Present address: ]Paul Scherrer Institut (PSI), CH-5232 Villigen, Switzerland

# Magnetic properties of the noncentrosymmetric tetragonal antiferromagnet
EuPtSi3

A. Bauer<EMAIL_ADDRESS>Physik-Department, Technische Universität
München, D-85748 Garching, Germany  Zentrum für QuantumEngineering (ZQE),
Technische Universität München, D-85748 Garching, Germany  A. Senyshyn  Heinz
Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748
Garching, Germany  R. Bozhanova  Physik-Department, Technische Universität
München, D-85748 Garching, Germany  W. Simeth  [  Physik-Department,
Technische Universität München, D-85748 Garching, Germany  C. Franz  Physik-
Department, Technische Universität München, D-85748 Garching, Germany  Jülich
Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),
D-85748 Garching, Germany  S. Gottlieb-Schönmeyer  Physik-Department,
Technische Universität München, D-85748 Garching, Germany  M. Meven  Jülich
Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),
D-85748 Garching, Germany  Institut für Kristallographie, RWTH Aachen, D-52056
Aachen, Germany  T. E. Schrader  Jülich Centre for Neutron Science (JCNS) at
Heinz Maier-Leibnitz Zentrum (MLZ), D-85748 Garching, Germany  C. Pfleiderer
Physik-Department, Technische Universität München, D-85748 Garching, Germany
Zentrum für QuantumEngineering (ZQE), Technische Universität München, D-85748
Garching, Germany  Munich Center for Quantum Science and Technology (MCQST),
Technische Universität München, D-85748 Garching, Germany

###### Abstract

We report a comprehensive study of single crystals of the noncentrosymmetric
rare-earth compound EuPtSi3 grown by the optical floating-zone technique.
Measurements of the magnetization, ac susceptibility, and specific heat
consistently establish antiferromagnetic order of localized Eu2+ moments below
the Néel temperature $T_{\mathrm{N}}=17$ K, followed by a second magnetic
transition at $T_{\mathrm{N1}}=16$ K. For a magnetic field along the easy
$[001]$ axis, the magnetic phase diagram is composed of these two phases. For
fields applied in the magnetically hard basal plane, two additional phases
emerge under magnetic field, where the in-plane anisotropy is weak with
$[100]$ being the hardest axis. At the phase transitions, the magnetic
properties exhibit hysteresis and discrepancies between differential and ac
susceptibility, suggesting slow reorientation processes of mesoscale magnetic
textures. Consistently, powder and single-crystal neutron diffraction in zero
field identify magnetic textures that are modulated on a length scale of the
order of $100~{}\textrm{\AA}$, most likely in the form of Néel-type
antiferromagnetic cycloids.

##  I Motivation

In rare-earth intermetallic compounds the interplay of localized moments with
itinerant electrons may give rise to a plethora of phenomena including complex
forms of magnetic order [1, 2, 3], quantum critical behavior [4, 5, 6], and
unconventional superconductivity [7, 8, 9]. While cerium-based compounds are
most commonly studied, due to the comparably simple electronic configuration
of Ce3+ with a single 4f electron, europium-based compounds offer an
intriguing alternative. For europium, the oxidation state Eu2+ is rather
stable due to the electron configuration 4f7 with a half-filled f shell,
resulting in a quenched orbital momentum and very weak spin–orbit coupling for
a rare-earth compound. In combination with crystal structures lacking
inversion symmetry, in which Dzyaloshinskii–Moriya interactions potentially
are leading-order in spin–orbit coupling [10, 11, 12], long-wavelength
modulated states may emerge in europium-based compounds in analogy to
transition-metal compounds [13, 14, 15, 16, 17, 18].

A candidate for such behavior is EuPtSi3, belonging to a class of europium-
based compounds crystallizing in the noncentrosymmetric tetragonal space group
$I4mm$ [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In a seminal
study, Kumar and colleagues investigated single-crystal platelets grown from a
tin solution [33]. Despite superconducting inclusions of the solvent obscuring
the intrinsic low-temperature transport properties, the authors established
that magnetism in EuPtSi3 is carried by local Eu2+ moments and that $[001]$ is
the direction of the magnetic easy axis. Signatures at 17 K and 16 K were
attributed to two subsequent phase transitions. Analyzing the size of the
specific heat anomaly and 151Eu Mössbauer spectra, the authors concluded that
as a function of decreasing temperatures a paramagnetic state, an
incommensurate amplitude-modulated antiferromagnetic state, and a commensurate
equal-moment antiferromagnetic state are observed. Electronic structure
calculations accounted for the easy axis and the size of the magnetic moments,
suggesting that the magnetic structure of EuPtSi3 is incommensurate or
noncollinear [34]. So far, however, the available data were scarce, in
particular for fields in the basal plane, and microscopic information on the
magnetic structure was lacking. Moreover, aspects such as the in-plane
anisotropy or the dependence of the magnetic properties on the temperature and
field history have not been addressed.

The present study reports the growth of a phase-pure single crystal of EuPtSi3
by means of the optical floating-zone technique, thus avoiding the issues
associated with high-temperature solution growth. Magnetic phase diagrams for
the major crystallographic axes are inferred from comprehensive measurements
of the magnetization, ac susceptibility, and specific heat. For magnetic field
parallel to the easy $[001]$ axis, two long-range ordered phases are observed.
For a magnetic fields applied in the hard basal plane, namely $[100]$ and
$[110]$, two additional phase pockets emerge at low temperature at elevated
fields. The in-plane anisotropy is weak with $[100]$ representing the hardest
axis. Hysteresis as well as discrepancies between differential and ac
susceptibility suggest the formation of complex mesoscale magnetic textures.
Consistently, powder and single-crystal neutron diffraction in zero magnetic
field indicate a long-wavelength modulation of local antiferromagnetic order,
in contrast to Ref. [33].

##  II Experimental Methods

For our study, a single crystal of EuPtSi3 was grown by means of the optical
floating-zone technique using an ultra-high vacuum compatible preparation
chain [35]. This procedure contrasts previous studies of related compounds
crystallizing in space group $I4mm$, in which poly-crystalline specimens were
typically prepared from the constituents by means of melting or sintering
followed by an annealing process [36, 20, 21, 22, 37] or small single crystals
were grown from high-temperature solutions using tin or indium as solvent [33,
19, 23, 25, 27, 29, 31]. While the growth from solution circumvents the
challenges arising due to the high vapor pressure of europium efficiently, at
least for EuPtSi3 inclusions of the solvent result in a superconducting
transition in electrical transport measurements, potentially masking the
intrinsic low-temperature behavior [33].

Such inclusions of the solvent are avoided when growing single crystals from
the melt as described in the following. Using our preparation chain as
optimized for intermetallic compounds, all furnaces were pumped to ultra-high
vacuum prior to each step of the growth process, before an inert atmosphere
was applied in the form of 6N argon additionally passed through a point-of-use
gas purifier [35]. Since europium is highly reactive to oxygen and moisture at
ambient conditions, as a first step in an argon glovebox stoichiometric
amounts of high-purity europium (4N) [38] and platinum (4N) were weighed in
for the synthesis of Eu5Pt4 [39]. Using a bespoke metal bellows load-lock, the
elements were transferred into the horizontal cold boat furnace and melted by
means of radio-frequency induction. Next, the resulting ingot, the additional
platinum, and the silicon (6N) required for stoichiometric EuPtSi3 were loaded
into the rod casting furnace in which a poly-crystalline feed rod with a
diameter of 6 mm and a length of ${\sim}25$ mm was cast, as shown in Fig. 1(a)
[40].

In total, four such rods were prepared and used for two growth attempts in the
image furnace [41]. The growth rates for the first and second attempt were 5
mm/h and 1 mm/h, respectively. In both cases, seed and feed rod were counter-
rotating at 6 rpm under an inert argon atmosphere at a pressure of 5 bar.
Despite evaporation losses, as observed by depositions on the inside of the
image furnace presumably by the europium, stable growth conditions were
obtained in both cases. The float-zoned ingots possess a shiny metallic
surface without visible oxidization, as depicted in Fig. 1(b) for the first
growth.

Figure 1: Metallurgical analysis of EuPtSi3. (a) Poly-crystalline feed rod.
(b) Float-zoned ingot. The black arrow marks the growth direction. The central
part yielding single-crystalline material is marked by the light gray box. (c)
X-ray powder diffraction. Taking into account the background (bg.) of the
setup, measured data (meas.) and a calculated Rietveld refinement (calc.)
using the space group $I4mm$ are in excellent agreement. (d) Schematic
depiction of the tetragonal unit cell. (e) X-ray Laue diffraction pattern from
a single-crystal sample showing the fourfold symmetry of the $[001]$ axis.

The structural properties of powder prepared from the first float-zoned ingot
were studied on a STOE STADI P X-ray diffractometer with copper $K_{\alpha 1}$
radiation. As shown in Fig. 1(c), the measured diffractogram is in excellent
agreement with phase-pure EuPtSi3 crystallizing in the noncentrosymmetric
tetragonal BaNiSn3-type structure with space group $I4mm$. A Rietveld
refinement with site occupancies fixed to their nominal values yields lattice
constants $a=4.286~{}\textrm{\AA}$ and $c=9.795~{}\textrm{\AA}$, as summarized
in Tab. 1. These values are in acceptable agreement with those reported by
Kumar and colleagues ($a=4.2660~{}\textrm{\AA}$, $c=9.8768~{}\textrm{\AA}$),
where discrepancies may hint toward the incorporation of tin into the
structure in Ref. [33]. The crystal structure of EuPtSi3 is illustrated in
Fig. 1(d), visualizing the breaking of inversion symmetry due to the distinct
stacking order of the platinum and silicon layers.

Using X-ray Laue diffraction, a large single-crystal grain of a few
millimeters in all dimensions was identified in the float-zoned ingot from the
first growth attempt, where Fig. 1(e) shows a typical Laue pattern of the
fourfold $[001]$ axis. The second growth attempt using the slower growth rate
did not yield single-crystalline sections of the ingot. A cube with an edge
length of 2 mm and surfaces perpendicular to $[001]$, $[110]$, and
$[1\bar{1}0]$ was cut from the single-crystal grain using a wire saw.
Measurements of the bulk properties were carried out in a Quantum Design
physical property measurement system. The magnetization was measured with the
standard extraction technique of the ACMS-II option. The ac susceptibility was
measured with an excitation frequency of 911 Hz at an excitation amplitude of
1 mT. The specific heat was measured using a quasi-adiabatic large heat pulse
technique, where typical pulses had a size of 30% of the temperature at the
start of the pulse [42].

The magnetic structure of EuPtSi3 was studied using the high-resolution powder
diffractometer SPODI [43, 44] and the 4-circle single-crystal diffractometer
HEiDi [45], both at the Heinz Maier-Leibnitz Zentrum (MLZ). On SPODI,
measurements were carried out in Debye–Scherrer geometry at an incident
neutron wavelength of $1.55~{}\textrm{\AA}$. The sample consisted of ground
float-zoned material with a grain size of less than $20~{}\mu\mathrm{m}$. Due
to the high absorption cross section of europium, the powder was filled into
an 0.5 mm wide gap between the walls of a hollow cylinder made of aluminum
with a wall thickness of 0.2 mm and an outer diameter of 20 mm. The cylinder
was mounted into a top-loading closed-cycle cryostat using helium as exchange
gas. Diffraction data were collected between 3.8 K and 300 K and corrected for
geometrical aberrations and the curvature of the Debye–Scherrer rings.

Full-profile Rietveld refinements of the diffraction data were carried out
using pseudo-Voigt peak profiles in the FullProf program package [46]. The
background contribution was determined in terms of a linear interpolation of
selected data points in nonoverlapping regions. The following parameters were
varied during the fitting: scale factor, zero-angular shift, profile shape
parameters, resolution (Caglioti) parameters, asymmetry and lattice
parameters, as well as fractional coordinates of atoms and their displacement
parameters. Since the site occupations deviated only slightly during fitting,
they were fixed to their nominal values in order to suppress correlations with
the displacement parameters.

On HEiDi, the single-crystal cube used for the measurements of the bulk
properties was studied by means of neutrons at an incident wavelength of
$1.1~{}\textrm{\AA}$. While cold neutrons would be favorable in terms of high
resolution at small scattering angles, the absorption cross section of
europium requires the use of faster neutrons in order to achieve sufficient
intensity. Rocking scans on various reflections revealed Gaussian profiles
with a full width at half maximum just above the experimental resolution of
${\sim}0.3^{\circ}$, confirming the single-crystalline nature and high
crystalline quality of the sample with a small mosaic spread of about
$0.2^{\circ}$. Using a closed-cycle cryostat, a set of specific reflections
was studied in the temperature range between 2 K and 20 K. Owing the high
absorption of the sample, the limited coverage of reciprocal space does not
allow for a structure refinement.

As the magnetic properties of EuPtSi3 depend on the temperature and field
history, the following measurement protocols are distinguished. For data as a
function of temperature, only one protocol was used. Prior to each
measurement, the sample was cooled to 2 K from high temperatures well above
$T_{\mathrm{N}}$ in zero magnetic field at a rate of about 5 K/min. Next, the
desired field value was applied before magnetization and ac susceptibility
data were collected alternately while increasing the sample temperature at a
rate of 0.3 K/min. For temperatures above 30 K, the rate was increased to 1
K/min. This protocol is referred to as zero-field cooling (zfc).

For data as a function of field, the sample was first cooled from high
temperatures well above $T_{\mathrm{N}}$ to the desired temperature in zero
magnetic field. In the subsequent field cycle, the field was changed in steps
of typically 100 mT. At each field value, the magnetization and subsequently
the ac susceptibility were measured, before the next field value was
approached at a rate of 10 mT/s. Five branches are distinguished: (i)
$0~{}\mathrm{T}\rightarrow+14~{}\mathrm{T}$, (ii) $+14~{}\mathrm{T}\rightarrow
0~{}\mathrm{T}$, (iii) $0~{}\mathrm{T}\rightarrow-14~{}\mathrm{T}$, (ii*)
$-14~{}\mathrm{T}\rightarrow 0~{}\mathrm{T}$, and (iii*)
$0~{}\mathrm{T}\rightarrow+14~{}\mathrm{T}$. Branch (i) corresponds to the
situation after zero-field cooling for data as a function of temperature.
Branches (ii) and (ii*) are measured under decreasing absolute field values
starting in high fields. Branches (iii) and (iii*) are measured under
increasing absolute field values starting from zero field, but coming from the
field-polarized state in the opposite field direction. This equivalence has
been checked for several temperatures (not shown).

Representing a key result of our study, magnetic phase diagrams are inferred
for fields applied along $[001]$, $[100]$, and $[110]$. In measurements as a
function of magnetic field, up to four signatures are observed and labeled in
increasing field strength $H_{1}$, $H_{2}$, $H_{3}$, and $H_{4}$. Signatures
in measurements as a function of temperature are labeled $T_{1}$, $T_{2}$,
$T_{3}$, and $T_{4}$. For clarity, transition fields and temperatures are
denoted with the same number when belonging to the same transition line in the
phase diagram. Furthermore, the transition temperatures observed in zero
magnetic field, $T_{\mathrm{N}1}=16$ K and $T_{\mathrm{N}}=17$ K, correspond
to the origin points of the transition lines (1) and (4), namely
$T_{\mathrm{N}1}=T_{1}(H=0)$ and $T_{\mathrm{N}}=T_{4}(H=0)$ 111In Ref. [33],
the two phase transition at zero-field were referred to as
$T_{\mathrm{N}2}=16$ K and $T_{\mathrm{N}1}=17$ K. Unfortunately, the change
of nomenclature was unavoidable in order to provide a consistent naming scheme
in the present study.. In fact, the positions of the transitions in phase
space depend not only on the temperature and magnetic field strength, but also
on the field direction. For convenience, these dependences are omitted in the
nomenclature. Although this naming scheme may lead to peculiarities when
considering single measurements only, it proofs to be intuitive when taking
into account the magnetic phase diagrams as a whole.

##  III Experimental results

The presentation of the experimental results starts in Sec. III.1 with the
zero-field susceptibility and an account of the anisotropy of the
magnetization at low temperatures. Next, for field parallel to $[001]$, in
Sec. III.2 magnetization and susceptibility are shown as a function of field
and temperature, complemented by the specific heat and the magnetic
contribution to the entropy. Following the same organizational structure, in
Sec. III.3 magnetization and susceptibility as a function of field and
temperature are shown for fields parallel to $[100]$ and $[110]$. As one of
the key results of this study, in Sec. III.4 magnetic phase diagrams are
presented for the three major crystallographic directions. In Sec. III.5 the
results of the powder and single-crystal neutron diffraction experiments are
delineated, before the magnetic structure is discussed in Sec. III.6.

###  III.1 Magnetic key characteristics and anisotropies

Figure 2: Magnetic properties of EuPtSi3. (a) Temperature dependence of the ac
susceptibility in zero static field for excitation fields along the major
crystallographic directions. (b) Inverse susceptibility as a function of
temperature. The solid lines are linear fits indicating Curie–Weiss-like
behavior. Inset: Close-up view of low temperatures. (c) Magnetization as a
function of increasing field after initial zero-field cooling at low
temperature.

Shown in Fig. 2(a) is the temperature dependence of the ac susceptibility for
excitation fields parallel to the major crystallographic axes $[100]$,
$[110]$, and $[001]$. As a function of decreasing temperature, a Curie–Weiss-
like increase of the susceptibility is observed until a distinct maximum at
$T_{\mathrm{N}}=17$ K marks the onset of magnetic order. The ac susceptibility
curves for excitation fields applied within the tetragonal basal plane, namely
$[100]$ (red symbols) and $[110]$ (orange symbols), are almost identical and
their absolute values are lower than for excitation fields along $[001]$ (blue
symbols). The crossing of the curves below $T_{\mathrm{N}}$ is associated with
the complex magnetic properties of EuPtSi3, as will become clear further
below.

The behavior in the paramagnetic state above $T_{\mathrm{N}}$ may be analyzed
in terms of the temperature dependence of the inverse ac susceptibility, shown
in Fig. 2(b). Excellent agreement between the measured data and linear
Curie–Weiss fits (solid lines) is observed up to room temperature. From the
slope of these fits, fluctuating moments of $\mu_{\mathrm{eff}}=(7.8\pm
0.1)~{}\mu_{\mathrm{B}}\,\mathrm{f.u.}^{-1}$ are inferred for all three
directions, consistent with the free-ion value of Eu2+,
$\mu_{\mathrm{eff}}=7.9~{}\mu_{\mathrm{B}}\,\mathrm{f.u.}^{-1}$. Curie–Weiss
temperatures of Curie–Weiss temperatures of $(4.1\pm 0.6)$ K, $(4.2\pm 0.5)$
K, and $(9.7\pm 0.9)$ K are inferred for excitation fields along $[100]$,
$[110]$, and $[001]$, consistent with Ref. [33]. These values are positive and
distinctly smaller than $T_{\mathrm{N}}$, contrasting the expectations for
both simple antiferromagnetism and ferromagnetism. Furthermore, as shown in
the inset of Fig. 2(b), the measured susceptibility starts to deviate from the
linear fits already well above $T_{\mathrm{N}}$. In combination with the
neutron scattering data indicating a long-wavelength modulation of the
magnetic structure, these observations imply that contributions beyond simple
exchange interactions, for instance in the form of Dzyaloshinskii–Moriya
interactions, are essential for the description of the magnetic properties of
EuPtSi3 already on the mean-field level.

The field dependence of the magnetization at low temperatures, shown in Fig.
2(c), provides further insight into the general character of the magnetism in
EuPtSi3. As a function of increasing field, the magnetization increases
monotonically until kinks, referred to as $H_{4}$, mark the onset of the
field-polarized state. At 2 K, value of $\mu_{0}H_{4}^{[100]}=9.0$ T,
$\mu_{0}H_{4}^{[110]}=8.8$ T, and $\mu_{0}H_{4}^{[001]}=5.5$ T are obtained
for fields applied parallel to the major crystallographic axes. In agreement
with the ac susceptibility, these values indicate an easy $[001]$ axis and a
hard basal plane with a small but finite in-plane anisotropy and $[100]$ as
the hardest axis. At high fields, the magnetization essentially saturates at
$\mu_{s}=6.8~{}\mu_{\mathrm{B}}\,\mathrm{f.u.}^{-1}$ independent of the
crystallographic direction, consistent with the free-ion value of Eu2+,
$\mu_{s}=7~{}\mu_{\mathrm{B}}~{}\mathrm{f.u.}^{-1}$. The good agreement of
both $\mu_{\mathrm{eff}}$ and $\mu_{s}$ with their respective free-ion values
indicates that the magnetism in EuPtSi3 is described well in terms of
localized Eu2+ moments. These observations stand in contrast to the reduced
values reported in Ref. [33] which were attributed to up to 8% of tin
inclusions in the sample. At fields below $H_{4}$, changes of slope in the
magnetization are characteristic of phase transitions of the magnetic order as
addressed in the following.

###  III.2 Properties for field along the easy $[001]$ axis

Figure 3: Field dependence of the magnetic properties of EuPtSi3 for field
parallel to $[001]$. (a) Differential susceptibility calculated from the
measured magnetization, $\mathrm{d}M/\mathrm{d}H$ (open symbols), and real
part of the ac susceptibility, $\mathrm{Re}\,\chi_{\mathrm{ac}}$ (solid
symbols), at 2 K for different field histories. Inset: Enlarged view of the
transition at low fields. (b)–(d) Magnetization as well as real and imaginary
part of the ac susceptibility for different temperatures.

For magnetic field along the easy $[001]$ axis, as shown in Fig. 3(a), the
susceptibility as a function of field at low temperature exhibits three
regimes of essentially constant values. Starting at zero field with a value of
0.07, a distinct maximum at 1.0 T, referred to as $H_{1}$, is followed by a
plateau at a susceptibility value of 0.16. Under further increasing field, a
point of inflection at 5.5 T defines the transition to the field-polarized
state, referred to as $H_{4}$. At high fields, the susceptibility exhibits a
vanishingly small value, reflecting the saturation of the magnetization as a
key characteristic of local-moment magnetism.

The differential susceptibility calculated from the measured magnetization,
$\mathrm{d}M/\mathrm{d}H$ (open symbols), is tracked by the real part of the
ac susceptibility, $\mathrm{Re}\,\chi_{\mathrm{ac}}$ (solid symbols), with
exception of the transition regime around $H_{1}$, where the maximum in
$\mathrm{d}M/\mathrm{d}H$ is more pronounced. Such discrepancies are abundant
in materials with mesoscale magnetic textures and a characteristic of a very
slow response of the system, suggesting that the transition at $H_{1}$ may
involve the reorientation of large magnetic domains [48, 49]. Consistent with
this conjecture, different field histories, labeled (i) through (iii), result
in very similar behavior with only minor changes of the shape of the anomaly
at $H_{1}$.

The evolution of $H_{1}$ and $H_{4}$ for increasing temperatures is
illustrated in Figs. 3(b), 3(c), and 3(d), where the magnetization as well as
the real and imaginary part of the ac susceptibility are shown for increasing
fields after zero-field cooling. Most prominently, with increasing temperature
both $H_{1}$ and $H_{4}$ shift to smaller field values and have vanished at 18
K. An upturn develops in the susceptibility just before the transition to the
field-polarized regime that is, however, not associated with an additional
phase transition. The imaginary part of the ac susceptibility, depicted in
Fig. 3(d), is small and its field dependence tracks the real part
qualitatively. Therefore, it can not be excluded that the imaginary part is
due to a tiny phase shift well below 1 deg, which would imply that it
represents an erroneous systematic contribution originating in the real part
of the ac susceptibility. The putative offset of the absolute values,
increasing with decreasing temperature, is due to eddy currents in the
metallic sample and thus also not intrinsic.

Figure 4: Temperature dependence of the magnetic properties of EuPtSi3 for
field parallel to $[001]$. (a) Real part of the ac susceptibility for selected
fields measured after zero-field cooling (zfc). Data are offset for clarity.
(b)–(d) Magnetization as well as real and imaginary part of the ac
susceptibility for a large number of fields up to 14 T. Inset: Real part of
the ac susceptibility at high fields.

As a function of temperature, the real part of the ac susceptibility exhibits
up to two transitions, as shown in Fig. 4(a) for four field values with
typical characteristics. Under decreasing temperature in zero magnetic field,
the susceptibility shows a Curie–Weiss-like dependence until a kink at $T_{4}$
marks the onset of long-range magnetic order. Below $T_{4}$, the
susceptibility is essentially constant until it decreases again below a second
anomaly, marked $T_{1}$. The zero-field values of the transition temperatures
$T_{1}$ and $T_{4}$ are also referred to as $T_{\mathrm{N}1}=T_{1}(H=0)$ and
$T_{\mathrm{N}}=T_{4}(H=0)$, respectively. The presence of two independent
transitions is more pronounced in small applied fields. In a field of 1 T, the
susceptibility exhibits a complex shape due to the weak temperature dependence
of $H_{1}$ at low temperatures. For larger fields, only the kink at $T_{4}$
remains. The transition temperatures and fields inferred from the field and
temperature dependence of the susceptibility are in excellent agreement.

The evolution of the magnetic properties for a large number of field values is
illustrated in Figs. 4(b) through 4(d). With increasing field, the transition
at $T_{4}$ shifts to lower temperatures until having vanished at fields of 6 T
and above. Additionally, as shown in the inset of Fig. 4(c), in high fields a
shallow maximum in the real part of the susceptibility, referred to as
$T_{m}$, is observed at temperatures well above $T_{\mathrm{N}}$. The maximum
shifts to higher temperatures with increasing field and marks the crossover
between the field-polarized regime at low temperatures and high fields and the
paramagnetic regime at high temperatures and low fields [50, 51].

Figure 5: Temperature dependence of the specific heat of EuPtSi3 for field
parallel to $[001]$. (a) Specific heat for different magnetic fields. The gray
solid line indicates the phonon contribution, $C_{\mathrm{ph}}$, using a Debye
temperature $\mathit{\Theta}_{\mathrm{D}}=356$ K. (b) Close-up view of the
transition regime. (c) Nonphonon contribution to the specific heat divided by
temperature as a function of temperature. (d) Estimate of the entropy
contribution not related to phonons.

Further information on the nature of the phase transitions may be inferred
from the specific heat, depicted in Fig. 5(a). In zero field, the specific
heat at low temperature is dominated by a pronounced lambda anomaly at
$T_{\mathrm{N}}$, characteristic of a second-order phase transition. Closer
inspection reveals that the anomaly exhibits fine structure in the form a two
narrowly separated maxima. As highlighted in Fig. 5(b), under small applied
fields along $[001]$ the low-temperature maximum distinctly shifts to lower
temperatures and vanishes in fields exceeding 1 T, tracking $T_{1}$ as
observed in the ac susceptibility. The high-temperature maximum associated
with the $\lambda$ anomaly shifts to lower temperatures at higher fields and
tracks $T_{4}$. Thus, the transition at $T_{4}$ features a substantial
reduction of the entropy characteristic of a second-order transition even
under large magnetic fields. In contrast, the anomaly at $T_{1}$ exhibits a
rather symmetric shape and is associated with a small change of entropy only,
hinting toward a first-order character of the transition.

Plotting $C/T$ as a function of $T^{2}$ (not shown) and fitting a straight
line for $T>T_{\mathrm{N}}$, we determine a Debye temperature
$\mathit{\Theta}_{\mathrm{D}}=356$ K. The resulting phonon contribution with
$C_{\mathrm{ph}}\propto T^{3}$ describes the behavior at intermediate
temperatures reasonably well. Consistent with the assumptions of the Debye
model, the fit starts to deviate from the measured specific heat for high
temperatures, $T>\mathit{\Theta}_{\mathrm{D}}/10$, providing an excellent
cross-check as demonstrated in detail for MnSi and related compounds in Refs.
[51, 42]. When subtracting the phonon contribution from the measured data, the
remaining specific heat may be attributed to electronic and magnetic
excitations. This contribution divided by temperature,
$(C-C_{\mathrm{ph}})/T$, is shown in Fig. 5(c).

For $T>T_{\mathrm{N}}$, a constant value of
$0.27~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-2}$ is observed in zero
magnetic field. This contribution may be interpreted as electronic specific
heat scaling linear in temperature with a relatively large Sommerfeld
coefficient that is characteristic of heavy-fermion behavior. However, the
slope at low temperatures suggests that $(C-C_{\mathrm{ph}})/T$ may become
vanishingly small at temperatures well below 2 K. This discrepancy may hint
toward an unusual breakdown of electronic correlations at low temperatures or
the presence of additional effects at higher temperatures that cause a
putative enhancement of the Sommerfeld coefficient. These additional
complexities prevent us from separating electronic and magnetic contributions
to the specific heat. Still, the numerical integration of
$(C-C_{\mathrm{ph}})/T$ as a function of temperature provides an estimate of
the associated entropies, as shown for zero magnetic field in Fig. 5(d).

For Eu2+, the quenched orbital momentum, $L=0$, implies a total momentum
$J=S=7/2$ corresponds to the spin momentum. Consequently, a total magnetic
entropy $S_{\mathrm{mag}}=R\ln 8$ is expected at high temperatures. Consistent
with the results of Kumar and colleagues, who used a measurement on poly-
crystalline LaPtSi3 for the subtraction of the phonon contribution, we obtain
a value of $0.8R\ln 8$ around $T_{\mathrm{N}}$. Possible explanations for this
reduced value include magneto-elastic coupling, resulting in an overestimate
of the phonon contribution, or magnetic correlations in the paramagnetic state
above $T_{\mathrm{N}}$. Also, the entropy contribution not related to phonons
clearly exceeds $R\ln 8$ for temperatures well above $T_{\mathrm{N}}$, which
may be attributed to electronic excitations as described above.

In previous studies, information on the magnetic structure of EuPtSi3 was
inferred from the jump of the specific heat at the magnetic phase transition.
Following these considerations, the size of the anomaly would allow to
distinguish between a magnetic structure with moments of the same size on all
sites (equal-moment, EM) and moments of periodically varying size (amplitude-
modulated, AM) [52, 33]. For $J=7/2$, one expects $\Delta
C_{\mathrm{EM}}=(63/26)R=20.15~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}$
and $\Delta
C_{\mathrm{AM}}=\frac{2}{3}C_{\mathrm{EM}}=13.43~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}$.
Kumar and colleagues estimated $\Delta
C=14.6~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}$ and interpreted this
value as a sign for an amplitude-modulated magnetic structure.

In this context, we note that the experimental value in Ref [33] was larger
than the calculated one, although tin inclusions may be expected to reduce the
size of the specific heat anomaly in analogy to the magnetic moment in
saturation. More importantly, however, the size of the jump as inferred from
the specific heat just at $T_{\mathrm{N}}$ may be erroneous. Instead, the
plateau well above the onset of magnetic order in $(C-C_{\mathrm{ph}})/T$
provides a reliable point of reference, permitting us to determine the size of
the jump unambiguously. In zero field, we find $\Delta
C_{\mathrm{el}}/T_{c}=(1.46-0.27)~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-2}$
and $T_{\mathrm{N}}=17$ K, translating to $\Delta
C=20.2~{}\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}$. This value is in
excellent agreement with a magnetic structure composed of equal moments.

###  III.3 Properties for fields in the hard basal plane

Figure 6: Field dependence of the magnetic properties of EuPtSi3 for field
parallel to $[100]$. (a) Differential susceptibility calculated from the
measured magnetization, $\mathrm{d}M/\mathrm{d}H$ (open symbols), and real
part of the ac susceptibility, $\mathrm{Re}\,\chi_{\mathrm{ac}}$ (solid
symbols), at 2 K for different field histories. (b)–(d) Magnetization as well
as real and imaginary part of the ac susceptibility for different
temperatures.

When applying the magnetic field in the hard basal plane, the magnetic
properties are qualitatively reminiscent of the situation for field parallel
to the easy $[001]$ axis. However, closer inspection reveals more complex
behavior. As shown in Fig. 6(a) for field along $[100]$, the differential and
the ac susceptibility exhibit four distinct anomalies as a function of
increasing field instead of two. Namely, the maximum at $H_{1}$ is followed by
a weak kink, denoted at $H_{2}$, and a shoulder, denoted $H_{3}$, before the
field-polarized regime is entered above the point of inflection at $H_{4}$.

Essentially identical behavior is observed for both histories measured under
increasing field, (i) and (iii), while decreasing field, history (ii), reveals
pronounced hysteresis at $H_{1}$, from $H_{1}$ to $H_{2}$, and around $H_{3}$.
For the construction of the magnetic phase diagrams, the transition $H_{3}$ is
defined as the center of the hysteresis. Discrepancies between the
differential susceptibility, $\mathrm{d}M/\mathrm{d}H$, and the real part of
the ac susceptibility, $\mathrm{Re}\,\chi_{\mathrm{ac}}$, accompany the phase
transitions at $H_{1}$, $H_{2}$, and $H_{3}$, being most pronounced under
decreasing field. Consistent with the results for field applied along the easy
$[001]$ axis, both the history dependence and the discrepancies between
differential and ac susceptibility suggest the presence of mesoscale magnetic
textures and the importance of slow reorientation processes in EuPtSi3.

With increasing temperature, the transition fields shift to lower values, as
illustrated in Figs. 6(b) through 6(d) where data are shown for increasing
fields after zero-field cooling. The signatures characteristic of $H_{2}$ and
$H_{3}$ vanish at lower temperatures than $H_{1}$ and $H_{4}$, enclosing
distinct phase pockets in the magnetic phase diagram, as highlighted below.

Figure 7: Temperature dependence of the magnetic properties of EuPtSi3 for
field parallel to $[100]$. (a) Real part of the ac susceptibility for selected
fields measured after zero-field cooling (zfc). Data are offset for clarity.
(b)–(d) Magnetization as well as real and imaginary part of the ac
susceptibility for a large number of fields up to 14 T.

As a function of temperature, four signatures may be distinguished, as shown
in Fig. 7(a) for four field values with typical characteristics. In analogy to
the corresponding anomalies in the field dependence of the susceptibility,
these temperatures are referred to as $T_{1}$, $T_{2}$, $T_{3}$, and $T_{4}$.
Their evolution for a large number of field values is illustrated in Figs.
7(b) through 7(d). Akin to the behavior for field parallel to $[001]$, a
shallow maximum is observed in the ac susceptibility at temperatures well
above $T_{\mathrm{N}}$ and associated with the crossover between the field-
polarized and the paramagnetic regime (not shown).

Figure 8: Field dependence of the magnetic properties of EuPtSi3 for field
parallel to $[110]$. (a) Differential susceptibility calculated from the
measured magnetization, $\mathrm{d}M/\mathrm{d}H$ (open symbols), and real
part of the ac susceptibility, $\mathrm{Re}\,\chi_{\mathrm{ac}}$ (solid
symbols), at 2 K for different field histories. (b)–(d) Magnetization as well
as real and imaginary part of the ac susceptibility for different
temperatures. Figure 9: Temperature dependence of the magnetic properties of
EuPtSi3 for field parallel to $[110]$. (a) Real part of the ac susceptibility
for selected fields measured after zero-field cooling (zfc). Data are offset
for clarity. (b)–(d) Magnetization as well as real and imaginary part of the
ac susceptibility for a large number of fields up to 14 T.

The behavior for field along $[110]$, shown in Fig. 8(a), is highly
reminiscent of the situation for field along $[100]$ with four distinct
anomalies being observed as a function of increasing field. Both the
hysteresis between different field histories and the discrepancy between
differential and ac susceptibility are more pronounced than for field parallel
to $[100]$, as elaborated on in the following.

The signatures associated with $H_{1}$ are observed only after zero-field
cooling, referred to as history (i), in contrast to the other field directions
studied. Even for this field history, the maximum in the differential
susceptibility at $H_{1}$ is not tracked by the ac susceptibility. Such a
dependence may hint at a long-wavelength modulated magnetic texture for which
the projection of the propagation direction into the basal plane is oriented
along one of the $\langle 110\rangle$ axes, in analogy to the helical-to-
conical transition for magnetic field along an easy axis in cubic chiral
magnets [49]. At $H_{2}$ a sharp peak in the differential susceptibility under
increasing field, histories (i) and (iii), compares with a kink in the ac
susceptibility that resembles the data for field along $[100]$. Under
decreasing field, history (ii), a broad asymmetric maximum is observed at
slightly lower field values in both differential and ac susceptibility. The
hysteretic loop around $H_{3}$ is wider than for field along $[100]$ and under
decreasing field, history (ii), at its low-field boundary a peak is observed
in the differential susceptibility that is not tracked by the ac
susceptibility.

The evolution of the anomalies with increasing temperature, shown in Figs.
8(b) through 8(d), as well as the magnetic properties as a function of
temperature, shown in Fig. 9, are highly reminiscent of the evolution for
field parallel to $[100]$. Consistent with the data as a function of field,
dependence on the temperature and field history is observed in particular in
scans at fields below $H_{1}$ (not shown). All things considered, however, the
magnetic phase diagrams for fields in the tetragonal basal plane turn out to
be essentially isotropic with a slightly harder $[100]$ axis.

###  III.4 Magnetic phase diagrams

Figure 10: Magnetic phase diagrams of EuPtSi3 for field along $[001]$,
$[100]$, and $[110]$ after zero-field cooling. Data inferred from measurements
as a function of field and temperature are shown in dark and light colors,
respectively. Paramagnetic (PM) and field-polarized (FP) regimes as well as
four ordered phases, labeled A through D, are distinguished. Arabic numbers
denote transition lines as defined by the transition fields and temperatures
denoted with that number.

The magnetic phase diagrams of EuPtSi3 for the three field directions studied
are shown in Fig. 10. The ordinate on the right-hand side displays an estimate
of the internal field values $H_{\mathrm{int}}$ when correcting for
demagnetization effects [53, 54]. Taking together the data for all field
directions, the magnetic phase diagrams feature four different phases, labeled
A through D, as well as the paramagnetic state (PM) at high temperatures and a
field-polarized regime (FP) at high fields. For zero magnetic field and
decreasing temperature, the paramagnetic state undergoes a transition to phase
B at $T_{\mathrm{N}}=T_{4}(H=0)=17$ K, followed by a transition to phase A at
$T_{\mathrm{N2}}=T_{1}(H=0)=16$ K.

Under increasing magnetic field along $[001]$, the transition between phases A
and B occurs at $H_{1}$, followed by a transition from phase B to the field
polarized-regime at $H_{4}$. The transition fields emanate from
$T_{\mathrm{N2}}$ and $T_{\mathrm{N}}$, respectively, and increase
monotonically with decreasing temperature. This phase diagram is consistent
with Ref. [33], apart from two noticeable exceptions. First, the presence of
phase B in zero field was not addressed. Second, for the transition line
denoted (1) a slightly convex shape was reported, implying reentrant behavior.
No such behavior is observed in our study.

For magnetic fields in the basal plane, two additional states emerge, labeled
C and D, before the field-polarized regime is entered. Hysteresis under
increasing and decreasing fields is observed at the transitions between phases
B and C, denoted transition line (2), as well as C and D, denoted transition
line (3). For field along $[110]$, the hysteresis is more pronounced and,
perhaps most notably, signatures associated with phase A are observed after
zero-field cooling only. These hysteretic effects as well as the discrepancies
between the differential and the ac susceptibility consistently suggest the
presence of mesoscale textures of a long-wavelength modulated magnetic state
with phase transitions that are governed by slow reorientation processes. The
zero-temperature extrapolation of the transition fields $H_{1}$, $H_{2}$,
$H_{3}$, and $H_{4}$ along $[110]$ are a few hundred millitesla smaller than
for field along $[100]$, characteristic of weak in-plane anisotropy featuring
$[100]$ as the hardest axis.

###  III.5 Neutron diffraction

Figure 11: Neutron powder diffraction on EuPtSi3. (a) Diffraction patterns for
different temperatures. Below the onset of magnetic order additional maxima
emerge. Data are offset for clarity. (b) Enlarged view of the triplet
structure around $2\mathit{\Theta}=9~{}\mathrm{deg}$ where an estimate of the
pure magnetic contribution is obtained by subtraction of high-temperature
data. Solid lines are Gaussian fits, the dashed line represents the sum of the
three Gaussians. (c) Rietveld refinement of low-temperature data. Measured
data (meas.), calculations (calc.), and their difference (diff.) are
distinguished. Angles dominated by scattering from the sample environment were
excluded (excl.). Short vertical lines indicate maxima related to the
crystalline structure (nucl.) and the magnetic ordering vectors (comm.,
incomm.).

Microscopic information on the magnetic structure of EuPtSi3 may be inferred
from neutron diffraction. As shown in Fig. 11, the powder diffractogram at
room temperature is consistent with a phase-pure sample crystallizing in space
group $I4mm$. The structural parameters obtained by means of a Rietveld
analysis are summarized in Table 1, establishing excellent agreement between
values extracted from neutron and X-ray diffraction, cf. Fig. 1(a).
Refinements considering Pt–Si mixed occupations on sites Pt1, Si1, and Si2
yield an upper limit for Pt–Si antisite disorder of less than 3%.

Table 1: Structural parameters of EuPtSi3 crystallizing in the tetragonal BaNiSn3-type structure with space group $I4mm$ (No. 107). The structural data were modeled for Eu1, Pt1, and Si1 occupying $2a$ sites $[0,0,z]$ and Si2 occupying $4b$ sites $[0,1/2,z]$ with Eu1 being fixed to the origin, $z=0$. Data from neutron and X-ray diffraction are in excellent agreement. Numbers in parentheses give statistical deviations of the last significant digit. Atomic site, | Neutron, $T=300$ K | X-ray, $T=297$ K | Neutron, $T=3.8$ K
---|---|---|---
Wyckoff | $a=4.2832(4)~{}\textrm{\AA},c=9.7932(7)~{}\textrm{\AA}$ | $a=4.28552(2)~{}\textrm{\AA},c=9.79466(6)~{}\textrm{\AA}$ | $a=4.2718(3)~{}\textrm{\AA},c=9.7747(8)~{}\textrm{\AA}$
position | $x/a$ | $y/a$ | $z/c$ |  $B_{\mathrm{iso}}$(Å)2 | $x/a$ | $y/a$ | $z/c$ |  $B_{\mathrm{iso}}$(Å)2 | $x/a$ | $y/a$ | $z/c$ |  $B_{\mathrm{iso}}$(Å)2
Eu1, $2a$ | 0 | 0 | 0 | 1.4(2) | 0 | 0 | 0 | 1.98(6) | 0 | 0 | 0 | 1.4(2)
Pt1, $2a$ | 0 | 0 | 0.646(1) | 1.1(1) | 0 | 0 | 0.6475(2) | 1.14(3) | 0 | 0 | 0.646(1) | 0.9(1)
Si1, $2a$ | 0 | 0 | 0.402(2) | 2.3(4) | 0 | 0 | 0.4009(8) | 2.4(2) | 0 | 0 | 0.398(2) | 2.2(3)
Si2, $4b$ | 0 | 1/2 | 0.262(2) | 1.2(2) | 0 | 1/2 | 0.2583(5) | 1.3(1) | 0 | 1/2 | 0.259(2) | 1.4(1)

As a function of decreasing temperature, the crystal structure stays unchanged
with the lattice constants decreasing by up to 0.2%. As shown in Fig. 11(a),
below $T_{\mathrm{N}}=17$ K additional intensity maxima emerge that are
attributed to the onset of long-range magnetic order. The present data focus
on low temperatures in zero field, corresponding to phase A in the magnetic
phase diagram. In order to obtain an estimate of the pure magnetic
contribution, the room-temperature diffractogram is subtracted from the low-
temperature data. As highlighted in Fig. 11(b), the most prominent
contribution is a characteristic triplet around
$2\mathit{\Theta}=9~{}\mathrm{deg}$, denoted (I), (II), and (III).

The description of these maxima requires the superposition of at least two
ordering vectors, notably in the conventional unit cell the commensurate wave
vector with $\bm{k}_{15}=(001)$ and an incommensurate wave vector
$\bm{k}_{10}=(00z)$ with $z=1.081$ r.l.u. at 4 K. Therefore, the triplet may
be indexed as $(002)-\bm{k}_{10}$, $(000)\pm\bm{k}_{15}$, and
$(000)\pm\bm{k}_{10}$. With increasing temperature, $z$ increases to 1.088
r.l.u. at 13 K, corresponding to a decrease of the real-space periodicity from
$121~{}\textrm{\AA}$ to $111~{}\textrm{\AA}$. As shown in Fig. 11(c), a
Rietveld refinement based on such antiferromagnetic order with multiple
ordering vectors is in very good agreement with the diffraction pattern up to
high scattering angles. Here, intervals of $2\mathit{\Theta}$ were omitted
(excl.) that were dominated by scattering from the sample environment.
Incommensurate order in phase A contrasts the findings of Kumar and
colleagues, which proposed a commensurate equal-moment antiferromagnetic
structure, based on specific heat measurements and Mössbauer spectroscopy
[33].

Unfortunately, the present data do not allow to determine the character of the
incommensurate order unambiguously. The refinement in Fig. 11(c) assumes an
incommensurate spin density wave with moments along the edge of the basal
plane, but calculations assuming Néel-type cycloidals and Bloch-type helices
describe the data equally well (not shown). Moreover, a search for
incommensurate ordering vectors was carried out in the basal plane of the
tetragonal crystal structure. When assuming $\bm{k}_{1}=(y+x,0,y-x)$ or
$\bm{k}_{3}=(x+y,x+y,2x)$, the fits converge at ordering vectors that
correspond to $\bm{k}_{15}$ and $\bm{k}_{10}$ on the resolution of our study.
Consequently, if there is a modulation within the basal plane of EuPtSi3 in
phase A, it will be either weak or its wavelength will be very long. No hints
for a reduction of the space group symmetry of the crystal structure were
observed, such as a loss of the centering translation.

Figure 12: Single-crystal neutron diffraction on EuPtSi3. (a) Intensity scans
along $(00L)$ for different temperatures. Subtracting high-temperature data
yields an estimate for the magnetic contribution at low temperatures (diff.).
(b) Enlarged view around the magnetic $(001)$ intensity maximum. A triplet
structure is clearly identified in high-resolution data. Solid lines are
Gaussian fits, the dashed line represents the sum of the three Gaussians.
(c),(d) Temperature dependence of the intensity and the position of maxima
around $(001)$.

Complementary single-crystal neutron diffraction data are presented in Fig. 12
in the form of two scans along $(00L)$ at temperatures above and below the
onset of magnetic order. In space group $I4mm$, intensity is expected only at
reflections $(HKL)$ where the sum of $H$, $K$, and $L$ is even, in excellent
agreement with high-temperature data in neutron and X-ray powder diffraction.
At low temperatures, the single-crystal diffraction data exhibit additional
maxima at positions with odd values of $L$. These maxima may be resolved best
when considering the difference (diff.) of the low-temperature and high-
temperature data and decrease in intensity with increasing $L$. Additional
contributions independent of temperature, for instance near $(005)$ or
$(007)$, appear to be spurious as they may attributed to scattering from the
sample environment.

As shown in Fig. 12(b), only a single broad maximum is resolved around $(001)$
on HEiDi in standard configuration (gray symbols). Placing an $\alpha_{2}$
Soller collimator with a horizontal beam divergence of $15^{\prime}$ in front
of the sample allows us to resolve a triplet structure. The experimental data
may be fitted with three Gaussians, yielding peak positions of 0.881 r.l.u.,
1.009 r.l.u., and 1.128 r.l.u., respectively. This splitting is qualitatively
consistent though larger than the splitting observed in the powder diffraction
data, which may be attributed to the nonoptimal alignment of the resolution
ellipsoids in our single-crystal diffraction experiment.

The integrated intensities of the three Gaussian maxima decrease
simultaneously as a function of increasing temperature, as shown in Fig.
12(c). This behavior clearly establishes the origin of the scattering as
magnetic. However, the data available do not allow us to discriminate whether
the intensity vanishes at $T_{\mathrm{N}1}$ or $T_{\mathrm{N}}$. As depicted
in Fig. 12(d), the position of the central maximum of the triplet remains
unchanged as a function of increasing temperature, consistent with a
commensurate ordering vector, while for the incommensurate maxima the
separation away from $(001)$ increases from 0.128 r.l.u. at 2 K to 0.157
r.l.u. at 15.5 K.

###  III.6 Magnetic structure

The crystal structure of EuPtSi3 sets the stage for potential magnetic
structures. Due to the lack of inversion symmetry, neighboring magnetic
moments, $\bm{S}_{i}$ and $\bm{S}_{j}$, may be subject to the
Dzyaloshinskii–Moriya interaction
$\bm{D}_{ij}\cdot\left(\bm{S}_{i}\times\bm{S}_{j}\right)$ with the
Dzyaloshinskii–Moriya vectors $\bm{D}_{ij}$ [10, 11, 12]. The corresponding
energy is minimized when $\bm{S}_{i}$, $\bm{S}_{j}$, and $\bm{D}_{ij}$ form an
orthogonal trihedron. In combination with interactions favoring collinear
alignment, such as ferromagnetic or antiferromagnetic exchange interactions,
the moments may form a long-wavelength modulated structure in the plane
perpendicular to $\bm{D}_{ij}$.

Table 2: Dzyaloshinskii–Moriya vectors as constrained by the crystal structure of EuPtSi3 in space group $I4mm$. Along specific types of crystallographic directions, for an europium moment at position $\bm{R}_{i}=(0,0,0)$ low-order nearest-neighbor europium moments at positions $\bm{R}_{j}$ are considered, where $a$ and $c$ are the tetragonal lattice constants. Applying the Moriya rules [11], allowed orientations of the Dzyaloshinskii–Moriya vector $\bm{D}_{ij}$ are inferred, thereby determining the favored type of long-wavelength modulation. direction | $\bm{R}_{j}$ | $\bm{D}_{ij}$ | Modulation
---|---|---|---
$\langle 100\rangle$ | $(a,0,0)$ | $\pm(0,1,0)$ | Néel-type
$\langle 111\rangle$ | $(a/2,a/2,c/2)$ | $\pm(1,\bar{1},0)$ | Néel-type
$\langle 110\rangle$ | $(a,a,0)$ | $\pm(1,\bar{1},0)$ | Néel-type
$\langle 001\rangle$ | $(0,0,c)$ | $(0,0,0)$ | None

The space group $I4mm$ constrains possible Dzyaloshinskii–Moriya vectors
$\bm{D}_{ij}$, as inferred from the Moriya rules [11] and summarized in Tab.
2. For moments neighboring along the basal edges $\langle 100\rangle$, the
tetragonal diagonals $\left\langle 111\right\rangle$, or the basal face
diagonals $\langle 110\rangle$, the Dzyaloshinskii–Moriya vectors are
perpendicular to the line connecting the moments. Thus, when considering long-
wavelength magnetic modulations, Néel-type cycloidals are favored over Bloch-
type helices. As Dzyaloshinskii–Moriya vectors with opposite sign are mapped
onto each other by mirror symmetries of space group $I4mm$, the magnetic
modulation is expected to inherit this achirality. For moments neighboring
along $\langle 001\rangle$, the Dzyaloshinskii–Moriya vector vanishes. As an
additional note, the space groups $I4mm$ belongs to point group $C_{4v}$, for
which magnetic vortices, so-called skyrmions, were predicted that are based on
Néel-type cycloidals [55, 56, 57].

In combination with the results of the bulk measurements and neutron
scattering, these considerations permit us to identify key characteristics of
the magnetic order in EuPtSi3. In zero magnetic field at low temperatures,
namely in phase A, our data are consistent with local-moment
antiferromagnetism of the europium moments that is modulated on a length scale
of ${\sim}100~{}\textrm{\AA}$ in terms of a Néel-type cycloidal. For this
conjecture, the following aspects are crucial.

First, the crystal structure belongs to space group $I4mm$ which lacks
inversion symmetry and therefore may support long-wavelength magnetic
modulations stabilized by means of Dzyaloshinskii–Moriya interactions,
favoring Néel-type cycloidals rather than Bloch-type helices. Second,
magnetization and specific heat data indicate that the magnetism is carried by
localized Eu2+ moments with quenched orbital momentum and equal size on all
sites. Third, the differential and the ac susceptibility exhibit a pronounced
history dependence, in particular for field along the basal face diagonals
$\langle 110\rangle$, suggesting the presence of mesoscale magnetic domains.
Fourth, neutron scattering is in agreement with commensurate antiferromagnetic
order that is superimposed by a long-wavelength incommensurate magnetic
texture propagating along $\langle 001\rangle$. Fifth, taken together, the two
previous aspects suggest either (i) propagation along $\langle 001\rangle$
with domains featuring projections of moments parallel to $\langle 110\rangle$
or (ii) propagation along directions slightly tilted away from $\langle
001\rangle$ toward $\langle 110\rangle$ such that the peak splitting is well
below the resolution of our neutron scattering study.

##  IV Conclusions

In summary, our study reported the growth of a single crystal of EuPtSi3,
crystallizing in the noncentrosymmetric tetragonal space group $I4mm$, by
means of the optical floating-zone technique. This single crystal was studied
by means of magnetization, ac susceptibility, and specific heat as well as
powder and single-crystal neutron diffraction. The magnetization and specific
heat are characteristic of localized Eu2+ moments. As a function of decreasing
temperature, two narrowly separated magnetic phase transitions are observed at
$T_{\mathrm{N}}=17$ K and $T_{\mathrm{N2}}=16$ K. Under applied magnetic field
along the easy $[001]$ axis, two phases are observed for which the phase
transition lines connect with $T_{\mathrm{N}}$ and $T_{\mathrm{N}1}$. At low
temperatures, the field-polarized state is entered above 5.5 T. Under applied
field in the hard basal plane, two additional phases emerge in larger fields.
The in-plane anisotropy is weak, where $[100]$ represents the hardest axis,
with the field-polarized state at low temperatures forming above 9.0 T and 8.8
T for field along $[100]$ and $[110]$, respectively.

Neutron scattering in zero field at low temperatures exhibits a splitting of
magnetic intensity maxima, characteristic of a long-wavelength modulated
antiferromagnetic texture, most likely in the form of a Néel-type cycloid.
These findings highlight that the quenched orbital momentum of intermetallic
europium compounds leads to relatively weak spin–orbit coupling, making them
prime candidates for long-wavelength magnetic modulations, in particular when
combined with crystal structures lacking inversion symmetry, thus permitting
Dzyaloshinskii–Moriya interactions.

###### Acknowledgements.

We wish to thank P. Böni, G. Causer, F. Haslbeck, J. Leiner, S. Mayr, and A.
Regnat for fruitful discussions and assistance with the experiments. Moreover,
we want to thank U. Keiderling for test measurements at the small-angle
scattering instrument V4 at the Helmholtz-Zentrum Berlin (HZB). We also want
to thank S. Mühlbauer and E. Blackburn for test measurements at the small-
angle scattering instrument SANS1 [58] using a 17 T superconducting magnet and
A. Ostermann for assistance with test measurements at the diffractometer
BIODIFF [59], both beamlines at the Heinz Maier-Leibnitz Zentrum (MLZ). Parts
of the data were collected on HEiDi, jointly operated by RWTH Aachen
University and the Jülich Centre for Neutron Science (JCNS) within the JARA
cooperation. This work has been funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under TRR80 (From Electronic Correlations to
Functionality, Project No. 107745057, Project E1) and the excellence cluster
MCQST under Germany’s Excellence Strategy EXC-2111 (Project No. 390814868).
Financial support by the European Research Council (ERC) through Advanced
Grants No. 291079 (TOPFIT) and No. 788031 (ExQuiSid) is gratefully
acknowledged.

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