# Magnetotransport induced by anomalous Hall effect

Jiaji Zhao  Bingyan Jiang  State Key Laboratory for Artificial Microstructure
and Mesoscopic Physics, Frontiers Science Center for Nano-optoelectronics,
Peking University, Beijing 100871, China  Jinying Yang  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Lujunyu Wang  Hengjie Shi
Guang Tian  State Key Laboratory for Artificial Microstructure and Mesoscopic
Physics, Frontiers Science Center for Nano-optoelectronics, Peking University,
Beijing 100871, China  Zhilin Li  Enke Liu  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Xiaosong Wu<EMAIL_ADDRESS>State Key Laboratory for
Artificial Microstructure and Mesoscopic Physics, Frontiers Science Center for
Nano-optoelectronics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing 100871, China  Shenzhen Institute
for Quantum Science and Engineering, Southern University of Science and
Technology, Shenzhen 518055, China  Peking University Yangtze Delta Institute
of Optoelectronics, Nantong, Jiangsu 226010, China

###### Abstract

In a magnetic metal, the Hall resistance is generally taken to be the sum of
the ordinary Hall resistance and the anomalous Hall resistance. Here it is
shown that this empirical relation is no longer valid when either the ordinary
Hall angle or the anomalous Hall angle is not small. Using the proper
conductivity relation, we reveal an unexpected magnetoresistance (MR) induced
by the anomalous Hall effect (AHE). A $B$-linear MR arises and the sign of the
slope depends on the sign of the anomalous Hall angle, giving rise to a
characteristic bowtie shape. The Hall resistance in a single-band system can
exhibit a nonlinearity which is usually considered as a characteristic of a
two-band system. A $B$-symmetric component appears in the Hall. These effects
reflect the fundamental difference between the ordinary Hall effect and the
AHE. Furthermore, we experimentally reproduce the unusual MR and Hall reported
before in Co3Sn2S2 and show that these observations can be well explained by
the proposed mechanism. MR often observed in quantum anomalous Hall insulators
provides further confirmation of the picture. The effect may also account for
the large MR observed in non-magnetic three-dimensional topological Dirac
semimetals.

xxx

##  I Introduction

Although the anomalous Hall effect was discovered over a century ago, there
has been a long-time controversy over its mechanism Nagaosa  _et al._ (2010).
Significant progress was made upon the introduction of the Berry phase Xiao
_et al._ (2010). It is now well understood that the intrinsic anomalous Hall
effect (AHE) is determined by the integral of the Berry curvature over
occupied states Nagaosa  _et al._ (2010). With the advent of the topological
band concept, the AHE has attracted revived interest. The strong Berry
curvature appearing in topological bands can give rise to a large AHE Chang
_et al._ (2013); Suzuki  _et al._ (2016); Kuroda  _et al._ (2017); Ghimire
_et al._ (2018); Kim  _et al._ (2018); Belopolski  _et al._ (2019); Noky  _et
al._ (2020); Deng  _et al._ (2020). Currently, great efforts have been made in
a quest for a strong AHE Suzuki  _et al._ (2016); Shekhar  _et al._ (2018);
Liu  _et al._ (2018); Wang  _et al._ (2018); Li  _et al._ (2020); Fujishiro
_et al._ (2021); Jiang  _et al._ (2021a). A related thermoelectric effect, the
anomalous Nernst effect (ANE), has also been actively pursued Guin  _et al._
(2019); Sakai  _et al._ (2018, 2020); Sakuraba  _et al._ (2020). Achieving a
large AHE and ANE will potentially lead to efficient spintronic and energy
conversion devices. The strength of AHE may be characterized by the anomalous
Hall angle, defined as
$\vartheta=\arctan(\sigma_{xy}^{\mathrm{A}}/\sigma_{xx})$, where
$\sigma_{xy}^{\mathrm{A}}$ is the anomalous Hall conductivity, and
$\sigma_{xx}$ is the longitudinal conductivity. In the past, $\tan\vartheta$
was less than 0.1 Kim  _et al._ (2018); Liu  _et al._ (2018); Li  _et al._
(2020). It was quickly boosted to 0.33 in topological materials Shen  _et al._
(2020). It is likely that we will witness a sharp increase of $\tan\vartheta$
in the near future.

The Hall resistivity measured in experiments is influenced by both ordinary
Hall effect (OHE) and AHE. An empirical relation is widely used to extract the
AHE resistivity, that is,

$\rho_{yx}=R_{\mathrm{H}}^{\mathrm{O}}B_{z}+\rho_{yx}^{\mathrm{A}},$ (1)

where $R_{\mathrm{H}}^{\mathrm{O}}$ is the ordinary Hall effect coefficient
and $B_{z}$ is the perpendicular field. The second term represents the
anomalous Hall resistivity, which is proportional to the perpendicular
magnetization. This relation applies to many materials Nagaosa  _et al._
(2010). It is shown in this paper that Eq. (1) is valid only when the ordinary
Hall angle and the anomalous Hall angle are small, which is well satisfied in
conventional materials. However, it is no longer the case in light of recent
progress in finding a large anomalous Hall angle. In a magnetic Weyl
semimetal, Co3Sn2S2, $\tan\vartheta=0.33$ has been achieved Shen  _et al._
(2020). In the meantime, unusual magnetoresistance (MR) and Hall effects were
observed and mechanisms were proposed to explain these effects Yang  _et al._
(2020); Zeng  _et al._ (2021). We reproduce these experimental observations
and show that they can be well explained by AHE, without invoking any
additional mechanism. Our results emphasize the necessity of employing the
correct conductivity relation for analyzing AHE data when the Hall angle is
not small. Moreover, they suggest an alternative explanation for the large MR
observed in topological Dirac semimetals Liang  _et al._ (2015); Novak  _et
al._ (2015); Feng  _et al._ (2015); Tang  _et al._ (2019).

##  II Experiments

Two types of Co3Sn2S2 single crystal were measured in this work. Both were
grown by a chemical vapor transfer method. One is bulk crystals with a typical
size of $1100\times 500\times 60\ \mathrm{\mu m}^{3}$ Jiang  _et al._ (2021b).
The other is a nanoflake with a thickness of 86 nm Yang  _et al._ (2020). Bulk
crystals were cut and polished. Silver paste was used for electrical contacts.
For the nanoflake sample, standard electron-beam lithography and argon plasma
etching were employed for the fabrication of Hall bars. Au films with Ti as an
adhesion layer were used in metallization for electrical contacts. A standard
low-frequency alternating-current method was employed for electrical transport
measurements using a lock-in amplifier.

##  III Results and discussion

Let us consider the simplest case, that is, an isotropic single band. In the
absence of an AHE, the resistivity is expressed as a second rank tensor

$\rho=\rho_{0}\begin{pmatrix}1&-\tan\theta\\\ \tan\theta&1\end{pmatrix},$ (2)

where $\rho_{0}$ and $\theta$ are the zero-field Drude resistivity and the
ordinary Hall angle, respectively. $\tan\theta=\mu B$, where $\mu$ is the
carrier mobility. To include the AHE, it is worth noting that the AHE gives
rise to a contribution to the conductivity rather than the resistivity. For
instance, the intrinsic anomalous Hall conductivity is determined by the
integral of the Berry curvature over occupied states and is thus independent
of the longitudinal resistivity Nagaosa  _et al._ (2010). Therefore, the total
conductivity is the sum of the conductivity given by Eq. (2) and the anomalous
Hall conductivity $\sigma_{xy}^{\mathrm{A}}$

$\sigma=\begin{pmatrix}\sigma_{0}/(1+\tan^{2}\theta)&\sigma_{0}\tan\theta/(1+\tan^{2}\theta)+\sigma_{xy}^{\mathrm{A}}\\\
-\sigma_{0}\tan\theta/(1+\tan^{2}\theta)-\sigma_{xy}^{\mathrm{A}}&\sigma_{0}/(1+\tan^{2}\theta)\end{pmatrix},$
(3)

where $\sigma_{0}=1/\rho_{0}$. Define
$\sigma_{xy}^{\mathrm{A}}\equiv\rho_{yx}^{\mathrm{A}}/\rho_{0}^{2}$ and let
$\theta=0$ (zero field). It can be seen that
$\tan\vartheta=\sigma_{xy}^{\mathrm{A}}/\sigma_{0}=\rho_{yx}^{\mathrm{A}}/\rho_{0}$,
as expected. Plugging it into Eq. (3), one finally obtains two elements of the
resistivity tensor in the presence of the AHE

$\displaystyle\rho_{xx}$
$\displaystyle=\frac{\rho_{0}}{(1+\tan\theta\tan\vartheta)^{2}+\tan^{2}\vartheta},$
(4) $\displaystyle\rho_{yx}$
$\displaystyle=\frac{\rho_{0}(\tan\vartheta\tan^{2}\theta+\tan\theta+\tan\vartheta)}{(1+\tan\theta\tan\vartheta)^{2}+\tan^{2}\vartheta}.$
(5)

When both $\tan\theta$ and $\tan\vartheta$ are small, Eq. (5) is reduced to
Eq. (1) after keeping terms up to the first order. However, if any one of two
Hall angles is not negligible, both $\rho_{xx}$ and $\rho_{yx}$ can be
substantially different from the Drude resistivity and Eq. (1), respectively.

Figure 1: Field dependence of MR and Hall calculated from Eqs. (4) , (5) , and
Eq. (1). The coercive field is set at $\mu B=0.5$. (a),(d) $\tan\vartheta=0.4$
when $\tan\theta>0$. The black dashed lines indicate the Drude resistivity
$\rho_{0}$, which is independent of $B$. (b),(e) $\tan\vartheta=-0.4$ when
$\tan\theta>0$.. The black dashed lines are calculated from Eq. (1). The inset
in (a) depicts the carrier velocity $\boldsymbol{v}$ and the electric field
$\boldsymbol{E}$. Here, $\boldsymbol{v}_{0}$ and $\boldsymbol{v_{k}}$ are the
band velocity with and without AHE, respectively. (c),(f) $\rho_{xx}$ and
$\rho_{yx}$ in a large $\tan\theta$ (field) range.

In Fig. 1, we plot the field dependence of $\rho_{xx}$ and $\rho_{yx}$
calculated from Eqs. (4) and (5). For comparison, the results described by Eq.
(1) are also presented. Several unexpected features stand out. Firstly, the
zero-field longitudinal resistivity deviates from the Drude resistivity.
Secondly, $\rho_{xx}$ is now field dependent, in sharp contrast to the field
independent $\rho_{0}$. Moreover, the field dependence exhibits a strong
$B$-linear component ($B$ antisymmetric). The linearity switches sign with
$\tan\vartheta$, hence magnetization. As a result, $\rho_{xx}$ exhibits a
discontinuity at the coercive field $B_{c}$. The bowtie feature in MR is
similar to that contributed by electron-magnon scatteringNguyen  _et al._
(2011). However, the magnon MR is usually negative. That is, $\rho_{xx}$ drops
when magnetization becomes parallel to field. Here, the sign of MR depends on
the sign of $\tan\vartheta$. When $\theta$ and $\vartheta$ have the same
signs, the MR is negative. Otherwise, it is positive. As for $\rho_{yx}$, it
is not linear in $B$ despite the fact that there is only one isotropic band.
The most prominent feature is that when the magnetization remains unchanged,
$\rho_{yx}(B)-\rho_{yx}(0)$ is not strictly $B$-antisymmetric, as opposed to
Eq. (1). The deviation from the $B$ antisymmetry is easily recognized from the
concave curvature below the coercive field. To see how these terms appear, we
show the Taylor series expansion of Eqs. (4) and (5) up to $\tan^{2}\theta$

$\displaystyle\rho_{xx}$
$\displaystyle=\rho_{0}\left(\frac{1}{1+\tan^{2}\vartheta}-\frac{2\tan\vartheta}{(1+\tan^{2}\vartheta)^{2}}\tan\theta+\frac{\tan\vartheta^{2}(3-\tan\vartheta^{2})}{{(\tan\vartheta^{2}+1)}^{3}}\tan^{2}\theta\right),$
(6) $\displaystyle\rho_{yx}$
$\displaystyle=\rho_{0}\left(\frac{\tan\vartheta}{1+\tan^{2}\vartheta}+\frac{1-\tan^{2}\vartheta}{(1+\tan^{2}\vartheta)^{2}}\tan\theta-\frac{\tan\vartheta(1-3\tan^{2}\vartheta)}{(1+\tan^{2}\vartheta)^{3}}\tan^{2}\theta\right).$
(7)

It can be seen that both the $B$-linear term of $\rho_{xx}$ and the $B^{2}$
term of $\rho_{yx}$ increase with the carrier mobility. When $\tan\vartheta$
is small, these terms are proportional to $\tan\vartheta$. Another noteworthy
observation is that all these unexpected components are an odd function of
$\tan\vartheta$, which is actually consistent with the Onsager reciprocity
relations Jiang  _et al._ (2021b).

Note that there is no MR in an isotropic single band system Pippard (1989), as
indicated by Eq. (2). This is because the Lorentz force is balanced by the
force exerted by the Hall electric field in a steady current state. In
contrast, AHE generates an anomalous velocity $\boldsymbol{v}_{\mathrm{A}}$,
rather than a force. This velocity, equal to
$\frac{e}{\hbar}\boldsymbol{E}\times\boldsymbol{\Omega_{k}}$ at zero field, is
perpendicular to the total electric field $\boldsymbol{E}$ Xiao  _et al._
(2010), which is tilted away from the direction of the electric current by
$\vartheta$. Here, $\boldsymbol{\Omega_{k}}$ is the Berry curvature. As
sketched in the inset of Fig. 1(a), the vector sum of this anomalous velocity
$\boldsymbol{v}_{\mathrm{A}}$ and the band velocity $\boldsymbol{v_{k}}$ is
larger than the velocity $\boldsymbol{v}_{0}$ in the absence of an AHE,
leading to a reduction of the longitudinal resistivity. When a magnetic field
is applied, the electric field will be gradually rotated, leading to a
negative or positive MR depending on the sign of $\vartheta$. When
$\tan\theta$ is large and has a sign opposite to $\tan\vartheta$, Eqs. (4) and
(5) predict a resistance peak and a sign reversal of the Hall resistance,
which will be discussed later.

Figure 2: Field dependence of MR and Hall calculated from Eqs. (4) and (5)
with $\tan\vartheta$ linear in $B$. $\alpha_{B}=0.002$ is used, such that when
a Zeeman field of 150 T is applied, the anomalous Hall angle is 0.3, close to
that in Co3Sn2S2. (a) MR. Data in a large $\tan\theta$ range are shown in the
inset. A strong peak appears at $\tan\theta\tan\vartheta\approx-1$. (b) Hall.
Data in a large $\tan\theta$ range are shown in the inset. An abrupt sign
reversal appears at $\tan\theta\tan\vartheta\approx-1$.

We now extend the above analysis further to a case in which $\tan\vartheta$ is
also linear in $B$. A $B$-linear anomalous Hall effect can appear in
paramagneticFert and Friederich (1976); Culcer  _et al._ (2003); Zhou  _et
al._ (2022) and antiferromagnetic materials Nakatsuji  _et al._ (2015); Suzuki
_et al._ (2016); Li  _et al._ (2022), but the AHE-induced transport is
probably more pronounced in three-dimensional (3D) topological Dirac
semimetals, because they often have extremely high carrier mobility. In these
materials, each Dirac point may be split into two Weyl cones by a magnetic
field along certain crystal directionsWang  _et al._ (2012). The separation
between two Weyl cones in momentum space is proportional to $B$. Since the
anomalous Hall conductivity in Weyl semimetals is proportional to the
separationBurkov (2014), one would expect that
$\tan\vartheta=\sigma_{xy}^{\mathrm{A}}/\sigma_{0}=\alpha_{B}B$, where
$\alpha_{B}$ is a coefficient. $\alpha_{B}$ is likely very small, as the
applied field in most cases is much smaller than the exchange field that is
responsible for AHE in magnetic materials. Owing to their extremely high
mobility, a Hall angle much greater than 1 is experimentally accessible. The
AHE-induced transport can be significant even when the anomalous Hall angle is
very small. Figures 2(a) and (b) display the simulation results. The MR is
quadratic in field, instead of linear at low fields for a constant
$\sigma_{xy}^{\mathrm{A}}$. The Hall resistance is nonlinear, mimicking a two-
band behavior. When $\tan\theta$ and $\tan\vartheta$ share the same sign, the
field dependence is sublinear, otherwise it is superlinear.

Figure 3: Field dependence of MR and Hall calculated from Eqs. (4) and (5)
with $\alpha_{B}=-0.06$. (a) MR. (b) Hall.

When $\tan\vartheta$ is small while $\tan\theta$ is large (strong field) such
that $\tan\theta\tan\vartheta\approx-1$, $\rho_{xx}$ may become gigantic, on
the order of $\rho_{0}/\tan^{2}\vartheta$. $\tan\theta$ can reach a large
value in many topological Dirac semimetals of high mobility. For instance,
ZrTe5 can have a mobility over $10^{5}\ \mathrm{cm^{2}V^{-1}s^{-1}}$ and reach
$\tan\theta=1$ at 0.067 T Tang  _et al._ (2019). Cd3As2 has a mobility of
$8\times 10^{6}\ \mathrm{cm^{2}V^{-1}s^{-1}}$ Liang  _et al._ (2015). With
further increase of $\tan\theta$, $\rho_{xx}$ is strongly suppressed.
Interestingly, a negative MR following a large positive MR with increasing
field has in fact been observed in ZrTe5 Zhou  _et al._ (2019). The MR
simulated from Eqs. (4) and (5) with $\alpha_{B}=-0.06$ is depicted in Fig. 3,
which qualitatively reproduces the MR peak observed in experiments. In the
meantime, $\rho_{yx}$ can abruptly change sign, which is in contrast to a
smooth change in a two-band system. Interestingly, an abrupt change in Hall
has been observed in TaP and attributed to Weyl node annihilation Zhang  _et
al._ (2017), but there was no concomitant MR peak. The bipolar behavior can be
traced back to the condition that $\theta$ and $\vartheta$ have opposite
signs. The overall behavior is similar to that shown in Figs. 1(c) and (f),
although the MR can be much stronger because of the smallness of $\alpha_{B}$.
Except for the nonlinear Hall and the quadratic MR at low fields, many
features are distinctive from the two-band transport, such as the negative MR,
MR peak and sharp reversal of the Hall resistance. The resistivity that
asymptotically approaches zero at high fields is particularly surprising. At
the same time, $\sigma_{xy}$ asymptotically approaches a plateau of
$\sigma_{xy}^{\mathrm{A}}$ for a constant $\tan\vartheta$. Although this may
look like a quantized Hall effect, the plateau value is not universal and
bears no relation to any topological invariant. At last, one should keep in
mind that the role of Landau levels in high fields is neglected in Eq. (3). It
is possible that these high-field features may be altered.

Co3Sn2S2 displays a record high anomalous Hall angle and relatively high
mobility compared to typical magnetic materials Liu  _et al._ (2018); Yang
_et al._ (2020); Tanaka  _et al._ (2020). A high mobility indicates a strong
increase of the Hall angle with magnetic field. In addition, the coercive
field in thin films was found to be greatly enhanced Yang  _et al._ (2020);
Tanaka  _et al._ (2020); Zeng  _et al._ (2021). These properties make Co3Sn2S2
an ideal place to study the AHE-induced transport described by Eqs. (4) and
(5). Indeed, unusual behaviors in both the MR and Hall resistivity have been
reported Yang  _et al._ (2020); Tanaka  _et al._ (2020); Zeng  _et al._
(2021).

At low temperatures, the longitudinal resistivity jumps up when the direction
of magnetization switches from $B$ antiparallel to parallel at $B_{c}$Yang
_et al._ (2020); Tanaka  _et al._ (2020); Zeng  _et al._ (2021). The
resistivity difference between antiparallel and parallel directions linearly
depends on field Zeng  _et al._ (2021). With increasing temperature, the jump
turns into a sudden drop. Yang et al. argued that the jump at low temperatures
was a Lorentz-like ordinary MR due to a fictitious magnetic field induced by
magnetization Yang  _et al._ (2020), while it was proposed by Zeng et al. that
possible nonlinear magnetic textures and the chiral magnetic field associated
with Weyl fermions accounted for the phenomenon Zeng  _et al._ (2021). We find
that Eq. (4) naturally explains the above observations, as depicted in Fig. 1.
The main contribution comes from the $\tan\theta$-linear term in Eq. (6).
Apparently, it is antisymmetric in magnetization ($\tan\vartheta$) and
linearly depends on $B$. Note that electron-magnon scattering gives rise to a
resistance drop at $B_{c}$ when the magnetization goes from antiparallel to
parallel. The effect increases with the magnon population, hence the
temperature. Consequently, the experimentally observed change from a jump to a
drop with increasing temperature is expected.

Since the AHE-induced MR is not strong, the bowtie feature depicted in Figs.
1(a) and (b) may not be obvious here. In quantum anomalous Hall insulators,
the anomalous Hall angle diverges as the system enters into the quantum Hall
state. One would expect a strong manifestation of Eq. (4). Indeed, we find
that the bowtie feature appeared in various quantum anomalous Hall insulators
Chang  _et al._ (2015); Deng  _et al._ (2020); Ovchinnikov  _et al._ (2021);
Mogi  _et al._ (2022). The feature may be overwhelmed by the resistivity peak
at quantum Hall plateau transitions that coincides with the reversal of the
magnetization. Fortunately, MnBi2Te4 thin layers exhibit multiple magnetic
transitions with increasing field. The anomalous Hall resistivity displays a
significant hysteresis loop only in the first transition at low fields, while
the resistivity peak occurs at the highest coercive field. Consequently, a
bow-tie-shaped MR was apparent Deng  _et al._ (2020); Ovchinnikov  _et al._
(2021). Furthermore, when the carrier density was tuned by a gate, the
relative sign between $\tan\theta$ and $\tan\vartheta$ can be changed. As a
result, the resistivity jump at the first coercive field became a drop, while
the MR changed from positive to negative (See Fig. S5B in the Supplementary
Materials of Ref. [Deng  _et al._ , 2020]). The same correlation between the
sign of the jump and the sign of MR was also observed in a semi-magnetic
topological insulator (See Fig. S7AB in the Supplementary Materials of Ref.
[Mogi  _et al._ , 2022]). All these observations are in excellent agreement
with Eq. (4).

We now go back to the Hall resistivity of Co3Sn2S2 nanoflakes. A remarkable
feature is that $\rho_{yx}(B)-\rho_{yx}(0)$ below $B_{c}$ is not strictly
antisymmetric in field Yang  _et al._ (2020); Tanaka  _et al._ (2020); Zeng
_et al._ (2021). It was found that the deviation of $\rho_{yx}$ from the
expected one is proportional to $B^{1.8}$ at low temperaturesZeng  _et al._
(2021). The sign of the deviation goes with the direction of magnetization. It
was speculated that a change in the magnetic texture might lead to a change of
the Weyl point separationYang  _et al._ (2020) or a gauge fieldZeng  _et al._
(2021), which was responsible for the observations. We point out that Eq. (5)
includes $B$-symmetric contributions. Moreover, the leading term is
proportional to $B^{2}$, as shown in Eq. (7). It is antisymmetric in
magnetization ($\tan\vartheta$). All features, including the power index, are
consistent with Eq. (5).

Having shown that previous experimental observations can be qualitatively
explained by the AHE, we further carried out transport measurements on
Co3Sn2S2 and performed quantitative analysis. Experiment results are shown in
Fig. 4. Benefiting from the large coercive field and high carrier mobility,
the nanoflake sample (N01) displays clear AHE-induced MR similar to those
reported before. The characteristic $B$-linear term in $\rho_{xx}$ and a
$B$-symmetric term in $\rho_{xx}$ are evident. To fit Eqs. (4) and (5) to our
data, a two-band model has to be used, as the MR and the nonlinearity in Hall
are too strong to be accounted for by a single-band model with AHE. After
substituting the one-band Drude conductivity with a two-band one, a reasonably
good agreement is obtained. All key features are captured by our model. For
comparison, the AHE-induced MR in the bulk crystal is small. Still, the drop
in $\rho_{xx}$ can be seen, shown in Fig. 4(c). This is due to the small
coercivity and lower carrier mobility of bulk crystals, which is consistent
with Eqs. (6) and (7).

Figure 4: Experimental data of MR and Hall for Co3Sn2S2. (a),(b) $\rho_{xx}$
and $\rho_{yx}$ of N01 at 10 K. A jump in $\rho_{xx}$ occurs at the coercive
field. The orange area at negative fields is enclosed by $\rho_{yx}(B)$ and
the one obtained by an inversion of $\rho_{yx}(B)$ through
$\left(0,\rho_{yx}(0)\right)$, highlighting the $B$-symmetric component of
$\rho_{yx}$. (c),(d) $\rho_{xx}$ and $\rho_{yx}$ of B04 at 20 K. Tiny drops in
$\rho_{yx}$ at the coercive field are discernible, in agreement with
$\tan\theta$ and $\tan\vartheta$ being of the same sign.

##  IV Conclusion

By pointing out the limitation of the empirical resistivity relation that is
widely used for AHE data analysis, we emphasize the necessity of employing the
proper conductivity relation when either the ordinary Hall angle or the
anomalous Hall angle is not small. It is shown that AHE gives rise to a
$B$-linear MR in the longitudinal resistivity. The sign of the linearity
depends on the sign of the anomalous Hall angle, yielding a characteristic
bow-tie-shaped MR. Additionally, AHE induces a nonlinearity and a $B^{2}$
component in the field dependence of Hall resistivity. The AHE-induced
transport can not only explain some MR features in quantum anomalous Hall
insulators but also quantitatively account for the unusual observations made
in Co3Sn2S2. We further extend the analysis to nonmagnetic 3D topological
Dirac semimetals and discuss the possible origin of the large MR observed in
these materials based on AHE.

###### Acknowledgements.

This work was supported by National Key Basic Research Program of China
(Grants No. 2022YFA1403700, No. 2020YFA0308800) and NSFC (Projects No.
11774009, No. 12074009, and No. 12204520). Zhilin Li is grateful for the
support from the Youth Innovation Promotion Association of the Chinese Academy
of Sciences (No. 2021008).

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