# Magnetic phase diagram of rare-earth orthorhombic perovskite oxides

Alireza Sasani  Physique Théorique des Matériaux, QMAT, CESAM, Université de
Liège, B-4000 Sart-Tilman, Belgium  Jorge Iñiguez  MaterialsResearch and
Technology Department, Luxembourg Institute of Science and Technology (LIST),
5 avenue des Hauts-Fourneaux, L-4362, Esch/Alzette, Luxemburg  Department of
Physics and Materials Science, University of Luxembourg, Rue du Brill 41,
L-4422 Belvaux, Luxembourg  Eric Bousquet  Physique Théorique des Matériaux,
QMAT, CESAM, Université de Liège, B-4000 Sart-Tilman, Belgium

###### Abstract

Spin reorientation and magnetisation reversal are two important features of
the rare-earth orthorhombic provskites ($RM$O3’s) that have attracted a lot of
attention, though their exact microscopic origin has eluded researchers. Here,
using density functional theory and classical atomistic spin dynamics we build
a general Heisenberg magnetic model that allows to explore the whole phase
diagram of the chromite and ferrite compounds and to scrutinize the
microscopic mechanism responsible for spin reorientations and magnetisation
reversals. We show that the occurrence of a magnetization reversal transition
depends on the relative strength and sign of two interactions between rare-
earth and transition-metal atoms: superexchange and Dzyaloshinsky-Moriya. We
also conclude that the presence of a smooth spin reorientation transition
between the so-called $\Gamma_{4}$ and the $\Gamma_{2}$ phases through a
coexisting region, and the temperature range in which it occurs, depends on
subtle balance of metal–metal (superexchange and Dzyaloshinsky-Moriya) and
metal–rare-earth (Dzyaloshinsky-Moriya) couplings. In particular, we show that
the intermediate coexistence region occurs because the spin sublattices rotate
at different rates.

##  I introduction

Rare earth orthorhombic perovskites ($RM$O3, where $R$ is an atom of the rare-
earth family and $M$ is a transition metal – Fe or Cr in this work) have been
studied for a long time due to their unique magnetic properties [1], the two
important magnetic behaviours being the spin reorientation (SR) and the
magnetisation reversal (MR). The SR involves the change of the spin direction
from one crystalline direction to another as a function of temperature (see
Fig. 1(a)) while MR refers to the inversion of the net magnetization of the
crystal as a function of temperature (see Fig. 1(b)). These materials are also
multiferroics (type II, i.e. the magnetic order induces a polarization) [2, 3]
with strong magnetoelectric (ME) response [2] surpassing most known ME
materials. All of these unique properties rely on the presence of two magnetic
sublattices, $R$ and $M$ with very different Néel temperatures such that for a
wide range of temperatures the $R$ spins are paramagnetic while the $M$ spins
are ordered. The associated magnetic interactions between these two
sublattices have been proved to be the key ingredients for the origin of the
SR, MR and multiferroic properties [4, 5, 6], hence for their use in
technological applications [7, 8, 9]. The SR can happen at high temperatures
(480 K in the case of SmFeO3) and this temperature can be lowered by doping
which makes it possible to have this behaviour at room temperature so that the
SR could be used in exchange bias devices [10, 11, 9, 12].

Figure 1: a) Schematic representation of the SR from $\Gamma_{4}$ (red color)
to $\Gamma_{2}$ (blue color) as a function of temperature where the transition
is smooth by passing through an intermediate mixed phase containing both
states ($\Gamma_{24}$). b) Schematic plot of the evolution of the total
magnetization of the crystal and showing two possible cases: (i) MR effect
(red line) where the magnetization changes sign below a critical temperature
due to the fact that the paramagnetic rare earth atom magnetizes in opposite
direction to the wFM of the transition metal atom. (ii) Absence of MR (blue
line) where the magnetization is amplified when temperature is reduced and
corresponding to the case where the rare earth atom magnetizes in the same
direction as to the wFM of the transition metal atom.

Magnetic structures of these materials have been determined from symmetry
analysis [13]. In this work we are going to use Bertaut notation for symmetry
adapted magnetic structures, namely, $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$
and $\Gamma_{4}$ (see Fig. 2) [14]. Two types of SR are observed, namely
$\Gamma_{4}$ to $\Gamma_{2}$ (PrFeO3, NdFeO3, SmFeO3, TbFeO3, HoFeO3, ErFeO3,
TmFeO3, YbFeO3) and $\Gamma_{4}$ to $\Gamma_{1}$ (CeFeO3, DyFeO3) [1].

During the $\Gamma_{4}$ to $\Gamma_{2}$ SR, the spins directions change from
the $a$ crystallographic direction to the $c$ direction, slowly rotating as a
function of temperature in $ac$ plane. The $\Gamma_{4}$ to $\Gamma_{2}$ SR can
develop at different speeds: for some materials it is rather fast (e.g. it
spans through a 3 K temperature range [15] for ErFeO3) while for others it can
occur over a large temperature range (e.g. 70K for NdFeO3 [16]). Tsymbal et
al. have shown that a mean field model can describe the $\Gamma_{4}$ to
$\Gamma_{2}$ SR and they observe a sudden jump at the start of the
reorientation and a smooth evolution afterwards [15] . Studies on TbFeO3 show
that there are two phase transitions, from $\Gamma_{4}$ to $\Gamma_{2}$ below
8.5 K and, then, at the ordering of Tb the Fe subsystem transforms back to
$\Gamma_{4}$ [17] which shows the importance of the $R$ site ordering in this
SR. It has also been shown that this SR is of second order [18, 19] and it
could be associated to the softening of a low frequency magnon mode [20, 21].
Bazaliy et al. have measured the magnetisation of Er in ErFeO3 in the SR
region and shown that there is 70% of change [5].

In the $\Gamma_{4}$ to $\Gamma_{1}$ transition, the SR is quite fast and the
spins change their directions from $a$ to $b$ crystallographic direction
sharply at a well defined transition temperature. It has been shown that the
anisotropic symmetric exchange interactions would be responsible for this SR
transition [22].

Several authors built models to describe the magnetic interactions in these
materials. Moskvin [23] has made a model to describe the magnetic interactions
in $R$FeO3 and $R$CrO3 and has also made use of Anderson exchange model [24]
to study exchange and DMI in these materials [25]. Yamaguchi [6] used a mean
field theory model and was more complete than Moskvin. He has shown that some
of the SR present in these materials can be explained by antisymmetric and
anisotropic symmetric exchange interactions between $R$ and Fe. More recently,
Bellaiche et al. have given some simple laws to explain the origin of
different cantings present in these structures [8]. In their work they have
shown that all the cantings on transition metals site can be described using
simple energy terms originating from the DMI between transition metals.
Regarding the interactions between $M$ and $R$ spins, Zhao et al. have shown
that the DMI between $M$ and $R$ can polarize the $R$ ion and hence could
explain the origin of the MR in these materials [7]; according to this work,
the MR is linked with a sign in the interaction between the two sublattices
that is material dependent and not fully understood.

Hence, although there has been a great effort to explain the magnetic
properties of these materials, a solid and complete understanding regarding SR
and MR in these materials and their origin is still missing.

In this article, we shed some light into the magnetic properties of the
$RM$O3s. We have used density functional theory (DFT) to fit a microscopic
Heisenberg model that includes the superexchange and the DMI interactions
between the magnetic cations $M$-$M$ and $M$-$R$ (where $M$ is $Cr$ or $Fe$,
and $R$ is Gd). This model is then used as starting point, and we tune the
different parameters to understand their specific role in magnetic bahaviours
of the material using classical spin dynamics. The spin dynamics results are
also compared with analytical solutions to confirm their consistency. Our work
allows to explain the origin of the SR and the parameters determining the SR
temperature interval and how the $R$ magnetism is affected while in its
paramagnetic regime. We find that the occurrence of a slow SR comes from an
original evolution of the $\Gamma_{4}$ and $\Gamma_{2}$ orders due to the
presence of two different interacting magnetic cations; this allows to have
two magnetic phases co-existing while no coupling exist between them in the
Hamiltonian.

##  II Technical details

The main goal of this paper is to give a qualitative picture of the magnetic
properties of $RM$O3’s. To understand these magnetic behaviours we have used
DFT calculations on GdFeO3 and GdCrO3, as reference materials, to have an
estimation of the order of magnitude of the magnetic interactions in these
crystals. We then tuned these parameters to study how they affect the overall
magnetic behaviour of the system. We build a Heisenberg model containing
$M$-$M$ and $M$-$R$ superexchange and DMI interactions. Because we will focus
on the temperature range where the $R$ sublattice is paramagnetic, we will
neglect the $R$-$R$ interactions (these interactions are nevertheless small as
compared to the $M$-$M$ and $M$-$R$ couplings). We fit this model against DFT
calculations [26, 27] done for the orthorhombic $Pnma$ phase of GdFeO3 and
GdCrO3. We used the VASP package [28, 29] and its projected augmented wave
implementation of DFT [30]. We used the so-called PBEsol-GGA [31] functional
for the exchange correlation part of the density functional; a Hubbard $U$
correction [32] on Fe, Cr and Gd of respectively 4, 2 and 5 eV have been used
with $J$ parameter of 1 and 0.5 eV on Fe and Cr. All the calculations were
done with a 6$\times$6$\times$4 mesh of k-points for sampling the reciprocal
space and a cut-off energy on the plane wave expansion of 700 eV to have a
good convergence on single ion anisotropic and DMIs (less than 5
$\mu\text{eV}$ convergence).

The calculations of the exchange interactions were done using Green’s function
method as implemented in the TB2J [33] code. In this method the maximally
localised Wannier function [34] as implemented in WANNIER90 [35] are
calculated using DFT (VASP interface to Maximally localised Wannier functions)
and using these Wannier functions and the Green’s function method, the
exchange parameters are calculated. Some of these superexchange interactions
were compared to the ones calculated using total energy to ensure the
consistency of the method. To calculate the DMI couplings, we calculated the
energy of different spin configuration and used the method given by Xiang et
al. [36]. We have checked that the results are qualitatively the same by using
different Hubbard $U$ and $J$ corrections while we have used the ones giving
the best Néel temperature for both sublattices. All of the fitted magnetic
interaction parameters were used to do spin dynamics with the VAMPIRE code
[37]. In this code the Landau-Lifshitz-Gilbert (LLG) equation for the spin
dynamics (Eq. (1)) is solved numerically.

$\begin{array}[]{l@{}l}\frac{\partial S_{i}}{\partial
t}=\frac{\gamma}{1+\lambda^{2}}\left[S_{i}\times B_{eff}^{i}+\lambda
S_{i}\times\left(S_{i}\times B_{eff}^{i}\right)\right]\\\ \end{array}$ (1)

In the temperature dependent spin dynamics simulations we have used a
simulation cell of 20 nanometers in each direction. The thermalisation step
was done in 50000 time steps of 1.5 fs and the measurement is done in 90000
time steps of 1.5 fs.

Before analyzing the fitted model, we start with an analytical model of the
magnetic interactions present in $RM$O3 systems.

##  III Analytical model

To understand the mechanism behind SR and MR, we develop in this section the
Heisenberg model and solve it analytically to understand the phase diagram of
$RM$O3 versus their microscopic magnetic interactions. This will also allow to
compare with the spin dynamics calculations to confirm that both give
consistent results.

###  III.1 Symmetry adapted spin representation:

We develop an analytical model of $RM$O3 using the symmetry adapted spin
representation. For each of the sub-lattices ($M$ or $R$) in the $Pnma$ unit
cell, we have four magnetic sites that results in four different magnetic
orders: $A$, $C$, $G$ and $F$ type as presented in Fig. 2(b). Using these four
magnetic orderings, we can define four symmetry adapted spin states, namely
$\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$, $\Gamma_{4}$ that are linear
combination of the $A$ ,$G$, $C$ and $F$ orderings in different directions
(Fig. 2)[14]. Because the ground state of the $M$ spin sub-lattice is a robust
$G$-type antiferromagnetic ordering in the $Pnma$ perovskite phase, the most
relevant $\Gamma_{j}$ states are those with $j$ = 1, 2 and 4, which present a
dominant $G$-type in one of the three crystallographic directions with the
presence of canted $A$, $C$ and $F$ type magnetic orders in the other
directions. We summarize in Table 1 the different $\Gamma_{j}$ states where
$\bar{G}$ shows the main magnetic order and the components without a bar are
small spin cantings.

Figure 2: a) Position of transition metal ions (gray spheres) and rare earth elements (blue spheres) in $Pnma$ structure, purple spheres represent oxygen atoms. b) Schematic representation of the $G$, $A$, $C$ and $F$ magnetic orders for transition metal sites present in $Pnma$ structure as highlighted by the red box in panel a. The arrows represent the positive (red arrows) or negative (purple arrows) value of the magnetic moment. c) Symmetry adapted representations $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$ and $\Gamma_{4}$ present in peroskites. Table 1: Irreducible representation of magnetic states present in the $Pnma$ phase of $RM$O3 for both transition metal M-site and $R$ site [14] Irrep |  M site |  $R$ site
---|---|---
[.5]0 $\Gamma_{1}$ |  (Ax,$\bar{G}_{y}$,Cz) | (0,0,Cz)
[.5]0 $\Gamma_{2}$ |  (Fx,Cy,$\bar{G}_{z}$) | (Fx,Cy,0)
[.5]0 $\Gamma_{3}$ |  ($\bar{C}_{x}$,Fy,Az) | (Cx,Fy,0)
[.5]0 $\Gamma_{4}$ |  ($\bar{G}_{x}$,Ay,Fz) | (0,0,Fz)

Using these notations, we can write the symmetry adapted magnetic states in
terms of their modulation vectors for magnetic sub-lattice a as follows (Eq.
(2) to Eq. (5)):

$\begin{array}[]{l@{}l}S_{i,a}^{\Gamma_{1}}&{}=A_{a,x}(-1)^{(n_{z}^{i})}+\bar{G}_{a,y}(-1)^{(n_{x}^{i}+n_{y}^{i}+n_{z}^{i})}+C_{a,z}(-1)^{(n_{x}^{i}+n_{y}^{i})}\\\
\end{array}$ (2)

$\begin{array}[]{l@{}l}S_{i,a}^{\Gamma_{2}}&{}=F_{a,x}+C_{a,y}(-1)^{(n_{x}^{i}+n_{z}^{i})}+\bar{G}_{a,z}(-1)^{(n_{x}^{i}+n_{y}^{i}+n_{z}^{i})}\\\
\end{array}$ (3)

$\begin{array}[]{l@{}l}S_{i,a}^{\Gamma_{3}}=C_{a,x}(-1)^{(n_{y}^{i}+n_{z}^{i})}+F_{a,y}+A_{a,z}(-1)^{(n_{z}^{i})}\\\
\end{array}$ (4)

$\begin{split}S_{i,a}^{\Gamma_{4}}=\bar{G}_{a,x}(-1)^{(n_{x}^{i}+n_{y}^{i}+n_{z}^{i})}+A_{a,y}(-1)^{(n_{z}^{i})}+F_{a,z}\\\
\end{split}$ (5)

Here $S_{i,a}^{\Gamma_{j}}$ is the spin of lattice site $i$ for magnetic sub-
lattice $a$ ($M$ or $R$) in irreducible representation $\Gamma_{j}$ and the
lattice site vector for lattice site $i$ can be written as
$n_{x}^{i}\hat{u}_{1}+n_{y}^{i}\hat{u}_{2}+n_{z}^{i}\hat{u}_{3}$ where
$\hat{u}_{1}$,$\hat{u}_{2}$ and $\hat{u}_{3}$ are unit cell vectors while the
coefficients $G$, $A$, $F$ and $C$ represent the magnitude of spin canting in
each direction, the $G$-type order being the main one. From now on, we will
use these spin representations in our Heisenberg model.

###  III.2 Heisenberg model

In this section, we develop the Heisenberg Hamiltonian for $RM$O3 in which we
include the magnetic interactions between all the magnetic species: transition
metal atoms ($M$) and rare earth atoms ($R$), which can be summarized as
follows if one stays at the second order of interactions ( higher order spin
interactions like biquadratic or four-spin couplings are neglected):

$H=H^{MM}+H^{RM}+H^{RR},$ (6)

where $H^{MM}$ is the Hamiltonian of M-M interactions, $H^{RR}$ the
Hamiltonian of R-R interactions and $H^{RM}$ the Hamiltonian of R-M
interactions. $H^{MM}$ can be written as follows:

$H^{MM}=H_{ex}^{MM}+H_{DMI}^{MM}+H_{SIA}^{MM},$ (7)

where $H_{ex}^{MM}$, $H_{DMI}^{MM}$ and $H_{SIA}^{MM}$ represent the
superexchange, DMI and single ion anisotropy (SIA) interactions of the $M$
cations. In our simulations we have neglected anisotropic symmetric exchange
interactions since our DFT calculations show that they are two orders of
magnitude smaller than DMIs (results not shown here).

For $H^{RR}$ we have neglected the $H_{ex}^{RR}$ and $H_{DMI}^{RR}$ since we
are interested in behaviours that take place at temperatures higher than the
Néel temperature of the $R$ spin sub-lattice. We will only keep the SIA
interactions for this site.

$H^{RR}=H_{SIA}^{RR}$ (8)

The Hamiltonian taking care of the $R$-$M$ interactions can be written as
follows:

$H^{RM}=H_{ex}^{RM}+H_{DMI}^{RM}$ (9)

The exchange, DMI and SIA terms can be developed as follows:

$H_{Ex}^{ab}=\frac{1}{2}\sum_{ij}^{N}\left(J_{ab,ij}S_{i,a}.S_{j,b}\right)\\\
$ (10) $H^{ab}_{dmi}=\frac{1}{2}\sum_{i,j}\left(D_{ab,ij}\times
S_{j,a}\right).S_{i,b}\\\ $ (11)
$H^{aa}_{SIA}=\sum_{i}K_{a}\left(S_{i,a}.\hat{e}_{i}\right)^{2},\\\ $ (12)

where $ab$ could be $a=b=M$, $a=b=R$ or $a=M$ and $b=R$ and $\hat{e}_{i}$ is a
unit vector pointing to the direction of the SIA axis, which, according to our
DFT calculation for GdFeO3 and GdCrO3, is the easy axis.

We can show that there is no interaction between the different $\Gamma_{j}$
magnetic orderings (see supplementary information (SI)) such that we can write
the total energy of the system as the sum of the energy of each state. In this
case we can write the total energy as:

$\begin{split}H=H^{\Gamma_{1}}+H^{\Gamma_{2}}+H^{\Gamma_{3}}+H^{\Gamma_{4}}\\\
\end{split}$ (13)

In our analytical derivations we have neglected the $\Gamma_{3}$ state since
this state is much higher in energy than $\Gamma_{1}$, $\Gamma_{2}$ and
$\Gamma_{4}$. This is related to the fact that the $\Gamma_{3}$ state does not
contain $G$-type order, which is the order driving the lowest energy in the
crystal through the strongest superexchange interactions between transition
metals.

By putting the spin states in the Hamiltonian we can derive the following
expressions for each of the states (see SI):

$\begin{split}H^{\Gamma_{1}}=&{}H_{ex}^{M}+H_{DMI}^{M}+H_{ex}^{RM}+H_{DMI}^{RM}\\\
=&{}NJ^{M}(A_{M,x})^{2}-3NJ^{M}(\bar{G}_{M,y})^{2}-NJ^{M}(C_{M,z})^{2}\\\
&{}-6Nd_{x}^{M}\bar{G}_{M,y}C_{M,z}-6Nd_{y}^{M}C_{M,z}A_{M,x}-6Nd_{z}^{M}A_{M,x}\bar{G}_{M,y}\\\
&{}-8Nd_{x}^{RM}C_{R,z}\bar{G}_{M,y}-8Nd_{y}^{RM}C_{R,z}A_{M,x}\end{split}$
(14)

$\begin{split}H^{\Gamma_{2}}=&{}H_{ex}^{M}+H_{DMI}^{M}+H_{ex}^{RM}+H_{DMI}^{RM}\\\
=&{}3NJ^{M}(F_{M,x})^{2}-NJ^{M}(C_{M,y})^{2}-3NJ^{M}(\bar{G}_{M,z})^{2}\\\
&{}-6Nd_{x}^{M}C_{M,y}\bar{G}_{M,z}-6Nd_{y}^{M}\bar{G}_{M,z}F_{M,x}-6Nd_{z}^{M}F_{M,x}C_{M,y}\\\
&{}-8NJ^{RM}F_{M,x}F_{R,x}-8Nd_{x}^{RM}\bar{G}_{M,z}C_{R,y}-8Nd_{y}^{RM}F_{R,x}\bar{G}_{M,z}\\\
&{}-8Nd_{z}^{RM}F_{R,x}C_{M,y}-8Nd_{z}^{RM}C_{R,y}F_{M,x}\end{split}$ (15)

$\begin{split}H^{\Gamma_{4}}=&{}H_{ex}^{M}+H_{DMI}^{M}+H_{ex}^{RM}+H_{DMI}^{RM}+H_{SIA}^{R}+H_{SIA}^{M}\\\
=&{}-3NJ^{M}(\bar{G}_{M,x})^{2}+NJ^{M}(A_{M,y})^{2}+3NJ^{M}(F_{M,z})^{2}\\\
&{}-6Nd_{x}^{M}A_{M,y}F_{M,z}-6Nd_{y}^{M}\bar{G}_{M,x}F_{M,z}-6Nd_{z}^{M}\bar{G}_{M,x}A_{M,y}\\\
&{}-8NJ^{RM}F_{M,z}F_{R,z}-8Nd_{x}^{RM}F_{z,R}A_{M,y}-8Nd_{y}^{RM}F_{R,z}\bar{G}_{M,x}\\\
&{}-NK^{M}(\bar{G}_{M,x})^{2}-NK^{R}(\bar{G}_{R,x})^{2}\end{split}$ (16)

Where $J^{M}$ and $J^{RM}$ are, respectively, the exchange interaction
magnitude for $M$ sublattice and between $R$ and $M$ spins(Ja is considered as
Jii for isotropic exchange interaction); $d_{i}^{a}$ is the magnitude of $i$th
component of DMI vector for magnetic sub-lattice $a$; and $N$ is the number of
magnetic atoms while $K^{a}$ represent the SIA magnitude of magnetic
sublattice $a$.

With Eqs. (14–16) we have decomposed the Hamiltonian in terms of three
independent representations $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{4}$,
themselves decomposed into the superexchange, DMI and SIA of their constituent
$A$, $C$, $G$ and $F$ magnetic orderings. This form allows us to decompose the
different microscopic contributions of the magnetic energy of the $RM$O3
systems.

##  IV DFT calculation of the magnetic interaction parameters:

In this section we will present the parameters that we have calculated using
DFT for GdFeO3 and GdCrO3, which will serve as a reference starting point in
our spin dynamics simulations. These values will guide us to scan the magnetic
phase diagram in regions that are relevant for these materials.

###  IV.1 Superexchange and DMI parameters

Table 2: Calculated magnetic interactions from DFT of GdFeO3 and GdCrO3 (units are meV). $J$ (average values between nearest neighbours) values are for the first nearest neighbours and $d_{i}$ are the DMI vector components along $i=$ $x$, $y$ and $z$. |  dx |  dy |  dz | J
---|---|---|---|---
[.5]0 Fe-Fe |  0.000 |  -1.805 |  -1.104 | 38
[.5]0 Cr-Cr |  -0.001 |  -0.810 |  -0.600 | 7.2
[.5]0 Gd-Fe |  0.008 |  -0.064 |  0.031 | 1.85
[.5]0 Gd-Cr |  -0.010 |  0.042 |  -0.019 | 2.15
[.5]0 Gd-Gd |  – |  – |  – | 0.19

The dominant interactions are the exchange interaction between transitions
metals. The DFT results for GdFeO3 and GdCrO3 show that the strongest exchange
interactions are between the nearest neighbours transition metals; going
further in distance gives very small values with respect to the nearest
neighbours such that they can be neglected. These interactions are 38 meV and
7.2 meV for nearest neighbours (see Table 2), in GdFeO3 and GdCrO3
respectively and 1 meV or below for the next nearest neighbours. The $R$-$M$
superexchange interactions are one order of magnitude smaller (approximately 2
meV) than the ones between transition metals. The $R$-$R$ superexchange
interactions are two orders of magnitude smaller than the transition metals
ones (around 0.2 meV) such that we have neglected the $R$-$R$ interactions in
our spin dynamics simulations. Calculated SIA for both sublattices in GdFeO3
shows that these parameters are small around 72 $\mu\text{eV}$ for Gd with
easy axis along $c$ direction and 75 $\mu\text{eV}$ for Fe with easy axis
along the $b$ in the $Pnma$ structure; in GdCrO3 the SIA constant for Cr in
along $c$ direction and it is 25 $\mu\text{eV}$.

The most relevant parameters for our behaviours of interest are the DMIs.
Table 2 shows the obtained results, where the relation $d_{y}>d_{z}\gg d_{x}$
always hold. It is the $\overset{-}{a}\overset{-}{a}\overset{+}{c}$ octahedral
rotation pattern that breaks the bond inversion center of symmetry and creates
the DMI [38]. Hence, we will have the biggest distortion in [110] cubic
direction (amplitude of the rotations in $y$ direction of the $Pnma$
structure) and the smallest one will be in the [$\overset{-}{1}$10] cubic
direction ($x$ direction of the $Pnma$ structure), (as shown in Fig. 3 the
oxygen octahedral rotation have the same sign in [110] direction and add up
while in [$\overset{-}{1}$10] direction they have opposite sign and subtract
from each other), while the distortion in the [001] cubic direction ($c$
direction of $Pnma$) will be almost half of the one in the [110] cubic
direction. The ratio between these distortions is close to be the same for any
$Pnma$ crystal and this structural ratio also drives the key magnetic
interactions as we will show below.

Figure 3: Schematic presentation of the cubic ([100] and [010]) and $Pnma$
([110] and [$\bar{1}$10] ) crystallographic directions with respect to each
other. The curved arrows represent the oxygen octahedra rotations that are in
the same direction when projected in the [110] direction while they are in
opposite direction when projected in the [$\bar{1}$10] direction.

At high temperatures there is no magnetic ordering on $R$ sites (paramagnetic
phase) and the interactions between the $R$ spins are negligible. Hence, at
high temperatures the SIA and DMI interactions of the $M$ sites determine the
magnetic equilibrium state. From the formulas 14,15 and 16 we can notice that
both $d_{y}^{M}$ and $d_{z}^{M}$, which have the biggest components compared
to $d_{x}^{M}$ (see table 2) in the $\Gamma_{4}$ state, are coupled with the
main magnetic order and spin direction (i.e $G_{M,x}$), making the energy of
this state lower compared to $\Gamma_{1}$ and $\Gamma_{2}$. When comparing
$\Gamma_{1}$ and $\Gamma_{2}$, we can observe that for the $\Gamma_{2}$ state
we have the $d_{y}^{M}$ terms that couple with the main spin direction, hence
stronger than the $d_{z}^{M}$ component present in $\Gamma_{1}$. This implies
that the $\Gamma_{2}$ state is lower in energy than the $\Gamma_{1}$ sate.
Hence we can have:

$E_{DMI}^{MM,\Gamma_{4}}<$ $E_{DMI}^{MM,\Gamma_{2}}<$
$E_{DMI}^{MM,\Gamma_{1}}$

where $E_{DMI}^{MM,\Gamma_{j}}$ is the energy from DMI between $M$ atoms in
the $\Gamma_{j}$ state, i.e. the DMI between $M$ cations favour the
$\Gamma_{4}$ state [23].

We can also see the effect of these interactions in $R$CrO3 structures.
According to table 2 the DMIs and exchanges in these structures are smaller
than for $R$FeO3, which makes the energy difference between different spin
orders ($\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{4}$) smaller. This is consistent
with the fact that the $\Gamma_{2}$ and $\Gamma_{4}$ states are both present
at high temperature for the $R$CrO3 crystal series [39].

Considering the DMIs between $R$ and $M$, we notice that in the $\Gamma_{2}$
state we have
$H^{RM,\Gamma_{2}}_{DMI}=-8Nd_{x}^{RM}\bar{G}_{M,z}C_{R,y}-8Nd_{y}^{RM}F_{R,x}\bar{G}_{M,z}-8Nd_{z}^{RM}F_{R,x}C_{M,y}-8Nd_{z}^{RM}C_{R,y}F_{M,x}$
terms in which $d_{y}^{RM}$ couples with the main spin directions and also
this state has more degrees of freedom compared to the other states making
energy of this state lower. As for $\Gamma_{4}$ we have
$H^{RM,\Gamma_{4}}_{DMI}=-8Nd_{x}^{RM}F_{z,R}A_{M,y}-8Nd_{y}^{RM}F_{R,z}\bar{G}_{M,x}$
and for $\Gamma_{1}$ we have
$H^{RM,\Gamma_{1}}_{DMI}=-8Nd_{x}^{RM}C_{R,z}\bar{G}_{M,y}-8Nd_{y}^{RM}C_{R,z}A_{M,x}$
which again due to having the coupling $8Nd_{y}^{RM}F_{R,z}\bar{G}_{M,x}$
compared to $8Nd_{x}^{RM}C_{R,z}\bar{G}_{M,y}$ terms the $\Gamma_{4}$ state is
lower than that of $\Gamma_{1}$ ($d_{y}^{RM}\gg d_{x}^{RM}$). Hence we can
write the order of the different energies due to DMI of $R$ and $M$ as:

$E_{DMI}^{RM,\Gamma_{2}}<$ $E_{DMI}^{RM,\Gamma_{4}}<$
$E_{DMI}^{RM,\Gamma_{1}}$.

From this analysis, we can see why a SR transition is possible when lowering
the temperature.

Indeed, as the temperature is lowered the interactions between $R$ and $M$
cations become stronger due to the magnetization of the $R$ site in the field
created by the $M$ spins, and the $\Gamma_{2}$ is more and more favoured
through the DMIs between $R$ and $M$ sites. Hence, we can explain the
$\Gamma_{4}$ to $\Gamma_{2}$ SR due to the DMIs between $R$ and $M$ sites as
discussed previously by Yamaguchi [6].

###  IV.2 Origin of ordering on $R$ site

The MR observed in $RM$O3 is the change of sign in the net magnetization of
the material. This property has been related to the polarisation of the $R$
site atoms as a result of interaction with transition metal atoms. In this
interaction $R$ site atoms could polarise in the direction of the weak
magnetic moment of the transition metal or in opposite direction, which would
result in the presence or the absence of the magnetisation reversal
respectively (see Fig. 1(b)). The remaining question is why the $R$
paramagnetic atoms magnetize in opposite direction for some $R$ elements (e.g.
NdFeO3, SmFeO3, DyFeO3, ErFeO3, TmFeO3, YbFeO3) and why they magnetize in the
same direction for others (e.g. PrFeO3, EuFeO3, GdFeO3, TbFeO3, HoFeO3) [40]
and what is the microscopic origin of this effect.

With our model we can have access to the detailed microscopic interaction
between $R$ and $M$ cations. Equations 14,15 and 16 show that there are two
types of interactions acting on $R$ sites: (i) the exchange interaction
between the weak ferromagnetic (wFM) order of the $M$ and $R$ sublattices
($J_{RM}F_{M}F_{R}$) and (ii) the DMI between the $G$-type orders on the $M$
and $R$ sublattices ($d_{y}^{RM}F_{R,(z,x)}\bar{G}_{M,(x,z)}$ and
$d_{x}^{RM}C_{R,z}\bar{G}_{M,y}$). These interactions can induce either $F$ or
$C$ type ordering on $R$ site. To check the validity of these possibilities,
we have used DFT calculations as computer experiments where we have replaced
the $R$ site by Cr in GdFeO3’s $Pnma$ structure to allow the study of full
non-collinear calculations and to have stronger $RM$ site interactions
compared to the Gd case. Our model is valid for two magnetic sublattices in
perovskites whatever the magnetic cations, such that replacing $R$ by Cr will
show the same qualitative trend.

We have done different calculations in which we constrained the magnetic
moments on the Fe site and relaxed the magnetic order of the $R$ site within
two different settings. In the first setting, we set the spin orbit coupling
(SOC) to zero to suppress the DMI such that the resulting magnetic order on
$R$ site would be due to superexchange interactions only. In the second
setting, we considered SOC, hence activating the second term
($d_{y}^{RM}F_{R,(z,x)}\bar{G}_{M,(x,z)}$) that couples the $G$ type order of
the $M$ sublattice to $F$ order on $R$ site. In Fig. 4 we show the result of
these two types of calculations where we can see that the magnetization line
jumps to higher values when the SOC is present. This shows that the DMI can
polarise the $R$ site as the superexchange, which is in agreement with the
results obtained by Zhao et al. [7] using DMI energetic expressions between
$R$ and $M$ sites.

We should also mention that in our simulations for $R$ = Cr, since the
superexchange interaction is AFM, it polarises the $R$ site in opposite
direction to the wFM direction of the $M$, while the DMI polarises the $R$
site in the same direction as the wFM of the $M$ site (see Fig. 4). Our
calculations for GdFeO3 also show that these interactions are in competition
with each other such that the final magnetization direction of the $R$ site
will be determined by the balance between them.

Considering GdCrO3 we can see in Table 2 that the DMI interactions between the
Gd and Cr spins has opposite sign compared to the DMI interaction between Gd
and Fe spins. This shows that we can also have the sign change of the DMI
depending on the electronic structure of the atoms in the structure. In this
case both the DMI and superexchange induce a polarisation in the same
direction and, indeed, the calculated magnetic ground state of the GdCrO3
shows that the wFM of Cr and polarisation of the Gd are in opposite
directions, while for GdFeO3 the wFM of the iron atoms and the polarisation
direction of the Gd atoms are in the same direction. Hence, depending on the
electronic structure of the atoms we can have the superexchange and DMI that
compete or cooperate that will result in the presence or the absence of the
MR.

Figure 4: Calculated magnetization of Cr as a function of magnetization of Fe
in CrFeO3 simulated at fixed atomic positions of relaxed $Pnma$ GdFeO3 (Gd is
replaced by Cr). Orange points are without spin orbit coupling (with a linear
fit orange line) and green points are with spin orbit coupling (with linear
fit green line).

##  V Spin dynamics

In this section we present the spin dynamics results obtained with the VAMPIRE
code through the Heisenberg model presented above (with $R-R$ superexchange
and DMI set to zero). First, we worked with the magnetic interactions
parameters obtained for GdFeO3. Then, we made additional spin dynamics
calculations by varying the values of these parameters to understand how the
phase diagram and associated SR transitions are affected by the change of the
magnetic interactions.

To verify that our model qualitatively respects the symmetry of the $Pnma$
phase of $RM$O3 compounds, we first simulated the ground state (0 K) of these
structures by tuning the SIA to obtain the magnetic moment direction along the
different $x$, $y$ and $z$ crystallographic directions. By doing so, we
verified that the obtained cantings actually correspond to the ones of the
$\Gamma_{4}$, $\Gamma_{1}$ and $\Gamma_{2}$ orders when magnetic moments lie
along $x$, $y$ and $z$ directions, respectively (see table 1).

In the following we will analyze both $\Gamma_{4}$ to $\Gamma_{2}$ and
$\Gamma_{4}$ to $\Gamma_{1}$ SR transitions.

###  V.1  $\Gamma_{4}$ to $\Gamma_{2}$ reorientation

As first step we have done temperature dependent SR. To have temperature
dependent SR, we have tuned the parameters obtained for GdFeO3 so as to induce
such behavior, since this effect is not present in GdFeO3. More precisely, we
increase the DMI interaction between $R$ and $M$ by 1 order of magnitude to
have the SR. Fig. 5(a) shows the evolution of the magnetic moment directions
with respect to the temperature when there is a SR, as obtained from our spin
dynamics simulations.

Figure 5: Temperature dependent SR as obtained from our spin dynamics
calculations. Panel (a) shows the transition-metal $M$ site spin projections
along the $x$ and $z$ directions. Panel (b) shows the evolution of the $x$ and
$z$ magnetic moment projection of the rare earth $R$ site in the same
temperature range as panel (a). The unis on y axis are spins normalised with
their moments($\frac{5}{2}$ and $\frac{7}{2}$ for Fe and Gd respectively.)

We can see a slow rotation of the spins from $x$ to $z$ direction as the
temperature decreases and that this reorientation is continuous in a range of
temperature where the two orders (associated to the $\Gamma_{4}$ and
$\Gamma_{2}$ states) are present together. Fig. 5(b) shows the temperature
evolution of the magnetic ordering of the $R$ spins due to its interaction
with the $M$ spins. Here, we can see that the SR happens around 15 K when the
normalized magnetic moment of the rare earth element is about 0.3. Below this
critical temperature, the $R$ site magnetic moment increases in a more
pronounced way. The magnetization of the $R$ sites creates a torque that
induces the rotation of the $M$ magnetic direction. Such an increase of the
ferromagnetic moment of the $R$ site has been observed experimentally for
ErFeO3 in the SR region [5]. This result shows that we need the ferromagnetic
ordering on the $R$ site to have this $\Gamma_{4}$ to $\Gamma_{2}$ SR
transition. This is also observed experimentally in, e.g., TbFeO3 where the
crystal goes from the $\Gamma_{2}$ state to the $\Gamma_{4}$ state when the Tb
atom orders into the AxGy magnetic phase (no ferromagnetic order) at very low
temperatures [17].

Figure 6: Temperature dependent spin dynamics results for GdFeO3: a)
normalized magnetic moment of the Fe site as projected along z and x
direction(Fz and Gx) and b) normalized magnetic moment of the Gd site
(ferromagnetic order along the $z$ direction).

Fig. 6(a) shows the evolution of the magnetic orders at higher temperature.
This figure shows a Néel temperature of 680 K (using the $M-M$ superexchange
parameters as obtained for GdFeO3) and that the wFM (Fz) appears at
temperatures below the Néel temperature and, after a jump at the phase
transition, stays constant (0.10 $\times\frac{5}{2}\mu_{B}$). In Fig. 6(b) we
show the evolution of the $R$ site magnetic moments where we can see that the
induced magnetization of the rare-earth spins is visible at temperatures as
high as 400 K.

Figure 7: Decomposition of magnetic ordering on different irreducible
representations ($\Gamma_{j}$) for $x$, $y$ and $z$ directions of the spins in
the $\Gamma_{4}$ to $\Gamma_{2}$ SR phase transition case. The calculations
are done at 0K (ground state). The horizontal axis shows the magnitude of the
$z$ component DMI between $R$ and $M$ sites ( we have multiplied the
calculated DMI component with these values to increase it) and the vertical
axis shows the normalised order parameter magnitude.  Figure 8: Plot of the
evolution of the $x$, $y$ and $z$ projections of the local $M$ cation spin
components versus the DMI strength between $R$ and $M$ cations. A schematic
representation of how the $M$ spins look like is also given for the three main
phases $\Gamma_{4}$, $\Gamma_{24}$ and $\Gamma_{2}$.

To further understand the SR, we have studied how the stability of the
magnetic orders is affected by the value of the DMI coupling between $R$ and
$M$. This allows us to determine how the strength of the interaction between
$R$ and $M$ spins influences the SR. Fig. 7 shows the equilibrium state of the
structure which is projected to the different irreducible magnetic orders
along $x$, $y$ and $z$ directions versus $d_{z}^{RM}$. (The figure presents
three components of the spin as projected to different irreducible
representations.)

For values of $d_{z}^{RM}<16$ (arb. units) we can see that we have the
$\Gamma_{4}$ state with the main direction of the spin along $x$ with G-type
AFM order (Gx) and small components (canting) of the spins along $y$ and $z$
directions with A-AFM (Ay) and FM (Fz) ordering respectively. For
$43\text{(arb. units)}>d_{z}^{RM}>16$ (arb. units) we have a coexisting region
that we denote $\Gamma_{24}$, where mostly $\Gamma_{2}$ and $\Gamma_{4}$
states are present. The system enters to this state through a sudden jump in
magnetic order (we also have a discontinuity in the energy of the system). As
we move towards higher values of $d_{z}^{RM}$ the $\Gamma_{4}$ contribution is
reduced while $\Gamma_{2}$ contribution increases up to $d_{z}^{RM}>43$ (arb.
units) where only the $\Gamma_{2}$ is present, the SR being completed. The
transition from $\Gamma_{2}$+$\Gamma_{4}$ to $\Gamma_{2}$ at
$d_{z}^{RM}=43$(arb. units) is continuous.

Figure 9: Decomposition of the total energy from spin dynamics to its
components i.e energy from exchange interaction (exch), energy from $y$
componenet of the DMI between M and RM in (d${}_{y}^{M}$,d${}_{y}^{RM}$) and
energy from $z$ componenet of the DMI between M and RM
(d${}_{z}^{M}$,d${}_{z}^{RM}$) .

To get further insight into this transition, in Fig. 8 we show the evolution
of the atomic site projection of the spins in $x$, $y$ and $z$ components.
Since the SR transition is due to the $d_{z}^{RM}F_{R,x}C_{M,y}$ term in the
Hamiltonian, we can observe an increase of the $C$-type canted order along the
$y$ direction as the interaction between $R$ and $M$ becomes stronger.
Additionally, since the magnitude of the canted ferromagnetic order on $M$ is
constant (FM order along $x$ and $z$ direction before and after SR), this
increase in $C$-type order can only come from a reduction of the $G$-type
order component of the spin.

Figure 10: Phase diagrams for $\Gamma_{4}$ to $\Gamma_{2}$ SR as a function of
$d_{z}^{RM}$ (panel a) and $d_{z}^{M}$ with constant exchange of 38 meV
(corresponding to the ferrites) and between $d_{z}^{RM}$ and exchange
interaction of the transition metals (panel b) with constant value of 1.1 meV
for $d_{z}^{RM}$ interaction. The dashed lines with green background showing
the DMI (panel a) and exchange (panel b) of Iron for the whole range of La
familly (La to Lu) in RFeO3)

In Fig. 9 we show how the different energy contributions of the system
(superexchange and DMI) evolve with respect to the $d_{z}^{RM}$ parameter. We
can see that the contributions coming from $d_{z}^{M}$ and superexchange
interactions between transition metals are positive and increase as we go from
$\Gamma_{4}$ to $\Gamma_{2}$ state, which means that they are against the SR.
In fact these interactions are determinant for how fast the SR happens. The
exchange interaction is the main interaction that resists against the SR and
this is due to the fact that the SR involves an increase of the $C$-type
ordering on $M$ sites (via $d_{z}^{RM}F_{R,x}C_{M,y}$, as mentioned above) and
the reduction of the $G$-type order, which costs some energy. Therefore, to
overcome this energy penalty we need larger interaction between $R$ and $M$ to
complete the SR which is provided by more ordering of the $R$ site atoms.

####  V.1.1 Parameters affecting the $\Gamma_{4}$ to $\Gamma_{2}$ SR

Figure 11: Phase diagrams for $\Gamma_{4}$ to $\Gamma_{2}$ SR as a function of
$d_{z}^{RM}$ (panel a) and $d_{z}^{M}$ with constant exchange of 9 meV
(coresponding to Cr) and between $d_{z}^{RM}$ and exchange interaction of the
transition metals (panel b) with constant value of 0.6 meV for $d_{z}^{RM}$
interaction . The dashed lines with green background showing the DMI (panel a)
and exchange (panel b) of Cr for the whole range of La familly (La to Lu) in
RCrO3)

One of the properties that is important to understand is the temperature range
in which the spins start and complete their reorientation. From our model, we
found that three parameters affect how fast the SR happens: the DMI
$d_{z}^{RM}$ between $R$ and $M$ cations (related to the ordering amplitude of
the $R$ sites), the DMI $d_{z}^{M}$ between $M$ cations, and the superexchange
interaction $J^{M}$ between $M$ cations. The ratios between these three
parameters drive and determine the energy difference between the $\Gamma_{4}$
and $\Gamma_{2}$ states and hence the temperature range where the SR takes
place.

To highlight these parameter effects we report in Fig. 10 and 11 a 2D plot
showing the presence of the $\Gamma_{4}$, $\Gamma_{2}$ and $\Gamma_{24}$
regions with respect to $d_{z}^{M}$ and $d_{z}^{RM}$ values at fixed $J^{M}$
as calculated for GdFeO3. Fig. 11 shows the same but for a fixed value of
$J^{M}$ corresponding to the one calculated for GdCrO3. As we can see, for too
small values of $d_{z}^{M}$ the system only experiences an abrupt transition
(first order) between $\Gamma_{4}$ and $\Gamma_{2}$ without any coexisting
region and the ratio between $d_{z}^{M}$ and $d_{z}^{RM}$ at which the
transition appears is rather constant. However, beyond a critical value of
$d_{z}^{M}$ a $\Gamma_{24}$ coexisting region appears and grows with the
amplitude of $d_{z}^{M}$. This means that, for a given value of $J^{M}$, if
$d_{z}^{M}$ is not large enough the system will never experience a slow SR.
Once the coexistence region opens, it grows very fast with $d^{M}$ such that
for large enough $d_{z}^{M}$ and $d_{z}^{RM}$ values a slow SR transition is
always guaranteed. On the other side, in Fig. 10(b) and 11 we can see how the
coexisting region area is affected by the value of $J^{M}$ at a fixed value of
$d^{M}_{z}$. Here we can remark that if $J^{M}$ is too large or too small then
the $\Gamma_{24}$ area is strongly reduced.

In Fig. 10 and 11 we also draw the maximum and minimum values of $d^{M}_{z}$
and $J^{M}$ as obtained for $R$FeO3 and $R$CrO3, respectively, for the whole
series of Lanthanides $R=$La to Lu (horizontal dashed lines). We can see that
the range of these parameters is not too large and that they cross small areas
of the coexistence region where SR is possible. We can remark that the SR area
for $M=$ Cr is particularly small while it is potentially larger for Fe.

These phase diagrams help if one wants to design engineering of SR speed in
these crystals. For example, if a slower SR is desired, the Fe case will be
more interesting through doping with atoms that will reduce the superexchange
interactions between irons and/or that will increase the DMI between irons
(increase of the wFM).

####  V.1.2 Origin of the slow $\Gamma_{4}$ to $\Gamma_{2}$ rotation

Figure 12: Decomposition of the energy of the $\Gamma_{2}$ and $\Gamma_{4}$ as
a function of $d_{z}^{RM}$. Panel a) shows the energy of $\Gamma_{2}$ and
$\Gamma_{4}$ and $\Gamma_{2}+\Gamma_{4}$ which is the sum of the energies of
the two states and $E_{tot}$ which is the totla energy from simulations. In
panel b), the energy of each state is decomposed into its pure $M$ sublattice
contributions and the interaction contribution between $R$ and $M$, the pure
$R$-$R$ interactions being neglected. In panel b), the zero energy reference
of $M$ and $R$ interactions are taken to be the one of $M^{\Gamma}_{2}$. We
can notice that $RM$ interacting term is the one that lowers the energy of
$\Gamma_{2}$ by becoming larger than the energy difference between pure
$M^{\Gamma}_{4}$ and $M^{\Gamma}_{2}$ Figure 13: Order parameter ($\Gamma_{2}$
with orange color and $\Gamma_{4}$ with blue color) decomposition into $M$ and
$R$ sublattices when crossing a SR region (here as a function of
$d_{z}^{RM}$). The inset shows the area where the SR starts and we can notice
the change of order of R site first that drags the M site order afterward.

As we show in the analytical part of our model, there is no interaction
between $\Gamma_{2}$ and $\Gamma_{4}$, and we have
$<\Gamma_{2}|H|\Gamma_{4}>=0$. This would mean that the transition should be
fast from our model since, without interaction between the two states, there
is apparently no reason why their coexistence will reduce the energy, and we
should have an sharp transition between them. However, our simulations based
on this non-interacting Hamiltonian show that a coexisting region exists where
both $\Gamma_{2}$ and $\Gamma_{4}$ are present together.

To figure out what is happening, we plot in Fig. 12(a) the energy change with
respect to the $d_{z}^{RM}$ as decomposed into a pure $\Gamma_{4}$, pure
$\Gamma_{2}$ , the sum of the energy of $\Gamma_{2}$ and $\Gamma_{4}$ and
total energy from our simulations Etot. To understand if there is hidden
coupling between $\Gamma_{2}$ and $\Gamma_{4}$ states, we have plotted the sum
of energies of $\Gamma_{2}$ and $\Gamma_{4}$ ($\Gamma_{2}+\Gamma_{4}$) and
total energy from our simulations Etot and as we can see, the two energies
match exactly which proves that although there is no coupling between the two
states, the SR is slow, .

In Fig. 12(b) we report the energy decomposition to $M$ sublattice only
($M^{\Gamma_{4}}$, $M^{\Gamma_{2}}$) and interaction between R and M in each
state (${RM}^{\Gamma_{4}}_{int}$, ${RM}^{\Gamma_{2}}_{int}$,). We can see that
the $M$ sublattice energy of the $\Gamma_{2}$ state ($M^{\Gamma_{2}}$, blue
line) is higher than the energy of the $M$ sublattice in the $\Gamma_{4}$
state ($M^{\Gamma_{4}}$, green line), as expected since the $\Gamma_{4}$ phase
is the ground state when only the $M$ sublattice is considered. The two
$RM_{int}^{\Gamma_{4}}$ (red line) and $RM_{int}^{\Gamma_{2}}$ (orange line)
interaction terms clearly show that $RM_{int}^{\Gamma_{2}}$ lowers the energy
of the $\Gamma_{2}$ phase with an amplitude that can compensate the energy
difference between $M^{\Gamma_{4}}$ and $M^{\Gamma_{2}}$, such that the
$\Gamma_{2}$ phase can be lower in energy than the $\Gamma_{4}$ phase. This
also proves that the $M$ sublattice alone prefers to stay in the $\Gamma_{4}$
state while the $R$ sublattice pushes the $M$ sublattice to be in the
$\Gamma_{2}$ state (the $RM_{int}^{\Gamma_{2}}$ energy is stronger and more
negative than the $RM_{int}^{\Gamma_{4}}$ energy).

In Fig. 13 we show the evolution of the order parameters but decomposed into
sublattice contributions ($\Gamma_{4}$_M, $\Gamma_{4}$_R , $\Gamma_{2}$_M and
$\Gamma_{2}$_R). We can observe that in the SR region, when going from
$\Gamma_{4}$ to $\Gamma_{2}$ the $R$ spins start to rotate first and they drag
the $M$ sublattice afterward (highlighted in the inset). Since the $M$
sublattice prefers to stay in the $\Gamma_{4}$ state while the $R-M$
interaction favors the $\Gamma_{2}$ state, the system ends up in a mixed state
even if no $\Gamma_{2}-\Gamma_{4}$ interaction is present in the Hamiltonian.
To understand this better, note that in this problem we do no really have two
competing orders ($\Gamma_{2}$ and $\Gamma_{4}$), but four (($\Gamma_{4}(M)$,
$\Gamma_{4}(R)$, $\Gamma_{2}(M)$ and $\Gamma_{2}(R)$). As we vary the key
Hamiltonian parameter in Fig. 13, the $\Gamma_{2}(R)$ order becomes favorable
over $\Gamma_{4}(R)$; we thus have a $\Gamma_{4}\rightarrow\Gamma_{2}$
rotation of the $R$ sublattice (accompanied by a relatively tiny
$\Gamma_{2}(M)$ component) that yields a reduction of the energy as compared
to a pure $\Gamma_{4}$ state. Eventually, the $\Gamma_{2}(R)$ order grows and
drags the $M$ spins to rotate as well, the final result being a pure
$\Gamma_{2}$ state.

####  V.1.3 Effect of SIA on $\Gamma_{4}$ to $\Gamma_{2}$ SR

Figure 14: Effect of SIA on the $\Gamma_{4}/\Gamma_{2}$ SR phase diagram.
Horizontal axis shows the amplitude of $z$ component of the DMI between $R$
and $M$ sites while the vertical axis shows the amplitude of the SIA of the
$M$ site.

Although the SIA amplitude on the $M$ site is not very large, we can probe it
from our model and have an estimate of its effect on SR. To that end, we
report in Fig. 14(a) the phase diagram of the $\Gamma_{4}$, $\Gamma_{2}$ and
$\Gamma_{24}$ presence with respect to $d^{RM}$ and SIA of $M$. We can see on
this plot that the SIA does not change the $\Gamma_{4}$ to $\Gamma_{24}$
transition position; in contrast, when the SIA increases, it has the tendency
to increase the $\Gamma_{24}$ SR area at the expense of the $\Gamma_{2}$
state. However, the effect of the SIA is much smaller than the ones of
$J^{M}$, $d^{M}$ or $d^{RM}$.

####  V.1.4 Summary for the $\Gamma_{4}$ to $\Gamma_{2}$ transition

In summary, in this section we have shown that our model well reproduces the
temperature dependent SR. This behaviour shows that the SR is directly linked
to the ordering of the rare earth in ferromagnetic order and proves that the
mechanism behind the SR in $RM$O3 compounds is not related to the SIA [41],
but it is the DMI between $R$ and $M$ that drives this SR.

A study by Vibhakar et al. [42] on triple A-site columnar-ordered quadruple
perovskites has shown that the mechanism behind SR in these structures is the
competition between DMI and SIA, which is similar to the mechanism that we
found to be at play in $RM$O3’s SR.

In our simulations, we have also studied how different parameters affect the
speed of SR. We can say that the presence of a smooth transition between
$\Gamma_{4}$ and $\Gamma_{2}$ phases through a coexisting region $\Gamma_{24}$
is very subtle and depends on the ratio between $J^{M}$, $d^{M}_{z}$ and
$d^{RM}_{z}$ interactions. If $d^{M}_{z}$ is zero, a transition between
$\Gamma_{4}$ and $\Gamma_{2}$ can exist but only through a first order abrupt
change; the $d^{M}_{z}$ interaction is mandatory to have a smooth SR
transition.

###  V.2  $\Gamma_{4}$ to $\Gamma_{1}$ reorientation:

Figure 15: Magnetic structure for $\Gamma_{4}$ to $\Gamma_{1}$ SR decomposed
to different irreducible representations for different components of the spin
in $x$, $y$ and $z$ directions. The horizontal axis showing the magnitude of
ASE in x direction between $R$ and $M$ and vertical axis showing the order
parameter magnitude normalised.

To explain the $\Gamma_{4}$ to $\Gamma_{1}$ SR we need a strong interaction
between the $R$ and $M$ sites within the $\Gamma_{1}$ state to allow the $M$
site order to go from its energetically favourable state $\Gamma_{4}$ to the
less energetically favourable state $\Gamma_{1}$. If not, we would have each
sublattice ordering in different direction like what is observed
experimentally in TbFeO3 [17]. However, according to our model, the sole
interactions between $R$ and $M$ atoms in the $\Gamma_{1}$ state are
$-8Nd_{x}^{RM}C_{R,z}\bar{G}_{M,y}$ and $-8Nd_{y}^{RM}C_{R,z}A_{M,x}$. The
second term is the coupling between $A_{M,x}$ and $C_{R,z}$, which is small
since the $A_{M,x}$ canting is very small compared to $\bar{G}_{M,y}$. Hence,
the only remaining term which can make this SR possible is
$-8Nd_{x}^{RM}C_{R,z}\bar{G}_{M,y}$. From our DFT calculations and from
symmetry analysis (since these parameters are originating from
$\overset{-}{a}\overset{-}{a}\overset{+}{c}$ oxygen octahedra rotations as
discussed in previous section) we know that $d_{x}^{RM}$ is very small (see
table 2) such that it is not possible to explain the $\Gamma_{4}$ to
$\Gamma_{1}$ SR using this interaction.

So far we have neglected the anisotropic spin exchange interactions (ASE) in
our model because the effects from these interactions are often negligible
with respect to the superexchange or DMI. Now that we have the DMI small too,
we will consider the ASE to check whether it can take some importance while
the DMI is small. The definition of DMIs and ASE vector components are as
follows:

$\begin{array}[]{l@{}l}d_{x}^{ab}=\frac{1}{2}(J_{yz}^{ab}-J_{zy}^{zb})\end{array}$
(17)
$\begin{array}[]{l@{}l}\varepsilon_{x}^{ab}=\frac{1}{2}(J_{yz}^{ab}+J_{zy}^{ab}),\end{array}$
(18)

where $\varepsilon_{x}^{ab}$($d_{x}^{ab}$) represents the ASE (DMI) vector
component in $x$ direction between atom $a$ and atom $b$ and $J_{yz}$ is the
exchange interaction between spins directing in $y$ direction on atom $a$ and
in $z$ direction on atom $b$ (other component, i.e in $y$ and $z$ directions,
can be obtained by cyclic permutation of the $xyz$ directions). We can see
that when a component of the DMI vector is small it is probable to have the
ASE vector for that component to be bigger depending on the magnitude of
$J_{yz}$.

In Table 3 we show the calculated ASE between Fe sites and Cr sites. The
calculated results show that the $x$ component of the ASE vector is the
largest with respect to the $y$ and $z$ components for Fe-Fe and Cr-Cr atom
pairs. In Table 3 we also report the calculated ASE vector for Fe-Gd and Cr-Gd
pairs. The biggest component of the ASE vector is in the $y$ direction for Gd-
Fe while it is along the $x$ direction for the Gd-Cr case. We note that the
ASE interaction in the Hamiltonian takes the same place as the DMI does, i.e
for the $\Gamma_{1}$ state we have:

$\begin{array}[]{l@{}l}-8N\varepsilon_{x}^{RM}C_{R,z}\bar{G}_{M,y}\end{array}$
(19) $\begin{array}[]{l@{}l}8N\varepsilon_{y}^{RM}C_{R,z}A_{M,x}\end{array}$
(20)

Considering these ASE interactions, we can say that the $\Gamma_{4}$ to
$\Gamma_{1}$ SR can happen through the $x$ component of the ASE (through
$-8N\varepsilon_{x}^{RM}C_{R,z}\bar{G}_{M,y}$ interaction), which will take
the place of the DMI when the latter is small. This conclusion is in agreement
with Zvezdin [22] who explained the origin of the $\Gamma_{4}$ to $\Gamma_{1}$
SR to originate from ASE.

Table 3: Calculated ASE components from DFT of GdFeO3 and GdCrO3 (units are meV). $\varepsilon_{i}$ are the ASE vector components along $i=$ $x$, $y$ and $z$. |  $\varepsilon_{x}$ |  $\varepsilon_{y}$ | $\varepsilon_{z}$
---|---|---|---
[.5]0 Fe-Fe |  -0.015 |  0.000 | 0.000
[.5]0 Cr-Cr |  0.051 |  0.000 | 0.003
[.5]0 Gd-Fe |  0.006 |  -0.016 | 0.007
[.5]0 Gd-Cr |  -0.030 |  0.010 | -0.009

We will now study this SR by tuning the $\varepsilon_{x}^{RM}$ ASE coupling in
our model. In Fig. 15 and 16 we report how the relative $\Gamma_{4}$ and
$\Gamma_{1}$ stability evolves with respect to the $\varepsilon_{x}^{RM}$
parameter at 0 K (ground state). In contrast to the $\Gamma_{4}$ to
$\Gamma_{2}$ transition we can see that there is no coexisting region between
$\Gamma_{4}$ and $\Gamma_{1}$ states, the transition is always abrupt with
respect to the $\varepsilon_{x}^{RM}$ amplitude. To confirm this, we also
explore in Fig. 15 how the superexchange parameter $J^{M}$ affects the
$\Gamma_{4}$ to $\Gamma_{1}$ SR transition. We can clearly see that whatever
value of $J^{M}$ we considered, the $\Gamma_{4}$ to $\Gamma_{1}$ SR is always
abrupt without any coexisting region. We can also remark that $J^{M}$ favours
the $\Gamma_{1}$ state with respect to the $\Gamma_{4}$ state, which can be
logically understood by the fact that in the $\Gamma_{1}$ phase all the
directions are AFM and, since the torque creating this SR is acting on
$G_{M,y}$ (in contrast to $\Gamma_{4}$ to $\Gamma_{2}$ SR where the torque is
acting on $C_{M,y}$), the cantings are smaller than in the $\Gamma_{4}$ state.
Hence, unlike the $\Gamma_{4}$ to $\Gamma_{2}$ case, the SR involving
$\Gamma_{1}$ happens as soon as the system overcomes the energy difference due
to ASE and SIA between the two states, making this transition abrupt.

We also need to mention that this mechanism explains the Ising-like (strongly
colinear) nature of the $\Gamma_{1}$ state [22]. Since the force creating this
SR is acting between $G$-type order of $M$ site and $C$-type order of $R$ site
(i.e $-8N\varepsilon_{x}^{RM}C_{R,z}\bar{G}_{M,y}$), the $M$ atoms in
$\Gamma_{1}$ state will have a very small canting compared to other states and
the spins will mainly order in the Gy type, hence closer to an Ising like
nature.

Figure 16: Phase diagrams of $\Gamma_{4}$ to $\Gamma_{1}$ transition by
plotting exchange on transition metal sites (JM) vs the anisotropic exchange
$\varepsilon_{x}^{RM}$. We can see that the SR transition is abrupt for the
whole range of exchange.

##  VI Conclusion

We have studied in this paper the microscopic mechanism behind the SR and MR
magnetic behaviours of the $RM$O3’s through a Heisenberg model where we
considered the superexchange interactions and DMI between the transition-metal
sites, as well as between the rare-earth ($R$) and transition-metal sites
($M$), and we neglected the superexchange and the DMI between the $R$ spins as
they are much smaller than the other interaction parameters.

We conclude that there are two interactions polarising the $R$ atom site, i.e
(i) the superexchange between $M$ sites (through its wFM) and $R$ sites and
(ii) the DMI between $R$ and $M$, which can results into two effects. Indeed,
we can have that both interactions polarise the $R$ element parallel to the
$M$ wFM canting direction such that there will be no MR but an amplification
of the total magnetization of the crystal (See Fig. 1 panel b blue curve). We
can also have that both interactions polarise the $R$ element in opposite
direction to the wFM of the $M$ cation such that the total magnetization
amplitude can be reduced up to a critical temperature below which its sign
change (See Fig. 1 panel b red curve). The change of sign appears when the
negative $R$ cation magnetization compensates the positive one of the $M$ site
(wFM).

Our analysis of the SR transitions have shown that the $\Gamma_{4}$ to
$\Gamma_{2}$ transition similarly comes mainly from the DMI interactions
between $M$ site and $R$ site but it can be weighted by the superexchange
between the $M$ sites. We found that within a relatively wide range of these
three interactions this SR transition is smooth and happens through a mixed
state where the $\Gamma_{4}$ and the $\Gamma_{2}$ phases coexist even though
they do not interact in our Hamiltonian. How broad is the temperature range in
which the SR takes place through the $\Gamma_{24}$ mixed state depends on a
subtle ratio between DMI and exchange interactions between $M$ sites, which
can vary depending on the rare earth and transition metal cation that is
present in the $Pnma$ perovskite structure. We also found that the
$\Gamma_{4}$ to $\Gamma_{1}$ SR transition depends on even more subtle
interactions (anisotropic superexchange that acts as the DMI) but, contrary to
the $\Gamma_{4}$ to $\Gamma_{2}$ SR, it never presents a coexisting region,
i.e. it always proceeds through an abrupt change.

The model we have presented can help in designing the strength and amplitude
of SR and MR in $RM$O3 through, e.g. doping, strain or pressure that would
tune the ratio between the key interactions as desired. Our model can also be
easily extended by including the interactions between the rare-earth spins to
study the complex magnetic phase diagrams below the Néel temperature of the
rare-earth sublattice. It can be enlarged too with the anisotropic exchanges
(important for the $\Gamma_{4}$ to $\Gamma_{1}$ SR) [22] or with a 4-spin
interaction term, which has been shown to be important in rare-earth
manganites [43]. Because it contains all the key interactions that allow to
describe most of the important magnetic properties of $RM$O3 compounds, the
model can be used to study dynamically magnetic domain walls. Going beyond,
the model can be coupled with a lattice model (second principles [44, 45]) to
have access to a full atom plus spin dynamics for the simulations of, e.g.,
recent ultrafast laser excitation experiments made on these crystals [46, 47,
48].

## Acknowledgements

The authors thank He Xu for his help in using the TB2J code. This work has
been funded by the Communauté Française de Belgique (ARC AIMED G.A. 15/19-09).
EB and AS thanks the FRS-FNRS for support. J.Í. thanks the support of the
Luxembourg National Research Fund through Grant No. FNR/C18/MS/12705883/REFOX.
The authors acknowledge the CECI supercomputer facilities funded by the
F.R.S-FNRS (Grant No. 2.5020.1), the Tier-1 supercomputer of the Fédération
Wallonie-Bruxelles funded by the Walloon Region (Grant No. 1117545) and the
OFFSPRING PRACE project.

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