# Anisotropic Nonsaturating Magnetoresistance Observed in HoMn6Ge6: A Kagome
Dirac Semimetal

Achintya Low  Tushar Kanti Bhowmik  Susanta Ghosh  Setti Thirupathaiah
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Material
Sciences, S. N. Bose National Centre for Basic Sciences, Kolkata-700106

###### Abstract

We report the magnetic and magnetotransport properties and electronic band
structure of the kagome Dirac semimetal HoMn6Ge6. Temperature-dependent
electrical resistivity demonstrates various magnetic-transition-driven
anomalies. Notably, a crossover from negative to positive magnetoresistance
(MR) is observed at around 150 K. While the linear nonsaturating positive MR
in the low-temperature region is mainly driven by the linear Dirac-like band
dispersions as predicted by the first-principles calculations, the negative MR
observed in the high-temperature region is due to the spin-flop type magnetic
transition. Consistent with anisotropic Fermi surface topology, we observe
anisotropic magnetoresistance at low temperatures. A significant anomalous
Hall effect has been noticed at high temperatures in addition to a switching
of the dominant charge carrier from electron to hole at around 215 K.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

The combination of frustrated magnetism, strong electronic correlations, and
quantum topology in the kogome systems immensely attracts researchers from the
fundamental science and potential technological applications points of view
[1, 2, 3, 4, 5, 6, 7, 8]. Particularly, the interplay among these peculiar
physical properties in the kagome lattice features intriguing quantum
phenomena, including the quantum spin liquid phase [9, 10, 11, 12], flat bands
[13, 14, 15, 16], Dirac and Weyl fermions [17, 18, 19, 20, 21, 15], Chern
quantum phase [22, 23], and Skyrmionic lattice [24, 25]. So far, the
transition metal-based kagome systems have been extensively explored to find
the Weyl fermions in Mn3Sn(Ge) [18, 26], giant anomalous Hall effect in
Co3Sn2S2 [19], Nernst effect in Fe3Sn [27], Skyrmion lattice in Fe3Sn2 [28].
On the other hand, the recent discovery of a nearly ideal quantum limit Chern
magnet with massive Dirac fermion in TbMn6Sn6 has ignited an intense search
for rare-earth-based kagome systems [23].

Following this discovery, several experimental reports appeared discussing the
tuning of topological properties in RMn6Sn6 (R = Gd-Tm, Lu) [29], large
anomalous Hall effect in RMn6Sn6 (R = Tb, Dy, and Ho) [30], Chern magnetic
state in TbMn6Sn6 [23], Dirac-like band crossings in HoMn6Sn6 [31], linear
unsaturated magnetoresistance in RMn6Sn6 (R = Gd, Tm, Lu) [29]. Among these,
HoMn6Sn6 is a ferrimagnetic metal in which the spins of the Ho-$4f$ sub-
lattice align antiferromagnetically with the spins of the Mn-$3d$ sub-lattice.
Thus, the interplay between the ferrimagnetism and the electronic band
structure plays a vital role in shaping the linear non-saturated
magnetoresistance of this system [32, 29, 31]. On the other hand, though
HoMn6Ge6 shares a similar crystal structure to HoMn6Sn6, minimal information
is available on the magnetotransport properties of HoMn6Ge6 discussing in
detail the relation between magnetic and magnetotransport properties [33].

HoMn6Ge6 crystallizes into HfFe6Ge6-type hexagonal structure with a space
group of $P6/mmm$, ordering antiferromagnetically at a N$\acute{e}$el
temperature of 466 K [34]. The magnetic moments of Mn and Ho are arranged in a
skew-spiral ($\widetilde{SS}$) fashion, and the plane of magnetic moments (Ho
and Mo) is angled at 60o with the $z$-axis in HoMn6Ge6 [35]. In contrast, in
$R$Mn6Sn6 systems, the moments of $R$ and Mn are aligned antiparallel to form
a ferrimagnetic structure with an easy-axis of magnetization angled with the
$z$-axis [23, 30, 29, 31]. The angle between the $z$-axis and the easy-axis of
magnetization strongly depends on the rare-earth element [35, 23, 30, 29, 31,
36]. Further, as discussed in Ref. [35], the magnetic structure of these
systems is very sensitive to the sample temperature. For instance, in the low-
temperature region ($<$ 55 K), the Ho and Mn magnetic moments form a skew-
spiral ($\widetilde{SS}$) structure, coupled antiferromagnetically, with a
plane of moments making an angle $\theta_{S}$ with the $z$-axis, producing
finite net magnetization along both the in-plane and out-of-plane directions.
However, as the temperature is increased, the skew-spiral structure gets
distorted, and the other magnetic transitions emerge at 220 and 260 K. Beyond
260 K, the $\widetilde{SS}$ structure completely gets destroyed, and the
system enters a cycloid state beyond 300 K [35].

Figure 1:  (a) The powder XRD pattern taken on the crushed single crystals of
HoMn6Ge6 overlapped with Rietveld refinement. The left inset in (a) shows an
optical image of as-grown single crystals. The right inset in (a) represents
the XRD pattern corresponding to the (000l) Bragg plane of the HoMn6Ge6 single
crystal. (b) EDXS data taken on HoMn6Ge6 single crystal. Insets in (b) show
the elemental mapping of Ho, Mn, and Ge atoms. (c) Schematic representation of
the primitive unit cell of HoMn6Ge6. (d) Kagome lattice plane formed by the Mn
atoms when projected onto the $ab$-plane.

In this work, high-quality single crystals of HoMn6Ge6 were grown using the Sn
flux to study the magnetic and magnetotransport properties. Electrical
resistivity demonstrates an overall metallic nature throughout the measured
temperature range with a few magnetic transition-driven anomalies. A crossover
from negative to positive magnetoresistance (MR) is observed at a critical
temperature of 150 K. While the Dirac-like linear band dispersion mainly
drives the linear nonsaturating positive MR in the low-temperature region, the
negative MR observed in the higher temperature region is due to the spin-flop
type magnetic transition. Most importantly, we identify a large anisotropy in
the magnetoresistance due to the anisotropic Fermi surface present in this
system. Further, we observe anomalous Hall effect in addition to a switching
of dominant charge carriers from electrons to holes at around 215 K, ingoing
from high to low temperatures. Also, we performed density functional theory
calculations on HoMn6Ge6 by considering the experimentally obtained skew-
spiral magnetic structure to understand our magnetotransport data better [35].

##  II Experimental and First-principles Calculation Details

Single crystals of HoMn6Ge6 were grown using Sn flux. Firstly, Holmium (Ho)
chunk (Alfa Aesar, $99.9\%$), Manganese (Mn) powder (Alfa Aesar, $99.95\%$),
Germanium (Ge) powder (Alfa Aesar, $99.99\%$), and Tin (Sn) shots (Alfa Aesar,
$99.995\%$) were weighed in the ratio of $1:6:6:24$. The mixture was kept in
an alumina crucible and sealed the crucible together with the mixture inside
an evacuated quartz ampoule. The mixture was heated up to 1000oC at a rate of
50o/hr and kept there for another 18 hrs before cooling the molten mixture
down to 600oC at a rate of 5oC/hr. Finally, the molten is annealed for another
two days at 600oC. Then, the ampoule was quickly transferred to a centrifuge
to separate the Sn flux from the samples. In this way, we obtained several
hexagonal-shaped HoMn6Ge6 single crystals with a typical size of 2mm $\times$
2mm $\times$ 0.4mm as shown in the inset of Fig. 1(a). X-ray diffraction (XRD)
was done on flat-shaped single crystals and as well on the powders of crushed
crystals using Rigaku SmartLab with 9 kW Cu Kα x-ray source. The exact
chemical composition of the as-grown single crystals was found using the
energy dispersive x-ray spectroscopy (EDXS). The electrical resistivity and
magnetotransport measurements were performed using the four-probe and Hall-
probe connections. Electrical, magnetic, and magnetotransport measurements
were performed in a 9 T physical properties measurement system (PPMS, DynaCool
of Quantum Design) using the VSM and ETO options. To eliminate the
longitudinal voltage contributions due to a possible misalignment of the Hall
connections, the Hall resistivity was calculated using the relation
$\frac{\rho_{H}(H)-\rho_{H}(-H)}{2}$.

To understand the magnetotransport properties, the electronic band structure
of HoMn6Ge6 was calculated using the density-functional theory (DFT) within
the Perdew-Burke-Ernzerhof-type generalized-gradient approximation (GGA) [37]
as implemented in the Quantum Espresso (QE) simulation package [38]. The
electronic wavefunction is expanded using plane waves up to a cutoff energy of
100 Ry. Brillouin zone sampling is done over a 10$\times$10$\times$6
Monkhorst-Pack $k$-grid. The crystal structure was optimized through the
variable-cell relaxation method as implemented in QE. Notably, the non-
collinear skew-spiral magnetic structure was considered to determine the
ground state electronic properties. The band structure was produced with and
without considering the spin-orbit coupling (SOC).

##  III Results and Discussions

Figure 1(a) depicts the x-ray diffraction (XRD) pattern of the crushed
HoMn6Ge6 single crystals overlapped with Rietveld refinement performed using
the Fullprof software. The Rietveld refinement confirms that HoMn6Ge6
crystallizes into the hexagonal kagome HfFe6Ge6-type crystal structure with a
space group of $P6/mmm$ (No. 191). The refined lattice parameters
$a=b=5.2415(5)~{}\AA$ and $c=8.1831(4)~{}\AA$ confirm that the studied sample
of HoMn6Ge6 is almost equivalent to the reported ones within the error-bars
[35, 33]. Also, see Table I in the Supplemental Material [39] for a comparison
of the structural parameters between our sample and the reported similar
compounds in Refs. [35, 33]. The right-inset in Fig. 1(a) shows the XRD
pattern taken on the flat surface of the single crystal, confirming that the
crystal growth plane is parallel to (0001)-plane direction. Fig. 1(b) depicts
the EDXS data conforming the exact chemical composition of the sample
Ho0.96Mn5.74Ge6. For convenience, hereafter, we represent it by the nominal
composition of HoMn6Ge6. Since the crystals were grown out of Sn flux, the
crystals were found with a maximum 2% of Sn impurity. The elemental mappings
done for Ho, Mn, and Ge, as shown in the top insets of Fig. 1(b), imply a good
homogeneity of the studied single crystals.

Figure 2:  (a) Longitudinal resistivity $\rho_{xx}$ plotted as a function of
temperature with current applied along $x$ $[2\bar{1}\bar{1}0]$-axis for both
0 T (red curve) and 9 T (blue curve) fields applied along $z$ $[0001]$-axis of
the crystal. (b) Magnetoresistance
MR(%)$=\frac{\rho_{xx}(9T)-\rho_{xx}(0T)}{\rho_{xx}(0T)}\times 100(\%)$
plotted as a function of temperature. (c) Magnetization plotted as a function
of temperature [$M(T)$] for both $H\parallel z$ and $H\perp z$ directions
measured under field-cooled and zero-field-cooled modes with $\mu_{0}H=0.05$
T.

Longitudinal electrical resistivity [$\rho_{xx}(T)$] is plotted as a function
of temperature, as shown in Fig. 2(a), measured without magnetic field (red-
colored data) and with $\mu_{0}H=9$ T of magnetic field (blue-colored data)
applied parallel to the $z$-axis [0001]. The low-temperature resistivity ($<$
100 K) can be reasonably fit with the quadratic equation of the Landau Fermi-
liquid theory, $\rho_{xx}(T)=\rho_{0}+\alpha T^{2}$. Here, $\rho_{0}=8.42$
$\mu\Omega cm$ is the residual resistivity due to impurity scattering and
$\alpha=2\times 10^{-9}$ $\mu\Omega cm/K^{2}$ is the electron-electron
scattering coefficient. Further, in the resistivity data ($\mu_{0}H=0$ T) we
observe $kinks$ at $T_{1}\approx$ 270 K and $T_{2}\approx$ 215\. The
transition temperatures $T_{1}$ and $T_{2}$ shift towards higher temperatures
under the external magnetic fields. For instance, at a field of 9 T, the
transition temperature $T_{2}$ shifted to 236 K, while we cannot identify
$T_{1}$ under the 9 T field as the measurements were done only up to 300 K.
This observation indicates the magnetic origin of the $kinks$. Further, the
observation of $T_{1}$ and $T_{2}$ transition temperatures under zero field is
consistent with a previous report on HoMn6Ge6 in which the temperature range
between $T_{1}$ and $T_{2}$ is referred by a mixed magnetic state
($\widetilde{SS}$+AFM) [33]. Fig. 2(b) depicts the magnetoresistance
percentage [$MR\%=(\rho_{xx}(T,H)-\rho_{xx}(T,0))/\rho_{xx}(T,0)$] plotted as
a function of temperature from which another transition is identified at
$T_{3}\approx$ 35 K, though it is not clearly visible from the $\rho_{xx}(T)$
data [see Fig. 2(a)]. Overall, the longitudinal electrical resistivity shows
the metallic nature throughout the measured temperature range with a residual
resistivity ratio (RRR) of $\rho_{xx}$(300K)/$\rho_{xx}$(2K) $\approx$ 20.

Figure 3:  (a),(b), and (c) Isothermal magnetization $M(H)$ measured at
different temperatures for the fields applied parallel to the $z$-axis
($H\parallel z$). Insets in (a)-(c) show same for $H\perp z$. (d),(e), and (f)
Field dependent magnetoresistance
[MR(%)$=\frac{\rho_{xx}(H)-\rho_{xx}(0)}{\rho_{xx}(0)}\times 100(\%)$]
measured for different temperatures.

Fig. 2(c) presents the magnetization plotted as a function of temperature
[$M(T)$] under the magnetic field $\mu_{0}H=$ 0.05 T applied parallel
($H\parallel z$) and perpendicular ($H\perp z$) to the $z$-axis, measured both
in the field-cooled (FC) and zero-field-cooled (ZFC) modes. Clear overlapping
between FC and ZFC data suggests that the magnetic moments are thermally
reversible in the system throughout the measured temperature range of 2 - 350
K. On the other hand, previous neutron diffraction study on similar system
revealed a complex magnetic structure with different magnetic states at
different temperatures [35]. For instance, in the low-temperature region ($<$
55 K), the Ho and Mn magnetic moments form a skew spiral ($\widetilde{SS}$)
structure, coupled antiferromagnetically, with a plane of moments making an
angle $\theta_{S}$ with the $z$-axis [see Fig. 6(b)], producing finite net
magnetization along both the in-plane and out-of-plane directions. Between 55
and 220 K, both wave vector ($q$) and phase angle ($\phi_{S}$) increase
monotonically with temperature. In the 220-260 K region, the $\widetilde{SS}$
structure gets distorted, accompanied by the Mn spin-reorientation, and thus
eventually leading to a weak ferromagnetism at 300 K. Beyond 300 K the skew-
spiral structure becomes a cycloid. In line with previous studies [35, 40, 41,
33] we also observe magnetic transitions in the magnetization data at
$T_{1}\approx$ 270 K, $T_{2}\approx$ 215 K, and at $T_{3}\approx$ 35 K.
Further, a weak ferromagnetic-like magnetization jump is noticed at
$T_{C}\approx$ 300 K. As can be seen from Fig. 2(c), the out-of-plane
magnetization ($H\parallel z$) is significantly higher than the in-plane
magnetization ($H\perp z$) below 300 K, while it is vice versa above 300 K.
This observation hints at the fluctuating magnetic moments across the $T_{C}$.
Also, note that a slight variation in the transition temperatures ($T_{1}$,
$T_{2}$ and $T_{3}$ ) is noticed between our data and Refs. [35, 33], possibly
due to different sample preparation methods. Specifically, the $M(T)$ of Ref.
[33] shows a cusp at around 25 K and a rapid increase in $M(T)$ below 77 K,
while no cusp is found but a rapid increase in $M(T)$ is found below 55 K in
Ref. [35]. On the other hand, from our data we observe a cusp at 33 K and a
rapid increase in $M(T)$ below 50 K [see Fig. 2(c)]. Similarly, the mixed
magnetic state of AFM+$\widetilde{SS}$ is reported within the temperature
range of 220-260 K in Ref. [35], 215-245 K in Ref. [33], while from our data
we find it in a broader temperature range of 215-270 K..

Next, Figs. 3(a), 3(b), and 3(c) depict magnetization isotherms [$M(H)$]
measured at different temperatures for $H\parallel z$ and $H\perp z$. We
observe weak ferromagnetic-like $M(H)$ data when measured at 2 K for both
$H\parallel z$ and $H\perp z$ at lower field regions without magnetization
saturation. Further field-induced hysteresis is visible for $H\parallel z$
between 2 and 4 T but not for $H\perp z$, possibly due to the out-of-plane
spin canting within this field range. This type of metamagnetic state is not
observed at higher temperatures. Instead, an out-of-plane spin-flop transition
has been found at a critical field ($H_{SF}$) that is temperature dependent,
due to which a sudden increase in magnetization was noticed [see Figs. 3(b)
and(c)]. However, the spin-flop transition is absent from the in-plane
magnetization ($H\perp z$) at any measured temperature. A previous report also
demonstrates similar spin-flop transitions in HoMn6Ge6 [33].

Isothermal magnetoresistance (MR) at different sample temperatures and with
the fields applied parallel to the $z$-axis are shown in Figs. 3(d), 3(e), and
3(f). Fig. 3(d) demonstrates the MR measured at 2 K, where the MR is linearly
dependent (red-dashed line) on the applied field in addition to the hysteresis
between the fields 2 and 4 T that originated from the field-induced
metamagnetic state [Fig. 3(a)]. From Fig. 3(e), we can see that the linear
dependency of MR is intact up to 50 K. However, at 100 K, we observe a
classical parabolic magnetoresistance. Though this parabolic nature is
sustained at higher temperatures (150 and 200 K) as well, beyond the spin-flop
($H_{SF}$) transition, the MR starts to decrease with increasing the field for
a given temperature. Eventually, beyond the $H_{SF}$ the MR becomes negative
above 150 K, consistent with the MR(T) data shown in Fig. 2(b) in which we can
notice negative MR above 150 K when measured with 9 T. However, the MR becomes
nearly field-independent as the temperature reaches 350 K.

Overall, HoMn6Ge6 shows a complex temperature dependent MR. Importantly, up to
50 K it shows a positive nonsaturating MR that is linearly depending on the
applied field. There exist several mechanisms explaining the linear MR, such
as (i) the presence of linear dispersive Dirac-like bands with very low
effective mass near the Fermi level in the case of topological materials [42,
43], (ii) quasi-random resistor network model in the case of
metal–semiconductor composites [44, 45], and (iii) carrier density
fluctuations due to irregular current paths from the inhomogeneous or grain
boundaries in the case of disordered systems [46, 47]. We can safely rule out
the (ii) mechanism as our system is not a metal-semiconductor composite. Also,
since our studied system is in the single crystalline form with no significant
grain boundaries and with good homogeneity, we can rule out the mechanism
(iii) also as the origin of linear MR. Previous reports on its sister
compounds RMn6Sn6 suggested that the linear MR could originate from the Dirac-
like linear bands [29]. In addition, previous band structure calculations on
these systems showed several Dirac-like linear band dispersions crossing the
Fermi level [36, 48]. To confirm that the linear MR originates from the linear
Dirac-like topological band structure, we performed density functional theory
(DFT) calculation, as discussed later explicitly.

Figure 4:  (a) Angle dependent magnetoresistance [ADMR(%)
$=\frac{\rho_{xx}(\theta)-\rho_{xx}(0^{o})}{\rho_{xx}(0^{o})}\times 100(\%)$]
measured for various applied fields at 2 K. Inset in (a) is a schematic
showing the ADMR measuring geometry. Here $\theta$ is the angle between
$z$-axis and the applied field direction. (b) is same as (a) but plotted in
polar coordinates for a better visualization. In (b) radius is the amplitude
of ADMR(%). (c) Field dependent magnetoresistance (MR) measured at 2 K for
different $\theta$ values. (d) is same as (c) but measured at 250 K of the
sample temperature.

Further, we measured the angle-dependent magnetoresistance (ADMR) with the
current and field directions always kept perpendicular to each other such that
only the field direction changes with the crystal axes as shown in the
schematic of Fig. 4(a). The angle-dependent magnetoresistance is calculated as
$\frac{\rho_{xx}(\theta)-\rho_{xx}(0^{o})}{\rho_{xx}(0^{o})}$(%) while the
applied magnetic field is kept constant. As seen from Fig. 4(a), at 2 K, the
MR is positive for all the applied magnetic fields but very sensitive to the
field angle, suggesting a strong anisotropic MR in this system. Specifically,
MR is minimum at the angles of 0∘, 180∘, and 360∘ and it is maximum at the
angles 60∘, 120∘, 240∘, and 300∘. Also, the saddle points in the ADMR are
observed at 90∘ and 270∘ for all the applied fields. For a better
visualization, we plotted the ADMR in polar coordinates as shown in Fig. 4(b),
from which we can clearly observe a butterfly pattern for the out-of-plane MR
at 2 K. It is worth to mention here that the ADMR measured at 250 K is
negligibly small (not shown) compared to the ADMR taken at 2 K. Next, Fig.
4(c) depicts the field-dependent MR plotted for different field angles (0∘,
45∘, and 80∘) measured at 2 K, from which we observe linear MR at all field
angles. Moreover, we see that the MR reaches almost 25% for
$\theta=80^{\circ}$, and the hysteresis, which was present for
$\theta=0^{\circ}$, has vanished at the other two angles. Fig. 4(d) depicts
filed dependent MR taken at $0^{\circ}$ and $90^{\circ}$ filed angles, from
which we observe the absence of spin-flop transition effect on the MR when
measured at $90^{\circ}$, consistent with $M(H)$ for $H\perp z$.

Figure 5:  (a) Field dependent Hall resistivity $\rho_{xy}$ measured for
different sample temperatures overlapped with the fits (black curves). (b)
Field dependent Hall conductivity $\sigma_{xy}$ plotted for 2, 50, and 100 K
of sample temperatures. Inset in (b) schematically shows the Hall effect
measuring geometry. (c) Normal Hall coefficient ($R_{0}$) plotted as a
function of temperature. (d) Anomalous Hall coefficient ($R_{S}$) [left axis]
and anomalous Hall scaling factor ($S_{H}$) [right axis] plotted as a function
of temperature. Figure 6:  (a) Hexagonal Brillouin zone showing the positions
of various high symmetry points. (b) Low temperature skew-spiral
($\widetilde{SS}$) magnetic structure of HoMn6Ge6 [35]. Here,
$\theta_{s}=60^{o}$ and $\phi_{s}=71.2^{o}$. (c) Electronic band structure of
HoMn6Ge6 calculated for the magnetic configuration shown in (b) with (black-
colored) and without (red-colored) including the spin-orbit coupling (SOC).
The right-side panels of (c) show the zoomed-in band structure to clearly see
various Dirac points. (d), (e), and (f) show the calculated Fermi surface maps
in three-dimensional. (g) and (h) show the constant energy contours taken at
the Fermi level in the $k_{x}-k_{y}$ and $k_{x}-k_{z}$ momentum planes,
respectively.

Next, Fig. 5(a) demonstrate field-dependent Hall resistivity [$\rho_{xy}(H)$]
measured at different sample temperatures. We see a deviation in the Hall
resistivity, leading to an anomalous Hall effect, at the spin-flop transition
temperature. Usually, in ferromagnets, the Hall resistivity can be expressed
by $\rho_{H}=R_{0}\mu_{0}H+R_{S}\mu_{0}M$ [49], where $R_{0}$ is the ordinary
Hall coefficient and $R_{S}$ is the anomalous Hall coefficient. Although the
anomalous Hall effect (AHE) mainly appears in ferromagnets, recently it has
been widely observed in noncollinear antiferromagnets as well [17, 50, 51,
52]. From our Hall resistivity data, we can see that the anomalous Hall effect
is induced by a spin-flop transition that resembles the $M(H)$ data shown in
Figs. 3(a)-(c). Moreover, the above formula can be rewritten as
$\frac{\rho_{H}}{\mu_{0}H}=R_{0}+R_{S}\frac{M}{H}$ which imitates a linear
equation and intercept on the $y$-axis gives $R_{0}$ and slope gives the
anomalous Hall coefficient $R_{S}$. Using this formalism, we have reasonably
fitted the Hall resistivity as depicted in Fig. 5(a). The anomalous scaling
factor ($S_{H}$) is calculated using the formula
$S_{H}=\mu_{0}R_{S}/\rho_{xx}^{2}$, where $\rho_{xx}$ is the longitudinal
resistivity. Fig. 5(c) depicts the normal Hall coefficient ($R_{0}$) plotted
as a function of temperature. From Fig. 5(c), we further notice $R_{0}$
changing from positive to negative at around 215 K, which indicates hole-type
(electron-type) carrier dominance below (above) 215 K.

As discussed above, a magnetic transition exists at around 215 K ($T_{2}$). It
is possible that the change in magnetic structure influences the electronic
structure near the Fermi level and thus the switching of charge carrier type
at 215 K. This observation suggests a strong correlation between the magnetic
and electronic structures in these systems. Moreover, the value of $R_{S}$
decreases rapidly with decreasing temperature and becomes negligibly small
below 100 K. Whereas the $S_{H}$ varies between 0.05 and 0.2 like in a typical
ferromagnetic metal [49, 53]. Further from the relation
$\mu_{0}R_{S}=S_{H}\rho_{xx}^{2}$, it is clear that for very low resistivity
$\rho_{xx}$ values the contribution from AHE ($R_{S}$) is negligible so that
only the ordinary Hall effect dominates at low temperatures. In Fig. 5(b), we
have plotted the Hall conductivity as a function of temperature calculated
using the formula,
$\sigma_{xy}=-\frac{\rho_{xy}}{{\rho_{xy}^{2}}+{\rho_{xx}^{2}}}$. In Fig.
5(b), we have fitted the Hall conductivity curves using the single band model,
$\sigma_{xy}=[\frac{n_{h}{\mu_{h}}^{2}}{1+(\mu_{h}B)^{2}}]eB$ as at low
temperatures the hole carriers dominate the total Hall transport. Here,
$n_{h}$ is hole carrier density, and $\mu_{h}$ is hole mobility. From the
fits, we derived the hole carrier concentration of $n_{h}$ is $1.06\times
10^{20}$ cm-3 and hole mobility of $\mu=0.049$ m2/Vs at 2 K. These values are
consistent with a previous report on these type of systems [29].

To uncover the mechanism of unsaturated linear MR, we performed density
functional theory calculations as illustrated in Fig. 6. For the DFT
calculations, we considered the magnetic structure as shown in Fig. 6(b),
depicted by the neutron diffraction study on this system [35]. Fig. 6(c)
displays the electronic band dispersion along the high-symmetry $k$-path of
$\Gamma-\mathrm{M-K}-\Gamma$ calculated without and with considering the spin-
orbit coupling (SOC). We notice several linear band crossings (Dirac-like)
from the electronic band structure at the $K$ point near the Fermi level. One
Dirac point (DP1) is observed at a binding energy of 90 meV below the Fermi
level ($E_{F}$) that is gapless without SOC, but under SOC, the Dirac point is
gapped by lifting the degeneracy. DP2 is found at a binding energy of 40 meV
below the $E_{F}$, which is always gaped. DP3 is found at a binding energy of
250 meV above $E_{F}$, which is gapless for both with SOC and without SOC. Our
calculations on HoMn6Ge6 align with previous calculations performed on
HoMn6Sn6 [31, 36]. Particularly, the calculations on HoMn6Sn6 demonstrated
that gap size at the Dirac points depends explicitly on the magnetic spin
orientation with respect to the crystal axis [31, 36]. This means, larger
(smaller) gaps were predicted for the fields applied along the out-of-plane
(in-plane) direction. Since our calculations were performed with an angle of
$\theta_{S}=60^{\circ}$ between the $z$-axis and plane of magnetic moments,
which is neither out-of-plane nor in-plane, we observe both gapped and gapless
Dirac points. This observation hints at the possibility of a critical angle of
magnetic moments at which the Dirac states are gapless and robust under SOC.

Apart from the linear band dispersions at the $K$ point, we also observe two
hole-like band dispersions ($h_{1}$ and $h_{2}$) crossing the Fermi level
around $K$ point and an electron-like ($e_{1}$) band dispersion crossing the
Fermi level around the $\Gamma$ point. No bands are found at the M-point
crossing the Fermi level. In addition, we found several flat bands ($fb_{1}$,
$fb_{2}$, and $fb_{3}$) that are dispersionless throughout the Brillouin zone.
Figs. 6(d)-(f) show the three-dimensional view of the Fermi surface maps,
which are mainly contributed by three types of Fermi pockets; one of them is
the electron-like ($e_{1}$) pocket with an almost cylindrical shape as shown
in Fig. 6(d), the second one is the hole-like ($h_{1}$) pocket, which is close
to one-third of a cylinder shared by each corner of the hexagon, as shown in
Fig. 6(e), and the last one is the hip-roof-shaped hole-like ($h_{2}$) pocket
at the $K$ point. With the help of these three types of Fermi pockets, we
estimated the hole carrier density $n_{h}$=$5.3\times 10^{20}$/cm3 and
electron carrier density $n_{e}$=$3.6\times 10^{20}$/cm3 to find the net hole
carrier density of $1.7\times 10^{20}$/cm3 using Luttinger’s theorem [54] [see
Supplemental Material [39]]. Interestingly, this value is close to the
experimentally calculated net carrier density ($1.06\times 10^{20}$ cm-3) from
the Hall data at 2 K. The estimated hole and electron carrier densities are
nearly equal, suggesting a possible charge compensation in this system [55,
56]. Further, the estimated net carrier density of $1.06\times 10^{20}$ cm-3
suggests HoMn6Ge6 to be a semimetal [57, 58]. Figs. 6(g) and 6(h) illustrate
the Fermi surfaces projected onto the $k_{x}-k_{y}$ and $k_{x}-k_{z}$ planes,
clearly showing the electron-like Fermi pocket at the $\Gamma$ and hole-like
Fermi pockets at the $K$ point.

As for the angle-dependent magnetoresistance (ADMR) measurements, the magnetic
field is always perpendicular to the current direction, which does not change
the Lorentz force acting on the charge carriers. The spin-charge scattering
can be neglected at low temperatures as the mean free path of the charge
carriers is much higher in this system at 2 K ($\approx$ 0.67 $\mu m$)
compared to the distance of $Mn-Mn$ (2.62 Å) or $Ho-Ho$ (5.24 Å) magnetic
moments. This leads us to conclude that the anisotropic ADMR has an electronic
band structure origin rather than a magnetism origin. The Fermi velocity can
be calculated as $v_{k}=\frac{1}{\hbar}\nabla_{k}\epsilon_{k}$, which depends
on the local curvature of the Fermi surface cross-section. The smaller the
orbit of the Fermi surface cross-section, the greater the local curvature,
leading to a higher Fermi velocity. Hence, a high microscopic Lorentz force
would act on the charge carriers [59, 60, 61]. Our calculations indicate that
the electron pocket $e_{1}$ and hole pocket $h_{1}$ exhibit nearly cylindrical
shapes, with their diameters and heights being quite similar in value, making
them nearly isotropic. Only the $h_{2}$ hole pocket has an anisotropic shape
(hip-roof-shaped), which could be the possible origin of this high anisotropic
ADMR. Because when the field is applied along the $z$-axis, the charge
carriers are subjected to orbit on the $k_{x}-k_{y}$ plane. Similarly, for the
fields applied along the $y$-axis the carriers would orbit on the
$k_{x}-k_{z}$ plane. As observed in Fig. 6(g), the Fermi sheets at $K$ point
are nearly circular for $H\parallel z$ [see Fig. 4(a) for $\theta=0^{o}$].
Only the out-of-plane Fermi sheets ($k_{x}-k_{z}$) have substantial curvature
[see Fig. 6(h)] which are detected experimentally for $H\parallel y$ [see Fig.
4(a) for $\theta=90^{o}$]. Therefore, we think the asymmetric MR originated
from the asymmetric out-of-plane Fermi pockets. Further, as evidenced by our
DFT band structure calculations, the linear nonsaturating MR is mainly
contributed by the linear Dirac-like bands near the Fermi level.

##  IV Summary

In summary, using the Sn flux, we have grown high-quality single crystals of
HoMn6Ge6. Electrical resistivity, with a few magnetic transition-driven
anomalies, demonstrates an overall metallic nature throughout the measured
temperature range. A crossover from negative to positive magnetoresistance
(MR) is observed at a critical temperature of 150 K. While the Dirac-like
linear band dispersions mainly drive the linear nonsaturating positive MR
exits in the low-temperature region, the negative MR observed in the higher
temperature region is due to the spin-flop type magnetic transition. We found
an anomalous Hall effect in addition to a dominant charge carrier switching
across 215 K. We performed electronic band structure calculations on HoMn6Ge6
by considering the skew-spiral magnetic structure of the system to realize
large anisotropy in the out-of-plane Fermi sheets. We suggest that the large
anisotropic out-of-plane magnetoresistance observed in HoMn6Ge6 originated
from the anisotropic out-of-plane Fermi surfaces. The band structure
calculations predict several Dirac-type band crossings at the $K$ point near
the Fermi level.

###### Acknowledgements.

This research has used the Technical Research Centre (TRC) Instrument
Facilities of S. N. Bose National Centre for Basic Sciences, established under
the TRC project of the Department of Science and Technology, Govt. of India.

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