Inner Relaxations in Equiatomic Single-Phase High-Entropy Cantor Alloy
Inner Relaxations in Equiatomic Single-Phase High-Entropy Cantor Alloy
Inner Relaxations in Equiatomic Single-Phase High-Entropy Cantor Alloy
165999
A. Smekhova, A. Kuzmin, K. Siemensmeyer, R. Abrudan, U. Reinholz, A. G. Buzanich, M.
Schneider, G. Laplanche, K. V. Yusenko, Inner relaxations in equiatomic single-phase high-entropy
Cantor alloy, J. Alloys Compd. 920 (2022) 165999.
1
Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), D-12489 Berlin, Germany
2
Institute of Solid State Physics, University of Latvia, LV-1063 Riga, Latvia
3
Bundesanstalt für Materialforschung und – prüfung (BAM), D-12489 Berlin, Germany
4
Institut für Werkstoffe, Ruhr-Universität Bochum, D-44801 Bochum, Germany
Abstract:
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Graphical abstract
1. Introduction
High-entropy and compositionally complex alloys based on 3d-block elements are
recognized to be promising functional materials that exhibit high potential for different
aspects in materials science [1-3]. Nowadays, they are also established to be very
attractive for practical applications in vital renewable energy technologies like energy
storage and conversion [4]. High-entropy alloys (HEAs) are known for their high
chemical complexity resulting in different local atomic environments [5],
crystallographic structures and mechanisms of phase formation [6], smaller diffusion
mobility of atoms [5, 7], and enhanced mechanical properties [8]. Their impressive
performance in the fields of hydrogen storage [9, 10], noble-metal-free electrocatalysts
[11, 12], oxygen evolution and reduction reactions [13, 14], carbon dioxide conversion
[15], and supercapacitors with virtually unlimited lifecycle [16, 17], etc has been already
uncovered. Besides, an enhanced radiation resistance [18] suggests the successful use of
HEAs in advanced nuclear applications [19].
Depending on composition and processing routes, HEAs exhibit variations in their
phase constitution and microscopic morphology [20]. Several HEA systems based on Cr,
Mn, Fe, Co, Ni and Al were proposed to be single-phase face-centered cubic (fcc)
structures in quite broad compositional and temperature intervals [21-23]. The equiatomic
CrMnFeCoNi HEA – the classical Cantor alloy – forms a stable fcc solid solution above
~800 °C that can be retained down to room temperature (RT), if the alloy is cooled at a
sufficiently high rate [24, 25], whereas AlxCrFeCoNi HEAs for x < 0.3 are fcc at RT
after cooling rapidly from above 1000 °C [26-30]. These features make these alloys
suitable model HEAs for the investigation of various fundamental
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aspects related to short- and long-range ordering. Moreover, they allow investigating the
relationships between local aggregations and local lattice distortions with macroscopic
properties such as phase stability, hardness, ductility, corrosion, and magnetism.
The equiatomic single-phase fcc CrMnFeCoNi HEA shows high-temperature
structural stability together with good strength and ductility at low temperatures [8, 31,
32]. High corrosion resistance due to the formation of a Cr2O3-rich passivation layer has
been reported for alkali and acidic media [33, 34]. Magnetic properties of the
CrMnFeCoNi HEA were recently reported in Ref. [35], where two magnetic
transformations below 100 K (the paramagnetic to spin-glass and the ferromagnetic one)
were found while the fcc structure was maintained down to 3 K. An extraordinary
mechanical and chemical stability of CrMnFeCoNi HEA under high-temperature and
high-pressure conditions were investigated in many details either experimentally or
theoretically in Refs. [24, 36-39].
To improve the mechanical properties of the Cantor alloy while retaining its fcc
structure, off-equiatomic compositions were investigated. For instance, Bracq et al. [40]
studied the isopleths of each consistent element (e.g. Mnx(CrFeCoNi)100-x) to identify the
compositional and temperature ranges in which fcc solid solutions are formed, and the
mechanical properties of these alloys were systematically investigated [41-43]. It was
found that these systems with a modified stoichiometry can form single-phase fcc solid
solutions within a broad composition range as 0–25 at.% for Cr, 0–50 at.% for Mn, 0–50
at.% for Fe, 10–50 at.% for Co and 10–100 at.% for Ni. Nevertheless, the influence of
each element on the stability and constitution of HEAs needs to be further addressed for
the efficient design of new materials.
A large number of different local atomic configurations in HEAs results from the
multitude of how the constituent elements are distributed around a particular atom within
the several coordination shells. For the fcc lattice, these local arrangements within the
first two coordination shells affect strongly the central atom. Indeed, different atomic
sizes and different local atomic environments of atoms, in turn, lead to lattice distortions
that are commonly considered as significant: in the original Cantor alloy, lattice
distortions are expected to be on a scale of up to ~0.03 Å [44] whereas, for the modified
Cantor alloys containing, for example, Al and Pd, such distortions could be larger [45,
46]. Therefore, a large number of different local atomic configurations in multicomponent
single-phase fcc HEAs play an important role in properties that depend strongly on local
atomic interactions [44]: among them, vacancy migration and diffusion, dislocation slip,
and magnetic properties could be mentioned.
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Local lattice distortions are responsible for the solid solution strengthening in
HEAs since they can introduce energy barriers against dislocation motion due to random
fluctuations [47-50]. Recently, lattice distortions in the Cantor alloy were studied by
neutron diffraction at ambient conditions [51], by element-specific extended X-ray
absorption fine structure (EXAFS) and by density functional theory (DFT) calculations
[52-54]. In Ref. [51] the datasets were well described by considering only a single atom
type with a compositionally weighted average scattering length of all the constituent
elements (a grey atom model) while in Ref. [54], the main attention was paid already to
element-resolved distortions of each constituent. It was found that, while distortions
themselves are quite small on average (~0.1%), their fluctuations are an order of
magnitude larger (up to 2–3%), especially for Cr and Mn. The DFT calculations were
performed considering the magnetic ordering and were compared to the non-spin-
polarized one to especially validate the impact of magnetism because the results of recent
ab initio calculations [49] pointed out that in addition to the configurational entropy, other
entropy contributions like electronic, vibrational, and magnetic excitations are of similar
importance. It was concluded that magnetism is responsible for the unique character of
local bond fluctuations and is critically required for realistic DFT simulations. In Ref.
[53], the atomic pair distances between individual atoms of the alloy revealed that Mn
atoms have a slightly larger bond distance (∼0.4%) with their neighbors. An attempt to
distinguish the local structural disorder from the thermal contribution was done by
combining the low-temperature neutron diffraction and X-ray techniques. In Ref. [52]
using temperature-dependent EXAFS, it was demonstrated that atomic size, charge
transfer, magnetism, and local ordering are the important factors that affect the
displacement of an individual atom in a multi- component alloy, i.e. element-specific
structural relaxations. At the same time, an attempt to use neutron diffraction data and a
series of atomistic simulations together with ab initio calculations [55] has shown that the
short-range order in a four- component CrFeCoNi alloy is coupled with size mismatch of
different constituents and lattice distortions. A brief overview of other methods used for
lattice distortion studies in HEAs can be found in Refs. [56] and [57, 58]; the latter ones
include also the theory background for quantitative comparison of lattice strains (static
lattice distortions) revealed by different experimental techniques. It is important to notice
that microscopic and X-ray diffraction studies performed for various HEA systems cannot
reliably address the constituent arrangements at the atomic scale; thus, a demand for
different element-specific studies at the absorption edges supported by a proper data
evaluation suitable for the multicomponent systems is currently of increasing interest.
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using Cu Kα radiation (λ = 1.54 Å) and the PIXcel 1D detector (Fig. 1). The profiles were
refined by applying the LeBail model-free fitting routine. The XRD patterns from the
CrMnFeCoNi alloy confirm its single-phase character with the fcc lattice (Fm-3m space
group) and unit cell parameter a = 3.58214(3) Å. Its elemental composition (at.%) is
Cr20.4Mn20.1Fe19.6Co19.7Ni19.9 as determined by electron microprobe analysis.
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distribution functions, so, only the relative positions of atoms obtained during the RMC
simulation are important and not their absolute coordinates.
For the RMC simulations, a starting structural model (a simulation box) was
constructed in a form of a supercell with a size of 4a × 4a × 4a (a = 3.582 Å) employing
the periodic boundary conditions (Fig. 3). The experimental value of the lattice parameter
a refined from the XRD data was used and the size of the supercell was kept constant
during RMC simulations. The Cr, Mn, Fe, Co, and Ni atoms were randomly distributed
in a proper concentration at the Wyckoff positions of the fcc supercell including 256
atoms (52 Cr, 52 Mn, 50 Fe, 51 Co, 51 Ni).
In the current study, we followed the same procedure as described previously in
Ref. [59]. The number of atomic configurations simultaneously considered in the EA was
32, and the largest allowed displacement of atoms from their initial position was
0.4 Å, which is sufficient to describe both static and thermal disorder. The fixed supercell
size and small value of the allowed atom displacements play the role of constraints and
stabilize the structural model. As a result, the final model does not deviate much from the
crystallographic structure but takes into account both thermal and static disorders present
in the alloy.
At each RMC iteration, the configuration-averaged K-edge EXAFS spectra χ(k)k2
were calculated over all Cr, Fe, Co, and Ni atoms in the simulation box, and their Morlet
wavelet transforms (WTs) were compared with those of the experimental EXAFS spectra.
The best agreement between the Morlet WTs of the experimental and calculated EXAFS
spectra was used as a criterion for the model structure optimization.
The WT calculations were performed in the k-space range from 3.0 to 10 Å−1 (or
12 Å−1) and in the R-space range from 1.0 to 6.0 Å. The number of RMC iterations was
5000 to guarantee the convergence of the structural model; no significant improvements
in the residuals were observed beyond this number. Each RMC simulation resulted in a
set of atomic coordinates, which was subsequently used to calculate the pair distribution
functions (PDFs) g(r), the mean interatomic distances r, the mean square relative
displacements (MSRDs) for each pair of atoms, and the mean square displacements
(MSDs) for atoms of each type. Calculated in such a way PDFs are element-specific
projections of the three-dimensional structure of a material over a radial distance relative
to the chosen absorbing atom, where only the amount and the type of atoms located at the
particular distance from the absorber are important. So, there is no need to generate
billions of local configurations expected for the five-component system, because they will
very probably result in the same or very similar PDFs. Moreover, since only one structural
model (the same starting configuration) is used in the RMC simulation during the
simultaneous fit to four experimental EXAFS spectra collected at
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different absorption edges (these spectra act as the independent experimental data sets)
and the fit is performed in the wavelet space, the small differences in the scattering
amplitudes of 3d constituents are automatically taken into account, and the contributions
from all principal components are properly accounting. Thus, the final structural model
and the extracted PDFs represent a self-consistent solution that is in good agreement with
all available experimental EXAFS data in both k and R spaces. To improve statistics, 12
sets of different (independent) starting structural models were considered for final PDFs,
so the final PDFs describe the configuration-averaged local environment around a
particular absorber in the most unbiased manner. Despite a total number of local
configurations considered in the fit does not cover all possible relative arrangements of
five different constituents, the fit outcomes like interatomic distances within the first
coordination shell of absorbers and their mean square displacements can be still
considered as statistically averaged quantities.
During RMC simulations the configuration-averaged EXAFS spectra were
calculated for atoms of each type (all constituents) using ab initio real-space multiple-
scattering FEFF8.50L code [68, 69] including the MS effects up to the 4th order. The
scattering potential and partial phase shifts were calculated within the muffin-tin (MT)
approximation [68, 69] for each absorption edge only once, considering the cluster with
a radius of 4.8 Å centered at the required absorbing metal atom (Cr, Fe, Co, or Ni). Small
changes in the cluster potential induced by atom displacements during the RMC/EA
simulations were neglected. The photoelectron inelastic losses were accounted for within
the one-plasmon approximation using the complex exchange- correlation Hedin-
Lundqvist potential [70]. The amplitude reduction factor S 2 was included in the scattering
0
amplitude calculated by the FEFF code [68, 69]; no additional correction of the EXAFS
amplitude was performed. For each absorption edge, the values of the E0 energies used in
the definition of the photoelectron wavenumber k = [(2me/ħ2)(E − E0)]0.5 were set to those
determined carefully in advance by performing the RMC simulations of reference
compounds (pure metallic foils).
2.4 Magnetometry
Magnetometry measurements were carried out using a commercial magnetic properties
measurement system (MPMS, Quantum Design) in magnetic fields up to 5 T at 5 K. The
field-assisted cooling in ± 5 T was followed by the measurements of magnetic hysteresis
loops without additional setting of the magnetic field to zero. The vibrating sample
magnetometer (VSM) mode was used with a frequency of 14 Hz and 2 mm amplitude.
The sample was firmly pressed to avoid any movements of flakes kept in a plastic capsule.
The sample mass (22.33 mg) was measured with a microbalance.
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A qualitative comparison of EXAFS spectra for reference foils with those of the
HEA allows one to conclude that the majority of the elements adopt the fcc structure of
the considered Cantor alloy independently of their crystallographic structure as pure
metals.
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(the average distances and the mean square displacements) and allow one to correlate, for
example, local distortions with magnetic properties since even tiny changes in distances
between atoms drastically influence the value and the direction of their magnetic
moments for a particular local configuration. Some examples of such an influence
depending on HEAs structural arrangement and stoichiometry were demonstrated in Ref.
[49] by ab initio calculations.
However, due to the similarities of scattering amplitudes for 3d components of the
Cantor alloy, the discussions about the distribution of individual components over the
coordination shells of a particular absorber have to be skipped. Strong conclusions about
a possible short-range ordering could not be done based on the obtained final PDFs. It
might only be possible if principal components of the alloy have a significantly different
number of valence electrons resulting in easily distinguishable scattering amplitudes or if
a particular type of absorber is located in a significantly different local crystallographic
surrounding in a multi-phase system. In the latter case, strong differences in partial PDFs
should be visible, which is not the case in the present study.
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FTs. In particular, we expect that the final structural configuration, which is close to
random and is self-consistent, reproduces the arrangement of nearest atoms in the first
coordination shell of the corresponding 3d absorbers. This arrangement is responsible for
the first peak in FTs in the range from 1.0 to 3.0 Å. An uncertainty in distinguishing
between some of the neighboring 3d elements located in the first coordination shell of
absorbers of each type does not influence the averaged values of interatomic distances
and structural relaxations that will be extracted further.
A set of total and partial PDFs calculated from the atomic coordinates of the final
structural model for the HEA is shown in Fig. 5. All constituents exhibit very similar total
PDFs averaged over different 3d neighbor atoms (Fig. 5a) with the same number and
positions of peaks within the first five coordination shells: this indicates the similarity of
their environment in the alloy with a distribution of all atoms over the fcc lattice sites of
HEA (within a sensitivity of the applied RMC method). Compared to PDFs of pure
metallic foils (see next section), it becomes clear that independently from the initial
crystallographic ordering characteristic for pure metals (fcc, bcc or α-phase), all atoms
reproduce the fcc structure of the studied Cantor alloy on the local scale.
Nevertheless, there is a well-pronounced splitting of the peak at ~3.5 Å in the total
PDF of Fe atoms which is also partly visible for some other elements. Such peak splitting
is not observed in pure fcc foils of Co and Ni, and, thus, is specific for the considered
sample. It could be due to local structure relaxation (off-center displacement) caused by
an existing chemical disorder (atomic heterogeneity) that results in two groups of shorter
and longer interatomic distances around Fe atoms. Another explanation could be related
to a dynamic effect (double potential), which is not detectable by X-ray diffraction.
Considering the partial PDFs (Fig. 5b), it could be noted that only PDFs involving Fe
atoms demonstrate such a feature. Thus, we relate it to a specific property of the internal
structure of the considered HEA sample, and it may not be observed in other Cantor alloys
prepared by different processing routes.
The mean interatomic distances r and the mean square relative displacements
(MSRD) σ2 were numerically obtained for atoms of each type as the first and second
moments of PDFs (see Table 1). In addition, the mean square displacements (MSD) were
calculated directly from the final atomic coordinates of atoms in the simulation box
relative to their ideal starting positions in the fcc lattice: such information is not typical
for EXAFS data analysis and is usually extracted from diffraction data. As could be seen,
the mean distances to the neighboring atoms calculated for absorbers of each type are
very similar to each other without showing any preferences. Moreover, compared to
distances calculated for the hcp/fcc Co and fcc Ni foils, the Cantor alloy demonstrates
noticeably larger values of the interatomic distances (compare Tables 1
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and 3). This reflects one of the important aspects of atomic heterogeneity in HEAs: Cr
and Fe atoms have larger sizes and hence force the atomic volume of the alloy to expand.
Such changes, in turn, provoke variations in the interatomic interactions and, thus, in
many macroscopic and local properties of this and similar HEAs. It is worth mentioning
that pure Fe, Co and Ni are ferromagnetic at room temperature while the Cantor alloy
presents a paramagnetic state, this may also be related to the observed difference in
interatomic distances.
Besides, it was found quantitatively that Cr atoms have the largest MSRD and
MSD values among all constituents of the studied Cantor alloy; and this trend was
observed by considering either total or partial PDFs (see Tables 1 and 2, respectively).
Earlier, the same conclusion was established for the Al-deficient single-phase fcc
Al0.3CrFeCoNi alloy in Ref. [59]. This suggests that the distribution of distances around
Cr atoms is larger compared to other constituents, and it may be a reason for distinctive
magnetic properties found experimentally in Ref. [35] and shown later in the current work
as well.
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[74] rather than to the sample itself. Besides, the hysteresis loops recorded right after FC
are shifted in the magnetization direction (vertical shift) up to ± 0.06 emu/g as compared
to the symmetric ZFC field-dependence (Fig. 10a). The direction of these vertical shifts
is governed by the direction of the applied magnetic field during field- assisted cooling.
The observed shift of the hysteresis loops is much smaller than the one reported in
Ref. [35] for the bulk slab of equiatomic Cantor alloy that was several mm thick.
However, it clearly demonstrates the presence of frozen magnetic contributions enforced
by the direction of FC. These contributions are not fully reversed by the small external
magnetic fields of ± 0.5 T and could be associated with the disordered regions around Cr
atoms where a strong competition between ferromagnetic and antiferromagnetic
couplings of nearest-neighbors is expected. Since all individual PDFs involving Cr atoms
demonstrate an increased structural displacement of Cr (Table 2), a fraction of regions
with such competing couplings leading to frustrated magnetism could be very large.
Taking into account the high heterogeneity of the studied HEA due to the five principal
components, these regions are expected to be distributed along the whole volume of the
flakes.
This result agrees with previous theoretical findings in Refs. [49] and [35] where
individual magnetic moments, as well as a distribution of ferromagnetic and
antiferromagnetic interactions, were found to be very sensitive not only to the size of the
fcc unit cell but also to the local configurations of the nearest neighbors. It was also shown
that Cr atoms prefer to align antiferromagnetically to the cumulative magnetic moment
of their first coordination shell regardless of the initial local arrangements [35] and, in
general, the antiferromagnetic alignment of Cr with respect to ferromagnetically coupled
Fe, Co and Ni is energetically favorable [49]. The same alignment was theoretically
predicted for the four-component fcc CrCoFeNi [75] and found experimentally in the sub-
surface volume of the Al0.3CrCoFeNi HEA [59].
In the case of performing additional field-dependent measurements in the broad
field range of ± 5 T right after the field-assisted cooling, the hysteresis loops recorded
afterward in the narrow field range do not show any shifts in vertical directions and
demonstrate the symmetric behavior as in the case of ZFC (Fig. 10b). This supports the
assumption about the presence of initially frustrated or non-collinear magnetic states
which can be reduced by applying higher magnetic fields. Further experimental
clarifications of relative magnetic coupling of each constituent could be explicitly made
only by the synchrotron-based element-specific X-ray magnetic circular dichroism
(XMCD) technique and are beyond the scope of the current work.
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4. Conclusions
The possibility to probe local ordering and inner relaxations in complex
multicomponent systems at the atomic scale in an element-specific way opens a path to
exhaustive structural studies of materials needed for modern technologies and
applications. In this regard, X-ray absorption spectroscopy in conjunction with the reverse
Monte Carlo method offers opportunities unattainable by other techniques commonly
used to investigate multicomponent alloys. Still, only approximate averaged atomic
distributions can be provided without fully probing the details of the enormous number
of different actual local atomic configurations in such multicomponent systems. EXAFS
spectra recorded at the K absorption edges of several principal components of the alloy
can be fitted simultaneously in an unbiased manner using one structural model to reveal
the pair distribution functions of the individual elements. Accordingly, quantitative
information regarding the number of the nearest neighbors, their distances to the absorber,
structural/thermal relaxations of each constituent and peculiarities of the local
coordination for each element can be extracted. This information can be further used to
infer how individual atoms fit into the crystallographic structure of the complex system
which in turn would underline the role of individual components in certain macroscopic
properties. Some natural limitations of this approach exist in the case when the
multicomponent system consists of elements with very close scattering amplitudes as for
the neighboring elements in the periodic table; nevertheless, the most important
information regarding absorbers of each type can be still extracted.
Our work demonstrates that all five components of the equiatomic CrMnFeCoNi
Cantor alloy are distributed at the nodes of the fcc crystallographic structure
independently of their initial ordering in the pure state, a presence of component- specific
structural relaxations, and that the alloy is single-phase at the atomic scale. Quantitative
analysis reveals statistically averaged interatomic distances for all types of constituents,
and as expected these distances are enlarged compared to the distances in pure hcp/fcc
foils of Co and Ni. Meanwhile, a much greater disorder was found around Cr atoms
suggesting their strong involvement in the formation of magnetically competing
couplings over the entire volume of the alloy, which are responsible for the magnetic
behavior observed by conventional magnetometry. Surface oxidation probed by XANES
reveals the general trend anticipated from previously published results.
Inner relaxations uncovered by element-specific structural findings in this work
support the outcomes of previous studies where stronger lattice distortions were
associated, in particular, with the large atomic size of Cr atoms while magnetic
characterization indicates a strong sensitivity of macroscopic magnetic properties to the
synthesis approach. Thus, further activity in developing functional multicomponent
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materials possessing a high atomic heterogeneity and their detailed studies at the atomic
scale will expand our understanding of individual elements relationship with properties
of high-entropy systems beyond the limits of the classical Cantor alloy. Besides, further
efforts in analysis and interpretation of available experimental data as well as extensive
theory modelling are needed to advance the field.
Acknowledgements
The authors thank the Helmholtz-Zentrum Berlin for the provision of access to
synchrotron radiation facilities and allocation of synchrotron radiation at the BAMline
and UE52 beamlines of BESSY II at HZB. The use of ALICE chamber (BMBF project
no. 05K19W06) and time for magnetometry measurements at the HZB CoreLab for
Quantum Materials is acknowledged as well. Dirk Schröpfer and the workshop from
BAM are acknowledged for flakes preparation by mechanical milling; Christiane
Stephan-Scherb is acknowledged for providing XRD data. A. Smekhova acknowledges
also personal funding from CALIPSOplus project (the Grant Agreement no. 730872 from
the EU Framework Programme for Research and Innovation HORIZON 2020). Institute
of Solid State Physics, University of Latvia as the Center of Excellence has received
funding from the European Union’s Horizon 2020 Framework Programme H2020-
WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508,
project CAMART2. G. Laplanche acknowledges the German Research Foundation
(Deutsche Forschungsgemeinschaft: DFG) for financial support through project LA
3607/3-2 of the Priority Program SPP 2006 “Compositionally Complex Alloys - High
Entropy Alloys”.
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1200
1000
600
400
200
30 40 50 60 70 80 90
2 (degrees)
Cantor alloy
2.5 3.0
reference
EXAFS (FY, arb. units)
RT foils
2.5
2.0 RT
2.0
1.5
1.5
1.0 1.0
0.5 0.5
0.0
0.0
-100 0 100 200 300 400 -100 0 100 200 300 400
E - E0 (eV) E - E0 (eV)
Fig. 2 X-ray absorption spectra of the single-phase fcc CrMnFeCoNi HEA (a) together
with the reference foils (b) recorded at the K absorption edges of Cr (E0 = 5987 eV), Fe
(E0 = 7108 eV), Co (E0 = 7708.5 eV), and Ni (E0 = 8333 eV) at RT by FY. The inset
shows enlarged EXAFS oscillations to underline the difference between the bcc (Cr and
Fe) and fcc (Co and Ni) lattices. EXAFS at the Mn K-edge was measured in transmission
(E0 = 6538 eV). E0 was determined as the energy corresponding to the first maximum of
the first derivative of each particular spectrum. The spectra are normalized to unity and
shifted vertically for clarity.
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Fig. 3 Starting and final atom configurations (supercells) used in the RMC simulations
for simultaneous fit to EXAFS spectra of the CrMnFeCoNi HEA at four absorption edges.
The supercell was randomly filled with constituent atoms according to the stoichiometry
of the Cantor alloy; periodic boundary conditions were applied. Color scheme: Cr (green),
Mn (gray), Fe (brown), Co (blue), Ni (red). The top views of the supercells are shown
along the z-direction in chosen xyz coordinates as depicted on the figures. The schematic
border of the fcc unit cells is displayed at the bottom left of each figure.
Fig. 4 Experimental and RMC-calculated Cr, Fe, Co, and Ni K-edge EXAFS spectra
χ(k)k2 and their Fourier and Morlet wavelet transforms for the fcc CrMnFeCoNi HEA at
300 K.
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Cr-Cr
120 120
Cr-Fe
Cr-Co
100 100
PDF g(r) (atoms/Å)
Fe-Mn
80 80
Fe-Fe
Fe-Co
60 60
Fe-Ni
Co-Mn
40 40
Co-Co
Co-Ni
20 20
Ni-Mn
Ni-Ni
0 0
1 2 3 4 5 6 0 1 2 3 4 5 6
Fig. 5 Pair distribution functions g(r) – (a) total and (b) partial – for the fcc CrMnFeCoNi
HEA at 300 K extracted from K-edge EXAFS spectra of Cr, Fe, Co, and Ni using the
RMC method.
Table 1 Structural parameters extracted from the total PDFs of the fcc CrMnFeCoNi
HEA.
Phase in r [Å] MSRD σ2 [Å2] MSD [Å]
pure metal (±0.02 Å) (±0.003 Å2) (±0.02 Å)
Cr bcc 2.55 0.039 0.24
Mn α-Mn 2.55 0.028 0.18
Fe bcc 2.54 0.028 0.17
Co hcp/fcc 2.54 0.028 0.19
Ni fcc 2.54 0.027 0.19
(All average distances are about 2.54-2.55 Å; Cr atoms have larger MSRD and MSD)
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Table 2 Structural parameters extracted from the partial PDFs of the studied Cantor
alloy.
r [Å] /
Cr Mn Fe Co Ni
MSRD [Å2]
Cr 2.55 / 0.050 2.55 / 0.038 2.55 / 0.038 2.54 / 0.033 2.55 / 0.037
Mn 2.55 / 0.026 2.54 / 0.026 2.55 / 0.028 2.54 / 0.025
Fe 2.55 / 0.028 2.54 / 0.026 2.54 / 0.023
Co 2.54 / 0.025 2.54 / 0.025
Ni 2.54 / 0.023
(All average distances are about 2.54-2.55 Å; Cr atoms have the largest MSRD)
Table 3 Structural parameters extracted from the PDFs of pure metallic foils.
r [Å] MSRD σ2 [Å2] MSD [Å] MSD σ2
phase N [Å2]
(±0.02 Å) (±0.003 Å2)
Cr bcc 2.67 8+6 0.048 0.13 0.017
Fe bcc 2.65 8+6 0.045 0.12 0.014
Co fcc 2.52 12 0.020 0.15 0.023
Co hcp 2.51 12 0.016 0.13 0.017
Ni fcc 2.50 12 0.016 0.13 0.017
Fig. 6 Experimental and RMC-calculated Cr, Fe, Co, and Ni K-edge EXAFS spectra
χ(k)k2 of the pure metallic foils and their Fourier transforms at 300 K.
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160
140
120
80
bcc Cr
60
bcc Fe
40
fcc Co
20
fcc Ni
0
1 2 3 4 5 6
Distance (Å)
Fig. 7 Pair distribution functions g(r) for the pure metallic foils at 300 K extracted from
K-edge EXAFS spectra of Cr, Fe, Co, and Ni using the RMC method.
6
(a) Co K-edge (c)
EXAFS (k)k2 (Å-2)
in hcp Co 80
3 fcc Co
hcp Co
0
PDF g(r) (atoms/Å)
60
-3
Exper.
RMC
-6
4 5 6 7 8 9 10 11 12 40
Wavenumber k (Å-1)
2 20
FT (k)k2 (Å-3)
0
0
1 2 3 4 5 6
-1 Distance (Å)
-2
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0.30
EXAFS (k)k2 (Å-2)
0.6
0.15
FT (k)k2 (Å-3)
0.3
0.00
0.0
-0.15
-0.3
-0.6 -0.30
3 4 5 6 7 8 9 10 0 1 2 3 4 5
Wavenumber k (Å-1) Distance (Å)
(c)
RDF g(r) (atoms/Å)
40
Fig. 9 Experimental and RMC-calculated
Mn K-edge EXAFS spectra χ(k)k2 (a)
20 together with their Fourier transforms (b)
and radial distribution function g(r) (c) for
0 the cubic α-Mn foil (space group I- 43m
1 2 3 4 5 6
Distance (Å) (217), a = 8.894 Å) at 300 K.
1.5 1.5
(a)
1.0 1.0
Magnetisation (emu/g)
Magnetisation (emu/g)
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-0.50 -0.25 0.00 0.25 0.50 -0.50 -0.25 0.00 0.25 0.50
0H (T) 0H (T)
Fig. 10 Field-dependences of the HEA flakes recorded at 5 K after the zero-field cooling
and field-assisted cooling in ± 5 T performed right after the cooling process (a) and after
an additional field-dependence in a broad field range of ± 5 T (b). The vertical shift of the
hysteresis loops is observed in (a).
23
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0.021 O K edge
0.018
0.015
530 540 550 560 570
0.024
Cr3+ Cr L2,3 edges
0.020
0.02
0.025 Fe3+-like
Fe L2,3 edges
0.020
0.015
700 710 720 730 740
0.015
770 780 790 800 810
0.024
RT
0.016
Fig. A1 Raw XANES spectra recorded from the fcc CrMnFeCoNi HEA with horizontally
polarized X-rays at RT by TEY at the L2,3 absorption edges of Cr, Mn, Fe, Co, and Ni,
and at the K edge of oxygen.
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported in this
paper.
Highlights
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