# Coexistence of Antiferromagnetic Cubic and Ferromagnetic Tetragonal
Polymorphs in Epitaxial CuMnSb

A. Ciechan  Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow
32/46, PL-02668 Warsaw, Poland.  P. Dłużewski  Institute of Physics, Polish
Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland.  S. Kret
Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46,
PL-02668 Warsaw, Poland.  K. Gas  Institute of Physics, Polish Academy of
Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland.  Center for Science
and Innovation in Spintronics, Tohoku University, Sendai 980-8577, Japan  L.
Scheffler  Physikalisches Institut (EP3), Universität Würzburg, 97074
Würzburg, Germany.  Institute for Topological Insulators, Universität
Würzburg, 97074 Würzburg, Germany.  C. Gould  Physikalisches Institut (EP3),
Universität Würzburg, 97074 Würzburg, Germany.  Institute for Topological
Insulators, Universität Würzburg, 97074 Würzburg, Germany.  J. Kleinlein
Physikalisches Institut (EP3), Universität Würzburg, 97074 Würzburg, Germany.
Institute for Topological Insulators, Universität Würzburg, 97074 Würzburg,
Germany.  M. Sawicki  Institute of Physics, Polish Academy of Sciences, Aleja
Lotnikow 32/46, PL-02668 Warsaw, Poland.  L.W. Molenkamp  Physikalisches
Institut (EP3), Universität Würzburg, 97074 Würzburg, Germany.  Institute for
Topological Insulators, Universität Würzburg, 97074 Würzburg, Germany.  P.
Bogusławski<EMAIL_ADDRESS>Institute of Physics, Polish Academy of
Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland.

###### Abstract

High-resolution transmission electron microscopy and superconducting quantum
interference device magnetometry shows that epitaxial CuMnSb films exhibit a
coexistence of two magnetic phases, coherently intertwined in nanometric
scales. The dominant $\alpha$ phase is half-Heusler cubic antiferromagnet with
the Néel temperature of 62 K, the equilibrium structure of bulk CuMnSb. The
secondary phase is its ferromagnetic tetragonal $\beta$ polymorph with the
Curie temperature of about 100 K. First principles calculations provide a
consistent interpretation of experiment, since (i) total energy of
$\beta$–CuMnSb is higher than that of $\alpha$–CuMnSb only by 0.12 eV per
formula unit, which allows for epitaxial stabilization of this phase, (ii) the
metallic character of $\beta$–CuMnSb favors the Ruderman-Kittel-Kasuya-Yoshida
ferromagnetic coupling, and (iii) the calculated effective Curie-Weiss
magnetic moment of Mn ions in both phases is about $5.5~{}\mu_{\mathrm{B}}$,
favorably close to the measured value. Calculated properties of all point
native defects indicate that the most likely to occur are
$\mathrm{Mn}_{\mathrm{Cu}}$ antisites. They affect magnetic properties of
epilayers, but they cannot induce the ferromagnetic order in CuMnSb. Combined,
the findings highlight a practical route towards fabrication of functional
materials in which coexisting polymorphs provide complementing functionalities
in one host.

CuMnSb, antiferromagnetism, ferromagnetism, polymorphism, phase coexistence,
high-resolution transmission electron microscopy, first principles
calculations

##  I Introduction

One of the most challenging and long-standing problems in fundamental
magnetism is a competition between ferromagnetic and antiferromagnetic phases.
Their interplay at the interface results in a well known effect of the
exchange bias, Meiklejohn and Bean (1957); Nogues and Schuller (1999) which
fuels now a rapid development of spintronics Xiong  _et al._ (2022) and
unconventional computing. Borders  _et al._ (2019) The material class of
Heusler alloys was previously used to study the origin of the transition
between magnetic phases because it offers a wide spectrum of functionalities.
Wollmann  _et al._ (2017) Indeed, Heusler alloys exhibit ferromagnetic (FM),
antiferromagnetic (AFM), and canted ferromagnetic order. This indicates that
different types of magnetic coupling are competing in this family. Moreover,
some of its members display structural polymorphism, which allows studying
relationships between the crystalline phase, the magnetic phase, and the
corresponding electronic structure.

Heusler alloys incorporate full-Heusler (X2YZ) and half-Heusler (XYZ)
variants, where X and Y stand for transition metals, whereas Z denotes anions
from the main group. In this class, qualitative changes in material
characteristics can be achieved by chemical substitution on either the
transition metal cation or on the anion sublattice. Typically, the change of
the cation does not change the crystal structure, but it can induce a
crossover between the AFM and the FM magnetic phases. A rarely met complete
solubility with only marginally affected crystallinity of the otherwise
chemically homogenous systems allowed to study the FM-AFM phase competition in
detail. The prominent examples are quaternary solid solutions such as
Ru2Mn1-xFexSn Douglas  _et al._ (2016); Decolvenaere  _et al._ (2019); McCalla
_et al._ (2021) Heuslers, and Co1-xNixMnSb, Cong  _et al._ (2010); Yuan  _et
al._ (2016) Cu1-xNixMnSb, Halder  _et al._ (2011); Ren  _et al._ (2005a, b)
Co1-xCuxMnSb, Duong  _et al._ (2007) and Cu1-xPdxMnSb Endo (1970) half-
Heuslers. In the latter case, the crossover between AFM to FM phases is
related to a change in the electronic structure from semimetallic to half-
metallic. Galanakis  _et al._ (2008); Şaşıoğlu  _et al._ (2008)

Cu–based CuMnZ compounds are antiferromagnets. This feature attracts attention
given the recent progress achieved in the AFM spintronics. Jungwirth  _et al._
(2018) Of particular interest is CuMnAs, with a high Néel temperature
$T_{\mathrm{N}}=480$ K. Wadley  _et al._ (2015) In this case, features
essential for applications, such as anisotropic magnetoresistance, Wang  _et
al._ (2020); Poole  _et al._ (2022) current-induced electrical switching of
the Néel vector Wadley  _et al._ (2016) and of the magnetic domains,
Grzybowski  _et al._ (2017) have been demonstrated.

The AFM order of CuMnZ is independent of the actual crystalline structure. The
equilibrium structure of bulk CuMnP and CuMnAs is orthorhombic, while that of
CuMnSb is half-Heusler cubic, referred to below as the $\alpha$ phase. On the
other hand, epitaxial growth can stabilizes metastable phases. This is the
case of epitaxial layers of CuMnAs, grown on both GaP Wadley  _et al._ (2015);
Reimers  _et al._ (2022); Wang  _et al._ (2020); Poole  _et al._ (2022); Linn
_et al._ (2023) and GaAs Grzybowski  _et al._ (2017) substrates, which
crystalize in the tetragonal structure, referred to below as the $\beta$
phase. Theoretical investigations of the crystalline properties of CuMnZ
series show that the total energy difference between the cubic and
orthorhombic phase is about 1 eV per f.u. (formula unit) for CuMnP, and about
0.5 eV per f.u. for CuMnAs. Máca  _et al._ (2012) This suggests that the
orthorhombic phase of CuMnSb, the last member of the CuMnZ series, may not be
stable, and indeed the stable structure is the $\alpha$ phase. However, as we
show here, epitaxial stabilization of CuMnSb in the $\beta$ phase is in
principle possible, because the calculated energy difference between
$\alpha$–CuMnSb and $\beta$–CuMnSb is small, about 0.12 eV per f.u. Moreover,
the $\beta$–CuMnSb polymorph becomes stable at pressures above 7 GPa. Malavi
_et al._ (2018)

Concerning the magnetic properties, the Néel temperature of both orthorhombic
CuMnAs and $\beta$–CuMnAs is well above the room temperature, Wadley  _et al._
(2015) whereas that of $\alpha$–CuMnSb is lower, about 60 K. Endo  _et al._
(1968); Scheffler  _et al._ (2022) Theory agrees with experiment, since
according to Ref. Máca  _et al._ , 2011, in the orthorhombic CuMnP and CuMnAs,
the AFM order is more stable than the FM by about 250 meV/Mn. This energy
difference is smaller in the cubic phase of CuMnZ compounds, for which the AFM
order is lower in energy than FM one by about 50 meV per f.u. Jeong  _et al._
(2005); Máca  _et al._ (2011) Finally, the AFM order of $\alpha$–CuMnSb is
stable under applied magnetic field, as $T_{\mathrm{N}}$ does not change up to
50 Tesla. Doerr  _et al._ (2004)

Turning to the electronic structure of the CuMnP-CuMnAs-CuMnSb series we
observe that the character of the energy band gap depends on the anion.
Similar to the case of e.g. zinc blende semiconductors, the band gap decreases
with the increasing atomic number of the anion. Máca  _et al._ (2011) Indeed,
CuMnP is a semiconductor, CuMnAs has a practically vanishing band gap, and
CuMnSb is a semimetal. com (a)

Here we experimentally confirm a puzzling coexistence of AFM and FM phases in
epitaxial stoichiometric CuMnSb films, observed by us previously, Scheffler
_et al._ (2020) and explain the underlying mechanism responsible for this
effect. A fine analysis of transmission electron microscopy (TEM) images, Sec.
II.2, points to the formation of tetragonal $\beta$–CuMnSb inclusions embedded
coherently within the cubic $\alpha$–CuMnSb host. The tetragonal structure of
these inclusions is the same as that of the tetragonal $\beta$–CuMnAs.
Magnetic properties of our films, Sec. II.3, demonstrate coexistence of two
magnetic phases: apart from the dominant AFM one, expected for CuMnSb, the
measurements reveal the presence of a FM contribution. This is an unexpected
feature within the CuMnZ series, exhibiting the AFM order.

In Sec. III, we employ calculations based on the density functional theory to
assess properties of CuMnSb films. In agreement with the experiment,
$\beta$–CuMnSb is weakly metastable, but its magnetic ground state is FM. Band
structures of $\alpha$ and $\beta$ polymorphs are close, but changes in the
density of states at the Fermi level account for the change of the dominant
mechanism of the magnetic coupling from AFM superexchange to FM Ruderman-
Kittel- Kasuya-Yoshida (RKKY). Finally, in Sec. III.5 native point defects in
CuMnSb are examined to assess their possible influence on the magnetic
properties. Máca  _et al._ (2016) Our results indicate that the dominant
native defects in $\alpha$–CuMnSb are Mn antisites, and their presence in the
films can possibly account for small differences between the measured and the
calculated magnetic characteristics, but they do not stabilize the FM order of
$\alpha$–CuMnSb.

##  II Experimental Results

Figure 1: (a) High-angle annular dark-field scanning transmission electron
microscopy image of a CuMnSb layer in the [100] zone axis. The inset in the
top-right corner brings up a part of the image in atomic resolution, where
bright dots represent columns of Mn and Sb atoms. (b) Electron diffraction
pattern of the layer. (c) Schematics of the positions of Bragg’s spots from
(b). Big bullets represent the main reflections from the cubic CuMnSb
structure, whereas the open triangles mark the positions of the weak extra
reflections. The orientation of the triangles follows from the analysis of the
data in panels (d-g). (d-e) Blown up two regions from (a), in which either
vertical or horizontal strips dominate. The corresponding Fourier transforms
are showed in the top right corners of both panels. (f) Selected area electron
diffraction pattern taken at the regions dominated by the vertically oriented
strips. (g) Diffraction intensity profiles taken along the horizontal [010]*
and the vertical [001]* lines passing through the center the diffraction
pattern. The solid line corresponds to the horizontal [010]* direction and the
dashed one to the vertical [001]* one in panel (f). Stars denote directions in
the reciprocal space. The arrows $\alpha$ and $\beta$ indicate the length of
$\alpha$–2g(002) and $\beta$–2g(002) diffraction vectors, respectively.

###  II.1 Experimental Methods

Growth conditions.

CuMnSb layers about 200 nm thick are grown by molecular beam epitaxy. Separate
growth chambers connected by an ultra-high vacuum transfer system are used for
the growth of the individual layers. Low tellurium-doped epi-ready GaSb (001)
wafers are used as substrates. Prior to the growth, the natural oxide layer is
desorbed in an Sb atmosphere. Then, 150 nm thick GaSb buffer layers are grown
on the substrates to ensure a high-quality interface for the growth of CuMnSb.
The GaSb buffer layers are grown at a substrate temperature of
$530^{\circ}\mathrm{C}$ and a beam equivalent pressure of $4.0\times 10^{-6}$
mbar and $5.3\times 10^{-7}$ mbar for Sb and Ga, respectively. Sb supply is
facilitated by a single-filament effusion cell, while Ga is provided by a
double-filament effusion cell.

A substrate temperature of $250^{\circ}\mathrm{C}$ is used for the growth of
CuMnSb films. The corresponding beam equivalent pressures are as follows:
$\mathrm{BEP}_{\mathrm{Cu}}=5.80\times 10^{-9}$ mbar,
$\mathrm{BEP}_{\mathrm{Mn}}=9.03\times 10^{-9}$ mbar, and
$\mathrm{BEP}_{\mathrm{Sb}}=4.23\times 10^{-8}$ mbar. Cu is supplied by a
double filament effusion cell, while Mn and Sb are supplied by single filament
effusion cells. Following the growth of CuMnSb, a 2.5 nm thick layer of Al2O3
is deposited on the samples through a sequential process of aluminum DC
magnetron sputtering and oxidation. Please, refer to Ref. Scheffler  _et al._
, 2022 for a comprehensive analysis of the growth process and physical
properties of the CuMnSb layers produced using the methodology outlined above.

Transmission Electron Microscopy.  Specimens for the transmission electron
microscopy (TEM) investigations are prepared by the focused ion beam method in
the form of lamellas cut along the [100] and [110] directions, i.e.,
perpendicularly to the surface (001) plane. Titan Cubed 80-300 electron
transmission microscope operating with accelerating voltage 300 kV and
equipped with energy-dispersive X-ray spectrometer (EDXS) is used for the
study. Most of the investigations are done on Cu grids, but for EDXS elemental
analysis a Mo grid is used to avoid interference of Cu fluorescence signal
from the grid. This analysis yields percentage atomic concentration at 37(3) :
32(5) : 31(7) for Cu, Mn, and Sb, respectively, which, within the experimental
errors (given in the parentheses), correspond to the expected stoichiometric
ratio of 33 : 33 : 33.

SQUID Magnetometry.  Magnetic characterization is performed in a commercial
superconducting quantum interference device (SQUID) magnetometer MPMS XL7. The
magnetic moment of antiferromagnetic layers is generally very weak and by far
dominated by the magnetic response of the bulky semiconductor substrates.
Therefore, to counter act the typical shortcomings of commercial magnetometers
built around superconducting magnets Sawicki  _et al._ (2011) and to minimize
subtraction errors during data reduction we actively employ the in situ
compensation. Gas and Sawicki (2022) It allows us to reduce the coupling of
the signal of the substrates to about 10% of their original strength. The
actual effectiveness of the compensation depends on the mass of the sample and
its orientation with respect to the SQUID pick-up coils. Sawicki  _et al._
(2011); Stamenov and Coey (2006) We also strongly underline the importance of
a thorough mechanical removal of the metallic MBE glue from the backside of
the samples for any magnetic studies. Its strongly nonlinear magnetic
contribution can be of the same magnitude as that of the layer of interest.
Gas  _et al._ (2021) To accurately establish the magnitude of magnetic moment
specific to CuMnSb we measure a reference sample grown without the CuMnSb
layer Scheffler  _et al._ (2022) using the same sample holder and following
exactly the same experimental sequence as that executed for the investigated
samples.

###  II.2 Structural characterization

An exemplary atomic resolution high-angle annular dark-field scanning
transmission electron microscopy (HAADF/STEM) image obtained for the [100]
zone axis (the direction of the projection) is in Fig. 1 (a). It confirms a
high quality cubic constitution of the material, as it is underlined in the
inset. However, at the contrast chosen here, the image in this field of view
reveals the presence of stripe-like features, which are the main subject of
this analysis. In this image, the apparent lengths and widths of the strips
are about 40 nm and about 4 nm, respectively, running predominantly either
vertically or horizontally in this particular projection. On other images, the
strips exhibit a relatively wide distribution of lengths in the 10-100 nm
window. Since similarly distributed shadowy stripes are observed also in the
[110] zone axis, we conclude that they form along all three principal
crystallographic directions without any particular preferences. The expected
F$\overline{4}3$m cubic structure of $\alpha$–CuMnSb is clearly confirmed by
the fourfold symmetry of the dominant (bright) spots seen on electron
diffraction pattern presented in Fig. 1 (b).

Importantly, the diffraction pattern in Fig. 1 (b) contains also a second set
of much fainter reflections, situated halfway between two adjacent reflections
of the main pattern. This indicates the presence of a second crystallographic
$\beta$ phase, which periodicity in the corresponding direction is doubled
relative to that of $\alpha$–CuMnSb, but otherwise coherent with this host
structure. We bring all the Bragg’s spots up in Fig. 1 (c), in which the
bullets represent the main reflections from $\alpha$–CuMnSb, whereas the open
triangles mark the positions of the weak ones, which are forbidden for this
structure.

The presence of $\beta$–CuMnSb is further substantiated by the inspection of
the two close-ups from Fig. 1 (a), shown in Fig. 1 (d) and (e). At this
magnification they reveal that, on top of the otherwise cubic arrangement of
atomic columns, the strips’ brightness alternates every second $\\{002\\}$
plane along the direction perpendicular to strip’s long axis. The modulation
is vertical in Fig. 1 (d), whereas it goes horizontally in Fig. 1 (e). The top
right corners of these figures contain the corresponding Fourier transform of
the parent image, and, similarly to Fig. 1 (b), both patterns are dominated by
the main reflections of $\alpha$–CuMnSb. The additional spots are embedded
either along vertical [Fig. 1 (d)] or horizontal [Fig. 1 (e)] lines, i.e., the
presence of vertical and horizontal orientations is mutually exclusive. This
feature is reflected in Fig. 1 (c), where the additional spots are marked by
differently oriented triangles. The triangles with apexes directed vertically
correspond to the vertical orientation of the brightness modulation in Fig. 1
(d), whereas the horizontal direction of apexes corresponds to the horizontal
modulation.

Figure 2: Crystal structures of (a) $\alpha$–CuMnSb with the cubic lattice
constant $a$, (b) tetragonal $\beta$–CuMnSb with the lattice constants $a$ in
the ($x$, $y$) plane and $c$ in the [001] direction, and (c) Cu3Mn2Sb2.

Based on the data shown above we propose that the second phase of CuMnSb,
present in our films in the form of strips, is a tetragonal structure, which
also is the structure of epitaxial CuMnAs, Wadley  _et al._ (2015); Reimers
_et al._ (2022); Wang  _et al._ (2020); Poole  _et al._ (2022); Linn  _et al._
(2023); Grzybowski  _et al._ (2017) and of CuMnSb at high pressures. Malavi
_et al._ (2018) This $\beta$–CuMnSb polymorph is shown in the panel (b) of
Fig. 2. The difference between $\alpha$ and $\beta$ phases consists in the
location of Cu ions: in the $\alpha$ phase every (001) plane between two
consecutive MnSb planes is half-occupied by Cu, whereas in the $\beta$ phase
Cu ions completely fill up every second (001) plane, and the overall
stoichiometry of the material is preserved.

Details regarding $\beta$–CuMnSb can be inferred from selected area electron
diffraction (SAED) patterns taken at regions with different orientations of
the strips. Diffraction pattern of an area dominated by the vertically
oriented strips is shown in more detail in Fig. 1 (f). In agreement with the
Fourier transforms, SAED shows the occurrence of specific reflections
corresponding to this particular orientation. The reflections common to both
the cubic $\alpha$ and the tetragonal $\beta$ polymorphs are split along the
[010]* direction, i.e., orthogonal to the strip’s axis, whereas the weak spots
of the $\beta$ phase are not split and are commensurate with the cubic phase.
(A star denotes a direction in the reciprocal space.)

We quantify the effect analyzing intensity profiles taken along lines passing
through the center of diffraction. The profiles are superimposed, and
presented in Fig. 1 (g). The profile along the [001]* direction reflects the
periodicity of $\alpha$–CuMnSb, while that along [010]* is additionally split.
From the Figure it follows that in our specimens the $c$ lattice parameter of
the $\beta$–CuMnSb strips is equal to that of the host $\alpha$–CuMnSb, 6.2(1)
Å, whereas the $a$ and $b$ parameters of the $\beta$ phase, 5.8(1) Å, are
smaller by about 7%. Analogous features are observed for the [010]-oriented
strips.

The existence of such a significant strain is confirmed by the calculation of
strain maps. We apply the geometrical phase analysis method Du (2018) for the
main image presented in Fig. 1 (a), and the results are presented in Fig. 3
(a) and (b) for the horizontal, $\epsilon_{xx}$, and the vertical,
$\epsilon_{zz}$, components of strain, respectively. It is seen that stripes’
strain is negative (dark shade) perpendicular to strips and almost zero along
the strips. For example, on the horizontal strain map [Fig. 3 (a)] only
vertical strips are visible because they are compressed horizontally, whereas
the horizontal strips are invisible because they are not deformed in the
horizontal direction.

Figure 3:  Strain maps of image shown in Fig. 1 (a). (a) The horizontal
component of strain $\epsilon_{xx}$, and (b) the vertical one,
$\epsilon_{zz}$. Geometrical phase analysis method has been applied. Du (2018)

The calculated properties of $\beta$–CuMnSb, such as its lattice parameters,
stability, and magnetic properties, are discussed in detail in Sec. III.3.
Anticipating, we mention that they are consistent with experiment. We have
also considered a second possible structure which is (almost) compatible with
the TEM data, Cu3Mn2Sb2, depicted in Fig. 2 (c). However, this compound is
higher in energy than the $\beta$ phase, and was dropped from further
considerations.

###  II.3 Magnetic properties

The temperature $T$ dependence of magnetization, $M(T)$, of the 200 nm thick
layer of CuMnSb, is depicted in Fig. 4 (a). The clear kink on $M(T)$ at
$T_{\mathrm{N}}=62$ K marks the position of the paramagnetic to
antiferromagnetic Néel transition in the layer. This value corresponds
precisely to the values of $T_{\mathrm{N}}$ established previously for
CuMnSb/GaSb layers of the thickness $t\geq 200$ nm, what, indirectly,
indicates stoichiometric material composition of this layer. Scheffler  _et
al._ (2022)

Figure 4:  Results of temperature dependent magnetic studies of 200 nm thick
CuMnSb layer (green bullets). (a) Magnetization $M$ established in a bias
field of $H=10$ kOe, and (b) the corresponding inverse of the molar magnetic
susceptibility $\chi_{m}^{-1}$. The solid and dashed orange lines indicate the
Curie-Weiss behavior of $\chi_{m}^{-1}(T)$ for $T>200$ K and for
$T_{\mathrm{N}}<T<200$ K, respectively. The arrows indicate the position of
the Néel temperature, $T_{\mathrm{N}}=62$ K, on both panels.

More specific information about the magnetic state of that sample is obtained
from the examination of the temperature dependence of the inverse magnetic
susceptibility, $\chi^{-1}(T)$, shown in Fig. 4 (b). We take here
$\chi(T)=M(T)/H$, where $H=10$ kOe is the external magnetic field applied
during the measurements. $\chi^{-1}(T)$ can be approximated by two straight
lines. The abscissa of the first one, which approximates $\chi^{-1}(T)$ above
200 K (the solid orange line in Fig. 4), yields exactly the same magnitude of
the Curie-Weiss temperature $T_{\mathrm{CW}}=-100(5)$ K as that established
previously for a thicker 510 nm layer, for which $\chi^{-1}(T)$ formed a
single straight line above $T_{\mathrm{N}}$ at the same experimental
conditions. Scheffler  _et al._ (2022) Also the slope of this line yields the
value of the effective magnetic moment
$m_{\mathrm{eff}}=5.4(1)\mu_{\mathrm{B}}$ per f.u., which is very close to
that found previously, $m_{\mathrm{eff}}=5.6\mu_{\mathrm{B}}$ per f.u.
Scheffler  _et al._ (2022) This correspondence indicates that the high
temperature part of $\chi^{-1}(T)$ is determined predominantly by AFM
excitations in the paramagnetic matrix of CuMnSb.

The abscissa of the second straight line, which approximates the experimental
data between $T_{\mathrm{N}}$ and about 200 K (marked as the dashed orange
line in Fig. 4), yields a more positive value of the Curie-Weiss temperature,
$T_{CW}^{\prime}=-10(10)$ K. This clear positive shift of $T_{CW}$ indicates
the existence of a ferromagnetic contribution to the overall antiferromagnetic
phase of the material, and that these FM excitations gain in importance below
about 200 K. Interestingly, a somewhat stronger effect, characterized by a
change of sign of $T_{CW}$ to $T_{\mathrm{CW}}^{\prime}=+60(10)$ K, was noted
in 40 nm CuMnSb layer grown on InAs. Scheffler  _et al._ (2020) In accordance
with the findings of structural characterization we propose that the by far
stronger AFM component originates from the dominant $\alpha$ phase, whereas
the FM one is brought about by $\beta$–CuMnSb polymorph.

Turning now to the magnetic characteristics established here for
$\alpha$–CuMnSb we note that they are close to those reported previously, as
shown in Tab. 1. The published data exhibit a certain distribution, which may
indicate that other factors, such as a weak crystalline disorder, may be at
work. In particular, either additional Mn interstitial ions or CuMn-MnCu
antisite pairs are likely to form. Máca  _et al._ (2016) The presence of such
defects was suggested to stabilize the experimentally observed AFM
{111}-oriented phase of $\alpha$–CuMnSb. Máca  _et al._ (2016) Finally, we do
not observe a canted AFM order at low temperatures Regnat  _et al._ (2018) in
any of our samples.

Table 1: Experimental Néel temperature $T_{\mathrm{N}}$, effective Curie-Weiss magnetic moment of Mn ions $m_{\mathrm{eff}}$(Mn), and Curie-Weiss temperature $T_{\mathrm{CW}}$ of $\alpha$–CuMnSb. Measured orientation of the AFM axis is also given (n.e. = not established). Refs. Forster  _et al._ , 1968 and Regnat  _et al._ , 2018 report the saturation Mn moment.  $T_{\mathrm{N}}$ (K) |  $m_{\mathrm{eff}}$(Mn) ($\mu_{\mathrm{B}}$) |  $T_{\mathrm{CW}}$ (K) | AFM order | Ref.
---|---|---|---|---
- | 3.9(1) | - | [111] | Forster  _et al._ ,1968
55 | 3.95 | -160(8) | [111] | Regnat  _et al._ ,2018
55 | 5.4 | -160 | [111] | Endo  _et al._ ,2021
62 | 5.2 | -120 | n.e. | Helmholdt  _et al._ ,1984
50 | 6.3 | -250 | n.e. | Bœuf  _et al._ ,2006
50 | - | - | n.e. | Máca  _et al._ ,2012
62 | 5.9 | -65 | n.e. | Scheffler  _et al._ ,2020
62 | 5.6 | -100 | n.e. | Scheffler  _et al._ ,2022
62 | 5.5 | -100 | n.e. | this work

##  III Theory

###  III.1 Theoretical Methods

Calculations are performed within the density functional theory Hohenberg and
Kohn (1964); Kohn and Sham (1965) in the generalized gradient approximation of
the exchange-correlation potential proposed by Perdew, Burke and Ernzerhof.
Perdew  _et al._ (1996) To improve description of $3d$ electrons, the Hubbard-
type $+U$ correction on Mn is added. Anisimov  _et al._ (1991, 1993);
Cococcioni and de Gironcoli (2005) The parameter $U(\mathrm{Mn})=1$ eV
reproduces the known formation energy of the intermetallic CuMn alloy and
gives a reasonable value of the Mn cohesive energy. We use the pseudopotential
method implemented in the Quantum ESPRESSO code, ESPRESSO  with the valence
atomic configuration $4s^{1.5}p^{0}3d^{9.5}$ for Cu,
$3s^{2}p^{6}4s^{2}p^{0}3d^{5}$ for Mn and $5s^{2}p^{3}$ for Sb ions. The
plane-waves kinetic energy cutoffs of 50 Ry for wave functions and 250 Ry for
charge density are employed. Finally, geometry relaxations are performed with
a 0.05 GPa convergence criterion for pressure. In defected crystals ionic
positions are optimized until the forces acting on ions become smaller than
0.02 eV/Å.

The properties of defected $\alpha$–CuMnSb are examined using cubic $2a\times
2a\times 2a$ supercells with 96 atoms (i.e., 32 f.u.), while magnetic order of
ideal crystals are checked using the smallest possible supercells. Here $a$ is
the equilibrium lattice parameter. The $k$-space summations are performed with
a $6\times 6\times 6$ $k$-point grid for the largest supercell, and
correspondingly denser grids are used for smaller cells.

Magnetic interactions and magnetic order depend on several factors, such as
the exchange spin splitting of the $d$(TM) shells, charge states of TM ions,
concentration of free carriers and their spin polarization, and the density of
states (DOS) at the Fermi energy $E_{\mathrm{F}}$. These factors are
interrelated, and are calculated self-consistently within ab initio approach.

Considering first the localized magnetic moments we note that spin
polarization of Co, Ni, and Cu ions in XMnZ compounds practically vanishes,
while that of the $d$(Mn) shell is substantial. Jeong  _et al._ (2005);
Galanakis  _et al._ (2008); Şaşıoğlu  _et al._ (2008) The robustness of the Mn
magnetic moment results from the large, 3 – 5 eV, spin splitting of the
$3d$(Mn) states. In fact, in XMnZ the $d$(Mn) spin up channel is occupied,
while most of the spin down $d$(Mn) states lay above the Fermi level. Here,
one can observe that spin polarization of the $d$(TM) electrons in free atoms
depends on the difference in the number of spin up and spin down electrons,
which is the highest in the case of Mn. Consequently, the Mn spin polarization
persists in XMnZ. On the other hand, spin splitting of $d$ electrons of Co and
Ni atoms is smaller, and thus it vanishes in XMnSb hosts, see the analysis for
TM dopants in ZnO. Ciechan and Boguslawski (2018)

In CuMnSb, the magnetic sublattice consists of Mn ions, which are second
neighbors distant by 4.3 Å. Therefore, the direct exchange coupling between
two Mn ions, given by overlaps of their $d$(Mn) orbitals, is negligibly small.
The remaining indirect exchange coupling is the sum of two contributions, and
the exchange constant $J_{indirect}=J_{sr}+J_{RKKY}$. da Silva and Falicov
(1972); Galanakis  _et al._ (2008); Şaşıoğlu  _et al._ (2008) The first term
$J_{sr}$ has a short-range AFM character, and it is inversely proportional to
the energy distance between the unoccupied $d$(Mn) states and
$E_{\mathrm{F}}$. The second coupling channel is of RKKY type mediated by free
carriers. This channel depends on the detailed electronic structure in the
vicinity of $E_{\mathrm{F}}$, and $J_{RKKY}$ is proportional to
DOS$(E_{\mathrm{F}})$. In particular, CoMnSb and NiMnSb half-metals are FM,
while CuMnP and CuMnAs insulators are AFM. As we show here, CuMnSb is the
border case.

###  III.2 Crystal and magnetic properties of $\alpha$–CuMnSb

Figure 5: Side view of the considered magnetic cells: (a) antiferromagnetic
with {001} planes with the same Mn spins shown in the $a\times a\times a$ cell
(AFM001), (b) antiferromagnetic with double {001} planes of the same spins on
Mn ions in the $a\times a\times 2a$ cell (AFM002), (c) antiferromagnetic with
ferromagnetic {111} planes realized in the $a\sqrt{2}\times a\sqrt{2}\times
a\sqrt{2}$ cell (AFM111), and (d) ferromagnetic in primitive rhombohedral cell
$a/\sqrt{2}\times a/\sqrt{2}\times a/\sqrt{2}$ (FM). Mn atoms with different
spin directions are indicated as Mn$\uparrow$ and Mn$\downarrow$.

A rhombohedral primitive cell of $\alpha$–CuMnSb contains one formula unit.
This structure consist in four interpenetrating fcc sublattices, one of them
being empty. The consecutive (001) MnSb planes are followed by the ”half-
empty” Cu planes, in which the planar atomic density is twice lower. The cubic
unit cell is presented in Fig. 2 (a). Local coordination of Mn ions can be
relevant from the point of view of magnetic interactions. With this respect we
notice that the magnetic coordination of an Mn ion consists in 12 equidistant
Mn atoms at $a/\sqrt{2}$. Moreover, the short-range coupling between two Mn
nearest neighbors is mediated through the closest ions, and it occurs through
one Mn-Cu-Mn ”bridge” and two Mn-Sb-Mn ”bridges”.

We consider four magnetic phases of $\alpha$–CuMnSb. The corresponding
supercells are shown in Fig. 5. Antiferromagnetic order with parallel Mn spins
in the (001) planes, AFM001, is calculated using the cubic $a\times a\times a$
cell with 4 f.u. (12 atoms), and shown in Fig. 5 (a). The AFM order with a
period doubled in the [001] direction with parallel Mn spins in each (001)
plane, denoted as AFM002, is shown in Fig. 5 (b). The corresponding $a\times
a\times 2a$ cell contains 8 f.u., and is one of the possible supercells in
which this phase can be realized. In the AFM111 phase, the Mn spins are
parallel in each (111) plane, but the consecutive (111) planes are AFM, as
shown in Fig. 5 (c), and the corresponding rhombohedral unit cell
$a\sqrt{2}\times a\sqrt{2}\times a\sqrt{2}$ contains 8 primitive cells with 24
atoms. Finally, the FM phase requires a primitive cell $a/\sqrt{2}\times
a/\sqrt{2}\times a/\sqrt{2}$ with 1 f.u., presented in Fig. 5 (d).

Figure 6: Volume dependence of the total energy relative to the ground state $E_{0}=E_{AFM001}(V_{0})$ of $\alpha$–CuMnSb in the AFM001, AFM002, AFM111 and FM phases. Both volume and energy are per formula unit. Lines are fitted to the calculated values (symbols).  Table 2:  The calculated lattice parameter $a$, the saturation Mn magnetic moment, $m_{\mathrm{sat}}$, and the energy of the given magnetic order relative to $\alpha$–CuMnSb in the AFM001 ground state, $\Delta E_{tot}$. All energies are per one formula unit. Our measured TEM values are also given.  |  $a$ (Å) |  $c$ (Å) | $m_{\mathrm{sat}}(\mu_{\mathrm{B}})$ |  $\Delta E_{tot}$ (meV)
---|---|---|---|---
$\alpha$–CuMnSb |  |  |  |
AFM001 | 6.10 | - | 4.59 | 0
AFM002 | 6.09 | - | 4.61 | 3
AFM111 | 6.10 | - | 4.55 | 7
FM | 6.11 | - | 4.70 | 19
TEM | 6.2(1) | - | - | -
$\beta$–CuMnSb |  |  |  |
AFM | 5.83 | 6.40 | 4.48 | 113
FM | 5.88 | 6.28 | 4.52 | 102
TEM | 5.8(1) | 6.2(1) | - | -

The obtained results are collected in Tab. 2 and in Fig. 6. The ground state
structure is AFM001, but the experimentally observed AFM111 is only 7 meV per
f.u. higher in energy. The least stable is the FM order, higher in energy than
AFM001 by about 20 meV per f.u. The equilibrium lattice parameters
$a\approx~{}6.1$ Å are practically independent of the magnetic order, and
close to the experimental value 6.088 Å. Forster  _et al._ (1968) Some phases
are characterized by a small distortion of the cubic symmetry caused by
different bond lengths between ferromagnetically and antiferromagnetically
oriented Mn ions. Differences in the lattice parameters between various
magnetic phases are below 0.01 Å, and are not reported in the Table. Similar
results for the AFM001 order were obtained in Ref. Máca  _et al._ , 2016,
while in Refs Jeong  _et al._ , 2005; Galanakis  _et al._ , 2008 the AFM order
is more stable than FM by 50 and 90 meV per Mn, respectively.

The last property reported in Tab. 2 is the saturation magnetic moment of Mn,
which also is similar in all phases, and equal to about $4.6\mu_{\mathrm{B}}$.
This value corresponds to the Curie-Weiss moment of $5.5(1)\mu_{\mathrm{B}}$,
and compares favorably with the experimental values given in Tab. 1.

The obtained results allow estimating the relative roles of the short- and
long-range contributions to the magnetic coupling. To this end, we assume the
hamiltonian in the form $H_{ex}=-J/2\sum_{i,j}\vec{s}_{i}\vec{s}_{j}$, where
the short range interaction is limited to the Mn NNs neighbors, and the long-
range term is neglected. The spin value, $s_{i}\approx 2.3$, is one half of
the calculated magnetic moment of Mn.

The exchange constant $J$ is positive (negative) for the FM (AFM) coupling,
and is obtained by comparing energies of various magnetic orders. In the
AFM001 phase, each Mn ion has 4 ferromagnetically oriented Mn NNs in the (001)
plane and 8 antiferromagnetically oriented Mn NNs in the two adjacent planes.
For the remaining magnetic phases, the energies calculated relative to the
ground state $E_{0}\equiv E_{\mathrm{AFM001}}$ depend on the magnetic order as
shown in Tab. 2. These results give the coupling constant in the range
$-0.6\geq J_{sr}\geq-0.2$ meV. This spread is quite large and cannot be
explained by (negligible) changes in atomic distances in cells with different
magnetic ordering. Therefore, we conclude that the Heisenberg nearest neighbor
model does not describe magnetic properties of $bulk$ phases. Indeed, such a
model is not appropriate for metallic or semimetallic systems such as
$\alpha$–CuMnSb, where the long-range RKKY coupling is present.

An opposite conclusion comes from the analysis of $\it single\ spin$
excitations from the AFM001 ground state. We use a $2a\times 2a\times 2a$
supercell to calculate the energy differences $\Delta E$ for the following
cases, in which we change (i) spin of one Mn ion, $1\mathrm{Mn}\\!\uparrow\to
1\mathrm{Mn}\\!\downarrow$, called a single spin-flip, (ii)
$2\mathrm{Mn}\\!\uparrow\to 2\mathrm{Mn}\\!\downarrow$ for spins of two
nearest Mn ions belonging to one layer and (iii) $2\mathrm{Mn}\\!\uparrow\to
2\mathrm{Mn}\\!\downarrow$ for two distant Mn ions. In these processes the
long-range coupling is not important, and indeed the calculated exchange
constant consistently is $J_{sr}\approx-0.4$ meV.

###  III.3 Crystal and magnetic properties of $\beta$–CuMnSb

We now consider two possible structures of the secondary phase proposed based
on the experimental results. They are characterized by doubling the
periodicity in the [001] direction. The unit cell of $\beta$–CuMnSb, shown in
Fig. 2, is tetragonally deformed relative to that of $\alpha$–CuMnSb, with the
corresponding lattice parameters $a=5.88$ Å and $c=6.275$ Å. They differ by
about 3 per cent from our calculated cubic $a$($\alpha$–CuMnSb) $=6.105$ Å.
The two interlayer spacings between the consecutive MnSb planes in the [001]
direction in the unit cell, shown in Fig. 2 (b), are quite different, namely
$d_{inter1}=2.80$ Å (no Cu), and $d_{inter2}=3.48$ Å (with Cu). Turing to the
magnetic order of $\beta$–CuMnSb, we find that the FM phase constitutes the
ground state with $m_{\mathrm{sat}}=4.6\mu_{\mathrm{B}}$ and is lower than the
AFM phase by 11 meV per f.u., as indicated in Tab. 2. Thus, the two
crystalline phases of CuMnSb are in two different magnetic phases.

The experimental Malavi  _et al._ (2018) lattice parameters of $\beta$–CuMnSb
reasonably agree with our values, i.e., the calculated $a=6.28$ Å and
$c/a=1.87$ are about 2% larger than those measured for the $compressed$
crystal at the critical pressure of 7 GPa. On the other hand, the calculations
of Ref. Malavi  _et al._ , 2018 predict that the magnetic order of the $\beta$
phase is AFM, in striking contrast with our results. Also their calculated
$m_{\mathrm{sat}}(Mn)=3.8\mu_{\mathrm{B}}$ is substantially smaller than our
$4.6\mu_{\mathrm{B}}$. The origin of these discrepancies is not clear, but it
may be due to the different exchange-correlation functionals used, and/or to
application of the $+U$(Mn) correction in our calculations (which can affect
the results. Jeong  _et al._ (2005))

Figure 7: Bands and partial DOSs for (a) the AFM001 of $\alpha$–CuMnSb and for
(b) the FM state of $\beta$–CuMnSb obtained using the $a\times a\times c$
cell. The right panels show the partial DOSs for Cu and Mn ions, and thus
different contributions to spin-up and spin-down density of Mn$\uparrow$ are
exposed also in AFM case. In (b) the spin degeneracy is lifted.

The calculated total energy of the FM $\beta$–CuMnSb relative to the AFM
$\alpha$–CuMnSb is higher by 102 meV per f.u. This energy difference is not
large, being comparable to the growth temperature, which implies that the
$\beta$–CuMnSb polymorph can indeed form during epitaxy. We also stress that
stoichiometry of the $\alpha$ and $\beta$ phases is the same, which
facilitates formation of $\beta$–CuMnSb. Finally, the observed $\beta$–CuMnSb
inclusions are coherent, i.e., lattice matched, with the host structure. This
agrees with the fact that the calculated excess elastic energy of matching the
lattice parameters of the $\beta$ phase to the host $\alpha$ phase is very low
and ranges from 3 meV per f.u. (when the tetragonal $a$ parameter constrained
to the cubic $a=6.105$ Å) to 20 meV per f.u. (the tetragonal $c$ parameter
constrained to the cubic $a$).

The second considered possibility, Cu3Mn2Sb2 shown in Fig. 2 (c), is higher in
energy by 0.37 eV per f.u. in the Cu–rich conditions than the ideal CuMnSb,
i.e., by 0.27 eV per f.u. than $\beta$–CuMnSb, its stoichiometry is markedly
different, and thus we can eliminate this structure from considerations.

###  III.4 Energy band structures of $\alpha$–$\mathbf{CuMnSb}$ and
$\beta$–$\mathbf{CuMnSb}$

Figure 7 (a) shows the energy bands and DOS of the AFM001 $\alpha$–CuMnSb. We
see that this phase has a metallic character, however DOS at the Fermi level
is low. The states close to $E_{\mathrm{F}}$ are built from $s$, $p$ and $d$
states of all ions with similar weights. The low DOS($E_{\mathrm{F}}$) makes
CuMnSb almost semimetallic with a low electrical conductivity. Compatible with
the small DOS$(E_{\mathrm{F}})$ is the high resistivity measured in Ref.
Regnat  _et al._ , 2018; com, b.

Since the system is antiferromagnetically ordered, the total DOSs of spin-up
and spin-down states are the same. In Fig. 7 only contributions of the
$3d$(Mn) and $3d$(Cu) orbitals are presented to reveal magnetic properties. We
see that the exchange spin splitting of the $d$(Mn) shell is large, about 5
eV. The closely spaced levels contributing to the DOS maxima centered at 4 eV
below the Fermi energy are composed mainly of the $d$ states of both Cu and
Mn. Spin-up and spin-down $3d$(Cu) orbitals are almost completely occupied,
and thus Cu ions are non-magnetic. In turn, the majority spin states of the
$3d$(Mn) orbitals are completely occupied, while the minority spin states at 1
eV above the Fermi energy are partially filled thanks to a small overlap with
spin up states. As a result, a single Mn ion is in between the $d^{5}$ and
$d^{6}$ configuration, with the saturation magnetic moment of
$4.6\mu_{\mathrm{B}}$ consistent with Tab. 2. Our results for $\alpha$–CuMnSb
are close to those of Ref. Jeong  _et al._ , 2005. A similar electronic
configuration takes place in CuMnAs, where the spin-down Mn states are
partially filled. Máca  _et al._ (2017)

The overall band structure of the FM $\beta$–CuMnSb displayed in Fig. 7 (b) is
close to that of $\alpha$–CuMnSb, which is particularly clear when comparing
partial DOS of both phases. In particular, $m_{\mathrm{sat}}$(Mn) is about
$4.5\mu_{\mathrm{B}}$ in both phases, and energies of both $d$(Mn)- and
$d$(Cu)-related bands are largely independent of the actual crystal structure.
This similarity can be due to the fact that the MnSb (001) planes play a
dominant role, and the exact locations of the Cu ions are less important.

On the other hand, the calculated DOS($E_{\mathrm{F}}$) for the $\alpha$ phase
is 0.35 states per spin and f.u., while for the $\beta$ phase we find 1.26
states per spin and f.u., which is 3.6 times higher. As a consequence,
$\alpha$–CuMnSb is semimetallic, and the AFM order is dominant, while $\beta$
phase is more metallic in character, which in turn favors the RKKY-type
coupling and the FM order. This feature can explain the different magnetic
phases of the $\alpha$ and $\beta$ polymorphs.

Analysis of the calculated electronic structure of Heusler and half-Heusler
CuMnZ led Sasioglu et al. Şaşıoğlu  _et al._ (2008) to the conclusion that
when the spin polarization of conduction electrons is large, and the $d$(Mn)
spin down states are far above $E_{\mathrm{F}}$, then the RKKY coupling is
dominant, and one should expect the FM order, otherwise the short range AFM
coupling is dominant. Our results do not confirm this conclusion, and indicate
that the important factor determining the magnetic order is the
DOS($E_{\mathrm{F}}$).

###  III.5 Point native defects in $\alpha$–CuMnSb

Formation energy of a defect D is given by

$E_{form}(\mathrm{D})\\!=\\!E(\mathrm{CuMnSb\\!:\\!D})\\!-\\!E(\mathrm{CuMnSb})\\!+\\!\sum_{i}\\!{n_{i}\mu_{i}},$
(1)

where $E(\mathrm{CuMnSb})$ and $E(\mathrm{CuMnSb:D})$ are the total energies
of a supercell without and with a defect, and $n_{i}=+1(-1)$ corresponds to
the removal (addition) of one $i$th atom. $\mu_{i}$s are the variable chemical
potentials of atoms in the solid, which in general are different from the
chemical potentials $\mu_{i}(\mathrm{bulk})$ of the standard state of
elements, i.e., Cu, Mn and Sb bulk. Details of calculations of chemical
potentials are given in Appendix A.

The point native defects considered here are vacancies $V_{X}$, interstitials
Xi, and antisites XY (where X and Y are Cu, Mn, or Sb) for all three
sublattices. As it was mentioned above, the Cu sublattice is ”half- empty”
compared to the MnSb sublattice. Consequently, we consider here formation of
interstitials at the empty sites of the Cu sublattice only, and neglect other
possibilities, expected to have higher formation energies $E_{form}$. Thus,
the set of defects considered here only partially overlaps with that of Ref.
Máca  _et al._ , 2016. Of particular interest to the present study are defects
involving Mn ions, since they can influence magnetic properties of
$\alpha$–CuMnSb .Máca  _et al._ (2016) This is why we consider them more
extensively, after briefly analyzing the non-magnetic defects. The calculated
formation energies are summarized in Tab. 3. Because of the magnetic coupling,
formation energies of the Mn-related defects depend on the spin direction
relative to the spins of the host Mn neighbors. We consider possible spin
configurations shown in Fig. 8 (b).

Table 3: Formation energies (in eV) of isolated point defects in the Mn-rich conditions. In parentheses are Mn-related values corrected for $\Delta H_{f}(\mathrm{MnSb})=0.48$ eV, which correspond to the Mn-poor case. Cu defects |  |  Sb defects |  |  Mn defects |
---|---|---|---|---|---
$V_{\mathrm{C}u}$ |  -0.01 |  $V_{\mathrm{Sb}}$ |  2.45 |  $V_{\mathrm{Mn}}$ |  -0.07 (-0.55)
$\mathrm{Cu}_{i}$ |  1.00 |  $\mathrm{Sb}_{i}$ |  3.64 |  $\mathrm{Mn}_{i}$ |  0.74–0.95 (1.22–1.43)
$\mathrm{Cu}_{\mathrm{M}n}$ |  0.48 (0.00) |  $\mathrm{Sb}_{\mathrm{Mn}}$ |  1.60 (1.12) |  $\mathrm{Mn}_{\mathrm{Cu}}$ |  0.00–0.62 (0.48–1.10)
$\mathrm{Cu}_{\mathrm{Sb}}$ |  1.20 |  $\mathrm{Sb}_{\mathrm{Cu}}$ |  2.82 |  $\mathrm{Mn}_{\mathrm{Sb}}$ |  1.18 (1.65)

Figure 8: All possible spin orientations of Mn ions in $\alpha$–CuMnSb AFM001.
In the ground state configuration, the Mn spins are parallel within each MnSb
(001) layer, and the consecutive MnSb (001) layers are AFM, as shown in (a).
(b) Mn $\mathrm{Mn}_{\mathrm{Cu}}$ antisites and $\mathrm{Mn}_{i}$
interstitials can assume 5 different local spin configurations. The
corresponding spin-dependent formation energies in eV are given by the numbers
below.

Formation energy determines the corresponding equilibrium concentration [$D$]
of a defect $D$ according to

$[D]=N_{0}\exp[-E_{form}(D)/k_{\mathrm{B}}T],$ (2)

where $k_{\mathrm{B}}$ is the Boltzmann constant and $N_{0}$ is the density of
the relevant lattice sites. Details of the calculations of $E_{form}$ are
provided in Supporting Information. To put the calculated formation energies
into a proper context, we note that if the growth temperatrure
$T_{growth}=250^{0}$C and $E_{form}=0.1$ eV, then
$\exp(-E_{form}/k_{\mathrm{B}}T_{growth})=0.1$, which corresponds to a high 10
atomic per cent concentration of this defect on the considered sublattice. On
the other hand, if $E_{form}=1$ eV, then
$\exp(-E_{form}/k_{\mathrm{B}}T_{growth})=9\times 10^{-11}$, which implies a
negligible defect concentration.

Sb sublattice. The prohibitively high values of $E_{form}$ demonstrate that
$V_{\mathrm{Sb}}$ and $\mathrm{Sb}_{i}$ should not form. Similarly, formation
energies of $\mathrm{Sb}_{\mathrm{C}u}$, $\mathrm{Sb}_{\mathrm{Mn}}$,
$\mathrm{Cu}_{\mathrm{Sb}}$ and $\mathrm{Mn}_{\mathrm{Sb}}$ antisites exceed 1
eV, and those defects are not expected to be present at high concentrations.
Consequently, the Sb sublattice is thermodynamically stable, robust, and
constitutes a defect-free back-bone of CuMnSb.

Cu and Mn sublattices. The properties of both Cu and Mn sublattices are
opposite to those of the Sb sublattice, as they are susceptible to contain
defects. In particular,

(i) Both the Cu and Mn vacancies can be present at high concentrations, since
their $E_{form}$ are low.

(ii) Formation energy of Cu interstitials at the Cu sublattice,
$E_{form}(\mathrm{Cu}_{i})=1$ eV, is relatively high, and their concentrations
are negligible. Additionally, the high formation energy of $\mathrm{Cu}_{i}$
interstitials is consistent with the sparse character of the Cu sublattice in
$\alpha$–CuMnSb.

(iii) Formation of $\mathrm{Mn}_{i}$ interstitials at the Cu sublattice is
characterized by $E_{form}$ = 0.7-1.4 eV, depending on the spin direction and
conditions of growth, and therefore they are not expected to be present at
high concentrations, especially in the Mn-poor conditions.

(iv) Finally, $\mathrm{Mn}_{\mathrm{Cu}}$ antisites, with formation energies
ranging from 0 to about 1 eV, can be present at high concentrations,
comparable to those of $V_{\mathrm{Mn}}$ and $V_{\mathrm{Cu}}$.

In brief, low formation energies are found for three defects, namely the
$V_{\mathrm{Cu}}$ and $V_{\mathrm{Mn}}$ vacancies and the
$\mathrm{Mn}_{\mathrm{Cu}}$ antisite, particularly at the Mn-rich growth
conditions. This indicates that a Cu deficit on the Cu sublattice is possible,
affecting stoichiometry. Significantly, $\mathrm{Mn}_{\mathrm{Cu}}$ antisites
make the Cu sublattice magnetic, and also they can participate in the magnetic
coupling between the adjacent MnSb (001) planes, thus influencing magnetic
properties, as it will be discussed in more detail below. In contrast,
$\mathrm{Sb}_{\mathrm{Cu}}$ antisites are present in negligible
concentrations. Our results are in a reasonable agreement with those of Ref.
Máca  _et al._ , 2016, especially given their neglect of spin effects and a
somewhat different theoretical approach. Interestingly, formation energies of
native defects in CuMnAs calculated in Ref. Máca  _et al._ , 2017 are close to
the present results in spite of the different anion.

###  III.6 Defect-induced magnetic coupling

There are two Mn-related point defects, $\mathrm{Mn}_{i}$ and
$\mathrm{Mn}_{\mathrm{Cu}}$, both situated on the Cu sublattice. When present
at high concentrations, they affect magnetism of $\alpha$–CuMnSb. Their
coupling with host Mn ions is different than the Mn-Mn coupling between the
host Mn because of the different local coordination. Energetics of both
defects is complex and rich, since the total energy of the system (and thus
formation energies) depends on their spin orientations relative to the
neighborhood. At both substitutional and interstitial sites in the Cu layer, a
Mn ion has 4 Mn nearest neighbors arranged in a tetrahedral configuration, 2
in the upper and 2 in the lower MnSb layer. The
$\mathrm{Mn}_{i}$–$\mathrm{Mn}_{\mathrm{Mn}}$ distance is shorter than that of
$\mathrm{Mn}_{\mathrm{Mn}}$–$\mathrm{Mn}_{\mathrm{Mn}}$, and equal to
$(\sqrt{3}/4)a$.

The possible local spin configurations are reduced to small clusters of 5 Mn
ions, shown in Fig. 8. The Mn spin-up and spin-down (001) MnSb layers are
denoted by in pink and blue, respectively, reflecting the calculated (001) AFM
magnetic ground state. The central $\mathrm{Mn}_{\mathrm{Cu}}$ (or
$\mathrm{Mn}_{i}$) ion of such a cluster provides an additional channel of
magnetic coupling between two adjacent MnSb layers. The corresponding
formation energies are given in Fig. 8.

As it was pointed out, in ideal $\alpha$–CuMnSb, the Mn ions are second
neighbors only, separated either by Sb (i.e., the Mn-Sb-Mn ”bridge” in the
MnSb(001) plane), or by Cu (forming a Mn-Cu-Mn ”bridge” linking 3 consecutive
(001) planes.) Thus, the short range magnetic coupling in ideal
$\alpha$–CuMnSb is successfully modelled in Sec. III.2 by the interaction
between two Mn second neighbors, situated either in the same MnSb layer, or in
two adjacent ones. In contrast, the 4 host Mn ions in the cluster are the
first neighbors of a $\mathrm{Mn}_{i}$ or a $\mathrm{Mn}_{\mathrm{Cu}}$
defect. Thus, one can expect that this coupling is stronger than the intrinsic
one in the ideal host, and indeed, the differences in energy between various
configurations in Fig. 8 are about 100 meV, which is too high to be explained
by the estimated $J_{sr}=0.4$ meV.

As it follows from Fig. 8, 5-atom clusters are magnetically frustrated. In
particular, the lowest energy case denoted as 4AFM favors the local FM
orientation of spins in two adjacent (001) planes, which is opposite to the
global host magnetic order. Our results do not confirm the conclusion of Ref.
Máca  _et al._ , 2017 who find that the 3AFM configuration has the lowest
energy, and thus it promotes the global AFM111 order. Instead, we rather
expect that Mn-related point defects induce disorder of the host AFM phase,
possibly leading to formation of a spin glass. Kroder  _et al._ (2019)

##  IV Summary

CuMnSb films were epitaxially grown on GaSb substrates. Magnetic measurements
reveal the presence of two magnetic subsystems. The dominant magnetic order is
AFM with the Néel temperature of 62 K, which is the same as in bulk CuMnSb. It
co-exists with a FM phase, characterized by the Curie temperature of about 100
K.

These findings go in hand with transmission electron microscopy and selective
area diffraction measurements, which demonstrate coexistence of two structural
polymorphs of the same stoichiometry. The dominant one is the cubic half-
Heusler $\alpha$–CuMnSb, which is the equilibrium structure of bulk samples.
The second component is a tetragonal $\beta$–CuMnSb polymorph, which forms
10-100 nm long elongated inclusions.

The results of our ab-initio calculations provide a consistent interpretation
of the experimental data, and in particular

(i) The $\beta$–CuMnSb phase is metastable, and its total energy is higher by
0.1 eV per f.u. only than that of the equilibrium $\alpha$–CuMnSb. Lattice
parameters of the $\beta$ phase differ from those of $\alpha$–CuMnSb by about
4 per cent. This lattice misfit between the two structures does not prevent
the pseudomorphic coexistence of both phases, since the calculated misfit
strain energy is below 20 meV per f.u.

(ii) In agreement with experiment, $\alpha$–CuMnSb is AFM, and the FM order is
19 meV per f.u. higher in energy. In contrast, the magnetic ground state of
$\beta$–CuMnSb is FM, which is more stable than AFM by 11 meV per f.u. This
indicates that indeed the $\beta$–CuMnSb inclusions are responsible for the FM
signal.

(iii) The different magnetic orders of the $\alpha$ and $\beta$ phases
originate in their somewhat different band structures. In particular, critical
for magnetic order is the DOS at the Fermi level, which is about 4 times
higher in $\beta$–CuMnSb than in the $\alpha$ phase. This shows that the
$\beta$ phase is more metallic in character, which in turn favors the FM order
driven by the Ruderman-Kittel-Kasuya-Yoshida interaction.

(iv) Our calculations predict the saturated magnetic moment of Mn
$m_{\mathrm{sat}}=4.6\mu_{\mathrm{B}}$ and $4.5\mu_{\mathrm{B}}$ for the
$\alpha$ and the $\beta$ phase, respectively. This corresponds to the
effective moment of $5.6\mu_{\mathrm{B}}$, in good agreement with the measured
$5.5\mu_{\mathrm{B}}$.

(v) The calculated formation energies of point native defects indicate that
the most probable are the MnCu antisites with low formation energies of 0–0.2
eV. However, their presence is expected to disorder the host magnetic AFM
phase rather than to induce a transition to the FM configuration.

(vi) Regarding the properties of the CuMnX series we see that their structural
stability is relatively weak, as they crystallize in a variety of structures.
In particular, unlike the bulk orthorhombic CuMnAs, epitaxial films of CuMnAs
are tetragonal, but both structures are AFM. In the case of CuMnSb,
polymorphism comprises also the equilibrium magnetic structure, AFM in the
bulk specimens, and FM in epitaxial films.

###### Acknowledgements.

LS, CG, JK and LWM thank M. Zipf for technical assistance. Our work was funded
by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) No.
397861849, by the Free State of Bavaria (Institute for Topological Insulators)
and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy - EXC2147 ct.qmat (Project-Id 390858490).

## Appendix A

The difference between chemical potential $\mu_{i}$ considered in formation
energy, eq. 1, and the chemical potential of bulk Cu, Mn or Sb,
$\mu_{i}(\textrm{bulk})$, is denoted by $\delta\mu_{i}$:

$\mu_{i}=\mu_{i}(\textrm{bulk})+\delta\mu_{i}.$ (3)

The highest possible value of $\mu_{i}$ is $\mu_{i}(\textrm{bulk})$, which
implies that the studied system is in equilibrium with the given bulk source
of atoms and $\delta\mu_{i}=0$, otherwise $\delta\mu_{i}<0$.

Chemical potentials of the components in the standard state are given by the
total energies per atom of elemental solids. The calculated cohesive energies
$E_{coh}$ of the face centered cubic Cu, the face centered cubic Mn with the
AFM magnetic order, and the triclinic Sb are, respectively, 3.40 (3.49), 2.65
(2.92) and 2.68 (2.75) eV/atom. They compare reasonably well with the
experimental values given in parentheses. Kittel (1986)

Chemical potentials of the involved atomic species depend on possible
formation of compounds. The ranges of variations of chemical potentials are
determined by conditions of equilibrium between various phases, i.e., Cu2Sb,
MnSb and CuMnSb. Thermodynamic equilibrium requires that

$\displaystyle\delta\mu(\textrm{Cu})+2\delta\mu(\textrm{Sb})=\Delta
H_{f}(\textrm{Cu}_{2}\textrm{Sb}),$
$\displaystyle\delta\mu(\textrm{Mn})+\delta\mu(\textrm{Sb})=\Delta
H_{f}(\textrm{MnSb}),$ (4)
$\displaystyle\delta\mu(\textrm{Cu})+\delta\mu(\textrm{Mn})+\delta\mu(\textrm{Sb})=\Delta
H_{f}(\textrm{CuMnSb}),$

where $\Delta H_{f}$ is the enthalpy of formation per formula unit (negative
for a stable compound).

The calculated values $\Delta H_{f}(\textrm{Cu}_{2}\textrm{Sb})=-0.03$ eV per
f.u., $\Delta H_{f}(\textrm{MnSb})=-0.48$ eV per f.u., and $\Delta
H_{f}(\textrm{CuMnSb})=-0.42$ eV per f.u. The very low $\Delta
H_{f}(\textrm{Cu}_{2}\textrm{Sb})$ is somewhat unexpected, since
$\textrm{Cu}_{2}\textrm{Sb}$ is a stable compound which crystallizes in the
tetragonal phase. Endo  _et al._ (2021) Next, our result $\Delta
H_{f}(\textrm{MnSb})=-0.48$ eV per f.u. agrees well with both the previous
value -0.52 eV per f.u. calculated in Ref. Podloucky and Redinger, 2018, and
the experimental -0.52 eV per f.u. Shchukarev  _et al._ (1961) Assuming that
the accuracy of the calculated values is 0.03 eV per f.u., the set of Equation
A is consistent if we assume $\Delta H_{f}(\textrm{Cu}_{2}\textrm{Sb})=0$, and
$\Delta H_{f}(\textrm{MnSb})=\Delta H_{f}(\textrm{CuMnSb})=-0.45$ eV per f.u.
This in turn implies that $\delta\mu(\textrm{Cu})=\delta\mu(\textrm{Sb})=0$,
and $\delta\mu(\textrm{Mn})=-0.45$ eV. Consequently, the allowed window of the
Mn chemical potential is

$-0.45~{}\textrm{eV}<\delta\mu(\textrm{Mn})<0,$ (5)

where the two limiting values correspond to the Mn-poor and Mn-rich
conditions, and an analogous window holds for Sb.

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