# Features of magnetization and spin reorientation
in weak ferrimagnets of the YFe1-xCrxO3 type

Alexander Moskvin<EMAIL_ADDRESS>Ural Federal University, 620083
Ekaterinburg, Russia  M.N. Mikheev lnstitute of Metal Physics of Ural Branch
of Russian Academy of Sciences, 620108 Ekaterinburg, Russia  Evgenii
Vasinovich<EMAIL_ADDRESS>Ural Federal University, 620083
Ekaterinburg, Russia

###### Abstract

A brief critical review is given of the 50-year history of experimental and
theoretical studies of the magnetic properties of a new promising class of
weak ferrimagnets such as RFe1-xCrxO3 with a non-magnetic R-ion (R = La , Y,
Lu), i.e. systems with competing signs of the Dzyaloshinsky vectors Fe–Fe,
Cr–Cr and Fe–Cr. The spin Hamiltonian of the system is considered taking into
account isotropic exchange, antisymmetric Dzyaloshinsky-Moriya exchange, and
second- and fourth-order single-ion anisotropy. Within the framework of the
molecular field approximation, calculations were made of the Neel
temperatures, the average magnetic moments of $3d$ ions, total and partial
magnetizations, and effective anisotropy constants. The existence in the model
system YFe1-xCrxO3 of two regions of negative magnetization $0.25\leq x\leq
0.5$ and $x\approx 0.8$ with the corresponding magnetization reversal points
reaching room temperature at $x\approx 0.45$. The phenomenon of spin
reorientation observed for single-crystal samples in a wide range of
concentrations is explained by a sharp decrease in the contribution of
antisymmetric exchange to magnetic anisotropy with increasing deviation from
the parent compositions and competition between the contributions of single-
ion anisotropy of Fe and Cr ions. It has been suggested that the spatial
orientation of the Neel ${\bf G}$ vector and the $G_{xyz}$ configuration are
the reason for the small value of saturation magnetization observed
experimentally for compositions inside or near the region of negative
magnetization.

##  I Introduction

Rare earth orthoferrites RFeO3 and orthochromites RCrO$3$ (R = Y, or rare
earth ion) have attracted and continue to attract special attention of
researchers for many decades due to the combination of unique magnetic,
magnetooptical, magnetoelastic and magnetic resonance properties, primarily
weak ferro- and antiferromagnetism, spin-reorientation transitions, negative
magnetization phenomena (magnetization reversal), magnetoelectric effects.
Orthoferrites and orthochromites remain a relevant object not only of
fundamental theoretical and experimental, but also of applied research related
to the perspective for using these materials in various innovative spintronics
devices.

The features of $4f$–$3d$ interaction and, first of all, antisymmetric
Dzyaloshinsky-Moriya exchange remain at the center of fundamental research on
orthoferrites and orthochromites. The development of the microscopic theory of
the Dzyaloshinsky-Moriya interaction [1, 2, 3, 4, 5], in particular the
orientation and sign of the Dzyaloshinsky vector for pairs of $3d$ ions, made
it possible not only to give a quantitative description weak ferro- and
antiferromagnetism of orthoferrites and orthochromites, but also to predict a
new class of magnets with competing signs of the Dzyaloshinsky vector based on
mixed weak ferromagnets RFe1-xCrxO3 [6, 7, 8, 9, 10, 11], Fe1-xCrxBO3 [12],
Mn1-xNixCO3 [14, 13], Fe2-xCrxO3 [15], named weak ferrimagnets.

The first experimental studies of mixed orthoferrites-orthochromites
YFe1-xCrxO3 were carried out at Moscow University by A. M. Kadomtseva and co-
workers more than 50 years ago on both polycrystalline and single-crystalline
samples [17, 18, 19, 6, 7, 16]. These studies were initiated, in particular,
by the predictions of Goodenough and Kanamori [20], according to which the
superexchange integral Fe3+–O2-–Cr3+ should have a “ferromagnetic sign”, which
could lead to a fundamental restructuring of the magnetic structure of mixed
compositions. However, both experimental and theoretical studies have shown
that in these systems we are not dealing with the competition of isotropic
exchange interactions, but with the peculiarities of the Dzyaloshinsky-Moriya
interaction for various $3d$ ions.

Magnetic (torsion magnetic balances, vibrating magnetometer, torques) and
neutron diffraction measurements have revealed a number of unique magnetic
properties of YFe1-xCrxO3, such as a sharp drop of one or two orders of
magnitude in spontaneous magnetization in substituted compositions,
magnetization reversal, the appearance of spin reorientation, which is absent
in the “parent” compositions YFeO3 and YCrO3. These features received a
natural explanation within the framework of theoretical concepts about the
competition of Dzyaloshinsky-Moriya (DM) interactions Fe–Fe, Cr–Cr and Fe–Cr
in orthoferrites-orthochromites of the YFe1-xCrxO3 type, in particular,
different signs of the Dzyaloshinsky vectors ${\bf d}_{FeFe}$, ${\bf
d}_{CrCr}$ and ${\bf d}_{FeCr}$ [1, 6, 2], which led to the antiparallel
orientation of the average weak ferromagnetic moments of the Fe and Cr
subsystems in a wide range of concentrations, that is, to the weak
ferrimagnetic behavior of YFe1-xCrxO3. Thus, a new phenomenon was
experimentally discovered and theoretically explained, i.e. weak
ferrimagnetism (WFIM) in mixed $3d$ systems with competing signs of the
Dzyaloshinsky vector. Just as different signs of the ordinary exchange
integral determine different (ferro-antiferro) magnetic order, different signs
of Dzyaloshinsky vectors create the possibility of inhomogeneous (ferro-
antiferro) ordering of local weak ferromagnetic moments.

Weak ferrimagnets based on orthoferrites-orthochromites of the RFe1-xCrxO3
type (R = Nd, Gd, Dy, Y, Lu) [8, 9, 10, 11], as well as Fe1-xCrxBO3 [12],
Mn1-xNixCO3 [14], Fe2-xCrxO3 [15] were the subject of intensive fundamental
theoretical and experimental research in the late 20th century. A new surge of
interest in these systems already in the 21st century [21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33] is associated with the opened prospects for the
practical use of the magnetization reversal phenomenon and the associated
effects of “negative” magnetization, exchange bias for the creation of various
multifunctional spintronics devices, detection of specific magnetoelectric and
magnetocaloric properties [29, 34, 35].

However, in many “new” works devoted to the study of weak ferrimagnetic
orthoferrites-orthochromites, and carried out exclusively on polycrystalline
samples, mainly of composition YFe0.5Cr0.5O3, we are faced with both the
ambiguity of experimental data and the ambiguity in their interpretation. In
particular, this concerns the choice of the calculation scheme for the
molecular field approximation [24], incorrect interpretation of the nature of
the magnetization reversal phenomenon, for example, as a result of competition
between single-ion spin anisotropy and the Dzyaloshinsky-Moriya interaction
[23, 25, 30, 32, 33], neglect of many interactions that are fundamental for
these systems, described in sufficient detail in review articles [3, 4, 5].
The unusual effect of spin reorientation in weak ferrimagnets with a
nonmagnetic R-ion (Y, Lu), discovered back in 1972 [18], has also not yet
received an adequate description. Moreover, some authors [24] associate spin
reorientation in the weak ferrimagnet YFe1-xCrxO3 with the simple effect of
changing the canting angle of magnetic sublattices.

These circumstances stimulated this work, in which we give a critical review
of the literature data, consider the description of the magnetic structure, in
particular the magnetization reversal phenomena and spin reorientation in weak
ferrimagnets of the YFe1-xCrxO3 type, within the framework of a more rigorous
approach developed earlier for “parent” orthoferrites and orthochromites in
the works [6, 3, 4, 5]. Section 2 presents the general form of the Hamiltonian
for a weak ferrimagnet, with a brief analysis of the role of the various
contributions. Section 3 is devoted to the formulation of the molecular field
model (MFA) taking into account the multi-sublattice structure of RFe1-xCrxO3
with a non-magnetic R-ion (La, Y, Lu). Section 4 discusses experimental data
on the concentration dependence of the Neel temperature of YFe1-xCrxO3 and the
corresponding MFA calculations, and provides a comparison of the experimental
and calculated temperature dependence of the average magnetic moment in
YFe1-xCrxO3. Section 6 is devoted to the nature of spin reorientation in weak
ferrimagnets. Section 7 discusses experimental data on the concentration and
temperature dependence of magnetization and “hidden” noncollinearity in
YFe1-xCrxO3 and the magnetization reversal phenomenon, and compares it with
MFA calculations. A brief conclusion is presented in Section 8.

##  II Basic spin interactions and the Hamiltonian of a weak ferrimagnet
YFe1-xCrxO3

Like the “parent” orthoferrites YFeO3 and orthochromites YCrO3, mixed systems
YFe1-xCrxO3 are orthorhombic perovskites with space group $Pbnm$
($D_{2h}^{16}$) [21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33]. As shown in Fig.
1, Fe3+ (Cr3+) ions occupy positions 4b in the unit cell: $1\,(1/2,0,0)$,
$2\,(1/2,0,1/2)$, $3\,(0,1/2,1/2)$, $4\,(0,1/2,0)$. The classical basis
vectors of the magnetic structure for the $3d$ sublattice are defined as
follows:

$\displaystyle 4S{\bf F}={\bf S}_{1}+{\bf S}_{2}+{\bf S}_{3}+{\bf S}_{4}\,;$
$\displaystyle 4S{\bf G}={\bf S}_{1}-{\bf S}_{2}+{\bf S}_{3}-{\bf S}_{4}\,;$
$\displaystyle 4S{\bf C}={\bf S}_{1}+{\bf S}_{2}-{\bf S}_{3}-{\bf S}_{4}\,;$
$\displaystyle 4S{\bf A}={\bf S}_{1}-{\bf S}_{2}-{\bf S}_{3}+{\bf S}_{4}\,.$
(1)

Here the vector ${\bf G}$ describes the main antiferromagnetic component of
the magnetic structure, ${\bf F}$ is the vector of ferromagnetism (“overt
canting” of the sublattices), weak antiferromagnetic components ${\bf C}$ and
${\bf A}$ describe the canting of the magnetic sublattices without the
formation of a total magnetic moment (“hidden canting” of the sublattices).
“Allowed” spin configurations for the $3d$ sublattice, compatible with the
antiferromagnetic sign of the main isotropic superexchange, are denoted as
$\Gamma_{1}\,(A_{x},G_{y},C_{z})$, $\Gamma_{2}\,(F_{x},C_{y},G_{z})$,
$\Gamma_{4}\,(G_{x},A_{y},F_{z})$, where the only non-zero components of the
basis vectors appear in parentheses. Note that in the literature one can find
different options for numbering the positions of Fe3+ ions (see, for example,
the works [36, 37]), so that the basis vectors ${\bf G}$, ${\bf C}$, ${\bf A}$
may differ in sign.

Figure 1:  Structure of superexchange bonds; large balls – Fe3+, Cr3+ ions,
small ones – O2-; 1, 2, 3, 4 – magnetic ions in four nonequivalent positions

Within the framework of the theoretical approach, developed primarily for
“parent” orthoferrites in the works [2, 3, 4, 5], we present the Hamiltonian
of the weak ferrimagnet YFe1-x CrxO3 in the form

$\hat{H}={\hat{H}}_{ex}+{\hat{H}}_{DM}+{\hat{H}}_{SIA}^{(2)}+{\hat{H}}_{SIA}^{(4)}+{\hat{H}}_{TIA}$
(2)

of the sum of contributions from isotropic exchange interaction, antisymmetric
Dzyaloshinsky-Moriya exchange, single-ion and two-ion spin anisotropy. A rough
estimate of the various contributions gives $H_{ex}\sim 1$ meV, $H_{DM}\sim
0.1$ meV, $H_{SIA}^{(2)}\sim H_{TIA}\sim 0.01$ meV, $H_{SIA}^{(4)}\sim 0.001$
meV.

###  II.1 Isotropic superexchange

Below we restrict ourselves to the bilinear isotropic Heisenberg superexchange
nearest neighbor interaction in the form

${\hat{H}}_{ex}=\frac{1}{2}\sum_{\langle mn\rangle}I_{mn}\,({\hat{\bf
S}}_{m}\cdot{\hat{\bf S}}_{n}),$ (3)

determining the formation of the main $G$-type antiferromagnetic structure,
experimentally established using magnetic neutron diffraction for both parent
ferrites and weak ferrimagnets of the YFe1-xCrxO3 type (see, for example, [21,
24, 26]). The value of the exchange integrals $I_{FeFe}$ and $I_{CrCr}$ can be
found from data on the Neel temperatures of YFeO3 and YCrO3 based on a simple
mean field approximation

$T_{N}=\frac{zS(S+1)}{3k_{B}}I\,,$ (4)

although these “experimental” exchange integrals ($I_{FeFe}$ = 36.8 K in
YFeO3, $I_{CrCr}$ = 18.7 K in YCrO3), can be one and a half to two times less
than those obtained by other methods [2, 38, 39]. Thus, recent experimental
data give for YFeO3 $I^{(21)}=58.2$ K, $I^{(41)}=53.6$ K [40] or
$I^{(21)}\approx I^{(41)}=51.5$ K [37].

The only source of experimental information on the value of the exchange
integral $I_{FeCr}$ in orthoferrites-orthochromites are experimental data [41]
from a study of the Mössbauer spectra of 57Fe in RCr0.99Fe0.01O3 (R $=$ La,
La0.5Nd0.5, Nd, Sm, Gd, Dy, Y, Er, Yb, Lu), according to which in all systems
$I_{FeCr}$ has an “antiferromagnetic sign” $I_{FeCr}>0$, decreasing during the
transition from “heavy” to “light” R-ions. Thus, $I_{FeCr}=13.6\pm 0.8,\
13.4\pm 0.4,\ 7.2\pm 0.4$ K for R = Lu, Y, La, respectively. The model theory
of the angular dependence of superexchange integrals well describes the
experimental data for the exchange integrals $I_{FeFe}$, $I_{CrCr}$ in
orthoferrites and orthochromites [3, 4, 5] and predicts the change sign for
$I_{FeCr}$ at the Fe–O–Cr superexchange bond angle $\theta_{12}$ $\approx$
170∘. In other words, superexchange (Fe3+ – O2- – Cr3+) becomes ferromagnetic
for bond angles significantly exceeding angles typical for orthoferrites-
orthochromites (140∘–155∘). Unfortunately, at present there are no reliable
estimates of “non-Heisenberg” isotropic interactions, in particular,
biquadratic exchange

${\hat{H}}_{bq}=\frac{1}{2}\sum_{\langle mn\rangle}j_{mn}\,({\hat{\bf
S}}_{m}\cdot{\hat{\bf S}}_{n})^{2}\,.$ (5)

###  II.2 Antisymmetric Dzyaloshinsky-Moriya exchange

Antisymmetric Dzyaloshinskii-Moriya exchange [42, 43]

${\hat{H}}_{DM}=\frac{1}{2}\sum_{\langle mn\rangle}{\bf d}_{mn}\cdot[{\hat{\bf
S}}_{m}\times{\hat{\bf S}}_{n}]\,$ (6)

determines, as a rule, the main, antisymmetric contribution to the
Dzyaloshinskii interaction [42]. Microscopic theory of bilinear antisymmetric
exchange ${\hat{H}}_{DM}$ was discussed in detail in a number of works by one
of the authors [1, 2, 3, 4, 5]. In particular, he established a microscopic
formula for the Dzyaloshinsky vector in a pair of superexchange-coupled $S$
type magnetic ions with an orbitally nondegenerate ground state under the
assumption of local cubic symmetry, that is, for $3d$ ions (Fe3+, Mn2+, Cr3+,
Mn4+, Ni2+) with half-filled $t_{2g}$\- and/ or $e_{g}$-shells ($t_{2g}^{3}$,
$t_{2g}^{3}e_{g}^{2}$, $t_{2g}^{6}e_{g}^{2}$) [44, 2, 3, 4, 5]

$\mathbf{d}_{mn}=d_{mn}(\theta)[\mathbf{r}_{m}\times\mathbf{r}_{n}]\,,$ (7)

where ${\bf r}_{m,n}$ are unit radius vectors for bonds O – Mm,n with
presumably equal bond lengths, and the factor $d_{mn}(\theta)$ has a simple
angular dependence

$d_{mn}(\theta)=d_{1}+d_{2}\,\cos\theta_{mn}\,,$ (8)

where $\theta_{mn}$ is the angle connections Mm – O – Mn. The sign of the
scalar parameter $d_{mn}(\theta)$ can be considered as the sign of the
Dzyaloshinsky vector. Theoretically predicted sign relationships Dzyaloshinsky
vectors in pairs of $3d$ ions of $S$ type with local octahedral symmetry and
superexchange coupling angle $\theta>\theta_{cr}$ ($\cos\theta_{cr}$ =
$-d_{1}/d_{2}$) are given in Table 1.

Table 1: Theoretical predictions of the signs of Dzyaloshinsky vectors in pairs of $3d$ ions of $S$ type with local octahedral symmetry and superexchange coupling angle $\theta>\theta_{cr}$. 3$d^{n}$ | Cr3+, Mn4+ | Mn2+, Fe3+ | Ni2+, Cu3+
---|---|---|---
Cr3+, Mn4+ | + | – | +
Mn2+, Fe3+ | – | + | +
Ni2+, Cu3+ | + | + | –

The anisotropic antisymmetric Dzyaloshinsky-Moriya interaction, the magnitude
and orientation of the Dzyaloshinsky vectors, is responsible for the formation
of relatively weak overt (F-mode) and hidden (A and C-mode) spin
noncollinearity in the model of four magnetic sublattices, that is, weak
ferromagnetism and weak antiferromagnetism, respectively – “fine structure” of
three possible magnetic configurations $\Gamma_{1}\,(A_{x},G_{y},C_{z})$,
$\Gamma_{2}\,(F_{x},C_{y},G_{z})$, $\Gamma_{4}\,(G_{x},A_{y},F_{z})$. The weak
ferromagnetic moment, or the F-mode of noncollinearity, is determined by the
$y$-component of the Dzyaloshinsky vectors, the weak antiferromagnetic A- and
C-modes are determined by the $z$\- and $x$-components of the Dzyaloshinsky
vectors, respectively [3, 4, 5].

In 1975, based on a simple formula for the Dzyaloshinsky vector (7), a
connection was established between the crystallographic and “canted” magnetic
structures for four-sublattice orthoferrites RFeO3 and orthochromites RCrO3
[45, 2]:

$F_{x}=\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{I}G_{z};\
F_{z}=-\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{I}G_{x}\,;\ $
$A_{x}=\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{I}G_{y};\
A_{y}=-\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{I}G_{x};$
$C_{y}=\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{I}G_{z};\
C_{z}=-\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{I}G_{y}\,;$ (9)

where $a,b,c$ are unit cell parameters, $x_{1,2},y_{1,2},z_{2}$ are oxygen
(OI,II) parameters, $l$ – average cation-anion bond length. These relations
assume averaging over Fe3+ – O2- – Fe3+ bonds in the $ab$ plane and along $c$
axis. It is obvious that in this approximation, the relative magnitude of the
components of the “small” basis vectors that describe the explicit and hidden
non-collinearity of the spins is determined only by the structural parameters.

Taking into account (9), we arrive at a simple relationship between
crystallographic parameters and the magnetic moment of the Fe sublattice: in
units of emu/g

$M_{Fe}=\frac{4g\mu_{B}S}{\rho V}|F_{x,z}|=\frac{2g\mu_{B}Sac}{3l^{2}\rho
V}(x_{1}+2z_{2})\frac{d(\theta)}{I(\theta)}\,,$ (10)

where $\rho$ and $V$ are the density and volume of the unit cell,
respectively. The explicit skew of magnetic sublattices, or the ferromagnetism
vector $F_{x,z}$, can be calculated through the ratio of the Dzyaloshinsky
field ($H_{D}$) to the exchange field ($H_{E}$)

$F=H_{D}/2H_{E}\,.$ (11)

If we know the Dzyaloshinsky field, we can calculate the factor $d(\theta)$ in
orthoferrites as follows

$H_{D}=\frac{S}{g\mu_{B}}\sum_{i}|d_{y}(1i)|=\frac{S}{g\mu_{B}}(x_{1}+2z_{2})\frac{ac}{l^{2}}|d(\theta)|\,,$
(12)

which gives $|d(\theta)|\cong 3.2$ K $\approx 0.3$ meV in YFeO3 at $H_{D}=140$
kOe [46]. This value is in good agreement with data from recent experiments
[40, 37], which made it possible to find information about the Dzyaloshinsky
vector based on measurements of the spin wave spectrum. Note that despite the
fact that $F_{z}\approx$ 0.01 the parameter $d(\theta)$ is only an order of
magnitude smaller than the exchange integral in YFeO3.

The connection between the orientation of the Dzyaloshinsky vector and the
geometry of superexchange of magnetic cations (7) made it possible to find the
relationship between the quantities $F_{x,z},A_{x,y},C_{y,z}$, which determine
the weak non-collinearity of spins in configurations
$\Gamma_{1},\Gamma_{2},\Gamma_{4}$, which is in good agreement with
experimental data from nuclear magnetic resonance and magnetic neutron
diffraction [3, 4]. Note that $|A_{x,y}|>|F_{x,z}|>|C_{y,z}|$.

Figure 2:  Contribution of antisymmetric exchange to anisotropy constants for
different planes in RFeO3

These same relations make it possible to find the contribution of
antisymmetric exchange to magnetic anisotropy. Fig. 2 shows the calculated
contribution of antisymmetric exchange to the second-order anisotropy
constants for various planes in RFeO3 [2, 3, 4, 5]. These constants determine
the effective anisotropy energy $E_{an}=k_{1}\,\cos 2\theta$ for rotation of
the antiferromagnetism vector ${\bf G}$ in the $ac$ or $bc$ plane, or
$E_{an}=k_{1}\,\cos 2\phi$ for rotating the vector ${\bf G}$ in the $ab$
plane, where $\theta,\phi$ are the polar and azimuthal angles of the vector
${\bf G}$. It is obvious that antisymmetric exchange stabilizes the $G_{x}$,
or $\Gamma_{4}$ configuration in all orthoferrites [3, 4, 5] and, most likely,
in orthochromites, given the generally similar geometry bonds Cr–O–Cr and
Fe–O–Fe, and the energies of the $G_{z}(\Gamma_{2})$ and $G_{y}(\Gamma_{1})$
configurations are close. In other words, the contribution to magnetic
anisotropy in the $bc$ plane is small, since the constant
$k_{1}(bc)=k_{1}(ab)+k_{1}(ac)$ is almost an order of magnitude smaller than
the constants $k_{1}(ac)$ and $k_{1}(ab)$.

###  II.3  Symmetrical spin anisotropy

Single-ion spin anisotropy $H_{SIA}$ includes terms of the second and fourth
order: $H_{SIA}=H_{SIA}^{(2)}+H_{SIA}^{(4)}$. Due to the low symmetry of the
local environment of $3d$ ions in the structure of orthoferrites and
orthochromites, we represent second-order single-ion anisotropy in the most
general form as

${\hat{H}}_{SIA}^{(2)}=\sum_{n}K_{n}\Big{[}({\hat{\bf S}}_{n}\cdot{\bf
a}_{n})({\hat{\bf S}}_{n}\cdot{\bf b}_{n})-\frac{1}{3}({\bf a}_{n}\cdot{\bf
b}_{n}){\hat{\bf S}}_{n}^{2}\Big{]}\,,$ (13)

where $K_{n}$ are some anisotropy parameters, ${\bf a}_{n},{\bf b}_{n}$ are
unit vectors that define two characteristic preferred directions for $n$ site,
the second term ensures the “traceless” of the tensor anisotropy constants.
For uniaxial anisotropy ${\bf a}_{n}={\bf b}_{n}$.

Often, especially in spin resonance theory, the expression $H_{SIA}^{(2)}$ is
used for an individual ion in the form

$\displaystyle{\hat{H}}_{SIA}^{(2)}=D({\hat{S}}_{z}^{2}-\frac{1}{3}S(S+1))+E({\hat{S}}_{x}^{2}-{\hat{S}}_{y}^{2})+$
$\displaystyle+\frac{p}{2}\\{{\hat{S}}_{y},{\hat{S}}_{z}\\}+\frac{q}{2}\\{{\hat{S}}_{x},{\hat{S}}_{z}\\}+\frac{r}{2}\\{{\hat{S}}_{x},{\hat{S}}_{y}\\}\,,$
(14)

$D,\ E,\ p,\ q,\ r$ are anisotropy constants,
$\\{{\hat{S}}_{x},{\hat{S}}_{y}\\},\ \dots$ are anticommutators. The first two
terms in ${\hat{H}}_{SIA}^{(2)}$, or the so-called “rhombic” anisotropy,
determine the contribution to the first anisotropy constants for $ac$, $ab$,
and $bc$ planes in orthoferrites and orthochromites. The last three terms
determine the symmetric contribution to the Dzyaloshinsky interaction, taking
into account which we obtain a symmetric contribution to the explicit and
hidden noncollinearity of spins:

$F_{x,z}=\frac{q}{6I}G_{z,x}\,;\ A_{x,y}=\frac{r}{4I}G_{y,x}\,;\
C_{y,z}=\frac{p}{2I}G_{z,y}\,.$ (15)

Naturally, taking into account the signs in the relationships between small
basis vectors ${\bf F},{\bf A},{\bf C}$ with the “large” antiferromagnetism
vector ${\bf G}$ in (9) and (15) antisymmetric and symmetric contributions add
or subtract during reorientation of the $G_{x}\leftrightarrow G_{z}$ type, in
particular, thereby violating the orthogonality of the ferro- and
antiferromagnetism vectors (${\bf F}$ and ${\bf G}$) in the angular $G_{xz}$
phase, and leading to a change in the magnetization value during spin
reorientation. Both theoretical estimates [2] and experimental ESR data of
Fe3+ and Cr3+ ions in LaAlO3 orthoaluminates isostructural with orthoferrites
and orthochromites [47] and YAlO3 [48] indicate that the values of the
parameters $p,q,r$ do not exceed 0.1 K.

Taking into account the experimental values of exchange integrals, we expect
for a “symmetric” contribution to the ferromagnetism vector for orthoferrite
YFeO3 and orthochromite YCrO3: $|F_{x,z}|\leq 5\cdot 10^{-4}$ and
$|F_{x,z}|\leq 10^{-3}$ respectively, that is, 5 and 10% of the value of
$F\approx 0.01$, typical for the main “antisymmetric” contribution. Let’s
note, that experimental estimates of work [46] give slightly lower estimates,
2 and 5%, respectively. It is obvious that for weak ferrimagnets of the
YFe1-xCrxO3 type, especially in the region of compensation of the main
“antisymmetric” contribution to magnetization, the small “symmetric”
contribution of single-ion anisotropy can play an important role, making a
noticeable contribution to the total magnetic moment. At the same time, the
possibility of competition between the signs of the contributions of the Fe
and Cr sublattices cannot be excluded.

Second-order single-ion spin anisotropy turns out to be very sensitive to
distortions in the local environment of $3d$ ions. Thus, in the orthoferrites
YFeO3 and LuFeO3, the contribution $H_{SIA}^{(2)}$ to the first anisotropy
constant in the $ac$ plane $k_{1}(ac)$ differs in sign, so that in YFeO3 this
contribution partially compensates for the contribution of the Dzyaloshinsky-
Moriya interaction, and in LuFeO3 both contributions add up, resulting in a
sharp increase in $k_{1}(ac)$ [49]. The difference in the electronic structure
of the Fe3+($t_{2g}^{3}e_{g}^{2}$) and Cr3+($t_{2g}^{3}$) ions leads to a
significantly different reaction of the contribution of the single-ion second-
order anisotropy on local structure distortions. It is possible that the sharp
change in the $H_{SIA}^{(2)}$ contribution is responsible for the transition
from the basic $\Gamma_{4}$ configuration in YFeO3 to the $\Gamma_{2}$
configuration in LuCrO3 [38]. Let us pay attention to the close value, but
different sign of the effective second-order single-ion anisotropy constant
for the impurity ions Fe3+ and Cr3+ in the isostructural to orthoferrites and
orthochromites orthoaluminate LaAlO3 [47].

In the fourth-order single-ion anisotropy, which is nonzero only for Fe3+ ions
($S=5/2$), we highlight the main term, which in the local coordinate system
for weak distorted FeO6 octahedra in orthoferrites can be represent in the
form

${\hat{H}}_{cub}=\frac{a}{6}\sum_{n}\Big{[}{\hat{S}}_{x_{n}}^{4}+{\hat{S}}_{y_{n}}^{4}+{\hat{S}}_{z_{n}}^{4}-\frac{1}{5}S(S+1)(3S^{2}+3S-1)\Big{]}\,,$
(16)

the so-called “cubic” anisotropy [50, 51]. By the way, the value $a=4\cdot
10^{-3}$ cm-1, obtained from ESR data of Fe3+ ions in LaAlO3 [47] can be
considered a reasonable estimate of the cubic anisotropy constant of Fe3+ ions
and in YFe1-xCrxO3.

Two-ion anisotropy $H_{TIA}$ includes both the classical bilinear magnetic-
dipole interaction and more complex exchange-relativistic contributions [52].
The relatively small contribution of the magnetic dipole interaction, as well
as the contribution of $H_{DM}$, stabilizes the magnetic configuration of
$\Gamma_{4}$, and its value, decreasing with the transition to “light” R-ions,
practically becomes negligible in LaFeO3 [2, 51].

##  III Molecular field models for weak ferrimagnets of the YFe1-xCrxO3 type

The molecular field approximation (MFA) provides the basis for the simplest
and most physically visual, albeit semi-quantitative, description of spin
systems. In many cases, especially for such complex systems as weak
ferrimagnets such as YFe1-xCrxO3, the advantage of this method is the ability
to describe the fundamental features of the concentration and temperature
dependences of the main magnetic characteristics, i.e. the Neel temperature,
magnetization, magnetic anisotropy.

However, it should be taken into account that the approximations used in any
MFA method can significantly affect the results. To demonstrate this, below we
will consider two scheme of the molecular field theory for describing bilinear
interactions, i.e. isotropic exchange and antisymmetric Dzyaloshinsky-Moriya
interaction in weak ferrimagnets such as YFe1-xCrxO3.

We also note that the classical Monte Carlo (MC) method [28] has an advantage
over MFA in the ability to consistently take into account all possible
configurations of the distribution of Fe and Cr ions in the lattice, but
unlike MFA within the framework of the classical spin MC method ions are
replaced by classical vectors, which undoubtedly introduces significant errors
into the results.

###  III.1 MFA-I

In the simplest model of weak ferrimagnets of the RFe1-xCrxO3 type (R = La, Y,
Lu), which assumes a single magnetic ordering in the Fe–Cr subsystem with
molecular fields common to all Fe3+ (Cr3+) ions, the bilinear part of the
Hamilton operator is represented as

${\hat{H}}_{ex}+{\hat{H}}_{DM}=\sum_{n}\,({\bf h}_{n}\cdot{\hat{\bf
S}}_{n})-\frac{1}{2}\sum_{n}\,({\bf h}_{n}\cdot\langle{\bf S}_{n}\rangle)\,,$
(17)

where for the molecular field ${\bf h}_{n}$, taking into account the leading
contributions of isotropic exchange and the Dzyaloshinsky-Moriya interaction,
we have

${\bf h}_{n}=\sum_{m}\,\left(I_{mn}\langle{\bf S}_{m}\rangle+[{\bf
d}_{mn}\times\langle{\bf S}_{m}\rangle]\right)\,,$ (18)

where $\langle{\bf S}_{m}\rangle$ is the thermodynamic average spin of an
arbitrary ion (Fe3+ or Cr3+)

$\langle{\bf S}_{m}\rangle=-\frac{{\bf
h}_{m}}{h_{m}}\,S\,B_{S}\left(\frac{Sh_{m}}{k_{B}T}\right)\,,$ (19)

where $B_{S}$ is the function Brillouin, ${\bf h}_{m}$ is an effective field
for $m$-th site, $h_{m}=|{\bf h}_{m}|$.

It is obvious that, in contrast to the homogeneous parent systems YFeO3 and
YCrO3, for weak ferrimagnets of the YFe1-xCrxO3 type we are forced to
introduce a number of additional assumptions and approximations to solve the
molecular field equations (19):

1) Fe3+ and Cr3+ ions occupy lattice sites with equal probability;

2) the parameters of the spin Hamiltonian do not depend on either the local
configuration or the concentration of Fe3+ and Cr3+ ions;

3) long-range crystalline and magnetic (spin) order is preserved, in other
words, the classification of possible magnetic structures ($\Gamma_{1,2,4}$)
and the corresponding relationships between the average values of spin moments
in positions 1, 2, 3 and 4 are also preserved (see Table 2), which allows us
to consider the molecular field equations only for one position of $3d$ ions.

Table 2: Connection between spin components on different sublattices in phases $\Gamma_{1}(A_{x},G_{y},C_{z})$, $\Gamma_{2}(F_{x},C_{y},G_{z})$, $\Gamma_{4}(G_{x},A_{y},F_{z})$ | $S_{x}^{(1)}=-S_{x}^{(2)}=-S_{x}^{(3)}=S_{x}^{(4)}$
---|---
$\Gamma_{1}$ | $S_{y}^{(1)}=-S_{y}^{(2)}=S_{y}^{(3)}=-S_{y}^{(4)}$
| $S_{z}^{(1)}=S_{z}^{(2)}=-S_{z}^{(3)}=-S_{z}^{(4)}$
| $S_{x}^{(1)}=S_{x}^{(2)}=S_{x}^{(3)}=S_{x}^{(4)}$
$\Gamma_{2}$ | $S_{y}^{(1)}=S_{y}^{(2)}=-S_{y}^{(3)}=-S_{y}^{(4)}$
| $S_{z}^{(1)}=-S_{z}^{(2)}=S_{z}^{(3)}=-S_{z}^{(4)}$
| $S_{x}^{(1)}=-S_{x}^{(2)}=S_{x}^{(3)}=-S_{x}^{(4)}$
$\Gamma_{4}$ | $S_{y}^{(1)}=-S_{y}^{(2)}=-S_{y}^{(3)}=S_{y}^{(4)}$
| $S_{z}^{(1)}=S_{z}^{(2)}=S_{z}^{(3)}=S_{z}^{(4)}$

Thus, for the molecular field ${\bf h}_{Fe}$ in position 1 we obtain

${\bf
h}_{Fe}=P_{Fe}(x)\Big{\langle}4I_{FeFe}\,{\bf\hat{S}}^{(4)}_{Fe}+2I_{FeFe}\,{\bf\hat{S}}^{(2)}_{Fe}+\\\
+4[{\bf d}^{(41)}_{FeFe}\times{\bf\hat{S}}^{(4)}_{FeFe}]+2[{\bf
d}^{(21)}_{FeFe}\times{\bf\hat{S}}^{(2)}_{FeFe}]\Big{\rangle}+\\\
+P_{Cr}(x)\Big{\langle}4I_{FeCr}\,{\bf\hat{S}}^{(4)}_{Cr}+2I_{FeCr}\,{\bf\
hat{S}}^{(2)}_{Cr}+\\\ +4[{\bf
d}^{(41)}_{FeCr}\times{\bf\hat{S}}^{(4)}_{Cr}]+2[{\bf
d}^{(21)}_{FeCr}\times{\bf\hat{S}}^{(2)}_{Cr}]\Big{\rangle},$ (20)

where $P_{Fe}(x)=1-x$, $P_{Cr}(x)=x$ are the concentrations of Fe3+ and Cr3+
ions, respectively, the brackets $\langle\dots\rangle$ denote the
thermodynamic mean, and the components of the vectors ${\bf S}^{(2)}$ and
${\bf S}^{(4)}$ are expressed in terms of ${\bf S}^{(1)}$ in accordance with
the Table 2. The field ${\bf h}_{Cr}$ is obtained by replacing FeFe and FeCr
with, respectively, FeCr and CrCr on the right side. Let us note that in (20)
nonequivalent contributions of bonds 1-2 and 1-4 are highlighted, which is
especially important given the different orientation and magnitude of the
Dzyaloshinsky vectors for these bonds.

The works [24, 28] consider the classical approximation for weak ferrimagnets
of the YFe1-xCrxO3 type with “limiting” averaging over configurations, in
which, for example, the Hamiltonian of isotropic exchange is represented as
energy (per ion in position 1)

$E=z\sum_{\alpha\beta}P_{\alpha}(x)P_{\beta}(x)I_{\alpha\beta}\,({\bf
S}_{\alpha}\cdot{\bf S}_{\beta})=\\\ =z\big{[}(1-x)^{2}I_{FeFe}({\bf
S}_{Fe}\cdot{\bf S}_{Fe})+x^{2}I_{CrCr}({\bf S}_{Cr}\cdot{\bf S}_{Cr})+\\\
+2x(1-x)I_{FeCr}({\bf S}_{Fe}\cdot{\bf S}_{Cr})\big{]}\,.$ (21)

###  III.2 MFA-II

A more accurate version of the molecular field model for weak ferrimagnets
(MFA-II), which takes into account the dependence of small components of the
local molecular field, which describe the effects of overt (F-type) and hidden
(A-, C-type) spin non-collinearity, on the configuration of the nearest
environment, was proposed and implemented earlier in the works [1, 2, 8, 10,
12, 14].

Let us consider the Fe3+ ion in the $S^{(1)}$ position; for the Cr3+ ion the
formulas will be similar. Depending on the configuration of the environment,
the molecular fields will have the following form:

${\bf h}_{Fe}^{klr}=k\,\left(I_{FeFe}\,\langle{\bf
S}^{(4)}_{Fe}\rangle+\big{[}{\bf d}^{(41)}_{FeFe}\times\langle{\bf
S}^{(4)}_{Fe}\rangle\big{]}\right)+\\\ +l\,\left(I_{FeFe}\,\langle{\bf
S}^{(4)}_{Fe}\rangle+\big{[}{\bf d}^{\prime(41)}_{FeFe}\times\langle{\bf
S}^{(4)}_{Fe}\rangle\big{]}\right)+\\\ +r\,\left(I_{FeFe}\,\langle{\bf
S}^{(2)}_{Fe}\rangle+\big{[}{\bf d}^{(21)}_{FeFe}\times\langle{\bf
S}^{(2)}_{Fe}\rangle\big{]}\right)+\\\ +(u-k)\,\left(I_{FeCr}\,\langle{\bf
S}^{(4)}_{Cr}\rangle+\big{[}{\bf d}^{(41)}_{FeCr}\times\langle{\bf
S}^{(4)}_{Cr}\rangle\big{]}\right)+\\\ +(u-l)\,\left(I_{FeCr}\,\langle{\bf
S}^{(4)}_{Cr}\rangle+\big{[}{\bf d}^{\prime(41)}_{FeCr}\times\langle{\bf
S}^{(4)}_{Cr}\rangle\big{]}\right)+\\\ +(u-r)\,\left(I_{FeCr}\,\langle{\bf
S}^{(2)}_{Cr}\rangle+\big{[}{\bf d}^{(21)}_{FeCr}\times\langle{\bf
S}^{(2)}_{Cr}\rangle\big{]}\right),$ (22)

where $k$, $l$ and $r$ are the number of nearest neighbors of type Fe3+, and
$0\leq k\leq u$, $0\leq l\leq u$ and $0\leq r\leq u$, respectively, where
$u=2$. The need to use three numbers $k$, $l$ and $r$ is due to the fact that
the $S^{(1)}$ ion is connected to its neighbors by three different
Dzyaloshinsky vectors (see Table 3). The number of neighbors of type Cr3+ is
determined similarly by the inverse numbers $u-k$, $u-l$ and $u-r$.

Table 3:  Components $x,\ y,\ z$ of structural factors $[{\bf r}_{m}\times{\bf r}_{n}]$ that determine orientation vectors Dzyaloshinsky in YFe0.5Cr0.5O3, calculated by data neutron diffraction [32] | $x$ | $y$ | $z$
---|---|---|---
$[{\bf r}_{2}\times{\bf r}_{1}]$ | $0.216$ | $0.562$ | $0$
$[{\bf r}_{4}\times{\bf r}_{1}]$ | $\pm 0.303$ | $0.287$ | $0.397$

Thus, any average value, such as the magnetic moment, is calculated according
to the following rule:

$\langle{\bf
S}_{Fe}\rangle=\sum_{k=0}^{u}\sum_{l=0}^{u}\sum_{r=0}^{u}p^{klr}\,\langle{\bf
S}_{Fe}^{klr}\rangle=\\\ =-\sum_{klr}p^{klr}\,\frac{{\bf
h}_{Fe}^{klr}}{h_{Fe}^{klr}}\,S_{Fe}\,B_{S_{Fe}}\left(\frac{S_{Fe}h_{Fe}^{klr}}{k_{B}T}\right)\,,$
(23)

where $p^{klr}=\binom{u}{k}\binom{u}{l}\binom{u}{r}(1-x)^{k+l+r}x^{z-k-l-r}$
is the probability of occurrence of a configuration with numbers $k$, $l$ and
$r$ at concentration $x$, $\binom{u}{l}$ is the binomial coefficient, $z=6$.
Here and below, summation $\sum_{klr}$ will be denoted by
$\sum_{k=0}^{u}\sum_{l=0}^{u}\sum_{r=0}^{u}$.

Equations (23) can be reduced to linear form if in (22) we discard small terms
and leave only those that contain exchange $I$ or a component of the vector
${\bf G}$ in the corresponding phase. In most cases, such an approximation
does not noticeably affect the results, but, for example, for the vector ${\bf
C}$ in phase $\Gamma_{2}$ the error can reach 10%.

Taking into account the assumptions made, the equations for the transverse
components of magnetic moments in the $\Gamma_{4}$ phase have the form

$\displaystyle
m^{z}_{Fe}(\alpha_{Fe}^{+}+1)+m^{z}_{Cr}\beta_{Fe}^{+}=-\gamma_{Fe}^{F}\,;$
$\displaystyle
m^{z}_{Fe}\alpha_{Cr}^{+}+m^{z}_{Cr}(\beta_{Cr}^{+}+1)=-\gamma_{Cr}^{F}\,;$
$\displaystyle
m^{y}_{Fe}(\alpha_{Fe}^{-}+1)+m^{y}_{Cr}\beta_{Fe}^{-}=\gamma_{Fe}^{A}\,;$
$\displaystyle
m^{y}_{Fe}\alpha_{Cr}^{-}+m^{y}_{Cr}(\beta_{Cr}^{-}+1)=\gamma_{Cr}^{A}\,.$
(24)

The constants $\alpha$, $\beta$, $\gamma$ are given in Appendix A.

###  III.3 Effective field method for taking into account single-ion
anisotropy

To calculate the energy of single-ion anisotropy, you can use the effective
field approximation [51], in which the second-order spin anisotropy energy for
an individual ion takes the form

$\displaystyle
E_{SIA}^{(2)}=D(T)\Big{(}\cos^{2}\theta-\frac{1}{3}\Big{)}+E(T)\sin^{2}\theta\cos
2\phi+$ $\displaystyle+\frac{p(T)}{2}\sin 2\theta\sin\phi+\frac{q(T)}{2}\sin
2\theta\cos\phi+\frac{r(T)}{2}\sin^{2}\theta\sin 2\phi\,,$ (25)

Moreover, the temperature dependence of all anisotropy constants is determined
by a single factor $\rho_{2}(T)$:

$\displaystyle\rho_{2}(T)=\frac{\langle
3\hat{S}_{z}^{2}-S(S+1)\rangle}{S(2S-1)}\,,$ (26)

where the thermodynamic average appears, calculated taking into account the
molecular field acting on the ion.

The “cubic” contribution to the energy of fourth-order anisotropy can be
represented as [50, 51]

$E_{cub}=k_{cub}(T)C^{4A_{1}}_{0}({\bf S})\,,$ (27)

Where

$C^{4A_{1}}_{0}({\bf S})=\sqrt{\frac{7}{12}}C^{4}_{0}({\bf
S})+\sqrt{\frac{5}{24}}\Big{[}C^{4}_{4}({\bf S})+C^{4}_{-4}({\bf S})\Big{]}$

is the so-called invariant cubic tensor harmonic, the argument of which is the
orientation angles $\theta,\phi$ of the classical ion spin vector,

$k_{cub}(T)=k_{cub}(0)\frac{\langle
35\hat{S}_{z}^{4}-(30S^{2}+30S-25)\hat{S}_{z}^{2}+3S^{4}+6S^{3}-3S^{2}-6S\rangle}{2\sqrt{(2S+5)(2S+3)(2S+1)(2S-1)(2S-3)(S+2)(S+1)S(S-1)}}$
(28)

is the temperature-dependent cubic anisotropy constant, with
$a=\,\frac{5\sqrt{3}}{12}k_{cub}(0)$ at $S=5/2$.

In practice, it is important to take into account cubic spin anisotropy for
different crystal planes by replacing ${\bf S}\rightarrow{\bf G}$ in the
argument of the spherical harmonic, which limits the rotation of the $\bf G$
vector in a certain plane and highlighting the fourth-order contribution:

$\Phi_{an}^{(4)}=k_{2}\,\cos 4\theta$ (29)

for $ac$, $bc$ planes or

$\Phi_{an}^{(4)}=k_{2}\,\cos 4\varphi$ (30)

for the $ab$ plane.

In general, the constants $k_{2}$ in RFeO3 decrease quite smoothly in absolute
value [50, 51], changing by no more than a factor of two when going from La to
Lu. The difference between the constants $k_{2}(ac)$ and $k_{2}(bc)$ can serve
as a measure of deviation from the ideal cubic perovskite structure, for which
$k_{2}(ac)=k_{2}(bc)=-\frac{3}{4}k_{2}(ab)$. The different signs of these
constants, positive for the $ac$, $bc$ planes and negative for the $ab$ plane,
indicate the different nature of spin-reorientation transitions in the
corresponding planes, that is, second-order transitions in the $ac$, $bc$
planes and first-order transitions in the $ab$ plane. Indeed, all currently
known spin-reorientation transitions of the $\Gamma_{4}-\Gamma_{2}$
($G_{x}-G_{z}$) type in RFeO3 orthoferrites (R = Sm, Nd, Er, Tm) are smooth,
with two characteristic temperatures phase transitions of the second order,
which are the beginning and end of spin reorientation, and the only transition
of the $\Gamma_{4}-\Gamma_{1}$ ($G_{x}-G_{y}$) type (DyFeO3) known for these
crystals is a stepwise transition of the first-order [16]. A unique example
that confirms our conclusions about the sign of the second anisotropy constant
is the mixed orthoferrite Ho0.5Dy0.5FeO3 [16] in which two spin-reorientation
transitions $G_{x}-G_{y}$ ($T=46$ K) and $G_{y}-G_{z}$ ($T=18\div 24$ K),
realized through one first-order phase transition in the $ab$ plane and two
second-order phase transitions in the $bc$ plane, respectively [16].

##  IV Neel temperature of weak ferrimagnets

MFA methods can be used to calculate the concentration dependence of the Neel
temperature of weak ferrimagnets. Fig. 3 shows such dependences for
YFe1-xCrxO3 according to data from various authors. First of all, we note that
only the first experimental studies of weak ferrimagnets in the laboratory of
A.M. Kadomtseva were performed on single-crystalline samples obtained in the
laboratory of B. Wanklyn, while all later studies were performed on
polycrystalline samples obtained by various methods [22, 24, 28, 31]. Let us
pay attention to the large scatter of experimental data, especially near the
“half” composition $x=0.5$, where the values of $T_{N}$ from different sources
differ by almost 100 K. On the one hand, this is due to large fluctuations of
local configurations with a large deviation from the parent compositions, on
the other hand, to the extremely small value of magnetization for such
compositions [6] and the large contribution of short-range order at $T\geq
T_{N}$. Some authors argue that an accurate determination of Neel temperatures
for these compositions is generally impossible [22].

Figure 3:  Concentration dependence of the Neel temperature in a weak
ferrimagnet YFe1-xCrxO3. Experiment: $\square$ and $\blacksquare$ – [6, 19]
(poly- and single crystals, respectively); $\blacklozenge$ – [32],
$\blacktriangledown$ – [29], $\bullet$ – [31], $\star$ – [26],
$\blacktriangle$ – [24, 34, 35], $\times$ – [22]. Theoretical curve –
molecular field model (see Hashimoto formula (32))

In molecular field theory, the $T_{N}(x)$ dependence is quite easy to obtain
within the framework of a kind of “zero” or “exchange” approximation, in which
we neglect the weak effects of the Dzyaloshinsky-Moriya interaction and
symmetric spin anisotropy compared to the large isotropic exchange:

${\bf
h}_{Fe}=P_{Fe}(x)I_{FeFe}\Big{\langle}4\,{\bf\hat{S}}^{(4)}_{Fe}+2\,{\bf\hat{S}}^{(2)}_{Fe}\Big{\rangle}+\\\
+P_{Cr}(x)I_{FeCr}\Big{\langle}4\,{\bf\hat{S}}^{(4)}_{Cr}+2\,{\bf\hat{S}}^{(2)}_{Cr}\Big{\rangle},$
(31)

in accordance with (20). Using the condition of small exchange molecular
fields near $T_{N}$ and the expansion of the Brillouin function

$B_{S}(x\ll 1)\approx\frac{S+1}{3S}x\,,$

we get instead of (19) system linear equations for average spins $\langle{\bf
S}_{Fe}\rangle$ and $\langle{\bf S}_{Cr}\rangle$, the condition solvability
which is ratio [53]

$\big{[}T_{N}(x)-(1-x)T_{N}(Fe)\big{]}\big{[}T_{N}(x)-xT_{N}(Cr)\big{]}=\\\
=\frac{z^{2}}{9}x(1-x)S_{Fe}(S_{Fe}+1)S_{Cr}(S_{Cr}+1)I_{FeCr}I_{CrFe},$ (32)

which can be used either to describe the dependence $T_{N}(x)$, or to find the
value of the integrals of the superexchange interaction Fe–O–Cr (Cr–O–Fe) from
the known experimental dependence $T_{N}(x)$. Naturally, the relation (32) is
satisfied under the assumption that exchange integrals are independent of
concentration; in addition, it is natural to assume that $I_{FeCr}=I_{CrFe}$.

Exchange integrals $I_{FeFe}=36.6$ K and $I_{CrCr}=18.7$ K are found using the
well-known MFA expressions for the relationship of exchange integrals with the
Neel temperatures of orthoferrite YFeO3 $T_{N}(0)=640-655$ K [54] and
orthochromite YCrO3 $T_{N}(1)=140-159$ K [31].

Using the experimental value of the exchange integral $I_{FeCr}=13.4\pm 0.4$
K, found in the work [41], and assuming $I_{FeCr}=I_{CrFe}$, in Fig. 3 the
theoretical dependence $T_{N}(x)$ is presented, which quite satisfactorily
describes the “older”, but more reliable experimental results obtained on both
single-crystalline and polycrystalline samples [6]. Let us note the systematic
deviation of the “new” data [24] obtained on polycrystalline YFe1-xCrxO3
samples from the experimental data of the work [6].

Let us also pay attention to the work [24], in which, to describe their
experimental data, the authors used strange expressions for calculating
average spin values, where in the argument of the Brillouin functions, instead
of natural expressions for effective molecular fields (31) (see also formula
(2) from [24]) other expressions have appeared, multiplied by $P_{Fe}(x)$ for
${\bf h}_{Fe}$ and $P_{Cr}(x)$ for ${\bf h}_{Cr}$ respectively. Apparently,
the authors motivate this by the need to take into account “the probabilistic
aspect of the occupancy of the site on which the mean field is acting”. In our
opinion, this deviation from the classical canons of mean field theory led to
a number of incorrect conclusions, for example, the dependence $T_{N}(x)$, the
use of which for processing experimental data led to a nonphysical conclusion
about the value of the exchange integral $I_{FeCr}\approx 48$ K (!?),
significantly exceeding not only $I_{CrCr}$, but also $I_{FeFe}$.

##  V Orientation and magnitude of the Neel vector in weak ferrimagnets

The positive (antiferromagnetic) sign and rather large value of the exchange
integral $I_{FeCr}$ indicates the preservation of the basic $G$-type
antiferromagnetic structure in weak ferrimagnets of the YFe1-xCrxO3 type
observed in parent orthoferrites and orthochromites. This is confirmed by both
direct neutron diffraction data [24, 26, 27], and indirect data from
magnetostriction measurements [7].

Fig. 4 shows the temperature dependence of the average magnetic moment

$\langle M\rangle=g\mu_{B}\big{[}0.5\,\langle S_{x}(Fe)\rangle+0.5\,\langle
S_{x}(Cr)\rangle\big{]}$ (33)

in YFe1-xCrxO3 ($x=0.50$), obtained from neutron diffraction data [26], as
well as the result of calculation within the MFA-I approximation taking into
account $\sim$10% covalent spin reduction typical of $3d$ oxides. Let us note
the completely satisfactory agreement between theory and experiment. Small
quantitative discrepancies are naturally explained by the magnetic
inhomogeneity of polycrystalline samples.

Figure 4:  Temperature addiction average magnetic moment in YFe0.5Cr0.5O3
($\mu_{B}/$ion), obtained from neutron diffraction data: $\bullet$ – [26];
$\square$ – [35]. Solid line – theory

The data in Fig. 4 indicate a significantly higher real Neel temperature for
the samples YFe0.5Cr0.5O3 studied in the work [26] than the value $T_{N}=275$
K declared by the authors, based on magnetic measurements under extremely low
magnetization, especially since the authors themselves point out the
difference between the ZFC and FC curves at temperatures above $275$ K.

The most reliable results of a neutron diffraction study of the
antiferromagnetic $G$-structure of polycrystalline YFe0.5Cr0.5O3 samples were
carried out in a recent work [35], the authors of which discovered an
“angular” structure with the orientation of the antiferromagnetism vector at
$T=2$ K at an angle $\approx$ 22∘ to the $a$ axis ($\langle
M\rangle=3.4\,\mu_{B}$, see Fig. 4), although to determine the exact
“azimuthal” orientation of the ${\bf G}$ vector in the $bc$ plane, the authors
refer to the need for single-crystal samples. These results actually confirmed
the “old” data from neutron diffraction measurements of the orientation of the
antiferromagnetism vector of polycrystalline YFe0.5Cr0.5O3 [19] samples,
according to which the vector ${\bf G}$ when the temperature decreases below
$260$ K and down to $6$ K is oriented at an angle $(30\pm 15)^{\circ}$ to the
$a$ axis.

By the way, the authors [35] did not find any features of dielectric and
pyroelectric properties near the Neel temperature $T_{N}=275$ K declared by
them, which, as in the case of the work data [26], most likely indicates a
significantly higher real Neel temperature in this composition is closer to
$T_{N}=335$ K according to the work [17].

Magnetic measurements of the torque curves of single-crystal YFe0.5Cr0.5O3
samples showed that as the temperature decreases below $T_{N}=363$ K the
magnetic moment is unusually small ($\approx 0.022$ emu/g) and up to $T\approx
300$ K is oriented along the $c$ axis, but with a further decrease in
temperature a component of the magnetic moment appears along the $a$ axis, the
value of which increases to $\approx 0.17$ emu/g at $T=78$ K. Measurements of
torque curves in various planes in a field of 6.25 kOe showed that at $78\leq
T\leq 171$ K the total magnetic moment is oriented in the $ac$ plane at an
angle of $60^{\circ}$ to $c$ axis, which, generally speaking, does not agree
with neutron diffraction data [35] for a polycrystal, and indicates an angular
phase $G_{xz}$ with the orientation of the antiferromagnetism vector at an
angle $60^{\circ}$ to $a$ axis, but does not exclude a more complex spatial
orientation of the antiferromagnetism vector. A similar situation with a
“flat” angular or spatial orientation of the antiferromagnetism vector was
observed in single crystals of almost all compositions YFe1-xCrxO3 ($x=0.15,\
0.38,\ 0.5,\ 0.65,\ 0.85$) [6], and also, according to the authors of the work
[19], and in polycrystalline samples at $x=0.3,\ 0.5,\ 0.7$. In all cases, one
must also remember the small contribution of single-ion anisotropy to the
magnetization; taking this into account at a small total magnetization value
disrupts the relationship between the orientation of the magnetic moment and
the antiferromagnetism vector, which is typical for the antisymmetric
contribution to the Dzyaloshinsky interaction.

Figure 5:  Concentration dependence of the effective anisotropy constant in
the $ac$ plane: a) contribution of the DM interaction at different values of
the parameter $d_{FeCr}$; b) the result of taking into account single-ion
anisotropy at $d_{FeCr}=-2.5$ K

##  VI The nature of spin reorientation in weak ferrimagnets of the
YFe1-xCrxO3 type

Unlike YFeO3 and YCrO3, which are weak ferromagnets with a basic magnetic
structure of the $G_{x}F_{z}$ type below $T_{N}$, weak ferrimagnetic
orthoferrites-orthochromites YFe1-xCrxO3, according to magnetic measurement
data, reveal complete or partial spin reorientation of the $G_{x}F_{z}$–
$G_{z}F_{x}$ type in a wide substitution range [6]. This unexpected behavior,
usually typical of orthoferrites with magnetic rare-earth ions (Er, Tm, Dy, …)
[56, 57], is explained mainly by a strong decrease in the contribution of the
DM interaction to the magnetic field. anisotropy.

Figure 6:  Calculated temperature dependencies effective constants anisotropy
second order $k_{1}(ac)$ for YFe1-xCrxO3 at $d_{FeCr}=-2.5$ K,
$D_{Fe}=D_{Cr}=-0.018$ K and various concentrations $x$: red thick line –
MFA-I, blue thin line – MFA-II

The contribution of competing antisymmetric exchange to the magnetic
anisotropy of weak ferrimagnets has an unusual concentration dependence. Thus,
if in pure orthoferrite YFeO3 and orthochromite YCrO3 antisymmetric exchange
makes a decisive contribution to the stabilization of the magnetic
configuration $\Gamma_{4}$, then in a weak ferrimagnet YFe1-xCrxO3 it can
cause a spin-reorientation transition $\Gamma_{4}$ – $\Gamma_{2}$,
characteristic of some orthoferrites RFeO3 with magnetic rare earth ions (R =
Nd, Sm, Tb, Ho, Er, Tm, Yb). Figure 5a shows the concentration dependence of
the contribution of the DM interaction to the first anisotropy constant for
YFe1-xCrxO3 in the $ac$ plane for different values of the parameter
$d_{FeCr}$, calculated within the framework of the simple MFA-I approximation
in the low temperature limit. A characteristic feature of this dependence is
the appearance of several extrema with a sharp decrease in the contribution in
the region of intermediate concentrations near $x\sim 0.6-0.7$.

However, the appearance of spontaneous spin-reorientation transitions in weak
ferrimagnets of the YFe1-xCrxO3 type with a non-magnetic R-ion is not only the
result of competition between the Dzyaloshinsky-Moriya interactions Fe–Fe,
Cr–Cr and Fe–Cr, but also comparable in magnitude, but “opposite” in sign,
contributions from single-ion anisotropy of the Fe and Cr sublattices. For
illustration, Fig. 5b shows the result of competition between DM interaction
and single-ion anisotropy in the $ac$ plane at $d_{FeCr}=-2.5$ K and various
single-ion anisotropy constants in the form $H_{SIA}^{(2)}=D\,S_{z}^{2}$ [47].
It is obvious that with an increase in the single-ion contribution, the region
where the sign of the effective anisotropy constant changes in the $ac$ plane
and the appearance of both “flat” angular phases of the $G_{xz}$ type and
spatial ones of the $G_{xyz}$ type appears and expands.

Effective anisotropy constants in weak ferrimagnets have a specific
temperature dependence [61]. Figure 6 shows the calculated temperature
dependences of the effective second-order anisotropy constant $k_{1}(ac)$ for
YFe1-xCrxO3 at $d_{FeCr}=-2.5$ K, $D_{Fe}=-0.018$ K, $D_{Cr}=-0.018$ K and
various concentrations $x$. Let us note the significant difference in the
results when using the MFA-I and MFA-II models for a number of concentrations.

The nature of spin-reorientation transitions significantly depends on the
magnitude and sign of the fourth-order single-ion anisotropy constants [56,
57]. Fig. 7 shows the temperature dependence of the fourth-order single-ion
anisotropy constants $k_{cub}$ (28) for a number of concentrations in
YFe1-xCrxO3.

In contrast to the yttrium system, lutetium orthoferrites-orthochromites
LuFe1-xCrxO3 ($x=0.0,\ 0.1,\ 0.2,\ 0.5,\ 0.6,\ 1.0$) retain the main magnetic
structure of the $G_{x}F_{z}$ type without signs of a spontaneous spin-
reorientation transition [8]. As noted above, in YFeO3 and LuFeO$3$ the
contribution of single-ion anisotropy to the first anisotropy constant
$k_{1}(ac)$ differs in sign, so that in YFeO3 this contribution partially
compensates for the contribution of the Dzyaloshinsky-Moriya interaction, and
in LuFeO3 both contributions add up, resulting in a sharp increase in
$k_{1}(ac)$ [2, Mukhin, 49] and greater stability of the $\Gamma_{4}$
configuration.

##  VII Features of magnetization of weak ferrimagnets RFe1-xCrxO3 with non-
magnetic R-ion

###  VII.1 Compensation points and negative magnetization

One of the most interesting properties of weak ferrimagnets is the specific
temperature and concentration dependences of magnetization, which are a
consequence of the competition between Dzyaloshinsky Fe–Fe, Cr–Cr and Fe–Cr
(Cr–Fe) interactions. Thus, back in 1977, it was predicted [6], and in 1978,
for the first time experimentally discovered [7], the phenomenon of
temperature compensation of the magnetic moment in a single-crystal sample of
a weak ferrimagnet YFe1-xCrxO3 ($x=0.38$) at $T_{comp}=225$ K. Much later,
this effect (magnetization reversal, negative magnetization) was discovered in
polycrystalline samples of yttrium orthoferrite-orthochromite at $x=0.5$
($T_{comp}=248$ K [23], $T_{comp}\approx 230$ K [24], $T_{comp}=245$ K [31],
$T_{comp}=175$ K [32]), at $x=0.4$ ($T_{comp}=170$ K [24]), as well as
lutetium orthoferrite-orthochromite at $x=0.5$ ($T_{comp}=224$ K [27],
$T_{comp}=230$ K [55], $T_{comp}=142$ K [32]).

Figure 7:  Temperature dependence of fourth-order single-ion anisotropy
$k_{2}(ac)$ at concentrations $x=0,\ 0.3,\ 0.6,\ 0.9$

It is interesting that the appearance of two points of concentration, and
therefore temperature compensation, previously predicted by Kadomtseva and co-
authors in weak ferrimagnets of the RFe1-xCrxO3 type with a non-magnetic R-ion
(R = La, Y , Lu) [6] was experimentally confirmed in LaFe1-xCrxO3, where the
magnetization inversion effect was observed both for compositions with
$x=0.50$ and $x=0.55$, and at concentration $x$ in the range $0.85-0.95$ [25].
The results of the work [22] also indicate the detection of temperature
compensation of magnetization in YFe1-xCrxO3 as at $x=0.50$ ($T_{comp}\approx
250$ K), and for $x=0.75$ ($T_{comp}\approx 50$ K). The authors of the work
[24] briefly mention the possible compensation of magnetization for
YFe1-xCrxO3 in the composition range $x\sim 0.8$.

Note that the compensation temperature $T_{comp}$, more precisely, the
magnetization reversal temperature $T_{mr}$ significantly depends on the
magnitude and method of switching on (ZFC, FC) the external field (see, for
example, work [25]), and for polycrystalline samples heat treatment can play a
major role, which generally leads to both a scatter in the experimental values
of $T_{mr}$ and ambiguity in quantitative theoretical analysis. It is
interesting that even in the first experimental works attention was drawn to
the ambiguity of data on magnetization at $T=77$ K even for single-crystal
samples with a nominal composition of YFe0.5Cr0.5O3 and close Neel
temperatures [6].

Observation of the compensation point significantly depends on the magnitude
of the external magnetic field (FC mode). Thus, the compensation temperature
observed in LuFe0.5Cr0.5O3 at $T_{comp}=224$ K in a field of $100$ Oe is
suppressed in higher fields. In low fields, the total magnetic moment of a
weak ferrimagnet in the FC mode, initially oriented in the direction of the
applied field, is reoriented in the direction opposite to the applied magnetic
field as the temperature decreases below $T_{comp}$. However, a sufficiently
strong external magnetic field can hold the magnetic moment of the sample in
the direction of the field [23, 27].

By the way, due to the presence of concentration and temperature compensation
points in weak ferrimagnets, the problem of the “sign” of the experimentally
measured magnetization appears, ignoring which can lead to an erroneous
interpretation of the experimental data.

###  VII.2 Competition of signs of Dzyaloshinsky vectors $d_{FeFe}$,
$d_{CrCr}$ and $d_{FeCr}$ and features of concentration and temperature
dependences of magnetization in YFe1-xCrxO3

We used the MFA-I and MFA-II methods outlined above to calculate the
concentration and temperature dependences of the total magnetization of weak
ferrimagnets and weak ferromagnetic moments of the sublattices
$m_{Fe},m_{Cr}$, or in other words, small components of the basis vector of
ferromagnetism ${\bf F}$ (overt canting) for Fe- and Cr-sublattices in
magnetic configurations $\Gamma_{4}$ and $\Gamma_{2}$. In a similar way, small
components of the basis vectors ${\bf A}$ and ${\bf C}$ can be calculated
(hidden canting) for Fe and Cr sublattices in magnetic configurations
$\Gamma_{1,2,4}$.

Figure 8:  $T$–$x$ phase diagram of magnetization compensation regions of a
weak ferrimagnet YFe1-xCrxO3 for different values of the parameter $d_{FeCr}$.
The “+” sign indicates the region of “positive” magnetization. The dotted line
highlights the room temperature $T=300$ K

When solving the molecular field equations numerically, we assume that the
parameters of both the isotropic superexchange interaction and the
Dzyaloshinsky-Moriya interaction do not depend on the concentration $x$, so
that $I_{FeFe},I_{CrCr},d_{FeFe},d_{CrCr}$ are close to their values in the
parent compositions YFeO3 and YCrO3. A definite argument in favor of this
assumption is the data from a neutron diffraction study of the YFe0.5Cr0.5O3
system [26, 32], indicating the similarity of the structural parameters in
this mixed orthoferrite-orthochromite with data for the parent YFeO3 and
YCrO3. Indeed, the values of the structure factor that determines the
orientation of the Dzyaloshinsky vector, calculated from the data [32] (Table
3), coincide with good accuracy with the corresponding data for YFeO3 (see,
for example, Table 3 in the work [51]), which justifies the possibility of
using constant, concentration-independent, values of Dzyaloshinsky vectors. In
accordance with the predictions of microscopic theory, the Dzyaloshinsky
vector ${\bf d}_{FeCr}$ has a comparable magnitude and opposite direction to
the vectors ${\bf d}_{FeFe}$ and ${\bf d}_{CrCr}$ (see Table 1 [1, 3, 4, 5]).
For the exchange integral $I_{FeCr}=I_{FeCr}$ we use the data from the work
[41]: $I_{FeCr}=13.4\pm 0.4$. The scalar factors $d_{FeFe}(\theta)=2.0$ K and
$d_{CrCr}(\theta)=1.7$ K were determined from low-temperature magnetization
data of the parent compositions.

The calculation results showed that when $d_{FeCr}$ differs in sign from
$d_{FeFe}$ and $d_{CrCr}$, the magnetization of YFe1-xCrxO3 drops sharply with
the deviation from the parent compositions, but at
$|d_{FeCr}|\geq|d_{FeCr}^{(cr)}|$, where $d_{FeCr}^{(cr)}\approx-1.55$ K, on
the $T$–$x$ phase diagram, a region of negative magnetization with two
compensation points appears and grows with increasing $|d_{FeCr}|$. Fig. 8
shows the $T$–$x$ phase diagram of the weak ferrimagnet YFe1-xCrxO3 (MFA-II),
where the $T_{N}(x)$ curve limits region of magnetic ordering, and thin lines
indicate lines of compensation points, that is, changes in the sign of
magnetization at different values of the parameter $d_{FeCr}$.

Figure 9 shows the results of calculating the temperature dependences of the
magnetization of a weak ferrimagnet YFe1-xCrxO3 at certain concentration
values from $x=0$ to $x=1$ within the framework of the MFA-I and MFA-II models
and under the assumption that the magnetic configuration $\Gamma_{4}(G_{x})$
is conserved throughout the concentration range. The scalar factor
$d_{FeCr}(\theta)=d_{CrFe}(\theta)=-2.5$ K was chosen from the condition that
the compensation point for the composition $x=0.38$ appears near $T\approx
225$ K. Naturally, both model approaches give identical or close results for
compositions with $x\leq 0.5$ and $x\geq 0.9$, but the deviation increases as
we approach compositions with $x\sim 0.7$. As one would expect, the theory
predicts two regions of temperature compensation – near compositions with
$x\sim 0.4$ (wide “high-temperature” region) and $x\sim 0.8$ (narrow “low-
temperature” region). For a composition with $x\sim 0.45$, compensation is
expected near room temperature.

Figure 9:  Temperature dependence of magnetization in a model weak ferrimagnet
YFe1-xCrxO3 at $x=0.0,\ 0.15,\ 0.4,\ 0.5,\ 0.6,\ 0.65,\ 0.8,\ 0.9,\ 1.0$:
dotted lines – experimental data, solid lines – theory (thick red – MFA-I,
thin blue – MFA-II)

Unfortunately, there are no data in the literature on regular measurements of
the temperature dependences of the magnetization of single-crystal samples of
weak ferrimagnets of the YFe1-xCrxO3 type in a wide range of concentrations,
with the exception of compositions with $x=0.15,\ 0.38,\ 0.5,\ 0.65$, for
which some information is available [6, 7]. Comparison of the calculated data
with the measurement data of the total magnetization for the composition
$x=0.15$ (see Fig. 9) shows quite satisfactory agreement, despite the presence
of spin reorientation $G_{x}\leftrightarrow G_{z}$ in the range of $250-400$
K. The saturation magnetization of single-crystal samples with $x=0.38,\ 0.5,\
0.65$ is significantly, two to three times, less than the predictions of the
molecular field theory, which, in light of the experimentally discovered
transitions for these compositions with a change in the orientation of the
weak ferrimagnetic moment in the $ac$ plane [6] indicates the possible
implementation of the spatial orientation of the antiferromagnetism vector,
that is, the $G_{xyz}$ configuration.

Magnetic measurements on polycrystalline samples, especially under conditions
of spin-reorientation states, extremely small values of magnetization near the
compensation region and negative magnetization, cannot provide reliable values
of Neel temperatures, compensation temperatures and absolute values of
magnetization. Nevertheless, the data of calculations within the framework of
molecular field theory with only one fixed parameter $d_{FeCr}$, presented in
Fig. 9 provide a semi-quantitative explanation of the experimental data of the
work [24], obtained on polycrystals of the weak ferrimagnet YFe1-xCrxO3 in a
wide range of concentrations $x=0-1$.

Figure 10:  Temperature dependences of partial magnetizations $m_{Fe}$ and
$m_{Cr}$ in the weak ferrimagnet YFe1-xCrxO3 for compositions $x=0.5,\ 0.65,\
0.75$

However, the model approaches and formulas used to describe the temperature
dependences of the magnetization of weak ferrimagnets of the YFe1-xCrxO3 type
in the works [24, 28] (see formulas (6) and (12) in [28]) do not follow from
any rigorous MFA approach, but are merely the result of a “plausible”
phenomenology using a number of, sometimes hidden, parameters. Thus, the
apparently impressive agreement between theory and experiment in the work [24]
(see Figures 5-7 of this work) is based on a fit with several parameters,
including, as noted above, the unphysically large value of the exchange
integral $I_{FeCr}$, as well as a non-physically strong concentration
dependence of the $d_{FeCr}$ parameter of the Dzyaloshinsky-Moriya
interaction, with a sharp drop from $d_{FeCr}=-1.3$ K at $x=0.1$ to almost
zero at $x=0.9$. There are a number of inconsistencies in the work, for
example, the “theoretical” Neel temperatures for the composition with $x=0.3$
in Fig. 2 and in Fig. 5 differ by 50 K. But moreover, the authors incorrectly
attributed a positive sign to the magnetization of compositions with $x=0.6$
and $x=0.7$, and were unable to explain the observed compensation of
magnetization at $x=0.8$ and the “return” to the positive sign of
magnetization at $x=0.9$. The authors do not consider the effects of spin
reorientation and the “symmetrical” contribution of single-ion anisotropy to
magnetization at all. In general, the authors’ statement “Thus, we are able to
obtain quantitative agreement between theory and agreement for the whole range
of doping and temperature with a very simple, consistent and transparent
approach” is more than controversial.

Let us emphasize once again that the features of the concentration and
temperature dependences of the magnetization of weak ferrimagnets of the
YFe1-xCrxO3 type, in particular the compensation phenomenon, are determined
primarily by the competition of the Dzyaloshinsky-Moriya Fe–Fe interactions,
Cr–Cr and Fe–Cr, although the contribution of single-ion spin anisotropy can
have a noticeable effect on both the offset of the compensation temperature
and the magnitude and nature of the temperature dependence of magnetization
near $T_{comp}$.

It is interesting that not only the total magnetization, but also the partial
magnetization of the Fe and Cr sublattices, $m_{Fe}$ and $m_{Cr}$ have unusual
concentration and temperature dependences. For illustration, Fig. 10 shows the
temperature dependences of $m_{Fe}$ and $m_{Cr}$ for compositions $x=0.5,\
0.65,\ 0.75$ calculated in the MFA-II approximation. If for the composition
$x=0.5$ (as well as for $x<0.5$) ferrimagnetic ordering of weak ferromagnetic
moments of the Fe-Cr sublattices is realized, then for compositions with
$x=0.65,\ 0.75$ we discover the phenomenon of temperature compensation for
partial contributions.

##  VIII Conclusion

Weak ferrimagnets with competing Dzyaloshinsky-Moriya interactions of the
RFe1-xCrxO3 type with magnetic and non-magnetic R-ions represent a new fairly
wide class of promising magnetic materials with a number of unique magnetic,
magnetocaloric, and magnetoelectric properties.

In this paper, we present a brief critical review of the 50-year history of
experimental and theoretical studies of the magnetic properties of weak
ferrimagnets of the YFe1-xCrxO3 type. The spin Hamiltonian of the system is
considered taking into account the main isotropic and anisotropic
interactions. Within the framework of the molecular field approximation,
calculations were made of the Neel temperatures, the average magnetic moments
of $3d$ ions, total and partial magnetizations, and effective anisotropy
constants. It is shown that in the model system YFe1-xCrxO3 there are two
regions of negative magnetization $0.25\leq x\leq 0.5$ and $x\approx 0.8$ with
corresponding compensation points, reaching room temperature at $x\approx$
0.45. The phenomenon of spin reorientation observed for single-crystal samples
in a wide range of concentrations is explained by a sharp decrease in the
contribution of antisymmetric exchange to magnetic anisotropy with increasing
deviation from the parent compositions and competition between the
contributions of single-ion anisotropy of Fe and Cr ions. It has been
suggested that the spatial orientation of the antiferromagnetism vector and
the $G_{xyz}$ configuration are the reason for the small value of saturation
magnetization observed experimentally for compositions inside or near the
region of negative magnetization.

Understanding the mechanisms of formation of the peculiarities of
concentration and temperature dependences of magnetization, primarily the
phenomenon of compensation and negative magnetization, the phenomenon of spin
reorientation, is the basis for a scientifically based approach to controlling
these properties, predicting and synthesizing new promising materials.

###### Acknowledgements.

The work was carried out within the framework of the Russian Science
Foundation project No. 22-22-00682.

##  Appendix A Magnetic moment equations in the MFA-II model

Phase $\Gamma_{4}(G_{x},A_{y},F_{z})$:

$\displaystyle
m^{z}_{Fe}(\alpha_{Fe}^{+}+1)+m^{z}_{Cr}\beta_{Fe}^{+}=-\gamma_{Fe}^{F}\,;$
$\displaystyle
m^{z}_{Fe}\alpha_{Cr}^{+}+m^{z}_{Cr}(\beta_{Cr}^{+}+1)=-\gamma_{Cr}^{F}\,;$
$\displaystyle
m^{y}_{Fe}(\alpha_{Fe}^{-}+1)+m^{y}_{Cr}\beta_{Fe}^{-}=\gamma_{Fe}^{A}\,;$
$\displaystyle
m^{y}_{Fe}\alpha_{Cr}^{-}+m^{y}_{Cr}(\beta_{Cr}^{-}+1)=\gamma_{Cr}^{A}\,;$

phase $\Gamma_{2}(F_{x},C_{y},G_{z})$:

$\displaystyle
m^{x}_{Fe}(\alpha_{Fe}^{+}+1)+m^{x}_{Cr}\beta_{Fe}^{+}=\gamma_{Fe}^{F}\,;$
$\displaystyle
m^{x}_{Fe}\alpha_{Cr}^{+}+m^{x}_{Cr}(\beta_{Cr}^{+}+1)=\gamma_{Cr}^{F}\,;\ $
$\displaystyle
m^{y}_{Fe}(\alpha_{Fe}^{-}-1)+m^{y}_{Cr}\beta_{Fe}^{-}=\gamma_{Fe}^{C}\,;$
$\displaystyle
m^{y}_{Fe}\alpha_{Cr}^{-}+m^{y}_{Cr}(\beta_{Cr}^{-}-1)=\gamma_{Cr}^{C}\,;\ $

phase $\Gamma_{1}(A_{x},G_{y},C_{z})$:

$\displaystyle
m^{x}_{Fe}(\alpha_{Fe}^{-}+1)+m^{x}_{Cr}\beta_{Fe}^{-}=-\gamma_{Fe}^{A}\,;$
$\displaystyle
m^{x}_{Fe}\alpha_{Cr}^{-}+m^{x}_{Cr}(\beta_{Cr}^{-}+1)=-\gamma_{Cr}^{A}\,;\ $
$\displaystyle
m^{z}_{Fe}(\alpha_{Fe}^{+}-1)+m^{z}_{Cr}\beta_{Fe}^{+}=-\gamma_{Fe}^{C}\,;$
$\displaystyle
m^{z}_{Fe}\alpha_{Cr}^{+}+m^{z}_{Cr}(\beta_{Cr}^{+}-1)=-\gamma_{Cr}^{C}\,;\ $

where

$\displaystyle\alpha_{Fe}^{\pm}=\sum_{klr}{(k+l\pm
r)\,I_{FeFe}}\,P^{klr}_{Fe};$
$\displaystyle\alpha_{Cr}^{\pm}=\sum_{klr}{(k+l\pm
r)\,I_{FeCr}}\,P^{klr}_{Cr}\,;\ $
$\displaystyle\beta_{Fe}^{\pm}=\sum_{klr}{(2u-k-l\pm u\mp
r)\,I_{FeCr}}\,P^{klr}_{Fe};$
$\displaystyle\beta_{Cr}^{\pm}=\sum_{klr}{(2u-k-l\pm u\mp
r)\,I_{CrCr}}\,P^{klr}_{Cr}\,;\ $
$\displaystyle\gamma_{Fe}^{F}=\sum_{klr}\Big{\\{}\big{[}(k+l)\,d_{FeFe}^{\,y_{4}}+r\,d_{Fe}^{\,y_{2}}\big{]}\langle
S_{Fe}\rangle+$
$\displaystyle+\big{[}(2u-k-l)\,d_{FeCr}^{\,y_{4}}+(u-r)\,d_{FeCr}^{\,y_{2}}\big{]}\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Fe};\ $
$\displaystyle\gamma_{Cr}^{F}=\sum_{klr}\Big{\\{}\big{[}(k+l)\,d_{FeCr}^{\,y_{4}}+r\,d_{FeCr}^{\,y_{2}}\big{]}\langle
S_{Fe}\rangle+$
$\displaystyle+\big{[}(2u-k-l)\,d_{CrCr}^{\,y_{4}}+(u-r)\,d_{CrCr}^{\,y_{2}}\big{]}\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Cr};\ $
$\displaystyle\gamma_{Fe}^{A}=\sum_{klr}\Big{\\{}(k+l)\,d_{FeFe}^{\,z_{4}}\,\langle
S_{Fe}\rangle+$ $\displaystyle+(2u-k-l)\,d_{FeCr}^{\,z_{4}}\,\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Fe};\ $
$\displaystyle\gamma_{Cr}^{A}=\sum_{klr}\Big{\\{}(k+l)\,d_{FeCr}^{\,z_{4}}\,\langle
S_{Fe}\rangle+$ $\displaystyle+(2u-k-l)\,d_{CrCr}^{\,z_{4}}\,\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Cr};\ $
$\displaystyle\gamma_{Fe}^{C}=\sum_{klr}\Big{\\{}\big{[}(k-l)\,|d_{FeFe}^{\,x_{4}}|+r\,d_{FeFe}^{\,x_{2}}\big{]}\langle
S_{Fe}\rangle+$
$\displaystyle+\big{[}(l-k)\,|d_{FeCr}^{\,x_{4}}|+(u-r)\,d_{FeCr}^{\,x_{2}}\big{]}\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Fe};\ $
$\displaystyle\gamma_{Cr}^{C}=\sum_{klr}\Big{\\{}\big{[}(k-l)\,|d_{FeCr}^{\,x_{4}}|+r\,d_{FeCr}^{\,x_{2}}\big{]}\langle
S_{Fe}\rangle+$
$\displaystyle+\big{[}(l-k)\,|d_{CrCr}^{\,x_{4}}|+(u-r)\,d_{CrCr}^{\,x_{2}}\big{]}\langle
S_{Cr}\rangle\Big{\\}}P^{klr}_{Cr};\ $ $\displaystyle
P^{klr}_{Fe}=\frac{p^{klr}}{h^{klr}_{Fe}}\,S_{Fe}\,B_{S_{Fe}}\left(\frac{S_{Fe}h^{klr}_{Fe}}{k_{B}T}\right);$
$\displaystyle
P^{klr}_{Cr}=\frac{p^{klr}}{h^{klr}_{Cr}}\,S_{Cr}\,B_{S_{Cr}}\left(\frac{S_{Cr}h^{klr}_{Cr}}{k_{B}T}\right);\
$ $\displaystyle h^{klr}_{Fe}=(k+l+r)I_{FeFe}\langle S_{Fe}\rangle+(z-k-
l-r)I_{FeCr}\langle S_{Cr}\rangle;\ $ $\displaystyle
h^{klr}_{Cr}=(k+l+r)I_{FeCr}\langle S_{Fe}\rangle+(z-k-l-r)I_{CrCr}\langle
S_{Cr}\rangle;\ $ $\displaystyle\langle
S_{Fe}\rangle=\sum_{klr}p^{klr}\,S_{Fe}\,B_{S_{Fe}}\left(\frac{S_{Fe}h^{klr}_{Fe}}{k_{B}T}\right);$
$\displaystyle\langle
S_{Cr}\rangle=\sum_{klr}p^{klr}\,S_{Cr}\,B_{S_{Cr}}\left(\frac{S_{Cr}h^{klr}_{Cr}}{k_{B}T}\right).$

Additional index at Dzyaloshinsky parameters indicates on number sublattices,
for example $d^{\,y_{2}}_{FeFe}=d_{FeFe}\,[{\bf r}_{2}\times{\bf r}_{1}]^{y}$
and $d^{\,z_{4}}_{FeCr}=d_{FeCr}\,[{\bf r}_{4}\times{\bf r}_{1}]^{z}$.

## References

  * [1] A. S. Moskvin and I. G. Bostrem, Fizika Tverdogo Tela, 19, 2616 (1977) [Sov. Phys. Solid State 19, 1532 (1977)].
  * [2] Moskvin, A.S. Antisymmetric Exchange and Magnetic Anisotropy in Weak Ferromagnets. D. Sc. Thesis, Lomonosov Moscow State University, Moscow, Russia, 1984. (In Russian)
  * [3] A. S. Moskvin, JMMM, 400, 117, (2016).
  * [4] Moskvin, A.S. Dzyaloshinskii–Moriya Coupling in 3d Insulators. _Condens. Matter._ 2019, _4_ , 84.
  * [5] A. S. Moskvin, JETP, 132, (4), 517-547, 2021.
  * [6] A. M. Kadomtseva, A. S. Moskvin, I. G. Bostrem, B. M. Wanklyn, and N. A. Khafizova, Nature of the anomalous magnetic properties of yttrium ferrite chromites, Zh. Eksp. Teor. Fiz. 72, 2286 (1977).
  * [7] A. M. Kadomtseva, V. N. Milov, A. S. Moskvin, and M. Pardavi-Khorvat, Compensation of a weak ferrimagnetic moment in yttrium ferrites–chromites, Sov. Phys. Solid State 20, 474 (1978).
  * [8] E. V. Sinitsyn, A. M. Kadomtseva, A. S. Moskvin, I. G. Bostrem, V. A. Timofeeva, and O. V. Kazakevich, Weak ferrimagnetism of mixed yttrium and lutetium chromite orthoferrites, Sov. Phys. Solid State 25, 161 (1983)
  * [9] A. M. Kadomtseva, JETP, 57, 833-837, 1983.
  * [10] A. M. Kadomtseva, A. S. Moskvin, Acta Physica Polonica, A 68, 303 (1985).
  * [11] A. S. Moskvin, G. G. Artem’ev, A. M. Kadomtseva, T. M. Ledneva, M. M. Lukina, and T. L. Ovchinnikova, Weak ferrimagnetism of neodymium ferrite–chromites, Sov. Phys. Solid State 33, 366 (1991).
  * [12] A. S. Moskvin and M. A. Vigura, Sov. Phys. Solid State 28, 1268 (1986).
  * [13] A. M. Kadomtseva, I. G. Bostrem, L. M. Vasil’eva, I. B. Krynetskii, M. M. Lukina, and A. S. Moskvin, Sov. Phys. Solid State 22, 1146 (1980).
  * [14] A. S. Moskvin, M. A. Vigura, and A. P. Agafonov, Sov. Phys. Solid State 28, 1631 (1986).
  * [15] G. P. Vorob’ev, A. M. Kadomtseva, A. S. Moskvin, Yu. F. Popov, and V. A.Timofeeva, Aspects of magnetic ordering in the system (FexCr1- x)2O3, Physics of the Solid State, 39, 97 (1997).
  * [16] Belov, K.P.; Zvezdin, A.K.; Kadomtseva, A.M.; Levitin, R.Z. _Orientational Transitions in Rare-Earth Magnetics_ ; Nauka: Moscow, Russia, 1979. (In Russian)
  * [17] K.P. Belov, M.A. Belyanchikova, A.M. Kadomceva, T.M. Ledneva, M.M. Lukina, T.L. Ovchinnikova, L.P. SHlyahina, Magnitnye svojstva smeshannyh ortoferritov-hromitov, FTT, 14, 944-946 (1972) [In Russian].
  * [18] K.P. Belov, A.M. Kadomceva, I.B. Kryneckij, T.L. Ovchinnikova, G.N. Ronami, V.A. Timofeeva, Osobennosti magnitnyh svojstv monokristallov YFe1-xCrxO3, FTT, 14, 1524-1527 (1972) [In Russian].
  * [19] K.P. Belov, A.M. Kadomceva, T.L. Ovchinnikova, M.M. Lukina, M. Pardavi-Horvat, V.A. Timofeeva, E. SHvab, Magnitnoe i nejtronograficheskoe issledovanie pereorientacionnyh perekhodov v YFe1-xCrxO3, Kristallografiya, 26(3), 602-604 (1976) [In Russian].
  * [20] J. B. Goodenough, Magnetism and the Chemical Bond, Wiley, 1963 (Russ. transl., Metallurgiya, 1966).
  * [21] Maxim V. Kuznetsov, Quentin A. Pankhurst, Ivan P. Parkin and Yury G. Morozov, Self-propagating high-temperature synthesis of chromium substituted lanthanum orthoferrites LaFe1-xCrxO3 (0¡x¡1), J. Mater. Chem., 2001,11, 854-858
  * [22] A. Dahmani, M. Taibi, M. Nogues, J. Aride, E. Loudghiri, A. Belayachi, Magnetic properties of the perovskite compounds YFe1-xCrxO3 (0.5 ¡ x ¡ 1), Materials Chemistry and Physics, 77, 912-917 (2003). https://doi.org/10.1016/S0254-0584(02)00188-8.
  * [23] Mao Jinhua et al., App. Phys. Lett., 98 (2011) 192510.
  * [24] N. Dasari, P. Mandal, A. Sundaresan and N. S. Vidhyadhiraja, Europhys. Lett. 99, 17008 (2012).
  * [25] T. Bora, S. Ravi, Study of magnetization reversal in LaCr1-xFexO3 compounds, J. Appl. Phys. 114 (2013), 033906.
  * [26] Nair, V., Subramanian, V., Santhosh, P. (2013). Magnetic structure and magnetodielectric effect of YFe0.5Cr0.5O3. Journal of Applied Physics, 113 (21). doi: 10.1063/1.4808459
  * [27] F. Pomiro, R. D. S$\acute{a}$nchez, G. Cuello, A. Maignan, C. Martin, and R. E. Carbonio, Phys. Rev. B 94, 134402 (2016).
  * [28] O. V. Billoni, F. Pomiro, S. A. Cannas, C. Martin, A. Maignan, and R. E. Carbonio, J. Phys.: Condens. Matter 28, 476003 (2016).
  * [29] L. H. Yin, J. Yang, P. Tong, X. Luo, W. H. Song, J. M. Dai, X. B. Zhu, and Y. P. Sun, Appl. Phys. Lett. 110, 192904 (2017) doi.org/10.1063/1.4983363
  * [30] A.P.G. Rodrigues, M.A. Morales, R.B. Silva, D.R.A.B. Lima, R.L.B.A. Medeiros, J.H. Araújo, D.M.A. Melo, Positive exchange bias effect in LaCr0.5Fe0.5O3 perovskite, Journal of Physics and Chemistry of Solids, 141, 2020, 109334, doi.org/10.1016/j.jpcs.2020.109334.
  * [31] Salazar-Rodriguez, R.; Aliaga Guerra, D.; Greneche, J.-M.; Taddei, K.M.; Checca-Huaman, N.-R.; Passamani, E.C.; Ramos-Guivar, J.A. Presence of Induced Weak Ferromagnetism in Fe-Substituted YFexCr1-xO3 Crystalline Compounds. Nanomaterials 2022, 12, 3516. https://doi.org/10.3390/nano12193516
  * [32] Jiayun Yang, Huanhuan Cao, Zeyi Lu, Jiajun Mo, Yang Zhou, Kaiyang Gao, Xinhao Xue, Yanfang Xia, and Min Liu, Mössbauer and Magnetic Properties Investigation of REFe0.5Cr0.5O3 (RE=La, Y) Perovskite, Phys. Status Solidi B, 2300145 (2023).
  * [33] Liu, W., Kan, X., Feng, S. et al. Negative magnetization and magnetic switching effect in R2FeCrO6 (R = La, Nd) perovskites. J Mater Sci: Mater Electron 34, 793 (2023). https://doi.org/10.1007/s10854-023-10196-1
  * [34] B. Rajeswaran, P. Mandal, R. Saha, E. Suard, A. Sundaresan, and C. N. R. Rao, Chem. Mater. 24, 3591 (2012).
  * [35] J. Y. Yang, X. D. Shen, V. Pomjakushin, L. Keller, E. Pomjakushina, Y. W. Long, and M. Kenzelmann, Characterization of magnetic symmetry and electric polarization of YCr0.5Fe0.5O3, Phys. Rev. B 101, 014415 (2020).
  * [36] E. Holzschuh, A. B. Denison, W. Kundig, P. F. Meier, and B. D. Patterson, Phys. Rev. B 27 5294 (1983).
  * [37] K. Amelin, U. Nagel, R. S. Fishman, Y. Yoshida, Hasung Sim, Kisoo Park, Je-Geun Park, and T. Rõõm, Phys. Rev. B 98, 174417 (2018).
  * [38] R. M. Hornreich, S. Shtrikman, B. M. Wanklyn, and I. Yaeger, Phys. Rev. B 13, 4046 (1976).
  * [39] S. E. Hahn, A. A. Podlesnyak, G. Ehlers, G. E. Granroth, R. S. Fishman, A. I. Kolesnikov, E. Pomjakushina, and K. Conder, Phys. Rev. B 89, 014420 (2014)
  * [40] K. Park, H. Sim, J. C. Leiner, Y. Yoshida, J. Jeong, S. Yano, J. Gardner, P. Bourges, M. Klicpera, V. Sechovský, M. Boehm, and J.-G. Park, J. Phys.: Condens. Matter 30, 235802 (2018).
  * [41] A. S. Moskvin, N. S. Ovanesyan, and V. A. Trukhtanov, Hyperfine Interactions, 1, 265 (1975).
  * [42] I. E. Dzyaloshinskii, Soviet Physics JETP, 5 1259 (1957); I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).
  * [43] T. Moriya, Phys. Rev. Lett. 4, 228 (1960); Phys. Rev. 120, 91 (1960).
  * [44] A. S. Moskvin, Fizika Tverdogo Tela, 12, 3209 (1970)(Soviet Physics Solid State, USSR 12, 2593 (1971)).
  * [45] A. S. Moskvin, E. V. Sinitsyn, Fizika Tverdogo Tela, 17, 2495 (1975) [Soviet Physics Solid State, 17, 2495 (1975)].
  * [46] S. Jacobs, H. F. Burne, and L. M. Levinson, J. Appl. Phys. 42, 1631 (1971).
  * [47] D. R. Taylor, J. Owen, B. M. Wanklyn, J. Phys. C.: Sol. St. Phys. 6, 2592 (1973).
  * [48] R. L. White, G. F. Herrmann, J. W. Carson, and M. Mandel, Paramagnetic Resonance of Fe3+ and Gd3+ in Yttrium Orthoaluminate, Phys. Rev. 136, A231 (1964). doi.org/10.1103/PhysRev.136.A231
  * [49] Kadomtseva, A.M.; Agafonov, A.P.; Lukina, M.M.; Milov, V.N.; Moskvin, A.S.; Semenov, V.A.; Sinitsyn, E.V. Nature of the Magnetic Anisotropy and Magnetostriction of Orthoferrites and Orthochromites.. _JETP_ 1981, _81_ , 700–706.
  * [50] A. S. Moskvin and I. G. Bostrem, Sov. Phys. Solid St. 21, 628 (1979).
  * [51] Moskvin, A. Structure–Property Relationships for Weak Ferromagnetic Perovskites. Magnetochemistry 2021, 7, 111. https://doi.org/10.3390/magnetochemistry7080111
  * [52] A.S. Moskvin, I.G. Bostrem, M.A. Sidorov, JETP 77, 127-137 (1993).
  * [53] T. Hashimoto, J. Phys. Soc. Jpn. 18, 1140 (1963).
  * [54] Mingyu Shang, Chenyang Zhang, Tingsong Zhang, Lin Yuan, Lei Ge et al., The multiferroic perovskite YFeO3, Appl. Phys. Lett. 102, 062903 (2013); doi: 10.1063/1.4791697
  * [55] I. Fita, V. Markovich, A. S. Moskvin, A. Wisniewski, R. Puzniak, P. Iwanowski, C. Martin, A. Maignan, Raúl E. Carbonio, M. U. Gutowska, A. Szewczyk, and G. Gorodetsky, Reversed exchange-bias effect associated with magnetization reversal in the weak ferrimagnet LuFe0.5Cr0.5O3, Phys. Rev. B 97, 104416 (2018) - Published 23 March 2018 DOI:https://ezproxy.urfu.ru:4262/10.1103/PhysRevB.97.104416
  * [56] Moskvin, A.; Vasinovich, E.; Shadrin, A. Simple Realistic Model of Spin Reorientation in 4f-3d Compounds. Magnetochemistry 2022, 8, 45. ttps://doi.org/10.3390/magnetochemistry8040045
  * [57] E.V. Vasinovich, A.S. Moskvin, Simple spin-reorientation model for rare-earth orthoferrites and orthochromites, Physics of the Solid State, 65, 889-892 (2023).
  * [58] Wei Zhang, Materials Chemistry and Physics 137 (2013) 877-883
  * [59] Anhua Wu, Bull. Mater. Sci., Vol. 35, No. 2, April 2012, pp. 259–263
  * [60] Jiajun Mo, Ceramics International Volume 48, Issue 21, 1 November 2022, Pages 31309-31314
  * [61] A. P. Agafonov, I. A. Zorin, A. M. Kadomtseva, A. S. Moskvin, and V. A. Timofeeva, Sov. Phys. Solid State 26, 1292 (1984).