# Low-temperature antiferromagnetic order in orthorhombic CePdAl3

Vivek Kumar<EMAIL_ADDRESS>Physik-Department, Technische Universität
München, D-85748 Garching, Germany  Andreas Bauer  Physik-Department,
Technische Universität München, D-85748 Garching, Germany  Zentrum für
QuantumEngineering (ZQE), Technische Universität München, D-85748 Garching,
Germany  Christian 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  Jan Spallek  Physik-
Department, Technische Universität München, D-85748 Garching, Germany  Rudolf
Schönmann  Physik-Department, Technische Universität München, D-85748
Garching, Germany  Michal Stekiel  Physik-Department, Technische Universität
München, D-85748 Garching, Germany  Astrid Schneidewind  Jülich Centre for
Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), D-85748
Garching, Germany  Marc Wilde  Physik-Department, Technische Universität
München, D-85748 Garching, Germany  Zentrum für QuantumEngineering (ZQE),
Technische Universität München, 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 the magnetization, ac susceptibility, and specific heat of optically
float-zoned single crystals of CePdAl3. In comparison to the properties of
polycrystalline CePdAl3 reported in the literature, which displays a
tetragonal crystal structure and no long-range magnetic order, our single
crystals exhibit an orthorhombic structure ($Cmcm$) and order
antiferromagnetically below a Néel temperature $T_{\rm N}$ = 5.6 K. The
specific heat at zero-field shows a clear $\lambda$-type anomaly with a broad
shoulder at $T_{\rm N}$. A conservative estimate of the Sommerfeld coefficient
of the electronic specific heat,
$\gamma=121~{}\mathrm{mJ~{}K^{-2}~{}mol^{-1}}$, indicates a moderately
enhanced heavy-fermion ground state. A twin microstructure evolves in the
family of planes spanned by the basal plane lattice vectors ao and co, with
the magnetic hard axis bo common to all twins. The antiferromagnetic state is
characterized by a strong magnetic anisotropy and a spin-flop transition
induced under magnetic field along the easy direction, resulting in a complex
magnetic phase diagram. Taken together our results reveal a high sensitivity
of the magnetic and electronic properties of CePdAl3 to its structural
modifications.

##  I Introduction

Cerium-based intermetallic compounds exhibit a variety of ground states and
various underlying exotic physical phenomena, such as unconventional
superconductivity [1, 2, 3, 4, 5, 6, 7, 8], heavy-fermion states [9, 10], non-
Fermi liquid behavior [11], vibronic hybrid excitations [12, 13, 14, 15, 16],
and complex magnetic order [17, 18, 19, 20, 21, 22, 23, 24]. On the
phenomenological level, the origin of this remarkable diversity of ground
states has been attributed to the competition of narrow f-electron bands and
strong electronic correlations together with spin-orbit interaction, crystal
electric field (CEF) effects, and strong magneto-elastic coupling. An
overarching theme connecting much of the research in f-electron compounds
concerns the condition of the emergence of magnetic order.

A class of compounds with the general formula CeTX3 (T is a transition metal
and X is a $p$-block element) crystallizing in subgroups of the BaAl4-type
($I4/mmm$) tetragonal structure has received special attention [25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 32, 22, 24, 23, 36, 37, 38, 39, 3, 6, 40, 41,
42]. In these compounds, a large number of structural variants and diverse
magnetic and electrical properties can be obtained by changing the transition
metal T. Many members of this class such as CeRhGe3, CeAuAl3, CeCuAl3, and
CeCoGe3 adopt a non-centrosymmetric tetragonal structure (BaNiSn3-type $I4mm$)
and exhibit antiferromagnetic behavior [25, 26, 27, 28, 29, 30, 31, 32, 33].
Other members such as CeAgAl3 display ferromagnetism with a centrosymmetric
orthorhombic crystal structure [34, 35]. A spin-glass state was reported in
non-centrosymmetric tetragonal CePtAl3 below 0.8 K [32]. Complex magnetic
phases have been observed in antiferromagnetic CeNiGe3[22, 24], CeCoGe3 [23,
36, 37, 38] and CePtSi3 [39]. The discovery of pressure-induced unconventional
superconductivity in the non-centrosymmetric tetragonal heavy-fermion
antiferromagnets CeRhSi3, CeIrSi3, CeCoGe3, CeIrGe3, and CeRhGe3 even suggests
a new direction in condensed matter physics [3, 6, 40, 41, 42].

An important aspect is the structural stability of these systems and the
emergence of different electronic ground states. As one of the first examples,
CePd2Al2 [13, 15], which is closely related to the class of CeTAl3 of
materials, was found to undergo a structural phase transformation from a
tetragonal to an orthorhombic lattice at 13.5 K. An inelastic neutron
scattering study revealed three magnetic excitations in the paramagnetic
phase. However, according to Kramer’s theorem, only two CEF excitations are
expected due to the splitting of ground state J = 5/2 of the Ce3+ ion into
three doublets in tetragonal/orthorhombic point symmetry suggesting strong
coupling between the crystal fields and the crystal structure. Later, Adroja
_et al._ found a similar anomaly in CeCuAl3 [14], where a structural
instability manifests itself in terms of a drastic change in lattice
parameters of the tetragonal structure around 300 ∘C [29]. These anomalous
excitations have been interpreted by means of Thalmeier and Fulde’s model of
bound states between phonons and CEF excitations as generalized to the
tetragonal point symmetry. Recently, Čermák _et al._ confirmed related hybrid
CEF-phonon excitations even for weak magnetoelastic coupling in isostructural
CeAuAl3 [43]. Moreover, CePd2Al2, CeCuAl3 and CeAuAl3 order
antiferromagnetically at low temperatures and exhibit incommensurate
amplitude-modulated magnetic structures [44, 45, 46, 15]. The presence of
multi-step magnetism and complex magnetic phase diagrams suggests the possible
existence of topologically non-trivial multi-k structures akin to skyrmion
lattices [47]. This raises the question, if and how the formation of magnetic
order depends on the stabilization of specific crystal structure.

In this paper we focus on CePdAl3. A study of as-cast polycrystalline CePdAl3
by Schank _et al._ in 1994 revealed a tetragonal $I4mm$ structure with lattice
constants a = 4.343 Å and c = 10.578 Å [48], where the heat treatment at high
temperature results in a structural phase transformation with an
antiferromagnetic order below $T_{\rm N}$ $\simeq$ 6 K. In contrast, no
magnetic order was found down to 0.1 K in a recent investigation by Franz _et
al._ on single crystalline tetragonal CePdAl3 grown by optical float zoning
with a growth rate of 6 mm/h [49]. For the work reported in the following, a
single crystal was prepared by optical float zoning using a much lower growth
rate of 1 mm/h. Under these conditions we found that CePdAl3 crystallizes in
an orthorhombic as opposed to a tetragonal structure [50]. In this paper, we
report comprehensive magnetization, ac susceptibility, and specific heat
measurements on single crystalline orthorhombic CePdAl3. As our main result we
find the characteristics of antiferromagnetic order below $T_{\rm N}$ = 5.6 K.
We determine the magnetic phase diagram upto 14 T, where we find the emergence
of complex magnetic phases under magnetic fields applied along the easy
direction. The presence of different structural and magnetic configurations of
CePdAl3 identifies a new example of a material in which to search for hybrid
excitations and new magnetic phases in the future.

Our paper is organized as follows. After a brief account of the experimental
methods in Sec. II, we present our experimental results in Sec. III. We start
with the structural properties and notation in Sec. III A, followed by the
specific heat results in Sec. III B and magnetic susceptibility data in Sec.
III C. The temperature- and field-dependence of the magnetization is presented
in Sec. III D. We find that the magnetic field-driven transitions for fields
applied along the easy direction are consistent with the specific heat as a
function of temperature as presented in Sec. III E. In Sec. III F, we examine
the magnetic transitions in more detail by analyzing the hysteresis of the
field-dependent magnetic susceptibility. Comprehensive datasets allow to infer
the magnetic phase diagram presented in Sec. III G. The conclusions are
summarized in Sec. IV.

##  II Experimental methods

A single-crystal of CePdAl3 was grown using the optical floating-zone
technique following a process similar that described in Ref. [49, 51, 52]. As
the main difference, the growth rate was reduced from 6 mm/h [49] to 1 mm/h
which resulted in the formation of an orthorhombic crystal.

The crystal structure of CePdAl3 was determined by means of single-crystal
x-ray diffraction (SCXRD). A platelet-shaped crystal with dimensions 50 $\mu$m
$\times$ 40 $\mu$m $\times$ 10 $\mu$m was cleaved of the CePdAl3 crystal as
grown. The platelet was investigated at a Rigaku XtaLAB Synergy-S
diffractometer, using a Mo x-ray source with $\lambda$ = 0.71 Å and a two-
dimensional HyPix-Arc 150∘ detector. Bragg reflections were indexed using
CrysAlisPro [53] as integrated with the diffractometer.

The single crystals were oriented by Laue x-ray diffraction and a cuboidal
sample was cut with orientations a${}_{\rm o}^{\star}$, c${}_{\rm o}^{\star}$
and bo as introduced below for the measurement of the bulk properties. The ac
susceptibility, magnetization, and specific heat were measured in a Quantum
Design physical property measurement system (PPMS) at temperatures down to 2 K
under magnetic fields up to 14 T. In order to determine the temperature
dependence of the bulk properties, the sample was first cooled from a high
temperature, well above $T_{\rm N}$, to the lowest attainable temperature in
the absence of a magnetic field. Subsequently, the field was set to the
desired value and data were collected for increasing temperature. This
protocol was repeated for different target magnetic fields. The ac
susceptibility was measured at an excitation amplitude of 1 mT and an
excitation frequency of 911 Hz. The specific heat was measured down to 2 K
using a large heat-pulse method [54]. For temperatures between 0.08 K and 4 K
the specific heat was measured in a Dryogenic adiabatic demagnetization
refrigerator using a conventional heat-pulse method.

The field dependence of the magnetization and the ac susceptibility was
measured using the following temperature versus field protocol. First, the
sample was cooled from a high temperature well above $T_{\rm N}$ to the target
temperature in the absence of a magnetic field. Second, data as a function of
magnetic field were recorded in a sequence of field sweeps from zero-field to
14 T, 14 T to -14 T, and -14 T to 14 T.

The bulk properties recorded on different pieces cut from the large single
crystal ingot were consistent. The temperature and field dependent features
along a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$ were qualitatively
identical. Therefore, comprehensive data focused on one of these directions,
c${}_{\rm o}^{\star}$, were recorded. Summarizing the key result of our study,
the magnetic phase diagrams of CePdAl3 were inferred. Signatures detected in
measurements as a function of temperature and magnetic field are labelled as
$T_{j}$ and $H_{j}$, respectively. For clarity, the same subscript $j$ is
assigned to the transitions corresponding to the same line in the phase
diagram.

##  III Experimental results

###  III.1 Crystal structure and twinning

Different crystal growth conditions favor a tetragonal ($I4mm$) [49] or
orthorhombic crystal structures of CePdAl3. By means of single crystal x-ray
diffraction, we determined that the orthorhombic lattice stabilizes in the
$Cmcm$ space group. The lattice parameters at room temperature are $a_{o}$ =
6.379 Å, $b_{o}$ = 10.407 Å and $c_{o}$ = 5.975 Å. The orthorhombic phase
exhibits a pseudo-tetragonal twinning in the basal plane, evident, for
instance, by the splitting of the Bragg reflections shown in Fig. 1(a). The
twinning law was determined by indexing all measured reflections with
components of the four twins presented in Fig. 1(b). An illustration of the
twin orientation is shown in Figs. 1(c) and (d). The three perpendicular
cartesian directions of twins for $i=1,2,3,4$ are denoted by a${}_{\rm
o}^{i}$, b${}_{\rm o}^{i}$ and c${}_{\rm o}^{i}$, where a${}_{\rm o}^{i}$ and
c${}_{\rm o}^{i}$ construct an effective basal plane and b${}_{\rm o}^{i}$
mutually represents the long axis. The volume fraction of the four twins
labelled $i$ = 1, 2, 3, and 4 are 0.38, 0.26, 0.23, and 0.13, respectively.
The mismatch angle between the twins numbered 1 and 2, as well as 3 and 4, are
around 3∘.

Measurements on different pieces cleaved of the single crystalline ingot
demonstrate the same twinning scheme with minor differences in twin fractions
of different twins. An attempt to detwin the crystals by means of high-
temperature treatment, etching, or cleaving of micrometer-sized crystals
neither affected the twinning as such nor the twinning fractions.

In turn, measurements in any direction in the effective basal plane reflect
effectively an admixture of a${}_{\rm o}^{i}$ and c${}_{\rm o}^{i}$ directions
due to the four twins. We define, therefore, two mutually perpendicular
effective _sample_ directions a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$,
explicitly taking into account the volume fractions of the four twins. This
definition is schematically depicted in Figs. 1(c) and (d) where a${}_{\rm
o}^{\star}$ is nearly aligned along a${}_{\rm o}^{1,2}$ and c${}_{\rm
o}^{3,4}$, while c${}_{\rm o}^{\star}$ is aligned to that of c${}_{\rm
o}^{1,2}$ and a${}_{\rm o}^{3,4}$. The third crystal direction, corresponding
to the long axis bo, remains unaffected by the twin deformations.

Figure 1: Twin scheme in the basal plane of orthorhombic CePdAl3 as derived
from single crystal x-ray diffraction. (a) X-ray scattering intensity
reconstructed in the $H0L$ plane. The splitting of the reflections is
characteristic of twin formation. (b) Indexed reflections of panel (a) with
the colors corresponding to different twin domains. Schematics of the lattice
vectors a${}_{\rm o}^{i}$ and c${}_{\rm o}^{i}$ of twin $i$ in the basal plane
$H0L$ of the orthorhombic crystal are depicted in the lower panels (c) and
(d). Four twins labelled $i$ = 1, 2, 3, and 4 were identified. a${}_{\rm
o}^{\star}$ and c${}_{\rm o}^{\star}$ are defined as mutual perpendicular
sample directions comprising the admixtures of twin lattice vectors.

###  III.2 Temperature-dependence of the specific heat

The temperature dependence of the specific heat C(T) of single-crystalline
tetragonal ($I4mm$) and orthorhombic ($Cmcm$) CePdAl3, as well as nonmagnetic
polycrystalline tetragonal ($I4mm$) LaPdAl3 measured in the absence of a
magnetic field are shown in Fig. 2. No evidence suggesting magnetic order was
observed in tetragonal CePdAl3 [49]. In orthorhombic CePdAl3, a $\lambda$-type
anomaly comprising a peak at 5.4 K followed by a shoulder closely above the
transition temperature $T_{\rm N}$ = 5.6 K is observed, where the
magnetization is characteristic of antiferromagnetism as reported below. The
behavior observed is consistent with a previous study of polycrystalline
CePdAl3 [48]. Moreover, the properties are reminiscent of the commensurate to
incommensurate magnetic transition reported of other strongly correlated
systems [55, 56].

A pronounced shoulder in the specific heat has also been seen in other
systems, notably, the chiral cubic magnet MnSi [57, 54], where it reflects a
change of character of the critical spin-fluctuations when approaching long-
range helimagnetic order and a concomitant fluctuation-induced first-order
transition. Details of the low-temperature specific heat of orthorhombic
CePdAl3 at zero-field are presented in Sec. III E below, which also includes
data collected at different magnetic fields.

Above $T_{\rm N}$, the expression ${C}/{T}=\gamma+\beta{T}^{2}$, where
$\gamma$ and $\beta$ are the electronic and phononic contributions to the
specific heat, respectively, has been fitted to the specific heat data in the
range $\sim$18 to $\sim$23 K of orthorhombic CePdAl3. The values obtained for
$\gamma$ and $\beta$ are 234 $\mathrm{mJ~{}mol^{-1}~{}K^{-2}}$ and 3.437
$\times$ $10^{-4}$ $\mathrm{J~{}mol^{-1}~{}K^{-4}}$, respectively. The Debye
temperature, $\Theta_{\rm D}$ = 305 K, associated with $\beta$ may be derived
using the relation $\beta=(12/5)\pi^{4}n{\rm R}/\Theta_{\rm D}^{3}$, where n
is the number of atoms per formula unit and R is the gas constant. The phonon
contribution to the specific heat in the Debye model [orange line in Fig. 2]
is given by

$\begin{split}C_{\rm ph,Debye}=&~{}9n{\rm R}\left(\frac{T}{\Theta_{\rm
D}}\right)^{3}\int_{0}^{x_{\rm
D}}{\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}}dx\end{split}$ (1)

where $x_{\rm D}=\Theta_{\rm D}/T$. At high temperatures the experimental data
of tetragonal LaPdAl3 and CePdAl3, as well as orthorhombic CePdAl3 approach
the Dulong-Petit limit, 3nR = 15R = 124.7 $\mathrm{J~{}mol^{-1}~{}K^{-1}}$,
where n = 5.

The large value of $\gamma$ = 234 $\mathrm{mJ~{}mol^{-1}~{}K^{-2}}$ obtained
from the low-temperature specific heat above $T_{\rm N}$ is typical for a
heavy-fermion system. It has to be borne in mind, however, that evaluating
$\gamma$ at the relatively high-temperature range above $T_{\rm N}$ is
associated with substantial uncertainties. A lower bound of $\gamma$, fitting
the experimental data in the antiferromagnetic state at temperatures between
$\sim$0.9 K and $\sim$3.7 K, yields a value of $\gamma$ = 121
$\mathrm{mJ~{}mol^{-1}~{}K^{-2}}$ still characteristic of heavy-fermion
behaviour.

At high temperatures ($T$ $>$ 100 K), the specific heat of all three compounds
exhibits essentially the same temperature dependence. However, the specific
heat of orthorhombic CePdAl3 is slightly smaller than for tetragonal CePdAl3,
suggesting reduced electronic and phononic contributions associated with the
reduced crystal symmetry. Compared to nonmagnetic LaPdAl3, the specific heat
of orthorhombic CePdAl3 is also slightly smaller, yet within the experimental
error of experiment. Indeed, a multiplication with a fraction of 0.99 to the
total signal of LaPdAl3 fully superimposes the data of CePdAl3 as shown in
Fig. 3(a) of $C/T$ vs $T$. The corresponding difference in specific heats may
be attributed to the magnetic contribution of the specific heat of
orthorhombic CePdAl3.

Shown in Fig. 3(b) is a sharp peak at $T$ = 5.4 K in the magnetic contribution
to the specific heat following subtraction of the phonon contribution
signaling an antiferromagnetic transition. In addition, a broad maximum around
30 K may be discerned as characteristic of a Schottky anomaly due to crystal
electric field contributions.

In the tetragonal as well as the orthorhombic symmetry of the lattice, the
degeneracy of the sixfold ground state multiplet of the Ce3+ ion splits into
three doublet states. These lift the first and second excited state with
respect to the ground state resulting in a contribution to the specific heat
which can be expressed as [58]

$\begin{split}C_{\rm CEF}=&\frac{\rm
R}{Z}\sum_{l=0}^{2}g_{l}\left(\frac{E_{l}}{kT}\right)^{2}{\rm
exp}\left(-\frac{E_{l}}{kT}\right)\\\ &-\frac{\rm
R}{Z^{2}}\left(\sum_{l=0}^{2}\frac{E_{l}}{kT}g_{l}{\rm
exp}\left(-\frac{E_{l}}{kT}\right)\right)^{2}\end{split}$ (2)

where

$\begin{split}Z=\sum_{l=0}^{2}g_{l}{\rm
exp}\left(-\frac{E_{l}}{kT}\right)\end{split}$ (3)

is the partition function, and $l$ = 0, 1 and 2 denote the ground, first and
second excited states, respectively. The degeneracy of the three doublet
states is $g_{0}=g_{1}=g_{2}=2$.

The energy difference $E_{1}-E_{0}=\Delta_{1}$ and $E_{2}-E_{0}=\Delta_{2}$
represent the levels of the first and the second excited states, respectively.
A fit of the data to Eqn. (2) between 20 K and 100 K yields $\Delta_{1}$ =
25.4 K and $\Delta_{2}$ = 76.0 K, respectively. Note that, the normalized
subtraction of the LaPdAl3 signal may introduce systematic errors in the
determination of the precise values of the excited states. For instance,
subtraction of the signal of LaPdAl3 after multiplication with a fraction of
0.98 yields $\Delta_{1}$ = 28.6 K and $\Delta_{2}$ = 95.5 K.

Furthermore, we have calculated the magnetic entropy $S=\int{(C/T})dT$
presented in Fig. 3(c). At the magnetic transition temperature, the entropy
reaches the theoretical value of R$\ln{2}$ for a doublet ground state expected
of Ce3+ ions. When increasing the temperature, the entropy increases and
reaches Rln4 around 30 K, approaching saturation above 100 K consistent with
the scheme of crystal electric field levels.

Figure 2:  (a) Zero-field specific heat of single-crystalline orthorhombic
(black) and tetragonal (blue) [49] CePdAl3 as a function of temperature. Data
of orthorhombic CePdAl3 were measured in a Dryogenic system between 0.08 K and
4 K, and in a PPMS between 2 K and 200 K. Also shown are the specific heat of
nonmagnetic polycrystalline LaPdAl3 (Gray line) and the Debye fit (orange
line) calculated from the low-temperature specific heat of the $Cmcm$
structure. The Debye temperature is $\Theta_{\rm D}$ = 305 K. The Dulong-Petit
limit for all three compounds, 15R = 124.7 $\mathrm{J~{}mol^{-1}~{}K^{-1}}$ is
depicted by a dashed line. (b) The low-temperature part of the specific heat
of orthorhombic CePdAl3 shows a pronounced $\lambda$-type anomaly with a broad
shoulder at the magnetic transition at $T_{\rm N}$. Figure 3: Magnetic
contribution to the specific heat and crystal electric field levels. (a)
Specific heat per unit temperature, $C/T$, of orthorhombic CePdAl3 and
tetragonal LaPdAl3 (with a multiplication of a fraction of 0.99) as well as
their difference. (b) Magnetic specific heat, $C_{\rm mag}$, and the fit to
the expression for the crystal electric field contribution to the specific
heat yields $\Delta_{1}$ = 25.4 K and $\Delta_{2}$ = 76.0 K (c) Magnetic
contribution to the entropy. The inset shows the entropy at low temperatures.

###  III.3 Temperature-dependence of the magnetic susceptibility

Figure 4: (a) Temperature dependence of the real part of the ac
susceptibility, Re $\chi$ac of orthorhombic CePdAl3 measured along a${}_{\rm
o}^{\star}$, c${}_{\rm o}^{\star}$ and bo at an excitation amplitude of 1 mT
and a frequency of 911 Hz. The inset shows the low-temperature part of Re
$\chi$ ac, reflecting the characteristics of an antiferromagnetic transition
at $T_{\rm N}$ = 5.6 K. (b) Susceptibility, H/M, as a function of temperature
for H $\parallel$ a${}_{\rm o}^{\star}$, H $\parallel$ c${}_{\rm o}^{\star}$
and H $\parallel$ bo measured in a field of 0.1 T. Gray lines are Curie-Weiss
fits. The inset shows the data for temperatures below 50 K.

The real part of the ac susceptibility, Re $\chi$ac of orthorhombic CePdAl3 as
a function of temperature is shown in Fig. 4(a) for a${}_{\rm o}^{\star}$,
c${}_{\rm o}^{\star}$ and bo. A clear magnetic transition is observed at
$T_{\rm N}$ = 5.6 K in the low temperature range, characteristic of the onset
of antiferromagnetic order as indicated by arrows in the inset. Namely, below
$T_{\rm N}$, Re $\chi$ac monotonically decreases along a${}_{\rm o}^{\star}$
and c${}_{\rm o}^{\star}$ with decreasing temperature, while slightly
increasing along bo. The magnitude of R e$\chi$ac along different axes differs
significantly for T <100 K, indicating sizeable magnetic anisotropy.

Figure 4(b) shows the normalized susceptibility, H/M, as a function of
temperature in a field of 0.1 T for H $\parallel$ a${}_{\rm o}^{\star}$, H
$\parallel$ c${}_{\rm o}^{\star}$ and H $\parallel$ bo. In the paramagnetic
state well above $T_{\rm N}$, a Curie-Weiss dependence is observed. A linear
fit to the data above 100 K yields Weiss temperatures $\Theta_{\rm
W}^{a^{\star}}$ = -0.8 K, $\Theta_{\rm W}^{c^{\star}}$ = -13.5 K and
$\Theta_{\rm W}^{b}$ = -33.0 K for H $\parallel$ a${}_{\rm o}^{\star}$, H
$\parallel$ c${}_{\rm o}^{\star}$ and H $\parallel$ bo, respectively,
characteristic of an antiferromagnetic coupling. Moreover, the effective
moments of 2.39, 2.49 and 2.44 $\mu$B per ion obtained under magnetic field
along a${}_{\rm o}^{\star}$, c${}_{\rm o}^{\star}$ and bo, respectively, are
close to the value of 2.54 $\mu$B expected for a free Ce3+ ion. This might
suggest a localized nature of the Ce moments in CePdAl3. The deviation of H/M
from the Curie-Weiss dependence for TN < T < 100 K shown in the inset of Fig.
4(b) may be related to CEF effects and electronic correlations. Furthermore,
despite the twin deformations, a significant difference between the
susceptibilities along a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$ in the
paramagnetic state indicates a large anisotropy in the basal plane,
characteristic of an easy-axis system.

###  III.4 Magnetization

The magnetic field dependence of the isothermal magnetization of orthorhombic
CePdAl3 at 2 K for H $\parallel$ a${}_{\rm o}^{\star}$, H $\parallel$
c${}_{\rm o}^{\star}$ and H $\parallel$ bo is shown in Fig. 5(a). No
hysteresis is observed. The magnetization varies linearly in the low-field
region up to 1 T as shown in the inset of Fig. 5(a) consistent with
antiferromagnetic order. For fields along a${}_{\rm o}^{\star}$ and c${}_{\rm
o}^{\star}$, an $S$-shaped rise is observed in the magnetization when further
increasing field. A kink around 5.5 T suggests a field-driven transition. The
magnetization values at 5.5 T are 0.85 $\mu$B for H $\parallel$ a${}_{\rm
o}^{\star}$, 0.44 $\mu$B for H $\parallel$ c${}_{\rm o}^{\star}$, and 0.18
$\mu$B for H $\parallel$ bo. The magnetization increases monotonically above
this transition where the moments along a${}_{\rm o}^{\star}$ and c${}_{\rm
o}^{\star}$ at 14 T, the highest field strength of studied, are 1.3 and 0.7
$\mu$B per Ce atom, respectively. In comparison, the magnetization increases
linearly with field for H $\parallel$ bo. The moment at 14 T is 0.4 $\mu$B per
Ce atom.

Figure 5: (a) Isothermal magnetization of orthorhombic CePdAl3 at 2 K measured
in a field along a${}_{\rm o}^{\star}$, c${}_{\rm o}^{\star}$ and bo up to 14
T. The arrow indicates the direction of increasing magnetic field. The inset
shows the linear variation of the magnetization below 1 T. Typical field
dependence of isothermal magnetization at various temperatures for (b) H
$\parallel$ a${}_{\rm o}^{\star}$, (c) H $\parallel$ c${}_{\rm o}^{\star}$ and
(d) H $\parallel$ bo. A field-driven spin-flop transition at $\sim$5.5 T is
observed below $T_{\rm N}$ for H $\parallel$ a${}_{\rm o}^{\star}$ (blue
arrow) and H $\parallel$ c${}_{\rm o}^{\star}$ (red arrow).

Keeping in mind the twinned crystal structure, the magnetization along
a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$ represent a mixture of the
crystallographic ao and co axes. A large quantitative difference in the
magnetization at 5.5 T along a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$
makes it unlikely, that a metamagnetic transition occurs at the same field
value in the ao and co directions in a single twin domain. Instead, it appears
most likely that the increase in the magnetization corresponds to a spin-flop
in the a${}_{\rm o}^{i}$ easy direction of each twin only.

Shown in Fig. 5(b), (c) and (d) are the isothermal magnetization at various
temperatures for H $\parallel$ a${}_{\rm o}^{\star}$, H $\parallel$ c${}_{\rm
o}^{\star}$ and H $\parallel$ bo, respectively. The spin-flop transition in
M(H) for a${}_{\rm o}^{\star}$ and c${}_{\rm o}^{\star}$ shifts to lower
fields under increasing temperature and vanishes above $T_{\rm N}$. In
contrast, the variation in M(H) along bo is essentially temperature
independent at and above $T_{\rm N}$.

Figure 6: (a) Susceptibility, dM/dH, calculated from the measured
magnetization of orthorhombic CePdAl3 for (a) H $\parallel$ a${}_{\rm
o}^{\star}$, (b) H $\parallel$ c${}_{\rm o}^{\star}$ and (c) H $\parallel$ bo.
Data are shifted by 0.025 for clarity. Peaks correspond to field-induced
transitions marked by arrows at $H_{3}$ (pink) and $H_{4}$ (sky blue). The
peaks disappear above $T_{\rm N}$.

In order to trace the field-driven magnetic transition, we have calculated the
differential susceptibility, dM/dH from the isothermal magnetization at
various temperatures presented in Fig. 6(a), (b), and (c) for H $\parallel$
a${}_{\rm o}^{\star}$, H $\parallel$ c${}_{\rm o}^{\star}$ and H $\parallel$
bo, respectively. For fields along a${}_{\rm o}^{\star}$ and c${}_{\rm
o}^{\star}$, the transition is characterized by a broad peak at $\sim$5.2 T at
2 K, which resolves into two peaks at elevated temperatures. These peaks exist
below $T_{\rm N}$ as marked by arrows at the transition fields $H_{3}$ and
$H_{4}$, following the labelling scheme described in Sec. II. With increasing
temperature the field range between the peaks increases and both peaks shift
to lower field values. No indication exists of a field-induced transition in
dM/dH for field along bo.

The evolution of the field-induced transitions may be traced in further detail
by the temperature dependence of the magnetization M(T) and the ac
susceptibility Re$\chi$ac(T). Shown in Fig. 7 is M(T) and Re$\chi$ac(T) at
various fields up to 14 T. By decreasing the temperature, orthorhombic CePdAl3
undergoes a phase transformation from paramagnetism to antiferromagnetic order
at a transition temperature $T_{1}$ (marked by red lines). This transition is
visible in Re$\chi$ac(T) in all crystallographic directions. The transition at
$T_{1}$ shifts to lower temperatures under increasing field but does not
vanish upto the highest field of 14 T studied. In the intermediate field range
from 2 T to 6 T, clear changes in M(T) and Re$\chi$ac(T) for field along
a${}_{\rm o}^{\star}$ [Fig. 7(a) and (d)]) and c${}_{\rm o}^{\star}$ [Fig.
7(b) and (e)] point to two additional phase transitions at temperatures
denoted $T_{3}$ (blue line) and $T_{4}$ (green line). These transitions
disappear at fields above 6 T. For H $\parallel$ bo, only the first transition
at $T_{1}$ is observed in M(T) and Re$\chi$ac(T) [Fig. 7(c) and (f)].

Figure 7: Temperature dependence of magnetization, M(T) and real part of ac
susceptibility, Re$\chi$ac(T) of orthorhombic CePdAl3 in magnetic fields up to
14 T. M(T) is shown in panels (a), (b) and (c), and Re$\chi$ac(T) in (d), (e)
and (f) for H $\parallel$ a${}_{\rm o}^{\star}$, H $\parallel$ c${}_{\rm
o}^{\star}$ and H $\parallel$ bo, respectively. Re$\chi$ac(T) is shifted for
clarity. Magnetic transitions are marked by vertical lines at temperatures
$T_{1}$ (red), $T_{3}$ (blue) and $T_{4}$ (green). A complex behavior with
multiple transitions is present for field along a${}_{\rm o}^{\star}$ and
c${}_{\rm o}^{\star}$ between 2 T and 6 T.

###  III.5 Field-dependence of the specific heat

The specific heat of orthorhombic CePdAl3 as a function of temperature at
different magnetic fields for H $\parallel$ c${}_{\rm o}^{\star}$ is presented
in Fig. 8. At zero magnetic field [Fig. 8(b)], a broad shoulder with a point
of inflection is observed at $T_{1}$ followed by a sharp peak at $T_{2}$.
Increasing the applied field results in a broadening of the peak at $T_{2}$
[Fig. 8(c)] and a splitting with an additional peak emerging at a lower
temperature $T_{4}$. For even higher fields up to 6 T [Fig. 8(d) to (h)], the
position of $T_{4}$ continues to shift to lower temperatures with the
emergence of another peak at $T_{3}$. The emergence of the peaks at $T_{3}$
and $T_{4}$ in the specific heat in the intermediate field range from 2 to 6 T
is consistent with the phase transitions deduced from the magnetization and
the ac susceptibility (see Figs. 6 and 7). For fields above 6 T, a noticeable
shift of $T_{1}$ and $T_{2}$ to lower temperatures is observed.

Figure 8: Specific heat of orthorhombic CePdAl3 as a function of temperature
under selected magnetic fields up to 14 T applied along the c${}_{\rm
o}^{\star}$ axis. Data measured in the Dryogenic system between 0.08 K and 4 K
are combined with data measured in the PPMS above 2 K. At $H=0$ the magnetic
transition displays a peak at $T_{2}$ preceded by a broad shoulder with a
point of inflection at $T_{1}$. Additional peaks emerge at $T_{3}$ and $T_{4}$
for magnetic fields between 2 T and 6 T.

###  III.6 Field-dependence of the magnetic susceptibility

In order to investigate the qualitative difference between the transitions
labelled as $H_{3}$ and $H_{4}$ in dM/dH (see Fig. 6), we have measured the
magnetic susceptibility as a function of magnetic field between 0 and 14 T.
Figure 9 shows the real part of the ac susceptibility, Re$\chi$ac, and the
susceptibility calculated from the magnetization, dM/dH, as a function of
increasing and decreasing field. At 2 K, dM/dH exhibits two peaks under
increasing field, first, a pronounced peak at 5.15 T, followed by a second
broad peak at 5.3 T for both H $\parallel$ c${}_{\rm o}^{\star}$ [Fig. 9(a)
and (d)] and H $\parallel$ a${}_{\rm o}^{\star}$ [Fig. 9(c) and (f)]. The
first peak shifts to 5 T resulting in a hysteresis, while the second peak
remains at the same field value under decreasing field. Similar effects exists
in Re$\chi$ac where the first peak becomes less pronounced with a smaller
hysteresis and a slightly lower field of 5.05 T. Also, the value of Re$\chi$ac
is slightly lower around the transition. At higher temperatures, both peaks
are shifted to lower field values. The hysteresis in Re$\chi$ac decreases
significantly and drops below the noise level at 5 K [Fig. 9(b) and (e)].
Here, the magnitude of Re$\chi$ac matches well with dM/dH except around the
first peak. The difference in character of the transitions labelled as $H_{3}$
and $H_{4}$ suggest their intrinsic origin rather than being related to the
twinned microstructure.

On the one hand, the hysteresis observed in dM/dH and Re$\chi$ac is
reminiscent of changes of population of multidomain states. On the other hand,
the smaller amplitude of Re$\chi$ac as compared to dM/dH indicates the
presence of slow relaxation processes around the phase transition. Similar
features are known to trace spin textures like helimagnetic disclination or
skyrmions in magnetic materials [59, 60, 61]. Further experimental
investigations are needed to explore such a possibility in orthorhombic
CePdAl3.

Figure 9:  Details of the magnetic transitions labelled as $H_{3}$ and
$H_{4}$. Shown are the real part of ac susceptibility, Re$\chi$ac, and the
susceptibility calculated from the magnetization, dM/dH of orthorhombic
CePdAl3 as a function of increasing and decreasing field for (a) H $\parallel$
c${}_{\rm o}^{\star}$ at 2 K, (b) H $\parallel$ c${}_{\rm o}^{\star}$ at 5 K,
and (c) H $\parallel$ a${}_{\rm o}^{\star}$ at 2 K. (d), (e) and (f) show the
magnetic transition regions corresponding to the blue rectangles in (a), (b)
and (c), respectively. Colors denote dM/dH for increasing (orange) and
decreasing (green) magnetic field. Black circles correspond to Re$\chi$ac for
increasing (filled symbols) and decreasing (open symbols) field, respectively.
dM/dH was calculated after smoothing the data.

###  III.7 Magnetic phase diagram

Combining the features detected in the magnetization and the specific heat, we
infer the magnetic phase diagrams for field parallel to c${}_{\rm o}^{\star}$
and bo shown in Fig. 10(a) and (b), respectively. Due to the twinned
microstructure, the response of the magnetization, specific heat, and ac
susceptibility are qualitatively alike for H $\parallel$ a${}_{\rm o}^{\star}$
and H $\parallel$ c${}_{\rm o}^{\star}$. In addition, the enhanced signal
observed along a${}_{\rm o}^{\star}$ as compared to c${}_{\rm o}^{\star}$
indicates that a${}_{\rm o}^{\star}$ reflects a larger fraction of the easy
axis, ao. Therefore, the transitions along both a${}_{\rm o}^{\star}$ and
c${}_{\rm o}^{\star}$ reflect equally the phenomenon belonging to the easy ao
axis of the untwinned single domain.

Four magnetic regions may be distinguished for field along c${}_{\rm
o}^{\star}$, denoted AF-I, AF-II, AF-III and AF-IV. At low temperature and
zero-field, the ground state is denoted as AF-I. With increase temperature,
AF-II appears at 5.4 K before entering in the paramagnetic (PM) state above
5.6 K. Signatures of the AF-II region are detected only in the specific heat.
The application of a magnetic field at low temperature drives a spin-flop
transition from AF-I to AF-IV with an intermediate region AF-III in a narrow
field range only. For finite field applied along the hard axis, i.e., H
$\parallel$ bo [Fig. 10(b)] only the AF-I and PM phases were observed,
possibly due to the lack of specific heat data for finite fields along the
hard axis. However, the AF-II transition was observed in zero field and the
AF-II regime is shown in the phase diagram in Fig. 10(b) for consistency.

While the magnetization suggests a collinear antiferromagnetic structure along
ao in the AF-I phase, and AF-IV shows the characterisics of a spin-flop phase,
the nature of AF-II and AF-III remain completely unknown. Neutron scattering
studies under magnetic field are needed to determine the nature of the four
antiferromagnetic phases we observed in orthorhombic single crystal CePdAl3.

Figure 10:  Magnetic phase diagram of orthorhombic CePdAl3 for (a) H
$\parallel$ c${}_{\rm o}^{\star}$ and (b) H $\parallel$ bo as inferred from
the magnetization and specific heat. Due to crystal twinning, the phase
diagram for H $\parallel$ a${}_{\rm o}^{\star}$ qualitatively resembles the
phase diagram for H $\parallel$ c${}_{\rm o}^{\star}$ shown in (a). Phase
transitions are guided by the lines which are denoted by numerals $j$ = 1, 2,
3, and 4. The associated temperature and field values are labelled as $T_{j}$
and $H_{j}$, respectively. Four magnetically ordered phases may be
distinguished as discussed in the text.

##  IV Conclusions

In summary, we measured the magnetization, ac susceptibility, and specific
heat of a single crystal of CePdAl3 grown by optical float-zoning. A highly
anisotropic behavior with a twinned orthorhombic crystal symmetry was
observed. Antiferromagnetic order with $T_{\rm N}=5.6$ K was observed in terms
of transitions in the ac susceptibility and specific heat. The magnetization
is characteristic of antiferromagnetic order with an easy ao direction in the
basal plane. Field-driven transitions were detected in the magnetization along
the easy direction, consistent with the ac susceptibility and specific heat.
Taken together, our study reveals a strong interplay of electronic
correlations, complex magnetic order and structural modifications in CePdAl3.

###### Acknowledgements.

We wish to thank A. Engelhardt, S. Mayr, and W. Simeth for fruitful
discussions and assistance with the experiments. We thank T. E. Schrader on
measurements with the Rigaku single-crystal diffractometer in the x-ray labs
of the Jülich Centre for Neutron Science (JCNS). This study has been funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
TRR80 (From Electronic Correlations to Functionality, Project No. 107745057,
Project E1), SPP2137 (Skyrmionics, Project No. 403191981, Grant PF393/19), 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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