche photodiode (ID Quantique, ID100). The laser is
pulsed using an acousto-optical modulator (AOM) (Gooch and Housego, 15210) set
up in a double pass configuration, scanned across the sample via a fast
scanning mirror (Optics in Motion, FSM-101), and focused on the sample using a
$100\times 0.9$ numerical aperture (NA) objective lens (Nikon, $100\times$
Plan Fluor $0.9$NA objective). During the measurements, we apply a microwave
magnetic field , which drives both NV spin transitions and magnetization
dynamics, via a microwave signal generator (Stanford Research, SG396), +43 dBm
amplifier (Mini-Circuits, ZHL-16W-43-S+), and a 25 µm diameter gold wire
placed across the sample.

###  4.2 Micromagnetic Simulations

We perform micromagnetic simulations to design our nanowires and to eventually
compare to our experimental results. These simulations are performed with the
Mumax3 software package, which uses the Landau-Lifshitz-Gilbert equation:

$\frac{\partial\vec{M}}{\partial
t}=\gamma_{\text{LL}}\frac{1}{1+\alpha^{2}}(\vec{m}\times\vec{B_{\text{eff}}}+\alpha(\vec{m}\times(\vec{m}\times\vec{B_{\text{eff}}}))$
(1)

to calculate the evolution of the magnetization $\vec{M}$ of finite
ferromagnetic cells. In Eq. 1, $\alpha$ is the Gilbert damping of the
material, $\gamma_{\text{LL}}$ is the gyromagnetic ratio of the material, and
$B_{eff}$ is the effective magnetic field at that cell, which includes
contributions from external, demagnetization, exchange, and anisotropy
fields[20].

The simulations use 5 nm x 5 nm x 10 nm cells with a saturation magnetization
$M_{\text{s}}$ of 8 x $10^{5}$ A/m, an exchange constant $A_{\text{ex}}$ of
1.3 x $10^{-11}$ J/m, and a Gilbert damping parameter $\alpha$ of 0.0063,
which are standard for Py.

After defining the wire geometry, initial simulations of domain wall
nucleation are performed by giving the system a constant (1,0,0) magnetization
(pointing tangential to the wire direction at the defect site) with an applied
magnetic field along the (0,1,0) direction. The system is then allowed to
relax into a low energy state. At low fields, the magnetization continues to
lie along the wire direction, but past a certain field, a transverse domain
wall nucleates in the wire due to shape anisotropy. The magnetization of the
nucleated state is used as the initial state for the domain wall dynamics
simulations. In these simulations, the nucleated magnetization from the
previous simulations is loaded, and then allowed to relax in an applied field,
like before.

To excite the system, we then apply a Gaussian pulse with a 20 ps full width
half maximum (FWHM) and a 0.5 mT amplitude. The system is allowed to freely
evolve in time for 20 ns and the average magnetization is sampled every 5 ps.
Finally, we use a discrete fast Fourier transform (FFT) to analyze the data in
the frequency domain, providing a resolution of 50 MHz and window from 0 to
100 GHz, well beyond the range of interest in this work.

For spatially resolved simulations, a shorter 1 ps FWHM pulse is used, and the
magnetization is sampled in discrete 50 x 50 nm blocks composed of 100 cells.
The average magnetization of each block is analyzed using an FFT as before and
the image is composed of each block’s FFT amplitude at the desired frequency.

To determine the stray field produced by a domain wall, we drive the system at
a constant microwave frequency for 20 ns and sample the magnetization of every
cell within 500 nm of the pinning site every 5 ps. Then, treating every cell
as an individual dipole, we solve for the magnetic field at the desired
location by summing the individual dipole fields at each time point. An FFT
can then be used to determine the microwave field’s frequency distribution.

Acknowledgments

We acknowledge support from the University of Chicago, the U.S. Department of
Energy Office of Science National Quantum Information Science Research Centers
(Q-NEXT), and a seed grant from the Penn State Materials Research Institute.
The authors would like to thank Eric Kamp for technical discussions and
Michael Labella for sample fabrication advice.

## Declarations

  * •

Competing interests: The authors declare no competing interests.

  * •

Availability of data and materials: all raw data are avaiable upon request.

  * •

Authors’ contributions: J.R. and N.S. conceived the idea for this project.
J.R. and J. D. carried out the experiments and analyzed the data with input
from N.S. who also supervised the project. All authors discussed the results.
J.R. and N.S. wrote the manuscript with contributions from all authors.

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# Abnormal planar Hall effect in quasi-1D Kondo chain CeCo2Ga8 and its
implications for hybridization dynamics

Shuo Zou1, Hai Zeng1, Zhuo Wang1, Guohao Dong2, Xiaodong Guo1, Fangjun Lu1,
Zengwei Zhu1, Youguo Shi2111Electronic address<EMAIL_ADDRESS>and Yongkang
Luo1222Electronic address<EMAIL_ADDRESS>1Wuhan National High Magnetic
Field Center and School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China;  2Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China.

The process how heavy-electron state is established in Kondo-lattice compounds
remains an unsolved issue. Recent angle-resolved photoemission spectroscopy
and ultrafast optical spectroscopy imply an intermediate regime with
hybridization fluctuations prior to the establishment of Kondo coherence,
which appears at odds with traditional transport measurements. Extensive
experimental works are highly demanded to both reconcile this dichotomy and
delineate the intrinsic features in this special regime. Here, on the example
of quasi-one-dimensional Kondo lattice compound CeCo2Ga8, we investigated
angular dependent magnetotransport properties by planar Hall effect and planar
anisotropic magnetoresistance measurements. Upon cooling from $T_{K}^{on}$ (an
onset of incoherent Kondo scattering ) to below $T^{*}$ (where coherent
$c$-$f$ hybridization comes into play), the two-fold symmetrical pattern of
planar Hall effect changes sign gradually (i.e. $180^{\circ}$ phase shift);
most strikingly, as a crossover, additional oscillations appear and persist
until the heavy-electron state is stabilized below $T^{*}$. These results
provide new insights for the regime of hybridization dynamics which might be
deemed as a precursor state of the heavy-electron state.

INTRODUCTION

The transformation of localized-itinerant electronic states is one of the key
ingredients in condensed matter physics, bridging a commonality among a
variety of prototypical strongly correlated electronic systems including but
not limited to the high-$T_{c}$ superconductors, heavy-fermion metals,
irradiates, organic superconductors, and twisted-bilayer graphene BrandowB-
JAC1992 ; ImadaM-RMP1998 ; SiQ-localQCP ; Coleman-QCP2005 ; Pepin-Mott2008 ;
KimBJ-Sr2IrO4PRL2008 ; OikeH-PRL114_067002(2015) ; CaoY-MAGNature556_80_2018 .
In heavy-fermion systems, such a transformation is usually connected to the
Kondo hybridization between itinerant conduction ($c$) and localized $f$
electrons (viz. $c$-$f$ hybridization), the contribution of delocalized $f$
electrons to Fermi surface (FS), and hence the emergence of composite heavy-
electron state. A classic description is that $c$-$f$ hybridization is
fulfilled below coherence temperature $T^{*}$, and this scenario was well
supported by traditional transport measurements in a series of Ce-based Kondo
lattice compounds AlievFG-CeCu2Si2JLTP1984 ; HeggerH-CeRhIn5PRL2000 ;
NakatsujiS-CeCoIn5PRL2002 ; LuoY-CeNiAsONM2014 . Spectroscopically, however,
recent angle-resolved photoemission spectroscopy (ARPES) and ultrafast optical
spectroscopy measurements revealed that hybridization already appears at much
higher temperature and no remarkable signature is detectable across $T^{*}$
ChenQY-PRB2017 ; KoitzschA-PRB2008 ; LiuY-PRL2020 ; JangaS-CeCoIn5PNAS2020 .
These contradicting results imply that a precursor-like state where a short-
range noncollective hybridization regime may exist above $T^{*}$. To reconcile
this dichotomy and to unveil the intrinsic features in this special regime as
well, extensive theoretical and experimental works are both highly demanded.

Angular dependent magnetotransport properties provide a useful probe to Fermi
surface topology and scattering process. In this work, we systematically
investigated planar Hall effect (PHE) and planar anisotropic magnetoresistance
(PAMR) on a quasi-one-dimensional (quasi-1D) Kondo lattice compound CeCo2Ga8.
The schematic of PHE and PAMR is depicted in Fig. 1(a). Unlike ordinary Hall
effect that is caused by an out-of-plane magnetic field, PHE represents an in-
plane-field-rotation-induced modulation of the transverse resistivity
($\rho_{yx}$). A classic origin of PHE arises from anisotropic longitudingal
resistivity ($\rho_{xx}$) under field, and can be described by the empirical
formula TangHX-PRL2003 ; ZhongJ-CPB2023 ; LiL-npjQM2023 :

$\displaystyle\rho_{xx}(\varphi)$
$\displaystyle=\rho_{\perp}-\Delta\rho\cos^{2}\varphi,$ (1a)
$\displaystyle\rho_{yx}(\varphi)$
$\displaystyle=\Delta\rho\sin\varphi\cos\varphi,$ (1b)

where $\Delta\rho\equiv\rho_{\perp}-\rho_{\parallel}$ stands for the
resistivity anisotropy, and $\rho_{\parallel}$ ($\rho_{\perp}$) is the
longitudinal resistivity when $\mathbf{B}$ is parallel (perpendicular) with
electrical current $\mathbf{j}$. Since magnetoresistance probes deviations of
FS from sphericity or the variance of local Hall mobility around the FS
PippardAB-MRinMetal ; HarrisJM-PRL1995 , hopefully, studies of PHE and PAMR
will uncover more details in electronic structure and scattering mechanism.

Herein, the two-fold oscillatory PHE in CeCo2Ga8 deeply inside the heavy-
electron state ($T\ll T^{*}\approx 12$ K) is found to exhibit additional
oscillations when temperature rises across $T^{*}$. Although the two-fold
oscillation finally gets recovered when approaching $T_{K}^{on}\sim 100$ K,
the onset temperature of incoherent Kondo scattering, the phase of PHE is
switched by $\sim 180^{\circ}$. These abnormal behaviors in PHE are inferred
as a consequence of the coexisting incoherent and coherent Kondo scatterings
that are also mediated by the magnetic anisotropy. Our results suggest
strongly the existence of an intermediate regime in which the $c$-$f$
hybridization is noncollectively short-ranged and thus the heavy-mass Fermi
surface is fluctuating.

RESULTS

The compound CeCo2Ga8 crystallizes in the YbCo2Al8-type orthorhombic structure
(Pbam, No. 55) Koterlin-Ce128 , as shown in Fig. 1(b). The cerium atoms form
individual chains along [001], and these chains are well isolated by the CoGa9
cages in the $\mathbf{ab}$ plane, rendering it a candidate of quasi-1D Kondo
lattice (namely Kondo chain) WangL-CeCo2Ga8 , and this was supported by our
earlier resistivity measurements where Kondo coherence is only observed for
electric current parallel to the chains ($\rho_{c}$), while the profiles of
$\rho_{a}$ and $\rho_{b}$ remain incoherent down to 2 K ChengK-CeCo2Ga8Q1D ;
ChengK-CeCo2Ga8Stress . Magnetic susceptibility, specific heat, and $\mu$SR
measurements manifested that it does not exhibit any trace of long-range
magnetic ordering down to 0.07 K WangL-CeCo2Ga8 ; Bhattacharyya-CeCo2Ga8uSR ,
but rather, remarkable non-Fermi-liquid behavior characterized by linear
resistivity [$\rho_{c}(T)\propto T$] and logarithmic specific heat
[$C/T\propto-\log T$] appears at low temperature, implying in the vicinity of
a quantum critical point.

To carry out the PHE and PAMR experiments, electrical contacts were made in a
Hall-bar geometry on a piece of CeCo2Ga8 single crystal whose orientation was
verified by X-ray diffraction. To eliminate spurious contamination between the
longitudinal ($\rho_{xx}$) and transverse ($\rho_{yx}$) voltages, focused-ion-
beam (FIB) technique was exploited to make the microstructured device, as
shown in Fig. 1(a). In our experiments, the electrical current ($\mathbf{j}$)
flows along [001], whereas the magnetic field ($\mathbf{B}$) was rotated
within the $\mathbf{ac}$ plane. The angle spanned by $\mathbf{j}$ and
$\mathbf{B}$ is denoted by $\varphi$. The temperature dependence of
$\rho_{xx}$ in the absence of $\mathbf{B}$ is shown in Fig. 1(c). It decreases
upon cooling from room temperature as a metallic behavior, but turns up near
$T_{K}^{on}\sim 100$ K where the incoherent Kondo scattering sets in. A broad
peak is observed in $\rho_{xx}(T)$ around $T^{*}\sim 12$ K below which
collective $c$-$f$ hybridization is believed to establish and the system
enters a coherent heavy-electron state. $-\log T$ behavior featuring single-
impurity Kondo effect is seen between $T_{K}^{on}$ and $T^{*}$, as expected.
It should be noticed that $T^{*}$ in this sample is relatively lower than in
our previous works ($\sim 17$ K) WangL-CeCo2Ga8 ; ChengK-CeCo2Ga8Q1D ; ChengK-
CeCo2Ga8Stress , probably due to different sample quality or affected by the
FIB treatment.

Turning now to the PHE and PAMR of CeCo2Ga8. The $\varphi$ dependent
$\rho_{yx}$ and $\rho_{xx}$ measured under 5 T and 1.6 K (well below $T^{*}$)
are presented in Fig. 1(d) and (e), respectively. Two characteristic features
commonly seen in planar Hall measurements are also evident here. First, both
$\rho_{yx}$ and $\rho_{xx}$ are sinusoidal on the angle $\varphi$, exhibiting
two-fold oscillations and obeying well to Eq. (1). Second, a 45 ∘ phase shift
is present between $\rho_{yx}$ and $\rho_{xx}$, which is better seen in the
polar-coordinate plots in Fig. 1(f-g). These features are akin to the PHE and
PAMR observed in many other topological semimetals, such as Cd3As2 LiH-PRB2018
, NiTe2 LiuQQ-PRB2019  . Since PHE usually originates from anisotropic
magnetoresistance (i.e. $\Delta\rho$), previously the observed PHEs were
generally interpreted due to either the presence of chiral anomaly in
topological Dirac / Weyl semimetals NielsenHB-PhysLettB1983 ; SonDT-PRB2013 ;
XiongJ-Science2015 ; LuoY-SciRept2016  or spin-momentum locked surface state
in topological insulators TaskinAA-NC2017 ; WuB-APL2018 ; ZhouL-PRB2019 . In
our case, anisotropic magnetoresistance is not surprising considering its
quasi-1D crystalline structure. First-principles calculations also pointed to
flat FSs for the itinerant $f$-electron bands WangL-CeCo2Ga8 , it then is
reasonable that the magnetoresistance is highly anisotropic. Besides, our
earlier magnetic susceptibility measurements and crystalline electric field
(CEF) analysis manifested $\mathbf{c}$ as the magnetic easy axis ChengK-
CeCo2Ga8Q1D , which may cause anisotropic spin scatterings. All these may
contribute to the observation of PHE in CeCo2Ga8.

To further look into the PHE and PAMR of CeCo2Ga8, we also measured
$\rho_{yx}(\varphi)$ and $\rho_{xx}(\varphi)$ under various fields, and the
results (at 1.6 K) are displayed in Fig. 2. Generally, the oscillating
amplitude of both $\rho_{xx}(\varphi)$ and $\rho_{yx}(\varphi)$ increases as
field strengthens. A major feature for the longitudinal channel is that the
phase of the two-fold oscillation switches by 180∘ when magnetic field exceeds
9 T, and meanwhile some sub-feature appears around $\varphi=0$ and 180 ∘
($\mathbf{B}\parallel$ [001]). This motivates us to look also into the field
dependent magnetoresistance
[MR$\equiv\frac{\rho_{xx}(B)-\rho_{xx}(0)}{\rho_{xx}(0)}\times 100\%$] for
$\mathbf{B}\parallel$ [100] and [001], seeing Fig. 2(b). A prominent feature
is that the two MR curves cross near 11.8 T, meaning $\Delta\rho$ changes sign
at this field. We infer that such a sign change in $\Delta\rho(B)$ is the
reason for the phase shift in the PAMR curve between 9 and 13 T. Meanwhile, we
also notice that while MR for $\mathbf{B}\parallel[100]$ is negative and
decreases monotonically all through the field range 0-15 T, the MR for
$\mathbf{B}\parallel[001]$ increases with field and reaches a maximum $\sim
1\%$ near 6 T before it turns down and becomes negative for field larger than
10.5 T. A low-field positive MR is common for Kondo-lattice compounds in the
coherent regime, e.g. CeAl3 RoeslerGM-PRB1992 , Ce(Co,Rh)In5 ChristiansonAD-
PRB2002 ; MalinowskiA-PRB2005 , YbNi2B2 YatskarA-PRB1999 , etc. The $B^{2}$
behavior of MR shown in Fig. 2(b) for both $\mathbf{B}\parallel$ [100] and
[001] ($B>6$ T) also resemble that of CeCoIn5 in the coherent regime
MalinowskiA-PRB2005 . According to the phenomenological two-fluid model for
heavy-fermion systems Nakatsuji-PRL2004 ; Yang-PRL2008 , $f$ electronic state
below $T^{*}$ is a mixture of the localized $f$ electron and itinerant
composite heavy-fermion fluid, and as $T$ decreases, the weight of the latter
enhances. Based on this framework and taking also account of Matthiessen’s
rule, we infer that the MR maximum probably arises from the competition
between the Kondo-impurity liquid resistivity that increases under field and
the heavy Fermi liquid resistivity that behaves in an opposite trend. In
particular, since in the localized phase, the spins are unscreened, they
interact with the crystalline electric field and thus exhibit magnetic
anisotropy; it then is reasonable that the positive MR contribution is larger
when external field is applied along the easy axis [001] ChengK-CeCo2Ga8Q1D .

Getting back to PHE. Figure 2(c) shows PHE under diverse magnetic field
strengths. It is interesting to note that although the phase of PAMR get
reversed between 9-13 T, the phase of PHE remains unchanged, and its shape is
not distorted, either. For all the fields measured 0-15 T, the curves of PHE
display nice sinusoidal shapes. The amplitude of PHE increases monotonically
with field and a tiny saturation trend is visible when field surpasses 13 T,
seeing Fig. 2(d). Furthermore, unlike what was reported in other examples,
there is no linear relationship between the amplitudes of PHE and PAMR [cf.
inset to Fig. 2(d)]. An intuitive expectation is that in compensated metals
the amplitude of PHE increases with field strength following a $B^{2}$ law
because of the quadratic $\rho_{xx}(B)$ as well as $\Delta\rho(B)$ Nandys-
PRL2017 . Such a rule seems obeyed in CeCo2Ga8 under low field $B\leq 5$ T.
High-field saturation of PHE has been observed in Dirac semimetal ZrTe5-δ and
discussed in the framework of chiral anomaly with
$L_{a}<L_{x}<L_{c}^{2}/L_{a}$ where $L_{x}$, $L_{a}$ and $L_{c}$ are
respectively sample length, “chiral” magnetic length ($L_{a}\propto 1/B$) and
chiral charge diffusion length P.Li ZrTe ; BurkovAA-PRB2017 . However, this
mechanism is unlikely in CeCo2Ga8 due to the irrelevance of band topology.
Regardless of the unclear mechanism to quantitatively understand the high-
field saturation of PHE, we notice another salient feature, i.e., the maximum
of $\rho_{xx}(B)$ and the deviation of PHE from the $B^{2}$-law seem to be
synchronous. This reminds us that the two may have the same origin - a
competition between incoherent and coherent Kondo scatterings. To see this
more clearly, temperature dependence of PHE investigation will be helpful.

Figure 3 displays PHE of CeCo2Ga8 at different temperatures, under a fixed
field 15 T. We start with the results at 80 K, just below $T_{K}^{on}$. A two-
fold oscillation pattern, albeit weakly distorted from a perfect sinusoidal
shape, is clearly seen at this temperature, cf. the left panel of Fig. 3(a). A
polar plot is also presented in the right panel. It should be mentioned that
the plume orients along $\varphi\approx 45~{}^{\circ}$, which happens to be
the angle where $\rho_{yx}$ minimizes at 1.6 K. This implies that a sign
reversal (or equivalently $180^{\circ}$ phase shift) may have taken place
across $T^{*}$. In reality, the evolution of PHE with decreasing $T$ is more
complicated than expected. Additional small peaks show up in
$\rho_{yx}(\varphi)$, and meanwhile, the major peak gradually moves. This
leads to severely distorted PHE pattern which is most apparent near 20 K in
Fig. 3(d). Further cooled down, a new peak positioned near 135 ∘ grows
rapidly, while the original peak shrinks and is gradually taken over by the
new peak. At 5 K (below $T^{*}$), $\rho_{yx}(\varphi)$ restores two-fold
oscillations, whereas the principal axis of the plume is shifted by about
$90~{}^{\circ}$ as compared with that of 80 K, seeing Fig. 3(h).

Note that although the sign of PHE reverses between incoherent and coherent
regimes, $\Delta\rho$
($=\rho_{xx}^{\mathbf{B}\parallel[100]}-\rho_{xx}^{\mathbf{B}\parallel[001]}$)
remains positive across the full window 1.6-50 K, as shown in Fig. 4(a-c).
This provides additional evidence for the nonlinearity between the amplitudes
of PHE and PAMR. Combining Figs. 2 and 4, a straightforward conclusion can be
drawn, i.e. the sign change of PHE here is not necessarily related to the sign
change of $\Delta\rho$; in other words, the PHE observed in CeCo2Ga8 seems far
beyond the classic description given by Eqs. (1).

DISCUSSION

To summarize the main experimental observations here, the abnormal PHE of
CeCo2Ga8 is featured by: (i) the additional oscillations for $T$ between
$T_{K}^{on}$ and $T^{*}$, (ii) the coincidence of sign change and the
establishment of coherent heavy-electron state, and (iii) the loss of
linearity between the oscillating amplitudes of PHE and PAMR. Natural
questions, then, concern the mechanism for this abnormal PHE and the reason
for its sign change between incoherent and coherent regime.

Abnormal PHEs with either distorted shapes or additional oscillations have
been reported in many systems, e.g. Bi4Br4 ZhongJ-Bi4Br4ACSNano2024 , TaP
YangJ-PRM2019 , LaAlO3/SrTiO3 (111) interface RoutPK-PRB2017 , bismuth YangSY-
PRR2020 , TbPtBi ZhuY-PRB2020 , MnBi2Te4 WuM-NanoLett2022 , KV3Sb5 LiL-
npjQM2023 , etc. In general, the mechanisms for abnormal PHE can be
categorized as follows: (i) an extrinsic factor caused by current jetting,
which is usually seen in high-mobility semimetals such as TaP YangJ-PRM2019 ;
(ii) phase coherence effect in topologically nontrivial materials with weak
antilocalization, as exemplified by Bi4Br4 ZhongJ-Bi4Br4ACSNano2024 ; (iii)
orbital anisotropy like in bismuth YangSY-PRR2020 ; (iv) competition in
transport between bulk and surface states mediated by orbital magnetic moment
in the magnetic topological insulator MnBi2Te4 WuM-NanoLett2022 ; (v) coupling
between fluctuations from lattice, spin, electronic and other degrees of
freedom, seeing KV3Sb5 for instance LiL-npjQM2023 .

In the case of CeCo2Ga8, firstly, no topological feature was reported in
first-principles calculations WangL-CeCo2Ga8 , and our previous transport
measurements did not find any signature of high mobility in this compound
either ChengK-CeCo2Ga8Q1D ; ChengK-CeCo2Ga8Stress . Therefore, interpretations
related to current-jetting or band topology are not applicable. The loss of
linearity between PHE amplitude and $\Delta\rho$ implies that there are
multiple scattering mechanisms in transport in CeCo2Ga8, and they contribute
differently to magnetoresistance and PHE. The commonest mechanism for
magnetoresistance arises from the bending of carrier trajectory under Lorentz
force, which typically results in Kohler’s rule in conventional metals, viz MR
measured at different fields and temperatures can be scaled into a single
function of $\rho_{xx}(0)/B$. However, such an attempt turns out to be a
failure for both $\mathbf{B}\parallel[100]$ and $[001]$, seeing Fig. 4(d).
This manifests that regular scattering process for orbital magnetoresistance
solely can hardly explain the results in CeCo2Ga8. On the other hand, the
coincidence of the sign change of PHE and the development of Kondo coherence
strongly suggests key roles played by Kondo scatterings and the associated
many-body physics. For this purpose, the scaling behavior based on the Bethe-
ansatz solution of the Coqblin-Schrieffer model was tested CoqblinB-PR1969 ;
SchlottmannP-PhysRep1989 ,

$\mathrm{MR}=f(\frac{B}{T+T_{K}}),$ (2)

where $T_{K}$ is the single-ion Kondo temperature. This model is found to fit
the MR$(T,B)$ curve rather well in the incoherent Kondo regime in that all the
curves collapse onto a common line [Fig. 4(e)], yielding $T_{K}=19(5)$ K. Note
that the gain of magnetic entropy recovers $R\ln{2}$ near 19 K WangL-CeCo2Ga8
, further validating our Coqblin-Schrieffer scaling analysis. Please be also
noted that this scaling law actually is not violated until below $T^{*}$,
indicating that incoherent Kondo scattering probably still plays some role
even after the onset of Kondo coherence. Of course, for $T$ well below
$T^{*}$, coherent Kondo scattering is dominant as evidenced by the
quasi-$B^{2}$ law of MR [inset to Fig. 2(b)]. These phenomena remind us of the
coexisting incoherent and coherent Kondo scatterings, and the weight of them
are temperature dependent and can be tuned by external parameters like
magnetic field. It is likely that the coexistence of them accounts for the
abnormal PHE here.

A schematic cartoon is proposed in Fig. 4(f) based on this idea. The phase
diagram is divided into three parts: Localized regime, Dynamic hybridization
regime, and Heavy-electron regime. At high temperature above $T_{K}^{on}$, the
$f$ electrons are fully localized, behaving as individual magnetic scattering
centers, and the FS is totally determined by the conduction electrons which in
CeCo2Ga8 are some cylindrical sheets along $\Gamma$-Y WangL-CeCo2Ga8 . At low
temperature below $T^{*}$, the $c$-$f$ hybridization is collective, giving
rise to the renormalized $f$ state near Fermi level and emergent large FSs
with heavy quasiparticle effective mass. In CeCo2Ga8, because cerium atoms are
in 1D chains that are well separated from each other, these new FSs are
probably flat sheets, affirmed by the first-principles calculations WangL-
CeCo2Ga8 . Due to the emergence of these flat sheets, the symmetry of the
whole Fermi surface changes substantially in that now the in-plane rotational
field will periodically be parallel or perpendicular to these flat sheets,
which probably bears consequence for the phase switch of PHE. The range
between $T^{*}$ and $T_{K}^{on}$, thus, determines an intermediate regime
wherein the $c$-$f$ hybridization is noncollective and dynamic. Heavy-mass FS
is not yet stabilized but remains “fluctuating” in this regime. Such an
intermediate regime may be deemed as a precursor state of the heavy-electron
state, and is likely responsible for the additional oscillations in PHE. In
addition, because the field dependent $T^{*}$ curve is rather anisotropic
[inset to Fig. 4(c)]. For temperatures $\sim 20$ K and field 15 K, one may
expect that the $c$-$f$ hybridization is collective for
$\mathbf{B}\parallel[001]$ but is noncollective for
$\mathbf{B}\parallel[100]$. This makes the PHE patterns particularly
complicated near 20 K. However, we must admit that to quantitatively reproduce
the PHE patterns is difficult here and requires additional theoretical inputs.

It should be pointed out that anisotropic magnetic scattering may also play
some role for the abnormal PHE ZhuY-PRB2020 . This seems conceivable
considering the high magnetic anisotropy based on our previous CEF analysis
ChengK-CeCo2Ga8Q1D . Such aniostropic magnetic scattering probably mediates
the Kondo scatterings and further complicates the angular magnetotransport. We
should also emphasize here that although resistive anomalies (e.g.
$\rho\propto-\ln T$) has been reported to characterize the incoherent Kondo
scattering, it is the first time that peculiar feature is captured in PHE in
this special regime. Whether such abnormal PHE is a common feature of the
dynamic hybridization regime in Kondo lattices is still an open question. More
related studies are encouraged in other prototypical heavy-fermion systems
such as Ce115s and so on. Theoretically, quantitative descriptions are also
required to interpret the microscopic nature of electronic structure and
scattering mechanism in this dynamic hybridization regime.

In all, longitudinal and transverse resistivities of the quasi-1D Kondo
lattice compound CeCo2Ga8 are systematically investigated in the presence of
rotational in-plane field, and abnormal PHE and PAMR that substantially
deviate from conventional empirical formulas are observed. Of particular
interest is that the two-fold oscillatory PHE in the high-$T$ incoherent phase
reverses its sign when entering the low-$T$ coherent phase, with additional
oscillations appearing in that between. These results shed new light on the
regime of hybridization dynamics which might be deemed as a precursor state of
the heavy electron state. Our work also implies that PHE - which used to be
mostly considered for the potential application in magnetic sensors - can be
also informative for a localized-itinerant transformation in strongly
correlated quantum materials.

METHODS

Single crystals preparation and characterization

Single crystalline CeCo2Ga8 was grown in a two-step method WangL-CeCo2Ga8 .
First, polycrystalline CeCo2Ga8 was prepared by arc-melting a stoichiometric
mixture of Ce, Co and Ga. Second, Gallium self-flux method was used to grow
CeCo2Ga8 single crystals. This was carried out by mixing up CeCo2Ga8
polycrystal and Ga in an alumina crucible in a molar ratio of 1:10, and sealed
in an evacuated quartz tube. The latter was heated up to 1100 ∘C in 10 h,
dwelt for 10 h, and slowly cooled at a rate of 2 ∘C/h down to 630 ∘C when the
excess flux was removed by centrifugation. Milimeter-sized CeCo2Ga8 single
crystals can be obtained. The chemical composition and crystalline structure
of the crystals were verified by energy dispersive spectrometer (EDS) and
X-ray diffraction (XRD).

Device fabrication

The microstructured Hall-bar device was fabricated by FIB technique (Scios 2
HiVac, Thermo Fisher Scientific Inc.) with effective dimensions $50\times
15\times 8$ $\mu$m3 along the principal axes [001], [100] and [010]
respectively. Electrical contacts were made via patterned mask with
photolithography subsequent growth of Ti/Au (10 nm / 100 nm) layers.
Electrical transport experiments were performed in an IntegraAC Mk II
recondensing cryostat equipped with a 16 T superconducting magnet (Oxford
Instruments). The electrical current was applied along [001]. Both in-plane
longitudinal resistivity ($\rho$xx) and transverse resistivity ($\rho$yx) were
measured by an AC Resistance bridge (372, Lake Shore Cryotronics). A home-
built rotation probe was exploited to conduct the in-plane 0-360 ∘ field
rotation with respect to the current.

Data availability

The authors declare that all source data supporting the findings of this study
are available within the paper.

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__
ACKNOWLEDGEMENTS

The authors thank Joe D. Thompson and Yifeng Yang for helpful discussions.
This work is supported by the National Key R&D Program of China
(2022YFA1602602, 2023YFA1609600), National Natural Science Foundation of China
(U23A20580, U22A6005, U2032204), the open research fund of Songshan Lake
Materials Laboratory (2022SLABFN27), Guangdong Basic and Applied Basic
Research Foundation (2022B1515120020), and the Synergetic Extreme Condition
User Facility (SECUF).

AUTHOR CONTRIBUTIONS

Y.L. conceived and designed the experiments. Y.S. and G.D. grew and
characterized the crystals. X.G. and Z.Z. fabricated the micro-structured
device with FIB. S.Z. performed most of the measurements with the aids from
H.Z., Z.W. and F.L. S.Z. and Y.L. discussed the data, interpreted the results,
and wrote the paper with input from all the authors.

COMPETING INTERESTS

The authors declare that they have no competing financial interests.

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Fig. 1 $|$ PHE and PAMR of CeCo2Ga8. a, Sketch of PHE [$\rho_{yx}(\varphi)$]
and PAMR [$\rho_{xx}(\varphi)$] measurements. $\varphi$ characterizes the
angle spanned by electrical current $\mathbf{j}$ and external magnetic field
$\mathbf{B}$. The sample investigated is prepared by FIB with
$\mathbf{j}\parallel$ [001], and the rotation of $\mathbf{B}$ is within
$\mathbf{ac}$ plane. b, Crystalline structure of CeCo2Ga8 which shows quasi-1D
cerium chain along [001]. c, Temperature dependence of resistivity
$\rho_{xx}$. The onset of Kondo effect is denoted by $T_{K}^{on}\approx 100$
K, and the Kondo coherence develops below $T^{*}\approx 12$ K where
$\rho_{xx}$ peaks. d, $\rho_{yx}(\varphi)$ at 1.6 K. The symbols represent
experimental data, while the solid line is the fitting curve. e, ibid, but for
$\rho_{xx}(\varphi)$. f and g respectively show $\rho_{yx}(\varphi)$ and
$\rho_{xx}(\varphi)$ in polar coordinates that exhibit two-fold oscillations.

Fig. 2 $|$ PAMR and PHE of CeCo2Ga8 under various magnetic field. a,
$\rho_{xx}(\varphi)$ for different field strengths. The two-fold oscillation
retains untill field larger than 9 T. b, Magnetoresistance as a function of
field for $\mathbf{B}\parallel[100]$ (red circles) and
$\mathbf{B}\parallel[001]$ (blue squares). The inset shows MR in the $B^{2}$
scale. c, $\rho_{yx}(\varphi)$ curve at 1.6 K for different field strength.
All the curves exhibit two-fold oscillation. d, PHE amplitude as a function of
magnetic field. A quadratic-$B$ law fits the curve rather well for field less
than 5 T, while for larger field, the amplitude of PHE tends to saturate. The
inset is a plot of PHE amplitude vs. $\Delta\rho$ where no systematic linear
relationship can be found between them.

Fig. 3 $|$ PHE of CeCo2Ga8 at various temperatures. Measurements were made
under fixed $B=15$ T. $\rho_{yx}(\varphi)$ profiles at representative
temperatures below $T_{K}^{on}$: a, 80 K; b, 40 K; c, 25 K; d, 20 K; e, 17 K;
f, 14 K; g, 10 K; h, 5 K. The plume orients along $\varphi\approx 45^{\circ}$
at 80 K. As $T$ decreases, additional oscillations appear, and when $T$ is
well below $T^{*}$, the two-fold oscillations restore with the principal axis
along $\varphi\approx 135^{\circ}$.

Fig. 4 $|$ Magnetoresistance of CeCo2Ga8. a, temperature dependence of
$\rho_{xx}$ under various fields 0, 4, 9 and 15 T, $\mathbf{B}\parallel[100]$.
b, ibid, but for $\mathbf{B}\parallel[001]$. c,
$\Delta\rho=\rho_{xx}^{\mathbf{B}\parallel[100]}-\rho_{xx}^{\mathbf{B}\parallel[001]}$
as a function of $T$, under external field 15 T. The inset shows a plot of
$T^{*}$ vs. $B$ for both field directions. d, Kohler’s plot. MR as a function
of $B/\rho_{xx}(0)$ for $\mathbf{B}\parallel[100]$ (upper panel) and
$\mathbf{B}\parallel[001]$ (lower panel). For both field directions, Kohler’s
rule is drastically violated. e, Coqblin-Schrieffer analysis for $S=1/2$ Kondo
impurity model. MR vs. $B/(T+T_{K})$. For both field directions, MR data in
the hybridization dynamic regime fall on a single line. Note that such a
scaling law holds even for below $T^{*}$. f, Cartoon illustration about the
establishment of heavy-electron state in CeCo2Ga8. The schematic FS is from
theoretical calculations in Ref. WangL-CeCo2Ga8 . The hatched balls signify
Kondo entanglement of $f$ (red) and conduction (blue) electrons. Above
$T_{K}^{on}$, the $f$ electrons are fully localized, and only light-mass
cylindrical sheets from Co/Ga orbitals are present along $\Gamma$-Y. Well
below $T^{*}$, $f$ electrons are delocalized due to coherent $c$-$f$
hybridization; consequently new flat sheets with heavy mass appear owing to
the quasi-1D nature of Kondo chains. For $T^{*}<T<T_{K}^{on}$ where the
$c$-$f$ hybridization is noncollective, the fluctuating heavy-mass FS may
survive to account for the abnormal PHE.