# Structure-property relationships for weak-ferromagnetic perovskites

A.S. Moskvin  Ural Federal University, 620083 Ekaterinburg, Russia  M.N.
Mikheev lnstitute of Metal Physics of Ural Branch of Russian Academy of
Sciences, 620108 Ekaterinburg, Russia

###### Abstract

Despite several decades of active experimental and theoretical studies of
rare-earth orthoferrites, the mechanism of the formation of specific magnetic,
magnetoelastic, optical, and magneto-optical properties remains a subject of
discussion. This paper provides an overview of simple theoretical model
approaches to quantitatively describing the structure-property relationships,
in particular, interplay between FeO6 octahedral deformations/rotations and
main magnetic and optic characteristics such as Néel temperature, overt and
hidden canting of magnetic sublattices, magnetic and magnetoelastic
anisotropy, optic and photoelastic anisotropy.

##  I Introduction

The rare-earth orthoferrites of the general formula RFeO3 (R = Y, or rare-
earth ion) have attracted and continue to attract the particular attention of
researchers for several decades owing to their weak ferromagnetism, remarkable
magneto-optical properties, spin-reorientation transitions between
antiferromagnetic phases, high velocity of domain walls, and many other
properties. Their physical properties remain a focus of considerable research
due to promising applications in innovative spintronic devices, furthermore,
they contribute to an emerging class of materials, i.e., multiferroics with
strong magnetoelectric (ME) coupling.

The rare-earth orthoferrites RFeO${}_{3}\,$ (space group D${}_{2h}^{16}$) have
a distorted perovskite structure with only one type of Fe3+ ion octahedrally
coordinated with six O2- ions [1]. Their complex non-collinear magnetic
structures and magnetic phase transitions are primarily due to the combination
of the antiferromagnetic (AFM) exchange interaction with the Dzyaloshinskii-
Moriya (DM) antisymmetric exchange interaction.

Despite the impressive advance of modern ab-initio band structure computation
techniques, as well as various quantum-chemistry cluster computation methods,
for the majority of practically important systems, these methods give only a
rather rough picture of the electronic structure and the energy spectrum which
can at the best serve only as a suggestion for building up an adequate theory.
One of the most recent and illustrative examples of ab-initio calculations is
given in Ref. [2], the authors of which performed first principles simulations
for the structural, elastic, vibrational, electronic and optical properties of
orthorhombic samarium orthoferrite SmFeO3 within the framework of density
functional theory. However, such calculations encounter great difficulties in
describing more "subtle" effects of magnetic, magnetoelastic, and optical
anisotropy [3].

Below, in the paper we apply simple theoretical approaches to quantitatively
evaluate the interplay between FeO6 octahedral deformations/rotations and main
magnetic and optic characteristics such as Néel temperature, overt and hidden
canting of magnetic sublattices, magnetic and magnetoelastic anisotropy, optic
and photoelastic anisotropy. These involve describing a distorted (low-
symmetry) structure as arising from a (high-symmetry) parent structure with
one or more static symmetry-breaking structural distortions

Knowledge of the structure-property relationships are required both to
elucidate the nature of physical properties and to accelerate materials
discoveries.

Accessibility to computational methods make the distortion-mode analysis
powerful, because it is possible to independently study various distortions
and directly assess their role in forming the electronic structure and
physical properties.

The structure–property relationships in ABO3 perovskites exhibiting octahedral
rotations and distortions were studied using group theoretical methods [4].
Relationship between the rotation angles of octahedra and bond-strength energy
in crystals with perovskite structure was studied in Ref. [5]. However, there
are practically no examples of using structure-property relationships to
describe subtle effects of magnetic and optical anisotropy in perovskites and,
in particular, orthoferrites.

The immediate motivation to write this work was the recent publication of an
article by Zhou et al. [6], in which, contrary to seemingly well-established
concepts, it is argued that single-ion anisotropy, and not the DM interaction,
is responsible for the formation of the weak ferromagnetic moment of rare-
earth orthoferrites. Moreover, the authors make this conclusion supposedly on
the basis of the structure–property relationships. We will consistently show
the incorrectness of their statements.

The rest part of the paper is organized as follows. In Section 2 we address
specific features of the crystal structure of rare-earth orthoferrites with a
focus on the FeO6 octahedra. Sections 3 and 4 are devoted to intersite
interactions, namely to Heisenberg superexchange interaction and
Dzyaloshinskii-Moriya antisymmetric interaction which are mainly controlled by
the FeO6 octahedral rotations. In Sec. 3 we shortly address main results of
the microscopic theory of the isotropic superexchange interactions for
$S$-type Fe3+ ions focusing on the angular dependence of the exchange
integrals. Most attention in Sec. 4 centers around the derivation of the
Dzyaloshinskii vector, its value, orientation, and sense (sign) under
different types of the (super)exchange interaction and crystal field.
Theoretical predictions of this section are compared with experimental data
for the overt and hidden canting in orthoferrites. Here, too, we consider a
weak ferrimagnetism, a novel type of magnetic ordering in systems with
competing signs of the Dzyaloshinskii vectors.

In Secs. 5-6, we apply simple theory to quantitatively explore the
relationship between structure and different magnetic and optic properties of
rare-earth orthoferrites. In Sec. 5 we address the "deformational" model of a
single-ion magnetic and magnetoelastic anisotropy. In Sec. 6 we develop the
deformation model of linear birefringeance and anisotropic photoelastic
effects for orthoferrites. Short summary is presented in Sec. 7.

##  II Crystal and magnetic structure of rare-earth orthoferrites

Orthoferrites are composed of relatively robust corner-shared FeO6 octahedra,
with nominally 12-coordinated rare earth (R) cations. These adopt low-symmetry
distortions from the ideal cubic perovskite Pm$\overline{3}$m symmetry. The Fe
atoms are located on inversion centres, the R atom and atom O1 lie on the
mirror planes perpendicular to the orthorhombic $b$-axis, and atom O2 occupy a
site of general symmetry.

The real FeO6 complex in orthoferrites can be represented as a homogeneously
deformed ideal octahedron. To find the degree of distortion, we introduce a
symmetric strain tensor according to the standard rules. In the local system
of cubic axes of the octahedron

$\varepsilon_{ij}=\frac{1}{4l^{2}}\sum_{n=1}^{6}(R_{i}(n)u_{j}(n)+R_{j}(n)u_{i}(n))\,,$
(1)

where ${\bf R}(n)$ is the radius-vector of the Fe-On bond, ${\bf u}(n)$ is the
On-ligand displacement vector, or

$\hat{\varepsilon}=\left(\begin{array}[]{ccc}1-\frac{l_{1}}{l}&\frac{1}{2}(\frac{\pi}{2}-\theta_{12})&\frac{1}{2}(\frac{\pi}{2}-\theta_{13})\\\
\frac{1}{2}(\frac{\pi}{2}-\theta_{21})&1-\frac{l_{2}}{l}&\frac{1}{2}(\frac{\pi}{2}-\theta_{23})\\\
\frac{1}{2}(\frac{\pi}{2}-\theta_{31})&\frac{1}{2}(\frac{\pi}{2}-\theta_{32})&1-\frac{l_{3}}{l}\end{array}\right)\,,$
(2)

where $l$ is the Fe-O separation in an ideal octahedron, $l_{i}$ are the Fe-Oi
interatomic distances $\frac{1}{3}(l_{1}+l_{2}+l_{3})=l$, and $\theta_{ij}$
are the bond angles Oi-Fe-Oj in a real complex. Local $x,y,z$ axes in
octahedron are defined as follows: the $z$-axis is directed along the Fe-OI,
the $x$-axis is along Fe-OII with the shortest Fe-O bond length. In general,
the deformations of octahedra in orthoferrites are small and do not exceed
0.02.

Diagonal components of the traceless strain tensor (2) (tensile/compressive
deformations) can be termed as $E$-type deformations since $\varepsilon_{zz}$
and $\frac{1}{\sqrt{3}}(\varepsilon_{xx}-\varepsilon_{yy})$ transform
according to the irreducible representation (irrep) $E$ of the cubic group Oh,
while off-diagonal components (shear deformations) can be termed as
$T_{2}$-type deformations since $var\epsilon_{yz}$, $\varepsilon_{xz}$, and
$\varepsilon_{xy}$ transform according to the irrep $T_{2}$ of the cubic group
Oh.

The vector $\mathbf{v}$ of rotation of the octahedron FeO6 , the direction and
value of which specifies the axis and the angle of rotation, respectively, is
related to the small displacements of oxygen ions as follows

$\mathbf{v}=\frac{1}{4l^{2}}\sum_{n=1}^{6}[{\bf R}(n)\times{\bf u}(n)]\,.$ (3)

Four Fe3+ ions occupy positions 4b in the orthorhombic elementary cell of
orthoferrites RFeO3 (space group $Pbnm$):

$1\,(1/2,0,0);2\,(1/2,0,1/2);3\,(0,1/2,1/2);4\,(0,1/2,0)\,.$

It is worth noting that another labeling of the Fe3+ positions than what was
used here is found in the literature (see, for example, Refs. [7, 8]), in
which case the basis vectors ${\bf G}$, ${\bf C}$, ${\bf A}$ may differ in
sign.

Classical basis vectors of magnetic structure for 3$d$ sublattice are defined
as follows:

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

where ${\bf S}_{i}$ is classical spin vector for Fe-ion in $i$-th position,
$S$ = 5/2 is the spin value. Here ${\bf G}$ describes the main
antiferromagnetic component (Néel vector), ${\bf F}$ gives the weak
ferromagnetic moment (overt canting), the weak antiferromagnetic components
${\bf C}$ and ${\bf A}$ describe a canting without net magnetic moment (hidden
canting). Allowed spin configurations for 3$d$-sublattice 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 components given in parentheses
are the only ones different from zero. The phase diagram of orthoferrites
indicates all but SmFeO3 adopt $\Gamma_{4}$ at room temperature.

Competition between magnetic anisotropy of the Fe-sublattice and R-Fe exchange
interaction leads to the spin-reorientational transitions
$\Gamma_{4}\rightarrow\Gamma_{2}$ or $\Gamma_{1}$ depending on R as
temperature decreases [9]. In all cases, the $G$-component is prevalent: $G\gg
F,C,A$.

Figure 1: (Color online) Structure of the Fe3+–O2-–Fe3+ superexchange bonding
in orthoferrites. $J_{ab}$ and $J_{c}$ are nearest-neighbor superexchange
integrals, $J^{\prime}$ superexchange integral for next-nearest-neighbors. 1,
2, 3, 4, are Fe3+ ions in four nonequivalent positions. Reproduced from Ref.
[8].

##  III Isotropic superexchange coupling and superexchange geometry

Figure 1 shows the intricate structure of the Fe3+–O2-–Fe3+ superexchange
bondings in orthoferrites.

Figure 2: Superexchange geometry and the Dzyaloshinskii vector.

The $G$-type of the magnetic structure of orthoferrites is determined by the
strong isotropic superexchange interaction:

${\hat{V}}_{ex}=\sum_{m>n}J_{mn}({\bf S}_{m}\cdot{\bf S}_{n})$ (5)

with an exchange integral, which depends primarily on the superexchange Fe-O-
Fe bonding angle, which, in turn, is determined by the angles of the rigid
rotation of the FeO6 octahedra which produce deviations of the Fe-O-Fe bonding
angle away from the ideal 180∘ found in the cubic perovskite aristotype
(Pm$\overline{3}$m symmetry).

First poor man’s microscopic derivation for the dependence of the
superexchange integral on the bonding angle (see Fig. 1) was performed by the
author in 1970 [10] under simplified assumptions for $S$-ions with
configuration $3d^{5}$ (Fe3+, Mn2+)

$J_{12}(\theta)=a+b\cdot\cos\theta_{12}+c\cdot\cos^{2}\theta_{12}\,,$ (6)

where parameters $a,b,c$ depend on the cation-ligand separation. A more
comprehensive analysis has supported validity of the expression.
Interestingly, the second term in (6) is determined by the ligand inter-
configurational $2p$-$ns$ excitations, while other terms are related with
intra-configurational $2p$-, $2s$-contributions.

Later on the derivation had been generalized for the $3d$ ions in a strong
cubic crystal field (see, e.g., Refs. [12, 13, 11, 14, 3, 15])

Orbitally isotropic contribution to the superexchange integral for pair of
Fe3+, Cr3+ ions with configurations $t_{2g}^{3}e_{g}^{2}$, $t_{2g}^{3}$,
respectively, can be written as follows

$J_{FeFe}=\frac{1}{25}(4J(e_{g}e_{g})+12J(e_{g}t_{2g})+9J(t_{2g}t_{2g}))\,,$
$J_{CrCr}=J(t_{2g}t_{2g})\,,$ $\displaystyle
J_{FeCr}=\frac{1}{5}(2J(e_{g}e_{g})+3J(t_{2g}t_{2g}))\,.$ (7)

Kinetic exchange contribution to partial exchange parameters
$J(\gamma_{i}\gamma_{j})$ related with the electron transfer to partially
filled shells can be written as follows [12, 11]

$J(e_{g}e_{g})=\frac{(t_{ss}+t_{\sigma\sigma}\cos\theta)^{2}}{2U};\,J(e_{g}t_{2g})=\frac{t_{\sigma\pi}^{2}}{3U}\sin^{2}\theta;\,$
$J(t_{2g}t_{2g})=\frac{2t_{\pi\pi}^{2}}{9U}(2-\sin^{2}\theta)\,,$ (8)

where $t_{\sigma\sigma}>t_{\pi\sigma}>t_{\pi\pi}>t_{ss}$ are positive definite
$d-d$ transfer integrals, $U$ is a mean $d-d$ transfer energy (correlation
energy). All the partial exchange integrals appear to be positive or
"antiferromagnetic", irrespective of the bonding angle value, though the
combined effect of the $ss$ and $\sigma\sigma$ bonds $\propto\cos\theta$ in
$J(e_{g}e_{g})$ yields a ferromagnetic contribution given bonding angles
$\pi/2<\theta<\pi$. It should be noted that the "large" ferromagnetic
potential contribution [16] has a similar angular dependence [17].

Figure 3: Dependence of the Fe3+-Fe3+, Cr3+-Cr3+, Fe3+-Cr3+ exchange integrals
(in K) on the superexchange bond angle in orthoferrites-orthocromites [13].

Some predictions regarding the relative magnitude of the
$I(\gamma_{i}\gamma_{j})$ exchange parameters can be made using the relation
among different $d-d$ transfer integrals as follows

$t_{\sigma\sigma}\,:\,t_{\pi\sigma}\,:\,t_{\pi\pi}\,:\,t_{ss}\approx\lambda_{\sigma}^{2}\,:\,\lambda_{\pi}\lambda_{\sigma}\,:\,\lambda_{\pi}^{2}\,:\,\lambda_{s}^{2}\,,$
(9)

where $\lambda_{\sigma},\lambda_{\pi},\lambda_{s}$ are covalency parameters.
The simplified kinetic exchange contribution (8) related with the electron
transfer to partially filled shells does not account for intra-center
correlations which are of a particular importance for the contribution related
with the electron transfer to empty shells. For instance, appropriate
contributions related with the transfer to empty $e_{g}$ subshell for the
Cr3+-Cr3+ and Fe3+-Cr3+ exchange integrals are

$\Delta J_{CrCr}=-\frac{\Delta
E(35)}{6U}\frac{t_{\sigma\pi}^{2}}{U}\sin^{2}\theta\,;\,$ $\Delta
J_{FeCr}=-\frac{\Delta
E(35)}{10U}\left[\frac{(t_{ss}+t_{\sigma\sigma}\cos\theta)^{2}}{U}+\frac{t_{\sigma\pi}^{2}}{U}\sin^{2}\theta\right]\,,$
(10)

where $\Delta E(35)$ is the energy separation between ${}^{3}E_{g}$ and
${}^{5}E_{g}$ terms for $t_{2g}^{3}e_{g}$ configuration (Cr2+ ion). Obviously,
these contributions have a ferromagnetic sign. Furthermore, the exchange
integral $J_{CrCr}$ can change sign at $\theta$ = $\theta_{cr}$:

$\sin^{2}\theta_{cr}=\frac{1}{\left(\frac{1}{2}+\frac{3}{8}\frac{\Delta
E(35)}{U}\frac{t^{2}_{\sigma\pi}}{t^{2}_{\pi\pi}}\right)}\,.$ (11)

Microscopically derived angular dependence of the superexchange integrals does
nicely describe the experimental data for exchange integrals $J_{FeFe}$,
$J_{CrCr}$, and $J_{FeCr}$ in orthoferrites, orthochromites, and
orthoferrites-orthochromites [13] (see Fig. 3). The fitting allows us to
predict the sign change for $J_{CrCr}$ and $J_{FeCr}$ at $\theta_{12}$
$\approx$ 133∘ and 170∘, respectively. In other words, the Cr3+-O2--Cr3+
(Fe3+-O2--Cr3+) superexchange coupling becomes ferromagnetic at
$\theta_{12}\leq 133^{\circ}$ ($\theta_{12}\geq 170^{\circ}$). However, it
should be noted that too narrow (141-156∘) range of the superexchange bonding
angles we used for the fitting with assumption of the same Fe(Cr)-O bond
separations and mean superexchange bonding angles for all the systems gives
rise to a sizeable parameter’s uncertainty, in particular, for $J_{FeFe}$ and
$J_{FeCr}$. In addition, it is necessary to note a large uncertainty regarding
what is here called the "experimental" value of the exchange integral. The
fact is that the "experimental" exchange integrals we have just used above are
calculated using simple MFA relation for the Néel temperature

$T_{N}=\frac{zS(S+1)}{3k_{B}}J\,,$ (12)

however, this relation yields the exchange integrals that can be one and a
half or even twice less than the values obtained by other methods [11, 18].

Above we addressed only typically antiferromagnetic kinetic (super)exchange
contribution as a result of the second order perturbation theory. However,
actually this contribution does compete with typically ferromagnetic potential
(super)exchange contribution, or Heisenberg exchange, which is a result of the
first order perturbation theory. The most important contribution to the
potential superexchange can be related with the intra-atomic ferromagnetic
Hund exchange interaction of unpaired electrons on orthogonal ligand orbitals
hybridized with the $3d$-orbitals of the two nearest magnetic cations.

Strong dependence of the $d-d$ superexchange integrals on the cation-ligand-
cation separation is usually described by the Bloch’s rule [19]:

$\frac{\partial\ln J}{\partial\ln R}=\frac{\partial J}{\partial
R}/\frac{J}{R}\approx-\,10\,.$ (13)

##  IV Crystal structure and the DM coupling in orthoferrites

Weak ferromagnetism is one of the most remarkable physical properties of
orthoferrites. A theoretical explanation and first thermodynamic theory for
weak ferromagnetism was provided by I. E. Dzyaloshinskii [20] in 1957 on the
basis of symmetry considerations and Landau’s theory of phase transitions of
the second kind.

Free energy of the two-sublattice uniaxial weak ferromagnet such as
$\alpha$-Fe2O3, MnCO3, CoCO3, FeBO3 was shown to be written as follows

$F=MH_{E}({\bf m}_{1}\cdot{\bf m}_{2})-M{\bf H}_{0}({\bf m}_{1}+{\bf
m}_{2})+E_{D}+E_{A}$ $=MH_{E}({\bf m}^{2}-{\bf l}^{2})-M{\bf H}_{0}{\bf
m}+E_{D}+E_{A}\,.$ (14)

In this expression ${\bf m}_{1}$ and ${\bf m}_{2}$ are unit vectors in the
directions of the sublattice moments, $M$ is the sublattice magnetization,
${\bf m}=\frac{1}{2}({\bf m}_{1}+{\bf m}_{2})$ and ${\bf l}=\frac{1}{2}({\bf
m}_{1}-{\bf m}_{2})$ are the ferro- and antiferromagnetic vectors,
respectively, $H_{0}$ is the applied field, $H_{E}$ is the exchange field,

$E_{D}=-MH_{D}[{\bf m}_{1}\times{\bf m}_{2}]_{z}=+2MH_{D}[{\bf m}\times{\bf
l}]_{z}=\\\ $ $+2MH_{D}(m_{x}l_{y}-m_{y}l_{x})$ (15)

is now called the Dzyaloshinskii interaction, $H_{D}$ $>$ 0 is the
Dzyaloshinskii field. The anisotropy energy $E_{A}$ is assumed to have the
form: $E_{A}=H_{A}/2M(m_{1z}^{2}+m_{2z}^{2})=2H_{A}/2M(m_{z}^{2}+l_{z}^{2})$,
where $H_{A}$ is the anisotropy field. The choice of sign for the anisotropy
field $H_{A}$ assumes that the $c$ axis is a hard direction of magnetization.
In a general sense the Dzyaloshinskii interaction implies the terms that are
linear both on ferro- and antiferromagnetic vectors. For instance, in
orthorhombic orthoferrites and orthochromites the Dzyaloshinskii interaction
consists of the antisymmetric and symmetric terms

$E_{D}=d_{1}m_{z}l_{x}+d_{2}m_{x}l_{z}=$
$\frac{d_{1}-d_{2}}{2}(m_{z}l_{x}-m_{x}l_{z})+\frac{d_{1}+d_{2}}{2}(m_{z}l_{x}+m_{x}l_{z})=$
$-2MH_{D}[{\bf m}\times{\bf
l}]_{y}+\frac{d_{1}+d_{2}}{2}(m_{z}l_{x}+m_{x}l_{z})\,,$ (16)

while for tetragonal fluorides NiF2 and CoF2 the Dzyaloshinskii interaction
consists of the only symmetric term. Despite Dzyaloshinskii supposed that weak
ferromagnetism is due to relativistic spin-lattice and magnetic dipole
interaction, the theory was phenomenological one and did not clarify the
microscopic nature of the Dzyaloshinskii interaction that does result in the
canting.

Later on, in 1960, T. Moriya [21] suggested a model microscopic theory of the
exchange-relativistic antisymmetric exchange interaction to be a main
contributing mechanism of weak ferromagnetism. He extended the Anderson theory
of superexchange to include spin-orbital coupling $V_{so}=\sum_{i}\xi({\bf
l}_{i}\cdot{\bf s}_{i})$, where $\xi$ is the coupling constant, and derived a
spin-Hamiltonian

$V_{DM}=\sum_{mn}({\bf d}_{mn}\cdot\left[{\bf S}_{m}\times{\bf
S}_{n}\right])\,,$ (17)

now called Dzyaloshinskii-Moriya (DM) spin coupling. Here, ${\bf d}_{mn}$ is
the axial Dzyaloshinskii vector.

Moriya found the symmetry constraints on the orientation of the Dzyaloshinskii
vector $\mathbf{d}_{ij}$. Let two ions 1 and 2 are located at the points A and
B, respectively, with C point bisecting the AB line:

1\. When C is a center of inversion: d=0.

2\. When a mirror plane $\perp$AB passes through C, ${\bf d}\parallel$ mirror
plane or ${\bf d}\perp$ AB.

3\. When there is a mirror plane including A and B, ${\bf d}\perp$ mirror
plane.

4\. When a twofold rotation axis $\perp$ AB passes through C, ${\bf d}\perp$
twofold axis.

5\. When there is an n-fold axis (n$\geq$2) along AB, ${\bf d}\parallel$ AB.

Presently Keffer [22] proposed a simple phenomenological expression for the
Dzyaloshinskii vector for two magnetic ions Mi and Mj interacting by the
superexchange mechanism via intermediate ligand O (see Fig. 1):

$\mathbf{d}_{ij}\propto[\mathbf{r}_{i}\times\mathbf{r}_{j}]\,,$ (18)

where ${\bf r}_{i,j}$ are unit radius vectors for O - Mi,j bonds with
presumably equal bond lenghts. Later on Moskvin [10] derived a microscopic
formula for Dzyaloshinskii vector in a pair of the $S$-type ions

$\mathbf{d}_{ij}=d_{ij}(\theta)[\mathbf{r}_{i}\times\mathbf{r}_{j}]\,,$ (19)

where

$d_{ij}(\theta)=d_{1}(R_{i},R_{j})+d_{2}(R_{i},R_{j})\cos\theta_{ij}\,.$ (20)

In other words, at variance with superexchange integral, the Dzyaloshinskii
vector depends both on the superexchange bonding angle and spatial Fe–O–Fe
bond orientation.

Note that the relation $d/J\approx\Delta g/g$, where $g$ is the gyromagnetic
ratio, $\Delta g$ is its deviation from the free-electron value, proposed by
Moriya for estimating the magnitude of the Dzyaloshinskii vector should be
used with extreme caution. So, in the case of $S$-ions such as Fe3+, Mn2+, it
is simply inapplicable.

Spin nondiagonality of the DM coupling implies very unusual features of the
${\bf d}$-vector somewhat resembling vector orbital operator whose
transformational properties cannot be isolated from the lattice. It seems the
${\bf d}$-vector does not transform as a vector at all.

Within simplest classical approximation the operator of symmetric and
antisymmetric $d$ \- $d$ exchange interactions in orthoferrites

${\hat{H}}=\sum_{i>j}J_{ij}({\bf S}_{i}\cdot{\bf S}_{j})+\sum_{i>j}({\bf
d}_{ij}\cdot[{\bf S}_{i}\times{\bf S}_{j}])$ (21)

can be written in terms of basis vectors as a free energy (see, e.g., Refs.
[11, 23, 15] and references therein). Neglecting the terms quadratic in the
components of small basis vectors ${\bf F}$, ${\bf C}$, ${\bf A}$, we obtain
for the free energy per ion

$\Phi=\frac{J_{F}}{2}{\bf F}^{2}+\frac{J_{G}}{2}{\bf
G}^{2}+\frac{J_{C}}{2}{\bf C}^{2}+\frac{J_{A}}{2}{\bf A}^{2}+$
$D_{x}(C_{y}G_{z}-C_{z}G_{y})+D_{y}(F_{z}G_{x}-F_{x}G_{z})+D_{z}(A_{x}G_{y}-A_{y}G_{x})\,,$
(22)

where

$J_{F}=-J_{G}=S^{2}(2J_{ab}+J_{c});J_{A}=-J_{C}=S^{2}(2J_{ab}-J_{c});$
$D_{x}=-S^{2}\sum_{2}d_{x}(12);$
$D_{y}=-S^{2}\left(\sum_{4}d_{y}(14)+\sum_{2}d_{y}(12)\right);$
$D_{z}=-S^{2}\sum_{4}d_{z}(14)\,,$ (23)

where $J_{ab}$ and $J_{c}$ are Fe-Fe exchange integrals in $ab$-plane and
along $c$-axis, respectively (see Fig. 1), $d_{x,y,z}(ij)$ are the components
of the Dzyaloshinskii vector for Fei-Fej bond. By minimizing the free energy
under condition ${\bf F}^{2}+{\bf G}^{2}+{\bf C}^{2}+{\bf A}^{2}=1$ and
$F,C,A\ll G$ we find

$F_{z}=-\frac{D_{y}}{J_{F}-J_{G}}G_{x};\,\,A_{y}=\frac{D_{z}}{J_{A}-J_{G}}G_{x};$
$F_{x}=\frac{D_{y}}{J_{F}-J_{G}}G_{z};\,\,C_{y}=-\frac{D_{x}}{J_{C}-J_{G}}G_{z};$
$A_{x}=-\frac{D_{z}}{J_{A}-J_{G}}G_{y};\,\,C_{z}=\frac{D_{x}}{J_{C}-J_{G}}G_{y};$
(24)

Hereafter we address the DM coupling for the S-type magnetic $3d$ ions with
orbitally nondegenerate high-spin ground state in a strong cubic crystal
field, that is for the $3d$ ions with half-filled shells $t_{2g}^{3}$,
$t_{2g}^{3}e_{g}^{2}$, $t_{2g}^{6}e_{g}^{2}$ and ground states ${}^{4}A_{2g}$,
${}^{6}A_{1g}$, ${}^{3}A_{2g}$, respectively. The strong crystal field
approximation seems to be more appropriate for the most part of $3d$ ions in
crystals.

Making use of expressions for spin-orbital coupling $V_{so}$ and main kinetic
contribution to the superexchange parameters, that define the DM coupling,
after routine algebra we have found that the DM coupling can be written in a
standard form (17) with Dzyaloshinskii vector (19), where $d_{12}$ can be
written as follows [24, 11, 14, 3, 15]

$d_{12}=X_{1}Y_{2}+X_{2}Y_{1}\,,$ (25)

where the exchange factors $X$ and dimensionless spin-orbital factors $Y$ do
reflect the exchange-relativistic structure of the second-order perturbation
theory and details of the electron configurations for S-type ions.

The factors $X$ and $Y$ are presented in Table 1 for S-type 3d-ions. There
$\xi_{3d}$ is the spin-orbital parameter, $\Delta E_{{}^{2S+1}\Gamma}$ is the
energy of the ${}^{2S+1}\Gamma$ crystal term,
$t_{\sigma\sigma}>t_{\pi\sigma}>t_{\pi\pi}>t_{ss}$ are positive definite $d$
\- $d$ transfer integrals, $U$ is the $d$ \- $d$ transfer energy (correlation
energy).

Note that the value of the Dzyaloshinskii vector in our approximation depends
on the parameters of the FeO6 octahedra rotation and does not depend at all on
octahedral distortions, or on the parameters of the low-symmetry crystal field
for Fe-centers, and hence on the $\delta g$ values, characterizing the
deviation of the $g$\- factor from its value in a free ion. Moriya’s
estimation $d/J\approx\Delta g/g$ in this case does not work at all. In
particular, the authors of Ref. [6] incorrectly associate the value of
$\delta{\hat{g}}=2\lambda{\hat{\Lambda}}$ (see expression (3) there) with the
spin canting angle for orthoferrites.

The signs for $X$ and $Y$ factors in Table 1 are predicted for rather large
superexchange bonding angles $|\cos\theta_{12}|>t_{ss}/t_{\sigma\sigma}$ which
are typical for many $3d$ compounds such as oxides and a relation $\Delta
E_{{}^{4}T_{1g}}(41)<\Delta E_{{}^{4}T_{1g}}(32)$ which is typical for high-
spin $3d^{5}$ configurations.

On the whole, the data in Table 1 allow us to evaluate both the numerical
value and sign of the $d_{12}$ parameters.

It should be noted that for critical angle $\theta_{cr}$, when the
Dzyaloshinskii vector changes its sign we have
$\cos\theta_{cr}=-d_{1}/d_{2}=\frac{\lambda_{s}^{2}}{\lambda_{\sigma}^{2}}$
for $d^{8}-d^{8}$ pairs and
$\cos\theta_{cr}=-d_{1}/d_{2}=\frac{\lambda_{s}^{2}}{\lambda_{\sigma}^{2}-\lambda_{\pi}^{2}}$
for $d^{5}-d^{5}$ pairs. Making use of different experimental data for
covalency parameters (see, e.g., Ref. [25]) we arrive at
$d_{1}/d_{2}\sim\frac{1}{5}-\frac{1}{3}$ and $\theta_{cr}\approx
100^{\circ}-110^{\circ}$ for Fe3+-Fe3+ pairs in oxides.

Relation among different $X$’s given the superexchange geometry and covalency
parameters typical for orthoferrites and orthochromites [11] is

$|X_{d^{8}}|\geq|X_{d^{3}}|\geq|X_{d^{5}}|\,,$ (26)

however, it should be underlined its sensitivity both to superexchange
geometry and covalency parameters. Simple comparison of the exchange
parameters $X$ (see Table 1) with exchange parameters
$I(\gamma_{i}\gamma_{j})$ (8) evidences their close magnitudes. Furthermore,
the relation (9) allows us to maintain more definite correspondence.

Theoretically predicted signs of the Dzyaloshinskii vector in pairs of the
S-type 3d-ions with local octahedral symmetry (the sign rules) are presented
in Table 2. The signs for $d^{3}-d^{3}$, $d^{5}-d^{5}$, and $d^{3}-d^{8}$
pairs turn out to be the same but opposite to signs for $d^{3}-d^{5}$ and
$d^{8}-d^{8}$ pairs. In a similar way to how different signs of the
conventional exchange integral determine different (ferro-antiferro) magnetic
orders the different signs of the Dzyaloshinskii vectors create a possibility
of nonuniform (ferro-antiferro) ordering of local weak (anti)ferromagnetic
moments, or local overt/hidden cantings. Novel magnetic phenomenon and novel
class of magnetic materials, which are systems such as solid solutions
YFe1-xCrxO3 with competing signs of the Dzyaloshinskii vectors are discussed
in Refs. [14, 26, 3, 15] in more detail.

Table 1: Expressions for the $X$ and $Y$ parameters that define the magnitude and the sign of the Dzyaloshinskii vector in pairs of the S-type 3d-ions with local octahedral symmetry. Signs for $X_{i}$ correspond to the bonding angle $\theta>\theta_{cr}$. | Ground state
---
configuration
$X$ | Sign $X$ | $Y$ | Sign $Y$ |  | Excited state
---
configuration
|  $3d^{3}$($t_{2g}^{3}$):${}^{4}A_{2g}$
---
$V^{2+}$, $Cr^{3+}$, $Mn^{4+}$
$-\frac{1}{3U}t_{\pi\pi}t_{\sigma\pi}\cos\theta$ | + | $\frac{2\xi_{3d}}{3\sqrt{3}}(\frac{1}{\Delta E_{{}^{4}T_{2g}}}+\frac{2}{\Delta E_{{}^{2}T_{2g}}})$ | + | $t_{2g}^{2}e_{g}^{1}$
|  $3d^{5}$($t_{2g}^{3}e_{g}^{2}$):${}^{6}A_{1g}$
---
$Mn^{2+}$, $Fe^{3+}$
|  $-\frac{1}{5U}(t_{\pi\pi}t_{\sigma\pi}\cos\theta$ -
---
$t_{\pi\sigma}\left(t_{ss}+t_{\sigma\sigma}cos\theta)\right)$
– | –$\frac{6\xi_{3d}}{5\sqrt{3}}(\frac{1}{\Delta E_{{}^{4}T_{1g}}(41)}-\frac{1}{\Delta E_{{}^{4}T_{1g}}(23)})$ | – |  $t_{2g}^{4}e_{g}^{1}$, $t_{2g}^{2}e_{g}^{3}$
|  $3d^{8}$($t_{2g}^{6}e_{g}^{2}$):${}^{3}A_{2g}$
---
$Ni^{2+}$, $Cu^{3+}$
$\frac{1}{2U}t_{\pi\sigma}(t_{ss}+t_{\sigma\sigma}\cos\theta)$ | – | $\frac{3\xi_{3d}}{2\sqrt{3}}(\frac{1}{\Delta E_{{}^{3}T_{2g}}}+\frac{1}{\Delta E_{{}^{1}T_{2g}}})$ | + | $t_{2g}^{5}e_{g}^{3}$
Table 2: Sign rules for the Dzyaloshinskii vector in pairs of the S-type 3d-ions with local octahedral symmetry and the bonding angle $\theta>\theta_{cr}$. $3d^{n}$ |  $3d^{3}$($t_{2g}^{3}$) |  $3d^{5}$($t_{2g}^{3}e_{g}^{2}$) |  $3d^{8}$($t_{2g}^{6}e_{g}^{2}$)
---|---|---|---
$3d^{3}$($t_{2g}^{3}$) | + | – | +
$3d^{5}$($t_{2g}^{3}e_{g}^{2}$) | – | + | +
$3d^{8}$($t_{2g}^{6}e_{g}^{2}$) | + | + | –
Figure 4: Basic vectors of magnetic structure for $3d$ sublattice in
orthoferrites and orthochromites

At variance with isotropic superexchange coupling the DM coupling has a much
more complicated structural dependence. Figure 3 shows the intricate structure
of the Fe3+–O2-–Fe3+ superexchange bondings in orthoferrites that points to a
complicated structural dependence of the Dzyaloshinskii vectors.

In Table 3 we present structural factors $\left[{\bf r}_{1}\times{\bf
r}_{2}\right]_{x,y,z}$ for the superexchange coupled Fe-O-Fe pairs in
orthoferrites with numerical values for YFeO3 [27]. In all cases, the vector
${\bf r}_{1}$ is oriented to the Fe ion in the position (1/2,0,0), the vectors
${\bf r}_{2}$ are oriented to the nearest Fe ions in the $ab$-plane (1a, 1b)
or along the $c$-axis (3a). It is easy to see that the weak ferromagnetism in
orthoferrites governed by the $y$-component of the Dzyaloshinskii vector does
actually make use of only about one-third of its maximal value.

Table 3: The structural factors $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{x,y,z}$ for the superexchange coupled Fe-O-Fe pairs in orthoferrites with numerical values for YFeO3. See text for detail. | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{x}$ | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{y}$ | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{z}$
---|---|---|---
1a |  $-\frac{z_{2}bc}{2l^{2}}$ = -0.31 |  $-\frac{z_{2}ac}{2l^{2}}$ = -0.29 |  $\frac{(y_{2}-x_{2}+\frac{1}{2})ab}{2l^{2}}$ = 0.41
1b |  $+\frac{z_{2}bc}{2l^{2}}$ = 0.31 |  $-\frac{z_{2}ac}{2l^{2}}$ = -0.29 |  $\frac{(y_{2}-x_{2}+\frac{1}{2})ab}{2l^{2}}$ = 0.41
3a |  $\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}$ = 0.20 | -$\frac{x_{1}ac}{2l^{2}}$ = -0.55 | 0

Simple formula for the Dzyaloshinskii vector (19) and structural factors from
Table 3 can be used to find a relation between crystallographic and canted
magnetic structures for four-sublattice’s orthoferrites RFeO3 and
orthochromites RCrO3 [28, 11, 14, 3, 15] (see Fig.3), where main G-type
antiferromagnetic order is accompanied by both overt canting characterized by
ferromagnetic vector ${\bf F}$ (weak ferromagnetism!) and two types of a
hidden canting, ${\bf A}$ and ${\bf C}$ (weak antiferromagnetism!):

$F_{z}=\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{J}G_{x}\,;\,F_{x}=-\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{J}G_{z}\,;$
$A_{y}=\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{J}G_{x}\,;\,A_{x}=-\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{J}G_{y}\,;$
$C_{y}=\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{J}G_{z}\,;\,C_{z}=-\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{J}G_{y}\,,$
(27)

where $a,b,c$ are unit cell parameters, $x_{1,2},y_{1,2},z_{2}$ are oxygen
($O_{I,II}$) parameters [27], $l$ is a mean cation-anion separation. These
relations imply an averaging on the Fe3+-O2--Fe3+ bonds in $ab$ plane and
along $c$-axis. It is worth noting that $|A_{x,y}|>|F_{x,z}|>|C_{y,z}|$.

First of all we arrive at a simple relation between crystallographic
parameters, canting angle, and magnetic moment of the Fe-sublattice: in units
of $G\cdot g/cm^{3}$

$M_{Fe}=\frac{4g_{S}\beta_{e}S}{\rho
V}|F_{x,z}|=\frac{2g\beta_{e}Sac}{3l^{2}\rho
V}(x_{1}+2z_{2})\frac{d(\theta)}{J(\theta)}\,,$ (28)

where $\rho$ and $V$ are the unit cell density and volume, respectively.

The theoretically predicted value of the spin canting angle along the
$c$-axis, or $F_{z}$-component, increases monotonically from LaFeO3 to LuFeO3
(see Fig. 5), which is in excellent agreement with latest experimental data
obtained on precisely oriented crystals of orthoferrites with a nonmagnetic
R-ion, R = La, Y, Lu [6]). The authors of Ref. [6] erroneously interpreted
this dependence as evidence that "… single-ion anisotropy effect is
responsible for the spin canting in the type-G antiferromagnets
orthoferrites".

Figure 5: Theoretical predictions for the overt canting in orthoferrites
(hollow circles) normalized on experimental data for YFeO3. Solid circles are
latest experimental data for orthoferrites with nonmagnetic R-ion [6].

The overt canting $F_{x,z}$ can be calculated through the ratio of the
Dzyaloshinskii ($H_{D}$) and exchange ($H_{E}$) fields as follows

$F=H_{D}/2H_{E}\,.$ (29)

If we know the Dzyaloshinskii field we can calculate the $d(\theta)$ parameter
in orthoferrites as follows

$H_{D}=\frac{S}{g\mu_{B}}\sum_{i}|d_{y}(1i)|=\frac{S}{g\mu_{B}}(x_{1}+2z_{2})\frac{ac}{l^{2}}|d(\theta)|\,,$
(30)

that yields $|d(\theta)|\cong$ 3.2 K in YFeO3 given $H_{D}$ = 140 kOe [29]. It
is worth noting that despite $F_{z}\approx$ 0.01 the $d(\theta)$ parameter is
only one order of magnitude smaller than the exchange integral in YFeO3.

Our results have stimulated experimental studies of the hidden canting, or
"weak antiferromagnetism" in orthoferrites. As shown in Table 4 the
theoretically predicted relations between overt and hidden canting nicely
agree with the experimental data obtained for different orthoferrites by NMR
[30] and neutron diffraction [31, 32].

In all cases, the magnitude of the Dzyaloshinskii vector ${\bf d}_{12}$ is
anticorrelated with the magnitude of the superexchange integral $J_{12}$ in
the sense that the superexchange geometry, favorable for the former, is
unfavorable for the latter. The specific supersensitivity of the DM coupling
to the superexchange geometry allows us to consider this interaction, first of
all, the value and orientation of the Dzyaloshinskii vector, as one of the
most important indicators determining the role of structural factors.

Table 4: Hidden canting in orthoferrites. Orthoferrite | Ay/Fz, theory [28] | Ay/Fz, exp | Ay/Cy, theory [28] | Ay/Cy, exp
---|---|---|---|---
YFeO$3$ | 1.10 |  | 1.10 $\pm$ 0.03[30]
---
1.4 $\pm$ 0.2[31]
1.1 $\pm$ 0.1[32]
2.04 | ?
HoFeO$3$ | 1.16 | 0.85 $\pm$ 0.10[32] | 2.00 | ?
TmFeO$3$ | 1.10 | 1.25 $\pm$ 0.05[30] | 1.83 | ?
YbFeO$3$ | 1.11 | 1.22 $\pm$ 0.05[31] | 1.79 | 2.0 $\pm$ 0.2[30]

Determination of the "sign" of the Dzyloshinskii vector is of a fundamental
importance from the standpoint of the microscopic theory of the DM coupling.
As was first shown in our paper [33] reliable local information on the sign of
the Dzyaloshinskii vector, or to be exact, that of the scalar Dzyaloshinskii
parameter $d_{12}$, can be extracted from the ligand NMR data in weak
ferromagnets. The procedure was described in details for 19F NMR data in weak
ferromagnet FeF3 [33]. Theoretically simulated NMR spectrum does nicely agree
with the experimental ones only for the "right" mutual orientations of the
${\bf F}$ and ${\bf G}$ vectors, that means $d(FeFe)>0$ in a full accordance
with our theoretical sign predictions (see Table 2). The same result,
$d(FeFe)>0$ follows from the the magnetic $x$-ray scattering amplitude
measurements in the weak ferromagnet FeBO3 [34].

###  IV.1 The DM coupling and effective magnetic anisotropy

At variance with the spin-symmetric single-ion anisotropy and anisotropic
exchange, the Dzyaloshinskii-Moriya interaction is a source of spin-
antisymmetric anisotropy. Hereafter we demonstrate a contribution of the DM
coupling into effective magnetic anisotropy in orthoferrites within a simple
classical approach. Taking into account the expression (22) for the classical
energy of orthoferrite and relations (24) for small basis vectors the
classical energies of the three spin configurations
$\Gamma_{1}(A_{x},G_{y},C_{z})$, $\Gamma_{2}(F_{x},C_{y},G_{z})$, and
$\Gamma_{4}(G_{x},A_{y},F_{z})$ given $|F_{x}|=|F_{z}|=F$,
$|C_{y}|=|C_{z}|=C$, $|A_{x}|=|A_{z}|=A$ can be written as follows [11, 3, 15]

$\displaystyle
E_{\Gamma_{1}}=J_{G}-48JS^{2}F^{2}\left[\frac{1}{3}(\frac{C}{F})^{2}+\frac{2}{3}(\frac{A}{F})^{2}\right]\,;$
(31) $\displaystyle
E_{\Gamma_{2}}=J_{G}-48JS^{2}F^{2}\left[1+\frac{1}{3}(\frac{C}{F})^{2}\right]\,;$
(32) $\displaystyle
E_{\Gamma_{4}}=J_{G}-48JS^{2}F^{2}\left[1+\frac{2}{3}(\frac{A}{F})^{2}\right]\,,$
(33)

with obvious relation $E_{\Gamma_{4}}<E_{\Gamma_{1}}\leq E_{\Gamma_{2}}$. The
energies allow us to find the constants of the in-plane magnetic anisotropy
$E_{an}=k_{1}\,\cos 2\theta$ ($ac$, $bc$ planes, $\theta$ is the polar angle
of the ${\bf G}$-vector), $E_{an}=k_{1}\,\cos 2\varphi$ ($ab$ plane, $\varphi$
is the azimutal angle of the ${\bf G}$-vector):
$k_{1}(ac)=\frac{1}{2}(E_{\Gamma_{2}}-E_{\Gamma_{4}})$;
$k_{1}(bc)=\frac{1}{2}(E_{\Gamma_{2}}-E_{\Gamma_{1}})$;
$k_{1}(ab)=\frac{1}{2}(E_{\Gamma_{4}}-E_{\Gamma_{1}})$. Detailed analysis of
different mechanisms of the magnetic anisotropy of the orthoferrites [11, 3,
15] points to a leading contribution of the DM coupling. Indeed, for all the
orthoferrites RFeO3 this mechanism does predict a minimal energy for
$\Gamma_{4}$ configuration which is actually realized as a ground state for
all the orthoferrites, if one neglects the R-Fe interaction. Furthermore,
predicted value of the constant of the magnetic anisotropy in $ac$-plane for
YFeO3 $k_{1}(ac)$ =2.0$\cdot 10^{5}$ erg/cm3 is close enough to experimental
value of 2.5$\cdot 10^{5}$ erg/cm3 [29]. Interestingly, the model predicts a
close energy for $\Gamma_{1}$ and $\Gamma_{2}$ configurations so that
$|k_{1}(bc)|$ is about one order of magnitude less than $|k_{1}(ac)|$ and
$|k_{1}(ab)|$ for most orthoferrites [11, 3, 15]. It means the anisotropy in
$bc$-plane will be determined by a competition of the DM coupling with
relatively weak contributors such as magneto-dipole interaction and single-ion
anisotropy. It should be noted that the sign and value of the $k_{1}(bc)$ is
of a great importance for the determination of the type of the domain walls
for orthoferrites in their basic $\Gamma_{4}$ configuration (see, e.g., Ref.
[35]).

In conclusion, we emphasize once again that we are considering the classical
theory of the magnetic state of orthoferrites, which is the result of a simple
MFA approximation. The applicability of this popular approximation to the
description of quantum antiferromagnets with the Dzyaloshinskii interaction,
in particular the anisotropy effects, raises natural doubts. For example, in
the author’s paper [36], the role of the DM interaction as a source of
magnetic anisotropy is considered in detail and it is shown that for quantum
$s$ = 1/2 antiferromagnets in contrast to the simple MFA approach, the DM
contribution to the energy of anisotropy for an exchange-coupled spin-1/2 pair
turns into zero. However, just as in the case of isotropic exchange, the use
of the classical description of the DM interaction for magnets with a large
spin $S$ = 5/2 seems quite reasonable.

##  V Magnetic and magnetoelastic anisotropy in orthoferrites

###  V.1 Second-order spin anisotropy

Free energy of the second-order spin anisotropy of the Fe-sublattice in
orthoferrites can be written as follows [23, 11]

$\Phi_{an}^{(2)}=DG_{z}^{2}+E(G_{x}^{2}-G_{y}^{2})+2p(G_{x}A_{y}+G_{y}A_{x})+$
$2q(G_{y}C_{z}+G_{z}C_{y})+2r(G_{x}F_{z}+G_{z}F_{x})\,,$ (34)

where we confined ourselves to terms that are linear and quadratic in the
components of the main antiferromagnetic vector $\bf G$. First, let us pay
attention to the appearance of three terms of the type of symmetric
Dzyaloshinskii interaction, the inclusion of which leads to "symmetric"
corrections in the expressions for the parameters of the overt and hidden
canting (24) and (27). In particular, taking into account the $r$-contribution
leads to the appearance of a difference between the $F_{x}$ and $F_{z}$ weak-
ferromagnetic components:

$\frac{F_{z}-F_{x}}{F}=\frac{4r}{D_{y}}.$ (35)

The main, quadratic in the components of the antiferromagnetic vector,
contribution to the energy of magnetic anisotropy is usually considered by
limiting the rotation of the vector $\bf G$ in a certain plane:

$\Phi_{an}^{(2)}=k_{1}\,\cos 2\theta$ (36)

for $ac$-, $bc$-planes or

$\Phi_{an}^{(2)}=k_{1}\,\cos 2\varphi$ (37)

for $ab$-plane.

The main mechanisms of second-order spin anisotropy for 3d-sublattice within a
two-sublattice model are associated with single-ion anisotropy (SIA), as well
as two-ion anisotropy, determined by the Dzyaloshinskii-Moriya coupling,
magnetic dipole and exchange-relativistic Fe-Fe interaction (TIA) (see, e.g.,
Refs. [37, 38].) Anisotropy parameters are not equal in the two-sublattice and
four-sublattice models because the weak antiferromagnetic order is absorbed
into renormalized anisotropy parameters. Therefore anisotropy parameters
should not be directly compared between two and four sublattice models. Two-
sublattice model interpretation is typical for conventional magnetic
"macroscopic" measurements, while spin waves excitations measured by the
method of submillimeter dielectric (THz) spectroscopy [39, 8], Raman
scattering [40] or inelastic neutron scattering [41, 42] should been analyzed
within a full four-sublattice model.

The second-order single-ion spin anisotropy for the $S$-type $3d$ ions is a
result of the third-order perturbation theory with a zero approximation
corresponding to either a free ion or a highly symmetric cubic environment,
taking into account the quadratic effects in the spin-orbit interaction and
linear in the low-symmetry crystal field (LSCF).

$V_{SIA}=\sum_{\gamma\nu}d_{\gamma}B_{\gamma\nu}^{*}{\hat{V}}^{2\gamma}_{\nu}(S)\,,$
(38)

where
${\hat{V}}^{2\gamma}_{\nu}(S)=\sum_{q}\alpha^{\gamma\nu}_{q}{\hat{V}}^{2}_{q}$
are combinations of the components of the rank-2 spin irreducible tensor
operator which are transformed according to the irreducible representation of
the $O_{h}$ point symmetry group, $\gamma$ = $E,T_{2}$, $B_{\gamma\nu}$ are
the low-symmetry crystal field parameters,
$d_{\gamma}\propto\frac{\lambda^{2}}{(\Delta E)^{2}}$ are dimensionless
parameters.

The low-symmetry crystal field can be represented as the sum of the local
“deformation” contribution associated with low-symmetry distortions of the
FeO6 octahedron and the nonlocal contribution of the rest of the lattice.
Within the framework of the "deformation" model, the LSCF parameters for
$S$-type $3d$ ions in weakly distorted octahedra in the linear approximation
are proportional to the components of the deformation tensor of the octahedron
of the corresponding symmetry

$B_{\gamma\nu}=b_{\gamma}\varepsilon^{\gamma}_{\nu}\,,$ (39)

where $b_{\gamma}$ are the parameters of the electron-lattice coupling.
Numerical estimates for $3d^{5}$ configuration [43] show that the $b_{E}$
parameters are about an order of magnitude higher than the $b_{T_{2}}$
parameters. Thus, the Hamiltonian of the single ion spin anisotropy of the
second order can be represented as follows

$V_{SIA}=\sum_{\gamma\nu}K_{\gamma}\left(\varepsilon^{\gamma}\cdot
V^{2\gamma}(S)\right)\,,$ (40)

where $K_{\gamma}=b_{\gamma}d_{\gamma}$. Within the mean-field approximation
(MFA) for the energy of magnetic anisotropy we arrive at

$E_{SIA}=\sum_{\gamma}\tilde{K}_{\gamma}\left(\varepsilon^{\gamma}\cdot
C^{2\gamma}({\bf S})\right)\,,$ (41)

where $C^{2\gamma}({\bf S})$ is a symmetrized combination of tensorial
spherical harmonics with classical vector ${\bf S}$ to be its argument,

$\tilde{K}_{\gamma}=K_{\gamma}\langle\langle
V^{2}_{0}(S)\rangle\rangle=K_{\gamma}\frac{\langle\langle
3S_{z}^{2}-\frac{35}{4}\rangle\rangle}{4\sqrt{3\cdot 5\cdot 7}}$

are in fact local temperature-dependent magnetoelastic constants. The latter
expression can be represented in a cartesian form as

$E_{SIA}=\sqrt{\frac{3}{2}}\tilde{K}_{E}(\varepsilon_{11}\alpha_{1}^{2}+\varepsilon_{22}\alpha_{2}^{2}\,+\varepsilon_{33})\alpha_{3}^{2})+$
$+\sqrt{6}\tilde{K}_{T_{2}}(\varepsilon_{23}\alpha_{2}\alpha_{3}+\varepsilon_{13}\alpha_{1}\alpha_{3}+\varepsilon_{12}\alpha_{1}\alpha_{2})\,,$
(42)

where $\alpha_{i}$ are direction cosines of the vector ${\bf S}$ in the local
system of cubic axes. For weakly distorted octahedral Fe3+O6 complexes in
orthoferrites RFeO3 ($|\varepsilon_{ij}|\sim 10^{-2}$): $\tilde{K}_{E}\approx$
20 $cm^{-1}$, $\tilde{K}_{T_{2}}\approx$ 2.5 $cm^{-1}$ [11, 44].

In the system of crystallographic axes $a,b,c$, we obtain an expression for
the free energy of the single-ion crystallographic anisotropy as follows

$E_{SIA}=\sum_{\gamma}\tilde{K}_{\gamma}\left(a^{2}(\gamma)\cdot C^{2}({\bf
S})\right)\,,$ (43)

where

$a^{2}_{q}(\gamma)=\sum_{\nu}D^{(2)}_{q;\gamma\nu}(\omega)\varepsilon^{\gamma}_{\nu}$
(44)

are structure factors which depend both on the FeO6 octahedron rotation and
deformation parameters,
$D^{(2)}_{q;\gamma\nu}(\omega)=\sum_{q_{1}}\alpha^{\gamma\nu}_{q_{1}}D^{(2)}_{qq_{1}}(\omega)$
are linear combinations of Wigner matrices,
$\omega=(\phi_{1},\theta,\phi_{2})$ are Euler angles, which determine
transformation between octahedron local coordinates and $abc$-system.

Table 5: Contributions of the main mechanisms to the first constants of the magnetic anisotropy of orthoferrites YFeO3 and LuFeO3 ($\times 10^{5}$ erg/cm3). See text for detail. Mechanism |  | $k_{1}(ac)$
---
Y | Lu
| $k_{1}(bc)$
---
Y | Lu
| $k_{1}(ab)$
---
Y | Lu
DM coupling | 3.1 3.1 | -0.8 -0.9 | -3.9 -4.0
Magnetodipole | 0.9 0.8 | -0.2 -0.5 | -1.1 -1.3
SIA | -1.9 1.0 | -5.6 -1.8 | -3.7 -2.8
Total | 2.1 4.9 | -6.6 -3.2 | -8.7 -8.1
Experiment | 2.1 $\sim$ 6.0 | -5.7 ? | -7.8 ?

Magnetodipole interaction in orthoferrites was considered in Refs. [45, 37].
First of all, it should be noted that due to the symmetry of the Fe sublattice
of orthoferrites, the magnetodipole interaction does not contribute to the
Dzyaloshinskii interaction. For all orthoferrites, the magnetodipole
interaction stabilizes the $\Gamma_{4}$ configuration, and the contribution to
the anisotropy constants for all planes decreases monotonically by a factor of
about 40 on going from LuFeO3 to LaFeO3, reflecting a decrease in orthorhombic
distortions. Magnetodipole contribution to $k_{1}(ac)$ for YFeO3 reaches a
value of the order of 40% of its experimental value.

Theoretical estimations [44, 11] for main contributions to the first constants
of the magnetic anisotropy of orthoferrites YFeO3 and LuFeO3 are presented in
Table 5. The SIA contribution includes taking into account both the main local
contribution calculated in the framework of the deformation model and a small
nonlocal lattice contribution calculated in the point charges model.

We do not attach much importance to the exact coincidence of the predicted and
experimental [9] values of the constant $k_{1}(ac)$ for YFeO3. More important
is the theoretical prediction of an unexpectedly strong increase in this
constant for LuFeO3. The SIA contribution to $k_{1}(ac)$ partially compensates
for the large contribution of the DM interaction in YFeO3, whereas in LuFeO3
they add up. This result is confirmed by experimental data on the measurement
of the threshold field $H_{SR}$ of spin reorientation
$\Gamma_{4}\rightarrow\Gamma_{2}$ in the orthoferrite Lu0.5Y0.5FeO3, in which
$H_{SR}$ = 15 T as compared to $H_{SR}$ = 7.5 T in YFeO3 [44]. Thus, one can
estimate $k_{1}(ac)$ in LuFeO3 as about three times as much as $k_{1}(ac)$ in
YFeO3.

The value of the ratio $k_{1}(ab)/k_{1}(ac)\approx$ 3.7 was estimated from the
experimental data of Raman spectroscopy in YFeO3 [40].

Unfortunately, despite numerous, including fairly recent, studies of the
magnetic anisotropy of orthoferrites, we do not have reliable experimental
data on the magnitude of the contributions of various anisotropy mechanisms.

Competition of various contributions in the temperature dependence of AFMR
(antiferromegnetic resonance) frequencies and anisotropy constants in YFeO3
was addressed in Ref. [46]. However, the authors neglected to take into
account the hidden canting modes in the thermodynamic potential (see
expression (1) in their article), which did not allow an adequate description
of the DM contribution to the anisotropy. Let us pay attention to recent works
on the determination of the parameters of the spin Hamiltonian in YFeO3 from
measurements of the spin-wave spectrum by the inelastic neutron scattering
[41, 42] and terahertz absorption spectroscopy [8]. However, these authors
started with a simplified spin-Hamiltonian that took into account only
Heisenberg exchange, DM interaction, and single-ion anisotropy. Obviously,
disregarding the magnetic dipole and exchange-relativistic anisotropy, the
"single-ion anisotropy" constants found by the authors are some effective
quantities that are not directly related to SIA.

Concluding the subsection let us note that the contribution of single-ion
crystallographic anisotropy to the Dzyaloshinskii interaction in orthoferrites
does not exceed 1% [11].

###  V.2 Magnetoelastic coupling

The common nature of the magnetic and magnetoelastic anisotropy leads to the
fact that we must require from microscopic theory a simultaneous explanation
of the numerical values both for the anisotropy constants and magnetoelastic
constants.

In the general case, magnetoelastic energy is understood as the part of the
crystal energy that describes the coupling of the magnetic (spin) subsystem of
the crystal with the crystal lattice and depends both on the macroscopic
deformation and latent displacements of the Bravais sublattices, and on the
parameters of the magnetic (spin) order – magnetization, antiferromagnetism
vectors, and other basis vectors of the structure. Magnetoelastic interactions
are manifested, for example, in a change in the size and shape of the sample
upon a change in the magnetic state (magnetostriction), as well as in a change
in the magnetic state upon deformation of the sample. The nature of
magnetoelastic interactions is associated with the dependence of the
parameters of exchange interactions, magnetic anisotropy on crystallographic
parameters - interatomic distances and bond angles.

The main role in the magnetoelastic effects is played by the terms of the
energy, which are quadratic in the components of the largest of the basis
vectors, the antiferromagnetic vector $\bf G$:

$\Phi_{me}\,=\,\Lambda_{ijkl}^{0}\varepsilon_{ij}G_{k}\,G_{l}\,+\,\Pi_{kl}^{n}(\Gamma_{\nu})\,u_{n}(\Gamma_{\nu})\,G_{k}\,G_{l}\>,$
(45)

where $G_{k}$, $G_{l}$ are components of the antiferromagnetic vector,
$\varepsilon_{ij}$ is the tensor of macroscopic deformations,
$u_{n}(\Gamma_{\nu})$ are components of the symmetrized vectors of "hidden"
displacements of the Bravais sublattices ("internal distortions"), that alone
do not lead to macroscopic deformation of the crystal, $\Lambda_{ijkl}^{0}$,
$\Pi_{kl}^{n}(\Gamma_{\nu})$ are the tensors of magnetoelastic constants. The
elastic energy of the crystal has a standard form

$\Phi_{e}\,=\,\frac{1}{2}\,C_{ijkl}^{0}\varepsilon_{ij}\,\varepsilon_{kl}\,+\,C_{ij}^{n}(\Gamma_{\nu})\,u_{n}(\Gamma_{\nu})\varepsilon_{ij}\,+$
$\frac{1}{2}\,C^{nm}(\Gamma_{\nu})\,u_{n}(\Gamma_{\nu})\,u_{m}(\Gamma_{\nu})\,.$
(46)

Generally speaking, hidden displacements $u_{n}(\Gamma_{\nu})$ can be
associated with deformations:

$u_{n}(\Gamma_{\nu})=A_{ij}^{n}(\Gamma_{\nu})\varepsilon_{ij}\,,$ (47)

where $A_{ij}^{n}(\Gamma_{\nu})$ is the so-called "inner stress" tensor, so
that as a result, it is possible to use the renormalized energies $\Phi_{e}$
and $\Phi_{me}$, where only the components of the strain tensor
$\varepsilon_{ij}$ will appear.

The equilibrium values of macroscopic deformations and displacements of the
sublattices are found by minimizing the elastic and magnetoelastic energies.

The magnetostriction effects are usually described by a simplified expression
for magnetoelastic energy in the form as follows

$\Phi_{me}=\lambda_{0}[G_{z}^{2}+\gamma(G_{x}^{2}-G_{y}^{2})]Tr{\hat{\varepsilon}}+\lambda_{1}G_{z}^{2}(\varepsilon_{zz}-\frac{1}{3}Tr{\hat{\varepsilon}})+$
$\lambda_{2}G_{z}^{2}(\varepsilon_{xx}-\varepsilon_{yy})+\lambda_{3}(G_{x}^{2}-G_{y}^{2})(\varepsilon_{zz}-\frac{1}{3}Tr{\hat{\varepsilon}})+$
$\lambda_{4}(G_{x}^{2}-G_{y}^{2})(\varepsilon_{xx}-\varepsilon_{yy})+$
$\mu_{1}G_{y}G_{z}\varepsilon_{yz}+\mu_{2}G_{x}G_{z}\varepsilon_{xz}+\mu_{3}G_{x}G_{e}\varepsilon_{xy}\,,$
(48)

where
$Tr{\hat{\varepsilon}}=(\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz})$.
For a spin-reorientation transition in a certain plane of the orthoferrite
this energy can be represented as follows [9]

$\Phi_{me}=(L_{a}\varepsilon_{aa}+L_{b}\varepsilon_{bb}+L_{c}\varepsilon_{cc})\cos
2\theta+\frac{1}{2}\mu\varepsilon_{\alpha\beta}\sin 2\theta\,,$ (49)

($\theta\rightarrow\varphi$ for $ab$-plane), where $L_{a,b,c},\mu$ are
magnetoelastic constants expressed in terms of parameters $\lambda$’s and
$\mu$’s from (48),
$\varepsilon_{\alpha\beta}=\varepsilon_{ac},\varepsilon_{ab},\varepsilon_{bc}$
are shear deformations for the corresponding planes.

Table 6 presents the results of theoretical estimates of the contribution of
various mechanisms to the magnetostriction constants in YFeO3, performed
within the framework of absolutely the same approximations and the same
parameters that were used to calculate the anisotropy constants in Table 5
[11].

Surprisingly, the DM interaction, being the main source of effective magnetic
anisotropy in orthoferrites, practically does not make any noticeable
contribution to the magnetostriction constants.

The symmetry of the magnetodipole interaction leads to the relationship
between the magnetoelastic parameters in (48) [37]:

$\lambda_{2}=\lambda_{3};\,\,\,\lambda_{1}+3\lambda_{4}=-\frac{3}{4}(\mu_{1}+\mu_{2}+\mu
3)\,.$

In the limit of an ideal perovskite structure

$\lambda_{2}=\lambda_{3}=0;\,\,\lambda_{1}:\lambda_{4}:\mu_{1}:\mu_{2}:\mu_{3}=(-9):2:8:8:(-12)\,,$

so that for the magnetoelastic parameters $L_{a,b,c}$ we have

$L_{a}:L_{b}:L_{c}=\begin{cases}(-1):(-5):6\,,\,(ac-plane);\\\
(-5):(-1):6\,,\,(bc-plane);\\\ (-4):4:0\,,\,(ab-plane)\,.\end{cases}$

The magnetodipole contribution to the magnetoelastic parameters, varying
relatively weakly in the series of orthoferrites, makes a noticeable, although
not determining, contribution to the magnetostriction constants.

For the nonlocal contribution of the lattice in the model of point charges,
the magnetoelastic parameters $L_{a,b,c},\mu_{i}$ in the ideal perovskite
limit satisfy the relations

$\begin{cases}L_{a}=-L_{b}\,,L_{c}=0\,,\mu_{2}=-8L_{a}\,,\,(ac-plane);\\\
L_{a}=-L_{b}\,,L_{c}=0\,,\mu_{1}=+8L_{a}\,,\,(bc-plane);\\\
L_{a}=-L_{b}\,,L_{c}=0\,,\mu_{3}=0\,,\,(ab-plane)\,,\end{cases}$ (50)

which are quite satisfactory for LaFeO3 [11]. On the whole, this mechanism,
like the magnetodipole one makes a noticeable, although not determining,
contribution to the magnetostriction constants.

The deformation model of spin anisotropy considered above provides the
simplest example of a microscopic mechanism for the formation of
magnetoelastic energy. Indeed, considering macroscopic crystal deformations
instead of octahedral deformations in the expression (42) for the single-ion
spin anisotropy energy, we arrive at the magnetoelastic energy

$E_{me}=\Lambda_{E}(\varepsilon_{11}\alpha_{1}^{2}+\varepsilon_{22}\alpha_{2}^{2}+$
$+\varepsilon_{33})\alpha_{3}^{2})+\Lambda_{T_{2}}(\varepsilon_{23}\alpha_{2}\alpha_{3}+\varepsilon_{13}\alpha_{1}\alpha_{3}+\varepsilon_{12}\alpha_{1}\alpha_{2})\,,$
(51)

where $\alpha_{i}$ are direction cosines of vector ${\bf S}$ in local system
of cubic axes, parameters $\Lambda_{E}$ = $\sqrt{\frac{3}{2}}\tilde{K}_{E}$
and $\Lambda_{T_{2}}$ = $\sqrt{6}\tilde{K}_{T_{2}}$ are magnetoelastic
constants.

Table 6: Contributions of the main mechanisms to the magnetostriction constants for YFeO3 . Mechanism |  |  $ac$-plane
---
La | Lb | Lc | $\mu_{2}$
|  $bc$-plane
---
La | Lb | Lc | $\mu_{1}$
|  $ab$-plane
---
La | Lb | Lc | $\mu_{3}$
DM coupling | 0.0 0.0 0.1 0.0 | 0.0 0.0 0.1 0.0 | 0.1 0.1 0.0 0.0
Magnetodipole | 0.0 -0.4 0.4 -1.1 | -0.4 -0.1 0.5 -1.1 | -0.3 0.3 0.0 1.8
SIA - E term | -1.6 -1.3 2.9 4.1 | -1.0 -2.2 3.2 1.9 | 0.6 -0.9 0.3 12.3
SIA - T2 term | -0.4 0.2 0.2 1.7 | 0.1 -0.2 0.1 2.0 | 0.5 -0.5 0.0 0.6
SIA - lattice | 0.4 -0.2 -0.2 -2.1 | -0.1 0.1 -0.0 -2.0 | -0.5 0.4 0.1 0.0
Total | -1.6 -1.7 3.3 2.6 | -1.4 -2.4 3.8 0.8 | 0.3 -0.7 0.4 14.7
Experiment | -1.6 -1.3 2.9 2.4 | ? ? ? ? | ? ? ? ?

The results of calculating the magnetoelastic constants performed at the same
values of $\tilde{K}_{E}\approx$ 20 $cm^{-1}$, $\tilde{K}_{T_{2}}\approx$ 2.5
$cm^{-1}$ as in the case of single-ion anisotropy, the deformation model of
single-ion magnetoelastic coupling, primarily the $E$-contribution, can be the
leading mechanism of magnetostriction for $3d$-system in orthoferrites. It is
the $E$-contribution of the deformation model that determines the anomalously
high value of the magnetoelastic parameter $\mu_{3}$, and hence the
anomalously large values of the shear deformation $\varepsilon_{ab}$ upon
spin-reorientation in the $ab$-plane.

Theoretical predictions of the magnetoelastic "shear" parameters $\mu$
stimulated experimental studies of shear strains accompanying spin-
reorientation transitions in orthoferrites [47]. A specific feature of such
deformations is the dependence on the antiferromagnetic domain structure, so
that in order to detect them during the $\Gamma_{4}-\Gamma_{2}$ transition
induced by an external field, it was necessary to "violate" the exact
orientation of the field along the $a$-axis of the crystal, thereby
highlighting a certain type of domains (see Fig. 6).

Figure 6: (Color online) The field dependence of magnetostriction in YFeO3 for
the external field with orientation near ${\bf H}\parallel{\bf a}$ [47]. Left
in the figure - an illustration of the nature of shear deformation in
antiferromagnetic domains with different orientations of the magnetic moment.

A model quantitative analysis of the role of hidden displacements in
magnetoelastic effects in orthoferrites was carried out in Ref. [48] based on
calculations of elastic energy parameters within the framework of the rigid
ion model.The authors showed that the contributions of the lattice strains and
the contribution of the sublattice displacements to the magnetoelastic energy
are comparable in magnitude, in agreement with the qualitative conclusions of
Refs. [49, 50].

###  V.3 Single-ion cubic anisotropy

The fourth-order single ion spin anisotropy appears at least in the fourth
order of the perturbation theory in the spin-orbit interaction and, in the
general case, can be represented by an effective spin Hamiltonian as follows

$\hat{V}_{SIA}^{\mbox{\scriptsize{(4)}}}=\sum_{\gamma\nu}k_{\gamma\nu}^{*}{\hat{V}}^{4\gamma}_{\nu}(S)\,,$
(52)

where we made use of cubic irreducible tensorial operators
${\hat{V}}^{4\gamma}_{\nu}(S)=\sum_{q}\alpha_{4q}^{\gamma\nu}{\hat{V}}^{4}_{q}(S)$,
that is linear combinations of irreducible tensorial operators for rotation
group acting in a spin space, which automatically "prohibits" the spin
anisotropy of the fourth order for $S$ < 2\. In general,
$\gamma=A_{1},E,T_{1},T_{2}$, however, for an ideal FeO6 octahedron, the
fourth-order spin anisotropy is actually reduced only to the cubic
contribution with $\gamma=A_{1}$, or cubic spin anisotropy:

$\hat{V}_{SIA}^{\mbox{\scriptsize{cub}}}=k_{A_{1}}{\hat{V}}^{4A_{1}}_{0}(S)=$
$=k_{A_{1}}\left[\sqrt{\frac{7}{12}}{\hat{V}}^{4}_{0}(S)+\sqrt{\frac{5}{24}}({\hat{V}}^{4}_{4}(S)+{\hat{V}}^{4}_{-4}(S))\right]\,,$
(53)

or in cartesian coordinates

$\hat{V}_{SIA}^{\mbox{\scriptsize{cub}}}=\frac{a}{6}\left[{\hat{S}}_{x}^{4}+{\hat{S}}_{y}^{4}+{\hat{S}}_{z}^{4}-\frac{1}{5}\,S\left(S+1\right)\left(3S^{2}+3S-1\right)\right]\,,$
(54)

where $a=\,\frac{5\sqrt{3}}{12}k_{A_{1}}$ given $S$=5/2. In the mean-field
approximation we obtain for the energy of magnetic cubic anisotropy

$E_{SIA}^{\mbox{\scriptsize{cub}}}=\tilde{k}_{A_{1}}C^{4A_{1}}_{0}({\bf
S})\,,$ (55)

where

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

is the invariant cubic tensor spherical harmonic,

$\tilde{k}_{A_{1}}=k_{A_{1}}\langle\langle{\hat{V}}^{4}_{0}(S)\rangle\rangle=k_{A_{1}}\frac{\langle\langle
35S_{z}^{4}-(30S^{2}+30S-25)S_{z}^{2}+3S^{4}+6S^{3}-3S^{2}\rangle\rangle}{2\sqrt{(2S+5)(2S+3)(2S+1)(2S-1)(2S-3)(S+2)(S+1)S(S-1)}}$
(56)

is the temperature-dependent anisotropy constant (S = 5/2). Cubic spin
anisotropy has simple form (53) or (54), (55) only in the coordinate system,
where the $xyz$-axes coincide with the principal axes of the cubic crystal
field, that is symmetry axes of the fourth order. In the system of $abc$-axes,
the energy of cubic anisotropy has more complex expression

$E_{SIA}^{\mbox{\scriptsize{cub}}}=\tilde{k}_{A_{1}}\sum_{\Gamma\mu}a_{\Gamma\mu}C^{4\Gamma}_{\mu}({\bf
S})\,,$ (57)

where

$C^{4\Gamma}_{\mu}({\bf S})=\sum_{q}\alpha_{4q}^{\Gamma\mu}C^{4}_{q}({\bf S})$

is the combination of spherical tensor harmonics to be a basis of the irrep
$\Gamma$,

$a_{\Gamma\mu}=\sum_{qq^{\prime}}\alpha_{4q}^{\Gamma\mu}{}^{*}D^{(4)}_{qq^{\prime}}(\omega)\alpha_{4q^{\prime}}^{A_{1}0}\,,$
(58)

are structure factors which depend on the FeO6 octahedron rotation parameters,
$D^{(4)}_{qq^{\prime}}(\omega)$ are Wigner matrices,
$\omega=(\phi_{1},\theta,\phi_{2})$ are Euler angles, which determine
transformation between $abc$\- and octahedron systems, that is octahedron
rotation parameters.

It is practically important to consider the cubic spin anisotropy for
different crystal planes by replacing ${\bf S}\rightarrow{\bf G}$ in argument
of spherical harmonic in (57), limiting the rotation of the vector $\bf G$ in
a certain plane, and highlighting the fourth-order contribution:

$\Phi_{an}^{(4)}=k_{2}\,\cos 4\theta$ (59)

for $ac$-, $bc$-planes or

$\Phi_{an}^{(4)}=k_{2}\,\cos 4\varphi$ (60)

for $ab$-plane.

Figure 7 shows the calculated values of the fourth-order anisotropy constants
for orthoferrites [51] with the parameter $\tilde{k}_{A_{1}}$ normalized to
the experimental value $k_{2}(ac)$ for YFeO3: $k_{2}(ac)$ = 1.35$\cdot
10^{4}erg/cm^{3}$ [9]. On the whole, the constants $k_{2}$ rather smoothly
decrease in absolute value (Fig. 7), changing by no more than two times on
going from La to Lu. The difference between the constants $k_{2}(ac)$ and
$k_{2}(bc)$ can serve as a measure of the 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 a different character 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 [9]. Indeed, all currently known spin-reorientation transitions of
the $\Gamma_{4}-\Gamma_{2}$ ($G_{x}-G_{z}$) type in orthoferrites RFeO3 (R =
Sm, Nd, Er, Tm) are smooth with two characteristic temperatures of the second
order phase transitions to be a start and finish of the spin-reorientation,
and the only known for these crystals a transition of the type
$\Gamma_{4}-\Gamma_{1}$ ($G_{x}-G_{y}$) (DyFeO3) is a jump-like transition of
the first type. A unique example that confirms our conclusions about the sign
of the second anisotropy constant is a mixed orthoferrite Ho0.5Dy0.5FeO3 [9]
in which two spin-reorientation transitions $G_{x}-G_{y}$ ($T$ = 46 K) and
$G_{y}-G_{z}$ (18 $\div$ 24 K), realized through one phase transition of the
first order in $ab$-plane and two phase transitions of the second order in
$bc$-plane, respectively.

Figure 7: The YFeO3-normalized ($ac$-plane) $k_{2}$ constants for
orthoferrites

The microscopic expression for the cubic anisotropy constant $k_{A_{1}}$ for
the ground state of ions Mn2+ or Fe3+ with $3d^{5}$ configuration, obtained in
the scheme of a strong crystal field, looks as follows [51]:

$k_{A_{1}}=\sum_{iS\Gamma}\left(\sum_{j}\frac{\lambda^{(1T_{1})}_{{}^{6}A_{1g};j{}^{4}T_{1g}}\lambda^{(1T_{1})}_{j{}^{4}T_{1g};i{}^{2S+1}\Gamma}}{\Delta
E(j{}^{4}T_{1g})}\left\\{{\begin{array}[]{ccc}{1}&{1}&{2}\\\
{S}&{\frac{5}{2}}&{\frac{3}{2}}\end{array}}\right\\}\right)^{2}\times$
$\times(-1)^{\Gamma}\Delta
E^{-1}(i{}^{2S+1}\Gamma)\left\\{{\begin{array}[]{ccc}{2}&{2}&{4}\\\
{\frac{5}{2}}&{\frac{5}{2}}&{S}\end{array}}\right\\}\,,$ (61)

where $i,j$ distinguish cubic terms, the ${}^{2S+1}\Gamma$ are excited terms
${}^{4}E_{g}(\times 2)$, ${}^{2}E_{g}(\times 7)$, ${}^{2}T_{2g}(\times 10)$,
${}^{4}T_{2g}(\times 3)$ (the number of identical terms is indicated in
brackets), $E(i{}^{2S+1}\Gamma)$ is the term energy measured from the energy
of the ground ${}^{6}A_{1g}$ term,
$\lambda^{(1T_{1})}_{{}^{6}A_{1g};j{}^{4}T_{1g}},\,\lambda^{(1T_{1})}_{j{}^{4}T_{1g};i{}^{2S+1}\Gamma}$
are spin-orbital parameters,
$\left\\{{\begin{array}[]{ccc}{\cdot}&{\cdot}&{\cdot}\\\
{\cdot}&{\cdot}&{\cdot}\end{array}}\right\\}$ are 6$j$-symbols. The main
difficulty in calculating $k_{A_{1}}$ is associated with a very large number
of terms in the sums over $i,j,S,\Gamma$ as well as the complexity of
calculating the reduced matrix elements of the spin orbit taking into account
the mixing of terms of the same symmetry. Numerical calculations performed in
the strong cubic field scheme for the Fe3+ ion with crystal-field parameter
10D$q$ = 12 200 $cm^{-1}$, Racah parameters B = 700 $cm^{-1}$, C = 2600
$cm^{-1}$, which correspond the Fe3+ ion in orthoferrite YFeO3 [52, 11], yield

$k_{A_{1}}=(0.678\zeta^{4}_{\pi\sigma}+0.091\zeta^{3}_{\pi\sigma}\zeta_{\pi\pi}-0.460\zeta^{2}_{\pi\sigma}\zeta^{2}_{\pi\pi}-$
$-\,0.045\zeta_{\pi\sigma}\zeta^{3}_{\pi\pi}-0.002\zeta^{4}_{\pi\pi})\cdot
10^{-13}\,\,cm^{-1}\,,$ (62)

given $\zeta_{\pi\sigma}=-3\sqrt{2}\zeta_{3d}$, $\zeta_{\pi\pi}=3\zeta_{3d}$
and for the spin-orbital coupling constant $\zeta_{3d}=500\,cm^{-1}$ yields

$k_{A_{1}}=0.78\,\,cm^{-1}\,.$

Introducing a single reduction factor for the parameters of the spin-orbital
coupling $\kappa$ = 0.86, we obtain $k_{A_{1}}^{*}\approx$ 0.43 $cm^{-1}$,
which nicely agrees with the value calculated from experimental data for
$k_{2}(ac)$ in YFeO3 and other orthoferrites [9].

##  VI Optical anisotropy and anisotropic photoelastic effects in
orthoferrites

Hereafter we will show that simple models of the structure-property
relationships, which were well proved above in the analysis of magnetic and
magnetoelastic anisotropy, can be successfully used to analyze the optical and
photoelastic anisotropy of orthoferrites.

###  VI.1 Natural birefringence of orthoferrites

The analysis of the absorption spectra [52], optical and magnetooptical
anisotropy [53, 54] of orthoferrites in a wide spectral range shows strong
evidence for the key role of the dipole-allowed charge transfer (CT) $p-d$
transitions ${}^{6}A_{1g}\rightarrow{}^{6}T_{1u}$ in the slightly distorted
octahedral complexes FeO6.

Optically, the orthoferrites are biaxial crystals showing a relatively large
natural birefringence [55]. The comparative analysis of the numerical values
and the frequency characteristics of birefringence for rare-earth
orthoferrites shows that the large natural birefringence in orthoferrites at T
= 300 K is mainly due to the $\,3d\,$-sublattice [56]. In particular, the
wavelength dependence of the $\,ab\,$-plane birefringence ($\Delta
n_{ab}\,=\,n_{a}\,-\,n_{b}$) is basically the same in all orthoferrites
including YFeO3 [56]. Optical axes in Eu, Tb, Dy, Yb orthoferrites, and the Y
orthoferrite are inclined in fact at the same angle of $\,\pm\>50\,^{\circ}\,$
to the $\,c\,$-axis ($\lambda\,=\,0.68\,\mu m$) [56, 57].

However, the natural birefringence in the $\,ab\,$-plane of orthoferrites at T
$\approx$ 300 K exhibits a puzzling behavior with a change in sign when
passing from LaFeO3 to LuFeO3 with a more or less regular change in the value
from -4$\cdot 10^{-2}$ to +4$\cdot 10^{-2}$ [55, 56] (see Fig. 8). Such a
behavior can be related with the specific behavior of distortions of the FeO6
octahedra in the series of orthoferrites [58]. Indeed, the linear
birefringence is determined by the anisotropic part of the permittivity
tensor, which, in turn, for the contribution of $p-d$ CT transitions is
determined by the anisotropic part of the FeO6 octahedron polarizability
tensor. Within the "deformation model" the anisotropic symmetric part of the
polarizability tensor for the FeO6 octahedron can be written as follows:

$\alpha_{ij}\,=\,\left\\{\begin{array}[]{cc}p_{E}\,\varepsilon_{ij}\,,&\>i\,=\,j\\\
p_{T_{2}}\,\varepsilon_{ij}\,,&\>i\,\not=\,j\>,\end{array}\right.$ (63)

where $\,\varepsilon_{ij}\,$ is the FeO6-octahedron deformation tensor
($Tr\,\hat{\varepsilon}\,=\,$0); $\>p_{E,\,T_{2}}\,$ are the photoelastic
constants, relating the polarizability to $E\,,T_{2}$ -deformations,
respectively. The relation (63) is valid in the local coordinate system of the
FeO6-octahedron. In the $\,abc\,$-axes system, it can be rewritten as

$\alpha_{ij}\>=\>p_{E}\,\varepsilon_{ij}^{E}\>+\>\,p_{T_{2}}\varepsilon_{ij}^{T_{2}}\>,$
(64)

where $\,\varepsilon_{ij}^{E}\,$ and $\,\varepsilon_{ij}^{T_{2}}\,$ are the
components of the tensor of the $\,E\,$\- and $\,T_{2}\,$-deformations of the
octahedron in the $\,abc\,$\- system, respectively.

Proceeding to the permittivity tensor $\,\hat{\epsilon}\,$ and summing over
all Fe-ions sites, we arrive at nonzero diagonal components of
$\,\hat{\epsilon}\,$:

$\epsilon_{ii}\>=\>P_{E}\varepsilon_{ii}^{E}\>+\>P_{T_{2}}\varepsilon_{ii}^{T_{2}}\>,$
(65)

where $\,P_{E,T_{2}}\,=\,4\pi
N\Bigl{(}\frac{n_{0}^{2}\,+\,2}{3}\Bigr{)}^{2}p_{E,T_{2}}\,$; $\,\>N\,$ is the
number of Fe3+ ions per 1 $\,cm^{3}\,$. Components of
$\,\hat{\varepsilon}^{E},\>\hat{\varepsilon}^{T_{2}}\,$ tensors serve as the
structure factors and may be calculated taking into account the known
components of the tensor of FeO6 octahedron local deformations and the
Eulerian angles relating the local axes to the $\,abc\,$ ones.

Thus, we have a two-parameter formula (65) for the birefringence of
orthoferrites as a function of rhombic distortions of their crystal structure.
The photoelastic constants $\,P_{E},\>P_{T_{2}}\,$ can be found from the
comparison of experimental data [55, 56] with the theoretical structure
dependence of the $\,ab\,$-plane birefringence :

$\Delta
n_{ab}\,=\,n_{a}-n_{b}\,=\,\frac{1}{2n_{0}}\left[P_{E}(\varepsilon_{xx}^{E}\,-\,\varepsilon_{yy}^{E})\,+\,P_{T_{2}}(\varepsilon_{xx}^{T_{2}}\,-\,\varepsilon_{yy}^{T_{2}})\right]$
(66)

treated as a dependence on the type of the orthoferrite. The Figure 8 shows
both experimental and calculated $\,\Delta n_{ab}\,$ given
$\,P_{E}\,=\,6.2\,n_{0}\,,\>P_{T_{2}}\,=\,4.0\,n_{0}\,$ (values obtained from
the least-squares fitting). A cogent agreement of the two-parameter formula
(66) with experiment testifies to the validity of the deformation model of the
birefringence.

Figure 8: Linear birefringeance $\Delta n_{ab}$ for orthoferrites RFeO3 in
$ab$-plane: solid circles are predictions of the deformation model, hollow
circles are experimental data [55] Figure 9: The orientation angles
($\pm\theta$) of optical axes in respective planes of orthoferrites predicted
by the deformation model. The solid black circles are scarce experimental data
for $bc$-plane (see text for detail).

Using the found parameter $\,P_{E,T_{2}}\,$ values, we are able to describe
all the peculiarities of the orthoferrite birefringence. In particular, Fig. 9
shows the theoretical predictions for the orientation angles $\pm\theta$ of
optical axes, measured from the $c$-axis for the $ac$\- and $bc$-planes and
from the $a$-axis for the $ab$-plane, together with scarce experimental data
on Eu, Tb, Dy, Y, Yb orthoferrites [56, 57]. Quite good agreement with the
available experimental data is another confirmation of the validity of the
deformation model of birefringence of orthoferrites. In general, for all its
simplicity, the deformation model reflects quite correctly the main
peculiarities of the natural birefringence of orthoferrites. Moreover, the
deformation model enables us to analyze the photoelastic effects in
orthoferrites.

###  VI.2 Photoelastic effects in orthoferrites

The elastic state of a lattice is characterized by the macroscopic
deformations tensor $\,\hat{\varepsilon}\,$ and by the sublattice
displacements not related to a change of macroscopic crystal sizes, so-called
hidden displacements. The importance of hidden displacements was pointed out
in Refs. [59, 60].

Within a linear approximation the permittivity tensor related to macroscopic
deformations and hidden displacements of Bravais sublattices as follows:

$\epsilon_{ij}\,=\,\epsilon_{ij}^{o}\,+\,P_{ijkl}^{o}\varepsilon_{kl}\,+\,P_{ij}^{n}(\Gamma_{\nu})u_{n}(\Gamma_{\nu})\>,$
(67)

where $\,P_{ijkl}^{o}\,$ and $\,P_{ij}^{n}\,$ are tensors of the photoelastic
constants, $\>\epsilon_{ij}^{o}\,$ is the permittivity tensor in absence of
deformations and displacements.

To compute $\,P_{ijkl}^{o}\,$ and $\,P_{ij}^{n}\,$, one can use the
deformation model of birefringence. The procedure is as follows :

1) local $\,E\,$\- and $\,T_{2}\,$-type deformations of the FeO6 complex as
functions of the macrodeformations $\,\hat{\varepsilon}\,$ and displacements
$\,u_{n}\,$ are to be found ;

2) obtained $\,\varepsilon_{ij}^{E}\,$ and $\,\varepsilon_{ij}^{T_{2}}\,$
values pertaining to the local octahedron axes are to be recalculated for the
$\,abc\,$-system and substituted in (65) ;

3) the resulting linear relation of $\,\epsilon_{ij}\,$ to macrodeformations
and displacements must be compared with (67), all photoelastic constants
$\,P_{ijkl}^{o}\,$ and $\,P_{ij}^{n}\,$ as functions of two parameters –
$\,P_{E}\,$ and $\,P_{T_{2}}\,$ – being hereby determined.

The relation of the tensor of FeO6 complexes microdeformations to
displacements of $\,O^{2-}\,$\- ions is given by Exp. (1):

Photoelastic constants $\,P_{ijkl}^{0}\,$ for YFeO3, calculated in the way
described above are given in Ref. [61]. Assuming that the photoelastic
constants for NdFeO3 and YFeO3 are close in magnitude and vary slightly with
temperature, the authors have evaluated the change of permittivity tensor
components for NdFeO3 as the temperature lowers from 293 K to 8 K. Data of
neutron diffraction study of the NdFeO3 crystal structure [62] and values of
magnetoelastic constants for YFeO3 [61] have been used. The authors have
obtained the following values
:$\quad\Delta\epsilon_{xx}\,=\,-(0.1\,-\,0.6)\cdot
10^{-3}\>n_{0}\,;\quad\Delta\epsilon_{yy}\,=\,+(8.5\,+\,8.6)\cdot
10^{-3}\>n_{0}\,;\quad\Delta\epsilon_{zz}\,=\,-(8.4\,+\,9.3)\cdot
10^{-3}\>n_{0}\,$. Here, the first term is due to the macroscopic
deformations, and the second term is the contribution of hidden displacements
of $\,O^{2-}\,$ ions . Note that in all $\,\hat{\epsilon}\,$ components, the
second term is larger than the first one, i.e., one may not neglect the hidden
displacements contribution.

When the direct action (external with respect to the elastic subsystem) on the
hidden displacements is lacking, i.e., in the free energy of the harmonic
crystal there exist no terms linear in $\,u_{n}\,$, the hidden displacements
are related to macrodeformations (see Exp. (47)). In this case, the third term
in (67) can be reduced to the second term, the photoelastic constants
$\,P_{ijkl}\,$ being thereby renormalized :

$P_{ijkl}\,=\,P_{ijkl}^{0}\,+\,\Delta P_{ijkl}\,;\quad\Delta
P_{ijkl}\,=\,\sum_{n}P_{ij}^{n}(\Gamma_{\nu})A_{kl}^{n}(\Gamma_{\nu})\,.$ (68)

The values of $\,\Delta P_{ijkl}\,$ for YFeO3 were estimated in Ref. [61] with
making use of the results of the model calculation of inner stress tensor
components for TmFeO3. The hidden displacements make an appreciable, and
sometimes leading, contribution to the photoelastic constants.

###  VI.3 Photomagnetoelastic effects in orthoferrites

Minimizing $\,\Phi_{me}\,+\,\Phi_{e}\,$ (see Exps. (45) and (46)) in
$\,\hat{\varepsilon}\,$ and $\,u_{n}\,$, one can determine their equilibrium
values. Substituting these values in (67), we obtain the basic formula for the
analysis of photomagnetoelastic effects [61] :

$\displaystyle\epsilon_{ij}\,=\,\epsilon_{ij}^{0}\,+\,P_{ijkl}\varepsilon_{kl}\,+\,Q_{ij\alpha\beta}G_{\alpha}\,G_{\beta}\>,$
$\displaystyle
Q_{ij\alpha\beta}\,=\,-\,\sum_{\Gamma_{\nu}mn}P_{ij}^{n}(\Gamma_{\nu})\left[C^{-1}(\Gamma_{\nu})\right]^{nm}\Pi_{\alpha\beta}^{m}(\Gamma_{\nu})\,.$
(69)

According to this formula, the photomagnetoelastic effect includes two terms :
the first one, purely $\,magnetostrictive\,$ in nature, is defined by ordinary
photoelastic constants $\,P_{ijkl}\,$; another term is due to the
magnetoelastic displacements [61] and does not depend on the magnetostrictive
deformations. In other words, there may exist the photomagnetoelastic effect
even if the magnetostriction is absent! It is worth noting that, in fact, the
Exp. 69 describes one of the mechanisms of quadratic magnetooptic Cotton-
Mouton effect.

The quantitative evaluation of the photomagnetoelastic constants
$\,Q_{ij\alpha\beta}\,$ is rather complicated. The constants were evaluated in
Ref. [61] using the data of the model calculations of the elastic parameters
$\,C^{nm}\,$, the magnetoelastic parameters $\,B_{\alpha\beta}^{m}\,$, and the
photoelastic parameters $\,P_{ij}^{n}\,$. To show the actual significance of
the magnetoelastooptic effect in orthoferrites, the authors [61] have
calculated the birefringence change in YFeO3 at the spin-reorientation
$\,\Gamma_{4}\,\rightarrow\,\Gamma_{2}\quad(G_{x}\,\rightarrow\,G_{z})\,$
induced by the external magnetic field (${\bf H}\parallel a$-axis). The spin-
reorientation is completed at $\,H\,=\,$75 $\,kOe\,$ and accompanied with
magnetostrictive deformations
$\,\varepsilon_{ii}\,=\,\varepsilon_{ii}^{0}\sin^{2}\theta\,,\quad\varepsilon_{xz}\,=\,\varepsilon_{xz}^{0}\sin\,2\theta\,$,
where $\,\varepsilon_{xx}^{0}\,=\,\varepsilon_{yy}^{0}\,=\,1.8\cdot
10^{-5},\>\varepsilon_{zz}^{0}\,=\,-\,3.7\cdot
10^{-5}\,,\>|\varepsilon_{xz}^{0}|\,=\,0.3\cdot 10^{-5}\,$ [9, 47].

Table 7: Different contributions to the change of birefringeance at the $\Gamma_{4}-\Gamma_{2}$ spin-reorientation in YFeO3 ($\lambda$ = 0.63 $\mu$m, $T$ = 300 K, $\Delta n_{ij}=\Delta(n_{i}-n_{j})$). $\Delta n_{ij},\epsilon_{xz}^{(0)}$ ($\times 10^{-4}$) | $\Delta n_{ab}$ | $\Delta n_{ac}$ | $\Delta n_{bc}$ | $\epsilon_{xz}^{(0)}$
---|---|---|---|---
Magnetostriction | -0.1 | 1.7 | 1.8 |  $\pm$0.3
Hidden displacements | -0.7 | 0.1 | 0.8 |  $\pm$0.7
Total | -0.8 | 1.8 | 2.6 |  $\pm$1.0 ($\pm$0.4)
Experiment [63] | -0.6$\pm$0.4 | 2.0$\pm$0.2 | 2.6$\pm$0.2 |  $\pm$0.7

Table 7 shows the estimations of the purely magnetostrictive contribution, the
magnetoelastic displacements contribution, as well as the total contribution
to $\quad\Delta n_{ab}\,,\>\Delta n_{bc}\,,\>\Delta n_{ac}\>$ $\Bigl{(}$we
denote $\>\Delta n_{ij}\,=\,\Delta\,(n_{i}\,-\,n_{j})\Bigr{)}\,$) as well as
the maximal $\,\epsilon_{xz}\,$ value $\epsilon_{xz}^{0}\,$ at the
$\,\Gamma_{4}\,\rightarrow\,\Gamma_{2}\,$ transition in YFeO3 ($\lambda$ =
0.63 $\mu$m , $T$ = 300 K) as compared with experimental data [63]. Note that
two magnetoelastic terms are comparable in magnitude, and so, both mechanisms
of forming the orthoferrite birefringence should be taken into account. The
theoretical predictions for the total magnetoelastic contribution to the
birefringence change at the $\,\Gamma_{4}\,\rightarrow\,\Gamma_{2}\,$
transition reasonably agree with the experimental data enabling us to draw the
conclusion that the photomagnetoelastic effects are dominant in forming the
magnetic birefringence for the yttrium orthoferrite YFeO3.

Thus, a simple deformation model based on the relation of the FeO6 octahedron
polarization to its deformation permits to explain the observed peculiarities
of the natural birefringence of orthoferrites and to calculate all
photoelastic and photomagnetoelastic effects in orthoferrites. Thus, besides
the lattice macrodeformations, an important role in photoelasto- and
photomagnetoelastic effects belongs to the hidden displacements of
sublattices.

The analysis made can be extended to other compounds, too. Always one should
keep in mind that the information about the photoelastic constants
$\,P_{ijkl}\,$ and macrodeformations $\,\hat{\varepsilon}\,$ of the crystal
is, in general, $\,insufficient\,$ to consider the photoelastic and,
especially, photomagnetoelastic effects. Indeed, the same macrodeformation
ensuing from the external mechanical stress application; the temperature
change with/without the magnetic order alteration; the application of the
external magnetic field can be accompanied with different hidden displacements
of sublattices resulting in different birefringence. These circumstances may
seemingly be a cause of the opposite thermal and pressure behaviour of the
MnF2 birefringence [64] at the same macrodeformation.

##  VII Summary

We applied simple physically clear theoretical approach to evaluate the
interplay between FeO6 octahedral distortions/rotations in rare-earth
orthoferrites and main magnetic and optic characteristics such as
superexchange integral and Néel temperature, overt and hidden canting of
magnetic sublattices, magnetic and magnetoelastic anisotropy, optic and
photoelastic anisotropy. Most attention in the paper focused on the
Dzyaloshinskii vector, its value, orientation, and sense. Our analysis once
again confirms the unambiguous leading role of antisymmetric exchange in the
formation of overt and hidden canting in orthoferrites and the fallacy of the
argumentation of the authors of the recent paper [6].

The model approach developed in this work goes far beyond the scope of only
orthoferrites. Our analysis revealed previously underestimated relationships
that can be used not only to elucidate the mechanisms of formation of various
physical properties, but also to design electronic structures for advanced
materials. Importantly, the relationships established with these model
approaches may be cross-validated by the construction of hybrid data sets,
which combine theoretical results with experiment data, making it possible to
extract and validate new insight into the material physics of strongly
correlated oxides [4]. We anticipate that this approach will spawn a number of
additional studies for perovskites and other crystals since it is immediately
generalizable: the synergy of simple cluster models with subsequent first-
principles calculations provides a platform to achieve rational structure-
driven design of complex materials. A good understanding of the structure-
property relationships can be used to develop new functional materials and
devices.

###### Acknowledgements.

I thank E.V. Sinitsyn and I.G. Bostrem for very fruitful multi-year
collaboration, I.E. Dzyaloshinskii, V.I. Ozhogin, R.E. Walstedt, S.V. Maleev,
B.Z. Malkin, B.S. Tsukerblatt, S.-L. Drechsler, and V.E. Dmitrienko for
stimulating and encouraging discussions. This study was supported by the
Ministry of Science and Higher Education of the Russian Federation, project
FEUZ-2023-0017.

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