# Pressure driven magnetic order in Sr1-xCaxCo2P2

Ola Kenji Forslund<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal
Institute of Technology, SE-106 91 Stockholm, Sweden  Daniel Andreica  Faculty
of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania  Yasmine
Sassa  Department of Physics, Chalmers University of Technology, SE-41296
Göteborg, Sweden  Masaki Imai  Chishiro Michioka  Kazuyoshi Yoshimura
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto
606-8502 Japan  Zurab Guguchia  Zurab Shermadini  Rustem Khasanov  Laboratory
for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI,
Switzerland  Jun Sugiyama  Neutron Science and Technology Center,
Comprehensive Research Organization for Science and Society (CROSS), Tokai,
Ibaraki 319-1106, Japan  Martin Månsson<EMAIL_ADDRESS>Department of Applied
Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden

###### Abstract

The magnetic phase diagram of Sr1-xCaxCo2P2 as a function of hydrostatic
pressure and temperature is investigated by means of high pressure muon spin
rotation, relaxation and resonance ($\mu^{+}$SR). The weak pressure dependence
for the $x\neq 1$ compounds suggests that the rich phase diagram of
Sr1-xCaxCo2P2 as a function of $x$ at ambient pressure may not only be
attributed to solely chemical pressure effects. The $x=1$ compound on the
other hand reveals a high pressure dependence, where the long range magnetic
order is fully suppressed at $p_{\rm c2}\approx 9.8$ kbar, which seem to be a
first order transition. In addition, an intermediate phase consisting of
dilute ferromagnetic islands (FMI) is formed above $p_{\rm c1}\approx 8$ kbar
where they co-exist with a magnetically disordered state. Moreover, such FMI
phase seems to consist of an high- (FMI-\small{1}⃝) and low-temperature
(FMI-\small{2}⃝) region, respectively, separated by a phase boundary at
$T_{\rm i}\approx 20$ K.

Pressure, muon spin relaxation, ThCr2Si2, Sr1-xCaxCo2P2

††preprint: APS/PRB

##  I Introduction

The ThCr2Si2 layered structure type family of compounds typically exhibits
ground states ranging from superconductivity to long range magnetic order
Johrendt  _et al._ (1997); Reehuis  _et al._ (1998); Hoffmann and Zheng
(2002); Rotter  _et al._ (2008a); Tan  _et al._ (2017). The ground states are
determined by competition between magnetism and superconductivity, as in
CaFe2As2 or Ba1-xKxFe2As2 Rotter  _et al._ (2008b); Torikachvili  _et al._
(2008). In this family, the $AM_{2}X_{2}$ structure type with a metal $A$,
transition metal $M$, and metalloid $X$ atoms, are made up of edge-share
$T\textasciiacute MX_{4}$ tetrahedra layers (Fig. 1). The delicate ground
state is mostly dependent on the inter-layer $X$-$X$ bonding distance across
the intermediate $A$ sheets. For several $AM_{2}$P2 ($A$ = Ca, Sr, and Ba, and
$M$ = Fe, Co, and Ni) materials, the phase transitions are related to subtle
structural changes present in the crystals. In particular, these families have
a tendency to transform from uncollapsed tetragonal (ucT) to collapsed
tearagonal (cT) structure. This is driven by the X-X bonding acting between
the $M_{2}X_{2}$ layers, for which strong enough bonds pull the layers closer
and induces a lattice relaxation. Reehuis and Jeitschko (1990); Kreyssig  _et
al._ (2008); Jia  _et al._ (2009)

In the case of Sr1-xCaxCo2P2, the crystals transforms from ucT to cT when the
chemical composition changes from $x=0$ to $x=1$. A Curie-Weiss behaviour is
observed in all compounds at high temperature, but the high temperature
fluctuations changes from antiferromagnetic to ferromagentic type around
$x\sim 0.5$ Jia  _et al._ (2009). Such change seem to be correlated to the
finally ground state as it is transformed from paramagnetic ($x<0.45$) to
antiferromagnteic (AF) at $x\approx 0.5$ J. Sugiyama, et al (2015). In fact,
bulk magnetisation measured as a function of $x$ show a clear correlation
between the detailed crystalline structure and the magnetic properties Jia
_et al._ (2009).

Previous study based on muon spin rotation, relaxation and resonance
($\mu^{+}$SR) J. Sugiyama, et al (2015) have indicated Pauli-paramagnetic
phases for $x<0.45$ at temperatures as low as 1.8 K, consistent with De Haas-
van Alphen measurement Teruya  _et al._ (2014). Short-range AF ordered phases
appears for $0.48\leq x\leq 0.75$, which stabilizes into a long-range AF
ordered phase for $x>0.75$. The formation of magnetically ordered phases were
shown to have strong correlations with the nearest neighboring Co distance
($d_{\rm Co-Co}$) within the Co2P2 planes, implying the importance of also the
inter-plane interaction for the formation of long range order. The Co-Co
distance decreases only moderately with $x$ up until $\sim 0.5$, at which
point $d_{\rm Co-Co}$ experience an abrupt decrease until 0.9 where it finally
levels out to an almost constant value J. Sugiyama, et al (2015).

It is thus interesting to continue the investigation by further decreasing the
distance $d_{\rm Co-Co}$ through the application of a hydrostatic pressure.
Such study was performed on CaCo2P2 single crystals using resistivity
(Baumbach  _et al._ , 2014). These measurements suggested that the AF order is
destabilises with pressure, hinting towards a quantum critical point (QCP).
Moreover, the signature of a second unknown phase was also indicated in the
pressurised state, but the details of such phase remain unknown.

In order to confirm and further comprehend the high pressured states, we have
conducted pressure dependent $\mu^{+}$SR measurements on the series of
Sr1-xCaxCo2P2 powder samples, including $x=1$, 0.9, 0.8 and 0.7. Given that
$\mu^{+}$SR is highly sensitive to magnetic fields and to magnetic volume
fraction, any subtle magnetic transition can be detected and characterized in
detail. The current study clearly reveal that the AF order is suppressed with
pressure, consistent with macroscopic measurements. As we shall show, this
transition does however seem to be first order type, thereby ruling out the
possibility of a QCP. Moreover, the sample exhibits a co-existence of magnetic
order and disorder at higher pressures just below the critical pressure.

Figure 1: The crystal structure of Sr1-xCaxCo2P2 shown in two planes: (a) the
cb plane and (b) the ab plane. The green spheres represents Sr/Ca atoms while
the blue and pink spheres represents Co and P atoms, respectively. The
structure is stabilised in Tetragonal symmetry with space group I4/mmm (#139).

##  II Experimental Setup

Sr1-xCaxCo2P2 polycrystalline samples were synthesised in a two step reaction
from the base elements; P, Sr, Ca, and Co. At first, Sr, Ca, Co were
individually put together with P in an evacuated quartz tube to facilitate a
solid state reaction at 800∘C and 700∘C to produce SrP, CaP and Co2P.
Sr1-xCaxCo2P2 was then synthesized via a solid-state reaction between SrP,
CaP, and Co2P at 1000∘C for 20 hours in Ar atmosphere. Detailed information
about the synthesis protocol is found in Ref. Imai  _et al._ , 2014.

The $\mu^{+}$SR measurements were performed at the GPD instrument on the
$\mu$E1 beamline at the S$\mu$S muon source of Paul Scherrer Institute (PSI),
Switzerland. Hydrostatic pressures up to 23 kbar were achieved by using a
piston cylinder cell made of MP35 and CuBe alloys, respectively. Temperatures
down to $T=2$ K were achieved using a 4He flow cryostat. Three pressed pellets
of the powder samples were stacked (5.9 mm diameter and 13 mm total height)
for each measurement and inserted into the pressure cell. Daphne oil was used
as the pressure medium in order to achieve the hydrostatic pressure. The
pressure of the sample at low temperatures was accurately determined via AC
susceptibility measurements of the superconducting transition temperature for
a small indium wire located at the bottom of (inside) the pressure cell. R.
Khasanov, et al (2016); D. Andreica (2001) Finally, the free analysis software
musrfit A. Suter and B. M. Wojek (2012) was used to analyzed the $\mu^{+}$SR
data.

##  III Results

Series of $\mu^{+}$SR measurements on Sr1-xCaxCo2P2 are presented as a
function of pressure, temperature and chemical composition ($x=$ 1, 0.9, 0.8
and 0.7). In particular, measurements in weak transverse field (TF)
configuration are used in other to estimate the phase boundaries, while zero
field (ZF) measurements are employed to deduced the detailed magnetic ground
state. Transverse refers to the direction of the externally applied field,
which is perpendicular with respect to the initial muon spin polarisation.

Figure 2: Zero field time spectra collected at selected pressures at $T=5$ K
for the Sr1-xCaxCo2P2 compounds: (a) $x=1$, (b) $x=0.9$, (c) $x=0.8$ and (d)
$x=0.7$. The solid lines represents fits using Eq. (5). Each spectra have been
shifted vertically for clarity of display.

###  III.1 Zero field

Figure 2 displays the collected ZF $\mu^{+}$SR time spectra at $T=5$ K for the
compositions $x=1$, 0.9, 0.8 and 0.7 as a function of pressure. The time
spectra for the compounds at 0 kbar are fully consistent with previously
published ambient pressure results J. Sugiyama, et al (2015). A strong
pressure dependence is observed for the $x=1$ compound. Indeed, several
oscillations are present in the time spectra for $p\leq 9.2$ kbar, where the
number of frequencies as well as the amplitude of the oscillations decreases
with the applied pressure. Such oscillations are fully suppressed for $p_{\rm
c2}\geq 9.7$ kbar. A much more moderate pressure dependence is observed for
the other compositions ($x=$ 0.9, 0.8 and 0.7). Consequently, it is initially
clear that the ZF time spectra has a strong $x$ dependence, while the most
evident hydrostatic pressure effect seem to be limited to $x=1$ compound. In
order to more systematically characterize the detailed changes across $x$ and
$p$, the time spectra were fitted using a combination of several exponentially
relaxing oscillations, an exponential and a static Gaussian Kubo-Toyabe
(SGKT):

$\displaystyle A_{0}\,P_{\rm ZF}(t)$ $\displaystyle=$
$\displaystyle\sum^{n}_{i}A^{\rm AF}_{i}\cos(f^{\rm AF}_{i}2\pi t+\phi^{\rm
AF}_{i})e^{-\lambda^{\rm AF}_{i}t}$ (1) $\displaystyle+$ $\displaystyle A_{\rm
tail}e^{-\lambda_{\rm tail}t}+A_{\rm PC}G(t,\Delta_{\rm PC})e^{-\lambda_{\rm
PC}t}$ (3) $\displaystyle+$ $\displaystyle A_{\rm PM}e^{-\lambda_{\rm PM}t},$
(5)

where $A_{0}$ is the initial asymmetry, determined by the instrument, and
$P_{\rm ZF}$ is the muon spin polarisation function in ZF configuration. In
detail, the number of oscillations, $n=3$ and $n=2$, where used for the
compositions $x=1$ and $x=0.8$, respectively, while $n=1$ was used for both
$x=0.9$ and $x=0.7$. $A^{\rm AF}_{i}$, $f^{\rm AF}_{i}$, $\phi^{\rm AF}_{i}$
and $\lambda^{\rm AF}_{i}$ are the asymmetry, precession frequency, phase and
relaxation rate for the internal field component that is perpendicular with
respect to the initial muon spin polarisation. $A^{\rm tail}$ and
$\lambda^{\rm tail}$ on the other hand are the asymmetry and relaxation rate
of the so called tail component, $i.e.$ the internal field component that is
parallel to the initial muon spin polarisation. In a perfect powder that is
magnetically ordered, 2/3 of the internal field components are expected to be
perpendicular while 1/3 of the internal field components are parallel (due to
the so-called ’powder average’). $A_{\rm PM}$, and $\lambda_{\rm PM}$ are
accounting for the new high pressure paramagnetic (PM) phase of the $x=1$
compound that is not magnetically ordered [Fig. 2(a)]. $A_{\rm PC}$,
$\Delta_{\rm PC}$ and $\lambda_{\rm PC}$ on the other hand are the asymmetry,
the field distribution width and the corresponding exponential relaxation rate
of the static Gaussian KT, represented by $G(t,\Delta)$, of the pressure cell
(PC). Here, $G(t,\Delta)$ originate from isotropically distributed magnetic
moments while the exponential accounts for additional relaxation present on
top of it. Such description holds for when the internal field is composed of
two separate yet independent magnetic field origins. In such case, the Fourier
transform of the convolution of each field distribution is the product of each
polarisation functions. In other words, the KT accounts for isotropically
distributed nuclear moments while the exponential accounts for additional
relaxation posed by highly fluctuating electronic moments.

The constraint $A_{\rm tail}=\frac{1}{2}\sum A^{\rm AF}_{i}$ (i.e. 1/3 vs.
2/3) was set for the fitting procedures using Eq. (5), in order to separate
the various contributions present for $p>0$. Moreover, the compounds are known
exhibit commensurate magnetic order. Indeed, $\phi_{1}=-20.0(2.9)^{\circ}$,
$\phi_{2}=-10.6(10.4)^{\circ}$ and $\phi_{3}=-28.7(16.6)^{\circ}$ are obtained
for the compound $x=1$ at $p=0$, consistent with a commensurate order.
Therefore, a common phase was set, $i.e.$ $\phi_{i}=\phi$, across all
oscillations for all measured pressures. Using the described fitting
procedure, the pressure dependencies of the fit parameters for the $x=1$
compound are shown in Fig. 3. At ambient pressure $A_{\rm
tail}\simeq\frac{1}{2}\sum A^{\rm AF}_{i}$ is obtained, suggesting that the
constraint set for higher pressures is valid. For higher pressures, the
asymmetries $A^{\rm AF}_{i}$ decreases gradually as the $A^{\rm PM}$ component
increases.

It should be noted that the total sample asymmetry ($\sum A=\sum A^{\rm
AF}_{i}+A_{\rm tail}+A_{\rm PM}$) show a fairly constant behaviour up to about
6 kbar. At higher pressures, a sudden decreases is observed. The origin of
this behaviour is highlighted in Fig. 4. As clearly seen, a significant
missing fraction presents itself at higher pressures. Since a missing fraction
cannot be fitted (since it lies outside the accessible time frame of the
$\mu^{+}$SR instrument), the total sample asymmetry ($\sum A$) exhibit a
decrease as a function of pressure. The origin of this missing fraction is
discussed below. As mentioned, $A^{\rm PM}$ represents the volumic fraction of
non-magnetically ordered state, $e.g.$ paramagnetic or spin liquid. Such high
pressure state is discussed in Sec. IV.

Figure 3: Pressure dependent fit parameters, collected at $T=5$ K, of the
$x=1$ compound obtained through Eq. (5). For reference, $\sum A=\sum A^{\rm
AF}_{i}+A_{\rm tail}+A_{\rm PM}$ has been computed and is included as well.
The solid lines are guide to the eye while the vertical shaded area and dashed
line indicate the critical pressures, $p_{\rm c1}\approx 8$ kbar and $p_{\rm
c2}^{\prime}\approx 9.5$ kbar.

The pressure dependence of the precession frequencies for the $x=1$ compound
are shown in Fig. 3(b). Overall, the pressure seem to have a weak effect on
the value of the frequencies themselves (i.e. the local internal magnetic
field of the sample). The main frequency, $f_{2}$, maintains more or less the
value at ambient pressure ($\sim 34$ MHz) up to 9.2 kbar ($\sim 33$ MHz). We
may initially define the pressure point at which all precession frequencies
are absent as the critical pressure, $p_{\rm c2}^{\prime}\approx 9.5$ kbar. We
will further refine such critical point based on the TF measurement presented
below. It is noted that the 24 MHz ($f_{3}$) frequency was not present in the
previous ambient pressure study J. Sugiyama, et al (2015). Most likely, the
$x=1$ sample of the present study is both much larger and of higher quality,
resulting in that even the minor frequency can be distinguished. That being
said, the inclusion of such frequency does not change/alter the interpretation
of this or the previous study.

Further, it is evident that the $f_{1}$ and $f_{3}$ components drop to 0 MHz
already at $p_{\rm c1}\approx 8$ kbar, i.e. prior to the vanishing of the
$f_{2}$ component at $p_{\rm c2}^{\prime}\approx 9.5$ kbar. It is however
clear the asymmetry ($A^{\rm AF}_{1}$) still poses non-zero values. In other
words, the polarisation is a fast exponential instead of an oscillation. Such
behaviour suggests a widening of the internal field distribution width, in
which the oscillation becomes highly damped. Such behaviour would be
consistent with a spin reorientation, structural transition ($i.e.$ muon sites
changes) and/or it could reflect a difficulty in fitting small asymmetries.
However, we will later show (below) that this effect is indeed a true sample
effect and not related to a fitting problem.

While it is not shown, the relaxation rates, $\lambda^{\rm AF}_{i}$, show a
weak pressure dependence. Roughly put, a value of $\lambda^{\rm
AF}_{i}=9~{}\mu$s-1 is obtained for all oscillation across all pressures.
$\lambda_{\rm tail}$ on the other hand exhibits low or values close to 0
across all pressures, suggesting a static magnetic ground state. $\lambda_{\rm
PM}$ on the other hand increases relatively for $p>2.3$ kbar, suggesting that
the high pressure phase is dynamic in origin.

Figure 4: Zero field time spectra collected at $T=5$ and 35 K at $p=9.7$ kbar
for the $x=1$ compound. The solid lines represents fits using Eq. 5 with $n=1$
but with $f_{1}=0$ MHz for $T=35$ K and $n=0$ for $T=5$ K. The horizontal
solid black line is the estimated $A_{0}$ while the dashed black horizontal
line indicate the amount of asymmetry missing.

The ZF time spectra collected at 5 and 35 K for the $x=1$ at 9.7 kbar are
shown in Fig. 4. The 5 K time spectra is identical to the one shown in Fig.
2(a). While the 5 K time spectrum show no significant signature, the 35 K time
spectrum manifests an fast relaxing exponential. Perhaps a dip/minimum is
present in Fig. 4 around $0.5~{}\mu$s, suggesting that the polarisation
function is more of an oscillation rather than an exponential. However, a fit
with a cosine function yields unreasonably high asymmetry values. Regardless,
the fraction of this fast exponential decreases with decreasing temperature.
Such behaviour is the origin behind the complex temperature dependence of
$A_{\rm TF}$ at lower temperatures (Fig. 7(a)). In fact, resistivity
measurements under pressure indicated a sudden change in the derivative of the
resistivity around this temperature and pressure (Baumbach  _et al._ , 2014)
and is discussed in Sec. IV.

As already noted, a significant fraction is missing, $i.e.$ the total
asymmetry does not add up to values close to $A_{0}$. Typically, missing
fractions are associated with muonium formations Boekema  _et al._ (1986); Cox
_et al._ (2006); Forslund  or by a presence of quasi-static wide field
distribution. Since the $x=1$ compound is metallic even under pressure,
muonium formation is unlikely. Instead, the missing fraction originates from a
wide internal field distribution yielding oscillations and fluctuations
outside the time window of $\mu^{+}$SR. Such a scenario is consistent with
having muon sites close to CoP2 tetrahedra layers giving a wide field
distributions due to slowly fluctuating Co $d$-moments. Similar missing
fraction effect was observed e.g. in the 2D AF magnet NaNiO2 Forslund  _et
al._ (2020), which was shown to originate from quasi static wide field
distributions at the muon site close to the Ni-O octahedra. In other words,
the suppression of asymmetries [Fig. 3(a)] is consistent with changes in the
magnetic characteristics at high pressure, giving the wider field
distributions at the muon site ($i.e.$ the magnetic environment around the
muon site changes). Given the crystal flexibility of the $x=1$ compound, it
could also be that additional energetically favorable crystalline muon sites
becomes available under higher pressure instead ($i.e.$ the crystalline muon
site changes). Such assertion is ideally confirmed by a combination of high
pressure XRD and DFT calculations. Regardless of the driver for the missing
fraction, it is clear that the origin is related to magnetism of the sample
since the missing fraction is in fact absent in the $p=17.2$ kbar measurement.

For the sake of completeness, the pressure dependent precession frequencies
for all measured $x$ is presented in Fig. 5. The frequencies of the $x=1$
compound are the same as presented in Fig. 3(b) (and are therefore made partly
transparent). The main frequency of about 34 MHz is present for $x\geq 0.8$.
Such frequency is a consequence of the long range order present at ambient
pressure for $x\geq 0.8$ J. Sugiyama, et al (2015). At $x=0.8$ however, an
additional frequency appears around 2 MHz. Such frequency is concluded to be
associated with a short-range order based on the fact that the same frequency
but as a highly damped oscillation is present for the $x=0.7$ sample J.
Sugiyama, et al (2015). Perhaps $x=0.8$ exhibits a co-existence of short and
long range order, similar to in the isostructural compound LCo2P2 Forslund
_et al._ (2021). Regardless though, only weak shifts in the precession
frequencies are observed as a function of pressure for $x<1$. Similarly, the
asymmetries and depolarisation rates show only weak pressure dependence, as
already hinted directly from Fig. 2(b-d).

Figure 5: Pressure dependent precession frequencies, collected at $T=5$ K, of
Sr1-xCaxCo2P2 compounds obtained through Eq. (5): $x=1$ (light pink), 0.9
(black), 0.8 (blue) and 0.7 (green). The solid lines are guide to the eye.

###  III.2 Transverse field

In order to gain a more detailed insights of the temperature dependent
behaviour, the series of compounds were also studied under TF configuration
for selected pressures. Figure 6 shows the collected TF ($\sim 50$ Oe)
$\mu^{+}$SR time spectra from the $x=1$ compound at $p=0$ for selected
temperatures. Regardless of temperature, a single distinct oscillation of
about 0.7 MHz is observed, corresponding to the externally applied field of 50
Oe. A strong temperature dependence in the amplitude is seen, reflecting the
formation of static internal magnetic fields. Moreover, the time spectra
exhibits a positive shift in asymmetry at lower temperatures. Therefore, the
TF time spectra were fitted using a combination of one exponentially relaxing
oscillation together an non oscillating exponential:

$\displaystyle A_{0}\,P_{\rm TF}(t)$ $\displaystyle=$ $\displaystyle A_{\rm
TF}\cos(f_{\rm TF}2\pi t+\phi_{\rm TF})e^{-\lambda_{\rm TF}t}$ (6)
$\displaystyle+$ $\displaystyle A_{\rm S}e^{-\lambda_{\rm S}t},$ (8)

where $A_{0}$ is the initial asymmetry determined by the instrument and
$P_{\rm TF}$ is the muon spin polarisation function in TF configuration.
$A_{\rm TF}$, $f_{\rm TF}$, $\phi_{\rm TF}$ and $\lambda_{\rm TF}$ are the
asymmetry, frequency, phase and depolarisation rate resulting from the applied
TF, while $A_{\rm S}$ and $\lambda_{\rm S}$ are the asymmetry and the
relaxation rate resulting from internal magnetic field components, that is
mostly parallel to the initial muon spin. The perpendicular internal magnetic
field components are excluded from Eq. (8), since such contribution are
usually difficult to model in TF configuration. Excluding such contribution
does not affect the fit results nor the interpretation of the data.

Figure 6: Transverse field (TF = 50 Oe) $\mu^{+}$SR time spectra collected for
CaCo2P2 ($x=1$) at $p=0$ kbar for selected temperatures. The solid lines
represents fits with Eq. (8).

The temperature dependencies of the obtained TF asymmetry using Eq. (8) are
displayed in Fig. 7(a) for the $x=1$ compound. $A_{\rm TF}$ has a temperature
dependence expected for a magnetically ordered sample. At low temperatures
$A_{\rm TF}$ experiences a temperature independent behaviour. As the
temperature increases, an increase in $A_{\rm TF}$ is observed and full
asymmetry ($A_{0}$) is recovered. Ideally, the normalized $A_{\rm TF}$/$A_{0}$
should corresponds to the paramagnetic volume fraction of the system. In this
case, it is in fact the non-magnetically ordered volume fraction of the sample
together with the fraction of muons stopping inside the pressure cell. In
other words, the increase of $A_{\rm TF}$ corresponds to the transition
temperature ($T^{\rm TF}_{\rm N}$) in which the sample changes from a
magnetically ordered to disordered state. An accurate value of $T^{\rm
TF}_{\rm N}$ is obtained by employing sigmoidal fit function as a function of
temperature for each measured pressure. The pressure dependent values [i.e.
($T^{\rm TF}_{\rm N}(p,x)$] are then utilized to contruct the detailed phase
diagram presented in Fig. 8. Intriguingly, $A_{\rm TF}$ show a complex
temperature dependence below $T^{\rm TF}_{\rm N}$ for the measurements
performed at $p=9.2$ and 9.7 kbar. In fact some of the TF asymmetry ($A_{\rm
TF}$) seems to be recovered around $T_{\rm i}\approx 20$ K (Fig. 7). Such
complexity is reflecting the fact that an initial faster relaxation is present
at higher temperature in ZF (Fig. 4). That $A_{\rm TF}$ still exhibits a drop
at $T_{\rm N}$ is evidence of static internal field formation, which in ZF
configuration resulted into a missing fraction as described above. As
mentioned, this fraction stems from muons experiencing very broad field
distribution width. At highest pressure on the other hand, no significant
temperature dependence is observed, suggesting the absence of magnetic order
at lower temperatures. A similar complicated $A_{\rm TF}(T)$ behaviour is not
observed in any of the other ($x\neq 1$) compounds.

Figure 7: Temperature and pressure dependence of $A_{\rm TF}$ for the $x=1$
compound, obtained using Eq. (8). The solid lines are fits using the sigmoidal
function, while the vertical shaded area indicate an approximative region for
the transition temperature $T_{\rm i}\approx 20$ K (see text and Fig. 8 for
further details).

###  III.3 Phase diagram

Based on the presented pressure dependent results, and from previous ambient
pressure study J. Sugiyama, et al (2015), a $T/p/x$ phase diagram for
Sr1-xCaxCo2P2 can be constructed [Fig. 8(a)]. In general, the detailed ground
state is estimated from the ZF measurements while the temperature boundary is
estimated from TF configuration. The transition temperatures, based on TF
measurements, are presented in Fig. 8 as a function of pressure. As already
pointed out, a strong pressure dependence is observed for the $x=1$ compound.
The transition temperature decreases linearly with pressure, until it is
completely and suddenly suppressed around $p_{\rm c2}\approx 9.8$ kbar. Such
critical pressure is fully coherent with the ZF frequency dependence presented
above in Fig. 3(b). Further, both ZF (Fig. 3) and TF (Fig. 7) data show
signatures for additional phases (FMI-\small{1}⃝ and FMI-\small{2}⃝) appearing
already at pressures in the vicinity of $p_{\rm c1}\approx 8$ kbar and $T_{\rm
i}\approx 20$ K. The origin of such phases are further discussed below in Sec.
IV.

A linear like decrease in the transition temperature is also observed for the
$x=0.9$ and $x=0.8$ compounds [Fig. 8(b)]. While the magnetic order is
persistent within the current pressure range, most likely, the long range
order will be destroyed at higher pressures. More unexpectedly, the pressure
dependence of the $x=0.7$ compound is opposite from the other ones. Instead of
a decrease in transition temperature, the pressure slightly increases $T_{\rm
N}$, suggesting that the magnetic order is stabilized under pressure. Such
behaviour is consistent with the results obtained at ambient pressure, where
the $T_{\rm N}$ of $x=0.7$ is lower than that of $x=0.75$, at which point a
long range magnetic order is stabilised J. Sugiyama, et al (2015). In other
words, chemical pressure (and hydrostatic pressure) stabilises the magnetic
order for $x<0.75$ until a long range order is stabilised. That being said,
the ZF time spectra [Fig. 2(d)] and the fit results show no apparent change
with pressure. Therefore, a short range order can be expected up to at least
20 kbar for the $x=0.7$ compound.

Figure 8: (a) Phase-diagram ($P/T/x$) for Sr1-xCaxCo2P2. (b) Individual $P/T$
phase diagrams for different chemical compositions ($x$). Phases are labelled:
SR = short-range, LR = long-range, AFM = antiferromagnetic, FMI =
ferromagnetic islands, PM = paramagnetic. Further, $p_{\rm c1}\approx 8$ kbar
and $p_{\rm c2}\approx 9.8$ kbar are the critical pressures and $x_{\rm
c1}\approx 0.45$ and $x_{\rm c2}\approx 0.75$ critical chemical compositions
J. Sugiyama, et al (2015). Finally, $T_{\rm i}\approx 20$ K is an approximate
transition temperature between the FMI-\small{1}⃝ and FMI-\small{2}⃝ phases.

##  IV Discussion

Given that no significant pressure dependence is observed for $x\neq 1$
compounds, we may attribute the high pressure ground state (up to 20 kbar) to
be the same as in ambient pressure. Previous ambient pressure $\mu^{+}$SR
study J. Sugiyama, et al (2015) as a function of $x$ determined the formation
of short range AF order for $0.45<x\leq 0.75$, which develops into an long
range AF order for higher $x$. Such findings is consistent with this study,
where distinct oscillations are present for $x>0.7$ compounds, but only a
single highly damped oscillation is present for the $x=0.7$ compound. The
magnetic order formation was found to be strongly correlated with the distance
of Co ions and the adjacent Co2P2 planes ($d_{\rm Co-Co}$). In detail, a
decrease in $d_{\rm Co-Co}$ is observed from about $x=0.4$ and decreases
linearly up to about $x=0.8$.

If the magnetic order is truly only dependent on $d_{\rm Co-Co}$, then one
would expect the formation of long range order at higher pressures for the
$x=0.7$ compound. Instead, only small pressure dependence is observed, even
though the $x=0.7$ seems slightly more stable at higher pressures. This would
suggest that the exchange mechanism that stabilises the magnetic order does
not only dependent on $d_{\rm Co-Co}$. Of course, we should acknowledge the
fact that the pressure applied in this study is hydrostatic and not uni-axial,
even though chemical pressure can be considered equivalent to hydrostatic
pressure. However, it is of course possible that a LRO is in fact stabilised
at even higher pressures. Such premise may be confirmed by a pressure study on
the $x=0.75$ compound, which is closer to the LRO phase and should yield lower
critical pressures. It is noted that while the frequency of the short range
ordered phase is small, the highly damped nature does not stem from magnetic
inhomogeniousity. Instead, the highly damped nature stems from a broad field
distribution due to SRO formation, whereas the value of the frequency itself
suggests that the internal field at the muon site is small.

The $x=1$ compound on the other hand exhibits a strong pressure dependence,
where the external pressure destabilizes the magnetic order. Such
destabilizing with pressure was also observed for $x=0.9$ and 0.8 compounds,
even though it is much weaker. A simple and rough extrapolation would suggest
that the long range order is fully suppressed at around 50 and 100 kbar for
$x=0.9$ and 0.8, respectively. The much weaker pressure dependence of the
$x=0.9$ and 0.8 compounds supports the fact that the magnetic order may be
established by exchanges other than simply the inter-plane Co ion interaction.

The pressure clearly induces a transition in the $x=1$ compound, from
magnetically long-range order to a magnetically disordered state. Suppression
of magnetic order under applied external pressure is a signature of quantum
criticality Forslund  _et al._ (2019); Sugiyama  _et al._ (2020). Of course,
for such scenario one would expect that $T_{\rm N}$ would be more smoothly
driven towards $T=0$ K, which is not really seen here. That being said, the
related compound CrRh2Si2 Movshovich  _et al._ (1996) or the $d$-electron
antiferromagnet Cr1-xVx Lee  _et al._ (2004) were both shown to exhibit a
second order nature, despite a similar abrupt decrease. Figure 3(b) show the
order parameter evolution as a function of pressure for the $x=1$ compound.
Here, an abrupt decrease of the order parameter is observed between 9.2 and
9.7 kbar. In other words, the suppression of the magnetic order under pressure
is not likely to be quantum critical. This is also coherent with the fact that
Fermi liquid behaviour is observed through out the pressure range, based on
resistivity measurements Baumbach  _et al._ (2014). Instead, the transition is
first order in nature and thus most likely driven by a structural component.
Such driver is consistent with muon site re-population mentioned above. In
fact, similar compounds have been shown to undergo first order transition
under pressure, e.g. EuCo2P2 and SrNi2P2 but then at higher temperatures Huhnt
_et al._ (1997).

Since there is no XRD data under pressure, and at low temperature, available,
we should acknowledge the fact that a muon site re-population may originate
from magneto structural transition and not simply from a crystalline
structural transformation. In other words, the missing fraction may not stem
from a change of crystalline muon sites but a change in the magnetic
environment. Such behaviour would be consistent with the sample undergoing AF-
FM transition under pressure above 6 kbar ($p_{\rm c1}\approx 8$ kbar).
Typically, missing fractions are also more commonly observed for FM structures
rather than in AF. Although, this is not a general rule. One could perhaps
expect FM correlations to emerge under pressure at lower temperatures given
that FM correlations seem to be present at ambient pressure above $T_{\rm N}$
Imai  _et al._ (2014). Indeed, the related compound LaCo2P2 is an itinerant FM
at ambient pressure Mörsen  _et al._ (1988); Reehuis  _et al._ (1994). While
any pressure dependent measurement is missing for LaCo2P2, a second phase
emerged in the itinerant FM UGe2 at lower temperature close to $p_{C}$
Pfleiderer and Huxley (2002), similar to what is observed for the $x=1$
compound. If the title compound undergoes a AF-FM transition under pressure,
the discussions about the QCP and its relationship to the order parameters
(Fig. 3(b)) is not viable given the fact that the order parameter of the
main/new phase (the missing fraction) is not accessible.

Resistivity measurements under pressure revealed a change in its derivative at
8.9 kbar below 50 K Baumbach  _et al._ (2014), which was slowly suppressed in
temperature with pressure. While the details and the origin remained unsolved,
such feature is here clearly revealed to be of magnetic origin. This is based
on an additional exponential component manifested in the ZF time spectrum
(Fig. 4) and that $A_{\rm TF}$ (Fig. 7) reveals an anomaly around this
pressure ($p_{\rm c1}\approx 8$ kbar) and temperature (which in turn is a
consequence of such exponential). Since the amount of missing fraction is
temperature independent at $p=9.7$ kbar (Fig. 4), the initial faster
relaxation at $p=9.7$ kbar should yield some clues regarding the second phase
that manifests itself at low temperatures (below $T_{\rm i}\approx 20$ K) and
above $p_{\rm c1}\approx 8$ kbar.

In order to comprehend the second phase present at high pressure and low
temperature, it is imperative to unveil microscopic origin of the exponential.
Unfortunately, the presence of the missing fraction makes the determination
challenging. Regardless, lets first point out that there exist mixed fractions
at this pressure: a missing fraction, a disordered phase ($A_{\rm PM}$) and a
fraction of muons depolarising in a fast exponential manner, that also changes
with temperature. With this in mind, we shall propose few scenarios to the
origin of the exponential. (I) The missing fraction is resulting from dilute
(ferro)magnetic islands (FMI). In this case, part of the muons will be
situated in the disordered phase that is relatively close to two separate yet
correlated magnetic islands. The inter-island correlation should depolarise
the moun ensemble in an exponential manner that is different from the
disordered phase. This scenario implies that some islands merges at lower
temperatures (below $T_{\rm i}\approx 20$ K), or the inter-island correlations
are suppressed at lower temperatures for reasons unknown. (2) The exponential
is a reminiscence of the missing fraction. It might be that
oscillations/fluctuations hidden as a missing fraction is ”spilled over” to
the time window of $\mu^{+}$SR. This would imply that the correlation times
changes as the temperature is lowered. In other words, the missing fraction is
somewhat fluctuating at higher temperatures, that becomes more static at lower
temperature. Although, the change in fluctuations has to be driven by other
means than just simple thermal effects. (3) The muon site coordinates might
have temperature dependencies at this pressure. Similar to the pressure
dependence of the muon site, the temperature dependence of it could result in
re-population of different sites. This implies that the low temperature phase
and high pressure phase is driven by a structural component or a spin
structure reorientation.

Given the structural and magnetic degree of freedom present in the $x=1$
compound, it may be that several processes discussed above are viable. In
order to discern one scenario from another, it is of high interest to perform
high pressure XRD, high pressure magnetisation and neutron diffraction.
Although, since the magnetic phases seem to form island like structures,
detailed experimental study might prove difficult. We wish to stress that the
complex temperature dependence of $A_{\rm TF}(T)$ was not observed for the
$x\neq 1$ compounds. In other words, impure samples or chemical disorder are
not likely the underlying reason behind the features observed here. Especially
since the pure $x=1$ compounds are in general cleaner than chemically doped
samples. Here, we should emphasize that $\mu^{+}$SR is able to detect magnetic
volume fractions in a sample. To conclude, we propose that the intermediate
phase emerging above $p_{\rm c1}\approx 8$ kbar consists of dilute
ferromagnetic islands (FMI) existing within a disordered phase. Further, such
FMI undergo an additional transition between a high- (FMI-\small{1}⃝) and low-
temperature (FMI-\small{2}⃝) state at $T_{\rm i}\approx 20$ K. It should also
be emphasized that the values of $p_{\rm c1}\approx 8$ kbar and $T_{\rm
i}\approx 20$ K are approximated from a combination of ZF/TF $\mu^{+}$SR data
(with limited number of pressure points) in combination with the resistivity
data from Ref. Baumbach  _et al._ , 2014. The presented numbers should
therefore be taken as estimates and further detailed studies are necessary to
more accurately determine the phase boundaries. However, $p_{\rm c2}\approx
9.8$ kbar could be considered slightly more well defined.

Finally, we wish to discuss nature of the high pressure phase (above $p_{\rm
c2}\approx 9.8$ kbar). The resistivity above such critical pressure revealed a
hump in the data Baumbach  _et al._ (2014). Such ’hump temperature’ was
increasing with higher pressures. One suggestion was that it may be related to
a broad magnetic transition, like seen in the related compounds CaNi1-xCoxP2
Jia  _et al._ (2010), BaFe1-xCrxAs2 Marty  _et al._ (2011) and BaFe1-xMnxAs2
Thaler  _et al._ (2011). However, such conclusion can be readily excluded
based on this $\mu^{+}$SR study. Instead, it might be that strong electron
coupling is behind the such resistivity hump, like in Nb3Sn and Nb3Sb Fisk and
Webb (1976), which was also one of the suggestion of Ref. Baumbach  _et al._ ,
2014.

##  V Conclusions

The pressure and temperate dependence on the magnetic nature of Sr1-xCaxCo2P2
for $x=1$, 0.9, 0.8 and 0.7 has been investigated by muon spin rotation,
relaxation and resonance ($\mu^{+}$SR). The weak pressure dependencies for the
compounds $x\neq 1$ suggests that the rich phase diagram of Sr1-xCaxCo2P2 at
ambient pressure may not only be due to chemical pressure effects. The $x=1$
compounds on the other hand exhibits strong pressure effects, where the long
range magnetic order present at ambient pressure become fully suppressed at
$p_{\rm c2}\approx 9.8$ kbar. Intriguingly, two additional phase emerges
already just below the critical region, occupying the phase space above
$p_{\rm c1}\approx 8$ kbar and below $p_{\rm c2}\approx 9.8$ kbar. Such
results are fully coherent with previous single crystal resistivity
measurement under pressure Baumbach  _et al._ (2014). Since $\mu^{+}$SR is
sensitive to magnetic volume fractions, such phase was proposed to be dilute
(ferro)magnetic islands (FMI) co-existing within a disordered phase. It is
also revealed that such phase consists of a high- (FMI-\small{1}⃝) and a low-
temperature (FMI-\small{2}⃝) region, respectively, with a phase boundary at
$T_{\rm i}\approx 20$ K.

###### Acknowledgements.

This research was supported by the Swedish Research Council (VR) via a Neutron
Project Grant (Dnr. 2016-06955) and the Carl Tryggers Foundation for
Scientific Research (CTS-18:272). E.N. is funded by the Swedish Foundation for
Strategic Research (SSF) within the Swedish national graduate school in
neutron scattering (SwedNess). D.A. acknowledges financial support from the
Romanian UEFISCDI project PN-III-P4-ID-PCCF-2016-0112, Contract Nr. 6/2018.
Y.S. is funded by VR through a Starting Grant (Dnr. 2017-05078) as well as the
Chalmers Area of Advance-Materials Science. J.S. acknowledges support from
Japan Society for the Promotion Science (JSPS) KAKENHI Grants No. JP18H01863
and No. JP20K21149.

## References

  * Johrendt  _et al._ (1997) D. Johrendt, C. Felser, O. Jepsen, O. K. Andersen, A. Mewis, and J. Rouxel, Journal of Solid State Chemistry 130, 254 (1997).
  * Reehuis  _et al._ (1998) M. Reehuis, W. Jeitschko, G. Kotzyba, B. Zimmer, and X. Hu, Journal of Alloys and Compounds 266, 54 (1998).
  * Hoffmann and Zheng (2002) R. Hoffmann and C. Zheng, The Journal of Physical Chemistry 89, 4175 (2002).
  * Rotter  _et al._ (2008a) M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. Pöttgen, Phys. Rev. B 78, 020503 (2008a).
  * Tan  _et al._ (2017) X. Tan, V. O. Garlea, K. Kovnir, C. M. Thompson, T. Xu, H. Cao, P. Chai, Z. P. Tener, S. Yan, P. Xiong, and M. Shatruk, Phys. Rev. B 95, 024428 (2017).
  * Rotter  _et al._ (2008b) M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101, 107006 (2008b).
  * Torikachvili  _et al._ (2008) M. S. Torikachvili, S. L. Bud’ko, N. Ni, and P. C. Canfield, Phys. Rev. Lett. 101, 057006 (2008).
  * Reehuis and Jeitschko (1990) M. Reehuis and W. Jeitschko, Journal of Physics and Chemistry of Solids 51, 961 (1990).
  * Kreyssig  _et al._ (2008) A. Kreyssig, M. A. Green, Y. Lee, G. D. Samolyuk, P. Zajdel, J. W. Lynn, S. L. Bud’ko, M. S. Torikachvili, N. Ni, S. Nandi, J. B. Leão, S. J. Poulton, D. N. Argyriou, B. N. Harmon, R. J. McQueeney, P. C. Canfield, and A. I. Goldman, Phys. Rev. B 78, 184517 (2008).
  * Jia  _et al._ (2009) S. Jia, A. J. Williams, P. W. Stephens, and R. J. Cava, Phys. Rev. B 80, 165107 (2009).
  * J. Sugiyama, et al (2015) J. Sugiyama, et al, Phys. Rev. B 91, 69 (2015).
  * Teruya  _et al._ (2014) A. Teruya, A. Nakamura, T. Takeuchi, H. Harima, K. Uchima, M. Hedo, T. Nakama, and Y. Ōnuki, Journal of the Physical Society of Japan 83, 113702 (2014), https://doi.org/10.7566/JPSJ.83.113702 .
  * Baumbach  _et al._ (2014) R. E. Baumbach, V. A. Sidorov, X. Lu, N. J. Ghimire, F. Ronning, B. L. Scott, D. J. Williams, E. D. Bauer, and J. D. Thompson, Phys. Rev. B 89, 094408 (2014).
  * Imai  _et al._ (2014) M. Imai, C. Michioka, H. Ohta, A. Matsuo, K. Kindo, H. Ueda, and K. Yoshimura, Phys. Rev. B 90, 014407 (2014).
  * R. Khasanov, et al (2016) R. Khasanov, et al, High Pressure Research 36, 140 (2016).
  * D. Andreica (2001) D. Andreica, (2001), Ph.D. thesis, IPP/ETH-Zurich, 2001.
  * A. Suter and B. M. Wojek (2012) A. Suter and B. M. Wojek, Phys. Proc. 30, 69 (2012).
  * Boekema  _et al._ (1986) C. Boekema, K. C. Chan, R. L. Lichti, A. B. Denison, D. W. Cooke, R. H. Heffner, R. L. Hutson, and M. E. Schillaci, Hyperfine Interactions 32, 667 (1986).
  * Cox  _et al._ (2006) S. F. J. Cox, J. S. Lord, S. P. Cottrell, J. M. Gil, H. V. Alberto, A. Keren, D. Prabhakaran, R. Scheuermann, and A. Stoykov, Journal of Physics: Condensed Matter 18, 1061 (2006).
  * (20) O. K. Forslund, “Your muonium is $\mu$-drogen,” in  _Proceedings of the 14th International Conference on Muon Spin Rotation, Relaxation and Resonance ( $\mu$SR2017)_, https://journals.jps.jp/doi/pdf/10.7566/JPSCP.21.011066 .
  * Forslund  _et al._ (2020) O. K. Forslund, H. Ohta, K. Kamazawa, S. L. Stubbs, O. Ofer, M. Månsson, C. Michioka, K. Yoshimura, B. Hitti, D. Arseneau, G. D. Morris, E. J. Ansaldo, J. H. Brewer, and J. Sugiyama, Phys. Rev. B 102, 184412 (2020).
  * Forslund  _et al._ (2021) O. K. Forslund, D. Andreica, H. Ohta, M. Imai, C. Michioka, K. Yoshimura, M. Månsson, and J. Sugiyama, “Co-existence of short- and long-range magnetic order in laco2p2,” (2021), arXiv:2107.02560 [cond-mat.str-el] .
  * Forslund  _et al._ (2019) O. K. Forslund, D. Andreica, Y. Sassa, H. Nozaki, I. Umegaki, E. Nocerino, V. Jonsson, O. Tjernberg, Z. Guguchia, Z. Shermadini, R. Khasanov, M. Isobe, H. Takagi, Y. Ueda, J. Sugiyama, and M. Månsson, Scientific Reports 9, 1141 (2019).
  * Sugiyama  _et al._ (2020) J. Sugiyama, D. Andreica, O. K. Forslund, E. Nocerino, N. Matsubara, Y. Sassa, Z. Guguchia, R. Khasanov, F. L. Pratt, H. Nakamura, and M. Månsson, Phys. Rev. B 101, 174403 (2020).
  * Movshovich  _et al._ (1996) R. Movshovich, T. Graf, D. Mandrus, M. Hundley, J. Thompson, R. Fisher, N. Phillips, and J. Smith, Physica B: Condensed Matter 223-224, 126 (1996), proceedings of the International Conference on Strongly Correlated Electron Systems.
  * Lee  _et al._ (2004) M. Lee, A. Husmann, T. F. Rosenbaum, and G. Aeppli, Phys. Rev. Lett. 92, 187201 (2004).
  * Huhnt  _et al._ (1997) C. Huhnt, W. Schlabitz, A. Wurth, A. Mewis, and M. Reehuis, Phys. Rev. B 56, 13796 (1997).
  * Mörsen  _et al._ (1988) E. Mörsen, B. Mosel, W. Müller-Warmuth, M. Reehuis, and W. Jeitschko, Journal of Physics and Chemistry of Solids 49, 785 (1988).
  * Reehuis  _et al._ (1994) M. Reehuis, C. Ritter, R. Ballou, and W. Jeitschko, Journal of Magnetism and Magnetic Materials 138, 85 (1994).
  * Pfleiderer and Huxley (2002) C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett. 89, 147005 (2002).
  * Jia  _et al._ (2010) S. Jia, S. Chi, J. W. Lynn, and R. J. Cava, Phys. Rev. B 81, 214446 (2010).
  * Marty  _et al._ (2011) K. Marty, A. D. Christianson, C. H. Wang, M. Matsuda, H. Cao, L. H. VanBebber, J. L. Zarestky, D. J. Singh, A. S. Sefat, and M. D. Lumsden, Phys. Rev. B 83, 060509 (2011).
  * Thaler  _et al._ (2011) A. Thaler, H. Hodovanets, M. S. Torikachvili, S. Ran, A. Kracher, W. Straszheim, J. Q. Yan, E. Mun, and P. C. Canfield, Phys. Rev. B 84, 144528 (2011).
  * Fisk and Webb (1976) Z. Fisk and G. W. Webb, Phys. Rev. Lett. 36, 1084 (1976).