# Pressure-induced ferromagnetism in the topological semimetal EuCd2As2

Elena Gati1,2,3<EMAIL_ADDRESS>Sergey L. Bud’ko1,2 Lin-Lin Wang1
Adrian Valadkhani4 Ritu Gupta5 Brinda Kuthanazhi1,2 Li Xiang1,2 John M.
Wilde1,2 Aashish Sapkota1,2 Zurab Guguchia5 Rustem Khasanov5 Roser Valentí4
Paul C. Canfield1,2<EMAIL_ADDRESS>1 Ames Laboratory, US Department of
Energy, Iowa State University, Ames, Iowa 50011, USA  2 Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA  3 Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany  4 Institute
for Theoretical Physics, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1,
60438 Frankfurt am Main, Germany  5 Laboratory for Muon Spin Spectroscopy,
Paul Scherrer Institute, 5232 Villigen PSI, Switzerland

###### Abstract

The antiferromagnet and semimetal EuCd2As2 has recently attracted a lot of
attention due to a wealth of topological phases arising from the interplay of
topology and magnetism. In particular, the presence of a single pair of Weyl
points is predicted for a ferromagnetic configuration of Eu spins along the
$c$-axis in EuCd2As2. In the search for such phases, we investigate here the
effects of hydrostatic pressure in EuCd2As2. For that, we present specific
heat, transport and $\mu$SR measurements under hydrostatic pressure up to
$\sim\,2.5\,$GPa, combined with ab initio density functional theory (DFT)
calculations. Experimentally, we establish that the ground state of EuCd2As2
changes from in-plane antiferromagnetic (AFMab) to ferromagnetic at a critical
pressure of $\,\approx\,$2 GPa, which is likely characterized by the moments
dominantly lying within the $ab$ plane (FMab). The AFMab-FMab transition at
such a relatively low pressure is supported by our DFT calculations.
Furthermore, our experimental and theoretical results indicate that EuCd2As2
moves closer to the sought-for FMc state (moments $\parallel$ $c$) with
increasing pressure further. We predict that a pressure of $\approx$ 23 GPa
will stabilize the FMc state, if Eu remains in a 2+ valence state. Thus, our
work establishes hydrostatic pressure as a key tuning parameter that (i)
allows for a continuous tuning between magnetic ground states in a single
sample of EuCd2As2 and (ii) enables the exploration of the interplay between
magnetism and topology and thereby motivates a series of future experiments on
this magnetic Weyl semimetal.

###### pacs:

xxx

††preprint: APS/123-QED

##  I Introduction

Topological materials have been at the center of many research activities in
recent years due to the presence of a plethora of exotic phenomena, of
relevance not only from a fundamental perspective 1 but for applications 2, 3,
4, 5, 6 as well. In this context, Weyl semimetals 3, 7, 4 have emerged as
particularly multifaceted realizations of topological materials displaying
anomalous transport phenomena, including the anomalous Hall effect, 8, 9
negative longitudinal magnetoresistance 10 and non-local transport 11.

A Weyl semimetal can be created from a Dirac semimetal by breaking either
inversion symmetry 12 or time reversal symmetry 3, 7, 4, 13. Thus, Weyl nodes
can only be found either in non-centrosymmetric crystals or in magnetic
materials, providing a seemingly straight-forward path to identify new Weyl
semimetals. However, in real materials, the observation of the intriguing
features of Weyl physics is often complicated 14, 15 by (i) additional non-
topological Fermi surface pockets and (ii) the occurrence of multiple Weyl
nodes. To overcome the latter issue, research is recently focussing on
magnetic systems, since the breaking of time-reversal symmetry allows, in
principle, the realization of a single pair of Weyl nodes 7, which, in the
presence of inversion symmetry, have to be located at the same energy. In
contrast, when the crystal structure breaks inversion symmetry, the minimal
number of Weyl nodes is 4, although often is found to be higher, such as 24 in
the TaAs structural family of compounds 3, 16, 17.

Whereas several magnetic Weyl semimetals were proposed and studied
experimentally 18, 19, 20, 21, 22, 23, there is a continuing interest in
identifying new candidate materials, which display unambiguous signatures of
Weyl physics, free from the above-mentioned complexities. Surveys of space
groups 24 and band structure calculations 25 have conditionally identified
EuCd2As2 (centrosymmetric space group $P\bar{3}m$1 26) as a candidate for
magnetic Weyl semimetal if it orders ferromagnetically (FM) with the moments
parallel to the $c$ axis. In such a FM state, EuCd2As2 would show the minimal
number of Weyl nodes in close proximity to the Fermi level without any further
manipulation, such as the application of magnetic field.

However, experimental studies indicate that the Eu spins in EuCd2As2 order
antiferromagnetically (AFM) at ambient pressure below
$T_{\textrm{N}}\,\simeq\,9\,$K 27, 28. Although controversially discussed,
most papers refer to the AFM structure as the A-type, i.e., FM layers which
stack in an AFM fashion along the $c$ axis, with the moment direction confined
in the $ab$ plane 28, 29. Albeit this order breaks the in-plane three-fold
symmetry and thus hinders the formation of Weyl nodes, there are several
experimental observations that indicate that Weyl physics can manifest itself
in EuCd2As2 due to the proximity to ferromagnetism. Among those are, e.g., the
report of a single pair of Weyl nodes in EuCd2As2 in the $c$ polarized state
that is stabilized by very moderate magnetic fields 29, as well as the claim
of Weyl nodes in the paramagnetic state whose presence was attributed to
quasi-static and quasi-long range ferromagnetic fluctuations 30. Overall,
these results underline that EuCd2As2 is a promising host for investigations
of the interplay of topology and magnetism 31. This notion has been further
supported by transport measurements at temperatures around the
antiferromagnetic transition 32.
Actually, it was recently shown that small differences in the synthesis
procedure of this compound can result in samples of EuCd2As2 with FM order and
moments lying in the $ab$ plane 33, 34, 35. Whereas this explicitly
demonstrates that a FM state in EuCd2As2 is very close (in some compositional
space) to the AFM one, and suggests a high degree of tunability of the
magnetic properties of EuCd2As2, this also can cause severe complications in
comparing published results on different crystals grown via different routes.
Here, we report the effect of hydrostatic pressure, $p$, up to 2.5 GPa on the
magnetic properties of AFM EuCd2As2 by presenting specific heat, transport and
$\mu$SR measurements, combined with DFT calculations of total energies of
different magnetic configurations up to 25 GPa. Our experiments unambiguously
identify a transition to an order with a pronounced FM component, which we
refer to as FM order hereafter, at $p_{c}\,\approx\,2$ GPa. This FM order is
likely characterized by an in-plane orientation of the magnetic moments, as
supported by our DFT calculations. Further, our calculations predict a change
of the moment orientation at $\approx\,$23 GPa to an out-of-plane
configuration, which is the required ground state for realizing a single pair
of Weyl nodes in this material. Our results therefore clearly identify
pressure as an experimental tuning parameter that allows for studies of the
correlation of the topological properties of EuCd2As2 with a change of its
magnetic ground state in a single sample.

The paper is organized as follows. Section II summarizes the experimental
settings and theoretical methods used in this work. In Section III, we discuss
our results, starting with a discussion of experimental phase diagram (Sec.
III.1, followed by a presentation of our $\mu$SR data (Sec. III.2) and
anisotropic magnetoresistance data (Sec. III.3) and concluded by a discussion
of our results from DFT calculations (Sec. III.4). Finally in Section IV, we
present our conclusions and an outlook.

##  II Methods

Single crystals of EuCd2As2 were grown out of Sn flux with the following
procedure; The elements with an initial stoichiometry of Eu:Cd:As:Sn =
1:2:2:10 were put into a fritted alumina crucible 36 (sold by LSP Ceramics as
a Canfield Crucible Set 37) and sealed in fused silica tube under a partial
pressure of argon. The thus prepared ampoule was heated up to 900∘C over 24
hours, and held there for 20 hours. This was followed by a slow cooling to
550∘C over 200 hours, and decanting of the excess flux using a centrifuge 38.
The crystals were characterized by the means of powder x-ray diffraction as
well as magnetic measurements. The latter measurements confirmed that these
crystals undergo an AFM transition33 at $T_{\textrm{N}}\,\sim\,9.5$ K.

Specific heat under pressure was measured on a single crystal using the AC
calorimetry technique, as described in detail in Ref. [39]. Resistance under
pressure was measured in a four-point configuration. Contacts were made using
Epo-tek H20E silver epoxy. Unfortunately, despite using different contact
materials and routes of surface preparation, it was not possible to get two-
point resistances smaller than several tens of Ohms. Given that we are mostly
interested in tracking anomalies in the temperature- and field dependence of
$R(T)$, which we carefully measure by reducing the measurement current, we
refrain from correcting the presented transport data for geometrical factors.
For specific heat as well as transport measurements, the cryogenic environment
was provided by a Quantum Design Physical Property Measurement System.
Pressure was generated in a piston-cylinder double-wall pressure cell with the
outer cylinder made out of CuBe and the inner cylinder out of Ni-Cr-Al alloy
(see Ref. [40] for a very similar design). A mixture of 4:6 light mineral
oil:n-pentane was used as a pressure-transmitting medium. This medium
solidifies at $p\,\approx\,3-4\,$GPa at room temperature41, thus ensuring
hydrostatic pressure application over the available pressure range. Pressure
at low temperatures was determined from the shift of the superconducting
transition temperature of elemental lead (Pb)42.

$\mu$SR measurements under pressure were performed in a 3He cryostat at the
$\mu$E1 beamline at the Paul-Scherrer-Institute in Villigen, Switzerland, by
using the GPD spectrometer. Typically, $5-10\,\cdot\,10^{6}$ positron events
were counted for each data point. A large number of single crystals of total
mass of $\approx\,2\,$g were placed inside a pressure cell with arbitrary
orientations. Both, the inner and the outer cylinder of the pressure cell are
made out of MP35N alloy43. Daphne 7373 oil was used as a pressure-transmitting
medium, which solidifies at room temperature close to 2.5 GPa41. The pressure
at low temperatures was determined from the shift of the superconducting
transition of elemental indium44, which was also placed in the pressure cell
and measured in an independent ac susceptibility experiment.

Total energies for EuCd2As2 with various Eu spin configurations were
calculated in DFT 45, 46 including spin-orbit coupling (SOC) with the
Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional 47. We employed a
plane-wave basis set and projector augmented wave (PAW) 48, 49 method as
implemented in VASP 50, 51, 52. To account for the half-filled strongly
localized Eu 4f orbitals, a Hubbard-like $U$ parameter of 4.4 eV was used
which places the f-states in the region of -0.8 eV to -1.4 eV, as observed
experimentally in photoemission experiments 30. The analysis of the magnetism
as a function of pressure was done by performing equation of state (EOS)
calculations of both A-type antiferromagnetic and in-plane ferromagnetic
EuCd2As2 with the primitive hexagonal unit cell doubled along the c-axis (Note
that the energies of different moment directions in the plane were so close
that differences could not be resolved within DFT). The corresponding
reciprocal lattice was sampled on a $\Gamma$-centered Monkhorst-Pack 53
(11$\times$11$\times$3) k-point mesh with a Gaussian smearing of 0.05 eV. We
used a kinetic energy cutoff of 318 eV to relax the shape of the unit cell and
atomic positions at selected volumes until the absolute force on each atom is
below 0.01 eV/Å. Specifically for obtaining the crystalline magneto-anisotropy
calculations, we increased the $k$-point mesh to ($22\times 22\times 6$), the
kinetic energy cutoff was set to 500 eV and the absolute force threshold was
set to 0.001 eV/Å per atom. In Appendix VI.4 we present a discussion on the
effects that the choice of initial settings in the DFT calculations have on
the final relaxed structures.

##  III Results and Discussion

###  III.1 Zero-field temperature-pressure phase diagram

First, we discuss our determination of the temperature-pressure phase diagram
of EuCd2As2 in zero field from specific heat, $C/T$, and resistance, $R$,
measurements. Figure 1 shows $C/T$ data as a function of temperature in three
pressure ranges up to 2.43 GPa. Close to ambient pressure, as represented by
the 0.15 GPa data in Fig. 1 (a), we observe a sharp transition at $\sim\,9.4$
K which signals the onset of antiferromagnetic order, consistent with
literature 33. Upon increasing pressure up to 1.30 GPa (Fig. 1 (a)), this
feature shifts to higher temperatures, whereas almost no change can be found
in the shape of the feature, i.e., in the sharpness and the maximum value of
$C/T$ at the transition. Further increasing pressure beyond 1.30 GPa (Fig. 1
(b)) results in a clear decrease of the transition temperature up to
$\,\sim\,$1.94 GPa. Concurrently, the specific heat feature starts to broaden
slightly and the maximum value decreases, implying that the associated entropy
release with the magnetic ordering is distributed over a wider temperature
range. At even higher pressures, up to the maximum pressure of our experiment
of 2.43 GPa (Fig. 1 (c)), the transition temperature increases again and the
increase takes place at a faster rate, compared to lower pressures. The
specific heat feature is significantly broadened, indicating that fluctuations
might play a role in a wider temperature range above the transition
temperature. The observations described above, such as the sharp feature in
the temperature-pressure phase diagram at $\approx\,$2 GPa (see also Fig. 3
below), suggest a change of the magnetic structure of EuCd2As2 with
hydrostatic pressure.

Figure 1: Selected data sets of specific heat divided by temperature, $C/T$,
vs. $T$ (8 K $\leq\,T\,\leq\,$11 K) of EuCd2As2 under pressure, $p$, for
$0.13\,$GPa$\,\leq\,p\,\leq\,1.30$ GPa (a), for
$1.30\,$GPa$\,\leq\,p\,\leq\,1.94$ GPa (b) and
$1.94\,$GPa$\,\leq\,p\,\leq\,2.43$ GPa (c). The inset in (b) shows an example
data set of d$(C/T)$/d$T$, the minimum of which is used to determine the
transition temperature (see arrows in the inset and the main panel).

To complement our thermodynamic analysis of the phase diagram, we also
performed measurements of the resistance, $R$, as a function of temperature at
different pressures. The data of $R$, normalized to the room-temperature value
at zero pressure $R_{300,p=0}$, is displayed in Fig. 2 and split into three
pressure ranges, similar to the specific heat data. Overall, the $T$-dependent
behavior of $R/R_{300,p=0}$ is consistent with the literature at ambient
pressure27, 28, 30. Initially, upon cooling, $R$ decreases, consistent with a
semimetallic behavior27 (see inset of Fig. 2 (a)). Below $\approx\,80\,$K, $R$
starts to increase rapidly 27. Upon entering the magnetically-ordered state at
low temperatures, $R$ decreases due to loss of spin-disorder scattering,
resulting in a peak of $R$ at the transition temperature. Over the entire
pressure range up to 2.42 GPa, the peak value of $R$ increases with pressure
with the strongest increase observed in the intermediate pressure range 1.4
GPa $\lesssim\,p\,\lesssim\,1.95\,$GPa (see inset of Fig. 2 (b)), while the
room-temperature value of $R$ changes by less than 25 %. Similar to the
specific heat data, we find that the peak in $R$ remains sharp for
$p\,\lesssim\,$1.4 GPa (Fig. 2 (a)), starts to broaden for intermediate $p$
(Fig. 2 b) and remains broad while clearly shifting to higher temperatures
beyond 1.95 GPa (Fig. 2 c).

Figure 2: Selected data sets of resistance, normalized to the zero-pressure
room-temperature value, $R/R_{300,p=0}$, vs. $T$ (5 K $\leq\,T\,\leq\,$15 K)
of EuCd2As2 under pressure, $p$, for $0\,$GPa$\,\leq\,p\,\leq\,1.39$ GPa (a),
for $1.39\,$GPa$\,\leq\,p\,\leq\,1.95$ GPa (b) and
$1.95\,$GPa$\,\leq\,p\,\leq\,2.42$ GPa (c). Current was applied within the
$ab$ plane. The inset in (a) shows all data over the full temperature range up
to 300 K. The arrows in (b) indicate the criterion which is used to determine
the transition temperature (see text). The inset in (b) depicts the pressure
dependence of $R$, normalized to its zero-pressure value $R_{p=0}$, at
$T\,=\,300\,$K as well as of $R_{\textrm{{max}}}.$ The inset in (c) shows an
example data set of d$R$/d$T$ at zero pressure, which is used to determine the
transition temperatures (see arrow and text).

In order to construct the temperature-pressure phase diagram of EuCd2As2 from
the data above, we used the following criteria. For the specific heat data, we
determined the position of the minimum in d($C/T$)/d$T$, as exemplarily shown
in the inset of Fig. 1 (b). This criterion is close to the one obtained in
isentropic constructions (see arrow in the main panel of Fig. 1 (b)). For the
resistance measurements, we refer to the Fisher-Langer relation 54 for
magnetic transitions in metals and chose the maximum of d$R$/d$T$ (see arrow
in Fig. 2 (b) and inset in Fig. 2 (c)).

The resulting phase diagram is presented in Fig. 3. Pressure-dependent
transition temperature data from both $C$($T$) and $R$($T$) runs agree with
each other well and show an initial, slow increase in $T_{\textrm{N}}$
($p\,\lesssim\,$1.30 GPa) with pressure followed by a gradual decrease of
$T_{\textrm{N}}$ with further increase in pressure (1.30
GPa$\,\lesssim\,p\,\lesssim\,$2.0 GPa). For $p\,\gtrsim\,$2.0 GPa we observe a
comparatively sharp increase in the magnetic ordering temperature. This sharp
change in the pressure dependence of the ordering temperature strongly
suggests that a different magnetic phase has been stabilized above the
critical pressure $p^{\star}\,\simeq\,2\,$GPa. Indeed, our $\mu$SR data, shown
and discussed below, provide the basis for our determination that the high
pressure phase has a ferromagnetic (FM) component to it.

Figure 3: Temperature-pressure ($T$-$p$) phase diagram of EuCd2As2, determined
from specific heat, resistance and $\mu$SR measurements. Light purple symbols
correspond to the transition from the paramagnetic (PM) to the
antiferromagnetic (AFM) state. Blue symbols indicate the position of the PM-
ferromagnetic (FM) transition. The color shading in the intermediate pressure
range indicates that EuCd2As2 shows strong tendencies towards FM order (see
text for more details).

###  III.2 Microscopic study of magnetism under pressure

To probe the magnetic properties under pressure, we used $\mu$SR measurements.
In these experiments, approximately half of the muons stop in the sample and
half in the pressure cell wall. The latter muons are sensitive to the magnetic
fields that are generated by the sample inside the pressure cell. Thus, the
signal from the muons stopping in the pressure cell wall will be measurably
distorted 43, 55, 56 for ferromagnetic samples. A detailed description of the
analysis procedure of the $\mu$SR data under hydrostatic pressure can be found
in Appendix Sec. VI.1.1.

Figure 4: Evolution of the internal field of the two stopping sites in
EuCd2As2, $B_{\textrm{int,1}}$ and $B_{\textrm{int,2}}$, with temperature,
$T$, for pressures, $p$, of (a) 0 GPa, (b) 1.72 GPa, and (c) 2.56 GPa. Note
that the 0 GPa data were taken inside the hydrostatic pressure cell. Lines
represent fits to the empirical formula,
$B_{\textrm{int,i}}\,=\,B_{\textrm{int,i}}^{0}\left[1-\left(\frac{T}{T^{\star}}\right)^{\alpha}\right]^{\beta}$
(see text). The insets in (a) and (b) show zero-field spectra (open grey
symbols) taken at $T\,=\,3.55\,$K and 2.41 K, respectively, together with the
fit for two stopping sites (blue lines).

First, we present in Fig. 4 the evolution of the internal field of EuCd2As2
with temperature and pressure, which can be inferred from the zero-field
$\mu$SR spectra (see insets in (a) and (c) for the asymmetry data at 0 GPa and
2.56 GPa). In total, we took high-statistics zero field data sets for 0 GPa
(Fig. 4 (a)), 1.72 GPa (Fig. 4 (b)) and 2.56 GPa (Fig. 4 (c)). For all
pressures, clear oscillations were observed below the respective pressure-
dependent ordering temperature, signalling the presence of a finite internal
field below the respective transition temperature. In each case, our data is
best modeled by using two muon stopping sites with internal fields
$B_{\textrm{int,1}}$ and $B_{\textrm{int,2}}$. For the 0 GPa data, the best
fit was obtained by using a ratio of 60:40 for the two stopping sites, whereas
for higher pressures we fixed the ratio to 50:50. The transition temperatures
from $\mu$SR, which are included in the phase diagram in Fig. 3, were
determined by fitting the $B_{\textrm{int,i}}$ data with the empirical formula
$B_{\textrm{int,i}}^{0}\left[1-\left(\frac{T}{T^{\star}}\right)^{\alpha}\right]^{\beta}$,
with $B_{\textrm{int,i}}^{0}$, $T^{\star}$, $\alpha$ and $\beta$ being free
parameters ($T^{\star}$ corresponds to the respective transition temperature).
Interestingly, the ratio of internal fields
$B_{\textrm{int,2}}/B_{\textrm{int,1}}$ at $T\,\approx\,2\,$K increases
significantly with pressure from 1.5 at 0 GPa to 1.7 at 1.72 GPa to 2 at 2.56
GPa. Even though we lack precise information on the muon stopping sites in
EuCd2As2, the change of the ratio of internal fields suggests that there is
some change of the magnetic structure, for example by rotation of the moments
at a muon stopping site, with pressure 57.

Figure 5: Temperature dependence of the relaxation rate of the pressure cell,
$\lambda_{\textrm{PC}}$, for pressures of 0 GPa and 2.56 GPa. Lines are guide
to the eyes. The insets the weak-transverse field spectra at 0 GPa and at 2.56
GPa for temperatures below and above the phase transition.

Next, we discuss our results of the type of magnetic order under pressure. To
this end, we show in Fig. 5 the pressure cell relaxation rate,
$\lambda_{\textrm{PC}}$, at two different pressures, which were obtained from
measuring the muon asymmetry in a weak transverse field of 30 Oe. As already
evident from the raw data shown in the insets, there is no additional
depolarization below the ordering temperature $T_{\textrm{N}}\,\approx\,9.5$ K
at ambient pressure ($p\,<\,p^{\star}$), whereas there is a clear additional
depolarization below $T_{\textrm{C}}\,\approx\,10.4\,$K at 2.56 GPa
($p\,>\,p^{\star}$). As depicted in the main panel, the former data correspond
to a weak temperature dependence of $\lambda_{\textrm{PC}}$, the size of which
equals the known relaxation rate of the pressure cell 43 (see Appendix Sec.
VI.1.1). This behavior is expected for antiferromagnetic order (see also our
discussion of ambient-pressure $\mu$SR data on oriented single crystals in
Appendix Sec. VI.1.3). In contrast, for 2.56 GPa, $\lambda_{\textrm{PC}}$
suddenly increases below $T_{\textrm{C}}$ and levels off for lower
temperatures, which can only be attributed to the presence of an additional
field created by the sample. Importantly, this central result of our work
represents compelling evidence for the realization of a FM state at high
pressures. This conclusion is further supported by an increase of
ferromagnetic fluctuations above the ordering temperature with pressure (see
Appendix Sec. VI.1.2). Based on the presented $\mu$SR data on an aggregate of
randomly oriented single crystals, we cannot make any statement on whether the
FM state is fully polarized along a crystallographic direction or whether
there is only a ferromagnetic component to the order. Below, based on
anisotropic magnetoresistance data, we argue that the high-pressure FM state
is likely almost in-plane polarized, and thus we refer here to the notion of
FM order.

###  III.3 Anisotropic magnetoresistance data

Now that we have established a change of the magnetic ground state from AFM to
FM by hydrostatic pressure in EuCd2As2, we want to discuss transport
measurements under pressure in finite magnetic field in the ordered states.
The aim of these experiments is to identify the moment direction in the high-
pressure state.

Figure 6: Anisotropic magnetoresistance of EuCd2As2 at $T\,=\,2\,$K at
different pressures. (a) In-plane $R/R_{0}$, with $R_{0}$ being the resistance
in zero external field, as a function of in-plane field $\mu_{0}H$ at
different pressures $p$. Arrow indicates the criterion to determine
$H_{\textrm{ab}}^{\star}$; (b) In-plane $R/R_{0}$ as a function of out-of-
plane field $\mu_{0}H$ at different pressures $p$. Arrow indicates the
criterion to determine $H_{\textrm{c}}^{\star}$; (c) Ambient-pressure
anisotropic magnetization as a function of in-plane as well as out-of-plane
magnetic field 33. Arrows indicate the saturation fields
$H_{\textrm{ab}}^{\star}$ and $H_{\textrm{c}}^{\star}$; (d) Evolution of
$H_{\textrm{ab}}^{\star}$ and $H_{\textrm{c}}^{\star}$ with pressure. For
$p\,<\,p_{c}$, the ground state is AFM with moments lying in the $ab$-plane.
Thus, $H_{\textrm{ab}}^{\star}$ and $H_{\textrm{c}}^{\star}$ are defined as
saturation fields into fully polarized states with moments in the $ab$ and $c$
direction, respectively. For $p\,>\,p_{c}$, the ground state is FM with
moments in the $ab$ plane, thus only $H_{\textrm{c}}^{\star}$ can be defined.

In Figs. 6 (a) and (b), we show our data of the pressure-dependent
magnetoresistance (MR) data at $T\,=\,2\,$K, taken in a longitudinal and in a
transverse configuration, i.e., $\mu_{0}H\parallel I\parallel ab$, (a) and
$\mu_{0}H\parallel c,I\parallel ab$ (b). At lowest pressures (0.13 GPa and
0.52 GPa, respectively), the longitudinal $R/R_{0}$ goes through a maximum and
becomes almost field-independent above
$\mu_{0}H^{\star}_{\textrm{ab}}\,\sim\,$0.7 T (Fig. 6 (a)), whereas the
transverse $R/R_{0}$ shows a step-like change around
$\mu_{0}H^{\star}_{\textrm{c}}\,\sim\,$1.8 T (c,d). Those field scales
coincide very well with the saturation fields for the specific field
orientations, determined from magnetization measurements at ambient pressure
33 (see Fig. 6 (c)).

As clearly visible in the raw data (a), $H^{\star}_{\textrm{ab}}$ decreases
rapidly with increasing pressure. As shown in Fig. 6 (d),
$H^{\star}_{\textrm{ab}}$ extrapolates to zero at $\approx\,$2 GPa, i.e., at
$p_{c}$. For 2.3 GPa, the $R/R_{0}$ data is essentially field-independent.
These results strongly suggest that the high-pressure FM state has its moments
aligned in the plane, i.e., a pure FM state: With pressure, we suppress
$H^{\star}_{\textrm{ab}}$ to zero, implying full moment alignment along the
in-plane direction.

Whereas $H^{\star}_{\textrm{ab}}$ is suppressed to zero with increasing
pressure, $H^{\star}_{\textrm{c}}$ is also suppressed, but remains finite at 2
GPa. For $p\,\gtrsim\,$2 GPa, no step-like change can be observed. Instead, we
observe a broad crossover feature in the field-dependent $R/R_{0}$ data, the
position of which increases with increasing pressure (see Fig. 6 (d)). The
absence of a metamagnetic transition for out-of-plane fields is consistent
with the notion of FM order with moments oriented in the $ab$-plane. The
decrease in $H^{\star}_{\textrm{c}}$ for $p\,<\,p_{c}$ is nonetheless
remarkable, as it suggests that the energy difference to the FM state with
moments along the $c$ axis is also reduced with pressure. We will further
discuss the proximity to this FM${}_{\textrm{c}}$ state based on DFT
calculations below. We note that our Hall data under pressure, presented in
Appendix Sec. VI.2, indicate the possibility that the change of magnetic
structure across $p_{c}$ is associated with a change of charge carrier
density.

###  III.4 Results of DFT calculations under pressure

In this section we study the pressure-dependent evolution of magnetic order in
EuCd2As2 and the possible transition from in-plane AFM (denoted by AFMab in
the following) to in-plane FM order (denoted by FMab) for low pressures with
the help of DFT calculations. To this end, we constructed the EOS for both
AFMab and FMab spin configurations around the equilibrium volume and fitted it
to the Birch-Murnaghan equation58, 59. The optimized equilibrium lattice
constants using the PBE exchange-correlation functional and $U$ = 4.4 eV
without spin-orbit coupling (SOC) for A-type antiferromagnetic (i.e., AFMab)
spin configuration are $a\,$= 4.49 Å and $c$ = 7.37 Å. These values agree with
the experimental data of 4.4336 and 7.2925 Å at $T\,=\,11$ K respectively,
just above $T_{N}$ (see Appendix Sec. VI.3) within about 1%, which can be
considered as a very good agreement for DFT-PBE calculations 60. For EOS
calculations, at each volume, the cell shape and atomic positions are fully
relaxed for both AFMab and FMab with SOC, with the magnetic moment pointing in
the in-plane direction to the nearest-neighbor (NN) Eu or equivalently along
$a$\- or $b$-axis. The fitted equilibrium volumes from EOS are 129.78 and
129.76 Å3/f.u. for AFMab and FMab with SOC, respectively, having an energy
difference of 0.3 meV/f.u. in favor of the AFMab state. This agrees well with
the 0.22 meV/f.u. preference to AFMab from the direct DFT calculation at the
initial AFM equilibrium volume of 128.96 Å3/f.u. With such small differences
in equilibrium volume and energy, the two fully-relaxed EOS of the competing
magnetic phases are aligned very closely together to have almost the same bulk
modulus of 46.7 GPa. To obtain the critical pressure $p_{c}$ for the AFM-FM
magnetic phase transition, the difference between the two EOS, $\Delta
E_{FM,ab-AFM,ab}$ vs. volume is plotted in Fig.7. At the initial AFMab
equilibrium volume, AFMab is preferred by 0.22 meV/f.u. This preference is
increased at larger volume, but decreased for smaller volume, i.e., with
increasing hydrostatic pressure. This leads to a AFMab-FMab magnetic phase
transition under hydrostatic pressure, as observed in experiment. The
transition region is shown on enlarged scales in the inset of Fig.7 with the
x-axis converted to pressure using the calculated bulk modulus. The dashed
line of $\Delta E_{FM,ab-AFM,ab}$ = 0.0 gives the critical pressure
$p_{c}\,\approx\,$3.0 GPa, in good agreement with the experimental data of
$\approx\,$2.0 GPa. To firmly establish the pressure-induced AFMab-FMab in
EuCd2As2 within DFT, we have also tested different exchange-correlation
functionals, PAW potentials and computational settings, including using the
same relaxed crystal structures for AFMab and FMab at each volume from either
the non-magnetic or AFMab configuration. All calculations confirm a AFMab-
FMab, with $p_{c}$ values in the range from 2.7 to 4.2 GPa (see App. VI.4 for
a detailed discussion on different initial conditions).

Next, we calculated the crystalline magneto-anisotropy energy (MAE), defined
by the required energy to switch the crystal magnetization from the low
pressure basal plane magnetic order and direction to, in this case explicitly,
FMc order, at increasing pressure. Such a calculation provides information
about the proximity of EuCd2As2 to the FMc state, which, according to recent
work, is able to host a single pair of Weyl nodes 24, 25. This calculation was
done in various steps. First, we relaxed the unit cell of EuCd2As2 at
different volumes by treating the Eu2+ 4f states completely as core states and
by performing non-spin polarized calculations with the PBE exchange
correlation functional. The former warrants that no spurious Eu f states
appear near the Fermi level and the latter ensures a fully non-magnetic
setting since no magnetic moments/spins are invoked in the calculation. Such a
relaxation guarantees a uniform change in the lattice without being influenced
by the nature of the assumed magnetism for the Eu2+ 4f states. The resulting
optimized non-magnetic equilibrium lattice parameters are given by
$a\,=\,4.48\,$ Åand $c\,=\,7.27\,$ Å, which are closer to the experimental
data (see Appendix VI.3) than the magnetically relaxed structures, due to the
absence of long-range magnetic order at the measured $T\,=\,11\,$K. In a
second step, we took the optimized non-magnetic crystal structures and
calculated total energies for different Eu magnetic configurations, by
including the Eu2+ 4f states as valence electrons with $U\,=\,4.4\,$eV and SOC
and with the Eu moments aligned either antiferromagnetically or
ferromagnetically and moments pointing in the plane for AFMab and FMab,
respectively, as well as aligned ferromagnetically along the $c$-axis, denoted
as FMc. The DFT results were fitted to the Birch-Murnagham EOS, yielding the
MAE by subtracting the corresponding fits. After fitting, the equilibrium
volume increases from $126.4$ Å${}^{3}/$f.u. for the non-magnetic
configuration to $130.21$ Å${}^{3}/$f.u. for both FM orders and to $130.21$
Å${}^{3}/$f.u. for AFMab. Note that throughout all performed calculations we
keep the valence of Eu fixed to 2+, so that we cannot exclude the possibility
of a valence transition (Eu2+ to Eu3+) at the simulated higher pressures. This
goes beyond the scope of the present work.

In Fig. 8 we show the results of these MAE calculations as a function of
hydrostatic pressure. The MAE is directly proportional to the $H_{c}^{\star}$
field that is measured in experiment. At ambient pressure, the theoretical
calculations overestimate the experimental $H_{c}^{\star}$ value, which is
expected in the framework of DFT and Birch-Munagham EOS, which we use.
Nevertheless, the tendencies as a function of pressure are robust. Here, we
find that the MAE or $H_{c}^{\star}$ decrease rapidly with increasing pressure
for $p\,<\,p_{c}$, consistent with our experimental observations, shown in
Fig. 6 (d). Above $p_{c}$, the MAE still continues to decrease with increasing
pressure, however at a slower rate. This trend is opposite to what is observed
for our experimental data for pressures close to $p_{c}$. However, we note
that the details of our theoretical prediction close to $p_{c}$ depend on the
details of the relaxation (The result of a different relaxation is shown in
Fig. 14). Thus, experimental data to higher pressures is needed for a
meaningful comparison between theory and experiment in the FMab ground state
region.

The decrease of MAE with pressures beyond $\approx\,10$ GPa does not depend on
details of the relaxation. Thus, there will eventually be a transition to the
FMc state at sufficiently high pressures, as long as Eu does not change its
valency. We predict that this transition to an FMc state occurs at
$p^{\prime}_{c}\approx 23$ GPa. Whereas this pressure is larger than the
experimentally-used pressures in this work, it is nevertheless experimentally
feasible to realize those pressures. Thus, based on our theoretical
calculations, it might be possible to stabilize the desired FMc state in
EuCd2As2 in experiment as a result of pressure effects and without application
of an external magnetic field, and, to study key transport quantities with
respect to topology. At the same time, it will be very interesting to perform
further studies to understand the microscopic origin of the high tunability of
the magnetic properties of EuCd2As2.

Figure 7: Equation of state (EOS) energy difference ($\Delta E_{FM,ab-
AFM,ab}$) vs. volume ($V$) between the fully-relaxed FM and A-type AFM, where
in both cases the moment point in-plane to the nearest-neighbor Eu. The
initial AFMab equilibrium volume of 128.96 Å3/f.u. as well as the phase
transition at $\Delta E_{FM,ab-AFM,ab}$=0.0 are indicated by the vertical and
horizontal dashed line, respectively. The inset shows an enlarged view around
the AFMab-FMab transition region with the x-axis converted to pressure ($p$)
using the calculated bulk modulus (see text). The dashed line for $\Delta
E_{FM,ab-AFM,ab}$=0.0 gives the critical pressure $p_{c}\,\approx\,$3.0 GPa.
Figure 8: Energy difference of the FMc order to the ground states (left axis)
and corresponding magnetic field (right axis) vs. pressure. The solid lines
show the energy difference between the FMc state and the respective ground
states, i.e., AFMab for $p\,\lesssim\,$3 GPa (red) and FMab for
$p\,\gtrsim\,$3 GPa (green). Dashed lines represent the energy differences
between non-ground state magnetic configurations. The ground state transition
to FMc takes place at approximately $p^{\prime}_{c}\approx 23$ GPa.

##  IV Conclusions

In summary, we presented an experimental study of the temperature-pressure
phase diagram up to $\sim\,2.5$ GPa of the magnetic Weyl semimetal EuCd2As2 by
means of specific heat, transport and $\mu$SR measurements and performed DFT
calculations up to pressures of $\approx$ 25 GPa. Our results clearly show
that this compound undergoes a transition from an antiferromagnetic state at
low pressures to a ferromagnetic state above 2 GPa, in which moments are
dominantly oriented within the $ab$ plane. In addition, we find that pressure
also drives EuCd2As2 closer to the sought-for FM state, in which moments are
oriented along the $c$ axis. We predict that a hydrostatic pressure of
$\approx$ 23 GPa will stabilize this FMc state, if no valence transition of
Eu2+ intervenes. Overall, our study identifies a clear and experimentally-
accessible tuning route to change the magnetic ground state in EuCd2As2 in a
single sample and investigate the interplay of magnetism and topological non-
trivial phases. Our results motivate further experimental and theoretical
studies on EuCd2As2 under pressure, which is a very powerful parameter to tune
the properties of this magnetic Weyl semimetal.

##  V Acknowledgments

We thank Na Hyun Jo and Young-Joon Song for useful discussions. SLB and PCC
thank B. Schweinsteiger for inspiring research group members. Work at the Ames
Laboratory was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DEAC02-07CH11358. E.G. and L.X. were funded, in
part, by the Gordon and Betty Moore Foundation’s EPiQS Initiative through
Grant No. GBMF4411. B.K. and L.-L.W. were funded by the Center for Advancement
of Topological Semimetals, an Energy Frontier Research Center funded by the
U.S. Department of Energy Office of Science, Office of Basic Energy Sciences,
through the Ames Laboratory under its Contract No. DE-AC02-07CH11358. Research
of R.G. is supported by the Swiss National Science Foundation (SNF-Grant No.
200021-175935). AV and RV acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) for funding through
TRR 288 - 422213477 (project B05).

##  VI Appendix

###  VI.1 Additional discussion of $\mu$SR data under pressure

####  VI.1.1 Analysis of $\mu$SR data inside a pressure cell

Zero field data analysis - The zero-field $\mu$SR data was analyzed by taking
two independent contributions to the total asymmetry, $A(t)$, into account

$A^{\textrm{ZF}}(t)=A^{\textrm{ZF}}_{\textrm{S}}(t)+A^{\textrm{ZF}}_{\textrm{PC}}(t),$
(1)

with $A^{\textrm{ZF}}_{\textrm{S}}(t)$ ($A^{\textrm{ZF}}_{\textrm{PC}}(t)$)
being the contribution from the sample (the pressure cell). The sample
contribution for EuCd2As2 was found to be best described by considering two
stopping sites with different internal field, resulting in

$\begin{split}&A^{\textrm{ZF}}_{\textrm{S}}(t)=\\\ \
&A^{\textrm{ZF}}_{\textrm{S,0}}[\sum_{i=1}^{2}m_{i}\left(\frac{2}{3}\cos{(\gamma_{\textrm{m}}B_{\textrm{int,i}})\exp{(-\lambda_{\textrm{T,i}})}+\frac{1}{3}\exp{(-\lambda_{\textrm{L,i}})}}\right)\\\
\ &+(1-m_{1}-m_{2})\exp{(-\Lambda t)}]\end{split}$ (2)

with $m_{i}$ the magnetic fraction of the $i$-th component,
$\gamma_{\textrm{m}}$ the gyromagnetic ratio of the muon, $B_{\textrm{int,i}}$
the internal field, $\lambda_{\textrm{T,i}}$ the transverse relaxation rate,
$\lambda_{\textrm{L,i}}$ the longitudinal relaxation rate, and $\Lambda$ the
relaxation rate of the paramagnetic portion of the sample. The 2/3 and 1/3
components arise from averaging over a large aggregate of arbitrarily-oriented
single crystals.

The background contribution from the pressure cell can be determined in an
independent set of experiment and can be described by two depolarization
channels (one originating from nuclear moments and one from electronic
moments) following a damped Kubo-Toyabe form

$\begin{split}&A^{\textrm{ZF}}_{\textrm{PC}}(t)=\\\ \
&A^{\textrm{ZF}}_{\textrm{PC,0}}\left(\frac{1}{3}+\frac{2}{3}(1-\sigma_{\textrm{PC}}^{2}t^{2})\right)\exp{(-\sigma_{\textrm{PC}}^{2}t^{2}/2)}\exp{(-\lambda_{\textrm{PC}}t)},\end{split}$
(3)

with $\sigma_{\textrm{PC}}$ ($\lambda_{\textrm{PC}}$) the relaxation rate
associated with the nuclear (electronic) moments.

Weak transverse field data analysis - As mentioned in the main text, there
exists an additional depolarization of the muons stopping in the pressure cell
when the sample inside the cell exhibits a strong magnetization. We note that
the muons which stop in the pressure cell wall mostly stop inside the inner
cylinder, i.e., very close to the sample, so that they can be sensitive to
stray fields emerging from the sample, as shown in recent calculations of the
muon stopping profile for pressure-cell experiments with this particular muon
energy 61. In the case of weak-transverse field experiments with finite
$B_{\textrm{ex}}$, the pressure cell contribution thus reads as

$\begin{split}&A^{\textrm{wTF}}_{\textrm{PC}}(t)=\\\ \
&A^{\textrm{wTF}}_{\textrm{PC,0}}(t)\exp(-\sigma_{\textrm{PC}}^{2}t^{2}/2)\exp(-\lambda_{\textrm{PC}})\cos(\gamma_{\textrm{m}}B_{\textrm{ex}}t+\phi).\end{split}$
(4)

Here, $\sigma_{\textrm{PC}}$ refers to the relaxation rate caused by nuclear
moments, whereas $\lambda_{\textrm{PC}}$ is relaxation rate determined by the
electronic moments as well as the influence of the field that is created by a
sample with macroscopic magnetization. The electronic contribution to
$\lambda_{\textrm{PC}}$ is typically almost temperature-independent43 and
$\lambda_{\textrm{PC}}\,\simeq\,0.05-0.1\,\mu$s (see also data above the
ordering temperature in Fig. 5).

####  VI.1.2 Evolution of $mu$SR relaxation above the ordering temperature
with pressure

Figure 9: Temperature dependence of the dynamical relaxation rate, $\Lambda$,
of EuCd2As2 for pressures of 0 GPa and 2.56 GPa for $10\,$K$\,<\,T\,<\,300$ K
and $12\,$K$\,<\,T\,<\,300$ K, respectively, i.e., in the paramagnetic state
at each pressure. Lines are guide to the eyes.

We now discuss the relaxation rate, $\Lambda$, of EuCd2As2 above the ordering
temperature, as shown in Fig. 9. For ambient pressure, we find a moderate
increase of $\Lambda$ with lowering temperature, which is fully consistent
with the expectations for paramagnetism stemming from the large Eu2+ moments.
Importantly, we note that we do not find any indications for the step-like
change of $\Lambda$ at $\approx\,100$ K, which Ma et al. 30 observed and which
was taken as a strong evidence for quasi-long range and quasi-static magnetic
order below 100 K that stabilizes Weyl physics even without spontaneous
breaking of time-reversal symmetry (see Appendix VI.1.3 for supporting
$\Lambda$ data on a well-oriented single crystal at ambient pressure outside
the pressure cell). Instead, the large contribution from the Eu paramagnetism
does not allow for any conclusion about the presence or nature of magnetic
correlations. However, upon increasing pressure to $p\,=\,2.56$ GPa, we find
that $\Lambda$ increases much faster below $\approx\,$20 K, reflecting a
strong enhancement of FM fluctuations in the proximity of the FM ground state.

####  VI.1.3 $\mu$SR data at ambient pressure

Figure 10: Ambient-pressure zero-field $\mu$SR asymmetry of an oriented single
crystal of EuCd2As2 using the GPS spectrometer. The orientation of the sample
with respect to the detectors is indicated in the inset of (a). The asymmetry
is evaluated from the difference in counts of the up- and down-detectors (a)
as well as the forward- and backward-detectors (b).

So as to characterize the magnetic order at ambient pressure in more detail,
we performed $\mu$SR measurements in zero-field on an oriented single crystal
in the GPS spectrometer. Given that the sample is oriented, information on the
orientation of the magnetic order can be inferred from differences in counts
in the forward- and backward vs. the up- and down-detectors (see Fig. 10 for a
schematic sketch of the experimental setup). Clear oscillations are seen in
the asymmetry evaluated from the difference between the up- and down-detectors
(Fig. 10 (a)) as well as the forward-backward-detectors (Fig. 10 (b)). This
implies that the moments are not aligned out-of-plane at ambient pressure,
consistent with earlier literature results 28, 29. The data is best described
by taking two muon stopping sites into account and the respective internal
fields inferred from these ambient-pressure measurements agree very well with
the ones plotted in Fig. 4, which were inferred from the data inside the
pressure cell at 0 GPa.

Figure 11: Temperature dependence of the dynamical relaxation rate, $\Lambda$,
of EuCd2As2 above the magnetic ordering temperature. Data were taken inside
the pressure cell at ambient pressure on a arbitrarily-oriented aggregate of
single crystals (black symbols) as well as an oriented single crystal at
ambient pressure outside of the pressure cell (red symbols).

In Fig. 11, we show the relaxation rate, $\Lambda$ above the ordering
temperature inferred from the GPS experiment on the oriented single crystal
and compare it with the data inferred from the randomly-oriented aggregate of
single crystals inside the pressure cell at ambient pressure. Whereas the
overall magnitude of $\Lambda$ differs between the two experiments, which
likely can be attributed to a directional-dependent size of the relaxation,
none of the data reveal a clear feature at $\approx$ 100 K, as previously
stated in Ref. 30.

###  VI.2 Hall effect measurements under pressure

Figure 12: Hall resistance, $R_{\textrm{Hall}}$, of EuCd2As2 at different
pressures at $T\,=\,2\,$K (a,b) and $T\,=\,13\,$K (c,d) in low fields (a,c)
and high fields (b,d). Data were taken with magnetic field along in-plane
direction. The insets show the high-field slope of the Hall data as a function
of pressure at the respective temperatures.

Hall effect measurements were made by carefully painting two current contacts
to cover the two opposite ends of the crystal to ensure as uniform of a
current density as possible and two voltage contact on the remaining two side
surfaces of the crystal. Current was applied along an in-plane direction, the
magnetic field aligned perpendicular to it within the plane and voltage
measured along the $c$ axis.

Here, we want to discuss our results of changes of Hall resistance across the
critical pressure. In Fig. 12 we show the Hall resistance,
$R_{\textrm{Hall}}$, as a function of pressure for $T\,=\,2\,$K (a,b) and
$T\,=\,13\,$K (c,d), i.e., below and above the magnetic transition
temperatures $T_{\textrm{N}}(p)$ and $T_{\textrm{C}}(p)$. At 2 K and low
pressures, $R_{\textrm{Hall}}$ goes through a minimum followed by a maximum in
a very narrow field range up to 0.8 T. The ordinary Hall effect cannot account
for this behavior of $R_{\textrm{Hall}}$ below $\sim\,0.8\,$T as it is almost
linear (see Fig. 12 (b) for data up to higher fields). In addition, this field
scale matches with the one observed in the MR measurements above as well as
magnetization measurements at ambient pressure 33. Thus, this indicates that
the low-field Hall effect data are dominated by the anomalous Hall effect
contribution. At high pressures, in the FM state, the pronounced maxima and
minima are absent and the anomalous contribution to $R_{\textrm{Hall}}$ is
distinctly smaller, but nonetheless finite. A full disentanglement of the
various contributions to the Hall resistance in a magnetic material is
unfortunately not possible on the basis of the present data set, given the
lack of magnetization data under pressure.

For sufficiently large fields, $\mu_{0}H\,\gtrsim\,4\,$T, the
$R_{\textrm{Hall}}$ data at 2 K and 13 K are linear in field (see Figs. 12 (b)
and (d)), and thus reflects the ordinary Hall contribution. The slope,
$R_{H}$, obtained from linear fits of the Hall data up to maximum field, is
plotted as a function for pressure in the insets of Figs. 12 (b) and (d). For
each temperature (2 K and 13 K), $R_{H}$ is constant up to $\approx\,2$ GPa
and starts to decrease slightly above. Even though the number of data points
at very high pressures, where the decrease becomes pronounced, is limited, our
data might indicate a change of charge carrier density across the critical
pressure. It will be interesting to investigate the microscopic origin of the
change of the magnetic properties with pressure in the future from
experimental and theoretical point of view in more detail.

###  VI.3 Low-temperature lattice parameters of EuCd2As2 at ambient pressure

So as to better compare experimental and theoretical lattice parameters, we
determined the lattice parameters of EuCd2As2 experimentally at low
temperatures, given that our DFT calculations are performed at zero
temperature. Single crystal X-ray diffraction measurements were performed on
an in-house four-circle diffractometer using Cu $K_{\alpha 1}$ radiation from
a rotating anode X-ray source, using germanium (1 1 1) monochromator. A He
closed-cycle refrigerator was used for temperature dependence measurements
between 11 and 300 K. Three Be domes were used as a vacuum shroud, heat
shield, and the last innermost dome containing the sample. The innermost dome
was filled with a small amount of He gas to improve thermal contact to the
sample surface. Measurements were carried out on a single crystal of 0.087 g
attached to a flat copper sample holder that is attached to the cold finger.
The mosaicities of the sample were less than 0.04∘ for both the (0, 0, 5) and
(3, 0, 5) reflections at all measured temperatures. The positions of the
reflections were fit using a Lorentzian lineshape, and used to determine the
lattice parameters of the sample from 11-300 K, which are shown in Fig. 13.

Figure 13: Lattice parameters, $a$ (left axis) and $c$ (right axis) of
EuCd2As2 as a function of temperature, $T$, at ambient pressure, determined
from x-ray diffraction.

As discussed in the main text, the experimental lattice parameters at 11 K are
within 1 % of the calculated values for both the magnetic and the non-magnetic
relaxation. Based on previous works 60, this can be considered as a very good
agreement between experiment and theory.

###  VI.4 Effect of lattice parameter relaxation in the calculation of
energies

To justify the validity of our calculated data, we used several slightly
different calculation settings. Whereas the different calculations resulted in
slightly different estimates of the critical pressures, they always resulted
in the AFMab-FMab-FMc transition sequence with increasing pressure. Thus, the
key message of our paper is stable with respect to different calculation
settings. In the following, we discuss the spread of $p_{c}$ values obtained
from different calculation procedures. The order of magnitude for the
difference in energy of possible magnetic ground states is already below $1$
meV at ambient pressure, for the MAE even below $0.1$ meV. Therefore, tiny
structural changes, caused by different methods of relaxing the unit cell or
pseudo potential files, affect the resulting critical pressure $p_{c}$
measurably. Independent of all previously shown calculations, we performed a
relaxation using the magnetic space groups $C_{c}2/m$, corresponding to the
AFMab, and $C2/m$, FMab, as starting points for the initial crystal structure.
Fixing the Wyckoff positions with regard to the magnetic space groups
guarantees no loss of symmetry information. Thereby, we also included SOC and
a Hubbard $U=4.4$ eV. Apart from this, the input parameters are identical to
the non-magnetic case.

Figure 14 illustrates the outcome of this method. Similar to Fig. 8 in the
main text, this figure describes the difference of the corresponding ground
state energy to the FMc state as a function of pressure on the left axis and
the corresponding magnetic field. The critical pressure $p_{c}$ for the
transition from AFMab order (red curve) to FMab (green curve), given by the
crossing point of the red and green curve, is shifted to a slightly larger
pressure of $4.2$ GPa relative to what is shown in Fig. 8. The predicted
transition to FMc, occurs at a lower pressure of $p^{\prime}_{c}\approx 20$
GPa.

Figure 14: Energy difference of the ground state compared to FMc order (left
axis) and corresponding magnetic field (right axis) vs. pressure. The red
curve describes the difference of AFMab to FMc, whereas the green curve
describes the difference of FMab to FMc. Compared to the relaxation types in
Fig. 7 and Fig. 8 $p_{c}$ is slightly increased to $4.2$ GPa and
$p^{\prime}_{c}$, describing the transition to FMc ground state, decreased to
$20$ GPa.

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