# A-type antiferromagnetic order and magnetic phase diagram of the trigonal Eu
spin-7/2 triangular-lattice compound EuSn2As2

Santanu Pakhira  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
M. A. Tanatar  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Thomas Heitmann  The Missouri Research Reactor and Department of Physics
and Astronomy, University of Missouri, Columbia, Missouri 65211, USA  David
Vaknin  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  D. C. Johnston  Ames Laboratory, Iowa State University, Ames, Iowa 50011,
USA  Department of Physics and Astronomy, Iowa State University, Ames, Iowa
50011, USA

###### Abstract

The trigonal compound EuSn2As2 was recently discovered to host Dirac surface
states within the bulk band gap and orders antiferromagnetically below the
Néel temperature $T_{\rm N}\approx 24$ K. Here the magnetic ground state of
single-crystal EuSn2As2 and the evolution of its properties versus temperature
$T$ and applied magnetic field $H$ are reported. Included are zero-field
single-crystal neutron diffraction measurements versus $T$, magnetization
$M(H,T)$, magnetic susceptibility $\chi(H,T)=M(H,T)/H$, heat capacity $C_{\rm
p}(H,T)$, and electrical resistivity $\rho(H,T)$ measurements. The neutron-
diffraction and $\chi(T)$ measurements both indicate a collinear A-type
antiferromagnetic (AFM) structure below $T_{\rm N}=23.5(2)$ K, where the Eu2+
spins $S=7/2$ in a triangular $ab$-plane layer (hexagonal unit cell) are
aligned ferromagnetically in the $ab$ plane whereas the spins in adjacent Eu
planes along the $c$ axis are aligned antiferromagnetically. The
$\chi(H_{ab},T)$ and $\chi(H_{c},T)$ data together indicate a smooth crossover
between the collinear AFM alignment and an unknown magnetic structure at
$H\sim 0.15$ T. Dynamic spin fluctuations up to 60 K are evident in the
$\chi(T)$, $C_{\rm p}(T)$ and $\rho(H,T)$ measurements, a temperature that is
more than twice $T_{\rm N}$. The $\rho(H,T)$ of the compound does not reflect
a contribution of the topological state, but rather is consistent with a low-
carrier-density metal with strong magnetic scattering. The magnetic phase
diagrams for both $H\parallel c$ and $H\parallel ab$ in the $H$-$T$ plane are
constructed from the $T_{\rm N}(H)$, $\chi(H,T)$, $C_{\rm p}(H,T)$, and
$\rho(H,T)$ data.

##  I Introduction

Rare-earth intermetallic compounds are of considerable interest due to the
observations of complex magnetism, superconductivity, heavy-fermion behavior,
Kondo effect, valence fluctuation, large magnetocaloric effect,
magnetoresistance, and quantum criticality Fertig1977_IC ; Nagarajan1994_IC ;
Ghosh1995_IC ; Curro2000_IC ; Hundley2001_IC ; Sakai2011_IC ; Yamaoka2014_IC ;
Pecharsky1997_IC ; Pakhira2016_IC ; Pakhira2017_IC ; Buschow1969_IC ;
Gignoux1984_IC ; Ishida2002_IC ; Arndt2011_IC . Besides $4f$-element
magnetism, many rare-earth-based intermetallics have expanded in a new
direction by hosting novel electronic states through a complex interplay of
magnetism and band topology Hirschberger2016_GdPtBi ; Shekhar2018_RPtBi ;
Wang2016_YbMnBi2 ; May2014_EuMnBi2 ; Masuda2016_EuMnBi2 ; Soh2019_EuMnSb2 ;
Schellenberg2011_EuCd2As2 ; Jo2020_EuCd2As2 ; Xu2019_EuIn2As2 ;
Pakhira2020_EuMg2Bi2 ; Pakhira2021_EuMg2Bi2 . This interplay has generated
immense interest due to the discovery of nontrivial quantum phenomena such as
anomalous quantum Hall effect (AQHE), axion electrodynamics, and realization
of relativistic particles like Majorana fermions Hasan2010_TI ; Fu2007_TI ;
Tokura2019_TI ; Qi2006_QAHE ; Liu2008_QAHE ; Yu2010_QAHE ; Essin2009_AED ;
Li2010_AED ; Fu2008_Majorana ; Akhmerov2009_Majorana ; Cook2011_Majorana .
Topological states like Dirac surface points are expected to be gapped or
split in the presence of magnetic order due to time-reversal-symmetry
breaking. Thus studies of the nature of magnetic order and the associated
phase diagram in magnetic topological compounds is essential to reveal the
presence or absence of nontrivial electronic states.

EuSn2As2 is an intrinsic rare-earth-based topological insulator (TI) that has
been recently reported to exhibit Dirac surface states with energies within
the bulk band gap Li2019_EuSn2As2 . However, conventional Hall-effect
measurements reveal the presence of hole carriers with a concentration
$p=3.6\times 10^{20}$ cm-3 at 2 K, rendering the compound metallic up to at
least 300 K with an in-plane resistivity of order 0.4 m$\Omega$ cm
Chen2020_EuSn2As2 . The hole carriers might arise, for example, from low
concentrations of Sn vacancies.

EuSn2As2 crystallizes in the layered NaSn2As2-type crystal structure with
trigonal space group $R\overline{3}m$. In this crystal structure (Fig. 1), the
Eu atoms form a triangular lattice in the $ab$ plane (hexagonal unit cell)
with each Eu atom coordinated by six As atoms to form a trigonal prism. The Sn
and As atoms are covalently bonded in a honeycomb arrangement and the Sn atoms
between two adjacent Sn-As layers face each other with inversion symmetry. In
such a lattice geometry, magnetic frustration can arise when the nearest-
neighbor exchange interaction between the Eu ions is antiferromagnetic (AFM).
The compound is indeed reported to order antiferromagnetically below its Néel
temperature $T_{\rm N}\approx 24$ K Li2019_EuSn2As2 ; Arguilla2017_EuSn2As2 ;
Chen2020_EuSn2As2 . However, the nature of the AFM ordering and the phase
diagram in the magnetic field $H$-temperature $T$ plane is not yet known.
Interestingly, the magnetic susceptibility $\chi(T)$ below $T_{\rm N}$ is
found to be quite different in three earlier reports where single crystals
were grown in different ways. Comparing the $\chi(T)$ data reported in Refs.
Li2019_EuSn2As2 ; Arguilla2017_EuSn2As2 ; Chen2020_EuSn2As2 , the $ab$-plane
$\chi_{ab}(T)$ shows a clear Curie-like upturn at $T<10$ K and its magnitude
depends on the single-crystal growth procedure. In order to understand the
band topology and their magnetic field evolution in a magnetic TI, an
understanding of the nature of magnetic ordering and the magnetic $H$-$T$
phase diagram is required.

Herein we report room-temperature x-ray measurements of the crystallography,
zero applied magnetic field $H$ neutron-diffraction measurements of the
magnetic structure at $T\lesssim T_{\rm N}$, and measurements of the
magnetization $M(H,T)$, magnetic susceptibility $\chi(H,T)=M(H,T)/H$, heat
capacity $C_{\rm p}(H,T)$, and electrical resistivity $\rho(H,T)$. We find
that EuSn2As2 exhibits A-type AFM ordering below $T_{\rm N}=23.5(2)$ K with
the moments aligned ferromagnetically in the $ab$ plane with the spins in
adjacent planes along the $c$-axis aligned antiferromagnetically. Depending on
the crystal-growth procedure, different amounts of low-temperature Curie-
Weiss-like upturns in the low-field $\chi(T)$ are obtained, confirming
previous results. Low-field $\chi(T<T_{\rm N})$ measurements suggest the
presence of three-fold colliinear AFM domains. In magnetic fields applied in
the $ab$ plane the spins reorient or change the magnetic structure under a
small field of $H=0.12$ T. In addition to the drop in $\rho(T)$ on cooling
below $T_{\rm N}$, weak low-field transitions with a very small entropy change
were detected in the $\rho(H,T)$ data. The magnetic phase diagrams in the
$H$-$T$ plane associated with AFM-PM phase boundary constructed from the
observed magnetic, heat capacity and resistivity data are presented for both
$H\parallel ab$ and $H\parallel c$.

The experimental details are presented in Sec. II and the structural and
elemental composition characterization of our crystals in Sec. III. The
single-crystal neutron diffraction measurements of the magnetic structure are
presented in Sec. IV, the $\chi(H,T)$ and $M(H,T)$ measurements in Sec. V, the
$C_{\rm p}(H,T)$ measurements in Sec. VI, and the $\rho(H,T)$ measurements in
Sec. VII. A summary and the magnetic $H\parallel c$-$T$ and $H\parallel ab$
phase diagrams constructed from the various measurements are presented in Sec.
VIII.

##  II  Experimental details

EuSn2As2 single crystals were grown in three different ways using high-purity
elemental Eu (Ames Laboratory, 99.99%), Sn (Alfa Aesar, 99.999%) , and As
(Alfa Aesar, 99.9999%). One batch of crystals, labeled as “Crystal A” were
grown with a starting ratio of Eu:Sn:As = 1:3:20, as utilized previously
Li2019_EuSn2As2 . In this process, the starting elements were loaded into an
alumina crucible which was sealed in a silica tube under $\approx$ 1/4 atm of
Ar gas. The sealed tube was then heated to 1000 ∘C at a rate 100 ∘C/h where
the temperature was held for 12 h, followed by cooling to 750 ∘C/h at a
cooling rate of 2 ∘C/h. The silica tube containing the crucible was then
centrifuged to remove the crystals from the remaining flux. Clean shiny
crystals of typical dimensions $2\times 1\times 0.5$ mm3 were obtained by this
method.

Another batch of crystals denoted “Crystal B” was grown with a molar ratio
Eu:Sn:As = 1.1:2:20 where the silica tube containing the loaded alumina
crucible was initially heated to 600 ∘C at a rate 50 ∘C/h, held there for 24
h, and then heated to 1050 ∘C at a rate 50 ∘C/h followed by a dwell of 48 h at
that temperature. Finally the tube was cooled to 600 ∘C at a rate of 2 ∘C/h
and single crystals of typical size $1\times 1\times 0.5$ mm3 were obtained
after centrifuging the excess flux.

A third batch of EuSn2As2 single crystals denoted “Crystal C” were synthesized
with a stoichiometric starting ratio of the elements (Eu:Sn:As = 1:2:2) in an
alumina crucible that was sealed under $\approx 1/4$ atm of Ar in a silica
tube Arguilla2017_EuSn2As2 . In this growth, a sealed silica tube containing
the filled crucible was heated to 850 ∘C at a rate 50 ∘C/h and the temperature
held there for 12 h. Then the silica-tube assembly was slowly cooled to room
temperature in 100 h. The crystals were separated by mechanical cleaving. A
few thin single crystals were selected for measurements with dimensions of
$3\times 2\times 0.2$ mm3.

We grew EuSn2As2 crystals using the three methods described above in an
attempt to obtain the smallest paramagnetic Curie-like term at low $T$ which
we found to be the crystals C.

Room-temperature powder x-ray diffraction (XRD) measurements were carried out
on ground crystals using a Rigaku Geigerflex x-ray diffractometer with
Cu-$K_{\alpha}$ radiation. Crystal-structure analysis was performed by
Rietveld refinement using the FULLPROF software package Carvajal1993 . The
chemical composition and homogeneity of the crystals were measured with a JEOL
scanning electron microscope (SEM) equipped with an EDS (energy-dispersive
x-ray spectroscopy) analyzer. A Magnetic-Properties-Measurement System (MPMS)
from Quantum Design, Inc., was used for $\chi(T)$ and $M(H)$ measurements in
the $T$ range 1.8–300 K and with $H$ up to 5.5 T (1 T $\equiv 10^{4}$ Oe). A
physical properties measurement system (PPMS, Quantum Design, Inc.) was used
to measure $C_{\rm p}(H,T)$ and $\rho(H,T)$ in the $T$ range 1.8–300 K and in
the $H$ range 0–9 T. The $C_{\rm p}(H,T)$ was measured by a relaxation
technique using the PPMS.

Samples for electrical resistivity measurements were cut into rectangular
parallelepiped shapes with typical dimensions 2$\times$0.5$\times$0.1 mm3. The
long axis was parallel to the $ab$ plane and the short axis was along the $c$
axis. Contacts to the samples were made using silver wires soldered to the
sample with tin Tanataranisotropy ; SUST . For measurements in magnetic fields
in the $H\bot c$ direction, samples were attached to a plastic support using
Apiezon N grease. Four-probe resistivity measurements were performed using a
Quantum Design, Inc., Physical Property Measurement System (PPMS).

Single-crystal neutron-diffraction experiments were performed in zero applied
magnetic field using the TRIAX triple-axis spectrometer at the University of
Missouri Research Reactor (MURR). An incident neutron beam of energy
$E_{i}=30.5$ meV was directed at the sample using a pyrolytic graphite (PG)
monochromator. A pyrolytic-graphite (PG) analyzer was used to reduce the
background. Neutron wavelength harmonics were removed from the beam using PG
filters placed before the monochromator and between the sample and analyzer.
Beam divergence was limited using collimators before the monochromator;
between the monochromator and sample; sample and analyzer; and analyzer and
detector of $60^{\prime}-60^{\prime}-40^{\prime}-40^{\prime}$, respectively.

A 20 mg EuSn2As2 crystal (Crystal C) was mounted on the cold tip of an
Advanced Research Systems closed-cycle refrigerator with a base temperature of
4 K. The crystal was aligned in the $(HH0)$ and $(00L)$ planes. Due to strong
absorption by Eu, we could only identify a few peaks and the alignment relied
on the (1 1 0) and (0 0 15) Bragg peaks using the room-temperature EuSn2As2
lattice parameters $a=4.207(3)$ Å and $c=26.463(5)$ Å and the base-temperature
lattice parameters $a=4.178(3)$ Å and $c=26.233(5)$ Å.

##  III  Structural and elemental characterization

Figure 1: Unit cell and crystal structure of EuSn2As2. Eu atoms form a
hexagonal layer situated between two buckled SnAs honeycomb layers. The Eu
atoms are colored dark purple, the Sn atoms blue, and the As atoms green. The
crystal structure was drawn using the VESTA program Momma2011 . Figure 2:
Room-temperature XRD pattern of powdered EuSn2As2 Crystals C along with the
refinement of the data using the FULLPROF software. The red circles are the
experimental data points, the black solid line is the calculated refinement
based on the NaSn2As2 crystal structure (space group $R\overline{3}m$), the
blue solid curve is the difference between the experimental and calculated
diffraction patterns, and the green vertical bars represent the allowed Bragg
positions. The fit-quality parameters are given in the figure.

The crystal structure of EuSn2As2 is shown in Fig. 1 Momma2011 . The room-
temperature powder XRD pattern taken on ground EuSn2As2 single crystals
(Crystals C) is shown in Fig. 2. The experimental data were fitted using the
FULLPROF software package based on the symmetry of the NaSn2As2-type crystal
structure Asbrand1995  having a trigonal lattice with space group
$R\overline{3}m$. The best fit does not reveal any significant impurity
contribution in the studied crystals. The refined lattice parameters
$a=b=4.2006(3)$Å and $c=26.394(2)$ Å are in good agreement with reported
values Arguilla2017_EuSn2As2 . The SEM-EDS scans further confirm the
homogeneity of the crystals with average compositions EuSn1.95(4)As2.00(4) for
Crystal A, EuSn1.96(5)As2.00(5) for Crystal B, and EuSn1.95(5)As2.02(5) for
Crystal C.

##  IV  Single-crystal neutron-diffraction measurements

Figure 3: (a) Diffraction patterns along $(0~{}0~{}L)$ reciprocal-lattice
units of single-crystal EuSn2As2 (Crystal C) at 4 and 36 K as indicated. (b)
Difference between the $(0~{}0~{}L)$ patterns at 4 K and 36 K only showing
Bragg-diffraction reflections at $L=1.5+3n\ (n=1,2,3,\ldots)$.  Figure 4: (a)
Diffraction patterns along $(H~{}H~{}0)$ of single-crystal EuSn2As2 at 4 K and
36 K as indicated. The extra peaks are from the aluminum sample holder. (b)
The difference between the $(H~{}H~{}0)$ patterns taken at 4 K and 35 K. The
absence of emergent magnetic Bragg reflections at low temperature indicates
that $ab$-plane layers of Eu moments are aligned ferromagnetically in the $ab$
plane and are stacked antiferromagnetically along the $c$ axis.

Figure 3(a) shows diffraction scans of Crystal C along $(0~{}0~{}L)$ r.l.u.
(reciprocal-lattice units) at 4 and 35 K, where reflections at $L=1.5+3n\
(n=1,2,3,\ldots)$ emerge at $T=4$ K. For more clarity, Fig. 3(b) shows the
difference between these two scans, where within experimental uncertainty
there is no evidence for other reflections associated with a modulated
structure along the $c$ axis. A similar scan at 4 and 35 K with expected
magnetic reflections along $(1~{}1~{}L)$ (not shown) shows very weak new
signals at the magnetic Bragg reflections due to the decreasing form factor of
Eu2+ at larger momentum transfers $Q$.

Qualitatively, these newly emerging Bragg reflections indicate the doubling of
the unit cell along the $c$ axis, and establish that these Bragg reflections
are associated with A-type AFM ordering with propagation vector
$\vec{\tau}=\left(00\frac{1}{2}\right)$ r.l.u. The magnetic structure consists
of $ab$-plane layers of Eu spins with the moments ferromagnetically aligned in
the $ab$ plane that are stacked antiferromagnetically along the $c$ axis. The
in-plane direction of the FM moments cannot be determined from neutron
diffraction. This AFM structure is equivalent to a $c$-axis helix with a turn
angle between FM-aligned layers of $180^{\circ}$. Our model calculations of
the magnetic structure, shown in the inset of Fig. 5, predicts that whole and
half-integer $(0~{}0~{}L)$ reflections are allowed, but are too weak to be
detected with our small crystal. The calculations show that the most prominent
reflections are the $L=1.5+3n\ (n=1,2,3,\ldots)$ as observed experimentally.
Using this model we estimate the average ordered magnetic moment at $T=4$ K to
be $g\langle S\rangle=(6.7\pm 0.5)~{}\mu_{\rm B}$, where $g$ is the
spectroscopic splitting factor of the Eu spin and $\langle S\rangle$ is the
expectation value of the spin angular momentum in units of $\hbar$. This value
agrees within the error with the value of 7 $\mu_{\rm B}$ obtained using $g=2$
and $S=7/2$. We note that our modeling precludes a helical structure (in zero
magnetic field) assuming that each of the six FM layers is rotated by 60
degrees with respect to the previous layer, as such a structure predicts other
prominent half integer $(0~{}0~{}L)$ peaks that are not observed
experimentally.

Figure 4 shows that no additional magnetic peaks are observed in the
difference between scans taken at 4 and 35 K along $(H~{}H~{}0)$, consistent
with a single AFM propagation vector $\vec{\tau}$. Other scans along
$(\mbox{0.5~{}0.5~{}}~{}L)$ and ($H$ $H$ 4.5) (not shown) are consistent with
a single $\vec{\tau}$. Figure 5 shows the integrated intensity versus $T$ of
the (0 0 4.5) magnetic Bragg reflection. The solid curve is a power-law fit
$I(T)=C(1-T/T_{\rm N})^{2\beta}$ yielding $T_{\rm N}=23.5(2)$ K and
$\beta=0.33(1)$, where the latter value is consistent with three-dimensional
ordering of Heisenberg spins. The magnetic structure depicted in Figure 5
corresponds to half the magnetic unit cell.

Figure 5: Integrated intensity as a function of temperature $T$ of the (0 0
4.5) rlu magnetic Bragg reflection. The solid curve is a phenomenological
power-law fit by $I(T)=C|1-T/T_{\rm N}|^{2\beta}$, yielding $T_{\rm
N}=(23.5\pm 0.2)$ K and $\beta=0.33(1)$. The magnetic structure of the Eu2+
ordered moments depicts half of the AFM unit cell. The neutron-diffraction
data are not sufficient for determining the direction of the
ferromagnetically-aligned moment in each $ab$ plane.

##  V  Magnetic Susceptibility and Magnetization Measurements

###  V.1 Magnetic susceptibility in the high-temperature paramagnetic state

Figure 6: Inverse magnetic susceptibility $\chi^{-1}(T)$ of EuSn2As2 for (a)
$H=0.1$ T and (b) $H=1$ T, each with $H\parallel ab$ and $H\parallel c$, along
with the respective fits by the Curie-Weiss law (1).

The inverse susceptibilities $1/\chi_{ab}(T)$ and $1/\chi_{c}(T)$ of Crystal C
measured in $H=0.1$ T are plotted in Fig. 6 for applied fields of (a) 0.1 and
(b) 1 T. The data were analyzed in the paramagnetic state above $T_{\rm N}$
using the Curie-Weiss (CW) law

$\displaystyle\chi_{\alpha}(T)=\frac{C_{\alpha}}{T-\theta_{\rm
p\alpha}}\quad(\alpha=ab,\ c),$ (1)

where $\theta_{\rm p\alpha}$ is the Weiss temperature with $H$ in the $\alpha$
direction. The Curie constant $C_{\alpha}$ is given by

$\displaystyle C_{\alpha}$ $\displaystyle=$ $\displaystyle\frac{N_{\rm
A}{g_{\alpha}}^{2}S(S+1)\mu^{2}_{\rm B}}{3k_{\rm B}}=\frac{N_{\rm
A}\mu^{2}_{\rm{eff,\alpha}}}{3k_{\rm B}},$ (2) $\displaystyle\mu_{\rm
eff,\alpha}^{2}$ $\displaystyle=$ $\displaystyle
g_{\alpha}^{2}S(S+1)\mu^{2}_{\rm B}.$

where $N_{\rm A}$ is Avogadro’s number, $g_{\alpha}$ is the spectroscopic-
splitting factor ($g$ factor), $S=7/2$ is the Eu2+ spin quantum number,
$k_{\rm B}$ is Boltzmann’s constant, and $\mu_{\rm eff\alpha}$ is the
effective moment of an Eu spin. It is evident from Fig. 6 that $\chi^{-1}(T)$
for $H=0.1$ and 1 T are both well described by the CW law 1) in the $T$ range
60–300 K for both $H\parallel ab$ and $H\parallel c$. The fitted parameters
are summarized in Table 1. The $\mu_{\rm{eff}\alpha}$ values obtained from the
$C_{\alpha}$ values, $\mu_{\rm{eff\alpha}}$ ($\mu_{\rm B}/{\rm
Eu})=\sqrt{8C_{\alpha}}$, are close to the value 7.94 $\mu_{\rm B}$/Eu
expected for Eu2+ spins with $S$ = 7/2 and $g=2$. The presence of dominant FM
interactions between the Eu spins is reflected in the positive values of
$\theta_{\rm{p}\alpha}$. The FM interactions are responsible for the FM
alignment of the Eu spins in the $ab$ plane of the A-type AFM structure. The
values of $f_{\alpha}=\theta_{\rm p\alpha}/T_{\rm N}$ are listed for later
use.

Table 1:  Listed are the applied magnetic field $H$, field direction, molar Curie constant $C_{\alpha}$ ($\alpha=ab,c$), effective moment per Eu spin $\mu{\rm{}_{eff\alpha}(\mu_{B}/Eu)}=\sqrt{8C_{\alpha}}$ and Weiss temperature $\theta\rm_{p\alpha}$ obtained from the $\chi^{-1}(T)$ data for $H=0.1$ and 1 T in Fig. 6 from fits by Eq. (1) in the temperature range 60–300 K. Also listed are the ratios $f_{\alpha}=\theta_{\rm p\alpha}/T_{\rm N}$, where $T_{\rm N}=24$ K. $H$ | Field | $C_{\alpha}$ | $\mu_{\rm eff\alpha}$ | $\theta_{\rm p\alpha}$ | $f_{\alpha}$
---|---|---|---|---|---
(T) | direction |  | $\rm{\left(\frac{cm^{3}K}{mol}\right)}$ | $\rm{\left(\frac{\mu_{B}}{mol}\right)}$ | (K)
0.1 | $H\parallel ab$ | 7.84(1) | 7.91(1) | 20.8(1) | 0.87
0.1 | $H\parallel c$ | 7.91(1) | 7.95(1) | 19.2(2) | 0.80
1 | $H\parallel ab$ | 7.85(1) | 7.92(1) | 20.5(1) | 0.85
1 | $H\parallel c$ | 7.90(1) | 7.97(1) | 18.3(1) | 0.76

###  V.2  Low-temperature magnetic susceptibility

Figure 7: Temperature $T$ dependence of the in-plane ($H\parallel ab$)
magnetic susceptibility $\chi\equiv M/H$ measured in $H=0.01$ T for three
crystals (Crystal A, Crystal B, and Crystal C) grown in three different ways.
As discussed in Sec. II, the concentration of magnetic defects is expected to
decrease from Crystal A to Crystal B to Crystal C, as observed.

The temperature-dependent $\chi_{ab}(T)=M_{ab}(T)/H$ ($H\parallel ab$)
measured at $H=0.01$ T for the three differently-grown crystals A, B, and C
are shown in Fig. 7. Our data are consistent with those reported earlier
Li2019_EuSn2As2 ; Arguilla2017_EuSn2As2 ; Chen2020_EuSn2As2 , where EuSn2As2
crystals exhibit an AFM transition at $T_{\rm N}=24$ K revealed by a sharp
peak in $\chi_{ab}(T)$. In addition, we find low-$T$ Curie-like upturns in our
data below 10 K, which is strongest for Crystal A, moderate for Crystal B and
small for Crystal C. It was previously suggested that the upturn in EuSn2As2
crystals arises from an in-plane ferromagnetic component associated with
canting of the magnetic moments Li2019_EuSn2As2 . However, in this case,
$\chi_{ab}(T)$ should not be sample-dependent as found here. In order to
obtain the intrinsic $\chi(T)$ at low $T$, the Curie-like contribution must be
corrected for as discussed below.

Figure 8: (a) Temperature dependence of the zero-field-cooled in-plane
($\chi_{ab},H\parallel ab$) and out-of-plane ($\chi_{c},H\parallel c$)
magnetic susceptibilities $\chi_{\alpha}=M_{\alpha}/H$ of EuSn2As2 (Crystal C)
measured in a magnetic field $H=0.01$ T. The values of the anisotropic
susceptibilities at $T_{\rm N}=24$ K are $\chi_{ab}(T_{\rm N})=0.904~{}{\rm
cm^{3}/mol}$ and $\chi_{c}(T_{\rm N})=0.762~{}{\rm cm^{3}/mol}$. (b) The
Heisenberg susceptibility $\chi_{J}(T)$ in the paramagnetic state ($T>T_{\rm
N}$) calculated by taking the spherical average of the data in panel (a) at
$T>T_{\rm N}$ as shown by the green symbols on a scale normalized to that at
$T=T_{\rm N}$. $\chi_{ab}(T)$ (filled black circles) and $\chi_{c}(T)$ (filled
red squares) with $T\leq T_{\rm N}$ are obtained by vertical translations of
the respective $\chi$ data in panel (a) to match the $\chi_{J}(T=T_{\rm N})$
data. The solid green curve gives the fit by Eq. (6) to the $\chi_{ab}(T\leq
T_{\rm N})$ data to obtain the low-$T$ magnetic-impurity Curie-Weiss
contribution. The open circles give $\chi_{ab}/\chi(T_{\rm N})$ vs $T$ after
subtraction of the Curie-Weiss contribution which is consistent with A-type
AFM ordering with the ordered moments ferromagnetically aligned in the $ab$
plane with the moments in adjacent planes along the $c$ axis aligned
antiferromagnetically.

The $\chi(T)$ was measured for an EuSn2As2 crystal C under both zero-field-
cooled (ZFC) and field-cooled (FC) protocols and no thermal hysteresis was
observed down to the lowest measured temperature of 1.8 K. Figure 8(a) shows
the $T$ dependence of the ZFC in-plane ($\chi_{ab}$) and out-of-plane
($\chi_{c}$) data for Crystal C measured in a small $H=0.01$ T (100 Oe) field.
A sharp peak associated with the AFM transition is observed at $T_{\rm N}=24$
K for both $\chi_{ab}$ and $\chi_{c}$, in agreement with the earlier data. The
low-$T$ upturn in the $\chi_{ab}$ data is larger than in the $\chi_{c}$ data,
suggesting the presence of XY anisotropy for the magnetic impurity.

The $\chi_{ab}(T)$ and $\chi_{c}(T)$ data in Fig. 8(a) exhibit significant
anisotropy in the paramagnetic temperature range $T_{\rm N}<T\lesssim 40$ K
that likely arises from differences in the spin-fluctuation amplitudes in
these two directions. For further analysis, the data were spherically averaged
over this $T$ range yielding

$\displaystyle\chi_{J}(T\geq T_{\rm
N})=\frac{1}{3}[2\chi_{ab}(T)+\chi_{c}(T)].$ (3)

These $\chi_{J}(T)$ data normalized to $\chi_{J}(T=T_{\rm N})$ are plotted in
Fig. 8(b), where the data for $\chi_{ab}$ and $\chi_{c}$ for $T\leq T_{\rm N}$
are obtained by vertical translations of the respective $\chi$ data to agree
with $\chi_{J}(T=T_{\rm N})$. The $\chi_{c}(T)$ below $T_{\rm N}$ is nearly
independent of $T$ as expected from molecular-field theory (MFT) for
crystallographically-equivalent Heisenberg ordered moments aligned in the $ab$
plane Johnston2012 ; Johnston2015 . On the other hand, the $\chi_{ab}(T)$ data
decrease rapidly and attain a minimum value yielding the ratio $\chi_{ab,\rm
min}/\chi(T_{\rm N})\approx 1/2$, which is indeed the value expected for an
A-type Heisenberg antiferromagnet on a triangular lattice with moments aligned
in the $ab$ plane within equally-populated three-fold domains
Pakhira2021_EuMg2Bi2 .

We utilize the MFT in Refs. Johnston2012 ; Johnston2015  to calculate the low-
field $\chi_{ab}(T\leq T_{\rm N})$. Taking $f=\theta_{\rm p\,ave}/T_{\rm
N}=0.83$ for $H=0.1$ T from Table 1, we have

$\frac{\chi_{Jab}(T\leq T_{\rm N})}{\chi_{J}(T_{\rm
N})}=\frac{(1+\tau^{*}+2f+4B^{*})(1-f)/2}{(\tau^{*}+B^{*})(1+B^{*})-(f+B^{*})^{2}},$
(4a) where $B^{*}=2(1-f)\cos(kd)\,[1+\cos(kd)]-f,$ (4b) $t=\frac{T}{T_{\rm
N}},\quad\tau^{*}(t)=\frac{(S+1)t}{3B^{\prime}_{S}(y_{0})},\quad
y_{0}=\frac{3\bar{\mu}_{0}}{(S+1)t},$ (4c) the ordered moment versus $T$ for
$H=0$ is denoted by $\mu_{0}$, the reduced ordered moment
$\bar{\mu}_{0}=\mu_{0}/\mu_{\rm sat}$ is determined by numerically solving the
self-consistency equation $\bar{\mu}_{0}=B_{S}(y_{0}),$ (4d)
$B^{\prime}_{S}(y_{0})=[dB_{S}(y)/dy]|_{y=y_{0}}$, and the Brillouin function
$B_{S}(y)$ is $B_{S}(y)=\frac{1}{2S}\left\\{(2S+1){\rm
coth}\left[(2S+1)\frac{y}{2}\right]-{\rm
coth}\left(\frac{y}{2}\right)\right\\}.$ (4e)

For $T=0$, using Eq. (4a) yields Johnston2012 ; Johnston2015

$\frac{\chi_{Jab}(T=0)}{\chi_{Jab}(T_{\rm N})}=\frac{1}{2[1+2~{}{\rm
cos}(kd)+2~{}{\rm cos}^{2}(kd)]}$ (5)

Thus for an A-type antiferromagnet with $kd\to 180^{\circ}$ and the moments
aligned in the $ab$ plane, one obtains a value of 1/2 for this ratio, in
agreement with the above value $\chi_{ab,\rm min}/\chi(T_{\rm N})\approx 1/2$.

Now we fit $\chi_{ab}(T\leq T_{\rm N})$ in Fig. 8(b) by the sum of the AFM
contribution described by Eqs. (4) and a Curie-Weiss magnetic-defect
contribution according to

$\displaystyle\chi_{ab}(T\leq T_{\rm N})=\chi_{Jab}(T\leq T_{\rm
N})+\frac{C_{\rm imp}}{T-\theta_{\rm p\,imp}},$ (6)

where $C_{\rm imp}$ and $\theta_{\rm p\,imp}$ are the Curie constant and Weiss
temperature associated with the magnetic impurities/defects discussed
previously. The best fit of the data by Eq. (6) is shown in Fig. 8(b) as the
green solid curve with $C_{\rm imp}=0.0874$ cm3-K/mol f.u. (f.u. means formula
unit) and $\theta_{\rm p\,imp}=1.4$ K. The blue open circles show the data
after subtracting the low-$T$ CW contribution. The $C_{\rm imp}$ value is only
$\approx 1.1$% of the value $\rm{7.88\,cm^{3}}$ K/mol for Eu2+ with $g=2$ and
$S=7/2$. From Fig. 8(b), after correcting for the CW contribution the data
exhibit $\chi_{ab}(T=1.8~{}{\rm K})/\chi_{ab}(T=T_{\rm N})\approx 1/2$, as
expected for A-type AFM order with turn angle $kd\to 180^{\circ}$.

Figure 9: Temperature dependence of the magnetic susceptibilities (a)
$\chi_{ab}(H)\equiv M_{ab}(H)/H$ and (b) $\chi_{c}(H)\equiv M_{c}(H)/H$ for
Crystal C normalized by the respective values at $T=T_{\rm N}(H)$ in applied
magnetic fields $H$ from 0.01 T to 0.15 T. The $\chi_{ab}(H)$ data are
strongly dependent on $H$, while the $\chi_{c}(H)$ data are almost independent
of $H$. $T_{\rm N}$ is seen to decrease by about 40% at the highest field. The
data suggest a smooth evolution of the magnetic structure on increasing $H$
from 0.01 to $\sim 0.15$T.

The $\chi_{ab}(T)$ and $\chi_{c}(T)$ data normalized by the respective values
at $T_{\rm N}$ for different applied magnetic fields are shown in Figs. 9(a)
and 9(b), respectively. Neglecting the low-$T$ upturns, the extrapolated
$\chi_{ab}(T=1.8~{}{\rm K})/\chi_{ab}(T=T_{\rm N})$ value shows a smooth
increase from $\approx 0.5$ as discussed above to larger values with
increasing $H_{ab}$ and saturates at $\approx 1$, while the ratio
$\chi_{c}(T=1.8~{}{\rm K})/\chi_{c}(T=T_{\rm N})$ is nearly independent of $T$
and $H$ for $T\leq T_{\rm N}$.

These $\chi_{\alpha}(H,T\leq T_{\rm N})$ results are similar to those we
observed previously for the trigonal compound ${\rm EuMg_{2}Bi_{2}}$
containing stacked triangular lattices of Eu2+ spins $S=7/2$ Pakhira2020
where the magnetic structure in $H=0$ was also found by neutron-diffraction
measurements to be A-type AFM with the Eu moments aligned in the $ab$ plane
Pakhira2021 . In particular, we find here that with increasing $ab$-plane
field $M_{ab}$ becomes nearly independent of $H$ for $H\gtrsim 0.125$ T. A
similar result was interpreted in Ref. Pakhira2021  as possibly arising from
the spins rotating with increasing $H_{ab}$ to be perpendicular to $H$, which
gives $\chi_{ab}(T\leq T_{\rm N})/\chi(T_{\rm N})\approx\chi_{c}(T\leq T_{\rm
N})/\chi(T_{\rm N})\approx 1$ as also observed here.

The $ab$-plane anisotropy arising from magnetic-dipole interactions between
moments on a triangular lattice is small and with a strong preference of these
interactions for $ab$-plane moment alignment over $c$-axis alignment
Johnston2016 . For an isolated layer of Eu spins with the moments aligned
ferromagnetically in the $ab$ plane, $ab$-plane alignment is more stable than
$c$-axis alignment by $\Delta E=0.294$ meV, which corresponds to a magnetic
field anisotropy $H_{\rm anis}=\Delta E/(7~{}\mu_{\rm B})=7.25$ kOe. Thus the
small $ab$-plane magnetic field of 0.15 T discussed above is insufficient to
cause a spin-flop transition with the moments having a $c$-axis component at
$T\ll T_{\rm N}$.

Therefore we propose that with increasing $ab$-plane field $H_{ab}$ at $T=2$
K, the above three AFM domains initially at 120∘ to each other as dictated by
the Eu trigonal point symmetry $\bar{3}m$ begin to align perpendicular to
$H_{ab}$, which according to Fig. 9(a) is completed at $H_{ab}=0.15$ T. Then
with a further increase in $H_{ab}$ the collinear spins in each domain
progressively cant towards $H_{ab}$ until they reach FM alignment at the
critical field $H_{ab}^{\rm c}=3.90$ T listed in Table 2 Johnston2017 .

###  V.3 Magnetization versus applied magnetic field isotherms

Figure 10: Four-quadrant magnetization versus field $M(H)$ hysteresis data for
Crystal C with $ab$-plane and $c$-axis magnetic-field orientations at $T=2$ K.
No hysteresis is observed for either field direction.  Figure 11: (a, b)
Magnetization $M$ isotherms for Crystal C as a function of applied magnetic
field $H$. Low-field $H$-dependences of (c, d) $M/H$, and (e, f) $(dM/dH)$ at
different temperatures for $H\parallel ab$ and $H\parallel c$, respectively.

The field dependence of the isothermal magnetization $M$ was measured at
different temperatures for both in-plane and out-of-plane field directions.
Figure 10 shows four-quadant $M(H)$ hysteresis curves recorded at $T=2$ K for
$H\parallel ab$ and $H\parallel c$. No evidence of magnetic hysteresis was
observed. On the scale of the figure, the $M_{ab}(H)$ and $M_{c}(H)$ data are
linear in field and then saturate to values of $\approx 6.8~{}\mu_{\rm B}$ and
$\approx 7~{}\mu_{\rm B}$ at the critical fields $H_{ab}^{\rm c}=3.90(5)$ T
and $H_{c}^{\rm c}=5.00(5)$ T, respectively. The $T$ dependence of
$H_{ab}^{\rm c}$ and $H_{ab}^{\rm c}$ obtained from discontinuities in slope
of $M_{\alpha}(H)$ isotherms in Figs. 11(a) and 11(b), respectively, are
listed in Table 2.

Table 2:  The in-plane and out-of-plane critical fields $H_{ab}^{\rm c}$ and $H_{c}^{\rm c}$, respectively, at different temperatures obtained from Figs. 11(a) and 11(b), respectively. The critical field for a given field direction is the field for which the slope of $M(H)$ shows a discontinuity with increasing field. $T$ (K) |  $H_{ab}^{\rm c}$ (T) |  $H_{c}^{\rm c}$ (T)
---|---|---
2 | 3.63(7) | 4.70(7)
5 | 3.56(7) | 4.60(7)
10 | 3.28(7) | 4.32(7)
15 | 3.0(1) | 3.7(1)
20 | 2.5(1) | 3.0(1)

The $M_{ab}(H)$ and $M_{c}(H)$ data measured at different temperatures are
plotted for $H=0$–5.5 T in Figs. 11(a) and 11(b), respectively, and $M/H$ vs
$H$ for $H\parallel ab$ and $H\parallel c$ in the low-field region $H=0$–0.3 T
in Figs. 11(c) and 11(d), respectively. The $M_{c}(H)/H$ vs $H$ data are
almost independent of $H$ except for $H\rightarrow 0$, which is attributed to
the remanent field in our magnetometer.

The $M_{ab}(H)$ data in Figs. 11(c, e) exhibit a sharp transition-like feature
at $H=0.02$ T at the lowest temperature $T=2$ K, whereas for the $M_{c}(H)$
data this signature is weak as revealed in Figs. 11(d, f). No signature of
this anomaly is observed for $T\geq 5$ K. This feature is likely due to the
field-induced saturation of the magnetic defects. These saturations give rise
to a significant enhancement in the lowest-$T$ $\chi(T)$ causing a rapid
upturn in the low-field region. In this case the value $H\approx 0.02$ T is
the critical field value beyond which the magnetic defects tend to saturate.
Since the defect contribution is larger in $M_{ab}$ compared to $M_{c}$, the
jump in $dM_{ab}(H)/dH$ observed at $T=2$ K is sharper for the $ab$-plane data
than for the $c$-axis data.

The $M_{ab}(H)/H$ vs $H$ data at $T=2\,{\rm K}$ in Fig. 11(c) deviate from a
constant value for $H\lesssim 0.2$ T. This feature is reflected more clearly
in the derivative plot $dM_{ab}(H)/dH$ vs $H$ in Fig. 11(e). A clear maximum
is observed in $dM_{ab}(H)/dH$ at $H_{\rm p}\approx 0.12$ T for $T=2$ K, where
the maximum shifts to lower $H$ with increasing $T$, whereas no such feature
is observed in the $dM_{c}(H)/dH$ data in Fig. 11(f) in that $H$ region. This
feature in the $dM_{ab}(H)/dH$ data only persists up to $T\approx T_{\rm N}$,
indicating that it likely arises from a field-induced change in the A-type
ground-state spin configuration in $H=0$ as discussed at the end of the
previous section. Note that $H_{\rm p}(T=2\,{\rm K})$ is much smaller that the
critical field $H_{ab}^{\rm c}(T=2\,{\rm K})=3.90$ T in Table 2.

##  VI  Heat capacity measurements

Figure 12: Temperature dependence of zero-field heat capacity $C_{\rm p}$ for
Crystal C along with a fit by Eqs. (7) in the temperature range 50–300 K. The
data are fitted well for $T\geq 60$ K. The bump in the data at $\approx 290$ K
is an experimental artifact. Figure 13: Temperature $T$ dependence of (a) the
magnetic component of the heat capacity $C_{\rm{mag}}$, (b) $C_{\rm{mag}}/T$ ,
and (c) the magnetic entropy $S_{\rm{mag}}$ obtained from the $C_{\rm{mag}}/T$
data in panel (b) using Eq. (10) for EuSn2As2 (Crystal C). The solid black
lines in panels (a) and (b) represent the predictions of MFT in Eq. (9) for
$S=7/2$ and $T_{\rm N}=24$ K. $S_{\rm{mag}}$ attains its saturation value
$R\ln(2S+1)=R\ln 8$ by $T\approx 60$ K, demonstrating the presence of AFM
correlations above $T_{\rm N}=24$ K.

The temperature dependence of the zero-field heat capacity $C_{\rm p}(T)$ for
an EuSn2As2 crystal C is shown in Fig. 12 in the temperature range 1.8–300 K.
A $\lambda$-type peak is observed at $T_{\rm N}=24$ K associated with the AFM
ordering. The $C_{\rm p}(T)$ value of $\approx 123$ J/mol K at $T=300$ K is
close to the classical high-$T$ Dulong-Petit limit $C_{\rm V}=3nR=124.71$
J/mol K associated with lattice vibrations for this compound, where $n=5$ is
the number of atoms per formula unit and $R$ is the molar-gas constant. The
bump in the $C_{\rm p}(T)$ around $T=290$ K is an instrumental artifact
arising from melting of the Apiezon N grease used for making thermal contact
between the crystal and sample platform of the heat capacity puck. We fitted
the $C_{\rm p}(T)$ data in the measured temperature range using the sum of a
Sommerfeld electronic heat capacity contribution $\gamma T$ and the Debye
model for the lattice heat capacity given by

$\displaystyle C_{\rm p}(T)$ $\displaystyle=$ $\displaystyle\gamma
T+n~{}C_{\rm V\,Debye}(T/\Theta_{\rm D}),$ (7a) $\displaystyle C_{\rm
V\,Debye}(T)$ $\displaystyle=$ $\displaystyle 9R\left(\frac{T}{\Theta_{\rm
D}}\right)^{3}\int_{0}^{\Theta_{\rm D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx,$
(7b)

where $\gamma$ is Sommerfeld electronic heat capacity coefficient and
$\Theta_{\rm D}$ is the Debye temperature. The lattice heat capacity $C_{\rm
V\,Debye}(T)$ in Eq. (7b) was parameterized using an accurate Padé approximant
function Goetsch_2012 . A fit of the data in Fig. 12 in the temperature range
60–300 K by Eqs. (7) yielded $\gamma$ = 5.9(6) mJ/(mol K2) and $\Theta_{\rm
D}$ = 210(1) K.

The magnetic contribution $C_{\rm{mag}}(T)$ to the heat capacity is estimated
by subtracting the lattice contribution given by Eq. (7b) with the fitted
value of $\Theta_{\rm D}$ from the zero-field $C_{\rm p}(T)$ data. The
$C_{\rm{mag}}(T)$ is plotted in Fig. 13(a) which shows a clear $\lambda$-type
peak at $T_{\rm N}=24$ K followed by an extended tail in the higher-$T$
region, indicating the persistence of dynamic short-range AFM ordering at
$T>T_{\rm N}$. This feature has also been observed in other Eu2+ $S=7/2$ AFM
compounds, e.g., Pakhira2020_EuMg2Bi2 ; Sangeetha_EuCo2As2_2018 ;
Sangeetha_EuCo2P2_2016 ; Sangeetha_EuNi2As2_2019 .

The origin of the strong AFM correlations in EuSn2As2 above $T_{\rm N}$ may be
traced to the very soft response of $M_{ab}(H)$ at low fields in Figs. 11(c,
e). Due to the presence of the strong short-range AFM ordering above $T_{\rm
N}$, the heat-capacity jump $\Delta C_{\rm{mag}}(T_{\rm N})\approx 10$ J/mol K
in Fig. 13(a) is much smaller than $\Delta C_{\rm{mag}}(T_{\rm N})$ for
$S=7/2$ expected from MFT given by Johnston2015

$\displaystyle\Delta C_{\rm{}_{mag}}^{\rm
MFT}=R\frac{5S(S+1)}{1+2S+2S^{2}}=20.1~{}{\rm J/(mol~{}K)}.$ (8)

The MFT prediction of $C_{\rm{mag}}(T)$ is given by Johnston2015

$\displaystyle\frac{C_{\rm{mag}}(t)}{R}=\frac{3S\bar{\mu}_{0}^{2}(t)}{(S+1)t[\frac{(S+1)t}{3B^{\prime}_{S}(t)}-1]},$
(9)

where the parameters are defined in Eqs. (4). Figures 13(a) and 13(b) depict
$C_{\rm{mag}}$ and $C_{\rm{mag}}/T$ vs $T$ along with the MFT predictions.
Dramatic differences between the MFT predictions and the experimental data are
seen, again attributed to the strong influence of dynamic short-range AFM
fluctuations.

The magnetic entropy $S_{\rm{mag}}(T)$ for EuSn2As2 is obtained from the
$C_{\rm{mag}}(T)/T$ data in Fig. 13(b) according to

$\displaystyle S_{\rm
mag}(T)=\int_{0}^{T}\frac{C_{\rm{mag}}(T^{\prime})}{T^{\prime}}dT^{\prime},$
(10)

where the $C_{\rm{mag}}(T)/T$ data in the $T$ range 0–1.8 K are extrapolated
using the MFT prediction shown in Fig. 13(b). As can be seen from Fig. 13(c),
the temperature dependence of zero-field $S_{\rm{mag}}$ saturates at a value
of $\approx 18$ J/mol K, which is close to the theoretical high-$T$ limit
$S_{\rm mag}=R{\rm ln}(2S+1)=17.29$ J/(mol K) expected for EuSn2As2 with
$S=7/2$. The $S_{\rm{mag}}(T)$ saturates around $T=60$ K which is more than
twice $T_{\rm N}=24$ K of the compound. This again reflects the strong dynamic
short-range AFM ordering that occurs above $T_{\rm N}$. Similar observations
of entropy release above $T_{\rm N}$ were found for other Eu2+ and Gd3+
compounds with spins 7/2 in Refs. Sangeetha_EuCo2As2_2018 ;
Sangeetha_EuCo2P2_2016 ; Sangeetha_EuNi2As2_2019 ; Anand2015 ; Anand2014 ;
Singh2009 ; Pandey2017 ; Johnston2021 .

Figure 14: $C_{\rm p}(T)$ of Crystal C measured for different applied magnetic
fields with $H\parallel c$ showing the suppression of the AFM ordering
temperature $T_{\rm N}$ with increasing $H$. Inset: expanded plot of the data
between 15 and 27 K.

The $C_{\rm p}(T)$ data measured at different applied magnetic fields with
$H\parallel c$ are plotted in Fig. 14. The data show that $T_{\rm N}$ shifts
to lower $T$ with increasing $H$ as expected. These data are used later in
Sec. VIII to help construct the magnetic phase diagram in the $H$-$T$ plane.

##  VII  Electrical resistivity measurements

Figure 15: In-plane electrical resistivity of EuSn2As2, presented using a
normalized $\rho/\rho$(300 K) scale. The $T$ dependence over the whole $T$
range in $H=0$ is shown in the top-left inset. The main panel zooms in on the
$T$ range of the main AFM transition, with measurements made in $H{\parallel
ab}$ of 0 T (black, top curve), 1 T (red), 2 T (green), 3 T (blue), 4 T (cyan)
and 6 T (magenta). Note the negative magnetoresistance in the $T$ range below
40 K and the crossover to positive magnetoresistance above. The bottom-right
inset shows a zoom of the second feature in the $T$-dependent resistivity, a
slight decrease below $T_{L}\sim 6$ K in $H=0$ (black curve). This feature
broadens in $H=0.05$ T (red curve) and is suppressed with $H=0.1$ T (green
curve) and 0.2 T (blue curve). Figure 16: Temperature $T$ derivative
$d[\rho(T)/\rho(300~{}{\rm K})]/dT$ of EuSn2As2 up $T=50$ K. The derivative
shows a jump at $T_{\rm N}\approx 23$ K in zero field, with the maximum
denoted by a filled red circle. The jump remains sharp in a magnetic field
$H\parallel ab$ of 1 T (maroon curve), and an additional sharp peak occurs for
2 T and a step at 3 T with a broad crossover background, while the sharp peak
increases in magnitude. The feature at $T_{\rm N}$ further broadens in 4 T.
The inset shows a suppression of the zero-field anomaly in the derivative at 6
K with magnetic field, and magenta diamonds show the position of the feature
onset.  Figure 17: Temperature dependence of the normalized resistivity
$\rho(T)/\rho(300~{}\rm K)$of EuSn2As2 measured in magnetic fields $H\parallel
c$ as indicated. The data for successive fields are offset from each other for
clarity, where the bottom curve is for $H=0$ and the top one is for $H=9$ T.
Figure 18: Temperature dependence of the normalized resistivity derivative
$d[\rho(T)/\rho(300~{}{\rm K})]/dT$ of EuSn2As2 measured in magnetic fields
$H\parallel c$ as indicated. The filled diamonds and open pentagons at
temperatures $T_{\rm A}$ represent potential transitions of unknown origin.
The filled blue triangles represent maxima $d[\rho(T)/\rho(300~{}{\rm K})]/dT$
at $T_{\rm N}(H)$.

Electrical resistivity $\rho$ measurements were performed on a number of
samples from batch C; however, $\rho(T,H)$ of only two samples were studied in
detail. They revealed consistent behavior, so we present data for only one of
them. The resistivity value at room temperature showed a notable variation,
suggesting an important role of cracks Tanataranisotropy . The average
$\rho(300$ K) as determined on six samples was 131 $\mu\Omega$ cm with a
standard deviation of 46 $\mu\Omega$ cm. This value is nearly three times
smaller than reported previously Arguilla2017_EuSn2As2 ; Chen2020_EuSn2As2 .
Due to this large uncertainty we present normalized resistivity data
$\rho/\rho(300~{}\rm K)$.

To put the magnitude of $\rho(300$ K) in perspective, a similar value of 150
$\mu\Omega$ cm was found in the optimally-doped iron-based superconductors
Tanatarpseudogap . Within the error bars, a somewhat smaller but comparable
value of $(120\pm 30)~{}\mu\Omega$ cm was found in SrCo2As2 Sangeetha2019 .
The $\rho(300~{}{\rm K})$ value of EuSn2As2 and the monotonically-decreasing
resistivity on cooling (see the top left inset in Fig. 15) suggest a metallic
character of the electronic transport with no contribution of Dirac cone
carriers. This is natural considering that the Dirac crossing is located 0.1
eV above the Fermi energy Li2019_EuSn2As2 . The metallic character of
transport is in line with the very weak $T$ dependence of the Hall effect
Chen2020_EuSn2As2  suggesting hole carriers with concentration $p=3.6\times
10^{20}$ cm-3 at 2 K and with less than a 10% change on increasing $T$ to 200
K.

The $T$-dependent $\rho$ of EuSn2As2 was measured in $H=0$ and in constant
nonzero $H$ applied parallel to the conducting $ab$ plane. The resistivity in
$H=0$ over the whole $T$ range below room temperature is shown in the top-left
inset of Fig. 15. The resistivity is close to a $T$-linear dependence at high
$T$, as normally expected of metals BG , followed by a slight upturn below
$T\approx 50$ K and a sharp decrease below $T_{\rm N}\approx 24$ K as observed
previously Chen2020_EuSn2As2 . This dependence is consistent with a
significant contribution of magnetic scattering in the paramagnetic state
above $T_{\rm N}$ MagneticScattering , revealed also in the negative
magnetoresistance at temperatures at least as high as 40 K associated with
spin correlations as also observed in the $C_{\rm p}(T)$ data in Figs. 13(a)
and 13(b).

Below $T_{\rm N}$ the resistivity in $H=0$ shows a sharp decrease with a clear
anomaly in the $T$-dependent derivative $d\rho/dT$ in Fig. 16, as expected for
magnetic materials FisherdrhodT . At temperatures below $T_{L}\sim 6$ K, a
second anomaly is observed in $\rho(T)$ as a clearly-visible downturn in the
bottom-right inset in Fig. 15 and as a sharp upturn in $d\rho/dT$ in Fig. 16.
These features are rapidly suppressed in magnetic fields as low as 0.05 T.

On application of a magnetic field with $H\parallel ab$, the resistivity at
temperatures both above and below $T_{\rm N}$ is significantly suppressed.
Notably, this suppression persists up to $\approx 40~{}{\rm K}$, above which
the magnetoresistance changes sign from negative to positive, similar to
previous results for $H\parallel c$ Chen2020_EuSn2As2 . Positive
magnetoresistance is a characteristic of a usual metal, as also reveled in the
$T$-inear resistivity above 40 K. The onset temperature of a magnetic effect
on $\rho$ on cooling as notably revealed by the sign change of the
magnetoresistance correlates well with the temperature of complete entropy
release of the Eu2+ spins in the heat capacity measurements in Fig. 13, as
also found in CeRhIn5 PaglioneRh .

The negative sign of magnetoresistance below $\approx 40$ K suggests that
magnetic scattering is suppressed with magnetic field even in the
magnetically-ordered state, revealing significant residual disorder in the
magnetic subsystem. This is not expected in simple collinear FM or AFM
materials PaglioneQCP ; PaglioneRh , and suggests a more complicated type of
magnetic ordering. The zero-field neutron-diffraction and low-field magnetic
susceptibility data demonstrate that the AFM ordering is collinear A-type with
the moments aligned in the $ab$ plane. However, in an $ab$-plane field the
magnetic structure likely changes due to rotations of the collinear AFM-
ordered moments in the three domains to become perpendicular to the field as
discussed in Sec. V.

Another feature in the resistivity data supporting an $H_{ab}$-dependent
change in the magnetic structure of EuSn2As2 is splitting of the main
transition at $T_{\rm N}$. As shown in Fig. 16, the sharp transtion turns into
a broad crossover and a sharp peak develops at about 20 K. The onset of this
peak can be noticed already at 1 T. In a field of 3 T the peak transforms into
broad double humps, and the curve becomes featureless at $H=4$ T with a broad
crossover maximum in the resistivity derivative still centered around 20 K.
These results correlate with the change in the magnetic structure as the
ordered moments, initially perpendicular to the field in $h_{\rm ab}=0.15$ T,
cant towards the applied field, where the critical field for FM alignment of
the moments with the field from Fig. 10 is $\approx 3.7$ T.

Resistivity measurements versus $T$ with $H\parallel c$ are presented in Fig.
17, and Fig. 18 shows the evolution of the $T$-dependent derivative of the
resistivity versus $T$. The curves in both figures are offset to avoid
overlapping, where the offset is upwards in Fig. 17 but downwards in Fig. 18.
The $\rho(T)$ reveals a complcated transformation from a sharp anomaly at
$T_{\rm N}$ in zero field to a broad crossover field at $H=3$ T. A small
feature at $\sim 7$ K transforms into a tiny decrease in the resistivity
derivative, suggesting a nearly vertical line in the phase diagram of unclear
origin.

##  VIII  Summary

Figure 19: (a) Magnetic $c$-axis field $H_{c}$ vs $T$ phase diagram of
EuSn2As2 constructed from the $C_{\rm p}(H,T)$ (filled green circles),
$\chi(H,T)$ (filled black squares), and $\rho(H,T)$ (filled blue triangles)
data. The solid red curve is a fit of the critical field $H_{c}^{\rm c}(T)$
data obtained from the $C_{\rm p}(T)$ data in Fig. 14 by Eq. (11) that
separates the antiferromagnetic (AFM) phase from the paramagnetic (PM) phase.
We suggest that the filled magenta diamonds near zero field near 5 and 6 K
identified from the $\rho(H,T)$ measurements are due to lockin transitions of
the Eu moments in three AFM domains becoming perpendicular to the applied
field. The unfilled pentagons at temperatures $T_{\rm A}$ are unidentified
transitions suggested from the $\rho(H,T)$ measurements in Fig. 18. (b)
$ab$-plane magnetic field $H_{ab}$ vs $T$ phase diagram of EuSn2As2 as
determined from $\rho(H_{ab},T)$ measurements. The red filled circles
represent $H_{ab}(T_{\rm N})$ obtained from Fig. 16. The violet diamonds near
$H_{ab}=0$ correspond to a feature in the $T$-dependent resistivity derivative
which we suggest reflects a lock-in transition at which the Eu moments in the
$ab$ plane become perpendicular to the applied field. The solid magenta
diamonds at 3–7 K correspond to a small anomaly observed in $\rho(T)$ of
unknown origin. The critical fields $H^{\rm c}$ vs $T$ in (a) and (b) (half-
filled boxes) from Table 2 were determined as the intersection of the
extrapolated lines from larger $H$ and smaller $H$ with respect to the
approximate saturation field.

We investigated the ground state spin texture and the $H$-$T$ phase diagrams
of the AFM topological insulator material EuSn2As2 which has recently been
found to host Dirac surface states. The compound undergoes AFM ordering below
the Néel temperature $T_{\rm N}=24$ K as identified from neutron diffraction
measurements. The magnetic susceptibility $\chi(T)$ measurements find the same
$T_{\rm N}$ with an additional low-$T$ upturn below $\sim 10$ K that is
evidently due to magnetic impurities or defects in the crystals. By measuring
the $\chi(T)$ on three EuSn2As2 crystals grown differently, we showed that the
low-$T$ upturn in the $\chi(T)$ data is crystal-growth dependent. We
subsequently examined the physical properties of crystals with the lowest
concentration of magnetic impurities/defects (Crystals C).

By analyzing $\chi(T)$ data measured at a low magnetic field of 0.01 T on the
basis of molecular field theory together with zero-field neutron-diffraction
measurements, we established that the AFM structure of the Eu2+ spins $S=7/2$
below $T_{\rm N}$ is collinear A-type. Here the Eu spins within an $ab$ plane
are aligned ferromagnetically within the plane and aligned
antiferromagnetically between adjacent $ab$-plane layers along the $c$ axis.

The in-plane magnetic susceptibility $\chi_{ab}=M_{ab}/H$ at 2 K was found to
be sensitive to the applied magnetic field in the low-$H$ region, while the
out-of-plane $\chi_{c}=M_{c}/H$ is approximately independent of $H$ up to the
$c$-axis critical field $H_{c}^{\rm c}$. The former measurement exhibits a
smooth maximum in $dM_{ab}/dH$ at $H\approx 0.12$ T which we interpret as the
in-plane Eu moments in different threefold domains locking in to become
perpendicular to the applied field. At higher fields, the $ab$-plane
magnetization increases due to the usual canting of the moments toward the
field direction.

A $\lambda$ anomaly is observed in the heat capacity at $T_{\rm N}=24$ K that
is gradually suppressed with increasing $H$. The full magnetic entropy
associated with the Eu2+ spins is released by about 60 K, which is more than
twice $T_{\rm N}$, implying the presence of strong short-range dynamic AFM
spin correlations above $T_{\rm N}$.

The $H_{c}$-$T$ magnetic phase diagram shown in Fig. 19(a) was constructed
from the combined results of $T_{\rm N}(H)$ data obtained from $c$-axis
$\chi(H,T)$ (black solid squares), heat capacity $C_{\rm p}(T)$ (green solid
circles), and electrical resistivity$\rho(H,T)$ (blue up-triangles)
measurements. The phase boundary between the AFM and PM states is consistent
with the molecular-field-theory prediction for spin $S=7/2$. According to MFT,
at a fixed $T$ the critical field $H_{c}^{\rm c}$ responsible for an AFM to PM
phase transition in a Heisenberg antiferromagnet with $ab$-plane moment
alignment is given by Johnston2015

$\displaystyle H_{c}^{\rm c}(t)=H_{c}^{\rm c}(0)\bar{\mu}_{0}(t)$ (11)

where the symbols and their evaluation are described in Eqs. (4). The red
solid curve in Fig. 19(a) shows the fit by Eq. (11) of the $H_{c}^{\rm c}(T)$
vs $T_{\rm N}$ data obtained from $C_{\rm p}(H,T)$ measurements. The best fit
is obtained with $T_{\rm N}=24.00(1)$ K and $H_{c}^{\rm c}(T=0)=4.76(1)~{}{\rm
T}$ as listed in the figure.

The magnetic phase diagram in Fig. 19(b) for fields in the $ab$-plane was
constructed from $ab$-plane $\rho(H_{ab},T)$ measurements in Fig. 16. The
inset shows a features at $T=6$ and 5 K in $H=0.05$ and 0.1 T, respectively,
that are suppressed in higher fields and transform into a small resistivity
decrease feature. These features are suggested to be associated with the lock-
in transition of the Eu moments becoming perpendicular to $H_{ab}$. These
features complement the collinear AFM ordering found from the zero-field
neutron-diffraction measurements.

###### Acknowledgements.

This research was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering. Ames
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DE-AC02-07CH11358.

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