# Effect of Ni substitution on the fragile magnetic system
${\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}}$

Atreyee Das  Ames National Laboratory, U.S. DOE, Iowa State University, Ames,
Iowa 50011, USA  Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA  Tyler J. Slade  Ames National Laboratory, U.S. DOE,
Iowa State University, Ames, Iowa 50011, USA  Rustem Khasanov  Laboratory for
Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
Sergey L. Bud’ko  Ames National Laboratory, U.S. DOE, Iowa State University,
Ames, Iowa 50011, USA  Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA  Paul C. Canfield  Ames National Laboratory,
U.S. DOE, Iowa State University, Ames, Iowa 50011, USA  Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA

###### Abstract

$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ is an itinerant ferromagnet with a
Curie temperature, $T_{C}$, of $\sim$ 3.8 K and a remarkably small saturated
moment of 0.1 $\mu_{B}/\text{Co}$. Here we present the growth and
characterization of single crystals of the
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ series for
0.00 $\leq x\leq$ 0.186. We measured powder X-ray diffraction, composition as
well as anisotropic temperature dependent resistivity, temperature and field
dependent magnetization along with heat capacity on these single crystals. We
also measured muon-spin rotation/relaxation ($\mu\text{SR}$) for some Ni
substitutions ($x$ = 0.027, 0.036, 0.074) to study the evolution of internal
field with Ni substitution. Using the measured data we infer a low
temperature, transition temperature-composition phase diagram for
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$. We find that
$T_{C}$ is suppressed for low dopings, $x\leq 0.014$; whereas for
$0.036\leq{x}\leq 0.186$, the samples are antiferromagnetic with a Neel
temperature, $T_{N}$, that goes through a weak and shallow maximum
($T_{N}\sim$ 3.4 K for $x\sim$ 0.07) and then gradually decreases to 2.4 K by
$x$ = 0.186. For intermediate Ni substitutions, $0.016\leq{x}\leq 0.027$, two
transition temperatures are inferred with $T_{N}>T_{C}$. Whereas the $T-x$
phase diagram for
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ and the $T-p$
phase diagram determined for the parent
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ under hydrostatic pressure are
grossly similar, changing from a low doping or low pressure ferromagnetic (FM)
ground state to a high doped or pressure antiferromagnetic (AFM) state,
perturbation by Ni substitution enabled us to identify an intermediate doping
regime where both FM and AFM transitions occur.

##  I Introduction

Itinerant, metallic, magnetic systems with low transition temperatures have
attracted much attention in recent years. The reason can be attributed to the
fact that many of these materials allow for suppressing their transition
temperatures by application of pressure, changing the chemical composition, or
by magnetic field Canfield and Bud’ko (2016). Suppressing phase transitions to
zero temperature is often studied as several exotic physical phenomenon such
as unconventional superconductivity, non Fermi liquid, etc. are found in
proximity of the quantum critical point (QCP) for a second-order transition or
a quantum phase transition (QPT) in the case of a first order transition.
Canfield and Bud’ko (2016); Brando  _et al._ (2016); Pfleiderer  _et al._
(2001a); Huy  _et al._ (2007); Dagotto (1994); Paglione and Greene (2010);
Pfleiderer  _et al._ (2001b); Gegenwart  _et al._ (2008); Lévy  _et al._
(2007); Uemura  _et al._ (2007); Pfleiderer  _et al._ (2004); Ubaid-Kassis
_et al._ (2010); Pfleiderer (2009); Saxena  _et al._ (2000); Aoki  _et al._
(2001); Cheng  _et al._ (2015); Ran  _et al._ (2019). Intensive studies have
shown that although antiferromagnetic (AFM) transitions in many metallic
systems can be continuously suppressed to zero temperature by the non-thermal
tuning parameters discussed above Canfield and Bud’ko (2016); Shibauchi  _et
al._ (2014); Gegenwart  _et al._ (2008); Löhneysen  _et al._ (1994);
Friedemann  _et al._ (2009); Schmiedeshoff  _et al._ (2011); Tokiwa  _et al._
(2013); Mun  _et al._ (2013); Nandi  _et al._ (2010), the situation becomes
quite different for ferromagnetic (FM) transitions. The current theoretical
models suggest that stoichiometric systems with minimum disorder generally
avoid a FM QCP at zero field, and either the transition becomes a first order
through a tricritical point, or a long wavelength AFM phase appears. Belitz
_et al._ (1999); Chubukov  _et al._ (2004); Conduit  _et al._ (2009);
Karahasanovic  _et al._ (2012); Pedder  _et al._ (2013); Belitz and
Kirkpatrick (2017); Brando  _et al._ (2016); Rech  _et al._ (2006); Wysokiński
(2019). The evolution of a first order FM transition to a quantum phase
transition (QPT) has been experimentally verified in several systems, Brando
_et al._ (2016); Huxley  _et al._ (2000); Pfleiderer and Huxley (2002); Uhlarz
_et al._ (2004); Niklowitz  _et al._ (2005); Kaluarachchi  _et al._ (2017);
Gati  _et al._ (2021) and the transition from ferromagnetic to a modulated
magnetic phase has been experimentally observed as well Brando  _et al._
(2016); Kotegawa  _et al._ (2019); Niklowitz  _et al._ (2019); Jesche  _et
al._ (2017) . An exception to this scenario was reported in a recent
theoretical work, where it was proposed that a FM QCP is achievable even in
clean systems with non-centrosymmetric symmetry and sufficiently strong spin-
orbit interactions Kirkpatrick and Belitz (2020). Although hydrostatic
pressure, or in some AFM systems, applied magnetic field, are the preferred
choices for suppression of transition temperature, the reason being that they
do not introduce any additional disorder, chemical substitution is broadly
accepted as a viable tuning parameter to access QCP/QPT and, more generally,
tune the ground state in magnetic systems. The complex and intriguing physics
associated with FM-QPT and AFM QCP described above motivates the search for
new metallic ferromagnetic systems and the tuning of their magnetic
properties.

$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ is a recently discovered itinerant
ferromagnetic compoundSaunders  _et al._ (2020). It belongs to the family of
$\text{R}_{5}\text{Co}_{2}\text{Ge}_{3}$, (R = La-Sm), which crystallizes in a
monoclinic structure (space group C2/m) Lin  _et al._ (2017). Transport,
magnetization, and heat capacity measurements along with muon-spin
rotation/relaxation ($\mu$SR) studies at ambient pressure reveal that
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ undergoes a ferromagnetic transition
at 3.8 K, and has a very low saturated moment of $\sim$ 0.1
$\mu_{B}/\text{Co}$ Saunders  _et al._ (2020). With an effective moment of
$\sim$ 1.0 $\mu_{B}/\text{Co}$, it has a Rhodes-Wohlfarth ratio Rhodes and
Wohlfarth (1963) of 4.9, higher than several well established itinerant
ferromagnetic systems.Santiago  _et al._ (2017); Saunders  _et al._ (2020).
All of this makes $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ a very promising
candidate for tuning or modifying its magnetic transition.

Initial tuning of the magnetic transition temperature of
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ was done via the application of
hydrostatic pressure. The temperature-pressure (T-p) phase diagram up to 5.12
GPa was determined from resistivity, magnetization, and heat capacity
measurements Xiang  _et al._ (2021). It was revealed that up to $\sim$ 1.7
GPa, the system remains ferromagnetic and its $T_{C}$ is suppressed to 3 K.
Upon further increase of pressure, $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$
enters a different low temperature ground state where the transition
temperature depends non-monotonically on the applied pressure. This new state
is predominantly antiferromagnetic in nature which suggests that
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ may be another example of a metallic
ferromagnetic system which avoids quantum criticality by the stabilization of
a new non-ferromagnetic phase.

Motivated by the desire to further study the evolution of magnetic order in
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$, in this work, we report the
synthesis and physical properties of
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ with
${0.00\leq x_{EDS}\leq 0.186}$, where $x_{EDS}$ is the substitution value
determined from EDS measurements. Substitution by Ni changes the band filling
and clearly introduces a different type of perturbation than pressure into the
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ system.

Single crystals of
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ were grown
and their temperature and field dependent magnetization, temperature dependent
resistance and heat capacity were thoroughly investigated. Our studies reveal
that for the low substitution levels, between ${0.00\leq x_{EDS}\leq 0.014}$,
the system remains ferromagnetic and the transition temperature is suppressed
to 3.2 K. For higher substitution levels, ${0.036\leq x_{EDS}\leq 0.186}$, the
magnetization measurements indicate the appearance of an AFM state. We also
see the emergence of an upturn in the resistance that can be associated with
the opening of a super-zone gap at temperatures similar to that of the AFM
transition, which is found to first weakly increase to a shallow maximum
$\text{T}_{N}\sim$ 3.5 K near $x\sim$ 0.07 and then slowly decrease to T = 2.4
K for $x$ = 0.186. The intermediate Ni substitutions, $0.016\leq{x}\leq
0.027$, show both anti- and ferromagnetic transitions with
$\text{T}_{N}>\text{T}_{C}$. Muon-spin rotation/relaxation ($\mu\text{SR}$)
measurements were done on ${x_{EDS}=0.027,~{}0.036~{}\text{and}~{}0.074}$
samples to compare the evolution of the internal field due to Ni substitution
and compliment the results obtained from the above transport and thermodynamic
studies.

##  II Experimental Details

Single crystals of
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$; with
${x_{nominal}}$ = 0.0, 0.01, 0.0175, 0.02, 0.0225, 0.025, 0.03, 0.04, 0.06,
0.08, 0.10, 0.12 and 0.15 were synthesized using a self flux solution growth
method Canfield and Fisher (2001); Canfield (2020); Canfield and Fisk (1992)
in a manner similar to the growth of pure
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ Saunders  _et al._ (2020). Small
pieces of Lanthanum (Materials Preparation Center - Ames National Laboratory
$99.99\%$), Cobalt (American Elements $99.95\%$), Nickel (Alfa Aesar
$99.98\%$) and Germanium (Alfa Aesar $99.999\%$) with a stating composition of
${\text{La}_{45}\text{(Co}_{1-x}\text{Ni}_{x})_{45}\text{Ge}_{10}}$, were
weighed and then placed in a Tantalum crucible Canfield and Fisher (2001);
Canfield (2020), and sealed in a fused silica ampoule under a partial argon
atmosphere. The ampoule was heated in a box-furnace to $1180\tccentigrade$
over 5 hours, held at the temperature for 10 hours, quickly cooled down to
$900\tccentigrade$ followed by a slow cool to $800\tccentigrade$ over 80
hours. After dwelling at $800\tccentigrade$ for a few hours, the excess
solution was then decanted using a centrifuge Canfield and Fisk (1992);
Canfield (2020); Canfield and Fisher (2001). The typical dimensions of the
crystals are ${5~{}\text{mm}\times 3~{}\text{mm}}$ with an average thickness
of ${0.5~{}\text{mm}}$ and are of the same morphology as that of undoped
compound. (See the inset to Fig 2 below for a picture of a representative
crystal with the $b$\- and $c$-axes identified.) Higher Ni substitution levels
did not yield well formed, or usable, single crystals and hence we stopped
attempts of substitution at ${x_{nominal}}$ = 0.15.

The Ni substitution levels $x_{EDS}$ of the
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ crystals were
determined by Energy Dispersive Spectroscopy (EDS) quantitative chemical
analysis using a Thermo Noran Microanalytical system Model C10001 EDS
attachment to a JEOL JSM-5910LV scanning-electron microscope (SEM). An
acceleration voltage of 21 kV, working distance of 10mm and take off angle of
35${\degree}$ were used for measuring all standards and crystals with unknown
composition. Undoped, single crystalline,
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ was used as a standard for La, Co
and Ge whereas single crystalline $\text{La}\text{Ni}_{2}\text{Ge}_{2}$ was
the standard for Ni quantification. The composition of the plate like crystals
was measured at 3-4 different spots on the crystal’s face, revealing good
homogeneity of each crystal. The spectra were fitted and the average
compositions as well as the error bars were obtained using NIST-DTSA II
Lorentz 2020-06-26 software, accounting for both inhomogeneity and goodness of
fit of each spectra Newbury and Ritchie (2014).

Figure 1: Measured Ni substitution level, ($x_{EDS}$) values, for
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ crystals versus
$x_{nominal}$ values used in growth solution. The red line is the linear fit
across the data points with the intercept fixed to (0,0). (Inset : The lower
dopings, nominal ${0.01\leq x\leq 0.04}$ of the growths done, are zoomed in
for clarity).

The crystal structure was inferred using a Rigaku Miniflex-II powder
diffractometer using Cu ${K_{{\alpha}}}$ radiation
$(\lambda=1.5406~{}$\mathrm{\text{Å}}$)$. The crystals were ground to a fine
powder and sieved through a $90~{}{\mu}m$ mesh U.S.A. Standard Testing Sieve
to reduce preferential orientation. The sieved powder was then mounted and
measured on a single crystal Si, zero-background sample holder. The patterns
were refined and the lattice parameters were determined using GSAS II
software. Toby and Von Dreele (2013).

Temperature and field dependent magnetization measurements were done in a
Quantum Design Magnetic Property Measurement System (QD-MPMS) SQUID
magnetometer with samples mounted on a diamagnetic Poly-Chloro-Tri-Fluoro-
Ethylene (PCTFE) disk, which snugly fits inside a straw, using a small amount
of superglue. The signal from the disk was measured beforehand and subtracted
from the combined sample-disk magnetization so as to obtain the value of the
moment due to the sample. $M(H)$ measurements were done in magnetic fields
from 0 kOe to 50 kOe at a temperature of 1.8 K. Zero field cooled (ZFC) $M(T)$
measurements were done by first lowering the temperature to 1.8 K, and then
applying a magnetic field of 1 kOe, after which measurements were done up to
300 K. In addition, for a few substitutions (x = 0.027, 0.036 and 0.074), a 5
quadrant $M(H)$ was done at 1.8 K to a maximum field of 10 kOe as well a zero
field cooled (ZFC) and field cooled (FC) $M(T)$ upto 10 K, at a low field of
20 Oe. But before, both $M(H)$ and $M(T)$ data collection, the sample was
centered at a temperature of 20 K in a field of 1 kOe, followed by a
demagnetization procedure, to reduce remanent magnetic field in the
magnetometer. For both these measurements the magnetic field was applied
perpendicular to the face of the crystal, $H||a^{*}$, and in the plane of the
plate-like crystal, $H||b~{}\text{and}~{}H||c$ as done for the undoped
compound. Saunders  _et al._ (2020).

Temperature dependent resistance measurements were done in the standard four-
probe geometry in a Quantum Design Physical Property Measurement System (QD-
PPMS) using a 3 mA excitation with a frequency of 17 Hz. Plate like samples of
each $x$ of ${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$
were cut perpendicular to each other with a wire saw so as to apply current
either along the $b$ or $c$-axis of the crystal. Electrical contacts were made
using 25$\mu m$ diameter platinum wires attached to the bar shaped samples
using Epotek-H20E epoxy. $R(T)$ was measured in the range of
$1.8~{}\text{K}\leq T\leq 300~{}\text{K}$ on cooling under zero magnetic
field. Due to irregularities in the sample shape, there are uncertainties in
the measurements of the sample’s dimensions. For this reason, $R(T)$ data will
be presented as normalized resistance either by its value at 300 K or at 10 K,
for clarity in low temperature data.

Figure 2: (Color online) Powder X-Ray diffraction data for
${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$ for
$x_{EDS}=0.036$. The data for other $x$-values are qualitatively the same. The
vertical red lines represent the expected peak positions for the C2/m
${\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}}$ structural model. (Inset: Typical
crystals of ${\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}}$
for $x_{EDS}$ = 0.036 on a mm grid. The crystallographic $b-$ and $c-$
directions are indicated by red arrows next to the crystals. Crystals of the
same batch were used for obtaining the powder pattern shown here.)

Temperature dependent heat capacity measurements were performed in a QD-PPMS.
A plate-like sample was mounted on the micro-calorimeter platform using a
small mount of Apeizon N-grease and measured in the range of
$1.9~{}\text{K}\leq T\leq 25~{}\text{K}$ under zero magnetic field. The
addenda (contribution from the grease and the sample platform) was measured
separately and subtracted from the total data to obtain the contribution only
due to the sample using the PPMS software. For $x_{EDS}$ = 0.036, $C_{p}(T)$
was measured down to 0.5 K using a 3-He insert to the PPMS.

Figure 3: (Color online) Change in the monoclinic lattice parameters a,b,c of
the unit cell with change in $x_{EDS}$ in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ obtained from
the refinement of the powder x-ray diffraction patterns. Figure 4: (Color
online) (a) The results of the field dependent magnetization at T = 1.8 K for
all Ni substituted samples for H$||a^{*}$ plotted for $0\leq H\leq
25~{}\text{kOe}$. The samples were cooled to 1.8 K in the absence of an
external field (ZFC) and $M(H)$ measurements were then done in field
increasing from zero. (Inset: (i) $M(H)$ data up to 1 kOe for all the dopings.
(ii) Zoomed in $M(H)$ data for the AFM samples ($x_{EDS}\geq$ 0.036)). (b) ZFC
$M(T)$ measured at 1 kOe plotted for T $\leq$ 10 K. The feature around $\sim$
4.2 K is an artifact associated with the QD MPMS classic system used for data
collection. (Inset: (iii) $M(T)$ data for the AFM Ni substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ ($x_{EDS}\geq$
0.036).

$\mu\text{SR}$ experiments were done at the $\pi$M3 beam line using the GPS
(General Purpose Surface) spectrometer at the Paul Scherrer Institut,
Switzerland Amato  _et al._ (2017). The samples were glued on a oxygen free
copper (OFC) mount using GE varnish with care that all the single crystals are
oriented along the same direction. The crystals were secured with kapton tape
covering a 8x8 mm square grid. Both zero-field (ZF) and weak transverse-field
(wTF) measurements were done at temperatures ranging from $\simeq$ 1.5 K to
$\simeq$ 8 K. 100 ${\%}$ spin polarized muons were implanted into the sample
along the $a$* direction (perpendicular to the plate-like face of the
crystals). For wTF an external field of 30 Oe was applied along the $b-c$
plane of the crystals. The experiments were performed in a spin-rotated mode,
suitable for single crystal samples, allowing us to probe independently the
time evolution of the perpendicular and parallel components of the muon-spin
polarization $(P(t))$ and determine the direction of magnetic moments in the
samples.

##  III Results

### Composition and Lattice parameters

The Ni substitution levels for the different crystals, determined by the EDS
measurements $(x_{EDS})$ are plotted as a function of the nominal Ni levels
$(x_{nominal})$, used for the growth are shown in Fig 1. As $x_{nominal}$ is
increased, $x_{EDS}$ increases in a monotonic manner. The data in Fig 1 can be
fit well with a straight line, going through the origin, with a slope of
$1.15\pm 0.03$. The inset to Fig 1 shows the data for
$0.00\leq\text{x}_{nominal}\leq 0.04$. It should be pointed out that the
values of $x_{EDS}$ for $0.020\leq\text{x}_{nominal}\leq 0.025$ are
essentially the same. The detailed measurement results of all these Ni
substitution levels will be shown in the Appendix. We will show only the
results for one of the substitutions, $\text{x}_{nominal}$ = 0.0225
($\text{x}_{EDS}$ = 0.027) in the main text for clarity. From this point
onwards, in this paper, $x$ and $x_{EDS}$ refer to the $x$ values determined
by EDS measurements in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ when referring
to Ni substitutions. In the rare case when we need to refer to the nominal Ni
concentration in the growth melt we will explicitly use the $x_{nominal}$
notation.

Figure 2 shows a representative powder x-ray pattern for
${\text{La}_{5}\text{(Co}_{0.964}\text{Ni}_{0.036})_{2}\text{Ge}_{3}}$.
Similar x-ray patterns for crystals of each Ni substitution have been refined
and the lattice parameters $a$, $b$, $c$, and $\beta$ are obtained after
refinement of the powder diffraction data using GSAS IIToby and Von Dreele
(2013). The peaks are matched with the expected peaks of
$\text{Pr}_{5}\text{Co}_{2}\text{Ge}_{3}$ monoclinic structure with space
group C2/m Lin  _et al._ (2017). Figure 3 shows the change in the lattice
parameters $a$, $b$ and $c$ of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ versus $x$.
There is resolvable decrease of the $b$-lattice parameter, perhaps a modest
increase in the $a$-lattice parameter but essentially no resolvable change in
the $c$-lattice parameter. $\beta$, the angle between $a$ and $c$, remains
almost constant as $x$ increases. Given that the difference in ionic radii
between Co and Ni is small and the substitution level is less than 0.20, it is
not surprising that there is little resolvable change in the lattice
parameters. The exact values of the lattice parameters with change in $x$
values are shown in the Table 1 in the Appendix.

### Magnetization

Anisotropic field and temperature dependent magnetization measurements were
performed on single crystalline samples for each Ni substitution. As was the
case for the pure $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ ($x$ = 0.00), for
finite $x$, $a*$ is the easy magnetic axis, $b$ is the hard axis, and $c$ is
intermediate Saunders  _et al._ (2020). To compare the evolution of magnetism
with $x$ thoroughly, we will present a summary of the $M(H)$ and the $M(T)$
data for the field along the $a$* direction in the main text for clarity. For
each Ni substitution, the anisotropic $M(H)$ and $M(T)$ along all the three
directions is shown separately in the Appendix (Figs A1 \- A13). The T = 1.8
K, $M(H)$ data in Fig 4(a) suggests that there is a sudden change in the low
field response of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$, from clear
ferromagnetic (FM) behavior for $x\leq$ 0.027 to antiferromagnetic (AFM)-like
$M(H)$ for $x\geq$ 0.036. This is further evident as we look at the 5 quadrant
$M(H)$ taken at T = 1.8 K for $x$ = 0.027, 0.036 and 0.074 in Fig 5. A finite
spontaneous moment is present for $x$ = 0.027 only. We do not observe any
significant hysteresis associated with domain pining for any of the Ni
substitutions.

The H = 1 kOe, ZFC $M(T)$ data shown in Fig 4(b) reveal a change in behavior
from FM to AFM as $x$ increases. For $x\geq$ 0.036 there is, upon cooling, an
increase in $M(T)$ followed by a local maximum and then a decreasing $M(T)$
that is consistent with a antiferromagnetic transition. Low field $M(T)$ was
measured at H = 20 Oe for $x$ = 0.027, 0.036 and 0.074 (Fig 6) for H$||a$*.
Even though there is a slight irreversibility for $x$ = 0.036 and 0.074, the
ZFC-FC splitting is much clearer below the ordering temperature for $x$ =
0.027 again suggesting a transformation of the magnetic ground state. Thus,
all of the above evidences in the $M(H,T)$ data suggest that there is an
evolution from a low temperature ferromagnetic state for $x$ = 0.00 to an
antiferromagnetic state for $x\gtrsim$ 0.04.

Figure 5: The 5 quadrant $M(H)$ taken at T = 1.8 K for $x$ = 0.027, 0.036 and
0.074 for $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ for
H$||$a*. (Inset: The $M(H)$ data is zoomed in to show the low field feature).

Although the high temperature paramagnetic regime was fit to a Curie-Weiss
like behavior in the temperature range of $20~{}\text{K}\leq T\leq
100~{}\text{K}$ for the parent compound Saunders  _et al._ (2020), the doped
compounds do not follow a Curie-Weiss like behavior with a temperature
independent contribution, $\chi~{}=~{}\frac{C}{T-\Theta}+\chi_{0}$ for the
above temperature range at a field of 1 kOe, and hence, we did not fit our
data and study the change of $\mu_{eff}$ and $\Theta$ with $x$.

Figure 6: Temperature dependent magnetization for $x$ = 0.027, 0.036 and 0.074
in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ taken in a
low field of H = 20 Oe for H$||$a* collected in ZFC-FC mode. (Inset: The same
data for the AFM Ni substituted ($x$ = 0.036 and 0.074) samples.

Figure 7: (Color online) The anisotropic, low temperature resistance data
normalized at 10 K for
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ single
crystals. The left hand column, (a-c) presents I$||b$ data and the right hand
column (d-f) presents I$||c$ data. For each current direction the data are
divided into three groupings based on the Ni concentrations. Panels (a) and
(d) are for $x$ = 0.0 and 0.014. (b) and (e) for $x$ = 0.016, 0.027, 0.036 and
0.039. (c) and (f) for the remaining x upto 0.186 of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.

.

### Resistance

The results of the anisotropic, zero-field, in-plane resistance measurements
normalized to T = 10 K, plotted for the temperature range $1.8~{}\text{K}\leq
T\leq 10~{}\text{K}$ are shown in Fig 7 (Normalized resistance data
(${R(T)}/{R(300~{}\text{K})}$) is shown over the full
$1.8~{}\text{K}-300~{}\text{K}$ temperature range in the Appendix in Figs A1
\- A13). For all the measured samples,
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ shows metallic
behavior for current applied both along the -$b$ and the -$c$ axis for
temperatures above the magnetic ordering. The residual resistance ratio
$(RRR=\rho(300~{}\text{K})/\rho(1.8~{}\text{K}))$ ranges between 2 - 4, the
value being $\sim$ 4 for $x$ = 0.00 Saunders  _et al._ (2020) and decreases as
$x$ increases to 0.186. In all the 6 panels in Fig 7 we see a change in the
behavior in resistance below $\sim$ 4 K as seen at ambient and high pressure
in the parent $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ compound Saunders  _et
al._ (2020); Xiang  _et al._ (2021). For $x>$ 0.014 a very clear, qualitative
anisotropy develops in the magnetically ordered state, becoming even clearer
for $x\geq$ 0.036.

In Figs 7 (a) and (d), for $x$ = 0.00 and 0.014, we see a sharp drop in the
resistance that is associated with a loss of spin-disorder scattering below
the FM transition at around 3.8 K for current applied both along $b$ and $c$
axis. The situation drastically changes as we increase the value of 0.036
$\leq x\leq$ 0.186. When the current is along the $b$-axis (Fig 7(b) and (c)),
we still see a drop in resistance indicating loss of spin disorder scattering,
but when the current is along the $c$-axis for 0.036 $\leq x\leq$ 0.186, (Fig
7 (e) and (f)), there is an increase in resistance below the transition
temperature. This anisotropic behavior of loss of spin disorder scattering
along the $b-$ axis and increase of resistance along the $c-$ axis persists as
we increase the amount of $x$ up to 0.125 and finally for the highest doping
we studied, $x$ = 0.186, we can no longer resolve a resistive feature,
associated with a transition particularly along the $c$-axis (See Fig 7(f) and
Fig A13). For the intermediate Ni dopings, the situation however, is more
interesting. For, $x$ = 0.016 and 0.027, when the current is applied along the
$c$-axis, there is first an increase in resistance, followed by a decrease
upon further cooling (See Fig 7(e)).

The persistent loss of spin-disorder scattering for I$||b$ coupled with the
evolution of an increasing resistance just below the transition temperature
for I$||c$ strongly suggest that the magnetic transition changes from
ferromagnetic (FM) to antiferromagnetic (AFM). The increase of resistance at a
particular temperature is suggestive of the formation of super zone-gap due to
the nesting of the Fermi surface.Monceau (2012); Bud’ko and Canfield (2000);
Elliott and Wedgwood (1963). This in turn reveals that as we increase the
doping from 0.036 and beyond, the system enters a low-temperature (T $\sim$
1.8 K) state which is different from the FM state and this new state is most
likely characterized by an AFM component.
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ with $x$
between 0.016 and 0.027 show both features, suggesting competing interactions
characteristic to both FM and AFM behavior. The transformation from a FM
ground state to an AFM ground state and a distinct directional anisotropy in
the temperature dependent resistance measurements is similar to that of
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ under pressure for both $b$ and the
$c$ directions.Xiang  _et al._ (2021)

### Heat Capacity

To further investigate the low temperature phase transition(s) in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ with change of
$x$, zero-field, temperature dependent heat capacity was measured on
representative $x$ valued samples. We specifically chose $x$ = 0.00, 0.014,
0.016, 0.027, 0.036, 0.039 and 0.074 as these lie on the either side of the
transition boundary between FM and AFM ground state.

The results of the specific heat measurements are shown in Fig 8, where we see
a $\lambda-$ like feature for all the samples measured, which changes subtly
but clearly as we change $x$. The anomaly is consistent with the second order
nature of the transition. For $x$ = 0.00, the anomaly associated with the
ferromagnetic transition is sharp and seen at $\sim$ 3.9 K. Saunders  _et al._
(2020) For higher $x$ the anomaly is less steep and broader.

The evolution of the transition temperatures can be seen more clearly in the
$\frac{d(C_{p}(T)}{dT}$ data plotted in Fig 9. We can see a single transition
for $x$ = 0.00, 0.014 and 0.016 which splits into two transitions for $x$ =
0.027 and again becomes a single transition for the high dopings ($x$ =
0.074).

Figure 8: (Color online) Temperature dependent specific heat data for $x$ =
0.00, 0.014, 0.016, 0.027, 0.036, 0.039 and 0.074 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ shown in
different panels for clarity.

### Phase Diagram

Figure 9: (Color Online) $\frac{d(C_{p}/T)}{dT}$ versus T for x = 0.014,
0.016, 0.027, 0.036, 0.039 and 0.074. The minima of the curves are taken as a
measure of the transition temperature ($T_{mag}$). (Inset: The data has been
zoomed in clarity where the $T_{mag}$ has been marked with circles. The two
minima for x = 0.027 are evident which gives rise to the two transition
temperature for this Ni-substitution. Figure 10: The T-x phase diagram of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ is plotted with
the transition temperature obtained from different measurements. (a) $T_{mag}$
is plotted based on the results of temperature dependent resistance data, (b)
$T_{N}$ is plotted based on the results of temperature dependent magnetization
data. The magnetic ground state based from $M(H)$ at 1.8 K and the FM
transition temperature obtained from the Arrott plots for $x$ = 0.027 are also
included. (c) $T_{mag}$ is plotted based on the results of heat capacity
measurements. The details on determination of the transition temperatures and
the different regions marked in the phase diagram are discussed in detail in
the text.

From the resistance, magnetization and heat capacity measurements, the
temperature-composition (T-x) phase diagram of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ can be
constructed. The phase diagrams obtained from each of these measurements will
be presented separately first, for clarity, and then combined together to
summarize the effect of Ni substitution on this system.

In the T-x phase diagram obtained from the $R(T)$ measurements (Fig 10(a)),
for the low Ni dopings, (${0.00\leq x\leq 0.014}$) which have a loss of spin
disorder scattering in the data for measurements both along I$||b$ and I$||c$
directions (Fig 7)(a) and (d)) and manifest a FM ground state, (Fig 4) the
transition temperature, $T_{mag}$, is determined from the intersection of the
two dashed lines as shown in Fig A15 in the Appendix Saunders  _et al._
(2020); Xiang  _et al._ (2021). For the intermediate dopings, (0.016 $\leq$ x
$\leq$ 0.027), where features characteristic to both loss of spin disorder as
well as opening of a superzone gap are observed for measurement along I$||c$
(Fig 7 (b) and (e)), the $T_{mag}$ is determined following the same protocol
as above for both directions. The transition temperature ($\text{T}_{mag}$)
associated with the antiferromagnetic Ni substitutions which manifest an
opening of a superzone gap are determined from the $\frac{d\rho}{dT}$ data
Fisher and Langer (1968) and is shown in Fig A16 \- FigA22. Based on all this,
in the phase diagram (Fig 10(a)) the low Ni dopings having a loss of spin
disorder along both $b$ and $c$ direction are shown in blue and the high
dopings which show opening of a superzone gap along I$||$c and loss of spin
disorder along I$||$b, in green. The intermediate substitutions which show
both features along I$||$c are highlighted using a red box.

In order to understand the nature of the two transitions for the intermediate
dopings (0.016 $\leq x\leq$ 0.027) in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$, field
dependent magnetization was measured along H$||$a* at different constant
temperatures for $x$ = 0.027. The results of these measurements are shown in
Fig A23 and explained in Appendix. Based on the results, Arrott plots, shown
in Fig 11, were constructed to determine the ferromagnetic transition
temperature ($T_{C}$). ${M}^{2}$ vs $H/M$ plotted for $T<T_{C}$ have a
positive ${M}^{2}$ intercept, and have a negative intercept ${M}^{2}$ for
$T>T_{C}$, and the curve at $T\sim T_{C}$ passes through the origin Arrott
(1957). From our data (Fig 11), the FM $T_{C}$ is $\sim$ 2.6 K. This
temperature is consistent with the lower transition temperature obtained from
the $R(T)$ data. From all the above discussed reasons, we can say that the
lower temperature transition of the intermediate dopings (0.016 $\leq x\leq$
0.027) in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ is
ferromagnetic in nature.

Figure 11: (Color Online) $M^{2}$ as a function of H/M (Arrott Plot) obtained
by measuring field-dependent magnetization at different, constant temperatures
for x = 0.027 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. The data used
to make the Arrott Plot is shown in Fig A23 in the Appendix.

The T-x phase diagram obtained from the magnetization measurements is shown in
Fig 10(b), there is a clear demarcation of the AFM region, shown in green for
$x$ = 0.036 and higher Ni substitutions. The field dependent magnetization at
T = 1.8 K identifies the low temperature state of this system, which is also
shown. The blue points denotes a FM ground state or the state with FM
component, and the green data points are for the AFM ground state. The
transition temperature, $T_{N}$, for the AFM Ni-substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ is obtained
from the $M(T)$ data by using $\frac{d(\chi T)}{dT}$ as a proxy Fisher (1962),
following the same procedure as used to determine $\text{T}_{mag}$ from the
$R(T)$ data for the AFM samples by taking the point of maximum slope. Figures
A16(b) - A22(b) in Appendix, show the $\frac{d(\chi T)}{dT}$ data and the
criteria for $T_{N}$ for each direction of the applied field as well as the
polycrystalline average for the other AFM Ni substitutions. The FM transition
temperature for $x$ = 0.027, obtained from the Arrott plots is also shown. For
the lower x values, with a FM ground state, (${0.00\leq x\leq 0.027}$), the
analysis of the $M(T)$ data does not provide quantitative insight into the
value of the transition temperature and hence there are no data points in the
phase diagram for the FM Ni substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.

The phase diagram, Fig 10(c), is obtained from the heat capacity measurements.
The transition temperature $T_{mag}$ is obtained from the
$\frac{d(C_{p}/T)}{dT}$ data. The point of maximum slope for $C_{p}(T)$ and
the minima of $\frac{d(C_{p}/T)}{dT}$ are similar, within the error bar, and
hence for simplicity the minima point is taken as $T_{mag}$. Figure 9 shows
the $\frac{d(C_{p}/T)}{dT}$ data where the $T_{mag}$ is marked with circles
with each $x$. The two minima for $x$ = 0.027, observed at T = 2.7 K and 3.0 K
are similar to the two transitions observed in the resistivity data for the
same doping when measured along I$||$c direction, indicating the presence of
two transitions. Heat capacity for the x = 0.036 sample was also measured down
to T = 0.5 K using a He-3 inset, details are discussed in the Appendix. The
two transitions are observed at 3.3 K and $\sim$ 2 K (Fig A14) and has been
marked in the phase diagram as well.

From all the above discussed data, it is clear that the transition
temperatures obtained from magnetization, resistance and heat capacity are
similar and consequently the phase diagrams (Fig 10) are qualitatively and
quantitatively similar.

### Muon Spin Rotation/Relaxation ($\mu$SR)

#### Zero-field (ZF)

The data and the phase diagrams discussed above show that the magnetic
ordering in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$
evolves from a FM state to an AFM state with increasing x, with an
intermediate range of x where both transitions are observed. The magnetic
phases suggested by our phase diagrams can be further explored by conducting
both weak transverse field (wTF) and zero field (ZF) $\mu$SR experiments for
$x$ = 0.027, 0.036 and 0.074 single crystals by studying the time evolution of
the muon asymmetry Réotier and Yaouanc (1997). The results of the ZF
measurements will be discussed here whereas the results of the wTF data will
be shown in the Appendix. For completeness, in this work, we will compare the
results of our ZF measurements with the $\mu$SR data on the parent
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ samples Saunders  _et al._ (2020).
In both the measurements, muons were implanted into the sample along the $a$*
direction (perpendicular to the plane of the plates).

Figure 12: ZF-$\mu$SR time dependent asymmetry for $x$ = 0.00, 0.027, 0.036
and 0.074 in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$
below the transition temperature. The $x$ = 0.00 data has been taken from
Ref.Saunders  _et al._ (2020). The solid lines are fit to Eq.1 (see text for
details). Figure 13: The internal field ($B_{int}$) obtained by fitting Eq. 1
to the ZF-$\mu$SR time spectra for $x$ = 0.00, 0.027, 0.036 and 0,074 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. The vertical
dashed lines in panel for $x$ = 0.027, 0.036 and 0.074 are the average of the
transition temperatures obtained from the transport, magnetization and heat
capacity measurements discussed earlier. The data for $x$ = 0.00 is from
Ref.Saunders  _et al._ (2020). The two curves for $x$ = 0.00 is due to the
assumption of two muon stopping sites ($\sim 20\%$ at the higher internal
field site and $\sim 80\%$ at a lower field site).

In ZF-$\mu$SR experiments, the muon spins precess about the internal field at
the muon stopping site and oscillations are observed only when the initial
polarization is perpendicular to the field and is absent for a parallel
internal field Réotier and Yaouanc (1997). As muons were implanted along the
$a$* direction of the crystals, we will label the component of the
polarization perpendicular [parallel] to $a$* as P$\perp$a* [P$||$a*]
respectively. Figure A24 in the Appendix shows the time evolution of muon
asymmetry for $x$ = 0.027 both perpendicular and parallel to $a$*. As
oscillations are observed in both P$\perp$a* and P$||$a* sets of data, it can
be concluded that the internal field along the muon stopping sites has
components both parallel and perpendicular to the crystallographic $a$
direction of the
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ crystals.

Figure 12 show the ZF-$\mu$SR time spectra for $x$ = 0.00, 0.027, 0.036 and
0.074 in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ in the
ordered state (T $\sim$ 1.5 K). The data for $x$ = 0.00 has been published in
our previous work on this system Saunders  _et al._ (2020). The muon asymmetry
for the parent compound is visually different from the Ni substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ samples with
the oscillations becoming much broader for all the substituted ones.

The ZF asymmetry can be analyzed by taking contributions from oscillating as
well as non-oscillating components and the data can be fit using,

$\begin{split}A_{ZF}(t)&=A(0)[(1-f)e^{-\lambda_{L}t}\\\
&+f\sum\limits_{i=0}^{n}e^{-\lambda_{T,i}}\cos(\gamma_{\mu}B_{int,i}t)]\end{split}$
(1)

where, $A(0)$ is the initial asymmetry, $f$ is the oscillating fraction,
$\lambda_{L}$ is the longitudinal relaxation rate and $\gamma_{\mu}/2\pi$ =
135.5 MHz/T is the gyromagnetic ratio of the $\mu$SR signal. $B_{int,i}$ and
$\lambda_{T,i}$ are respectively the internal field and the transverse
relaxation rate of the $i^{th}$ component at the muon stopping site in the
sample. The ZF data set for the parent ($x$ = 0.00) compound was analyzed by
assuming two inequivalent muon stopping sites ($\sim 20\%$ at the higher
internal field site and $\sim 80\%$ at a lower field site) Saunders  _et al._
(2020). However, for the Ni substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ samples the
data are fit to Eq.1 by assuming one stopping site. When fit using two sites,
the obtained parameters are very close to each other and hence for simplicity
we use only a single site.

The value of the internal field at the muon stopping position is obtained by
fitting Eq. 1 to the ZF spectra. The internal field is determined by the
surrounding magnetic moments and is proportional to the value of ordered
moments for the different Ni substituted
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ samples
studied. Consequently, the temperature dependence of $B_{int}$ reflects the
evolution of the magnetic order parameter for each $x$ as shown in Fig 13.

For $x$ = 0.027, the internal field decreases with increasing temperature up
to T = 2.54 K after which we see a change in the behavior that continues until
T = 3.1 K, above which the internal field abruptly drops to zero, indicating
the onset of the paramagnetic phase. The temperatures at which we see abrupt
changes in the internal field are consistent with the two transition
temperatures (2.66 K and 3.05 K) observed in the $R(T)$ and ${C}_{p}(T)$
measurements. Thus, the $\mu_{SR}$ data clearly suggests that there is 1) a
low T state with the highest internal field , 2) a high T PM state with no
internal field and 3) an intermediate temperature magnetic state with a finite
internal field having a different temperature dependence. Combining this with
the previous data magnetization and transport data, it can be interpreted that
the lowest T state below ($\sim$ 2.66 K) is FM with emergence of spontaneous
moment in $M(H)$ isotherms (Fig A23 in the Appendix) whereas, the intermediate
T region in the $\mu_{SR}$ data (2.66 K $\leq x\leq$ 3.05 K) is AFM. For the
other two Ni substituted ($x$ = 0.036 and 0.074) measured samples, we see a
monotonic decrease of the internal field with increasing temperature up to the
transition. The vertical dashed lines in each of the panel in Fig 13 are
magnetic transition temperatures obtained from the previously discussed
transport, magnetization and heat capacity measurements. The T dependence of
the internal field for $x$ = 0.00 is also included for comparison which has
been published in our previous work Saunders  _et al._ (2020) where the two
internal fields, due to two muon stopping sites ($\sim 20\%$ at the higher
internal field site and $\sim 80\%$ at a lower field site) decreases
monotonically up to 3.8 K at the onset of the paramagnetic phase.

##  IV Conclusion and Discussion

Figure 14: (Color online) The T-x phase diagram of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ obtained from
anisotropic resistance and magnetization as well as specific heat and $\mu$SR
data. The three distinct regions observed are the low $x$ ferromagnetic (FM)
marked by blue, the high $x$ antiferromagnetic (AFM) marked by green and a
high temperature paramagnetic (PM) region. The intermediate $x$ $(0.016\leq
x\leq 0.027)$ have two transitions and exhibit properties characteristic to
both FM and AFM behavior.

Based on all the three separate phase diagrams (Fig 10) and the ZF $\mu$SR
data analysis, a composite T-x phase diagram is constructed to summarize the
behavior of $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ as Co is substituted
with Ni (Fig 14). The phase diagram has three regions namely, the high
temperature paramagnetic (PM) region $(T\gtrsim 4K)$ shown in white, the low
$x$ ferromagnetic (FM) region $(x\leq 0.016)$ in blue and the high $x$
$(0.036\leq x\leq 0.186)$ antiferromagnetic (AFM) region marked in the green
shaded area. Although we do not have enough low temperature data (T $<$ 1.8
K), we anticipate that the FM state may well exist below 1.8 K for $x\gtrsim$
0.05, but we do not know its precise fate. For the intermediate $x$
$(0.016\leq x\leq 0.027)$ we observe features with two transitions and
properties characteristic to both FM and AFM behavior, which is highlighted in
a red box. The FM transition temperature decreases with $x$, whereas the AFM
transition temperature increases very slightly to 3.4K for $x\sim 0.04$ and
then decreases to roughly 2.5K for $x\sim 0.19$.

$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$, an itinerant ferromagnet with a
small saturated moment of $0.1~{}\mu_{B}/\text{Co}$, a low transition
temperature of 3.8 K and a Rhodes-Wohlfarth ratio of 4.9 was presumed to be a
promising candidate to suppress the magnetic order to low temperatures
Saunders  _et al._ (2020). When perturbed with hydrostatic pressure it was
observed that the system remains FM up to a pressure of 1.5 GPa and the
$\text{T}_{C}$ is weakly suppressed to $\sim$ 3.0 K. After this the magnetic
state transforms to a new, non FM-state where the transition temperature
depends non monotonically on the applied pressure, Xiang  _et al._ (2021)
indicating that $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ is another FM system
that avoids the quantum critical point through the emergence of a new,
spatially modulated phase. Substitution of Co with Ni introduces a different
type of perturbation by introducing additional electrons; however, like what
is observed under applied pressure, we also find quantum criticality is again
avoided and that Ni substitution transforms the magnetic state from FM to AFM.
The T-x phase diagram of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ reported here
is qualitatively similar to the T-p phase diagram of
$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ under hydrostatic pressure Xiang
_et al._ (2021).

.

The zero field (ZF) $\mu$SR measurements support the temperature dependent
magnetization, resistance and specific heat measurements. Signature of two
transitions are detected for $x$ = 0.027 in the temperature evolution of
internal field. The transition temperatures obtained for $x$ = 0.036 and 0.074
agree our other measurement results. The ZF time spectra differs between the
parent and the Ni substituted samples. Based on the values of the obtained
internal field, we do not infer a large change in the size of the magnetic
moment with Ni doping.

To summarize, single crystals of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ were
synthesized with $x$ varying between 0.00 and 0.186. Powder X-Ray diffraction
and EDS measurements confirmed the phase and an estimate of the Ni going in
the system respectively. Magnetization, resistance and heat capacity
measurements were performed at ambient pressure and a T-x phase diagram of the
system was constructed. For the low dopings, between ${0.00\leq x\leq 0.014}$,
the system remains ferromagnetic and the transition temperature is suppressed
to 3.2 K. A change in the ground state occurs at ${x=0.036}$ when the
magnetization measurements indicate the appearance of an AFM state at around
3.3 K. The single AFM transition is found to first weakly increase to a
shallow maximum $\text{T}_{N}\sim$ 3.4 K near x $\sim$ 0.07 and then slowly
decrease to T = 2.4 K for x = 0.186. The intermediate Ni substitutions,
$0.016\leq x\leq 0.027$, show both transitions with
$\text{T}_{N}>\text{T}_{C}$. ZF $\mu$SR measurements and wTF measurements (in
the Appendix) on $x$ = 0.027, 0.036 and 0.074 give a measure of transition
temperature which agrees with the other results. Given that relatively a small
substitution level of Ni is needed to drive the magnetic ordering to the AFM
state ($x\sim$ 0.03), such lightly doped samples are in themselves promising
materials to study under pressure to see if an AFM QCP can be more readily
reached.

## ACKNOWLEDGEMENTS

We would like to thank E. Gati and D.H. Ryan for useful discussions regarding
this system and $\mu$SR analysis. This work was done at Ames National
Laboratory and supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division. Ames
National Laboratory is operated for the U.S. Department of Energy by Iowa
State University under Contract No. DE-AC02-07CH11358.

## APPENDIX

Temperature dependent normalized resistance, magnetization at H = 1 kOe as
well as field dependent magnetization at T = 1.8 K data are plotted separately
for each x in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$
as shown in Fig A1 \- Fig A13.

Figure A1: (Color online) (a): Zero field, temperature dependent normalized
resistance (Inset: The low temperature normalized $R(T)$ behavior shown for
1.8 K $\leq$ T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1
kOe for 1.8 K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a
base temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H
$\leq$ 4 kOe for $x$ = 0.00 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A2:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.014 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. Figure A3:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.016 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A4:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.026 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A5:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.027 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. Figure A6:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.027 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A7:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.036 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. Figure A8:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.039 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A9:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K) for x = 0.076, (b): Temperature dependent magnetization at H =
1 kOe for 1.8 K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a
base temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H
$\leq$ 4 kOe for $x$ = 0.074 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A10:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.092 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. Figure A11:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.120 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A12:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.125 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. Figure A13:
(Color online) (a): Zero field, temperature dependent normalized resistance
(Inset: The low temperature normalized $R(T)$ behavior shown for 1.8 K $\leq$
T $\leq$ 10 K), (b): Temperature dependent magnetization at H = 1 kOe for 1.8
K $\leq$ T $\leq$ 10 K, (c): Field dependent magnetization at a base
temperature of 1.8 K shown for H $\leq$ 4 kOe and in the inset for H $\leq$ 4
kOe for $x$ = 0.186 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$.  Figure A14:
The Specific heat data for x = 0.036 measured using a 3-He insert to PPMS
along with its derivative plotted in the right hand axis. A sharp transition
is observed at around 3.3 K, and another subtle one $\sim$ 2 K which is not
observes in any other measurement.

As $x$ = 0.036 lies at the border where we start seeing only a single AFM
transition in the $M(T)$ and opening of a super-zone gap in the $R(T)$ data,
it was probed to low temperatures by measuring specific heat down to 0.5 K
using a 3-He inset. Fig A14 shows transitions around $\sim$ 3.3 K, also
observed in specific heat data down to base temperature of 1.8 K and another
at $\sim$ 2 K, which is not observed in the data of previously discussed
magnetization and resistance measurements.

Turning to normalized resistance data, the transition temperature ($T_{mag}$)
of the Ni substituted samples manifesting a FM ground state is determined from
the intersection of the two dashed lines for measurements along I$||$b and
I$||$c. $T_{mag}$ is marked using a vertical dashed line for each of the
substitutions in Fig A15.

$T_{mag}$ for the AFM Ni substitutions is inferred from the $\frac{d\rho}{dT}$
plots and is determined by construction of three dashed lines in the low-,
intermediate-, and high-temperature regimes. The intermediate temperature line
goes through the point of maximum slope of the $\rho$(T). $\text{T}_{mag}$ is
determined as the midpoint of the intersection points of the three dashed
lines, and the uncertainty in $\text{T}_{mag}$ is obtained from the
temperature difference of the two intersection points Xiang  _et al._ (2021).
$\text{T}_{mag}$ for the AFM Ni substitutions is marked using a vertical line
in A16(a) - A22(a).

Figure A15:  (Color online) The temperature dependent normalized resistance
along both b- and c- axis of the low Ni substitutions which show ferromagnetic
behavior in $M(H)$ measurements at T = 1.8 K. The transition temperature
($T_{mag}$) is marked using a vertical line as per the criteria used (See text
for details). Figure A16:  (Color online) (a) $\frac{d\rho}{dT}$ and (b)
$\frac{d(\chi T)}{dT}$ for x = 0.036 which is used as a criteria to obtain the
transition temperature for the AFM states by taking the point of maximum
slope. In panel (a), the values for current along the b-direction is shown in
the left axis and values for current along the c-axis is in the right axis.
The criteria for obtaining the transition temperature is marked for I$||$c in
(a) and for H$||$a* in (b). The vertical line in each of the panel are the
($T_{mag}$) and ($T_{N}$) are the average transition temperatures obtained
from $\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots respectively. Figure
A17:  (Color online) (a) $\frac{d\rho}{dT}$ and (b) $\frac{d(\chi T)}{dT}$ for
x = 0.039 which is used as a criteria to obtain the transition temperature for
the AFM states by taking the point of maximum slope. In panel (a), the values
for current along the b-direction is shown in the left axis and values for
current along the c-axis is in the right axis. The vertical line in each of
the panel are the ($T_{mag}$) and ($T_{N}$) are the average transition
temperatures obtained from $\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots
respectively. Figure A18:  (Color online) (a) $\frac{d\rho}{dT}$ and (b)
$\frac{d(\chi T)}{dT}$ for x = 0.074 which is used as a criteria to obtain the
transition temperature for the AFM states by taking the point of maximum
slope. In panel (a), the values for current along the b-direction is shown in
the left axis and values for current along the c-axis is in the right axis.
The vertical line in each of the panel are the ($T_{mag}$) and ($T_{N}$) are
the average transition temperatures obtained from $\frac{d\rho}{dT}$ and
$\frac{d(\chi T)}{dT}$ plots respectively. Figure A19:  (Color online) (a)
$\frac{d\rho}{dT}$ and (b) $\frac{d(\chi T)}{dT}$ for x = 0.092 which is used
as a criteria to obtain the transition temperature for the AFM states by
taking the point of maximum slope. In panel (a), the values for current along
the b-direction is shown in the left axis and values for current along the
c-axis is in the right axis. The vertical line in each of the panel are the
($T_{mag}$) and ($T_{N}$) are the average transition temperatures obtained
from $\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots respectively. Figure
A20:  (Color online) (a) $\frac{d\rho}{dT}$ and (b) $\frac{d(\chi T)}{dT}$ for
x = 0.120 which is used as a criteria to obtain the transition temperature for
the AFM states by taking the point of maximum slope. In panel (a), the values
for current along the b-direction is shown in the left axis and values for
current along the c-axis is in the right axis. The vertical lines in each of
the panel are the ($T_{mag}$) and ($T_{N}$) are the average transition
temperatures obtained from $\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots
for each direction respectively. Figure A21:  (Color online) (a)
$\frac{d\rho}{dT}$ and (b) $\frac{d(\chi T)}{dT}$ for x = 0.125 which is used
as a criteria to obtain the transition temperature for the AFM states by
taking the point of maximum slope. In panel (a), the values for current along
the b-direction is shown in the left axis and values for current along the
c-axis is in the right axis. The vertical lines in each of the panel are the
($T_{mag}$) and ($T_{N}$) are the average transition temperatures obtained
from $\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots for each direction
respectively. Figure A22: (Color online) (a) $\frac{d\rho}{dT}$ and (b)
$\frac{d(\chi T)}{dT}$ for x = 0.186 which is used as a criteria to obtain the
transition temperature for the AFM states by taking the point of maximum
slope. For the $\frac{d(\chi T)}{dT}$ along H$||$b, we do not get any visible
feature. In panel (a), the values for current along the b-direction is only
shown. The vertical line in each of the panel are the ($T_{mag}$) and
($T_{N}$) are the average transition temperatures obtained from
$\frac{d\rho}{dT}$ and $\frac{d(\chi T)}{dT}$ plots respectively. Figure A23:
(Color online) $M(H)$ taken at different temperatures (1.8 K $\leq$ T $\leq$
4.2 K) for x = 0.027 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ to show the
evolution of ground state from a ferromagnetic (T $\leq$ 2.6 K) to a non
ferromagnetic state.

The $M(H)$ data on the $x$ = 0.027 sample measured for $T\leq$ 4.2 K are shown
in Fig A23. The Arrott plots ($M^{2}~{}\text{vs}~{}H/M$), shown in Fig 11, are
constructed based on Fig A23 data. A finite spontaneous moment clearly
indicates FM behavior. Data taken for T $\geq$ 2.6 K has no finite intercept
suggesting non-FM behavior (negative intercept in Fig 11).

The $\mu$SR experiments (both ZF and wTF) were performed in a spin-rotated
mode, where the initial muon spin polarization ($P_{\mu}$) is turned by a
certain angle relative to muon beam momentum Khasanov  _et al._ (2020). Such
geometry is used to measure single crystalline samples.

Figure A24: Muon asymmetry for x = 0.027 collected at T $\sim$ 1.5 K with the
component of the initial muon polarization perpendicular and parallel to a*.
Figure A25: wTF-$\mu$SR time dependent asymmetry for x = 0.027, 0.036 and
0.074 in $\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. For
each substitution we have data for temperatures above and below the magnetic
transition as shown in black and red symbols separately. The solid lines are
fit to Eq. 3 (see text for details).

Weak transverse field (wTF) $\mu$SR experiments were done by applying an
external field of $H_{ex}$ = 30 Oe perpendicular to the muon spin polarization
to determine the onset temperature of the magnetic transitions in the samples.
The polarized muons were implanted along the a* direction as discussed
earlier. In the paramagnetic state, muons experience the spatially invariant
external field and produce long-lived oscillations, reflecting the coherent
muon precession around the external field. In the ordered state, the muon
asymmetry has a complex precession due to vector combination of the internal
field of the sample and the externally applied field.

As the wTF muon spectra was measured both above and below the magnetic
transition ($\approx$ 3 K), the time evolution of the muon asymmetry can be
assumed to have non-magnetic (nm) and magnetic (m) contributions to the data
and can be written as a combination of the two contributions;

$\begin{split}A(t)&=A(0)P(t)\\\
&=A_{nm}(0)P_{nm}(t)+A_{m}(0)P_{m}(t)\end{split}$ (2)

where, $A_{nm}(0)[A_{m}(0)]$ are the initial asymmetry and
$P_{nm}(t)[P_{m}(t)]$, the time evolution of the muon spin polarization in the
non-magnetic and [magnetic] states respectively.

Figure A26: Temperature dependence of the initial asymmetry A(0) obtained by
fitting Eq.3 to the wTF muon spectra for x = 0.027, 0.036 and 0.074 in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$. The black
solid line is the fit of Eq.4 to A(0). The vertical black line in each panel
is the $T_{mag}$ and the grey shaded area is the $\Delta T_{mag}$ obtained
from the same fit.

The wTF data in the paramagnetic state can be analyzed using only the non-
magnetic sample contribution to the $\mu$SR asymmetry. The magnetic
contribution [$A_{m}(0)P_{m}(t)$] in the ordered state vanishes by $\sim 1\mu
s$ and thus we will also analyze the wTF ordered state data using the non-
magnetic contribution for simplicity after avoiding the early time data. The
non-magnetic wTF data is fit using;

$\begin{split}A_{wTF}(t)&=A(0)P(t)\\\
&=A(0)cos(\gamma_{\mu}B_{ext}t~{}+~{}\phi)e^{-\sigma^{2}t^{2}/2}\end{split}$
(3)

where, A(0) is the initial asymmetry, $\gamma_{\mu}/2\pi$ = 135.5 MHz/T is the
gyromagnetic ratio of the muons, $\phi$ the initial phase of the muon spin
ensemble and $\sigma$ is the Gaussian relaxation rate caused by nuclear
moments. The solid lines in Fig A25 are fits to this equation. The black
points for all the three substitutions represent data above the magnetic
transition showing coherent muon precession whereas the red points are data
taken with the sample in a magnetically ordered state. The fit lines for data
taken in the magnetically ordered state (red line) were not fit to and do not
pass through the early time data points (t $\sim$ 1 $\mu s$) as already
discussed.

$x$ | $a(\AA)$ | $b(\AA)$ | $c(\AA)$ | $\beta(\degree)$ | $V(\AA)^{3}$ | $wR$ | $GOF$
---|---|---|---|---|---|---|---
0.00 | 18.332(4) | 4.3421(2) | 13.256(3) | 103.99(1) | 1023.7(1) | 8.79 | 1.50
0.014(6) | 18.330(7) | 4.3409(4) | 13.254(5) | 103.96(2) | 1023.6(3) | 12.16 | 1.84
0.016(6) | 18.328(3) | 4.3421(2) | 13.260(2) | 103.99(1) | 1023.9(1) | 10.73 | 1.57
0.026(4) | 18.333(5) | 4.3409(3) | 13.253(3) | 104.27(1) | 1023.8(2) | 12.54 | 1.66
0.027(6) | 18.341(3) | 4.3417(2) | 13.258(2) | 103.96(1) | 1024.6(1) | 10.50 | 1.52
0.027(8) | 18.351(4) | 4.3415(3) | 13.253(3) | 103.89(2) | 1025.1(2) | 10.32 | 1.50
0.036(7) | 18.342(6) | 4.3411(4) | 13.249(4) | 103.95(2) | 1024.1(2) | 10.22 | 1.52
0.039(5) | 18.343(5) | 4.3417(3) | 13.253(3) | 103.94(2) | 1024.3(3) | 9.52 | 1.52
0.076(7) | 18.360(6) | 4.3386(4) | 13.259(4) | 103.93(3) | 1025.9(2) | 9.37 | 1.53
0.092(7) | 18.342(4) | 4.3385(3) | 13.258(3) | 103.99(2) | 1023.7(1) | 11.05 | 1.69
0.120(7) | 18.349(3) | 4.3389(2) | 13.265(2) | 104.21(1) | 1024.4(2) | 15.00 | 2.17
0.125(8) | 18.356(5) | 4.3378(3) | 13.255(3) | 103.92(2) | 1024.6(2) | 9.86 | 1.63
0.186(10) | 18.355(3) | 4.3364(2) | 13.262(2) | 104.05(1) | 1024.1(2) | 12.31 | 1.71
Table 1: The values of the lattice parameters a,b,c, angle $\beta$ between a
and c, and the volume of the unit cell V of the different x in
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ with the value
of x indicated in the leftmost column. The lattice parameters are obtained
from the refinement of the powder diffraction pattern. The wR and the GOF are
the goodness of the fit obtained.

To study the change of the wTF muon data with Ni substitution, we track the
temperature dependence of the initial time asymmetry A(0), inferred from Eq.
3, as a function of temperature for each x. Eq.4 determines the magnetic
ordering temperature, $T_{mag}$, and $\Delta T_{mag}$ is the width of the
magnetic transition.

$A(0,T)=A(0)\frac{1}{1+e^{([T_{mag}-T])/\Delta T_{mag})}}~{}+~{}A_{bg}(0)$ (4)

where, A(0) is the total initial asymmetry, $T_{mag}$ is the magnetic ordering
temperature and $\Delta T_{mag}$ is the width of the magnetic transition.
$A_{bg}(0)$ is the background contribution which represents the muons missing
the sample and stopping in the holder, cryostat walls, etc.

Figure A26 shows the temperature dependence of the initial asymmetry for the
measured Ni substituted samples. The transition temperatures obtained from
fitting the wTF data is similar to values obtained from the previously
discussed measurements. However, no sign of two transitions for x = 0.027 is
obtained from wTF $\mu$SR studies. The width of the transition for each of the
substitutions is small suggesting magnetic order sets in uniformly in the
samples. The value of $\Delta T_{mag}$ obtained from fitting Eq. 4 and is
comparable to the widths of transitions we obtain from our resistance,
magnetization and specific heat measurements.

Table 1 shows the exact values of the Ni substitution levels obtained from EDS
measurements on the single crystals of
$\text{La}_{5}\text{(Co}_{1-x}\text{Ni}_{x})_{2}\text{Ge}_{3}$ along with
lattice parameters a, b, c, the angle $\beta$, and volume of the unit cell
obtained from the refinement of the powder X-Ray diffraction patterns of
representative samples. The uncertainty associated with each parameter is also
shown in parenthesises.

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