# Inhomogeneous magnetic ordered state and evolution of magnetic fluctuations
in Sr(Co1-xNix)2P2 revealed by 31P NMR

Nao Furukawa  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Qing-Ping Ding  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Juan Schmidt  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Sergey L. Bud’ko  Ames National Laboratory, U.S. DOE, 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, Ames, Iowa
50011, USA  Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Yuji Furukawa  Ames National Laboratory, U.S. DOE, Ames, Iowa
50011, USA  Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

###### Abstract

SrCo2P2 with a tetragonal structure is known to be a Stoner-enhanced Pauli
paramagnetic metal being nearly ferromagnetic. Recently Schmidt et al. [Phys.
Rev. B 108, 174415 (2023)] reported that a ferromagnetic ordered state is
actually induced by a small Ni substitution for Co of $x$ = 0.02 in
Sr(Co1-xNix)2P2 where antiferromagnetic ordered phase also appears by further
Ni-substitution with $x=0.06-0.35$. Here, using nuclear magnetic resonance
(NMR) measurements on 31P nuclei, we have investigated how the magnetic
properties change by the Ni substitution in Sr(Co1-xNix)2P2 from a microscopic
point of view, especially focusing on the evolution of magnetic fluctuations
with the Ni substitution and the characterization of the magnetically ordered
states. The temperature dependences of 31P spin-lattice relaxation rate
divided by temperature ($1/T_{1}T$) and Knight shift ($K$) for SrCo2P2 are
reasonably explained by a model where a double-peak structure for the density
of states near the Fermi energy is assumed. Based on a Korringa ratio analysis
using the $T_{1}$ and $K$ data, ferromagnetic spin fluctuations are found to
dominate in the ferromagnetic Sr(Co1-xNix)2P2 as well as the antiferromagnets
where no clear antiferromagnetic fluctuations are observed. We also found the
distribution of the ordered Co moments in the magnetically ordered states from
the analysis of the 31P NMR spectra exhibiting a characteristic rectangular-
like shape.

##  I Introduction

Magnetic fluctuation is one of the important parameters to control the
physical properties of unconventional superconductors (SCs) such as high
$T_{\rm c}$ cuprates and iron-based SCs. The discovery of superconductivity in
iron pnictides in 2008 Kamihara2008  triggered considerable experimental and
theoretical attention on transition-metal (TM) pnictides Johnston2010 ;
Canfield2010 ; Stewart2011 . Among them, Co compounds with ThCr2Si2-type
structure, $A$Co2X2 ($A$ = alkaline earth or lanthanide, $X$ = pnictogen) have
been found to show a rich variety of magnetic properties with different
crystal structures. SrCo2P2 with an uncollapsed tetragonal (ucT) structure
shows no magnetic ordering and is a Stoner-enhanced Pauli paramagnet with
dominant ferromagnetic interactions Moresen1998 ; Jia2009 ; Imai2014 ;
Imai2015PP ; Imai2017 . In contrast, an A-type antiferromagnetic (AFM) state
has been reported in CaCo2P2 with a collapsed tetragonal (cT) structure below
a Néel temperature of 110 K, in which the Co moments are ferromagnetically
aligned in the $ab$ plane and the moments adjacent along the $c$ axis are
aligned antiferromagnetically Reehuis1998 ; Imai2017 ; Baumbach2014PRB . The
ordered Co moments in the AFM state are reported to be 0.32-0.35 $\mu_{\rm B}$
from the neutron diffraction (ND) measurements Reehuis1998  and nuclear
magnetic resonance (NMR) measurements Higa2018 . A ferromagnetic (FM) state is
also reported by replacing the divalent alkaline-earth ions by the trivalent
La ions in the metallic LaCo2P2 with an ucT structure below a Curie
temperature of $T_{\rm C}$ = 130 K where the saturated Co moment is reported
to be of 0.4 $\mu_{\rm B}$ Reehuis1994 ; Imai2015PRB ; Teruya_LaCo2P2 .

Magnetic ordering in Co-based compounds was also reported to be induced by Ca
substitution for Sr in (Sr1-xCax)Co2P2 Jia2009 ; Imai2017 ; Sugiyama2015PRB
and also by Ge substitution for P in SrCo2(P1-xGex)2 Moriyama2023 . In the
case of (Sr1-xCax)Co2P2, although the Stoner-enhanced Pauli paramagnetic state
is observed in the ucT phase with $x=0-0.5$, the A-type AFM ordering appears
in the cT phase with $x=0.6-1$ Jia2009 , indicating a strong relationship
between the magnetism and crystal structure. It is interesting to point out
that, in contrast to the Co compounds, no magnetic order has been observed in
the cT phase in Fe-based compounds such as CaFe2As2 Kreyssig2008 ; Pratt2009 ;
Prokes2010 ; Ran2011  where AFM fluctuations are also revealed to be
completely suppressed Kawasaki2010 ; Kawasaki2011 ; Soh2013 ; Furukawa2014 .
In the case of SrCo2(P1-xGex)2, a ferromagnetic ordered state has been
observed in $x$ = 0.42-0.61 which is an intermediate region between the ucT
(the SrCo2P2 side) and the cT (the SrCo2Ge2 side) phases and a ferromagnetic
quantum criticality has been discussed Moriyama2023 .

Figure 1: Phase diagram of Sr(Co1-xNix)2P2. $T_{\rm N}$ and $T_{\rm C}$ are
from Ref. Schmidt2023 . AFM and FM represent the antiferromagnetic and
ferromagnetic ordered states at a low magnetic field of 100 Oe, respectively.
Arrows indicate the positions of the representative of the Ni substitution
levels used in the present work.

Recently it was shown that, from the detailed investigation of the Ni
substituted SrCo2P2 compounds, Sr(Co1-xNix)2P2 can be systematically tuned to
have one of three different magnetic ground states: AFM, FM and paramagnetic
states by electron doping Schmidt2023 . Figure 1 shows a schematic phase
diagram of Sr(Co1-xNix)2P2 determined by magnetization and resistivity
measurements Schmidt2023 . Although $x$ = 1 (SrNi2P2) exhibits a one-third
collapsed orthorhombic structure (tcO) Schmidt2023 , the tcO phase is rapidly
suppressed with Co additions and the magnetically ordered phases appear only
in the ucT phase, which is different from the aforementioned other systems
(Sr1-xCax)Co2P2 and SrCo2(P1-xGex)2. With only a 2 % Ni substitution to the
nonmagnetic SrCo2P2, ordering with a ferromagnetic component is induced with
$T_{\rm C}$ = 4.3-6.6 K. With further Ni substitution, the AFM state appears
for 0.06 $\leq$ $x$ $\leq$ 0.35 with the highest $T_{\rm N}$ = 36 K around $x$
= 0.1-0.12. As the magnetic properties of Co compounds seem to largely depend
on crystal structure as well as the valence number of Co ions, Sr(Co1-xNix)2P2
is a good system to investigate only the carrier doping effect on the magnetic
properties without changing the ucT crystal structure. Up to now, no
investigation of the magnetic properties from a microscopic point of view or
magnetic fluctuations in Sr(Co1-xNix)2P2 has been reported. Since magnetic
fluctuations are considered to be one of the key ingredients to characterize
the physical properties of materials, it is crucial to reveal how magnetic
fluctuations vary with Ni substitution in Sr(Co1-xNix)2P2. NMR is an ideal
tool for the microscopic study of the magnetic properties and the low-energy
magnetic fluctuations in correlated electron systems. It is known that the
nuclear spin-lattice relaxation rate (1/$T_{1}$) reflects the wave vector
$q$-summed dynamical susceptibility. On the other hand, NMR spectrum
measurements, particularly the Knight shift ($K$), give us information on the
static magnetic susceptibility $\chi$. Thus from the temperature dependence of
1/$T_{1}$ and $K$, one can obtain valuable insights into magnetic fluctuations
in materials.

In this paper, we carried out 31P NMR measurements to examine the local
magnetic properties of Sr(Co1-xNix)2P2. The characteristic temperature
dependences of $K$ and 1/$T_{1}T$ exhibiting a double-maxima around 25 K and
125 K in the parent compound SrCo2P2 were reasonably explained by a model
where a double-peak structure of the density of states (DOS) near the Fermi
energy is assumed. Our analysis, based on the modified Korringa relation,
reveals electron correlations enhanced around FM wavenumber $q$ = 0 in
SrCo2P2. We find the FM spin fluctuations are strongly enhanced with the only
2 % Ni substitution in the ferromagnetic Sr(Co0.98Ni0.02)2P2. Similar strong
FM spin fluctuations are also found to dominate in an antiferromagnet
Sr(Co0.87Ni0.13)2P2 with $T_{\rm N}$ = 30 K. With further Ni substitution up
to $x$ = 0.57 through $x$ = 0.42, such FM spin fluctuations as well as the
magnetic ordering completely disappear, leading to a noncorrelated metal where
a typical Korringa behavior of 1/$T_{1}T$ = const. was observed without
showing any indication of quantum critical behavior around $x$ = 0.42 in
Sr(Co1-xNix)2P2. Before proceeding, we would like to point out that in Ref.
Schmidt2023  samples were presented as Sr(Ni1-xCox)2P2; in this work we
present our data based on the formula Sr(Co1-xNix)2P2 since the magnetic
ordering we are interested in develops for $x$ as small as 0.02 and it
disappears before $x$ = 0.50.

##  II Experimental details

The single crystals of Sr(Co1-xNix)2P2 ($x$ = 0, 0.02, 0.13, 0.42 and 0.57)
used in this study were grown out of Sn flux Schmidt2023 , using conventional
high-temperature growth techniques Canfield2020 . The sizes of the crystals
are similar ($\sim$$3\times 4\times 0.1$ mm3). The Ni substitution levels of
the single crystals used in this study were determined by wavelength
dispersive x-ray spectroscopy, and the crystals are characterized by
magnetization and resistivity measurements Schmidt2023 . No magnetic ordering
was observed for $x$ = 0 (pure SrCo2P2), $x$ = 0.42 or 0.57 for $T$ $>$ 2 K
from the magnetization measurements, while a ferromagnetic ordering at $T_{\rm
C}$ = 4.3 K and an antiferromagnetic ordering at $T_{\rm N}$ = 30 K were
reported under a weak magnetic field of 100 Oe for our crystals $x$ = 0.02 and
0.13, respectively Schmidt2023 . Details of the characterization of the
crystals are reported in Ref. Schmidt2023 .

31P (nuclear spin $I$ = 1/2 and gyromagnetic ratio $\frac{\gamma_{\rm
N}}{2\pi}$ = 17.237 MHz/T) NMR spectra were obtained either by fast Fourier
transform (FFT) of the NMR echo signals or by plotting spin-echo intensity
with changing resonance frequency under an external magnetic field $H$ of
7.4089 Tesla. When the 31P NMR spectra become very broad due to magnetic
ordering for $x$ = 0.02 and 0.13, we measured the 31P NMR spectra by sweeping
the external magnetic field $H$ at a constant resonance frequency. The NMR
spectra for $x$ = 0.42 were also measured by sweeping the $H$. The magnetic
field was applied parallel to either the crystal $c$ axis or the $ab$ plane,
and the direction of the magnetic field in the $ab$ plane was not controlled.
The origin of the Knight shift, $K$ = 0, of the 31P nucleus was determined by
the 31P NMR measurements of H3PO4.

The 31P spin-lattice relaxation rate $(1/T_{1})$ was measured at the peak
position of the the spectra with a saturation recovery method. $1/T_{1}$ at
each temperature ($T$) was determined by fitting the nuclear magnetization
$M_{\rm N}$ versus time $t$ using the exponential function $1-M_{\rm
N}(t)/M_{\rm N}(\infty)=e^{-t/T_{1}}$, where $M_{\rm N}(t)$ and $M_{\rm
N}(\infty)$ are the nuclear magnetization at time $t$ after the saturation and
the equilibrium nuclear magnetization at $t$ $\rightarrow$ $\infty$,
respectively. All the nuclear magnetization recovery curves measured for all
crystals of Sr(Co1-xNix)2P2 ($x$ = 0, 0.02, 0.13, 0.42 and 0.57) were well
fitted with the function as shown in the supplementary materials SM .

Figure 2: Frequency-swept 31P-NMR spectra for magnetic fields $H\parallel c$
axis (red) and $H\parallel ab$ plane (black) in Sr(Co1-xNix)2P2 measured near
room temperature (300 K for $x$ = 0 and 0.13, 290 K for $x$ = 0.02, 0.42 and
0.57). The vertical dashed line represents the zero-shift position ($K=0$).
Figure 3: Temperature dependence of 31P-NMR spectra under the magnetic fields
$H\parallel c$ and $H\parallel ab$ in Sr(Co1-xNix)2P2. (a) $x$ = 0, (b) $x$ =
0.02, (c) $x$ = 0.13, (d) $x$ = 0.42 and (e) $x$ = 0.57. The blue vertical
dashed lines in (b), (c), (d) and (e) represents the corresponding zero-shift
positions in the 31P NMR spectra ($K=0$). For (a) and (e), the frequency-swept
spectra were measured under the constant magnetic field of $H$ = 7.4089 T. The
field-swept spectra were measured with constant frequency of 127 MHz for (b)
and 131.1 MHz for (c) and (d), respectively. For (c), the spectra above 40 K
are replotted from the frequency-swept NMR spectra at $H$ = 7.4089 T where the
horizontal axis for the spectra was changed from frequency to magnetic field
by using the $\gamma_{\rm N}$ value of P nucleus. The red curves in (b) and
(c) are simulated spectra (see the text for details).

##  III Experimental results

###  III.1 31P NMR spectrum

Figure 2 shows the frequency-swept 31P-NMR spectra of Sr(Co1-xNix)2P2 at near
room temperature for magnetic fields parallel to the $c$ axis ($H\parallel c$)
and the $ab$ plane ($H\parallel ab$). Since 31P nucleus has $I$ = 1/2, a
single line was observed as shown in the figure. In the case of SrCo2P2 ($x$ =
0), relatively sharp lines are observed with the full-width at half maximum
(FWHM) values of 13.6 kHz and 19.6 kHz for $H\parallel c$ and $H\parallel ab$,
respectively, at room temperature. With the Ni substitution for Co in
Sr(Co1-xNix)2P2, the observed spectra become broader to 34.0 kHz and 36.0 kHz
for $x$ = 0.02, 92.0 kHz and 154 kHz for $x$ = 0.13, 132 kHz and 189 kHz for
$x$ = 0.42 and 99.0 kHz and 150 kHz for $x$ = 0.57, for $H\parallel c$ and
$H\parallel ab$, respectively. This is mainly due to a distribution of the
hyperfine coupling constant produced by the Ni substitution and also a
distribution of the Ni concentration. It is noted that the values of the FWHM
for $x$ = 0.42 are slightly greater than those of others. This originates from
a slightly larger distribution of Ni concentration for $x$ = 0.42 crystals
than others reported in Ref. Schmidt2023 . It is worth mentioning that the NMR
spectra indicate no obvious phase separation in our crystals, as we do not
observe any separated lines in the spectra.

The typical temperature dependence of 31P NMR spectra in Sr(Co1-xNix)2P2 is
shown in Fig. 3. For $x$ = 0 [Fig. 3(a)], although the NMR lines are
relatively sharp at high temperatures, the spectra become slightly broader
with decreasing temperature due to an inhomogeneous magnetic broadening. This
probably originates from a slight inhomogeneity of the hyperfine coupling
constant due to some defects. A similar inhomogeneous broadening was observed
in 59Co NMR spectra of the isostructural compound SrCo2As2 Wiecki2015 . For
the magnetically ordered compounds with $x$ = 0.02 and 0.13, the observed
spectra become very broad at low temperatures due to the magnetic ordering as
shown in Figs. 3(b) and 3(c), evidencing the the magnetic ordering in both
samples from a microscopic point of view. We also note that the spectra become
asymmetric. Since 31P nucleus has $I$ = 1/2 and we used single crystals for
our measurements, the asymmetric shape indicates a distribution of the
hyperfine coupling constants at the P sites. It is interesting to point out
that the shape of the NMR line in the magnetically ordered state for $x$ =
0.13 shows a characteristic rectangular-like shape [for example, see the
spectra at the bottom in Fig. 3(c)]. Such a rectangular shape of the NMR
spectrum is expected for powder samples of antiferromagnets Yamada1986 ;
Devi2022 . However, as we used singe crystals for our measurements, this
cannot be invoked to explain the NMR line shape observed. It is also noted
that, although the plateau region is less than that of $x$ = 0.13, a similar
rectangular-like shape is observed in $x$ = 0.02 at low temperatures. We will
discuss the shape of the NMR line below. For the non-magnetic compound with
$x$ = 0.42 and 0.58, although the observed spectra are asymmetric due to the
distribution of hyperfine coupling and Ni contents, no obvious change in the
peak positions was observed, consistent with the non-magnetic ground state
reported from the magnetization measurements Schmidt2023 .

Figure 4: Temperature dependence of full width at half maximum (FWHM) of
31P-NMR spectra under the magnetic field parallel to the $c$ axis ($H\parallel
c$) and to the $ab$ plane ($H\parallel ab$) in Sr(Co1-xNix)2P2.

Figure 4 shows the temperature dependence of FWHM of 31P-NMR spectra under
both magnetic field directions $H\parallel c$ and $H\parallel ab$ for
Sr(Co1-xNix)2P2. The values of FWHM of the field-swept NMR spectra for $x$ =
0.02, 0.13 and 0.42 were converted from Tesla to MHz by using the relation $f$
= $\frac{\gamma_{\rm N}}{2\pi}$$H$. As described above, in the case of $x$ =
0.13, the values of FWHM for both $H$ directions increase with decreasing
temperature and level off below $T_{\rm N}$ = 30 K due to the magnetic
ordering. Interestingly similar temperature dependence of FWHM, but with
different magnitudes, is observed in $x$ = 0.02. This suggests that the
magnetically ordered state at $x$ = 0.02 is well developed below $\sim$ 10 K
under a magnetic field of $\sim$ 7.4 T, although the compound is reported to
show a ferromagnetic ordered state below $T_{\rm C}$ = 4.3 K under $H$ = 0.01
T.

Figure 5: Left panels: Temperature dependence of Knight shift for magnetic
fields $H\parallel c$ axis ($K_{c}$) and $H\parallel ab$ plane ($K_{ab}$) for
Sr(Co1-xNix)2P2. The black curves are the calculated temperature dependence of
Knight shift assuming a double-peak structure in the density of states near
Fermi energy (see the text for details). Right panels: Temperature dependence
of magnetic susceptibility in units of cm3/mol per transition metal (TM) ions
under a magnetic field of 7 T for $H\parallel c$ axis ($\chi_{c}$) and
$H\parallel ab$ plane ($\chi_{ab}$) in Sr(Co1-xNix)2P2 except for the $\chi$
data for $x$ = 0.57 measured at 1 T.  Figure 6: NMR Knight shift $K$ versus
magnetic susceptibility $\chi$ plots for the corresponding $c$ (a) and $ab$
(b) components of $K$ in Sr(Co1-xNix)2P2 with temperature as an implicit
parameter. The solid lines are linear fits. Note that as the NMR shifts for
$x$ = 0.42 and 0.57 are nearly independent of temperature, we plot the
representative points (the squares in majenta for $x$ = 0.42 and the triangles
in blue for $x$ = 0.57) in (a) and (b) measured at $T$ = 290-300 K showing
that the points are on the lines and the values of $K_{0}$ are nearly
independent of $x$.

Figure 5 shows the temperature dependence of the NMR shift for $H\parallel c$
axis ($K_{c}$) and $H\parallel ab$ plane ($K_{ab}$) for Sr(Co1-xNix)2P2. For
the case of $x$ = 0, both $K_{c}$ and $K_{ab}$ show broad maxima around 25 K
and 110 K, consistent with the temperature dependence of the magnetic
susceptibility shown in the top panel in the right column. These results are
consistent with the previous NMR measurements reported by Imai ${\it
et~{}al.}$ Imai2017 ; Imai2014 . For $x$ = 0.02 and 0.13, both $K_{c}$ and
$K_{ab}$ increase with decreasing temperature showing the Curie-Weiss-like
behaviors, consistent with the temperature dependence of the magnetic
susceptibility Schmidt2023 . In contrast, a nearly temperature-independent
behavior of both $K_{c}$ and $K_{ab}$ is observed in $x$ = 0.42 by which
substitution value the AFM state has disappeared. This indicates that the
magnetic susceptibility of the dominant part of the crystal is nearly
independent of temperature, although the magnetic susceptibility data show an
increase with decreasing temperature. Similarly, both $K_{c}$ and $K_{ab}$ for
$x$ = 0.57 are nearly temperature independent expected for non-magnetic
metals, although again the magnetic susceptibility data show small increases
with decreasing temperature. These results indicate that the increase in
$\chi$ might be due to the distribution of Ni content and/or impurity spins .

The NMR shift $K$ is related to the local electron spin susceptibility
$\chi_{\rm spin}$ through the hyperfine coupling constant $A$ by
$K=K_{0}+\frac{A}{\mu_{\rm B}N_{\rm A}}\chi_{\rm spin}$, where $\mu_{\rm B}$
is the Bohr magneton, $N_{\rm A}$ is Avogadro’s number, and $K_{0}$ is the
temperature independent chemical shift originating from the core electrons of
P atoms. As shown in Fig. 5 (the right column), the magnetic susceptibilities
are nearly isotropic, although slight magnetic anisotropies can be seen below
$\sim$ 200 K and 70 K for $x$ = 0 and 0.02, respectively. Therefore the clear
difference between $K_{c}$ and $K_{ab}$ is due to the different hyperfine
coupling constants $A_{c}$ and $A_{ab}$ which can be estimated from the so-
called $K-\chi$ plot analysis.

Figures 6(a) and 6(b) plot $K_{c}$ and $K_{ab}$ against $\chi_{c}$ and
$\chi_{ab}$, respectively, for all compounds with $T$ as an implicit parameter
where $T$ is chosen to be well above magnetic ordering temperatures for $x$ =
0.02 and 0.13 to avoid the effects of the magnetic ordering. $K_{ab}$ and
$K_{c}$ vary with the corresponding $\chi$ as expected, although one can see a
slight deviation from the linear relationship. From the slopes of the fitted
lines for the high-temperature regions above $\sim$ 100 K shown by the lines,
we estimated the hyperfine coupling constants parallel to the $c$ axis
$A_{c}=(1.47\pm 0.1)$ T/$\mu_{\rm B}$ and parallel to the $ab$ plane
$A_{ab}=(2.25\pm 0.1)$ T/$\mu_{\rm B}$ for $x$ = 0, $A_{c}=(1.67\pm 0.15)$
T/$\mu_{\rm B}$ and $A_{ab}=(2.07\pm 0.15)$ T/$\mu_{\rm B}$ for $x$ = 0.02,
and $A_{c}=(1.48\pm 0.12)$ T/$\mu_{\rm B}$ and $A_{ab}=(2.01\pm 0.12)$
T/$\mu_{\rm B}$ for $x$ = 0.13. Similar, but slightly different values of
$A_{c}=1.68$ T/$\mu_{\rm B}$ and $A_{ab}=2.66$ T/$\mu_{\rm B}$ have been
reported for $x$ = 0 Imai2017 . The values of chemical shift $K_{0}$ were also
estimated to be $K_{0}$ = 0.06 (0.03) $\%$ for $H\parallel ab$ ($H\parallel
c$) for $x$ = 0, $K_{0}$ = 0.06 (0.035) $\%$ for $x$ = 0.02, and $K_{0}$ =
0.12 (0.06) $\%$ for $x$ = 0.13.

Figure 7: Typical field-swept 31P-NMR spectrum in the magnetically ordered
state of $x$ = 0.13 measured at 10 K under $H\parallel c$. The red curve is
the calculated spectrum based on the modulated magnetic ordered state
illustrated in the left inset with $B_{\rm int,0}$ = 0.22 T and $B_{\rm
int,1}$ = 0.14 T (see the text for details). The arrows in the left inset
represent the internal magnetic induction at the P sites. The right inset
shows the temperature dependence of magnetization of the crystal used in this
study for $H\parallel c$ and $H\parallel ab$ under $H$ = 7 T.

###  III.2 31P NMR spectrum in magnetically ordered state

As described above, we observed characteristic rectangular-like shape spectra
in the magnetically ordered states in $x$ = 0.02 and 0.13. For $x$ = 0.13, the
antiferromagnetic ground state has been reported under a small magnetic field
of 0.01 T Schmidt2023 . Although such a rectangular-like line shape is
expected for powder samples of antiferromagnets Yamada1986 ; Devi2022 , this
can not be applied here since we used single crystals. In addition, the AFM
state in $x$ = 0.13 is transformed into a forced ferromagnetic (saturated
paramagnetic) state under strong magnetic fields. In fact, the magnetization
measurements show a saturated magnetization of $\sim$ 0.15 $\mu_{\rm B}$ per
TM ions above 1.5 T under $H\parallel ab$ at 2 K and a similar saturated
magnetization of $\sim$ 0.15 $\mu_{\rm B}$ is observed above 3 T for
$H\parallel c$ Schmidt2023 . This indicates that all magnetic moments in the
magnetic ordered state are aligned along the magnetic field direction. This
can also be seen in the temperature dependence of magnetization $M$ of the $x$
= 0.13 crystal used in the present study at $H$ = 7 T shown in Fig. 5 (right
middle panel) and replotted in the right inset of Fig. 7 on a semi-log scale
where $M$ shows a saturation moment of 0.16 $\mu_{\rm B}$/TM below $\sim$ 10 K
with no obvious magnetic anisotropy.

Therefore, as our NMR spectrum measurements have been performed above $H$ = 7
T, we consider that the NMR spectra were measured in the state where the
magnetic moments are saturated in the $x$ = 0.13 compound by the application
of the magnetic field. The broad NMR line is mainly due to the distribution of
the hyperfine field at the P site due to the Ni substitution which generates
different environments of the P site with respect to the number of nearest
neighbors (NN) of a given Co occupied by Ni. From the estimate of the
probability of finding $N$ (= 1-4) Ni at NN Co sites with $P(N,x)$ =
$C_{N}^{4}x^{N}(1-x)^{4-N}$, we found that only the distribution of the
hyperfine field cannot explain the rectangular-like shape of the spectrum.
Therefore, one needs to introduce a distribution of Co ordered moments in
order to explain the characteristic shape of the NMR line. Thus we conclude
that the magnitude of the Co-ordered moments is not uniform and the magnetic
states are inhomogeneous in the magnetic ordered state of the Ni-substituted
SrCo2P2. To explain the observed spectra, we modeled that the Co ordered
moments are not uniform and the magnitude is modulated with a sinusoidal
function as shown in the inset of Fig. 7 which produces the uniform and
distributed internal magnetic inductions $B_{\rm int,0}$ and $B_{\rm int,1}$
at the P site, respectively. Here it is noted that the modulation of the
magnetic moment is assumed to be incommensurate with the lattice.

The red curve in Fig. 7 is a calculated result with $B_{\rm int,0}$ = 0.22 T
and $B_{\rm int,1}$ = 0.14 T with an appropriate inhomogeneous magnetic
broadening, which reasonably reproduces the observed spectrum (shown by the
black curve) measured at $T$ = 10 K with $H\parallel c$ for
Sr(Co0.87Ni0.13)2P2 ($x$ = 0.13). Thus we consider that the rectangular-like
NMR spectra observed in the magnetically ordered state are due to the
variation of the Co-ordered moments. Utilizing the values of $A_{c}$ = 1.48
T/$\mu_{\rm B}$, the uniform and the distribution components of the magnetic
moment are estimated to be 0.15$\pm$0.01 and 0.10$\pm$0.01 $\mu_{\rm B}$,
respectively. The value of the uniform magnetic moment is in great agreement
with a value of 0.16 $\mu_{\rm B}$ estimated from the $M$ measurements shown
in the right inset of Fig. 7.

The red curves in the left column of Fig. 3(c) show calculated results where
$B_{\rm int,0}$($B_{\rm int,1}$) = 0.22(0.14) T are nearly independent of
temperature below 10 K, and decrease gradually above 10 K to 0.19(0.11) T at
20 K and 0.17(0.10) T at 30 K for $H\parallel c$. The temperature-dependent
behavior is consistent with the temperature dependence of $M$ shown in the
inset of Fig. 7.

For the case of $H\parallel ab$ [the right column of Fig. 3(c)], the
calculated spectra also capture the characteristic shape of the observed
spectra where we used $B_{\rm int,0}$($B_{\rm int,1}$) = 0.33(0.19),
0.31(0.20), 0.25(0.16), and 0.20(0.13) T for $T$ = 4.2, 10, 20, and 30 K,
respectively, indicative of the variation of the Co ordered moments in the
magnetically ordered state. The uniform and the distribution components of the
Co ordered moments for the $H\parallel ab$ direction are estimated using
$A_{ab}$ = 2.01 T/$\mu_{\rm B}$ to be 0.16$\pm$0.01 and 0.10$\pm$0.01
$\mu_{\rm B}$ at 4.2 K. These values are very close to those for $H\parallel
c$. Again the estimated uniform magnetization from the NMR spectrum simulation
is in excellent agreement with the magnetization data, indicating that the
magnetization measurements picked up the average value of magnetization as
expected.

It is noted that the model can also roughly reproduce the NMR spectra at low
temperatures for $x$ = 0.02 as shown by the red curves in Fig. 3(b), although
the tails of the spectra at the lower magnetic field side cannot be well
reproduced. This is presumably due to the large inhomogeneous distribution of
hyperfine field coming from the Ni substitution as described above. For
example, $B_{\rm int,0}$($B_{\rm int,1}$) = 0.082(0.037) T for $H\parallel c$
and 0.12(0.065) T for $H\parallel ab$ were used for the calculation for the
spectra at 1.6 K. By using $A_{c}(A_{ab})$ = 1.67(2.07) T/$\mu_{\rm B}$, the
uniform and distribution components of the Co ordered moments are estimated to
be 0.049(0.022) and 0.058(0.031) $\mu_{\rm B}$ for $H\parallel c$ and
$H\parallel ab$, respectively, for $x$ = 0.02. These are also in good
agreement with 0.049 and 0.056 $\mu_{\rm B}$ for for $H\parallel c$ and
$H\parallel ab$ determined by the magnetization measurements under 7 T shown
in the right panel in Fig. 5 of the measured crystal of $x$ = 0.02. Thus we
conclude that the magnetic states in Sr(Co1-xNix)2P2 are non-uniform and the
distribution of the Co-ordered moments is incommensurate with the lattice.

We have tried to measure the 31P NMR spectrum around $H$ = 1 T in the AFM
state for $x$ = 0.13. However, the signal intensity was too poor to measure,
so we were not able to obtain the spectrum in the AFM state. Therefore, we
cannot discuss the magnetic structure in the AFM state in Sr(Co1-xNix)2P2.
Nevertheless, it is worth to mention that, in the isostructural As-system
Sr(Co1-xNix)2As2, the AFM state appearing in $x$ = 0.013-0.25 with a maximum
$T_{\rm N}$ = 26.5 K at $x$ = 0.10 has been determined to be a planar helical
magnetic state originating from FM interactions in the $ab$ plane and weaker
AFM interactions along the helix $c$ axis Sangeetha2019 ; Wilde2019 . As we
describe below, this magnetic structure would be consistent with our
observation of strong ferromagnetic spin fluctuations in $x$ = 0.13. Another
possible magnetic structure is the A-type antiferromagnetic state observed in
CaCo2P2 Reehuis1998 ; Imai2017 ; Baumbach2014PRB . Further experiments such as
ND measurements are highly required to determine the magnetic structure in the
AFM state in Sr(Co1-xNix)2P2.

###  III.3 31P spin-lattice relaxation time $T_{1}$

Figure 8:  Temperature and $x$ dependences of the 31P spin-lattice relaxation
rate divided by temperature (1/$T_{1}T$) for both field directions
[$H\parallel c$ (left panels), $H\parallel ab$ (right panels)] in
Sr(Co1-xNix)2P2. The 1/$T_{1}T$ data for $x$ = 0 shown by closed and open
stars are from Ref. Imai2017 . The black solid curves for $x$ = 0 are
calculated results with the model discussed in the text. Note that the
vertical scales for $x$ = 0.13 are different from others. Figure 9: $T$
dependence of the 31P spin-lattice relaxation rate divided by temperature
(1/$T_{1}T$) for $H\parallel c$ of SrCo2P2. The red curve is the calculated
result based on the the double peak structure in the density of states near
Fermi energy ($E_{\rm F}$) shown in the inset.

To investigate the dynamical magnetic properties, we have measured 1/$T_{1}$
versus $T$. Figure 8 shows the temperature and $x$ dependence of 1/$T_{1}T$
for both field directions (left: $H\parallel c$, right: $H\parallel ab$) in
Sr(Co1-xNix)2P2. At $x$ = 0, similar to the case of $K$, and thus $\chi$,
broad maxima are observed around $\sim$ 25 K and 120 K, as reported previously
Imai2017 ; Imai2014 . For comparison, we show the 1/$T_{1}T$ data for $x$ = 0
reported from Imai et al. Imai2017  by closed and open stars in the figures,
which show a relatively good agreement with our results. With Ni substitution
of $x$ = 0.02, 1/$T_{1}T$ still shows a similar temperature dependence with
those of 1/$T_{1}T$ for $x$ = 0 exhibiting a broad maximum around 120 K for
both magnetic field directions where the broad maximum around 25 K is not
clearly observed. No apparent anomaly due to ferromagnetic phase transition is
observed around $T_{\rm C}$, which is probably broadened into a high
temperature cross over into the saturated paramagnetic state by the
application of the magnetic field of $\sim$ 7.4 T. On the other hand, at $x$ =
0.13, 1/$T_{1}T$ is enhanced and shows clear peaks around $T_{\rm N}$ for both
magnetic field directions. However, the peaks in 1/$T_{1}T$ should not be due
to the AFM transition because the ground state of the compound is a forced
ferromagnetic (saturated paramagnetic) state under $H$ = $\sim$ 7.4 T. It will
be interpreted as a crossover from the paramagnetic to the forced
ferromagnetic states. With further Ni substitution of $x$ = 0.42 and to 0.57,
in contrast, the values of 1/$T_{1}T$ are suppressed and 1/$T_{1}T$ follows
the Korringa law, $T_{1}T$ = const. which is indicative of a paramagnetic
metallic state without significant magnetic fluctuations.

First, we analyze the characteristic temperature dependence of 1/$T_{1}T$
showing the double-maxima observed in $x$ = 0. Since the characteristic
temperature dependence of 1/$T_{1}T$ can be due to a band structure near the
Fermi energy $E_{\rm F}$, we calculated it based on a model where we adopt a
double-peak structure of the density of states (DOS) shown in the inset of
Fig. 9.

In this model, the Fermi energy ($E_{\rm F}$) is assumed to be just below the
double peaks having the widths of $W_{1}$ and $W_{2}$ away from the $E_{\rm
F}$ by $\Delta_{1}$ and $\Delta_{2}$, respectively. ${\cal D}_{i}$
($i=1,2,3,4,5)$ represent the DOS for the band structure near $E_{\rm F}$.
Using the formula,

$\frac{1}{T_{1}}\sim\int_{0}^{\infty}{\cal D}^{2}(E)f(E)(1-f(E)){\rm{d}}E$ (1)

where $f(E)$ is the Fermi distribution function, we calculated $1/T_{1}T$ with
a set of parameters of $\Delta_{1}$ = 300 K, $\Delta_{2}$ = 25 K, $W_{1}$ =
125 K, $W_{2}$ = 30 K, $D_{i}/D_{1}$ = 1, 0.82, 0.75, 0.55, and 0.15 for $i$ =
1, 2, 3, 4, and 5, respectively. The calculated result reasonably reproduces
the experimental data, as shown by the red curve in Fig. 9. This indicates
that the characteristic temperature dependence of $1/T_{1}T$ exhibiting double
maxima can be explained by the rigid peculiar band structure modeled. The
calculated results are also shown in Fig. 8. It should be noted that, since
the number of parameters is large (9 parameters) in the model, we cannot
precisely determine the value of each parameter from the fitting. Therefore,
there is relatively large uncertainty in the value of each parameter ($\sim$
25 %); however, again we point out that the model of the double-peak structure
of the DOS qualitatively explains the characteristic temperature dependence of
1/$T_{1}T$. It is interesting to point out that such a double-peak DOS
structure in SrCo2P2 has been reported from the density-function-theory band-
structure calculations which reproduces the experimental data of de Haas-van
Alphen effect experiments Gotze2021 . According to the calculations, the
positions of the double peaks in the DOS are largely influenced by the crystal
structure parameters such as lattice parameters and the Wyckoff position of P,
and also different approximations used in the calculations Gotze2021 .
Nevertheless, it is interesting to point out that the centers of the
calculated two peaks are located around 0.005-0.011 eV and 0.02-0.035 eV above
$E_{\rm F}$ which are not far from the values obtained by the fit with
$\Delta_{2}$ = 25 K (0.0022 eV) and $\Delta_{1}$ = 300 K (0.026 eV). In
contrast, it is noted that the widths of each peak from the theoretical
calculations are roughly estimated to be $\sim$ 0.011 eV and $\sim$ 0.005 eV
for the peak around 0.005-0.011 eV and 0.02-0.035 eV, respectively, which are
different from our fitting values of $W_{2}$ = 30 K (0.0026 eV) and $W_{1}$ =
125 K (0.011 eV).

We also calculated the temperature dependence of Knight shift using the same
band structure model. The solid curves in Fig. 5 (the top panel on the left)
are the calculated results with appropriate proportional constants utilizing
the same set of parameters used for the $1/T_{1}T$ fitting except for the
small change in ${\cal D}_{4}$/${\cal D}_{1}$ from 0.55 to 0.45. Although the
fits do not reproduce the data perfectly, the model seems to capture the
behavior of $K$ qualitatively. The reason for the slight change in ${\cal
D}_{4}$ is not clear, but it may be due to magnetic fluctuations which do not
follow the temperature dependence of the Knight shift.

From the calculations, it turns out that, as naturally expected, the broad
maximum around 125 K in $1/T_{1}T$ is mainly due to the peak in the DOS far
away from the $E_{\rm F}$ and the maximum around 25 K originates from the peak
close to the $E_{\rm F}$. Therefore, the similar temperature dependence of
1/$T_{1}T$ observed in $x$ = 0.02 suggests that the DOS structure is not
modified much by the Ni substitution although the system shows the
ferromagnetic ground state. At present we do not have a clear idea to explain
this. However, it may suggest that a very small change in DOS leads to the
ferromagnetic long-range magnetic ordering and SrCo2P2 is located close to
ferromagnetic instability. In contrast, for $x$ = 0.13, 1/$T_{1}T$ are
enhanced, which cannot be explained by the simple band model. For $x$ = 0.42
and 0.57, the 1/$T_{1}T$ = const. behavior can be explained by a flat
structure of the DOS near the $E_{\rm F}$ expected for a normal metallic
state. This indicates that the Ni substitution greater than $x$ = 0.42
completely smears out the double-peak structure, leading to a more itinerant
nature of Co 3d electrons.

###  III.4 Magnetic fluctuations

Figure 10: Left panels: Temperature dependence of the Korringa ratios
$1/T_{1,\bot}TK_{\text{spin},ab}^{2}$ (closed symbols) and
$1/T_{1,\|}TK_{\text{spin},c}^{2}$ (open symbols) for spin fluctuations in the
$ab$ plane and along the $c$ axis, respectively, in Sr(Co1-xNix)2P2. Right
panels: Semi-log plot of the temperature dependence of the parameter
$\alpha_{\bot}$ for spin fluctuations in the $ab$ plane (closed symbols) and
$\alpha_{\|}$ along the $c$ axis (open symbols).

Now we discuss the magnetic fluctuations in Sr(Co1-xNix)2P2 based on a
modified Korringa relation analysis using the $T_{1}$ and the electron spin
part of Knight shift ($K_{\rm spin}$) data. Within a Fermi liquid picture,
$1/T_{1}T$ is proportional to the square of the DOS near the Fermi energy
${\cal D}(E_{\rm F})$ and $K_{\text{spin}}(\propto\chi_{\text{spin}}$) is
proportional to ${\cal D}(E_{\rm F})$. In particular,
$T_{1}TK_{\text{spin}}^{2}$ = $\frac{\hbar}{4\pi k_{\rm
B}}\left(\frac{\gamma_{\rm e}}{\gamma_{\rm N}}\right)^{2}$ = $S$, which is the
Korringa relation. Deviations from $S$ can reveal information about electron
correlations in materials Moriya1963 ; Narath1968 , which are expressed via
the parameter $\alpha=S/(T_{1}TK_{\text{spin}}^{2})$. For instance, the
enhancement of $\chi(\mathbf{q}\neq 0)$ increases $1/T_{1}T$ but has little or
no effect on $K_{\text{spin}}$, which probes only the uniform $\chi$ with
$\mathbf{q}$ = 0. Thus $\alpha>1$ for AFM correlations and $\alpha<1$ for FM
correlations.

Since 1/$T_{1}T$ probes magnetic fluctuations perpendicular to the magnetic
field, it is natural to consider the Korringa ratio
1/($T_{1,\perp}TK^{2}_{{\rm spin},ab}$), where 1/($T_{1,\perp}T$) =
1/$(T_{1}T)_{H||c}$, when examining the character of magnetic fluctuations in
the $ab$ plane Wiecki2015-2 . We also consider the Korringa ratio
1/($T_{1,||}TK^{2}_{{\rm spin},c}$) for magnetic fluctuations along the $c$
axis. Here, 1/($T_{1,\parallel}T$) is estimated from 2/$(T_{1}T)_{H||ab}$ $-$
$1/(T_{1}T)_{H||c}$ Wiecki2015-2 .

The temperature and $x$ dependences of 1/($T_{1,\perp}TK^{2}_{{\rm spin},ab}$)
and 1/($T_{1,||}TK^{2}_{{\rm spin},c}$) above the magnetic ordering
temperatures in Sr(Co1-xNix)2P2 are shown in the left panels of Fig. 10. The
calculated $\alpha_{\parallel}$ and $\alpha_{\perp}$ are shown in the right
panels of Fig. 10.

For $x$ = 0, $\alpha_{\parallel}$ ($\alpha_{\perp}$) decrease from $\sim$0.4
(0.2) at 300 K to $\sim$0.08 (0.14) around 110 K and then become nearly
temperature independent below that temperature, indicating dominant
ferromagnetic spin correlations between Co spins at low temperatures in the
compound. Furthermore, $\alpha_{\perp}$ is less than $\alpha_{\parallel}$
suggesting the in-plane ferromagnetic fluctuations are stronger than those
along the $c$ axis. The lowest values of $\alpha_{\parallel}$ and
$\alpha_{\perp}$ are almost comparable, or slightly greater than those
isostructural compounds BaCo2As2 and SrCo2As2 in which dominant ferromagnetic
spin fluctuations have also been reported Wiecki2015 ; BaCo2As2 . It is
interesting to point out that the coexistence of ferromagnetic and
antiferromagnetic spin fluctuations have been reported in SrCo2As2 Wiecki2015
; Li2019 , although we do not have any clear evidence from the present 31P-NMR
measurements for AFM spin fluctuations in SrCo2P2.

For $x$ = 0.02, although the values of $\alpha_{\parallel}$ and
$\alpha_{\perp}$ are nearly the same with the $\alpha_{\perp}$ value (but
slightly smaller than $\alpha_{\parallel}$) in $x$ = 0 at 300 K ,
$\alpha_{\parallel}$ and $\alpha_{\perp}$ decrease rapidly down to 0.015,
almost one order in magnitude less than the case of $x$ = 0, indicating a
strong enhancement of ferromagnetic spin fluctuations at low temperatures in
$x$ = 0.02. It is also interesting to note that the ferromagnetic spin
fluctuations are nearly isotropic in $x$ = 0.02.

A similar enhancement of the isotropic ferromagnetic spin fluctuations at low
temperatures is also observed in $x$ = 0.13. The values of
$\alpha_{\parallel}$ and $\alpha_{\perp}$ for $x$ = 0.13 are almost comparable
with those in $x$ = 0.02. These results mean that the strong ferromagnetic
spin fluctuations are induced with only a 2 % Ni substitution and are kept
almost the same up to $x$ = 0.13 where the magnetic ground state changes from
ferromagnetic ($x$ = 0.02) to antiferromagnetic ($x$ = 0.13). It is also
interesting to point out that we do not observe any trace of AFM spin
fluctuations in $x$ = 0.13, indicating that the magnetic fluctuations in $x$ =
0.13 are dominated by the ferromagnetic fluctuations. At present, the reason
for no observation of the AFM spin fluctuations in $x$ = 0.13 is not well
understood. However, it is interesting to point out that, in the A-type
antiferromagnet CaCo2P2 where the Co moments are ferromagnetically aligned in
the $ab$ plane and the moments adjacent along the $c$ axis are aligned
antiferromagnetically Reehuis1998 ; Imai2017 ; Baumbach2014PRB , no clear AFM
spin fluctuations were observed and the magnetic fluctuations were found to be
dominated by ferromagnetic ones Higa2018 . In addition, as described above,
another possible AFM state of the planar helical magnetic state observed in
Sr(Co1-xNix)2P2 Sangeetha2019 ; Wilde2019  would also provide dominant
ferromagnetic spin fluctuations. Although the AFM structure in Sr(Co1-xNix)2P2
is still an open question, such magnetic states might lead to the dominant
ferromagnetic spin fluctuations.

With Ni substitution of $x$ = 0.42, $\alpha_{\parallel}$ and $\alpha_{\perp}$
increase and become temperature independent without showing any indication of
quantum critical behavior. At $x$ = 0.57, the nearly temperature independent
$\alpha_{\parallel}$ and $\alpha_{\perp}$ become closer to unity, indicating
that the magnetic fluctuations disappeared in the compound. Here we
tentatively used $K_{0}$ = 0.03 (0.06) for $H\parallel c$ ($H\parallel ab$)
since the data points for $x$ = 0.42 and 0.57 in the $K=\chi$ plots [Figs.
6(a) and (b)] are well on the fitting lines showing the nearly $x$
independence of $K_{0}$.

Figure 11:  Temperature dependences of $1/T_{1}TK_{\text{spin}}$ (closed
symbols) and $1/T_{1}TK_{\text{spin}}^{3/2}$ (open symbols) for the in-plane
(red) and the $c$ axis (black) directions for $x$ = 0 (a), 0.02 (b), and 0.13
(c).

It should be noted that, however, the Korringa analysis usually applies to
paramagnetic materials where electron-electron interaction is weak. Since the
compounds with $x$ = 0.02 and 0.13 exhibit magnetic orders in contrast to $x$
= 0, 0.42 and 0.57, we also analyze NMR data based on self-consistent
renormalization (SCR) theory, as Imai ${\it et~{}al.}$ have performed in
(Sr1-xCax)Co2P2 Imai2017 . As shown above, the magnetic fluctuations are
governed by ferromagnetic spin correlations. In this case, according to SCR
theory for weak itinerant ferromagnets, 1/$T_{1}T$ is proportional to $K_{\rm
spin}$ or to $K^{3/2}_{\rm spin}$ for three dimensional (3D) or two-
dimensional (2D) ferromagnetic spin fluctuations, respectively SCR1 ; SCR2 .

Figures 11(b) and 11(c) show the $T$ dependence of 1/($T_{1}TK_{\rm spin})$
and 1/($T_{1}TK_{\rm spin}^{3/2})$ for the two directions for $x$ = 0.02 and
0.13, respectively, together with those in $x$ = 0 (a). For both $x$ = 0.02
and 0.13, it seems that both the 1/($T_{1,\parallel}TK_{\rm spin,c}$) and
1/($T_{1,\perp}TK_{{\rm spin},ab}$) are nearly constant in the temperature
region of 150 - 300 K while both the 1/($T_{1,\parallel}TK_{{\rm s},c}^{3/2}$)
and 1/($T_{1,\perp}TK_{{\rm s},ab}^{3/2}$) slightly increase with increasing
temperature in that temperature region. This suggests the ferromagnetic spin
fluctuations are most likely characterized by 3D in nature in the high-
temperature region for both $x$ = 0.02 and 0.13. It is noted that both
1/($T_{1}TK_{\rm spin})$ and 1/($T_{1}TK_{\rm spin}^{3/2})$ decrease with
decreasing temperature below $\sim$150 K, suggesting that the SCR theory may
not be applicable to describe the dynamical properties of the Co moments in
$x$ = 0.02 and 0.13 exhibiting long-range magnetic ordered states. As for $x$
= 0, although the SCR theory for the 3D FM model seems slightly better than
that for the 2D model, it is difficult to conclude the dimensionality of the
FM fluctuations from the present NMR data.

Figure 12:  Contour plot of $\alpha_{\parallel}$ vs temperature showing the
magnitude of the spin fluctuations together with the magnetic phase transition
temperatures from Ref. Schmidt2023 . Note that the lower the magnitude of
$\alpha_{\parallel}$, the stronger in the ferromagnetic spin fluctuations. The
blue diamonds and white circles represent $T_{\rm C}$ and $T_{\rm N}$,
respectively, from Ref. Schmidt2023 . Note that, whereas the phase transition
temperature data Schmidt2023  were inferred from low field (zero field) data,
the $\alpha_{\parallel}$ data were inferred from $\sim$ 7.4 T data. The values
of $\alpha_{\parallel}$ are not shown in the magnetically ordered state (the
white area).

##  IV Summary

In conclusion, we conducted 31P NMR measurements on Sr(Co1-xNix)2P2 to
investigate its static and dynamic magnetic properties. The characteristic
temperature dependences of $K$ and 1/$T_{1}T$ exhibiting the double-maxima
around 25 K and 125 K in the Stoner-enhanced paramagnetic metal SrCo2P2 were
reasonably explained by a simple model where a double-peak structure of the
density of states near the Fermi energy is assumed. We analyzed our $T_{1}$
and $K$ data based on the modified Korringa relation and discussed the
evolution of magnetic fluctuations in the paramagnetic state in
Sr(Co1-xNix)2P2. Figure 12 summarizes the evolution of the magnetic
fluctuations where the $x$ dependence of the Korringa ratios of the
longitudinal magnetic fluctuation $\alpha_{\parallel}$ vs. temperature is
shown in the contour plot together with the magnetic phase transition
temperatures from Ref. Schmidt2023 . Here we tentatively plotted the data of
$\alpha_{\parallel}$, but the in-plane magnetic fluctuations $\alpha_{\perp}$
provide qualitatively the same behavior Korringa2 . The lower values in the
magnitude of $\alpha_{\parallel}$, much less than unity, qualitatively
indicate the stronger ferromagnetic spin fluctuations. For $x$ = 0,
$\alpha_{\parallel}$ is found to be much less than unity (the red-ish region
in Fig. 12) evidencing that electron correlations are enhanced around FM
wavenumber $q$ = 0 in the paramagnetic SrCo2P2. Such FM spin fluctuations are
enhanced with only 2 % Ni substitution in the ferromagnetic
Sr(Co0.98Ni0.02)2P2. Similar strong FM spin fluctuations are observed in the
antiferromagnet Sr(Co0.87Ni0.13)2P2 with $T_{\rm N}$ = 30 K. With further Ni
substitution up to 0.42 and to 0.57, $\alpha_{\parallel}$ gradually increases
and becomes close to unity (blue region in Fig. 12) indicating that such FM
spin fluctuations completely disappeared, leading to a non-correlated metal.
It is noted that no clear quantum critical behavior in Sr(Co1-xNix)2P2 can be
observed around $x$ = 0.42 where the long-range magnetic state is not observed
down to the lowest temperature (1.6 K) of our experiment. This is different
from the As-system Sr(Co1-xNix)2As2 where a non-Fermi liquid behavior has been
observed near $x$ = 0.3 when the long-range magnetic order disappeared
Sangeetha2019 . Based on the analysis of the characteristic shape of the NMR
spectra observed in the magnetically ordered state, we found that the ordered
state is inhomogeneous where the magnitude of the ordered Co moments is
modulated incommensurately with the lattice in Sr(Co1-xNix)2P2.

##  V Acknowledgments

We thank K. Rana for assistance in the initial stage of the NMR measurements.
The research was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering. Ames National
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DE-AC02-07CH11358.

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