# Weyl points and anomalous transport effects tuned by the Fe doping in Mn3Ge
Weyl semimetal

V. Rai  Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science
(JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-FIT, 52425 Jülich, Germany
RWTH Aachen, Lehrstuhl für Experimentalphysik IVc, Jülich-Aachen Research
Alliance (JARA-FIT), 52074 Aachen, Germany  Department of Physics and
Materials Science, University of Luxembourg, 162A avenue de la Faiencerie,
L-1511 Luxembourg, Grand Duchy of Luxembourg  S. Jana  Forschungszentrum
Jülich GmbH, Jülich Centre for Neutron Science (JCNS-2) and Peter Grünberg
Institut (PGI-4), JARA-FIT, 52425 Jülich, Germany  RWTH Aachen, Lehrstuhl für
Experimentalphysik IVc, Jülich-Aachen Research Alliance (JARA-FIT), 52074
Aachen, Germany  J. Perßon  Forschungszentrum Jülich GmbH, Jülich Centre for
Neutron Science (JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-FIT, 52425
Jülich, Germany  S. Nandi<EMAIL_ADDRESS>Forschungszentrum Jülich
GmbH, Jülich Centre for Neutron Science (JCNS-2) and Peter Grünberg Institut
(PGI-4), JARA-FIT, 52425 Jülich, Germany  RWTH Aachen, Lehrstuhl für
Experimentalphysik IVc, Jülich-Aachen Research Alliance (JARA-FIT), 52074
Aachen, Germany

###### Abstract

The discovery of a significantly large anomalous Hall effect in the chiral
antiferromagnetic system - Mn3Ge - indicates that the Weyl points are widely
separated in phase space and positioned near the Fermi surface. In order to
examine the effects of Fe substitution in Mn3Ge on the presence and location
of the Weyl points, we synthesized (Mn1-αFe${}_{\alpha})$3Ge ($\alpha=0-0.30$)
compounds. The anomalous Hall effect was observed in compounds up to
$\alpha=0.22$, but only within the temperature range where the magnetic
structure remains the same as the Mn3Ge. Additionally, positive longitudinal
magnetoconductance and planar Hall effect were detected within the same
temperature and doping range. These findings strongly suggest the existence of
Weyl points in (Mn1-αFe${}_{\alpha})$3Ge ($\alpha=0-0.22$) compounds. Further,
we observed that with an increase in Fe doping fraction, there is a
significant reduction in the magnitude of anomalous Hall conductivity, planar
Hall effect, and positive longitudinal magnetoconductance, indicating that the
Weyl points move further away from the Fermi surface. Consequently, it can be
concluded that suitable dopants in the parent Weyl semimetals have the
potential to tune the properties of Weyl points and the resulting anomalous
electrical transport effects.

##  I Introduction:

The field of condensed matter physics has entered a new era with the recent
discovery of topological phases of matter [1, 2]. A significant breakthrough
in this direction is the identification of chiral antiferromagnets that host
Weyl Fermions [3, 4]. The presence of Weyl nodes in chiral antiferromagnets
leads to anomalous transport effects, which are rarely observed in
conventional antiferromagnets, as mentioned in Refs. [5, 6, 7]. The
observation of a substantial anomalous Hall effect (AHE) in chiral
antiferromagnets is interesting, paving the way for research in the realms of
topological materials, spintronics, and high-efficiency energy harvesting
devices [4, 3, 8, 9, 10, 11, 12, 13]. The separation of Weyl points in phase
space and their positioning relative to the Fermi energy play a crucial role
in determining the magnitude of AHE and the electrical transport effects
induced by the chiral anomaly [14, 15, 10, 5, 7, 16, 17, 18].

The chiral antiferromagnetic hexagonal phase of Mn3Ge hosts multiple pairs of
Weyl Fermions situated in proximity to the Fermi surface [19, 7]. As a result,
Mn3Ge exhibits large anomalous Nernst effect, magneto-optical Kerr effect,
AHE, and positive longitudinal magnetoconductivity (LMC), all justifying the
presence of Weyl points near the Fermi surface [20, 10, 21, 3, 22, 6, 13].
Temperature-dependent Nernst effect measurements on Mn3Ge have demonstrated
that the position of Weyl points can be manipulated by changing the
temperature [10]. Recent efforts have focused on controlling the Weyl nodes in
various compounds through doping, external pressure, and other means [23, 24,
25, 26, 27, 28, 29, 30, 31]. However, the precise understanding of the role
played by external factors in the dynamics of Weyl Fermions is still
incomplete.

Our research focuses on investigating the characteristics of Weyl Fermions in
chiral antiferromagnets by studying the evolution of Weyl points and the
resulting electronic transport effects in hexagonal-(Mn1-αFeα)3Ge. While the
evolution of magnetism and trivial transport effects with Fe doping in
hexagonal-Mn3Ge has been studied in the past [32, 33, 34, 35, 36], their
recognition as potential Weyl semimetals has been reported recently [37, 38].
Previous studies [34, 35, 36] have shown that for low Fe doping levels
($\lesssim 15\%$), the compounds exhibit magnetization similar to Mn3Ge, even
at the lowest measured temperature. However, it was observed that the Néel
temperature ($T_{\text{N1}}$) decreases as Fe doping in Mn3Ge increases. For
higher Fe doping levels (approximately $0.15\lesssim\alpha\lesssim 0.25$), a
second magnetic transition appears at $T_{\text{N2}}$, which remains lower
than $T_{\text{N1}}$. The magnetization of the compounds in these two magnetic
regimes is distinct, indicating different magnetic structures in each
temperature regime. Finally, in the case of even higher Fe doping levels
($\alpha\gtrsim 0.30$), both antiferromagnetic transitions are suppressed by
the emergence of ferromagnetism in the sample.

In this report, we initially conducted a comprehensive analysis of the
magnetization properties of hexagonal-(Mn1-αFeα)3Ge compounds with varying Fe
doping fractions ($\alpha$) ranging from 0 to 0.30. Notably, we observed that
the magnetization behavior resembling Mn3Ge persists over a considerable
temperature range, specifically up to $\alpha$ = 0.22. For compounds with
$\alpha$ = 0.14 - 0.22, a change in magnetization behavior at low temperatures
allowed us to investigate the influence of such changes on the electrical
transport effects of the samples. Consequently, we performed a detailed
analysis of the AHE and various magnetoconductivity (MC) measurements on
single-crystal (Mn1-αFeα)3Ge compounds with $\alpha$ = 0 - 0.22. For low Fe
doping levels ($\alpha$ = 0.04, 0.10), the AHE and positive LMC were observed,
exhibiting similarities (albeit with lower magnitudes) to the parent sample,
below the Neél temperature ($T_{\text{N1}}$). As the Fe doping increased
($\alpha$ = 0.18, 0.22), the samples underwent a second magnetic transition at
$T_{\text{N2}}$ (which is lower than $T_{\text{N1}}$). In these cases, the AHE
and positive LMC were observed between $T_{\text{N2}}$ and $T_{\text{N1}}$.
Moreover, it was found that these compounds exhibited a magnetic structure
similar to Mn3Ge between $T_{\text{N2}}$ and $T_{\text{N1}}$ [37].
Interestingly, below $T_{\text{N2}}$, the magnetic structure changed to a
collinear antiferromagnetic (AFM) configuration, and the AHE vanished. This
observation suggests that the Weyl points exist in (Mn1-αFeα)3Ge compounds
solely within the temperature range of $T_{\text{N2}}$ to $T_{\text{N1}}$.

The chiral anomaly effect in Weyl semimetals is characterized by several
phenomena such as positive LMC, planar Hall effect (PHE), and angular
magnetoconductivity ($\theta$MC) [16, 18]. However, these effects can arise
due to other effects as well, for example, due to the magnetization, or
unequal spin density of states (sDOS) near the Fermi surface. Our analysis
suggests that the observed positive LMC (at $B=1$ T), $\theta$MC, and PHE in
the Fe-doped Mn3Ge compounds are likely a result of the chiral anomaly effect
in Fe-doped Mn3Ge. However, further investigations are necessary to support
our claim.

We have observed that the AHE and signatures of the chiral anomaly effect
(positive LMC, $\theta$MC, and PHE) are present in the compounds with Fe
doping levels ranging from $\alpha=0$ to $\alpha=0.22$, but only within the
temperature range where the magnetic structure remains similar to that of
Mn3Ge. This suggests that the Weyl points exist in the Fe-doped samples as
long as they retain a magnetic structure similar to Mn3Ge. Moreover, the
magnitude of AHE and LMC decreases with an increase in the Fe doping level
($\alpha$), indicating that the Weyl points move further away from the
chemical potential as $\alpha$ increases.

##  II Experimental methods

The synthesis of single-crystals of hexagonal-(Mn1-αFeα)3+γGe with varying Fe
doping levels was carried out using a similar method described in Refs. [37,
20]. The self-flux method was employed to synthesize the single-crystals under
nearly identical conditions. Since the hexagonal phase of Mn3Ge is stabilized
with an excess of Mn ($\gamma$) [39, 40], hexagonal-(Mn1-αFeα)3+γGe compounds
with $\gamma\approx 0.2$ were synthesized.

The synthesis of (Mn1-αFeα)3.2Ge starts with the melting of pure elements with
a stoichiometric composition using the induction melting technique. The
resulting samples were then sealed in quartz tubes and heated in the furnace
up to 1273 K (for $\alpha=0$) or 1323 K (for $\alpha=0.30$) for 5 hours,
followed by a slow cooling down to 1123 K, at the rate of 1 K/hr. It is
important to mention that the Mn3Ge and Fe3Ge stabilize in the hexagonal
crystal structure above 903 K and 973 K, respectively [39, 40]. Therefore, all
the samples were water-quenched at 1123 K to preserve the high-temperature
hexagonal phase.

High-quality single-crystals were successfully obtained for compounds with Fe
doping levels $\alpha=(0-0.22)$. However, the samples with $\alpha=0.26$ and
$\alpha=0.30$ remained polycrystalline. The Laue diffraction patterns of the
samples with $\alpha=0$ to $\alpha=0.22$ exhibited sharp 6-fold diffraction
spots, indicating the formation of crystals in the hexagonal phase. An example
of a Laue pattern for a selected sample with $\alpha=0.22$ is shown in Figure
11 in the Appendix.

Chemical analysis of the crystals was performed using the ICP-OES (inductively
coupled plasma optical emission spectroscopy) method. The analysis yielded the
empirical formulas for samples with Fe doping levels $\alpha=0$,
$\alpha=0.04$, $\alpha=0.10$, $\alpha=0.18$, $\alpha=0.22$, and $\alpha=0.30$
as Mn3.10(5)Ge, (Mn0.96(1)Fe0.04(1))3.25Ge, (Mn0.90(1)Fe0.10(1))3.18Ge,
(Mn0.83(1)Fe0.17(1))3.25Ge, (Mn0.79(1)Fe0.21(1))3.21Ge, and
(Mn0.68(1)Fe0.32(1))3.20Ge, respectively. For simplicity, we will refer to
these samples based on their initial stoichiometric composition.

Figure 1: Magnetic structure of $\alpha=0.22$ compound at 130 K, as determined
by the Ref. [37]. The crystallographic axes a and b, related by 120°, show
hexagonal coordinates. The x, y, and z axes are described in the Cartesian
coordinate system. The c and z axis remain the same in both coordinate
systems.

A small amount of $\alpha=(0,0.04,0.10,0.18,0.22,$ $0.30)$ compounds were
crushed and X-ray powder diffraction (XRPD) was performed using the Huber
Imaging plate Guinier Camera (G670). The obtained data were analyzed using the
FullProf software, and the results are presented in Figure 10(a-f) in the
Appendix. The analysis confirms that all the crystals were synthesized in the
hexagonal phase with $P6_{3}/mmc$ (no. 194) space group symmetry. A small
amount of tetragonal phase (2-4%) was also observed as an impurity in the
$\alpha=0$ and $\alpha=0.04$ compounds (Figure 10(a, b) in the Appendix). This
presence of a tetragonal phase has been commonly observed during the synthesis
of hexagonal-Mn3Ge [41, 22].

The lattice parameters of the samples in the hexagonal phase were compared and
are shown in Figure 10(g) in the Appendix. As expected, the lattice parameters
and lattice volume decrease monotonically with an increase in Fe
concentration, following Vegard’s law [42]. Furthermore, neutron diffraction
analysis of the $\alpha=0.22$ compound [37] confirmed the substitution of Mn
by Fe on the Mn sites, while the excess Mn ($\gamma$) occupies the Ge sites.
The nuclear and magnetic structure of the $\alpha=0.22$ compound at 130 K is
illustrated in Figure 1. The plot also includes crystallographic axes in both
hexagonal ($a,b$) and Cartesian ($x,y$) coordinates for convenience.

Figure 2:  The temperature-dependent magnetization (M-T) measurements were
performed under field-cooled (FC) conditions with a magnetic field of 0.01 T.
The measurements shown here were taken during the warming of the sample. (a)
Figure 2(a) shows the M-T curves for single-crystal (sc) (Mn1-αFeα)3.2Ge with
$\alpha=0-0.22$ along the x axis. The M-T curve for the polycrystalline (poly)
sample with $\alpha=0.26$ is also included for comparison. (b) Figure 2(b)
displays the M-T curves for selected single-crystals with the magnetic field
(B) parallel to the z axis. (c) Figure 2(c) shows the M-T curves for the
(Mn1-αFeα)3.2Ge samples with $\alpha=0.26$ and $\alpha=0.30$.

The magnetization and electrical transport measurements of all the samples
were conducted using Quantum Design-Physical Property Measurement System (QD-
PPMS) and Quantum Design-DynaCool (QD-DC) instruments. In the measurements,
the magnetic and electrical properties were probed along the crystallographic
axes $x$, $y$, and $z$ of the sample, as defined in Figure 1. This allows a
comprehensive understanding of the anisotropic behavior and electronic
transport characteristics of the samples.

The tetragonal and hexagonal phases of Mn3Ge exhibit different magnetic
behaviors, with the tetragonal phase being ferrimagnetic and the hexagonal
phase being paramagnetic at 400 K. The Fe doped samples demonstrate
paramagnetic behavior at 400 K, which is above the Néel temperature
($T_{\text{N1}}$) for $\alpha\leq 0.26$ compounds. Therefore, it can be
concluded that the observed transport and magnetic effects originate from the
hexagonal phase of the compound.

##  III Magnetization

Figure 3: Magnetic phase diagram of the (Mn1-αFeα)3.2Ge ($\alpha=0-0.30$).
$T_{\text{N1}}$, $T_{\text{N2}}$, and $T_{\text{c}}$ are the magnetic
transition temperatures shown in Figure 2. AF-I and AF-II show different
antiferromagnetic regimes, whose corresponding magnetic structure is also
shown in the same region. The samples with $\alpha>0.22$ show ferromagnetic
(FM) behavior below $T_{\text{c}}$, as indicated in the figure.

The temperature-dependent magnetization (M-T) measurements were performed on
single crystals and polycrystalline compounds with varying Fe doping levels
($\alpha=0-0.30$), and the results are shown in Figure 2.

When measuring the magnetization along the x axis, compounds with
$\alpha=(0.04-0.26)$ exhibit a magnetic phase transition at the Néel
temperature ($T_{\text{N1}}$), similar to Mn3Ge [20] (Figure 2(a)). Below
$T_{\text{N1}}$, the magnetization along the x axis starts to increase,
resembling the behavior of the parent compound ($\alpha=0$). However, the
compounds do not behave uniformly at low temperatures. In contrast to
$\alpha\leq 0.10$ compounds, compounds with $\alpha=(0.14-0.26)$ exhibit a
second magnetic phase transition at $T_{\text{N2}}$ ($<T_{\text{N1}}$), as
shown in Figure 2(a). The magnetization magnitude below $T_{\text{N2}}$
remains significantly smaller compared to the magnetization between
$T_{\text{N1}}$ and $T_{\text{N2}}$.

Magnetization measurements along the z axis were also conducted for selected
compounds ($\alpha=0,0.04,0.10,0.18,0.22$), as depicted in Figure 2(b). The
M-T curve along the z axis follows a similar pattern as that along the x axis.
However, the magnetization along the z axis is nearly 10 times smaller than
that along the x axis throughout the measured temperature range.

The magnetization of $\alpha=0.26$ and $\alpha=0.30$ samples displays
ferromagnetic (FM) behavior, and their respective Curie temperatures
($T_{\text{c}}$) are shown in Figure 2(c).

The derivative of the $M-T$ curve was used to determine various magnetic
transition temperatures. The evolution of $T_{\text{N1}}$, $T_{\text{N2}}$,
and $T_{\text{c}}$ with Fe doping fraction ($\alpha$) is presented in the
phase diagram shown in Figure 3. As mentioned earlier, the magnetization of
$\alpha=(0.04-0.10)$ and $\alpha=(0.14-0.26)$ compounds exhibits behavior
similar to that of Mn3Ge below $T_{\text{N1}}$ and between $T_{\text{N2}}$ and
$T_{\text{N1}}$, respectively. We define this magnetization regime as AF-I, as
indicated in Figure 3. The magnetic structure of the $\alpha=0.22$ compound in
the AF-I regime has already been reported to be the same as Mn3Ge [37].
Therefore, we can conclude that the magnetic structure of all compounds in the
AF-I regime remains the same as that of the parent compound Mn3Ge.

The $M(H)$ measurements for $\alpha=(0.14-0.26)$ compounds at 4 K reveal
antiferromagnetic (AFM) behavior in the absence of residual magnetization.
Previous studies [34, 37] have reported that the $\alpha=0.22$ compound
exhibits collinear AFM behavior with Mn moments aligned along the z axis.
Hence, it can be inferred that compounds with $\alpha=(0.14-0.26)$ possess a
collinear AFM structure with Mn moments oriented along the z axis below
$T_{\text{N2}}$, as depicted in Figure 3. This magnetic region is referred to
as the AF-II regime in the phase diagram.

Figure 4: Temperature dependent longitudinal resistivity of the
(Mn1-αFeα)3.2Ge ($\alpha=0-0.22$) along (a) x axis, and (b) z axis. Magenta
arrows in (a, b) lie at $T_{\text{N1}}$, below which the respective samples
lie in AF-I region. Magenta circles in both the figures lie at
$T_{\text{N2}}$, below which the respective samples lie in AF-II region.

##  IV Electrical Transport Results

The electrical transport properties of single-crystal compounds with
$\alpha=0,0.04,0.10,0.18,0.22$ were investigated. The crystal size
$\alpha=0.14$ sample was very small. Therefore no electrical transport
measurement was performed using this sample. The samples under measurements
have typical dimensions of length $\simeq$ 1.0 mm - 1.5 mm, width $\simeq$ 0.4
mm - 0.5 mm, and thickness $\simeq$ 0.1 mm - 0.2 mm. Measurements were
performed along the different crystallographic axes, and consistent results
were obtained upon repeated measurements, confirming the intrinsic nature of
the observed electrical transport effects.

The electrical resistivity of all the compounds was measured along the three
different axes. We have observed similar resistivity along the x and y axes,
as reported by Refs. [37, 20]. However, a significantly different resistivity
behavior was observed along the z axis, compared to the x axis. The
longitudinal resistivity along the x and z axes for $\alpha=0-0.22$ compounds
is shown in Figure 4. The resistivity increases with increasing Fe doping
fraction, which is expected due to impurity doping. The parent compound,
$\alpha=0$, shows metallic behavior, along the x axis, below 200 K. However,
it shows semimetallic behavior above 200 K, up to $T_{\text{N1}}$. In contrast
to this, all the Fe doped compounds exhibit semimetallic resistivity behavior
along the x axis in the entire AF-I regime. The resistivity along the z axis
shows metallic behavior for the parent compound. Whereas, it shows a mixture
of metallic and semimetallic behavior for Fe doped compounds. The increase in
resistivity, along both the axes, at low temperatures suggest strong spin
scattering as magnetic moments of the compound increases. For
$\alpha=(0.18,0.22)$ compounds, the resistivity along both the x and z axes
drops below $T_{\text{N2}}$ and exhibits metallic behavior in the AF-II
regime.

###  IV.1 Anomalous Hall effect

Figure 5: In Figures (a) and (b), the Hall resistivity measurements at 4 K and
150 K, respectively, are shown for the field applied along the y axis. In
Figures (c) and (d), the Hall resistivity measurements at 4 K and 150 K,
respectively, are shown for the field applied along the z axis. Inset of (c)
represents the magnified version of the Hall resistivity ($\rho_{xy}$) for
$\alpha=0.10$ compound. The Hall resistivity measurements were performed at
150 K, representing the AF-I regime for all the compounds. Hall resistivity at
4 K represents the AF-II regime for the $\alpha=0.18$ and $\alpha=0.22$
compounds only. In Figure (b) of the Hall resistivity measurements,
$\rho_{xz}^{A}$ represents the anomalous Hall resistivity (AHR). It is
determined by taking the intercept of the linear fit of the Hall resistivity
data between -9 T and -3 T.  Figure 6: (a) The temperature dependence of the
anomalous Hall conductivity (AHC) ($\sigma^{A}_{xz}$) is shown for the case
where the current (I) is parallel to the x axis and the magnetic field (B) is
parallel to the y axis. The data for $\alpha=0$ is obtained from Ref. [20].
(b) The evolution of the AHC ($\sigma^{A}_{xz}$) with respect to $\alpha$
(doping fraction) is presented at three different temperatures. At $T=4$ K,
the AHC is zero for $\alpha=0.18$ and $\alpha=0.22$, and therefore not shown
in this plot. The purple curves represent the exponential decay behavior
(expon.) of the AHC. (c) The Hall coefficient
($R_{H}=\frac{\partial\rho_{H}}{\partial B}$) corresponding to the
$\alpha=0.04$ and $\alpha=0.18$ compounds is shown, indicating their distinct
magnetization behavior.

Hall resistivity measurements were performed on compounds with various Fe
doping fractions, namely $\alpha=0,0.04,0.10,0.18,0.22$, with the magnetic
field oriented along the y and z axes. Figure 5(a-d) displays the Hall
resistivity values for these compounds at temperatures of 4 K and 150 K, with
the magnetic field parallel to the y and z crystallographic axes.

When the magnetic field was applied along the y axis, a sharp jump in Hall
resistivity was observed, in several cases, near zero fields, which clearly
suggests the presence of anomalous Hall resistivity (AHR). The magnitude of
the AHR was determined by performing a linear fit to the high-field Hall
resistivity data, as shown in Figure 5(b). As depicted in Figure 5(a, b), AHR
($\rho_{xz}^{A}$) is present at both 4 K and 150 K in compounds with Fe doping
fractions of $\alpha=(0-0.10)$ and $\alpha=(0-0.22)$, respectively.
Furthermore, the Hall hysteresis at 150 K, observed in both the parent and Fe-
doped compounds, is confined to a range of 0.02 T, as illustrated in Figure 13
in the Appendix.

In comparison to the y axis, when the magnetic field aligns parallel to the z
axis (B$\parallel$z), the observed Anomalous Hall Resistance (AHR) is much
reduced, as depicted in Figure 5(c, d), or even absent. Prior research on
Mn3Ge has proposed that the AHR would vanish when the magnetic field aligns
with the z axis [3, 20]. However, we have observed a slight non-zero AHR for
B$\parallel$z. Within this setup, the AHR magnitude is significantly prominent
for $\alpha=0.04$, while it is minimal for $\alpha=0.10$ (shown in the inset
of Fig. 5(c)). The presence of AHR for B$\parallel$z is not uniformly observed
across all compounds. This inconsistency may stem from minor misalignment
between the sample’s orientation, the magnetic field’s direction, and the
crystallographic x or y axes. Furthermore, the small out of the (a-b) plane
magnetization could also contribute to the minor AHR under the B$\parallel$z
configuration [22, 20, 37, 3]. However, additional experimental investigations
are necessary to precisely identify its underlying source.

Multiple Hall resistivity ($\rho_{xz}$) measurements were conducted with a
magnetic field applied along the y axis at various temperatures. For each
measurement, the corresponding AHR ($\rho^{A}_{xz}$) was determined using the
linear fitting of the high field Hall resistivity data, as illustrated in
Figure 5(b). In this context, $\rho^{A}_{xz}$ represents the AHR when the Hall
voltage is measured along the z axis, the current is applied along the x axis,
and the magnetic field is applied perpendicular to both axes, i.e. along the y
axis.

The Hall resistivity ($\rho_{H}$) can be expressed in a general form as
$\rho_{\text{H}}=R_{0}B+R_{M}M+\rho_{H}^{A}$, where $R_{0}$ and $R_{M}$
correspond to the ordinary and magnetization induced Hall coefficients,
respectively. $\rho_{H}^{A}$ denotes the anomalous Hall resistivity due to the
non-vanishing Berry curvature [22]. In compounds with ferromagnetic (FM)
properties, the existence of AHR is expected due to the presence of remanent
magnetization [43]. In such FM samples, the Hall resistivity exhibits a
monotonic relationship with the magnetization of the sample under varying
magnetic fields, as mentioned in Ref. [43].

In the case of Fe-doped Mn3Ge compounds, a small residual magnetization has
been observed in the antiferromagnetic (AF-I) regime, indicating the presence
of weak FM behavior (Figure 12 in the Appendix). However, Refs. [22, 37] have
reported that the Hall resistivity of compounds with $\alpha=(0,0.22)$ in the
AF-I regime does not exhibit a monotonic dependence on isothermal
magnetization. This concludes that the observed AHR does not originate from
the residual magnetization, but the non-vanishing Berry curvature, which leads
to $\rho_{H}^{A}>0$. The nature of Hall resistivity of all the compounds
between $\alpha=0$ and $\alpha=0.22$, within the AF-I regime, remains the same
as long as the magnetic field is applied along the y axis. Therefore, it can
be concluded that the observed AHR (for B$\parallel$y), in the entire AF-I
regime, originates from the non-vanishing Berry curvature [22, 44]. This also
suggests the existence of Weyl points in the entire AF-I regime.

The Fe doped Mn3Ge compound with an $\alpha$ value of 0.30, which exhibits
notable ferromagnetic behavior (refer to Figure 12 in the Appendix), also
demonstrates an AHR as depicted in Figure 16(a) in the Appendix. By analyzing
the relationship between AHR and magnetization of $\alpha=0.30$ compound, it
is evident that an AHR of approximately 1.8 $\mu\Omega$ cm can be generated by
a residual magnetization of 1.5 $\mu_{\textbf{B}}$/f.u.. In contrast to this,
lower Fe doped compounds in the AF-I temperature range, show AHR $\geq 0.8$
$\mu\Omega$ cm, even though their residual magnetization remains below 0.030
$\mu_{\textbf{B}}$/f.u.. Furthermore, examining the field-dependent
magnetization (refer to Figure 12 in the Appendix) of compounds with
$\alpha=0-0.22$, we observe that the magnitude of residual magnetization of
all compounds in the AF-I regime remains nearly equal at 150 K. However, the
AHR at 150 K exhibits significant variations among all the compounds (refer to
Figure 5(b)). These observations clearly indicate that the AHR observed in the
AF-I regime does not originate from the residual magnetization in these
compounds but the the non-vanishing Berry curvature caused by the existence of
Weyl points within these compounds.

The anomalous Hall conductivity (AHC), which is an intrinsic quantity, is
proportional to the AHR of the sample. The AHC ($\sigma^{A}_{xz}$) can be
determined using $\rho^{A}_{xz}$ using the relation:

$\displaystyle\sigma^{A}_{xz}\approx-\rho^{A}_{xz}/(\rho_{xx}\rho_{zz})$ (1)

The Hall resistivity for compounds with $\alpha=0-0.22$ was measured at
various temperatures, and the corresponding AHR was calculated. Using the AHR
values at different temperatures and the relation mentioned above, the
temperature dependence of the AHC ($\sigma^{A}_{xz}$) was determined for these
compounds, as depicted in Figure 6(a). It is noteworthy that the AHC remains
non-zero throughout the AF-I regime and becomes zero for $\alpha=0.18$ and
$\alpha=0.22$ below 110 K, which corresponds to the AF-II regime. This
observation indicates the existence of Weyl points in the entire AF-I regime,
which disappears in the AF-II regime. The vanishing AHC in the AF-II regime is
expected due to the underlying magnetic symmetry, as explained in Ref. [37],
suggesting a strong connection between the Weyl points and the magnetic
symmetry of the system.

In Figure 6(b), it is evident that the anomalous Hall conductivity (AHC) at
130 K and 200 K (in the AF-I regime) exhibits an exponential decay pattern as
the Fe doping fraction ($\alpha$) increases. The significant decrease in AHC
with small Fe doping indicates that the Weyl points can be substantially
modified by suitable dopants in a Weyl semimetal. Similar decrease in AHC with
increasing Mn concentration have also been reported for Mn3Ge and Mn3Sn
compounds [6, 22]. The strength of the Berry curvature in an ideal Weyl
semimetal, where the Weyl points are separated by $\Delta k$ in phase space,
can be described by the following equation [14, 45, 46]

$\displaystyle\sigma^{A}_{ij}=\frac{e^{2}}{2\pi h}\Delta k$ (2)

This implies that with an increase in Fe doping, the separation between a pair
of Weyl points decreases [14, 16]. However, since the anomalous Hall effect
(AHE) is also influenced by the relative position of the Weyl points with
respect to the Fermi surface [6], it is possible that the Weyl points move
away from the Fermi surface as Fe doping fraction increases in Mn3Ge. Similar
decrease in AHE due to increase in separation between Weyl points and Fermi
surface has been reported by Refs. [47, 22] as well. Therefore, it can be
concluded that the properties of Weyl points in Mn3Ge can be adjusted by
introducing Fe dopants. To accurately determine the precise changes in the
separation between Weyl points and their relative position to the Fermi
surface, a comprehensive theoretical calculation is required.

The Hall coefficient ($R_{H}$) can be determined by calculating the slope of
the Hall resistivity at high magnetic fields:
$R_{H}=\frac{\partial\rho_{H}}{\partial B}$. The values of $R_{H}$ for
$\alpha=0.04$ and $\alpha=0.18$ were calculated and are shown in Figure 6(c).
Interestingly, unlike Mn3Ge [20], the sign change in $R_{H}$ does not occur
below $T{{}_{\text{N1}}}$ (325 K) for $\alpha=0.04$ down to 4 K. Furthermore,
Figure 6(c) demonstrates that $R_{H}$ for $\alpha=0.18$ exhibits a sudden drop
near 110 K, similar to the behavior observed in temperature dependent
magnetization and resistivity of the same sample. This suggests a significant
increase in carrier concentration below $T_{{}_{\text{N2}}}$ as the Mn spins
flip towards the z axis.

Figure 7: LMC of single-crystal samples with $\alpha=0-0.22$. (a - d) LMC
along the x and z axis at 4 K and 150 K. (e, f) Variation of the LMC (at 1 T)
with the Fe doping fraction ($\alpha$).

###  IV.2 Magnetoconductance and planar Hall effect

The observation of AHE in the coplanar triangular antiferromagnetic (AF-I)
regime underpins the existence of Weyl points in this regime. In addition to
AHE, the chiral anomaly is also a well-known phenomenon in Weyl semimetals.
Therefore, to identify the characteristic signatures of the chiral anomaly
effect, we performed longitudinal magnetoconductivity, angular
magnetoconductivity, and planar Hall effect measurements on the Fe-doped Mn3Ge
compounds. We will discuss each of these measurements in detail below.

####  IV.2.1 Longitudinal magnetoconductivity (LMC)

The longitudinal magnetoresistivity (LMR) of compounds with different Fe
doping levels, represented by $\alpha=$ (0, 0.04, 0.10, 0.18, 0.22), was
measured over a temperature range starting above $T_{\text{N1}}$ and down to 4
K. LMR along the $i$ axis ($i=xz$ in our case) is denoted as
$\Delta\rho_{ii}$, and defined as:
$\Delta\rho_{ii}=\rho_{ii}(B)-\rho_{ii}(0)$, where $\rho_{ii}$ denotes the
longitudinal resistivity. By utilizing $\Delta\rho_{ii}$ and the conductivity
$\sigma_{ii}=1/\rho_{ii}$, the longitudinal magnetoconductivity (LMC), denoted
as $\Delta\sigma_{ii}$, can be calculated using the following relationship:

$\Delta\sigma_{ii}=\sigma_{ii}(B)-\sigma_{ii}(0)\approx\frac{-\Delta\rho_{ii}}{\rho^{2}_{ii}}$
(3)

LMC for various compounds, at different temperatures, was determined and shown
in Figure 7. The analysis of LMC is focused at 4 K and 150 K. For compounds
with $\alpha\leq 0.10$, both temperatures lie in the AF-I regime. However, for
compounds with $\alpha=(0.18,0.22)$, 4 K and 150 K correspond to the AF-II and
AF-I regimes, respectively. Since the presence of Weyl points is expected only
in the AF-I regime, the discussion of chiral anomaly-induced effects will be
limited to the samples and temperatures which belong to the AF-I regime. The
LMC in the AF-II regime ($\alpha=0.18,0.22$; $T=4$ K), where chiral anomaly-
induced effects are not expected, is addressed in the Appendix B.

Positive LMC was observed in compounds with $\alpha=(0-0.22)$, particularly
when a low magnetic field is applied in the AF-I regime. However, the origin
of this positive LMC is not easily discernible since it can arise from various
effects. The most likely factors contributing to the observed positive LMC
include the current jetting effect [48], magnetization [49], the unequal spin
density of states (sDOS) in halfmetallic/semimetallic compounds [50, 51, 52,
53, 54, 55], and the chiral anomaly effect [17]. Each of these possibilities
will be discussed in detail below.

Figure 8: $\theta$MC for $\alpha=0-0.22$ compounds with sample rotated from x
axis, towards the y axis as shown in (a). $\theta$MC for
$\alpha=(0,0.04,0.10,0.18,0.22)$, measured at 1 T, is compared at 4 K and 150
K in (a), and (b) respectively. (c) Evolution of magnitude of $\theta$MC
($\Delta\sigma_{xx}^{\text{MC}}$) with temperature and magnetic field. Here,
$\Delta\sigma_{xx}^{\text{MC}}$ signifies the magnitude of oscillation
determined by $\sigma(0$°$)-\sigma(90$°$)$, keeping the magnetic field and
temperature constant. (c) Inset: Evolution of $\theta$MC magnitude with Fe
doping fraction ($\alpha$), compared at 4 K, 150 K, and 200 K. (d) Evolution
of the magnitude of $\theta$MC with the magnetic field.

Current jetting effect: The influence of the current jetting effect was
investigated in both the AF-I and AF-II regimes of Fe-doped compounds, as
described in Figure 14 (Appendix B). Some of the data shown in Figure 7 (AF-I
regime) is shown (in raw form) in Figure 14 as well, to determine the effect
of current jetting in the AF-I regime. The raw data for $\alpha=0.18$ at 4 K,
which belongs to the AF-II regime, is also shown in the same figure to
determine the effect of current jetting in the AF-II regime. Our measurements
on samples with $\alpha=0.10$ and $\alpha=0.18$ did not reveal a significant
contribution of the current jetting effect to the observed LMR in AF-I and AF-
II regime. Consistent results were obtained from repeated measurements using
different sample pieces. These findings suggest that the current jetting
effect plays a negligible role in Fe-doped compounds, and the observed
electrical transport effects are intrinsic. Similar observation has been
reported for Mn3Ge as well [20].

Magnetization: The role of magnetization in the LMC of Mn3Ge has already been
dismissed in Ref. [20], as no correlation between the magnitude of LMC and
sample magnetization was observed. In Fe-doped compounds, the magnetization in
the AF-I regime remains similar and greater in magnitude compared to Mn3Ge.
However, the magnitude of LMC decreases significantly even with a small Fe
doping fraction of 4% in Mn3Ge. Furthermore, the magnitude of LMC at 1 T
continues to decrease significantly with an increase in Fe doping fraction
(Figure 7). Therefore, the role of magnetization in the observed LMC of Fe-
doped Mn3Ge compounds within the AF-I regime is considered negligible.

Unequal sDOS in semi/half-metallic compounds: According to Ref. [20], it is
important to consider the possibility of trivial magnetoconductance due to
unequal spin density of states (sDOS), near the Fermi surface, in semi/half-
metallic compounds, which lead to the unequal spin scattering, and may result
in positive LMC [50, 51, 52, 53, 54, 55, 56]. In the case of the Fe-doped
compounds with doping fractions $\alpha=0.04-0.22$ (Figure 4), it is
noteworthy that the resistivity along the x axis is semi/half-metallic within
the AF-I regime. Additionally, compounds with $\alpha=0.10-0.22$ exhibit
semi/half-metallic behavior along the z axis as well. The semi/half-metallic
nature of each compound can be confirmed through spin density of states (sDOS)
calculations.

Typically, metallic compounds exhibit negative LMC, while semi/half-metallic
compounds show positive LMC [55]. In the case of the Fe-doped compounds, they
exhibit semi/half-metallic resistivity along the x axis in the AF-I regime,
which justifies the observed positive LMC (Figure 7(a, c)). However, compounds
with $\alpha=0$ and $\alpha=0.04$, which are metallic along the z axis, also
display positive LMC along the same axis at both 4 K and 150 K (Figure 7(b,
d)). Consequently, the role of unequal sDOS alone cannot satisfactorily
explain the observed positive LMC within the AF-I regime. Therefore, it can be
concluded that the role of unequal sDOS in explaining the observed positive
LMC within the AF-I regime is not sufficient and requires further
investigation and consideration of other factors.

Chiral anomaly effect: Figure 7(a-d) provides clear evidence of a positive LMC
signature along both the x and z axes in all Fe-doped compounds within the
AF-I regime. The decrease in LMC along the x axis for $\alpha=0$, beyond 1.5 T
at 4 K and 150 K, can be attributed to the semi/half-metallic nature of Mn3Ge
[20]. However, this decrease is not observed beyond 4% Fe doping, indicating
that impurities can affect the spin density of states (sDOS) near the Fermi
surface, which typically leads to a negative slope in LMC for metallic
systems.

Figure 9: PHE for different compounds ($\alpha=0-0.22$) at 4 K and 150 K is
shown in (a) and (b), respectively. The measurements were performed at 1 T of
the applied magnetic field. The setup for the PHE measurement in the $x-y$
plane is illustrated in (a). (c, d) Variation of PHE magnitude
($\Delta\sigma_{xx}^{\text{Pl. mag.}}$) with temperature and magnetic field.
In this case, $\Delta\sigma_{xx}^{\text{Pl. mag.}}$ is determined by the data
fitting using the Eqn. 5. (c) Inset: Evolution of magnitude of PHE oscillation
($\Delta\sigma_{xx}^{\text{Pl. mag.}}$) with $\alpha$ (doping fraction).

Positive LMC induced by the chiral anomaly has been reported previously in
Mn3Ge and Mn3Sn [6, 4, 15]. Given that the doped compounds in the AF-I regime
also exhibit positive and increasing LMC, it is plausible to suggest that the
origin of the positive LMC within the range of $\alpha=0-0.22$ in the AF-I
regime is the chiral anomaly effect.

Figure 7(e, f) presents a comparison of the LMC evolution at 1 T with varying
Fe doping fractions ($\alpha$). It is evident that the LMC along the x and z
axes experience a sharp decrease as Fe doping increases at 4 K. Similarly,
even with a 4% Fe doping, the LMC exhibits a significant reduction at 150 K.
This abrupt decline in positive LMC, despite minor impurity doping, can be
attributed to the chiral anomaly-induced LMC phenomenon. Previous studies [4,
6] have suggested that the position of Weyl points relative to the chemical
potential ($\mu$) is sensitive to impurities. Furthermore, according to Refs.
[17, 18], the chiral anomaly-induced LMC ($\Delta\sigma_{\text{chiral}}$) is
inversely proportional to $\mu^{2}$, expressed as
$\Delta\sigma_{\text{chiral}}\propto\frac{1}{\mu^{2}}$. Thus, the sudden
decrease in LMC at 1 T following a small amount of Fe doping implies that the
LMC in doped compounds may originate from the chiral anomaly effect.

It is worth noting that positive LMC is also observed in the AF-II regime, at
low field, as depicted in Figure 15 in the Appendix. However, the chiral
anomaly-induced LMC is not expected to be present in this region because of
the lack of evidence of Weyl points in the AF-II regime [37]. Therefore, the
presence of the chiral anomaly effect cannot be solely justified based on the
positive LMC observed in the AF-I regime, and further investigations are
required to justify the evidence of the chiral anomaly effect in Fe doped
compounds.

####  IV.2.2 Angular magnetoconductivity ($\theta$MC)

The detection of anisotropy in angular magnetoconductivity ($\theta$MC)
measurements is a prominent indication of the chiral anomaly effect, as
discussed in Refs. [16] and [18]. Therefore, we carried out $\theta$MC
measurements on compounds doped with $\alpha=0-0.22$. As mentioned in Refs.
[37, 20, 3], the magnetization along the x and y axes remains nearly the same
for parent and 22% Fe doped Mn3Ge compounds, whereas, it differs significantly
along the z axes (section III). Along with the chiral anomaly effect, the
anisotropy in magnetization within a particular plane can also lead to the
observation of anisotropic magnetoconductivity if measured within the same
plane. Consequently, to eliminate the contribution of magnetoconductivity
arising from the difference in magnetization along the x and z axes, we
performed the $\theta$MC measurements exclusively within the x-y plane. This
choice was made due to the similar magnitude and behavior of magnetization
observed along the x and y axes.

The setup for the $\theta$MC measurement is illustrated in Figure 8(a), where
the longitudinal magnetoconductivity of the sample was measured along the x
axis while rotating the sample in the x-y crystallographic plane. The x axis
of the sample was oriented at an angle $\theta$ with respect to the applied
magnetic field. Here, $\theta=0$ represents the longitudinal
magnetoconductance (LMR), while $\theta=90^{\circ}$ corresponds to the
magnetoconductance where the applied magnetic field is perpendicular to the
electric current direction.

The $\theta$MC measurements were performed on the compounds with Fe doping
fractions $\alpha=0-0.22$ at constant temperatures and magnetic fields. Figure
8(a, b) shows the observed angular anisotropy in $\theta$MC for most of the
compounds. The $\theta$MC values at 1 T for different compounds are compared
at 4 K and 150 K. At 4 K, angular anisotropy in $\theta$MC was observed only
for $\alpha=0,0.04,0.10$ compounds. In contrast, at 150 K, significant angular
anisotropy in $\theta$MC was observed for all the compounds within the range
of $\alpha=0-0.22$.

For compounds with a small LMC
$(\Delta\sigma_{xx}(B)=(\sigma_{xx}(B)-\sigma_{xx}(0))<<\sigma_{xx}(0)))$, the
chiral anomaly-induced $\theta$MC can be described by the following equation
[57, 16, 18]:

$\displaystyle\sigma_{xx}(\theta)-\sigma_{xx}(\perp)\approx\Delta\sigma_{xx}^{\text{MC}}\text{cos}^{2}\theta$
$\displaystyle\Delta\sigma_{xx}^{\text{MC}}=(\sigma_{xx}^{\text{MC}}(\parallel)-\sigma_{xx}^{\text{MC}}(\perp))$
(4)

Where, $\Delta\sigma_{xx}^{\text{MC}}$ correspond to the magnitude of angular
variation of $\theta$MC. Ideally, at constant temperature,
$\Delta\sigma_{xx}^{\text{MC}}(B)=(\sigma_{xx}(B)_{\theta=0^{o}}-\sigma_{xx}(0)_{\theta=0^{o}})$.
However, it may differ due to the presence of a transverse
magnetoconductivity, which is generally non-zero in most cases.

From Figure 8(a, b), it can be observed that the $\theta$MC for almost all the
compounds fits well with Equation 4. Using this fitting,
$\Delta\sigma_{xx}^{\text{MC}}$ was determined for all the compounds at
various temperatures and compared in Figure 8(c). Notably, the non-zero
magnitude of $\theta$MC is observed only in the temperature and doping regime
where the magnetic structure is similar to Mn3Ge, i.e., in the AF-I regime.
$\theta$MC becomes negligible for $\alpha=(0.18,0.22)$ below approximately 110
K, which is close to the $T_{\text{N2}}$ of the corresponding compound.

The evolution of $\Delta\sigma_{xx}^{\text{MC}}$, at constant temperatures,
with Fe doping fraction is also shown in the inset of Figure 8(c), revealing a
significant decrease in $\Delta\sigma_{xx}^{\text{MC}}$ with Fe doping,
similar to the LMC of the same compound (Figure 7(e, f)). This suggests that
the origin of angular anisotropy in $\theta$MC is likely to be the chiral
anomaly effect, which is possibly responsible for the positive LMC observed in
these compounds in the AF-I regime. Additionally, it was observed that
$\Delta\sigma_{xx}^{\text{MC}}$ for $\alpha\geq 0.04$ increases almost
linearly with the increase in magnetic field (Figure 8(d)), which is expected
in the case of chiral anomaly effects in type-II Weyl semimetals [5]. This
suggests that the Fe-doped compounds may also host type-II Weyl points,
similar to Mn3Ge, within the AF-I regime [7]. The decrease in
$\Delta\sigma_{xx}^{\text{MC}}$ with magnetic field for $\alpha=0$ has been
explained in Ref. [20].

####  IV.2.3 Planar Hall Effect (PHE)

The Planar Hall conductivity, also known as the Planar Hall effect (PHE), was
measured for the compounds with Fe doping fractions $\alpha=0-0.22$ by
rotating the magnetic field within the x-y plane, as depicted in Figure 9(a).
The measurements were performed within this plane for the same reasons
mentioned earlier for the $\theta$MC measurements.

PHE measurements were conducted at 4 K and 150 K with an applied magnetic
field of 1 T. The PHE data for various compounds are presented in Figure 9(a,
b). Similar to the $\theta$MC results, PHE oscillations with a 180∘
periodicity are observed at 4 K for $\alpha=0-0.10$, but they disappear for
higher doping fractions. However, at 150 K, PHE oscillations are observed for
all the compounds within the measured range of $\alpha=(0-0.22)$.

The angular dependence of chiral anomaly-induced PHE, at a given magnetic
field and temperature, follows the relation [18]:

$\displaystyle\sigma_{xy}^{\text{PHE}}(\theta)\approx\Delta\sigma_{xx}^{\text{Pl.
mag.}}[\sin\theta\cos\theta]$ (5)

Here, $\Delta\sigma_{xx}^{\text{Pl. mag.}}$ represents twice the magnitude of
the PHE oscillations. Ideally, at a constant temperature,
$\Delta\sigma_{xx}^{\text{Pl.
mag.}}(B)=(\sigma_{xx}(B)_{\theta=0}-\sigma_{xx}(0)_{\theta=0})$, similar to
the magnitude of the $\theta$MC.

As shown in Figure 9(a, b), the PHE data fit reasonably well with Equation 5.
Furthermore, the evolution of $\Delta\sigma_{xx}^{\text{Pl. mag.}}$ for all
the doped compounds at 1 T is analyzed in Figure 9(c). It is interesting to
observe that $\Delta\sigma_{xx}^{\text{Pl. mag.}}$ varies with temperature and
doping fraction ($\alpha$) in a similar manner as observed in the case of
$\theta$MC (compare Figure 8(c) and Figure 9(c)). Once again, it is evident
that the PHE is present only in the AF-I regime. Additionally, the PHE
significantly weakens with an increase in Fe doping, similar to the
observations in LMC and $\theta$MC. Moreover, as depicted in Figure 9(d),
$\Delta\sigma_{xx}^{\text{Pl. mag.}}$ shows a roughly linear increase with the
magnetic field at 150 K, which aligns with the field dependence of $\theta$MC
(Figure 8(d)).

Analysis: The similarity in the magnitude of the PHE
$(\Delta\sigma_{xx}^{\text{Pl. mag.}}(B,T,\alpha))$ and the magnitude of the
LMC $(\Delta\sigma_{xx}^{\text{MC}}(B,T,\alpha))$ can be attributed to their
common origin from the chiral anomaly effect [57, 5, 18, 17, 16]. The
observation of both $\theta$MC and PHE only within the AF-I regime suggests
that their possible origin is the chiral anomaly effect. The significant
decrease in the magnitude of $\theta$MC, PHE, and LMC with Fe doping further
supports the role of the chiral anomaly in the observed behavior of $\theta$MC
and PHE. This weakening of the effects also implies that the Weyl points move
significantly further from the Fermi surface as the Fe doping fraction
increases in Mn3Ge.

##  V Conclusion

Extensive studies were conducted on the magnetization behavior of both the
parent Mn3Ge compound and Fe-doped Mn3Ge compounds. The results showed that
compounds with Fe doping fractions up to $\alpha=0.26$ exhibit magnetization
and magnetic structure similar to Mn3Ge, within the AF-I temperature regime.
Notably, intermediate Fe-doped compounds displayed magnetization behavior that
differed significantly from that of Mn3Ge.

Further investigations focused on the electrical transport properties of
compounds with Fe doping fractions ranging from $\alpha=0$ to $\alpha=0.22$.
The AHE was found exclusively within the AF-I regime for all these compounds.
Signatures of Weyl points were observed in a significantly large fraction of
Fe doping, which highlights the robust and topologically protected nature of
Weyl points in doped compounds. The AHE weakened considerably with increasing
Fe doping concentration, indicating changes in the separation of Weyl point
pairs and the position of Weyl points relative to the chemical potential as
the Fe doping levels increased [58, 14].

Furthermore, positive LMC, non-zero $\theta$MC, and PHE were observed in the
AF-I regime of the doped compounds. Analysis suggested that these effects
likely originated from the chiral anomaly phenomenon. However, additional
experimental and theoretical investigations are necessary to precisely
determine the contributions of the chiral anomaly and other effects in the
observed electrical transport features within the AF-I regime. The weakening
of LMC, $\theta$MC, and PHE with increasing Fe doping implied that the Weyl
points moved further away from the chemical potential as the Fe doping
fraction in Mn3Ge increased [58]. In conclusion, appropriate doping in the
parent Weyl semimetallic system can significantly control the characteristics
of Weyl points.

It is noteworthy that the AHE and all other effects, which can be associated
with the chiral anomaly, completely vanish in the AF-II regime, which
corresponds to a collinear antiferromagnetic magnetic structure, as shown in
the phase diagram. This clear absence of Weyl points in the AF-II regime
indicates that the existence of Weyl points in Fe-doped Mn3Ge compounds, as
well as similar compounds, is dictated by the magnetic symmetry of the
compound.

##  APPENDIX A SAMPLE CHARACTERIZATION

### X-ray diffraction and magnetization

Figure 10: (a-f) Room temperature X-ray powder diffraction (XRPD) of
$\alpha=0-0.30$ compounds. The arrows signify unknown impurity peaks. The
goodness of the fitting parameter, $\chi^{2}$, for each compound is mentioned
in each plot. Obs., cal., implies the observed and calculated XRPD pattern.
Diff. shows the difference between observed and calculated intensity. Hex. and
Tet. denote hexagonal and tetragonal phases, respectively. (g) Variation of
lattice parameters with the doping fraction ($\alpha$). Inset: Variation of
the lattice volume with $\alpha$. Figure 11: Laue diffraction pattern
corresponding to the X-ray beam along the (a) z $[0001]$ axis and (b) y
$[01\bar{1}0]$ axis, respectively, of the single-crystal $\alpha=0.22$.
Figure 12: (a-f) Field dependent magnetization of the single-crystal (sc)
compounds with $\alpha=0,0.10,0.22$ in the case of B applied along the x and z
axes. (g, h) Field dependent magnetization of the polycrystalline (poly)
$\alpha=0.26,0.30$ compounds.

X-ray powder diffraction of different compounds is shown in Figure 10(a-f),
and corresponding lattice parameters are shown in Figure 10(g). Along with the
hexagonal phase, a small amount of the tetragonal (impurity) phase ($2-4\%$)
was also observed in $\alpha=0,0.04$ compounds. The compound with
$\alpha=0.30$ contains two tiny impurity peaks, which could not be fitted with
the space group symmetry - $P6_{3}/mmc$.

Laue diffraction of the $\alpha=0.22$ compound corresponding to X-ray beam
incident along the z and y axes are shown in Figure 11. 6-fold diffraction
spots (in Figure 11(a)) implies that the single-crystal synthesized in a
hexagonal phase.

Field dependent magnetization, M(H), of a few selected compounds is shown in
Figure 12. Since the M(H) for $\alpha=0.04$ and $\alpha=0.18$ is similar to
the M(H) for $\alpha=0$ and $\alpha=0.22$, respectively, the M(H) for
$\alpha=(0.04,0.18)$ is not shown.

##  APPENDIX B ELECTRICAL TRANSPORT

Figure 13: Low field Hall resistivity of different compounds (mentioned in the
plot) at 150 K. Figure 14: LMC (raw) data for $\alpha=0.10,0.18$ samples. The
measurements were performed with different voltage contacts as shown in (a,
b). In (a, b), sky blue circles enclose the contacts at the ‘center’ of the
sample, and the rectangular (orange) region encloses the contacts at the
‘edge’ of the sample. (c), and (d) show LMC along the x axis at 4 K and 130 K,
respectively. At both temperatures, LMC was measured by the ‘edge’, and
‘center’ voltage contacts (shown in (a, b)). Figure 15: LMC of the compounds
with $\alpha=(0.18,0.22)$ in AF-II regime (4 K). Figure 16: (a) Combined plot
for the Hall resistivity and magnetization of the $\alpha=0.30$
(polycrystalline) sample, at $T=100$ K. ‘Mag’ and ‘Hall’ represent the
magnetization and Hall resistivity, respectively. (b) LMC (B$\parallel$I) for
the $\alpha=0.30$ (polycrystalline) compound. 100 K and 300 K data represent
LMC below and above the FM ordering temperature ($T_{\text{c}}$ $\approx$ 200
K), respectively.

Low field Hall resistivity: Low field Hall resistivity of $\alpha=0-0.22$, at
150 K, is shown in Figure 13. It can be observed that the Hall hysteresis
remains below 0.02 T for all the compounds. The Hall resistivity shows
saturation with the application of just 0.02 T of the magnetic field.

Current jetting effect in the LMC measurement: The current jetting effect in
the LMC of the Fe doped ($\alpha=0.10,0.18$) compounds was measured. We
mention that compounds with $\alpha=0.10$ and $\alpha=0.18$ lie in the AF-I at
130 K and 150 K. Therefore, the behavior of LMC at 150 K (shown in Figure 7 is
same as the behavior of LMC at 130 K shown in Figure 14. To determine the
effect of current jetting, the two different voltage contacts (near edge, and
center) were prepared on each sample, as shown in Figure 14(a, b). The LMC of
the sample, with the magnetic field applied along the current direction (x
axis), was measured corresponding to both the contacts, as shown in Figure
14(c, d). It can be observed that the LMC at 4 K remains very similar, even if
one of the voltage contacts is off-centered. A similar observation was made at
130 K as well. This justifies that similar to the parent compound, current
jetting does not have a significant effect in the Fe doped Mn3Ge compounds (up
to $\alpha=0.22$).

LMC in the AF-II regime: At 4 K, compounds with $\alpha=(0.18,0.22)$
correspond to the AF-II regime, where AHE vanishes. Also, we have observed in
the main text that the amplitude of $\theta$MC oscillation (at 4 K) for
$\alpha=0.18,0.22$ compounds is negligible. This implies that the conductivity
in these samples, at 4 K, is independent of the direction of the magnetic
field relative to the electric field (current). These observations clearly
suggest that the chiral anomaly effect and Weyl points are not present in the
AF-II regime of the Fe doped compounds. As shown in Figure 15, LMC along both
the x and z axes initially increases with an increase in the magnetic field.
However, LMC starts to decrease at higher fields. Since chiral anomaly is not
expected at this temperature (in this compound), the low field increasing LMC
might originate due to the weak localization in the compound [59, 60].
However, further studies are required to determined its true origin.

AHE and LMC in the FM regime ($\alpha=0.30$): It is important to note that
positive LMC and large AHE have been observed in the case of polycrystalline
$\alpha=0.30$ compound as well (Figure 16). Since the compound with
$\alpha=0.30$ is FM [34], the observation of positive magnetoconductivity and
AHE is expected [43, 49]. In FM compounds, AHE originates from the residual
magnetization of the sample [61, 43]. We have plotted field dependent
magnetization and Hall resistivity together in Figure 16(a), where it can be
observed that the Hall resistivity almost scales with the magnetization of the
compound. This justifies that the AHE observed in this compound originated
from the residual magnetization of the sample. Moreover, positive LMC is very
common in ferromagnetic materials. Therefore, the observed negative LMC in
Figure 16(b), at 100 K, is most likely driven by the magnetization of the
sample [49, 55].

## References

  * Ando [2013] Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).
  * Hasan et al. [2010] M. Z. Hasan, C. L. Kane, et al., Rev. Mod. Phys. 82, 3045 (2010).
  * Nayak et al. [2016] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. Kübler, et al., Sci. Adv. 2, e1501870 (2016).
  * Kuroda et al. [2017] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S. Akebi, et al., Nat. Mater. 16, 1090 (2017).
  * Nandy et al. [2017] S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Phys. Rev. Lett. 119, 176804 (2017).
  * Chen et al. [2021] T. Chen, T. Tomita, S. Minami, M. Fu, T. Koretsune, M. Kitatani, I. Muhammad, D. Nishio-Hamane, R. Ishii, F. Ishii, et al., Nat. Commun. 12, 1 (2021).
  * Yang et al. [2017] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and B. Yan, New J. Phys. 19, 015008 (2017).
  * Higo et al. [2018] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune, et al., Nat. Photonics 12, 73 (2018).
  * Ikhlas et al. [2017] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki, D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji, Nat. Phys. 13, 1085 (2017).
  * Wuttke et al. [2019] C. Wuttke, F. Caglieris, S. Sykora, F. Scaravaggi, A. U. Wolter, K. Manna, V. Süss, C. Shekhar, C. Felser, B. Büchner, et al., Phys. Rev. B 100, 085111 (2019).
  * Tsai et al. [2020] H. Tsai, T. Higo, K. Kondou, T. Nomoto, A. Sakai, A. Kobayashi, T. Nakano, K. Yakushiji, R. Arita, S. Miwa, et al., Nature 580, 608 (2020).
  * Jungwirth et al. [2016] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016).
  * Jeon et al. [2021] K.-R. Jeon, B. K. Hazra, K. Cho, A. Chakraborty, J.-C. Jeon, H. Han, H. L. Meyerheim, T. Kontos, and S. S. Parkin, Nat. Mater. 20, 1358 (2021).
  * Šmejkal et al. [2018] L. Šmejkal, Y. Mokrousov, B. Yan, and A. H. MacDonald, Nat. Phys. 14, 242 (2018).
  * Wu et al. [2023] M. Wu, K. Kondou, T. Chen, S. Nakatsuji, and Y. Otani, AIP Adv. 13, 045102 (2023).
  * Burkov [2014a] A. Burkov, Phys. Rev. Lett. 113, 247203 (2014a).
  * Burkov and Kim [2016] A. A. Burkov and Y. B. Kim, Phys. Rev. Lett. 117, 136602 (2016).
  * Burkov [2017] A. Burkov, Phys. Rev. B 96, 041110 (2017).
  * Soh et al. [2020] J.-R. Soh, F. de Juan, N. Qureshi, H. Jacobsen, H.-Y. Wang, Y.-F. Guo, and A. Boothroyd, Phys. Rev. B 101, 140411 (2020).
  * Rai et al. [2022] V. Rai, S. Jana, M. Meven, R. Dutta, J. Perßon, and S. Nandi, Phys. Rev. B 106, 195114 (2022).
  * Wu et al. [2020] M. Wu, H. Isshiki, T. Chen, T. Higo, S. Nakatsuji, and Y. Otani, Appl. Phys. Lett. 116, 132408 (2020).
  * Kiyohara et al. [2016] N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl. 5, 064009 (2016).
  * Liu et al. [2020] Z. Liu, Q. Zhang, Y. Zhang, H. Zhang, X. Ma, and E. Liu, J. Phys.: Condens. Matter. 33, 115803 (2020).
  * Zhou et al. [2020] H. Zhou, G. Chang, G. Wang, X. Gui, X. Xu, J.-X. Yin, Z. Guguchia, S. S. Zhang, T.-R. Chang, H. Lin, et al., Phys. Rev. B 101, 125121 (2020).
  * Sukhanov et al. [2018] A. Sukhanov, S. Singh, L. Caron, T. Hansen, A. Hoser, V. Kumar, H. Borrmann, A. Fitch, P. Devi, K. Manna, et al., Phys. Rev. B 97, 214402 (2018).
  * Dos Reis et al. [2020] R. Dos Reis, M. G. Zavareh, M. Ajeesh, L. Kutelak, A. Sukhanov, S. Singh, J. Noky, Y. Sun, J. Fischer, K. Manna, et al., Phys. Rev. Mater. 4, 051401 (2020).
  * Thakur et al. [2020] G. S. Thakur, P. Vir, S. N. Guin, C. Shekhar, R. Weihrich, Y. Sun, N. Kumar, and C. Felser, Chem. Mater. 32, 1612 (2020).
  * Ghimire et al. [2019] M. P. Ghimire, J. I. Facio, J.-S. You, L. Ye, J. G. Checkelsky, S. Fang, E. Kaxiras, M. Richter, and J. Van Den Brink, Phys. Rev. Res. 1, 032044 (2019).
  * Hu et al. [2022] Y. Hu, C. Yue, D. Yuan, J. Gao, Z. Huang, Z. Fang, C. Fang, H. Weng, and W. Zhang, Sci. China: Phys. Mech. Astron. 65, 297211 (2022).
  * Ikhlas et al. [2022] M. Ikhlas, S. Dasgupta, F. Theuss, T. Higo, S. Kittaka, B. Ramshaw, O. Tchernyshyov, C. Hicks, and S. Nakatsuji, Nat. Phys. 18, 1086 (2022).
  * Low et al. [2022] A. Low, S. Ghosh, S. Changdar, S. Routh, S. Purwar, and S. Thirupathaiah, Phys. Rev. B 106, 144429 (2022).
  * Lecocq et al. [1963] Y. Lecocq, P. Locecq, and A. Michel, Compt. Rend. 256, 4913 (1963).
  * Kanematsu [1967] K. Kanematsu, Propriétés Thermodynamiques Physiques et Structurales des Derivés Semi-Metalliques (CNRS, Paris) p. 135 (1967).
  * Hori et al. [1992] T. Hori, Y. Yamaguchi, and Y. Nakagawa, J. Magn. Magn. Mater. 104, 2045 (1992).
  * Hori et al. [1995] T. Hori, H. Niida, H. Kato, Y. Yamaguchi, and Y. Nakagawa, Phys. B: Condens. Matter 211, 93 (1995).
  * Niida et al. [1995] H. Niida, T. Hori, and Y. Nakagawa, J. Magn. Magn. Mater. 140, 129 (1995).
  * Rai et al. [2023] V. Rai, A. Stunault, W. Schmidt, S. Jana, J. Perßon, J.-R. Soh, T. Brückel, and S. Nandi, Phys. Rev. B 107, 184413 (2023).
  * Ghosh et al. [2023] S. Ghosh, A. Low, S. Gorai, K. Mandal, and S. Thirupathaiah, J. Phys.: Condens. Matter 35, 485701 (2023).
  * Berche et al. [2014] A. Berche, J.-C. Tedenac, and P. Jund, Intermetallics 47, 23 (2014).
  * Gokhale and Abbaschian [1990] A. Gokhale and R. Abbaschian, J. Ph. Equilibria Diffus. 11, 460 (1990).
  * Chen et al. [2020] Y. Chen, J. Gaudet, S. Dasgupta, G. Marcus, J. Lin, T. Chen, T. Tomita, M. Ikhlas, Y. Zhao, W. Chen, et al., Phys. Rev. B 102, 054403 (2020).
  * Denton and Ashcroft [1991] A. R. Denton and N. W. Ashcroft, Phys. Rev. A 43, 3161 (1991).
  * Sales et al. [2006] B. C. Sales, R. Jin, D. Mandrus, and P. Khalifah, Phys. Rev. B 73, 224435 (2006).
  * Nakatsuji et al. [2015] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
  * Burkov [2014b] A. Burkov, Phys. Rev. Lett. 113, 187202 (2014b).
  * Soh et al. [2019] J.-R. Soh, F. De Juan, M. Vergniory, N. Schröter, M. Rahn, D. Yan, J. Jiang, M. Bristow, P. Reiss, J. Blandy, et al., Phys. Rev. B 100, 201102 (2019).
  * Chen et al. [2014] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * Liang et al. [2018] S. Liang, J. Lin, S. Kushwaha, J. Xing, N. Ni, R. J. Cava, and N. P. Ong, Phys. Rev. X 8, 031002 (2018).
  * Ritchie et al. [2003] L. Ritchie, G. Xiao, Y. Ji, T. Chen, C. Chien, M. Zhang, J. Chen, Z. Liu, G. Wu, and X. Zhang, Phys. Rev. B 68, 104430 (2003).
  * Yang et al. [2012] F. Yang, Y. Sakuraba, S. Kokado, Y. Kota, A. Sakuma, and K. Takanashi, Phys. Rev. B 86, 020409 (2012).
  * Roth et al. [1963] H. Roth, W. Straub, W. Bernard, and J. Mulhern Jr, Phys. Rev. Lett. 11, 328 (1963).
  * Furukawa [1962] Y. Furukawa, J. Phys. Soc. Jpn. 17, 630 (1962).
  * Kawabata [1980] A. Kawabata, J. Phys. Soc. Jpn. 49, 628 (1980).
  * Endo et al. [1999] T. Endo, H. Kubota, and T. Miyazaki, J. Magn. Soc. Jpn. 23, 1129 (1999).
  * Kokado et al. [2012] S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J. Phys. Soc. Jpn. 81, 024705 (2012).
  * Bhattacharyya et al. [2019] B. Bhattacharyya, B. Singh, R. Aloysius, R. Yadav, C. Su, H. Lin, S. Auluck, A. Gupta, T. Senguttuvan, and S. Husale, Sci. Rep. 9, 7836 (2019).
  * Deng et al. [2020] M.-X. Deng, J.-Y. Ba, R. Ma, W. Luo, R.-Q. Wang, L. Sheng, and D. Xing, Phys. Rev. Res. 2, 033346 (2020).
  * Son and Spivak [2013] D. Son and B. Spivak, Phys. Rev. B 88, 104412 (2013).
  * Hikami et al. [1980] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980).
  * Lu and Shen [2014] H.-Z. Lu and S.-Q. Shen, Proc. SPIE 9167, 91672E (2014).
  * Armitage et al. [2018] N. Armitage, E. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).