Moskvin

# Dzyaloshinskii interaction and exchange-relativistic effects in
orthoferrites

A. S<EMAIL_ADDRESS>Ural Federal University, 620083, Ekaterinburg,
Russia  Institute of Metal Physics UB RAS, 620108, Ekaterinburg, Russia

###### Abstract

We present an overview of the microscopic theory of the Dzyaloshinskii-Moriya
(DM) coupling and related exchange-relativistic effects such as exchange
anisotropy, electron-nuclear antisymmetric supertransferred hyperfine
interactions, antisymmetric magnetogyrotropic effects, and antisymmetric
magnetoelectric coupling in strongly correlated 3$d$ compounds focusing on
orthoferrites RFeO3 (R is a rare-earth ion or yttrium Y). Most attention in
the paper 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. Microscopically derived
expression for the dependence of the Dzyaloshinskii vector on the
superexchange geometry allows one to find all the overt and hidden canting
angles in orthoferrites RFeO3 as well as corresponding contribution to
magnetic anisotropy. Being based on the theoretical predictions regarding the
sign of the Dzyaloshinskii vector we have predicted and study in detail a
novel magnetic phenomenon, weak ferrimagnetism in mixed weak ferromagnets with
competing signs of the Dzyaloshinskii vectors. The ligand NMR measurements in
weak ferromagnets are shown to be an effective tool to inspect the effects of
DM coupling in an external magnetic field. Along with orthoferrites RFeO3 and
weak ferrimagnets RFe1-xCrxO3, although to a lesser extent, we address such
typical weak ferromagnets as $\alpha$-Fe2O3, FeBO3, and FeF3.

##  1 Introduction

It is not often in the history of science that one paper of an author opens up
a novel field of theoretical and experimental research. This is exactly what
happened with the article by Igor Dzyaloshinskii [1], devoted to the
explanation of the phenomenon of weak ferromagnetism.

More than a hundred years have passed after T. Smith [2] found in 1916 a weak,
or $parasitic$ ferromagnetism in an "international family line" of different
natural hematite $\alpha$-Fe2O3 single crystalline samples from Italy,
Hungary, Brasil, and Russia (Schabry, Ural Mountains, near Ekaterinburg) that
was assigned to ferromagnetic impurities. Later the phenomenon was observed in
many other materials, in fluoride NiF2 with rutile structure, in the
orthorhombic orthoferrites RFeO3 (where R is a rare-earth element or Y), in
rhombohedral antiferromagnets MnCO3, NiCO3, CoCO3, and FeBO3. However, only in
1954 L. M. Matarrese and J. W. Stout for NiF2 [3] and in 1956 A. S. Borovik-
Romanov and M. P. Orlova for very pure synthesised carbonates MnCO3 and CoCO3
[4] have firmly established that weak ferromagnetism is observed in chemically
pure antiferromagnetic materials and therefore it is an intrinsic property of
some antiferromagnets, the connexion between the weak ferromagnetism and any
impurities or inhomogeneities seems very unlikely. Furthermore, Borovik-
Romanov and Orlova assigned the uncompensated moment in MnCO3 and CoCO3 to an
overt canting of the two magnetic sublattices in almost antiferromagnetic
matrix. The model of a canted antiferromagnet became generally adopted model
of the weak ferromagnet.

A theoretical explanation and first thermodynamic theory for weak
ferromagnetism in $\alpha$-Fe2O3, MnCO3, and CoCO3 was provided by I. E.
Dzyaloshinskii (Dzialoshinskii, Dzyaloshinsky) [1] 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

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

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}=\\\ $ $+2MH_{D}(m_{x}l_{y}-m_{y}l_{x})$ (2)

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

$E_{D}=d_{1}m_{z}l_{x}+d_{2}m_{x}l_{z}=$
$\frac{d_{1}-d_{2}}{2}(m_{z}l_{x}-m_{x}l_{z})+\frac{d_{1}+d_{2}}{2}(m_{z}l_{x}+m_{x}l_{z})=$
$-2MH_{D}[{\bf m}\times{\bf
l}]_{y}+\frac{d_{1}+d_{2}}{2}(m_{z}l_{x}+m_{x}l_{z})\,,$ (3)

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 [5] suggested a model microscopic theory
of the exchange-relativistic antisymmetric exchange interaction to be a main
contributing mechanism of weak ferromagnetism

$V_{DM}=\sum_{mn}({\bf d}_{mn}\cdot\left[{\bf S}_{m}\times{\bf
S}_{n}\right])\,,$ (4)

now called Dzyaloshinskii-Moriya (DM) spin coupling. Here, ${\bf d}_{mn}$ is
the axial Dzyaloshinskii vector. Presently Keffer [6] 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):

$\mathbf{d}_{ij}\propto[\mathbf{r}_{i}\times\mathbf{r}_{j}]\,,$ (5)

where ${\bf r}_{i,j}$ are unit radius vectors for O - Mi,j bonds with
presumably equal bond lenghts. Later on Moskvin [7] derived a microscopic
formula for Dzyaloshinskii vector

$\mathbf{d}_{ij}=d_{ij}(\theta)[\mathbf{r}_{i}\times\mathbf{r}_{j}]\,,$ (6)

where

$d_{ij}(\theta)=d_{1}(R_{i},R_{j})+d_{2}(R_{i},R_{j})cos\theta_{ij}$ (7)

with $\theta_{ij}$ being the Mi \- O - Mj bonding angle. The sign of the
scalar parameter $d_{ij}(\theta)$ can be addressed to be the sign (sense) of
the Dzyaloshinskii vector. The formula (6) was shown to work only for $S$-type
magnetic ions with orbitally nondegenerate ground state, e.g. for 3$d$-ions
with half-filled shells (3$d^{5}$ or $t_{2g}^{3}$, $t_{2g}^{3}e_{g}^{2}$,
$t_{2g}^{6}e_{g}^{2}$).

It should be noted that sometimes instead of (6) one may use another form of
the structural factor for the Dzyaloshinskii vector:

$\left[{\bf r}_{1}\times{\bf r}_{2}\right]=\frac{1}{2}\left[({\bf r}_{1}-{\bf
r}_{2})\times({\bf r}_{1}+{\bf r}_{2})\right]=\frac{1}{2l^{2}}\left[{\bf
R}_{12}\times{\boldsymbol{\rho}}_{12}\right]\,,$ (8)

where ${\bf R}_{12}={\bf R}_{1}-{\bf R}_{2}$, ${\boldsymbol{\rho}}_{12}=({\bf
R}_{1}+{\bf R}_{2})$.

Figure 1: (Color online) Superexchange geometry and the Dzyaloshinskii vector. Table 1: Main exchange and DM coupling parameters in orthoferrites compared with other weak ferromagnets (WFMs), $I$ is the exchange integral, $\alpha_{D}$ is the canting angle. See text for detail. WFM | RFeO, Å | $\theta$ | TN, K |  $I$, K (MFA) | HE, Tesla | $\alpha_{D}$ | HD, Tesla |  $d$($\theta$), K
---|---|---|---|---|---|---|---|---
YFeO3 | 2.001 (x2) | 145∘ | 640 | 36.6 | 640 | $1.1\cdot 10^{-2}$ | 14 | 3.2
$\alpha$-Fe2O3 | 2.111 | 145∘ | 948 | 54.2 | 870-920 | $1.1\cdot 10^{-3}$ | 1.9-2.2 | 2.3
FeBO3 | 2.028 | 126∘ | 348 | 19.9 | 300 | $1.7\cdot 10^{-2}$ | 10 | 2.3
FeF3 | 1.914 | 153∘ | 363 | 20.7 | 440 | $5.5\cdot 10^{-3}$ | 4.88 | 1.1

Starting with the pioneering papers by Dzyaloshinskii [1] and Moriya [5] the
DM coupling was extensively investigated in 60-80ths in connection with the
weak ferromagnetism focusing on hematite $\alpha$-Fe2O3 and orthoferrites
RFeO3 [8, 9, 10, 11] (see, also review articles Refs. [12, 13]). Typical
values of the canting angle $\alpha_{D}$ turned out to be on the order of
0.001-0.01, in particular, $\alpha_{D}=1.1\cdot 10^{-3}$ in $\alpha$-Fe2O3
[14], 2.2-2.9$\cdot 10^{-3}$ in La2CuO4 [15], $5.5\cdot 10^{-3}$ in FeF3 [16],
$1.1\cdot 10^{-2}$ in YFeO3 [17], $1.7\cdot 10^{-2}$ in FeBO3 [18] (see Table
1).

V. I. Ozhogin et al. [19] in 1968 first raised the issue of the sign of the
Dzyaloshinskii vector, however, only in 1990 the reliable local information on
the sign of the Dzyaloshinskii vector, or to be exact, that of the
Dzyaloshinskii parameter $d_{12}$, was first extracted from the 19F ligand NMR
(nuclear magnetic resonance) data in weak ferromagnet FeF3 [20]. In 1977 we
have shown that the Dzyaloshinskii vectors can be of opposite sign for
different pairs of $S$-type ions [10] that allowed us to uncover a novel
magnetic phenomenon, weak ferrimagnetism, and a novel class of magnetic
materials, weak ferrimagnets, which are systems such as solid solutions
YFe1-xCrxO3 with competing signs of the Dzyaloshinskii vectors and the very
unusual concentration and temperature dependence of the magnetization[21]. The
relation between Dzyaloshinskii vector and the superexchange geometry (6)
allowed us to find numerically all the overt and hidden canting angles in the
rare-earth orthoferrites [9] that was nicely confirmed in 57Fe NMR [22] and
neutron diffraction [23] measurements.

The stimulus to a renewed interest to the subject was given by the cuprate
problem, in particular, by the weak ferromagnetism observed in the parent
cuprate La2CuO4 [15] and many other interesting effects for the DM systems, in
particular, the "field-induced gap" phenomena [24]. At variance with typical
3D systems such as orthoferrites, the cuprates are characterised by a low-
dimensionality, large diversity of Cu-O-Cu bonds including corner- and edge-
sharing, different ladder configurations, strong quantum effects for $s=1/2$
Cu2+ centers, and a particularly strong Cu-O covalency resulting in a
comparable magnitude of hole charge/spin densities on copper and oxygen sites.
Several groups (see, e.g., Refs.[25, 26, 27]) developed the microscopic model
approach by Moriya for different 1D and 2D cuprates, making use of different
perturbation schemes, different types of low-symmetry crystalline field,
different approaches to intra-atomic electron-electron repulsion. However,
despite a rather large number of publications and hot debates (see, e.g.,
Ref.[28]) the problem of exchange-relativistic effects, that is of
antisymmetric exchange and related problem of spin anisotropy in cuprates
remains to be open (see, e.g., Refs.[13, 29, 30, 31] for recent experimental
data and discussion). Common shortcomings of current approaches to DM coupling
in 3$d$ oxides concern a problem of allocation of the Dzyaloshinskii vector
and respective "weak" (anti)ferromagnetic moments, and full neglect of spin-
orbital effects for "nonmagnetic" oxygen O2- ions, which are usually believed
to play only indirect intervening role. From the other hand, the oxygen 17O
NMR-NQR studies of weak ferromagnet La2CuO4[32, 33] seem to evidence
unconventional local oxygen "weak-ferromagnetic" polarization whose origin
cannot be explained in frames of current models.

All the systems described above were somehow or other connected with
insulating weak ferromagnets where DM coupling manifests itself in the canting
of a basic antiferromagnetic structure. However, DM coupling can induce
helimagnetic distortion of the ferromagnetic order as in caesium cupric
chloride, CsCuCl3 to be a unique screw antiferroelectric crystal (see, e.g.,
Ref. [13] and references therein). In fact, it is known for a long time that
the DM coupling can produce long-period magnetic spiral structures in
ferromagnetic and antiferromagnetic crystals lacking inversion symmetry. This
effect was suggested for metallic MnSi and other crystals with B20 structure,
and it has been carefully proved that the sign of the DM coupling, hence the
sign of the spin helix, is determined by the crystal handedness.

Phenomenologically antisymmetric DM coupling in a continual approximation
gives rise to so-called Lifshitz invariants, energy contributions linear in
first spatial derivatives of the magnetization ${\bf m}({\bf r})$ [34]

$m_{i}\frac{\partial m_{j}}{\partial x_{l}}-m_{j}\frac{\partial
m_{i}}{\partial x_{l}}$ (9)

($x_{l}$ is a spatial coordinate). These chiral interactions derived from the
DM coupling stabilize localized (vortices) and spatially modulated structures
with a fixed rotation sense of the magnetization [35]. In fact, these are the
only mechanism to induce nanosize skyrmion structures in condensed matter.
Despite a clear weakness of the typical DM coupling as compared with typical
isotropic exchange interactions the DM coupling can be a central ingredient in
the stabilization of complex magnetic textures.

The DM coupling contribution to a micromagnetic free energy density $F({\bf
r})$ is usually represented as follows

$F({\bf r})=\sum_{i}{\bf D}_{i}({\bf m}({\bf r}))\cdot\left[{\bf m}({\bf
r})\times\frac{\partial{\bf m}({\bf r})}{\partial r_{i}}\right]\,,$ (10)

where the Dzyaloshinskii vectors ${\bf D}_{i}({\bf m}({\bf r}))$ are
considered generally to depend on magnetization direction ${\bf m}({\bf r})$
[36]. Within ab initio density functional theory (DFT) methods, DM coupling is
often computed by adding spin-orbital interaction perturbatively to spirals
with finite wavevectors ${\bf q}$ and extracting $D_{j}$ from the $q$-linear
term in the dispersion $E({\bf q})$. However, nearly exclusively, theoretical
studies in this context were in the past bound to pure spin models without
itinerancy, leaving the impact of charge fluctuations aside.

In recent years interest has shifted towards other manifestation of the DM
coupling, such as the magnetoelectric effect [37, 38, 39], where reliable
theoretical predictions have been lacking. All this stimulated the critical
revisit of many old approaches to the spin-orbital effects in 3$d$ oxides,
starting from the choice of proper perturbation scheme and effective spin
Hamiltonian model, implied usually only indirect intervening role played by
"nonmagnetic" oxygen O2- ions.

In this paper we present an overview of the microscopic theory of the DM
coupling and other related exchange-relativistic effects focusing on
orthoferrites RFeO3. The rest part of the paper is organized as follows. In
Sec. 2 we shortly address main results of the microscopic theory of the
isotropic superexchange interactions for so-called $S$-type ions focusing on
the angular dependence of the exchange integrals. Sec. 3 is devoted to
microscopic theory of the DM coupling. Starting with Moriya’s theory we arrive
at a more comprehensive derivation of the Dzyaloshinskii vector for the
$S$-type 3$d$-ions, its value, orientation, and sense (sign) under different
types of the (super)exchange interaction and crystal field. Here we consider
the DM coupling with participation of rare-earth ions. Theoretical predictions
of this section are compared in Sec. 4 with experimental data for the overt
and hidden canting, as well as magnetic anisotropy in orthoferrites. In Sec. 5
we address unconventional properties of weak ferrimagnetism as a novel type of
magnetic ordering in systems with competing signs of the Dzyaloshinskii
vector, in particular, features of the 4$f$ \- 3$d$ interaction in weak
ferrimagnets RFe1-xCrxO3, and unconventional spin-reorientation transitions in
weak ferrimagnets. In Sec. 6 we discuss several experimental tools to examine
the sign of the Dzyaloshinskii vector, including $\mu$SR of positive muons and
ligand NMR in weak ferromagnets. Last part of the article is devoted to
related exchange-relativistic effects, in particular, to exchange-relativistic
two-ion anisotropy (Sec. 7), antisymmetric supertransferred hyperfine
interaction as electron-nuclear counterpart of DM coupling (Sec. 8),
antisymmetric exchange-relativistic spin-other-orbit coupling determining
unconventional magnetooptics of weak ferromagnets (Sec. 9), and antisymmetric
exchange-relativistic spin-dependent electric polarization (Sec. 10). A brief
conclusion is made in Sec. 11.

##  2 Microscopic theory of the isotropic superexchange coupling

DM coupling is derived from the off-diagonal (super)exchange coupling and does
usually accompany a conventional (diagonal) Heisenberg type isotropic
(super)exchange coupling:

${\hat{V}}_{ex}=J_{12}({\bf S}_{1}\cdot{\bf S}_{2})\,.$ (11)

The modern microscopic theory of the (super)exchange coupling had been
elaborated by many physicists starting with well-known papers by P. Anderson
[40], especially intensively in 1960-70th (see review articles [41]). Numerous
papers devoted to the problem pointed to existence of many hardly estimated
exchange mechanisms, seemingly comparable in value, in particular, for
superexchange via intermediate ligand ion to be the most interesting for
strongly correlated systems such as 3$d$ oxides. Unfortunately, up to now we
have no reliable estimations of the exchange parameters, though from the other
hand we have no reliable experimental information about their magnitudes. To
that end, many efforts were focused on the fundamental points such as many-
electron theory and orbital dependence [42, 43, 7, 44], crystal-field effects
[45], off-diagonal exchange [46], exchange in excited states [47], angular
dependence of the superexchange coupling [7]. The irreducible tensor operators
(the Racah algebra) were shown to be very instructive tool both for
description and analysis of the exchange coupling in the 3$d$\- and
4$f$-systems [42, 43, 7, 44, 45].

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 [7] under simplified assumptions. For the $S$-ions with
configuration 3$d^{5}$ (Fe3+, Mn2+) the angular dependence of the
superexchange integral is

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

where parameters $a$, $b$, $c$ depend on the cation-ligand separation.
Parameters $a$ and $c$ are related with the contribution of the intermediate
ligand 2$p$ shell, while the $\propto\cos\theta_{12}$ term is related with the
low-energy ligand inter-configurational $2p\rightarrow 3s$ excitations.

Later on the derivation had been generalized for the 3$d$ ions in a strong
cubic crystal field [11]. Orbitally isotropic contribution to the exchange
integral for pair of 3$d$-ions with configurations
$t_{2g}^{n_{1}}e_{g}^{n_{2}}$ can be written as follows

$J=\sum_{\gamma_{i},\gamma_{j}}J(\gamma_{i}\gamma_{j})\,(g_{\gamma_{i}}-1)\,(g_{\gamma_{j}}-1)\,,$
(13)

where $g_{\gamma_{i}},g_{\gamma_{j}}$ are effective "$g$-factors" of the
$\gamma_{i},\gamma_{j}$ subshells of ion 1 and 2, respectively:

$g_{\gamma_{i}}=1+\frac{S(S+1)+S_{i}(S_{i}+1)-S_{j}(S_{j}+1)}{2S(S+1)}\,.$
(14)

Kinetic exchange contribution to partial exchange parameters
$I(\gamma_{i}\gamma_{j})$ related with the electron transfer to partially
filled shells can be written as follows [45, 11]

$J(e_{g}e_{g})=\frac{(t_{ss}+t_{\sigma\sigma}cos\theta)^{2}}{2U};\,J(e_{g}t_{2g})=\frac{t_{\sigma\pi}^{2}}{3U}sin^{2}\theta;$
$J(t_{2g}t_{2g})=\frac{2t_{\pi\pi}^{2}}{9U}(2-sin^{2}\theta)\,,$ (15)

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 [48] has a similar angular dependence [47].

Figure 2: Dependence of the Fe3+ \- Fe3+, Cr3+ \- Cr3+, Fe3+ \- Cr3+ exchange
integrals on the superexchange bond angle in orthoferrites-orthocromites [49].

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

$t_{\sigma\sigma}\,:\,t_{\pi\sigma}\,:\,t_{\pi\pi}\,:\,t_{ss}\approx\lambda_{\sigma}^{2}\,:\,\lambda_{\pi}\lambda_{\sigma}\,:\,\lambda_{\pi}^{2}\,:\,\lambda_{s}^{2}\,,$
(16)

where $\lambda_{\sigma},\lambda_{\pi},\lambda_{s}$ are covalency parameters.
The simplified kinetic exchange contribution (15) 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

$\Delta J_{CrCr}=-\frac{\Delta
E(35)}{6U}\frac{t_{\sigma\pi}^{2}}{U}sin^{2}\theta\,;$ $\Delta
J_{FeCr}=-\frac{\Delta
E(35)}{10U}\left[\frac{(t_{ss}+t_{\sigma\sigma}cos\theta)^{2}}{U}+\frac{t_{\sigma\pi}^{2}}{U}sin^{2}\theta\right]\,,$
(17)

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 $\theta$ = $\theta_{cr}$:

$sin^{2}\theta_{cr}=\frac{1}{\left(\frac{1}{2}+\frac{3}{8}\frac{\Delta
E(35)}{U}\frac{t^{2}_{\sigma\pi}}{t^{2}_{\pi\pi}}\right)}\,.$ (18)

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 [49] (see Fig. 2). 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 (mean-field approximation) relation

$T_{N}=\frac{zS(S+1)}{3k_{B}}J\,,$ (19)

however, this relation yields the exchange integrals ($J$ = 36.8 K in YFeO3)
that can be one and a half or even twice less than the values obtained by
other methods [11, 50, 51]. Most recent precise experimental data yield for
YFeO3 $J_{c}$ = 58.2 K, $J_{ab}$ = 53.6 K [52] or $J_{c}$ = $J_{ab}$ = 51.5 K
[53].

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 $d$-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 [54]:

$\frac{\partial\ln J}{\partial\ln R}=\frac{\partial J}{\partial
R}/\frac{J}{R}\approx-\,10\,.$ (20)

##  3 Microscopic theory of the DM coupling

###  3.1 Moriya’s theory

First microscopic theory of weak ferromagnetism, or theory of anisotropic
superexchange interaction was provided by Moriya [5] who extended the Anderson
theory of superexchange to include spin-orbital coupling
$V_{so}=\sum_{i}\xi({\bf l}_{i}\cdot{\bf s}_{i})$. Moriya started with the
one-electron Hamiltonian for $d$-electrons as follows

${\hat{H}}=\sum_{fm\sigma}\epsilon_{m}{\hat{d}}^{\dagger}_{fm\sigma}{\hat{d}}_{fm\sigma}+\sum_{m\not=m^{\prime},\sigma}t_{fmf^{\prime}m^{\prime}}{\hat{d}}^{\dagger}_{fm\sigma}{\hat{d}}_{f^{\prime}m^{\prime}\sigma}+$
$\sum_{fm\not=f^{\prime}m^{\prime},\sigma\sigma^{\prime}}{\hat{d}}^{\dagger}_{fm\sigma}({\bf
C}_{fmf^{\prime}m^{\prime}}\cdot{\boldsymbol{\sigma}}){\hat{d}}_{f^{\prime}m^{\prime}\sigma^{\prime}}\,,$
(21)

where

${\bf
C}_{fmf^{\prime}m^{\prime}}=-\frac{\xi}{2}\sum_{m^{\prime\prime}}\left(\frac{{\bf
l}_{fmfm^{\prime\prime}}t_{fm^{\prime\prime}f^{\prime}m^{\prime}}}{\epsilon_{m^{\prime\prime}}-\epsilon_{m}}+\frac{t_{fmf^{\prime}m^{\prime\prime}}{\bf
l}_{f^{\prime}m^{\prime\prime}f^{\prime}m^{\prime}}}{\epsilon_{m^{\prime\prime}}-\epsilon_{m^{\prime}}}\right)$
(22)

is a spin-orbital correction to transfer integral. Then Moriya did calculate
the generalized Anderson kinetic exchange that contains both conventional
isotropic exchange and anisotropic symmetric and antisymmetric terms, that is
quasidipole anisotropy and DM coupling, respectively. The expression for the
Dzyaloshinskii vector

${\bf
d}_{ff^{\prime}}=\frac{4i}{U}\sum_{m\not=m^{\prime}}\left[t_{fmf^{\prime}m^{\prime}}{\bf
C}_{f^{\prime}m^{\prime}fm}-{\bf
C}_{fmf^{\prime}m^{\prime}}t_{f^{\prime}m^{\prime}fm}\right]\,.$ (23)

has been obtained by Moriya assuming orbitally nondegenerate ground states $m$
and $m^{\prime}$ on sites $f$ and $f^{\prime}$, respectively. It is worth
noting that the spin-operator form of the DM coupling follows from the
relation:

${\bf S}_{1}({\bf S}_{1}\cdot{\bf S}_{2})+({\bf S}_{1}\cdot{\bf S}_{2}){\bf
S}_{2}=-i[{\bf S}_{1}\times{\bf S}_{2}]\,.$ (24)

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.

Despite its seeming simplicity the operator form of the DM coupling (4) raises
some questions and doubts. First, at variance with the scalar product
$\left({\bf S}_{1}\cdot{\bf S}_{2}\right)$ the vector product of the spin
operators $\left[{\bf S}_{1}\times{\bf S}_{2}\right]$ changes the spin
multiplicity, that is the net spin ${\bf S}_{12}={\bf S}_{1}+{\bf S}_{2}$,
that underscores the need for quantum description. 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 [55]. It seems the ${\bf D}$-vector does not
transform as a vector at all.

Another issue that causes some concern is the structure and location of the
${\bf D}$ vector and corresponding spin cantings. Obviously, the ${\bf
D}_{12}$ vector should be related in one or another way to spin-orbital
contributions localized on sites 1 and 2, respectively. These components may
differ in their magnitude and direction, while the operator form (4) implies
some averaging both for ${\bf D}_{12}$ vector and spin canting between the two
sites.

Moriya did not take into account the effects of the crystal field symmetry and
strength and did not specify the character of the (super)exchange coupling,
that, as we’ll see below, can crucially affect the direction and value of the
Dzyaloshinskii vector up to its vanishing. Furthermore, he made use of a very
simplified form (22) of the spin-orbital perturbation correction to the
transfer integral (see Exp. (2.5) in Ref. [5]). The fact is that the structure
of the charge transfer matrix elements implies the involvement of several
different on-site configurations ($t_{kn}\propto\langle
N_{1}-1N_{2}+1|{\hat{H}}|N_{1}N_{2}\rangle$). Hence, the perturbation
correction has to be more complicated than (22), at least, it should involve
the spin-orbital matrix elements (and excitation energies!) for one- and two-
particle configurations. As a result, it does invalidate the author’s
conclusion about the equivalence of the two perturbation schemes, based on the
$V_{SO}$ corrections to the transfer integral and to the exchange coupling,
respectively. Another limitation of the Moriya’s theory is related to a full
neglect of the ligand spin-orbital contribution to DM coupling. Despite these
shortcomings the Moriya’s estimation for the ratio between the magnitudes of
the Dzyaloshinskii vector $d=|{\bf d}|$ and isotropic exchange $J$:
$d/J\approx\Delta g/g$, where $g$ is the gyromagnetic ratio, $\Delta g$ is its
deviation from the free-electron value, respectively, in some cases may be
helpful, however, only for a very rough estimation.

###  3.2 Microscopic theory of the DM coupling: superexchange interaction of
the $S$-ions

The first microscopic theory of the DM coupling for the superexchange bond
M1–O–M2 proposed by the author [7] (see also Refs. [11, 10, 56]) assumed the
interaction of "free" ions with the ground ${}^{6}S$ state of the 3$d^{5}$
configuration (Mn2+, Fe3+), interacting through an intermediate anion of the
O2- type. Final expression for the Dzyaloshinskii vector was obtained as
follows

$\mathbf{d}_{12}=d_{12}(\theta_{12})[\mathbf{r}_{1}\times\mathbf{r}_{2}]\,,$
(25)

with

$d_{12}(\theta)=d_{1}(R_{10},R_{20})+d_{2}(R_{10},R_{20})cos\theta_{12}\,,$
(26)

where the first and the second terms are determined by the superexchange
mechanisms related with the ligand inter-configurational 2$p\rightarrow 3s$
excitations and intra-configurational 2$p$–2$p$ effects, respectively. It
should be noted that given $\theta$ = $\theta_{cr}$, where

$cos\,\theta_{cr}=-d_{1}/d_{2}$ (27)

the Dzyaloshinskii vector changes its sign.

However, later it was shown [11, 10, 56] that the correct theory of the
Dzyaloshinskii interaction even for $S$-type ions should take into account the
crystal field effects.

As the most illustrative example we consider a pair of 3$d^{5}$ ions such as
Fe3+, or Mn2+ with the ground state ${}^{6}S$ in an $intermediate$ octahedral
crystal field which does split the ${}^{2S+1}L$ terms into crystal
${}^{2S+1}L\Gamma$ terms and mix the crystal terms with the same octahedral
symmetry, that is with the same $\Gamma$’s [57]. Spin-orbital coupling does
mix the ${}^{6}S$ ground state with the ${}^{4}PT_{1g}$ term, however the
${}^{4}PT_{1g}$ term has been mixed with other ${}^{4}T_{1g}$ terms,
${}^{4}FT_{1g}$ and ${}^{4}GT_{1g}$. Namely the latter effect appears to be a
decisive factor for appearance of the DM coupling. The $|4(L)T_{1g}\rangle$
wave functions can be easily calculated by a standard technique [57] as
follows [11]:

$|4(P)T_{1g}\rangle=0.679|4PT_{1g}\rangle-0.604|4FT_{1g}\rangle+0.418|4GT_{1g}\rangle\,;$
$|4(F)T_{1g}\rangle=0.387|4PT_{1g}\rangle+0.777|4FT_{1g}\rangle+0.495|4GT_{1g}\rangle\,;$
$|4(G)T_{1g}\rangle=-0.604|4PT_{1g}\rangle-0.169|4FT_{1g}\rangle+0.737|4GT_{1g}\rangle\,,$
(28)

given the crystal field and intra-atomic correlation parameters [57] typical
for orthoferrites [58]: 10Dq = 12200 $cm^{-1}$; B = 700 $cm^{-1}$; C = 2600
$cm^{-1}$. We see that due to the crystal-field mixing effect all the three
crystal terms ${}^{4}PT_{1g}$, ${}^{4}FT_{1g}$, and ${}^{4}CT_{1g}$ will
contribute to the DM coupling. Furthermore, the overall nonzero contribution
will be determined by the ${}^{4}P-{}^{4}G$ mixing [56].

Table 2: Wave functions and energies for the 4T1g terms for Fe3+ ion in orthoferrites Wave function | Energy, cm-1
---|---
$|{}^{4}T_{1g}(41)\rangle=0.988|t_{2g}^{4}e_{g}^{1}{}^{4}T_{1g}\rangle-0.123|t_{2g}^{3}e_{g}^{2}{}^{4}T_{1g}\rangle+0.088|t_{2g}^{2}e_{g}^{3}{}^{4}T_{1g}\rangle$ | $E(41)=0.96\cdot 10^{4}$
$|{}^{4}T_{1g}(32)\rangle=0.058|t_{2g}^{4}e_{g}^{1}{}^{4}T_{1g}\rangle+0.844|t_{2g}^{3}e_{g}^{2}{}^{4}T_{1g}\rangle-0.534|t_{2g}^{2}e_{g}^{3}{}^{4}T_{1g}\rangle$ | $E(32)=2.96\cdot 10^{4}$
$|{}^{4}T_{1g}(23)\rangle=-0.140|t_{2g}^{4}e_{g}^{1}{}^{4}T_{1g}\rangle-0.522|t_{2g}^{3}e_{g}^{2}{}^{4}T_{1g}\rangle+0.841|t_{2g}^{2}e_{g}^{3}{}^{4}T_{1g}\rangle$ | $E(23)=3.69\cdot 10^{4}$

However, it is more appropriate to consider the interaction of 3$d$ ions in a
$strong$ crystal field scheme. Hereafter we address the DM coupling for the
$S$-type magnetic 3$d$ ions with orbitally nondegenerate high-spin ground
state in a strong cubic crystal field, that is for the 3$d$ 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
[11, 10, 56]. In particular, for the ${}^{4}T_{1g}$ terms of the 3$d^{5}$ ion
in the strong cubic crystal field approximation instead of expressions (28) we
arrive at the wave functions of the $t_{2g}^{n_{1}}e_{g}^{n_{2}}$
configurations as shown in Table 2.

Making use of expressions for spin-orbital coupling $V_{SO}$ and main kinetic
contribution to the superexchange parameters, that define DM coupling after
routine algebra we have found that the DM coupling can be written in a
standard form (25), where $d_{12}$ can be written as follows [10, 11, 56]

$d_{12}=X_{1}Y_{2}+X_{2}Y_{1}\,,$ (29)

where the $X$ and $Y$ factors do reflect the exchange-relativistic structure
of the second-order perturbation theory and details of the electron
configuration for $S$-type ion. The exchange factors $X$ are

$X_{i}=\frac{(g^{(i)}_{e_{g}}-1)}{2U}t_{\pi\sigma}(t_{ss}+t_{\sigma\sigma}cos\theta)-\frac{(g^{(i)}_{t_{2g}}-1)}{3U}t_{\pi\pi}t_{\sigma\pi}cos\theta\,,$
(30)

where $g^{(i)}_{e_{g}}$, $g^{(i)}_{t_{2g}}$ are effective $g$-factors for
$e_{g}$, ${t_{2g}}$ subshells, respectively,
$t_{\sigma\sigma}>t_{\pi\sigma}>t_{\pi\pi}>t_{ss}$ are positive definite
$d$–$d$ transfer integrals, $U$ is a $d$–$d$ transfer energy (correlation
energy). The general form of the dimensionless factors $Y$ determined by spin-
orbital constants and excitation energies is more complicated (see, e.g., Ref.
[56]). The both factors $X$ and $Y$ are presented in Table 3 for $S$-type
3$d$-ions, where $\xi_{3d}$ is the spin-orbital parameter, $\Delta
E_{{}^{2S+1}\Gamma}$ is the energy of the excited ${}^{2S+1}\Gamma$ crystal
term interacting with the ground state due to $V_{SO}$.

The signs for $X$ and $Y$ factors in Table 3 are predicted for rather large
superexchange bonding angles $|cos\theta_{12}|>t_{ss}/t_{\sigma\sigma}$ which
are typical for many 3$d$ 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 3$d^{5}$ configurations. Excited configuration $t_{2g}^{2}e_{g}^{3}$ does
not contribute to the DM coupling for 3$d^{5}$ ions.

It is worth noting that earlier we have detected and corrected a casual and
unintentional error in sign of the $X_{i}$ parameters having made both in our
earlier papers [10, 11] and recent paper Ref. [12]. Hereafter we present
correct signs for $X_{i}$ in (30) and Table 3 [56].

Rather simple expressions for the factors $X_{i}$ and $Y_{i}$ do not take into
account the mixing/interaction effects for the ${}^{2S+1}\Gamma$ terms with
the same symmetry and the contribution of empty subshells to the exchange
coupling (see Ref. [11]). Nevertheless, the data in Table 3 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. [59]) 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

$|X_{d^{8}}|\geq|X_{d^{3}}|\geq|X_{d^{5}}|\,,$ (31)

however, it should be underlined its sensitivity both to superexchange
geometry and covalency parameters. Simple comparison of the exchange
parameters $X$(see (30) and Table 3) with exchange parameters
$I(\gamma_{i}\gamma_{j})$ (15) evidences their close magnitudes. Furthermore,
the relation (16) allows us to maintain more definite correspondence.

Given typical values of the cubic crystal field parameter $10Dq\approx$ 1.5 eV
we arrive at a relation among different $Y$’s [11]

$|Y_{d^{8}}|\geq|Y_{d^{5}}|\geq|Y_{d^{3}}|$ (32)

with $Y_{d^{8}}\approx 7.0\cdot 10^{-2}$, $Y_{d^{5}}\approx-2.5\cdot 10^{-2}$,
$Y_{d^{3}}\approx 1.5\cdot 10^{-2}$.

Table 3: Expressions for the $X$ and $Y$ parameters that define the magnitude and the sign of the Dzyaloshinskii vector in pairs of the $S$-type 3$d$-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
| 3$d^{3}$($t_{2g}^{3}$):${}^{4}A_{2g}$
---
V2+, Cr3+, Mn4+
$-\frac{1}{3U}t_{\pi\pi}t_{\sigma\pi}cos\theta$ | + | $\frac{2\xi_{3d}}{3\sqrt{3}}(\frac{1}{\Delta E_{{}^{4}T_{2g}}}+\frac{2}{\Delta E_{{}^{2}T_{2g}}})$ | + | $t_{2g}^{2}e_{g}^{1}$
| 3$d^{5}$($t_{2g}^{3}e_{g}^{2}$):${}^{6}A_{1g}$
---
Mn2+, Fe3+
|  $-\frac{1}{5U}(t_{\pi\pi}t_{\sigma\pi}cos\theta$ -
---
$t_{\pi\sigma}\left(t_{ss}+t_{\sigma\sigma}cos\theta)\right)$
– | –$\frac{6\xi_{3d}}{5\sqrt{3}}(\frac{1}{\Delta E_{{}^{4}T_{1g}}(41)}-\frac{1}{\Delta E_{{}^{4}T_{1g}}(23)})$ | – |  $t_{2g}^{4}e_{g}^{1}$, $t_{2g}^{2}e_{g}^{3}$
| 3$d^{8}$($t_{2g}^{6}e_{g}^{2}$):${}^{3}A_{2g}$
---
Ni2+, Cu3+
$\frac{1}{2U}t_{\pi\sigma}(t_{ss}+t_{\sigma\sigma}cos\theta)$ | – | $\frac{3\xi_{3d}}{2\sqrt{3}}(\frac{1}{\Delta E_{{}^{3}T_{2g}}}+\frac{1}{\Delta E_{{}^{1}T_{2g}}})$ | + | $t_{2g}^{5}e_{g}^{3}$

The greatest value of the $d_{12}$ factor is predicted for $d^{8}$ \- $d^{8}$
pairs, while for $d^{5}$ \- $d^{5}$ pairs one expects a much less (may be one
order of magnitude) value. The $d_{12}$ factor for $d^{3}$ \- $d^{3}$ pairs is
predicted to be somewhat above the value for $d^{5}$ \- $d^{5}$ pairs. For
different pairs: $d_{12}$($d^{3}$ \- $d^{5}$)$\approx$ -$d_{12}$($d^{3}$ \-
$d^{3}$); $d_{12}$($d^{8}$ \- $d^{5}$)$\approx$ $d_{12}$($d^{5}$ \- $d^{5}$);
$d_{12}$($d^{3}$ \- $d^{8}$)$\geq$ $d_{12}$($d^{3}$ \- $d^{3}$). Puzzlingly,
that despite strong isotropic exchange coupling for $d^{5}$ \- $d^{5}$ and
$d^{5}$ \- $d^{8}$ pairs, the DM coupling for these pairs is expected to be
the least one among the $S$-type pairs. For $d^{5}$ \- $d^{5}$ pairs, in
particular, Fe3+ \- Fe3+ we have two compensation effects. First, the
$\sigma$-bonding contribution to the $X$ parameter is partially compensated by
the $\pi$-bonding contribution, second, the contribution of the ${}^{4}T_{1g}$
term of the $t_{2g}^{4}e_{g}^{1}$ configuration is partially compensated by
the contribution of the ${}^{4}T_{1g}$ term of the $t_{2g}^{2}e_{g}^{3}$
configuration.

Theoretical predictions of the corrected sign of the Dzyaloshinskii vector in
pairs of the $S$-type 3$d$-ions with local octahedral symmetry (the sign
rules) are presented in Table 4. 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 will be addressed below (Sec. 5) in more detail.

Table 4: Theoretical predictions of the sign of the Dzyaloshinskii vector in pairs of the $S$-type 3$d$-ions with local octahedral symmetry and the bonding angle $\theta>\theta_{cr}$. 3$d^{n}$ | 3$d^{3}$($t_{2g}^{3}$) | 3$d^{5}$($t_{2g}^{3}e_{g}^{2}$) | 3$d^{8}$($t_{2g}^{6}e_{g}^{2}$)
---|---|---|---
3$d^{3}$($t_{2g}^{3}$) | + | – | +
3$d^{5}$($t_{2g}^{3}e_{g}^{2}$) | – | + | +
3$d^{8}$($t_{2g}^{6}e_{g}^{2}$) | + | + | –

###  3.3 DM coupling in trigonal hematite $\alpha$-Fe2O3 and borate FeBO3

Making use of our theory based on the bare ideal octahedral symmetry of
$S$-type ions to the classical weak ferromagnet $\alpha$-Fe2O3 we arrive at a
little unexpected disappointment, as the theory does predict that the
contribution of the three equivalent Fe3+ \- O2- \- Fe3+ superexchange pathes
for the two corner shared FeO${}_{6}^{9-}$ octahedrons to the net
Dzyaloshinskii vector strictly turns into zero. Exactly the same result will
be obtained, if we consider the direct Fe3+ \- Fe3+ exchange in the system of
two ideal FeO${}_{6}^{9-}$ octahedrons bonded through the three common oxygen
ions when ${\bf R}_{12}\parallel C_{3}$. Obviously, it is precisely this fact
that caused a tiny spin canting in hematite being an order of magnitude
smaller than, e.g., in orthoferrites RFeO3 or borate FeBO3. So what was the
real reason of weak ferromagnetism in $\alpha$-Fe2O3 as "opening a new page of
weak ferromagnetism"? What is a microscopic origin of nonzero Dzyaloshinskii
vector which should be directed along the $C_{3}$ symmetry axis according
Moriya rules? First of all we should consider trigonal distortions for the
FeO${}_{6}^{9-}$ octahedrons which have a $T_{2}$ symmetry and give rise to a
mixing of the ${}^{4}T_{1g}$ terms with ${}^{4}A_{2g}$ and ${}^{4}T_{2g}$
terms. The best way to solve the problem in principle is to proceed with a
coordinate system where $O_{z}$ axis is directed along the $C_{3}$ symmetry
axis rather than with the usually applied $O_{z}\parallel C_{4}$ geometry
[56]. Symmetry analysis shows that the axial distortion along the Fe3+ \- Fe3+
bond can induce the DM coupling with Dzyaloshinskii vector directed along the
bond, however, only for locally nonequivalent Fe3+ centers, otherwise we
arrive at an exact compensation of the contributions of the spin-orbital
couplings on sites 1 and 2 [56].

Trigonal hematite $\alpha$-Fe2O3 has the same crystal symmetry
$R\overline{3}c-D_{3d}^{6}$ as weak ferromagnet FeBO3. Furthermore, the borate
can be transformed into hematite by the Fe3+ ion substitution for B3+ with a
displacement of both "old" and "new" iron ions along trigonal axis. As a
result we arrive at emergence of an additional strong isotropic
(super)exchange coupling of three-corner-shared non-centrosymmetric FeO6
octahedra with short Fe-O separations (1.942 Å) that determines very high Néel
temperature $T_{N}$ = 948 K in hematite as compared with $T_{N}$ = 348 K in
borate. However, the $D_{3h}$ symmetry of these exchange bonds points to a
distinct compensation of the two Fe-ion’s contribution to Dzyaloshinskii
vector. In other words, weak ferromagnetism in hematite $\alpha$-Fe2O3 is
determined by the DM coupling for the same Fe-O-Fe bonds as in borate FeBO3.
However, the Fe-O separations for these bonds in hematite (2.111 Å) are
markedly longer than in borate (2.028 Å) that points to a significantly weaker
DM coupling. Combination of the weaker DM coupling and stronger isotropic
exchange in $\alpha$-Fe2O3 as compared with FeBO3 does explain the one order
of magnitude difference in canting angles.

Table 5: The structural factors $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{x,y,z}$ for the superexchange coupled Fe1 \- O - Fe2,4 pairs in orthoferrites with numerical values for YFeO3, $a,b,c$ are lattice parameters, $l$ is a mean Fe - O separation, $x_{1},y_{1}$, $x_{2},y_{2},z_{2}$ are oxygen parameters for the O(4c) and O(8d) positions, respectively.  | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{x}$ | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{y}$ | $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{z}$
---|---|---|---
2 |  $\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}$ = 0.20 | -$\frac{x_{1}ac}{2l^{2}}$ = -0.55 | 0
4 |  $\pm\frac{z_{2}bc}{2l^{2}}$ = $\pm$ 0.31 |  $-\frac{z_{2}ac}{2l^{2}}$ = -0.29 |  $\frac{(y_{2}-x_{2}+\frac{1}{2})ab}{2l^{2}}$ = 0.41

###  3.4 DM coupling with participation of rare-earth ions

Spin-orbital interaction for the rare-earth ions with valent $4f^{n}$
configuration is diagonalized within the $(LS)J$ multiplets hence the
conventional DM coupling

${\hat{H}}_{DM}^{ff}=\sum_{m>n}({\bf d}_{mn}\cdot[{\bf S}_{m}\times{\bf
S}_{n}])=$ $\sum_{m>n}(g_{m}-1)(g_{n}-1)({\bf d}_{mn}\cdot[{\bf
J}_{m}\times{\bf J}_{n}])$ (33)

($g_{m,n}$ are the Lande factors) can arise for the $f$–$f$ superexchange only
due to a spin-orbital contribution on intermediate ligands. Obviously, for the
rare-earth-3$d$-ion (super)exchange we have an additional contribution of the
3$d$-ion spin-orbital interaction. The rare-earth-3$d$-ion DM coupling R3+ \-
O2- \- Fe3+ (R=Nd, Gd)

${\hat{H}}_{DM}^{fd}=\sum_{m>n}({\bf d}_{mn}\cdot[{\bf J}_{m}\times{\bf
S}_{n}])$ (34)

has been theoretically and experimentally considered in Refs. [60, 61].

Effective field at the R ion can be written as a sum of ferro- and
antiferromagnetic contributions:

${\bf H}_{fd}=\alpha{\bf
F}+{\bf\stackrel{{\scriptstyle\leftrightarrow}}{{\beta}}}{\bf G}\,,$ (35)

where $\alpha$ determines the contribution of the isotropic $f$ \- $d$
exchange, while tensor
${\bf\stackrel{{\scriptstyle\leftrightarrow}}{{\beta}}}$ does the contribution
of symmetric and antisymmetric anisotropic $f$ \- $d$ interactions. These
interactions were studied in GdFeO3 [60], and the authors found that

$\alpha F_{z}=-0.19\,;$
$\beta_{zx}G_{x}=(\beta_{zx}^{s}+\beta_{zx}^{a})G_{x}=-0.05+0.21=0.16\,,$ (36)

(in Tesla). Quite unexpected, the antisymmetric antiferromagnetic contribution
$\beta_{zx}^{a}G_{x}$ to effective field on Gd3+ ion in $\Gamma_{4}$ magnetic
phase at T = 0 K, which is determined by the $f$ \- $d$ DM coupling appears to
be the leading one. Moreover, according to the data from Ref. [61], in GdCrO3

$\alpha F_{z}=-0.13\,;$
$\beta_{zx}G_{x}=(\beta_{zx}^{s}+\beta_{zx}^{a})G_{x}=+0.05-0.47=-0.42\,$ (37)

that is the $f$ \- $d$ DM coupling has value greater than in the GdFeO3,
however, with opposite sign, which is consistent with the different sign of
the factor $Y$ for Fe3+ and Cr3+ (see Table 3).

Figure 3: (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.
[53].

##  4 Theoretical predictions as compared with experiment

###  4.1 Dzyaloshinskii interaction in orthoferrites

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

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

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}\,;\\\ $ $4S{\bf A}={\bf S}_{1}-{\bf
S}_{2}-{\bf S}_{3}+{\bf S}_{4}\,.$ (38)

Here ${\bf G}$ describes the main antiferromagnetic component, ${\bf F}$ 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. 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. [62, 53]), in which
case the basis vectors ${\bf G}$, ${\bf C}$, ${\bf A}$ may differ in sign.

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

${\hat{H}}=\sum_{i>j}J_{ij}({\bf S}_{i}\cdot{\bf S}_{j})+\sum_{i>j}({\bf
d}_{ij}\cdot[{\bf S}_{i}\times{\bf S}_{j}])$ (39)

can be written in terms of basis vectors as free energy as follows (see, e.g.,
Ref. [11] and references therein)

$\Phi=\frac{J_{F}}{2}{\bf F}^{2}+\frac{J_{G}}{2}{\bf
G}^{2}+\frac{J_{C}}{2}{\bf C}^{2}+\frac{J_{A}}{2}{\bf A}^{2}+$
$D_{x}(C_{y}G_{z}-C_{z}G_{y})+D_{y}(F_{z}G_{x}-F_{x}G_{z})+D_{z}(A_{x}G_{y}-A_{y}G_{x})$
$d_{x}(A_{y}F_{z}-A_{z}F_{y})+d_{y}(C_{z}A_{x}-C_{x}A_{z})+d_{z}(C_{x}F_{y}-C_{y}F_{x})\,,$
(40)

where for the energy per ion

$J_{F}=-J_{G}=S^{2}(2J_{ab}+J_{c});J_{A}=-J_{C}=S^{2}(2J_{ab}-J_{c});$
$D_{x}=-S^{2}\sum_{2}d_{x}(12);$
$D_{y}=-S^{2}\left(\sum_{4}d_{y}(14)+\sum_{2}d_{y}(12)\right);$
$D_{z}=-S^{2}\sum_{4}d_{z}(14);$ (41)

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

$F_{z}=-\frac{D_{y}}{J_{F}-J_{G}}G_{x};\,\,A_{y}=\frac{D_{z}}{J_{A}-J_{G}}G_{x};$
$F_{x}=\frac{D_{y}}{J_{F}-J_{G}}G_{z};\,\,C_{y}=-\frac{D_{x}}{J_{C}-J_{G}}G_{z};$
$A_{x}=-\frac{D_{z}}{J_{A}-J_{G}}G_{y};\,\,C_{z}=\frac{D_{x}}{J_{C}-J_{G}}G_{y};$
(42)

###  4.2 Overt and hidden canting in orthoferrites

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.

Figure 4: Basic vectors of magnetic structure for 3$d$ sublattice in
orthoferrites RFeO3 and orthochromites RCrO3

In Table 5 we present structural factors $\left[{\bf r}_{1}\times{\bf
r}_{2}\right]_{x,y,z}$ for the superexchange coupling of the Fe3+ ion in
position (1/2, 0, 0) with nearest neighbors in orthoferrites with numerical
values for YFeO3. 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.

In 1975 we made use of simple formula for the Dzyaloshinskii vector (6) and
structural factors from Table 5 to find a relation between crystallographic
and canted magnetic structures for four-sublattice’s orthoferrites RFeO3 and
orthochromites RCrO3 [9, 11] (see Fig. 4), 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!):

$F_{x}=\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{J}G_{z}\,;\,F_{z}=-\frac{(x_{1}+2z_{2})ac}{6l^{2}}\frac{d}{J}G_{x}\,;\,$
$A_{x}=\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{J}G_{y};A_{y}=-\frac{(\frac{1}{2}+y_{2}-x_{2})ab}{2l^{2}}\frac{d}{J}G_{x};$
$C_{y}=\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{J}G_{z}\,;\,C_{z}=-\frac{(\frac{1}{2}-y_{1})bc}{2l^{2}}\frac{d}{J}G_{y}\,,$
(43)

where $a,b,c$ are unit cell parameters, $x_{1,2},y_{1,2},z_{2}$ are oxygen
($O_{I,II}$) parameters, $l$ 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 $|A_{x,y}|>|F_{x,z}|>|C_{y,z}|$.

Table 6: Hidden canting in orthoferrites. Orthoferrite | Ay/Fz, theory [9] | Ay/Fz, exp | Ay/Cy, theory [9] | Ay/Cy, exp
---|---|---|---|---
YFeO$3$ | 1.10 |  | 1.10 $\pm$ 0.03[22]
---
1.23 $\pm$ 0.2[23]
1.1 $\pm$ 0.1[63]
1.14 [52]
1.03 [53]
2.04 | ?
HoFeO$3$ | 1.16 | 0.85 $\pm$ 0.10[63] | 2.00 | ?
TmFeO$3$ | 1.10 | 1.25 $\pm$ 0.05[22] | 1.83 | ?
YbFeO$3$ | 1.11 | 1.22 $\pm$ 0.05[23] | 1.79 | 2.0 $\pm$ 0.2[22]

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

$M_{Fe}=\frac{4g\mu_{B}S}{\rho V}|F_{x,z}|=\frac{2g\mu_{B}Sac}{3l^{2}\rho
V}(x_{1}+2z_{2})\frac{d(\theta)}{J(\theta)}\,,$ (44)

where $\rho$ and $V$ are the unit cell density and volume, respectively. The
overt canting $F_{x,z}$ can be calculated through the ratio of the
Dzyaloshinskii ($H_{D}$) and exchange ($H_{E}$) fields as follows

$F=H_{D}/2H_{E}\,.$ (45)

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

$H_{D}=\frac{S}{g\mu_{B}}\sum_{i}|d_{y}(1i)|=\frac{S}{g\mu_{B}}(x_{1}+2z_{2})\frac{ac}{l^{2}}|d(\theta)|\,,$
(46)

that yields $|d(\theta)|\cong$ 3.2 K = 0.28 meV in YFeO3 given $H_{D}$ = 140
kOe [17]. This value is in good agreement with the data of recent experiments
[52, 53] which make it possible to obtain information on the Dzyaloshinskii
vectors based on measurements of the spin-wave spectrum. 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 6 the
theoretically predicted relations between overt and hidden canting nicely
agree with the experimental data obtained for different orthoferrites by NMR
[22], neutron diffraction, measurement of the low-energy spin excitations by
inelastic neutron scattering and by absorption of THz radiation [23, 63, 52,
53].

###  4.3 The DM coupling and effective magnetic anisotropy

Hereafter we demonstrate a contribution of the DM coupling into effective
magnetic anisotropy in orthoferrites. The classical energies of the three spin
configurations in orthoferrites $\Gamma_{1}(A_{x},G_{y},C_{z})$,
$\Gamma_{2}(F_{x},C_{y},G_{z})$, 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]

$\displaystyle
E_{\Gamma_{1}}=E_{G}-48JS^{2}F^{2}\left[\frac{1}{3}(\frac{C}{F})^{2}+\frac{2}{3}(\frac{A}{F})^{2}\right]\,;$
(47) $\displaystyle
E_{\Gamma_{2}}=E_{G}-48JS^{2}F^{2}\left[1+\frac{1}{3}(\frac{C}{F})^{2}\right]\,;$
(48) $\displaystyle
E_{\Gamma_{4}}=E_{G}-48JS^{2}F^{2}\left[1+\frac{2}{3}(\frac{A}{F})^{2}\right]\,,$
(49)

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
$E_{an}=k_{1}\,cos2\theta$ ($ac$, $bc$ planes), $E_{an}=k_{1}\,cos2\varphi$
($ab$ plane): $k_{1}(ac)=\frac{1}{2}(E_{\Gamma_{2}}-E_{\Gamma_{4}})$;
$k_{1}(bc)=\frac{1}{2}(E_{\Gamma_{2}}-E_{\Gamma_{1}})$;
$k_{1}(ab)=\frac{1}{2}(E_{\Gamma_{4}}-E_{\Gamma_{1}})$. Detailed analysis of
different mechanisms of the magnetic anisotropy of the orthoferrites [11, 64]
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 [17]. 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, 64]. 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.
[65]).

Figure 5: (Color online) a) Concentration dependence of the magnetization and
Fe-, Cr- partial contributions in YFe0.5Cr0.5O3; b) The MFA phase diagram of
weak ferrimagnet YFe1-xCrxO3 given $\delta$ = -4; left and right arrows
demonstrate the orientation and magnitude of the magnetization for Fe- and Cr-
sublattices, respectively. The outer and inner thin curves mark the
compensation points for the net and partial (Fe, Cr) magnetization,
respectively. Experimental values of $T_{N}$ for single crystalline and
polycrystalline samples are marked by light and dark circles, respectively. c)
Concentration dependence of the low-temperature magnetization in YFe1-xCrxO3:
experimental data (circles) [21], the MFA calculations given $\delta$ = -2 and
-4; c) ; d) Temperature dependence of magnetization in YFe1-xCrxO3: solid
curves – experimental data for x = 0.38 (Kadomtseva et al., 1977 [66] – curve
1) and for x = 0.4 (Dasari et al., 2012 [71] – curve 2), dotted curve – the
MFA calculation for x = 0.4 [71] given $d_{FeCr}$ = -0.39 K

##  5 Weak ferrimagnetism as a novel type of magnetic ordering in systems with
competing signs of the Dzyaloshinskii vector

First experimental studies of mixed orthoferrites-orthochromites YFe1-xCrxO3
[21] performed in Moscow State University did confirm theoretical predictions
regarding the signs of the Dzyaloshinskii vectors and revealed the weak
ferrimagnetic behavior due to a competition of Fe-Fe, Cr-Cr, and Fe-Cr DM
coupling with antiparallel orientation of the mean weak ferromagnetic moments
of Fe and Cr subsystems in a wide concentration range. In other words, we have
predicted a novel class of mixed 3$d$ systems with competing signs of the
Dzyaloshinskii vector, so called weak ferrimagnets. Transversal weak
ferromagnetic moment of the Cr3+ impurity ion in orthoferrite YFeO3 can be
evaluated as follows

${\bf m}_{Cr}=g\mu_{B}S_{Cr}(2\delta-1){\bf F}\,,$ (50)

where dimensionless parameter

$\delta=\frac{d_{CrFe}}{d_{FeFe}}\frac{I_{FeFe}}{I_{CrFe}}$ (51)

does characterize a relative magnitude of the impurity-matrix DM coupling. By
comparing $m_{Cr}$ with that of the matrix value ${\bf
m}_{Fe}=g\mu_{B}S_{Fe}{\bf F}$ we see that the weak ferromagnetic moment for
the Cr impurity is antiparallel to that of the Fe matrix even for positive but
small $\delta<1/2$. However, the effect is more pronounced for negative
$\delta$, that is for different signs of $d_{CrFe}$ and $d_{FeFe}$.

Figure 6:  (Color online) a) Concentration dependence of the low-temperature
($T$ = 77 K) magnetization in LuFe1-xCrxO3: experimental data (circles) [67],
the MFA calculations (solid curve) given $\delta$ = -1.5. (b) Temperature
dependence of magnetization in LuFe1-xCrxO3: circles – experimental data [67]
given $x$ = 0.6 (1), 0.5 (2), 0.2 (3), 0.1 (4), 0.0 (5), solid curves – the
MFA calculations given $\delta$ = -1.5.

Results of a simple mean-field calculation presented in Figs.5-7 show that the
weak ferrimagnets such as RFe1-xCrxO3, Mn1-xNixCO3, Fe1-xCrxBO3 [21, 66, 67,
68, 69, 70] do reveal very nontrivial concentration and temperature
dependencies of magnetization, in particular, the compensation point(s).

In Fig. 5a,b,c we do present the MFA "weak ferrimagnetic" phase diagram,
concentration dependence of the low-temperature net magnetization, and the Fe,
Cr partial contributions in YFe1-xCrxO3 calculated at constant value of
$\delta$ = -4. Comparison with experimental data for the low-temperature net
magnetization [21] and the MFA calculations with $\delta$ = -2 (Fig. 5c)
points to a reasonably well agreement everywhere except $x\sim$ 0.5, where
$\delta$ parameter seems to be closer to $\delta$ = -3. In Fig. 5d we compare
first pioneering experimental data for the temperature dependence of
magnetization $m(T)$ in weak ferrimagnet YFe1-xCrxO3 ($x$ = 0.38) (Kadomtseva
et al., 1978 [66] – curve 1) with recent data for a close composition with $x$
= 0.4 (Dasari et al., 2012 [71] – curve 2). It is worth noting that recent MFA
calculations by Dasari et al. [71] given $d_{FeCr}$ = -0.39 K provide very
nice description of $m(T)$ at $x$ = 0.4. Note that the authors [71] found a
rather strong dependence of the $d_{FeCr}$ parameter on the concentration $x$.
The concentration and temperature dependencies of magnetization in
LuFe1-xCrxO3 are nicely described by a simple MFA model under the assumption
of constant sign magnetization given constant value of $\delta$ = -1.5 (Fig.
6a,b [67]), which, strictly speaking, did not exclude the possibility of an
alternative description of the dependence $m(x)$ with two points of
concentration compensation of magnetization (see dotted line in Fig. 6a).
Furthermore, strictly speaking, the absence of compensation concentration
points for low-temperature magnetization $m(x,T=77K)$ does not mean the
absence of compensation points at higher temperatures. Indeed, much later, in
2016, Pomiro et al. [72] observed the spontaneous magnetization reversal in
polycrystalline LuFe0.5Cr0.5O3 below TN = 290 K at a compensation temperature
Tcomp = 224 K and Billoni et al. [73] have performed more advanced classical
Monte Carlo simulations for RFe1-xCrxO3 with R = Y and Lu, comparing the
numerical simulations with experiments and MFA calculations. In addition to
the dependence T${}_{N}(x)$, this model is able to reproduce the magnetization
reversal (MR) observed experimentally in a field cooling process for
intermediate $x$ values.

Figure 7: a) The MFA simulation of the T-x-phase diagram of the weak
ferro(ferri)magnet Fe1-xCrxBO3 [68] given $I_{FeFe}=I_{FeCr}$= -20.3 K,
$I_{CrCr}$ = 2.0 K, arrows point to orientation of the net weak ferromagnetic
moment. Curves 1, 2, 3 mark the compensation points given $d_{z}(FeFe)$ =
$d_{z}(CrCr)$ = 0.67 K, $d_{z}(FeCr)$ = -0.67 K (1), -0.75 K (2), -0.90 K (3),
respectively. b) The MFA simulation of the temperature dependence of the net
magnetization in Fe1-xCrxBO3 [68] given $d_{z}(FeFe)$ = $d_{z}(CrCr)$ =
-$d_{z}(FeCr)$ = 0.67 K at different compositions, the insert shows
experimental data from Ref. [77] taken at external magnetic field 1 T. c) The
MFA simulation of the concentration dependence of the low-temperature
magnetization in Mn1-xNixCO3 [70] given $d_{z}(MnNi)>d_{z}^{(0)}(MnNi)$ and
$d_{z}(MnNi)<d_{z}^{(0)}(MnNi)$, respectively.

At variance with YFeO3 and YCrO3 which are weak ferromagnets with main GxFz-
type magnetic structure below TN, the orthoferrites-orthochromites
YFe1-xCrxO3, which referred as weak ferrimagnets, reveal full or partial GxFz–
GzFx type spin-reorientation in a wide range of substitution. This unexpected
behavior which is usually typical for orthoferrites with magnetic rare-earth
ions (Er, Tm, Dy,) was attributed mainly to a strong decrease of the DM
contribution to magnetic anisotropy in the $ac$-plane for $x$ = 0.5–0.6 [74,
75]. In contrast to the yttrium system, the lutetium orthoferrite-
orthocromites LuFe1-xCrxO3 (x = 0.0, 0.1, 0.2, 0.5, 0.6, and 1.0) reveal the
main GxFz type magnetic structure without signatures of the spontaneous spin-
reorientation transition. This difference can be explained by the
significantly larger contribution of single-ion anisotropy to $k_{ac}$ in
LuFeO3 as compared to YFeO3 [11, 76].

Let us turn to the features of other weak ferrimagnets. Fig. 7b shows a
calculated phase diagram of the trigonal weak ferrimagnet Fe1-xCrxBO3 [68].
Temperature-dependent magnetization studies from 4.2 to 600 K have been made
for the solid solution system Fe1-xCrxBO3 where $0\leq x\leq$ 95 [77]. A rapid
decrease is observed in the saturation magnetization with increasing $x$ at
4.2 K up to 0.40, after which a broad compositional minimum is found up to
x=0.60. Compositions in the range of 0.40$\leq x\leq$0.60 display unusual
magnetization behavior as a function of temperature in that maxima and minima
are present in the curves below the Curie temperatures. Fig. 7b shows a nice
agreement between experimental data [77] and our MFA calculations.

At variance with the $d^{5}$ \- $d^{3}$ (Fe-Cr) mixed systems such as
YFe1-xCrxO3 or Fe1-xCrxBO3 the manifestation of different DM couplings Fe-Fe,
Cr-Cr, and Fe-Cr in (Fe1-xCrx)2O3 is all the more surprising because of
different magnetic structures of the end compositions, $\alpha$-Fe2O3 and
Cr2O3 and emergence of a nonzero DM coupling for the three-corner-shared FeO6
and CrO6 octahedra, "forbidden" for Fe-Fe and Cr-Cr bonding. All this makes
magnetic properties of mixed compositions (Fe1-xCrx)2O3 to be very unusual
[78].

Unlike the $d^{5}$ \- $d^{3}$ (Fe-Cr) mixed systems YFe1-xCrxO3 or
Fe1-xCrxBO3, where the two concentration compensation points do occur given
rather large $d_{FeCr}$ parameter, in the $d^{5}$ \- $d^{8}$ systems, the
nickel and fluorine substituted orthoferrites RFe1-xNixFyO3-y [69] or
Mn1-xNixCO3 with Mn2+ \- Ni2+ pairs [70] we have the only concentration
compensation point irrespective of the $d_{MnNi}$ parameter. However, the
character of the concentration dependence of the weak ferrimagnetic moment
$m(x)$ depends strongly on its magnitude. Given the increasing concentration
the $m(x)$ is first rising or falling with $x$ depending on whether $d_{MnNi}$
greater than, or less than
$d_{MnNi}^{(0)}=(1+\frac{S_{Mn}}{S_{Ni}})\frac{I_{MnNi}}{2I_{MnMn}}d_{MnMn}$.
Figure (7c) does clearly demonstrate this feature.

It should be noted that just recently Dmitrienko et al. [79] have first
discovered experimentally that in accordance with our theory (see Table 4) the
sign of the Dzyaloshinskii vector in MnCO3 ($d^{5}$ \- $d^{5}$) coincides with
that one in FeBO3 ($d^{5}$ \- $d^{5}$) , whereas NiCO3 ($d^{8}$ \- $d^{8}$)
demonstrates the opposite sign.

###  5.1 Features of the 4$f$ \- 3$d$ interaction in weak ferrimagnets
RFe1-xCrxO3

It is undoubtedly of interest to investigate the influence of the weak
ferrimagnetic ordering of the 3$d$-sublattice on the behavior of the rare-
earth subsystem in mixed ferrites-chromites RFe1-xCrxO3. The character of
polarization of the R-ions and its concentration and temperature dependencies
yield valuable information not only on the state of the $d$-subsystem, but
also on the 4$f$ \- 3$d$ interaction mechanisms, primarily on the relative
roles of the ferro- and antiferromagnetic contributions to the effective field
at the R-ions [61, 80, 75, 81]. Of particular interest, in our opinion, is the
GdFe1-xCrxO3 system with $S$-type 4$f$\- and 3$d$-ions, where it might seem
that it is precisely the ferromagnetic contribution due to the isotropic 4$f$
\- 3$d$ exchange should play the decisive role in the polarization of the Gd
sublattice. At the same time, a detailed analysis of the magnetic properties
of GdFeO3 and GdCrO3 [60, 61] has quite unexpectedly revealed the substantial
role of the anisotropic exchange of the $S$-ions Gd3+ with $S$-type ions Fe3+
and Cr3+ and accordingly of the antiferromagnetic contribution to the
polarization of the Gd sublattices with the predominance of antisymmetric term
which is determined by the 4$f$ \- 3$d$ DM coupling (see (36) and (37)).

Figure 8: (Color online) Left panel: Calculated concentration dependence of
the average effective field on the Gd3+ ions in the weak ferrimagnet at T = 0
K. Right panel: Calculated temperature dependencies of the spontaneous
magnetization for a number of weak ferrimagnets GdFe1-xCrxO3 and end
compositions, $\tau$ = T/TN. Solid circles are experimental data for single
crystals GdFe0.83Cr0.17O3 with TN = 550 K [61].

Knowledge of the numerical values of the parameters of isotropic and
anisotropic 4$f$ \- 3$d$ interaction has enabled us to calculate, within the
framework of the molecular-field theory, the concentration and temperature
dependencies of the average effective field $H_{z}$, the magnetization of the
Gd sublattice, and the total magnetization of GdFe1-xCrxO3 in the entire range
of concentrations $x$ (see Fig.8) [61]. The exchange integrals and the DM
coupling parameters in the $d$ sublattices were chosen equal to the
corresponding values for weak ferrimagnet YFe1-xCrxO3. The concentration
dependence $H_{z}(x)$ at T = 0 K has very unusual form with two compensation
points, at small and relatively large concentrations of the Cr3+ ions. Whereas
at $x\approx$ 0.05 the compensation of the total magnetic moment is still
observed, at $x$ = 0.10 the reversal of the sign of $H_{z}$ leads to a
hyperbolic increase of $m(T)$ in the low-temperature region. At the same time
the calculation shows that at $x\approx$ 0.27 and $\tau\approx$ 0.17 we arrive
at the compensation point, which then "bifurcates" with one (high-temperature)
compensation point moving towards TN, with increasing $x$, while the other
(low temperature) towards T = 0 K. At $x$ > 0.5 the compensation points
vanish. Only for compositions directly adjacent to pure gadolinium
orthochromite is the compensation again observed, and with increasing
concentration of the Fe3+ ions the compensation point shifts from Tcomp = 110
K in pure GdCrO3, to Tcomp =TN, at $x\approx$ 0.95. As a whole the calculated
concentration and temperature dependencies of the magnetization in
GdFe1-xCrxO3 agree satisfactorily with the experimental data [61]. Finally, we
note the need for further experimental investigation of the rare-earth weak
ferrimagnets such as GdFe1-xCrxO3 both from the viewpoint of studying various
$f$ \- $d$ interactions, and of the possibility of obtaining novel advanced
magnetic properties.

###  5.2 Unconventional spin-reorientation in weak ferrimagnets

The contribution of the competing antisymmetric exchange to the magnetic
anisotropy of weak ferrimagnets has an unusual concentration dependence. So,
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
induce a spin-reorientation $\Gamma_{4}$ – $\Gamma_{2}$ transition which are
typical for several orthoferrites RFeO3 with magnetic rare-earth ions (R = Nd,
Sm, Tb, Ho, Er, Tm, Yb). In Fig. 9 we demonstrate concentration dependence of
the DM coupling contribution to first anisotropy constant for YFe1-xCrxO3 in
$ac$-plane given different values of the parameter $\delta$, which was
calculated within simple mean-field approximation [74, 75] in the limit of low
temperatures. 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. Furthermore, similarly
magnetization, this contribution to anisotropy has a specific temperature
dependence [74, 75]. On the whole, both effects can lead to the appearance of
spontaneous spin-reorientation transitions in weak ferrimagnets of the
YFe1-xCrxO3 type with a nonmagnetic "R"-ion. Indeed, in full accordance with
the theory, such transitions have been observed experimentally, for example,
the $\Gamma_{4}$ – $\Gamma_{2}$ spin-reorientation transition in
YFe0.85Cr0.15O3 [74, 75] (see Fig. 9).

Figure 9: Concentration dependence of the DM coupling contribution to the
first anisotropy constant in $ac$-plane given different values of the
parameter $\delta$. Inset: Temperature dependence of the magnetization in the
weak ferrimagnet YFe0.85Cr0.15O3 [75] demonstrating the $\Gamma_{4}$ –
$\Gamma_{2}$ spin-reorientation transition in the temperature range 240 - 400
K.

A more surprising situation was found in the weak ferrimagnet DyFe1-xCrxO3 at
a relatively low concentration of Cr ions. The Dy3+ ions in DyFeO3 stabilize
the $\Gamma_{1}(G_{y})$ configuration, so that at T = 40 K, a jump-like Morin
transition $\Gamma_{4}$ – $\Gamma_{1}$ is observed. In all single crystals of
weak ferrimagnets DyFe1-xCrxO3 ($x$ = 0.07, 0.10, 0.13, 0.15, 0.36, 0.40)
synthesized and studied in the laboratory of A. M. Kadomtseva (Moscow State
University) [74, 75], the Morin $\Gamma_{4}$ – $\Gamma_{1}$ spin-reorientation
transition to the low-temperature phase $\Gamma_{1}$ was detected, with the
exception of the composition with $x$ = 0.36, where the phase $\Gamma_{2}$ was
unexpectedly found to be the high-temperature phase. Puzzlingly, in
compositions $x$ = 0.1 and $x$ = 0.13, the Morin transition proceeded
according to the $\Gamma_{4}$ – $\Gamma_{421}$ – $\Gamma_{21}$ – $\Gamma_{1}$
($x$ = 0.1) or $\Gamma_{4}$ – $\Gamma_{421}$ – $\Gamma_{1}$ ($x$ = 0.1) scheme
and was accompanied by the deviation of the antiferromagnetic vector ${\bf G}$
into space with the appearance in a narrow temperature range of the projection
of the magnetic moment on the $b$-axis. Never before has the state of the
mixed configuration $\Gamma_{421}(G_{x}G_{y}G_{z})$ with the spatial
orientation of the antiferromagnetism vector and the appearance of the
$b$-component of the magnetic moment ($M_{b}\propto G_{x}G_{y}G_{z}$) been
observed.

###  5.3 Recent renewal of interest to weak ferrimagnets

The systems with competing antisymmetric exchange were extensively
investigated up to the end of 80ths mainly in the laboratory of A. M.
Kadomtseva at Moscow State University. Recent renewal of interest to the
systems with the compensation point was stimulated by the perspectives of the
applications in magnetic memory (see, e.g., Refs. [82, 71] and references
therein). For instance, weak ferrimagnet YFe0.5Cr0.5O3 exhibits magnetization
reversal at low applied fields. Below a compensation temperature (Tcomp), a
tunable bipolar switching of magnetization is demonstrated by changing the
magnitude of the field while keeping it in the same direction. The compound
also displays both normal and inverse magnetocaloric effects above and below
260 K, respectively. These phenomena coexisting in a single magnetic system
can be tuned in a predictable manner and have potential applications in
electromagnetic devices [82]. Weak ferrimagnets can exhibit the tunable
exchange bias (EB) effect [83]. Recently the EB effect with reversal sign was
found in LuFe0.5Cr0.5O3 ferrite-chromite [84] which is a weak ferrimagnet
below TN = 265 K, exhibiting antiparallel orientation of the mean weak
ferromagnetic moments of the Fe and Cr sublattices due to opposite sign of the
Fe - Cr Dzyaloshinskii vector as compared to that of the Fe - Fe and Cr - Cr.
The weak FM moments of the studied compound compensate each other at
temperature Tcomp = 230 K, leading to the net magnetic moment reversal and to
observed negative magnetization, at moderate applied field, below Tcomp.
Variety of such extraordinary properties as high compensation temperature,
temperature-controlled positive/negative EB below/above Tcomp, and switching
the magnetization direction to the opposite one by magnetic field without of
changing its polarity makes weak ferrimagnet LuFe0.5Cr0.5O3 of promising
candidate for application in magnetic memories.

Combining magnetization reversal effect with magnetoelectronics can exploit
tremendous technological potential for device applications, for example,
thermally assisted magnetic random access memories, thermomagnetic switches
and other multifunctional devices, in a preselected and convenient manner.
Nowadays a large body of magnetic materials might be addressed as systems with
competing antisymmetric exchange [85], including novel class of mixed
helimagnetic B20 alloys such as Mn1-xFexGe where the helical nature of the
main ferromagnetic spin structure is determined by a competition of the DM
couplings Mn - Mn, Fe - Fe, and Mn - Fe. Interestingly, that the magnetic
chirality in the mixed compound changes its sign at $x_{cr}\approx$ 0.75,
probably due to different sign of the Dzyaloshinskii vectors for Mn - Mn and
Fe - Fe pairs [86].

##  6 Determination of the sign of the Dzyaloshinskii vector

Determining the sign of the Dzyaloshinskii vector and the relative orientation
of the ${\bf F}$ and ${\bf G}$ vectors in weak ferromagnets is of both
fundamental importance from the standpoint of the microscopic theory of the DM
coupling and practical importance for the reliable identification of the
parameters of various anisotropic interactions in these materials. In
particular, for the rare-earth orthoferrites RFeO3, this concerns the
parameters of the 4$f$ \- 3$d$ interaction [60], the parameters of transferred
and supertransferred hyperfine interactions [87], and the magnitude of the
effective magnetic field for $\mu$-mesons [62]. The sign of the Dzyaloshinskii
vector determines the handedness of spin helix in crystals with the
noncentrosymmetric B20 structure.

How to measure the sign of the DM interaction in weak ferromagnets? According
to Ref. [19], an answer to this question can be given by determining
experimentally the direction of rotation of the antiferromagnetism vector
${\bf l}$ around the magnetic field ${\bf H}$ in the geometry ${\bf
H}\parallel{\bf d}\parallel$easy axis. However, as was pointed out later (see
Ref. [88]), a Mössbauer experiment on easy-axis hematite did not give an
unambiguous result.

According to Dmitrienko et al. [89], first of all, a strong enough magnetic
field should be applied to obtain the single domain state where the DM
coupling pins antiferromagnetic ordering to the crystal lattice. Next, single
crystal diffraction methods sensitive both to oxygen coordinates and to the
phase of antiferromagnetic ordering should be used. In other words, one should
observe those Bragg reflections $hkl$ where interference between magnetic
scattering on Mn atoms and nonmagnetic scattering on oxygen atoms is
significant. There are three suitable techniques: neutron diffraction,
Mössbauer $\gamma$-ray diffraction, and resonant x-ray scattering. The sign of
the DM vector in weak ferromagnetic FeBO3 was deduced from observed
interference between resonant X-ray scattering and magnetic X-ray scattering
[89].

The authors in Ref. [88] claimed that the character of the field-induced
transition from an antiferromagnetic phase to a canted phase in cobalt
fluoride CoF2 is due to the "sign" of the Dzyaloshinskii interaction, and this
affords an opportunity to determine experimentally the sign of the
Dzyaloshinskii vector. However, in fact they addressed a symmetric
Dzyaloshinskii interaction that is magnetic anisotropy

$V_{sym}=-D(m_{x}l_{y}+m_{y}l_{x})$

rather than antisymmetric DM coupling.

In our opinion, the most reliable experimental method for determining the
mutual orientation of the vectors of ferromagnetism ${\bf F}$ and
antiferromagnetism ${\bf G}$, and hence the sign of the Dzyaloshinskii vector,
is to study the magnitude and sign of the effective magnetic field on ligands,
as well as $\mu$-mesons in weak ferromagnets.

###  6.1 Positive muons in orthoferrites as a tool to examine the sign of
Dzyaloshinskii vector

In muon spin rotation ($\mu$SR) experiments, spin-polarized positive
(anti)muons are used to probe the microscopic field distribution at the
interstitial site(s) where the $\mu^{+}$ stop inside the sample under
investigation. The extreme sensitivity of the muon to small magnetic fields as
well as the absence of quadrupolar coupling makes this technique very
promising in probing magnetic orders, offering a valuable alternative to
neutron scattering. This approach, which shares many similarities with nuclear
magnetic resonance, has the advantage of being applicable to virtually any
material, but it has the drawback that the interstitial sites where the muon
stops and the nature of muon interaction with the host are generally unknown
[90]. Site assignment is the key initial ingredient in the not infrequent
cases where the internal magnetic field is dominated by the distant dipole
contribution, which requires only the knowledge of the site in order to be
computed by a classical sum over the dipole moments of the host lattice. Thus,
the comparison between predicted and measured local field can validate the
muon site assignment, and in turn, this assessment yields, e.g., a measure of
the magnetic moment values. However, additional local field contributions
exist and these are not negligible in many cases [91].

The dipolar field can be approximated with good accuracy, assuming a classical
moment ${\bf M}$ centered at the atomic positions of the magnetic atoms, and
evaluating the total contribution as

${\bf H}({\bf r}_{\mu})=\sum_{j}\left(\frac{3{\bf r}_{\mu j}({\bf
M}_{j}\cdot{\bf r}_{\mu j})}{r_{\mu j}^{5}}-\frac{{\bf M}_{j}}{r_{\mu
j}^{3}}\right)\,,$ (52)

where ${\bf r}_{\mu}$ is the muon position, ${\bf M}_{j}$ is the magnetic
moment of $j$-th ion, ${r_{\mu j}}$ is the distance between $j$-th ion and the
muon site. The hyperfine Fermi contact field contribution, transferred or
supertransferred, can be written as follows

$H({\bf r}_{\mu})=\frac{8\pi}{3}\mu_{B}\rho_{s}({\bf r}_{\mu})\,,$ (53)

where $\rho_{s}$ is the spin density at the muon site [91].

First detailed investigation of positive muons in orthoferrites RFeO3 (R= Sm,
Eu, Dy, Ho, Y, Er) has been performed by Holzschuh et al. [62]. In their
presentation the hyperfine field at the muon site in the orthoferrites can be
explained in terms of dipolar fields only. By comparing measured internal
magnetic fields with calculated dipolar fields of Fe3+ ions the authors found
the position of stable muon site, furthermore, they established that in
configuration $\Gamma_{4}$ the sign of $G_{x}$ should be positive for
$F_{z}>0$, in accordance with our earlier theoretical predictions [10], since
only this assumption leads to a reasonable muon site.

However, these results were severely criticized in work [92], the authors of
which argued that the interpretation [62] contains some serious flaws:
important details have not been worked out correctly and their analysis is not
complete enough to support some of their conclusions. First of all it concerns
the supertransferred hyperfine field contribution, which must not be
disregarded. Furthermore, they drew attention to the need for strict
accounting of the sign convention, labeling of the Fe3+ ions and the
representation of the spin configurations which is not uniform in the
literature. This all casts doubt in using theoretical relations, in
particular, concerning the mutual orientation of the ferro- and
antiferromagnetic vectors, that is, in fact, the sign of the Dzyaloshinskii
vector.

###  6.2 The ligand NMR in weak ferromagnets and first reliable determination
of the sign of the Dzyaloshinskii vector

As was firstly shown in our paper [20] reliable local information on the sign
of the Dzyaloshinskii vector, or to be exact, that of the 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 [20].

The F- ions in the unit cell of FeF3 occupy the six positions [93]. In a
trigonal basis these are
$\pm(x,1/2-x,1/4),\quad\pm(1/2-x,1/4,x),\pm(1/4,x,1/2-x)$, that correspond to
i) $\pm(3p(x-1/4),\sqrt{3}p(1/4-x),c/4)$, ii)
$\pm(3p(1/4-x),\sqrt{3}p(1/4-x),c/4)$, and iii) $\pm(0,2\sqrt{3}p(x-1/4),c/4)$
in an orthogonal basis with $O_{z}\parallel C_{3}$ and $O_{x}\parallel C_{2}$.
Each F- ion is surrounded by two Fe3+ from different magnetic sublattices.
Hereafter we introduce basic ferromagnetic F and antiferromagnetic G vectors:

$2S{\bf F}={\bf S_{1}}+{\bf S_{2}},2S{\bf G}={\bf S_{1}}-{\bf S_{2}},{\bf
F^{2}}+{\bf G^{2}}=1,$ (54)

where Fe${}_{1}^{3+}$ and Fe${}^{3+}_{2}$ occupy positions (1/2,1/2,1/2) and
(0,0,0), respectively. FeF3 is an easy plane weak ferromagnet with F and G
lying in (111) plane with ${\bf F}\bot{\bf G}$. The two possible variants of
the mutual orientation of the F and G vectors in the basis plane, tentatively
called as "left" and "right", respectively, are shown in Fig. 10. The DM
energy per Fe3+ \- F- \- Fe3+ bond can be written as follows

$E_{DM}=-2S^{2}d_{z}(12)(F_{x}G_{y}-F_{y}G_{x})=$
$-\frac{4\sqrt{3}}{l^{2}}p^{2}(x+\frac{1}{4})d(\theta)=+0.78S^{2}d(\theta)(F_{x}G_{y}-F_{y}G_{x})\,.$
(55)

In other words, the "left" and "right" orientations of basic vectors are
realized at $d(\theta)<0$ and $d(\theta)>0$, respectively.

Absolute magnitude of the ferromagnetic vector is numerically equals to an
overt canting angle which can be found making use of familiar values of the
Dzyaloshinskii field: $H_{D}=48.8$ kOe and exchange field: $H_{E}=4.4\cdot
10^{3}$ kOe [16] as follows

$F=H_{D}/2H_{E}\simeq 5.5\cdot 10^{-3}.$ (56)

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

$H_{D}=\frac{6S}{g\mu_{B}}|d_{z}(12)|=\frac{6S}{g\mu_{B}}0.39|d(\theta)|=48.8\,kOe\,,$
(57)

that yields $|d(\theta)|\cong$ 1.1 K that is three times smaller than in
YFeO3.

Figure 10: Two mutual orientations of $\bf F$ and $\bf G$ vectors in basal
plane of FeF3.

The local field on the nucleus of the nonmagnetic F- anion in weak ferromagnet
FeF3, induced by neighboring magnetic $S$-type ion (Fe3+, Mn3+,…) can be
written as follows [94]

${\bf H}=-\frac{1}{\gamma_{n}}\hat{A}{\bf S}$ (58)

($\gamma_{n}$ is a gyromagnetic ratio, $\gamma_{n}$=4.011 MHz/kOe, ${\bf S}$
is the spin moment of the magnetic ion), where the tensor of the transferred
hyperfine interactions (HFI) $\hat{A}$ consists of two terms: i) an isotropic
contact term with $A_{ij}=A_{s}\delta_{ij}$

$A_{s}=\frac{f_{s}}{2S}A^{(0)}_{s},\qquad
A^{(0)}_{s}=\frac{16}{3}\pi\mu_{B}\gamma_{n}|\varphi_{2s}(0)|^{2}\,;$ (59)

ii) anisotropic term with

$A_{ij}=A_{p}(3n_{i}n_{j}-\delta_{ij}),$ (60)

where ${\bf n}$ is a unit vector along the nucleus-magnetic ion bond and the
$A_{p}$ parameter includes the dipole and covalent contributions

$A_{p}=A_{p}^{cov}+A_{d},$ (61)
$A_{p}^{cov}=\frac{(f_{\sigma}-f_{\pi})}{2S}A_{p}^{(0)},A_{p}^{(0)}=\frac{4}{5}\mu_{B}\gamma_{n}\langle\frac{1}{r^{3}}\rangle_{2p},$
$A_{d}=\frac{g_{s}\mu_{B}\gamma_{n}}{R^{3}}.$ (62)

Here $f_{s,\pi,\sigma}$ are parameters for the spin density transfer: magnetic
ion - ligand along the proper $s-,\sigma-,\pi$-bond [95];
$|\varphi_{2s}(0)|^{2}$ is a probability density of the $2s$-electron on
nucleus; $\langle\frac{1}{r^{3}}\rangle_{2p}$ is a radial average.

The transferred HFI for 19F in fluorides are extensively studied by different
techniques, NMR, ESR, and ENDOR [94]. For 19F one observes large values both
of $A_{s}^{(0)}$ and $A_{p}^{(0)}$; $A_{s}^{(0)}=4.54\cdot\leavevmode\nobreak\
10^{4}$, $A_{p}^{(0)}=1.28\cdot\leavevmode\nobreak\ 10^{3}$ MHz [94], together
with the 100% abundance, nuclear spin $I=1/2$, and large gyromagnetic ratio
this makes the study of the transferred HFI to be simple and available one.

Contribution of the isotropic and anisotropic transferred HFI to the local
field on the 19F can be written as follows

${\bf H}(iso)=-\frac{2S}{\gamma_{n}}A_{s}{\bf F}=a_{F}{\bf F},$ ${\bf
H}(an)={\bf\stackrel{{\scriptstyle\leftrightarrow}}{{a}}}{\bf
G},\hat{a}=-\frac{2S}{\gamma_{n}}({\bf\stackrel{{\scriptstyle\leftrightarrow}}{{A}}}(1)-{\bf\stackrel{{\scriptstyle\leftrightarrow}}{{A}}}(2)).$
(63)

The $A_{s}$ and $A_{p}$ parameters we need to calculate parameter $a_{F}$ and
the HFI anisotropy tensor ${\bf\stackrel{{\scriptstyle\leftrightarrow}}{{a}}}$
that is to calculate the "ferro-" and "antiferro-" contributions to $H$ one
can find in the literature data for the pair 19F - Fe3+. For instance, in
KMgF3:Fe3+ ($R_{MgF}$ = 1.987Å) [96] $A_{s}=+72,A_{p}=+18$ MHz, in K2NaFeF6
($R_{FeF}$ = 1.91Å), in K2NaAlF6:Fe3+ $A_{s}$ = +70.17, $A_{p}$ = +20.34 MHz
[97]. Thus, we expect in FeF3 $|a_{F}|\sim 350\div 360$ MHz $(a_{F}<0)$ and
$H(iso)\simeq 2$ MHz ($\simeq 0.5$ kOe).

In the absence of an external magnetic field the NMR frequencies for 19F in
positions 1,2,3 can be written as follows

$\nu^{2}=\gamma_{n}^{2}[(\hat{a}{\bf G})^{2}+(a_{f}{\bf F})^{2}+2a_{F}{\bf
F}\hat{a}{\bf G}]=$ $\gamma_{n}^{2}(a_{xy}^{2}+a_{F}^{2}F^{2}\pm
2a_{F}a_{xy}F)+$ $\gamma_{n}^{2}(a_{yz}^{2}\mp
4a_{F}a_{xy}F)\left\\{\parbox{85.35826pt}{$\cos^{2}\varphi\\\
\cos^{2}(\varphi+60^{o})\\\ \cos^{2}(\varphi-60^{o})$}\right.$ (64)

where the $a_{xy},a_{yz}$ components are taken for 19F in position 1;
$\varphi$ is an azimuthal angle for ferromagnetic vector F in basis plane. The
formula (64) does provide a direct linkage between the 19F NMR frequencies and
parameters of the crystalline $(p,c,x,l)$ and magnetic $(F,\varphi,\pm)$
structures. As of particular importance one should note a specific dependence
of the 19F NMR frequencies on mutual orientation of the ferro- and
antiferromagnetic vectors or the sign of the Dzyaloshinskii vector: upper
signs in (64) correspond to "right orientation" ($d(\theta)$ > 0) while lower
signs do to "left orientation" ($d(\theta)$ < 0) as shown in Fig. 10.

Figure 11: (Color online) Comparison of simulated (upper panels) and
experimental (bottom panel) zero-field 19F NMR spectra in FeF3.

For minimal and maximal values of the 19F NMR frequencies we have

$\displaystyle{}\nu^{\pm}_{min}=\gamma_{n}[a_{xy}^{2}+a_{F}^{2}\pm
2a_{F}a_{xy}F]^{1/2},$
$\displaystyle\nu^{\pm}_{max}=\gamma_{n}[a_{xy}^{2}+a_{yz}^{2}+a_{F}^{2}F^{2}\mp
2a_{F}a_{xy}F]^{1/2}.$ (65)

Taking into account the smallness of isotropic HFI contribution, signs of
$a_{F}$ and $A_{xy}$ we arrive at estimations

$\nu^{\pm}_{min}\simeq\gamma_{n}(|a_{xy}|\mp|a_{F}F|)=2.92A_{p}\mp|a_{F}F|,$
$\nu^{\pm}_{max}\simeq\gamma_{n}\left([a_{xy}^{2}+a_{yz}^{2}]^{1/2}\pm\frac{|a_{xy}}{[a_{xy}^{2}+a_{yz}^{2}]^{1/2}}|a_{F}F|\right)=$
$3.65A_{p}\pm 0.8|a_{F}F|.$ (66)

Thus

$(\nu_{max}-\nu_{min})^{\pm}=0.68A_{p}\pm 1.8|a_{F}F|.$ (67)

By using the $A_{s}$ and $A_{p}$ values, typical for 19F –Fe3+ bonds [96, 97]
we get (in MHz)

$\nu^{+}_{min}=57.6,\nu^{+}{max}=75.7,(\nu_{max}-\nu_{min})^{+}=18.1$ (68)

given "right hand side orientation" of F and G (Fig. 10) and

$\nu^{-}_{min}=61.4,\nu^{-}_{max}=72.7,(\nu_{max}-\nu_{min})^{-}=11.3$ (69)

given "left hand side orientation" of F and G (Fig. 10) .

The zero-field 19F NMR spectrum for single-crystalline samples of FeF3 we
simulated on assumption of negligibly small in-plane anisotropy [98] is shown
in Fig. 11 for two different mutual orientations of ${\bf F}$ and ${\bf G}$
vectors. For a comparison in Fig. 11 we adduce the experimental NMR spectra
for polycrystalline samples of FeF3 [99, 100], which are characterized by the
same boundary frequencies despite rather varied shape. Obviously, the
theoretically simulated NMR spectrum does nicely agree with the experimental
ones only for "right" mutual orientations of ${\bf F}$ and ${\bf G}$ vectors,
or $d(FeFe)$ > 0, in a full accordance with our theoretical sign predictions
(see Table 4).

The same result, $d(FeFe)$ > 0 follows from the the magnetic $x$-ray
scattering amplitude measurements in the weak ferromagnet FeBO3 [89].

###  6.3 The sign of the Dzyaloshinskii vector in FeBO3 and $\alpha$-Fe2O3

Making use of structural data for FeBO3 [101] we can calculate the
$z$-component of the Dzyaloshinskii vector for Fe1-O-Fe2 pair, with Fe1,2 in
positions (1/2,1/2,1/2), (0,0,0), respectively, as follows:

$d_{z}(12)=d_{12}(\theta)\left[{\bf r}_{1}\times{\bf
r}_{2}\right]_{z}=+\frac{1}{3}(\frac{1}{2}-x_{h})\frac{ab}{l^{2}}d_{12}(\theta)$
$\approx+0.61\,d_{12}(\theta)\,,$ (70)

where $a$ = 4.626 Å, $b$ = 8.012 Å are parameters of the orthohexagonal unit
cell, $x_{h}$ = 0.2981 oxygen parameter, $l$ = 2.028 Å is a mean Fe-O
separation [101].

Similarly to FeF3 the DM energy per Fe3+ \- O2- \- Fe3+ bond can be written as
follows

$E_{DM}=d_{z}(12)\left[{\bf S}_{1}\times{\bf
S}_{2}\right]_{z}=-2S^{2}d_{z}(12)(F_{x}G_{y}-F_{y}G_{x})$ $=+2\cdot 0.61\cdot
S^{2}d_{12}(\theta)(F_{x}G_{y}-F_{y}G_{x})\,.$ (71)

In other words, the "left" and "right" orientations of basic vectors are
realized at $d(\theta)<0$ and $d(\theta)>0$, respectively.

Absolute magnitude of the ferromagnetic vector equals numerically to an overt
canting angle which can be found making use of familiar values of the
Dzyaloshinskii field: $H_{D}\approx 100$ kOe and exchange field: $H_{E}\approx
3.0\cdot 10^{3}$ kOe [18, 101] as follows

$F=H_{D}/2H_{E}\simeq 1.7\cdot 10^{-2}.$ (72)

If we know the Dzyaloshinskii field we can calculate the $d_{12}(\theta)$
parameter as follows

$H_{D}=\frac{6S}{g\mu_{B}}|d_{z}(12)|=\frac{6S}{g\mu_{B}}0.61|d(\theta)|=100\,kOe\,,$
(73)

that yields $|d(\theta)|\cong$ 1.5 K that is two times smaller than in YFeO3.
The difference can be easily explained, if one compares the superexchange
bonding angles in FeBO3 ($\theta\approx 125^{\circ}$) and YFeO3
($\theta\approx 145^{\circ}$), that is
$cos\,\theta$(FeBO3)/$cos\,\theta$(YFeO3)$\approx$ 0.7, that makes the
compensation effect of the $p$ \- $d$ and $s$ \- $d$ contributions to the
$X$-factor (see Table 3) more significant in borate than in orthoferrite.
Interestingly that in their turn the structural factor $\left[{\bf
r}_{1}\times{\bf r}_{2}\right]_{z}$ in FeBO3 is 1.6 times larger than the mean
value of the factor $\left[{\bf r}_{1}\times{\bf r}_{2}\right]_{y}$ in YFeO3.

The sign of the Dzyaloshinskii vector in FeBO3 has been experimentally found
recently due to making use of a new technique based on interference of the
magnetic x-ray scattering with forbidden quadrupole resonant scattering [89].
The authors found that the the magnetic twist follows the twist in the
intermediate oxygen atoms in the planes between the iron planes, that is the
DM coupling induces a small left-hand twist of opposing spins of atoms at
(0,0,0) and (1/2,1/2,1/2). This means that in our notations the Dzyaloshinskii
vector for Fe1-O-Fe2 pair is directed along $c$-axis, $d_{z}(12)>$ 0, that is
$d_{12}(\theta)>$ 0 in a full agreement with theoretical predictions (see
Table 4).

##  7 Exchange-relativistic anisotropy: Unconventional features of
conventional two-ion exchange anisotropy

So called quasi-dipole two-ion exchange anisotropy (anisotropic exchange)

$V_{an}=\sum_{m,n,\alpha,\beta}K_{\alpha\beta}(mn)S_{m\alpha}S_{n\beta}$ (74)

with a traceless symmetric tensor $K_{\alpha\beta}(mn)$ of anisotropy
parameters was introduced by Van Vleck as early as in 1937 [102]. For
$S_{1}=S_{2}=1/2$ the anisotropy was considered in details by Moriya [5] and
Yoshida [103]. Since then the simple Hamiltonian (74) had been used
increasingly without good reason for any 3$d$ ions and any spins $S\geq 1/2$.
Simple square-law temperature dependence of the effective anisotropy constant
$K_{TIA}(T)\sim B_{S}^{2}(T)\sim m^{2}(T)$ was addressed to be a "smoking gun"
of the magneto-dipole or anisotropic exchange origin of the anisotropy (see,
e.g., Refs. [104, 105]). However, a detailed many-electron analysis of the
exchange-relativistic anisotropy to be a result of the third order
perturbation contribution [106, 107]

$V_{an}(1,2)\sim\frac{V_{so}(1)V_{ex}(12)V_{so}(2)}{\Delta
E^{2}}+\frac{V_{so}(1)V_{so}(2)V_{ex}(12)}{\Delta E^{2}}+$
$\frac{V_{so}(1)V_{ex}(12)V_{so}(1)}{\Delta
E^{2}}+\frac{V_{so}(1)V_{so}(1)V_{ex}(12)}{\Delta E^{2}}$ (75)

(plus terms with $1\leftrightarrow 2$) has revealed some novel features of the
two-ion anisotropy missed in traditional approaches. First of all, it concerns
the tensor form of the anisotropic spin Hamiltonian. Simple quasi-dipole form
(74) is justified only for ions with $S_{m}=S_{n}$ = 1/2 and orbitally
nondegenerate ground state, while for different spins the tensor form becomes
more complicated. So, for the $S$-type ions, that is ions with orbitally
nondegenerate ground state $A_{1g},A_{2g}$ in cubic crystal field (Cr3+, Mn2+,
Fe3+, Ni2+,…) we arrive at an effective spin Hamiltonian as follows [107]

$E_{an}=\sum_{k_{1}k_{2}}\rho_{k_{1}}\rho_{k_{2}}\left(K^{2}_{12}(k_{1},k_{2})\cdot\left[C^{k_{1}}({\hat{\bf
S}_{1}})\times C^{k_{2}}({\hat{\bf S}_{2}})\right]^{2}\right),$ (76)

where does appear the tensor product of spherical tensorial harmonics,
$\rho_{k}(T)$ are temperature factors [108]:

$\rho_{0}=1;\,\rho_{1}=B_{S}(T)=\frac{\left<S_{z}\right>}{S};\rho_{2}=\frac{\left<3S_{z}^{2}-S(S+1)\right>}{S(2S-1)};$
$\rho_{3}=-\frac{\langle[3S(S+1)-1]\,{S_{z}}+S\,{S_{z}^{3}}\rangle}{S(S-1)(2S-1)},$
(77)

($\rho_{k}(T=0)$ = 1).

Along with a quasidipole term $(k_{1}=k_{2}=1)$ in $V_{an}$ there is a number
of novel nondipole terms with $k_{1}k_{2}=20(02),22$ and $k_{1}k_{2}=13(31)$.
Beyond that, $k_{1,2}$ should obey the triangle rule: $k_{1,2}\leq 2S_{m,n}$.
It is worth noting that in addition to conventional spin-dependent exchange,
the purely orbital spinless exchange interaction does contribute to the
quasidipole exchange anisotropy [107].

Figure 12: (Color online) The temperature dependence of effective anisotropy
constants in $\alpha$-Fe2O3 (left) and Cr2O3 (right). Circles are experimental
data from Refs. [109] and [104] for $\alpha$-Fe2O3 and Cr2O3, respectively.
Curves 1 represent the result of the fitting with the four-parameter formula
(78), curve 2 for hematite shows the result of the fitting that implies only
conventional contributions of single-ion and quasi-dipole anisotropy. Curves
$a$, $b$, $c$, and $d$ represent the temperature dependence of the quasi-
dipole contribution ($a$) and of the "non-dipole" factors
$(\rho_{2}-\rho_{1}^{2})$, $(\rho_{2}^{2}-\rho_{1}^{2})$, and
$(\rho_{1}\rho_{3}-\rho_{1}^{2}))$, respectively.

Within mean-field approximation the temperature dependence of effective 2nd
order exchange-relativistic anisotropy constant(s) for magnets with equivalent
on-site spins can be represented as follows [107]:

$K(T)=K(0)\rho_{1}^{2}+K_{20}(\rho_{2}-\rho_{1}^{2})+K_{22}(\rho_{2}^{2}-\rho_{1}^{2})+$
$K_{13}(\rho_{1}\rho_{3}-\rho_{1}^{2}),$ (78)

where the temperature factors $(\rho_{2}-\rho_{1}^{2})$,
$(\rho_{2}^{2}-\rho_{1}^{2})$, and $(\rho_{1}\rho_{3}-\rho_{1}^{2}))$ turn
into zero both at T = 0 K and T = TN(Tc). The constant $K_{11}$ for
conventional quasi-dipole anisotropy is determined as
$K_{11}=K(0)-K_{20}-K_{22}-K_{13}$. It is worth noting that the addition of
the magneto-dipole and single-ion anisotropies gives rise only to a
renormalization of the $K(0)$ and $K_{20}$ constants, respectively, so that
the expression (78) is believed to be an universal four-parametric formula for
the temperature dependence of the 2nd order anisotropy constants. As we see in
Fig. 12 the formula allows us to nicely describe nontrivial temperature
dependence of effective anisotropy constants in $\alpha$-Fe2O3 and Cr2O3.
Later, this approach was used to describe the temperature dependence of the
anisotropy constants in YFeO3 [110].

##  8 Antisymmetric supertransferred hyperfine interaction as electron-nuclear
counterpart of DM coupling

Detailed analysis of the 57Fe NMR data in orthoferrites [22] allowed us to
reveal antisymmetric supertransferred hyperfine (ASTHF) coupling

${\hat{H}}_{ASTHF}=\sum_{m>n}({\bf a}_{mn}\cdot[{\bf I}_{m}\times{\bf
S}_{n}])$ (79)

as an electron-nuclear analogue of the DM antisymmetric exchange and to
determine its contribution to the local field: $H_{ASTHF}\approx$ 0.26 T as
compared with corresponding isotropic contribution of 5.8 T [87]. Here ${\bf
I}$ is a nuclear spin, ${\bf a}_{mn}$ electron-nuclear analogue of the
Dzyaloshinskii vector.

For the first time such a electron-nuclear coupling was considered by Ozhogin
[111], the microscopic theory was considered by Moskvin [87]. Furthermore, in
Ref. [87] we have shown that the experimentally known external field
dependencies of the 57Fe NMR in the orthoferrites [22] do allow us to find out
and estimate the ASTHF coupling.

Indeed, taking into account the four-sublattice magnetic structure of the
orthoferrite RFeO3 with nonmagnetic R-ions (La, Y, Lu) the local field on the
57Fei nuclei in one of the $4b$-sites can be written as follows

${\bf H}_{loc}(i)=a_{G}(i){\bf G}+a_{F}(i){\bf F}+a_{C}(i){\bf C}+a_{A}(i){\bf
A}+{\bf\stackrel{{\scriptstyle\leftrightarrow}}{{a}}}(i){\bf G}$ (80)

where ${\bf G},\leavevmode\nobreak\ {\bf F},\leavevmode\nobreak\ {\bf
C},\leavevmode\nobreak\ {\bf A}$ are the basis magnetic vectors normalized as
follows: ${\bf G}^{2}+{\bf F}^{2}+{\bf C}^{2}+{\bf A}^{2}=1$. Here, the first
four terms represent the contribution of the dominant isotropic on-site and
supertransferred inter-site hyperfine interactions, while the last term does
the contribution of the anisotropic hyperfine interactions. Hereafter we take
into account that F, C, A $\approx 10^{-2}$G, and assume that the anisotropic
contribution does not exceed values on the order of $\propto 1\%$ of main
isotropic contribution $H_{0}=a_{G}G$.

The external field dependence of the 57Fe NMR frequency for the
$\Gamma_{4}(G_{x},A_{y},F_{z})$ magnetic configuration is as follows:

$\nu_{\Gamma_{4}}({\bf h}\parallel{\bf
c})=1-a_{xx}+(a_{zx}G_{x}+a_{F}F_{z})h+h^{2}/2,$ (81)

where we make use all the quantities $a_{zx},\leavevmode\nobreak\
a_{F},\leavevmode\nobreak\ a_{xx},\leavevmode\nobreak\ h$ in units of $H_{0}$
(in YFeO3 given T = 4.2 K $H_{0}$ = 551 kOe [22]), while $\nu$ is in units of
$\nu_{0}=\gamma H_{0}/2\pi$ ($\gamma/2\pi=0.138$ MHz/kOe). The field
derivative $(\partial\nu/\partial h)_{h=0}$

$[\partial\nu_{\Gamma_{4}}({\bf h}\parallel{\bf c})/\partial
h]_{h=0}=a_{F}F_{z}+a_{zx}G_{x}\,,$ (82)

is a sum of a ferromagnetic($a_{F}F_{z}$) and antiferromagnetic
($a_{zx}G_{x}$) contributions, respectively. Application of the magnetic field
parallel to $a$-axis (${\bf h}\parallel{\bf a}$) does induce a spin-
reorientational transition
$\Gamma_{4}(G_{x},A_{y},F_{z})-\Gamma_{2}(F_{x},C_{y},G_{z})$ so that for
angular $\Gamma_{42}$ configuration

$\nu_{\Gamma_{42}}({\bf h}\parallel{\bf a})=1-(a_{xx}G_{x}^{2}\pm
2a_{zx}^{(s)}G_{x}G_{z}+a_{zz}G_{z}^{2})+$ $(a_{xz}G_{z}+a_{F}F_{x}\pm
a_{G}G_{x})h+h^{2}/2$ (83)

where $a_{zx}^{(s)}$ is a symmetrical part of $a_{zx}$:
$a_{zx}^{(s)}=(a_{zx}+a_{xz})/2$). The signs $\pm$ in (83) correspond to
nuclei in positions 1,3 and 2,4, respectively (see Ref. [112]). The
$\Gamma_{4}$ \- $\Gamma_{2}$ spin reorientation is accompanied by the 57Fe NMR
frequencies splitting whose magnitude

$\Delta\nu=2(2a_{zx}^{s}G_{z}+h)G_{x}$ (84)

allows us to find out the $a_{zx}^{(s)}$ parameter, or, strictly speaking, its
absolute value: $|a_{zx}^{(s)}|=3.2\cdot 10^{-3}$ in YFeO3 [22],
$|a_{zx}^{(s)}|=3.4\cdot 10^{-3}$ in ErFeO3, and $|a_{zx}^{(s)}|=2.9\cdot
10^{-3}$ in HoFeO3 [112].

Experimental value of the field derivative $[\partial\nu_{\Gamma_{4}}({\bf
h}\parallel{\bf c})/\partial h]_{h=0}$ = -10.2$\cdot 10^{-3}$ in YFeO3 [22]
with taking account of $a_{F}=2H_{STHF}/H_{0}-1$ = -0.79 ($H_{STHF}$ is the
contribution of the STHF interaction 57Fe–O2-–Fe3+ to the local field) [22]
and $F_{z}=1.1\cdot 10^{-2}$ [17] allows us to find out the value
$a_{zx}G_{x}=-1.6\cdot 10^{-3}$. Finally we obtain that for $a_{zx}^{(s)}=\pm
3.2\cdot 10^{-3}$, $a_{zx}^{(a)}=\mp 1.6\cdot 10^{-3}$ given $F_{z}>0,G_{x}<0$
($G_{x}\approx$ -1) and $a_{zx}^{(a)}=\mp 4.8\cdot 10^{-3}$ given
$F_{z}>0,G_{x}>0$ ($G_{x}\approx$ +1), that is for $d(\theta)>0$ and
$d(\theta)<0$, respectively. In any case antisymmetric and symmetric parts of
the anisotropic hyperfine interaction in YFeO3 are of a comparable magnitude.
The origin of the antisymmetric part
${\bf\stackrel{{\scriptstyle\leftrightarrow}}{{a}}}^{a}$ can be related only
with antisymmetric STHF interaction ${\hat{H}}_{ASTHF}$ (79) that is with
electron-nuclear analogue of the DM coupling. Then

$a_{zx}^{(a)}(i)=-\frac{S}{g_{n}\beta_{n}}\sum_{j}a_{y}(ij).$ (85)

The ASTHF interaction is a result of a combined action of the generalized STHF
interaction 57Fe–O2-–Fe3+ [113] and spin-orbital coupling for Fe3+ ion. As for
conventional spin-spin DM coupling the electron-nuclear analogue of the
Dzyaloshinskii vector depends on the superexchange 57Fe–O2-–Fe3+ bond geometry

${\bf a}(ij)=a(\theta)\left[{\bf r}_{i}\times{\bf r}_{j}\right],$ (86)

where ${\bf r}_{i},\leavevmode\nobreak\ {\bf r}_{j}$ are unit cation-anion
radius vectors, and

$a(\theta)=a_{1}+a_{2}\cos\theta,$ (87)

where $\theta$ is the cation-anion-cation bond angle.

For a rough estimate of the $a$ parameter one may use relation
$a/A_{STHF}\leq\xi/\Delta E$ where $\xi$ is a single electron spin-orbital
coupling parameter for 3$d$ electron; $\Delta E$ is the energy of the excited
terms of the ${}^{4}T_{1}$ type for Fe3+ ion; $A_{STHF}$ is a isotropic STHF
constant:

$\hat{V}_{STHF}=\sum_{i\neq j}A_{STHF}(ij)({\bf I}_{i}\cdot{\bf S}_{j}).$ (88)

In our case $\xi\leq 5\cdot 10^{2}\leavevmode\nobreak\ cm^{-1}$, $\Delta E\geq
10^{4}\leavevmode\nobreak\ cm^{-1}$, and we arrive at

$a/A_{STHF}\leq 5\cdot 10^{-2}$

that nicely agrees with estimate based on the experimental data [22]

$|a_{zx}^{(a)}/H_{STHF}|\approx 4.6\cdot 10^{-2}\,.$

The ratio is comparable with the ratio of the Dzyaloshinskii field $H_{D}$ to
the exchange field $H_{E}$: in YFeO3 $H_{D}/H_{E}\approx 2.2\cdot 10^{-2}$.
All that is quite natural as the Dzyaloshinskii field is of the exchange-
relativistic nature: $H_{D}/H_{E}\propto\xi/\Delta E$, hence,
$|a_{zx}^{(a)}/H_{STHF}|\propto H_{D}/H_{E}$. In other words, if $H_{STHF}$ is
an electron-nuclear analogue of the exchange field, the
$H_{ASTHF}=|a_{zx}^{(a)}|$ is an electron-nuclear analogue of the
Dzyaloshinskii field. In YFeO3 $H_{STHF}$ = 58 kOe, $H_{ASTHF}$ = 2.6 kOe
given $G_{x}>0$ or $H_{ASTHF}$ = 0.9 kOe given $G_{x}<0$. For rough estimate
of the electron-nuclear Dzyaloshinskii field one may use
$H_{ASTHF}\approx(H_{D}/H_{E})H_{STHF}$.

Antisymmetric STHF interaction should be observed in other weak ferromagnets.
It is worth noting that for an easy-plane phase of rhombohedral weak
ferromagnets such as FeBO3, FeF3, $\alpha$-Fe2O3, the antiferromagnetic
contribution to the field derivative $[\partial\nu({\bf h}\bot{\bf
C}_{3})/\partial h]_{h=0}$ is determined only by the ASTHF interaction:

$[\partial\nu({\bf h}\bot{\bf C}_{3})/\partial
h]_{h=0}=a_{xy}^{(a)}G_{y}+a_{F}F_{x},$ (89)

that makes its detection and estimation more easier than in orthoferrites.

It should be noted that the electron-nuclear double resonance (ENDOR)
measurements in Pb5Ge3O11:Gd3+ revealed antisymmetric 207Pb–O2-–Gd3+
supertransferred hyperfine interaction [114] whose origin can be related with
the ligand spin-orbital contribution.

##  9 Antisymmetric exchange-relativistic spin-other-orbit coupling and
unconventional magnetooptics of weak ferromagnets

Interestingly that circular magnetooptic effects in weak ferromagnets are
anomalously large and are comparable with the effects in ferrite garnets
despite two-three orders of magnitude smaller magnetization [58, 115, 116]. In
1989 the anomaly has been assigned to a novel type of magnetooptical
mechanisms related with so called spin-other-orbit coupling [117].

Combined effect of a conventional on-site spin-orbital coupling and orbitally
off-diagonal exchange coupling for an excited orbitally degenerated state can
give rise to a novel type of exchange-relativistic interaction, so called
spin-other-orbit coupling, whose bilinear form can be written as a sum of
isotropic, anisotropic antisymmetric, and anisotropic symmetric terms,
respectively

${\hat{V}}_{SoO}=\sum_{m>n}\lambda^{(0)}_{mn}({\bf L}_{m}\cdot{\bf
S}_{n})+\sum_{m>n}(\boldsymbol{\lambda}_{mn}\cdot[{\bf L}_{m}\times{\bf
S}_{n}])+$ $\sum_{m>n}({\bf
L}_{m}\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\lambda}}}_{mn}{\bf
S}_{n})\,.$ (90)

It is worth noting that $\boldsymbol{\lambda}_{mn}$ has the symmetry of the
Dzyaloshinskii vector, while the last term has the symmetry of the two-ion
quasidipole spin anisotropy. Generally speaking, all the three terms can be of
a comparable magnitude.

Interestingly, the contribution to the bilinear interaction ${\hat{V}}_{SoO}$
is made by both the spin-dependent exchange and spin-independent purely
orbital exchange. However, the spin-dependent exchange leads to the occurrence
of additional nonlinear spin-quadratic terms, the contribution of which can be
taken into account by the formal replacement of the linear spin operator ${\bf
S}_{n}$ in (18) for the nonlinear operator ${\bf S}_{mn}$

$\displaystyle\hat{S}_{q}(mn)\,=\,\hat{S}_{q}(n)\,+\,\gamma\,\left[\hat{V}^{2}\Bigl{(}S(m)\Bigr{)}\,\times\,S^{1}(n)\right]^{1}_{q}\,=$
$\displaystyle=\hat{S}_{q}(n)+\gamma\sum_{q_{1},q_{2}}\left[\begin{array}[]{ccc}2&1&1\\\
q_{1}&q_{2}&q\end{array}\right]\hat{V}^{2}_{q_{1}}\Bigl{(}S(m)\Bigr{)}S_{q_{2}}(n)\>,$
(93)

where $\;V^{2}_{q}(S)\;$ is the rank 2 spin irreducible tensor operator. In
particular,

$\hat{V}_{0}^{2}(S)\,=\,2\,\left[\frac{(2S\,-\,2)!}{(2S\,+\,3)!}\right]^{1/2}\Bigl{(}3\hat{S}_{z}^{2}\>-\>S(S\,+\,1)\Bigr{)}\>.$
(94)

The coefficient $\;\gamma\;$ in (93) can be calculated for specific terms. The
isotropic part of $\;V_{so}^{ex}\;$ can be presented, in the general case, as
follows

$V_{so}^{ex}\>=\>\sum_{mn}\,\lambda(mn)\left({\bf L}(m)\cdot{\bf
S}(n)\right)\,+$ $+\sum_{m\not=n}\,\lambda^{{}^{\prime}}(mn)\Bigl{(}{\bf
L}(m)\cdot{\bf S}(m)\Bigr{)}\Bigl{(}{\bf S}(m)\cdot{\bf S}(n)\Bigr{)}\>.$ (95)

Similarly to the Dzyaloshinskii vector, to estimate the parameters of the
spinother orbit coupling, we can use the simple relation

$\>\lambda(mn)\>\approx\>\frac{\lambda^{{}^{\prime}}I^{{}^{\prime}}}{\Delta
E_{S\Gamma}}\>,$ (96)

where $\lambda^{{}^{\prime}}\,$ and $\,I^{{}^{\prime}}\,$ are the spinorbital
constant for the $T_{1}$-, $T_{2}$-states and the nondiagonal exchange
parameter, respectively, $\Delta E_{S\Gamma}$ is a certain excitation energy.
The simple estimation shows that, due to ${\hat{V}}_{SoO}$, the effective
magnetic fields acting on the $T_{1}$ and $T_{2}$ orbital states, e.g., for
Fe3+ions in ferrites, can amount about 100 T and more.

We have shown that an antisymmetric exchange-relativistic spin-other-orbit
coupling gives rise to an unconventional "antiferromagnetic" contribution to
the circular magnetooptics for weak ferromagnets which can surpass
conventional "ferromagnetic" term [117, 118] (see, also Ref. [119]).

The circular magnetooptics is governed by an axial gyration vector ${\bf g}$,
which is dual to the permittivity tensor $\varepsilon_{ij}$. For instance the
Faraday rotation $\Theta_{F}$ in noncubic crystals can be written as follows

$\Theta_{F}=A({\bf g}\cdot{\bf n})\,,$ (97)

where ${\bf n}$ is a unit vector in the direction of light propagation ${\bf
k}$, and $A$ is a coefficient which depends on the direction of ${\bf k}$, on
the polarization of light, and on the principal values of the refractive
index. The gyration vector has the same symmetry properties as the
ferromagnetic moment that does justify a well known relation

${\bf g}=\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\alpha}}}{\bf
F}+\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\gamma}}}{\bf
H}_{ext}\,,$ (98)

where the gyration vector is a sum of so-called ferromagnetic and diamagnetic
terms, respectively. However, in weak ferromagnets where orthogonal components
of the ferromagnetic and antiferromagnetic vectors may transform identically
we arrive at an additional "antiferromagnetic" contribution. For example, in
the case of orthoferrites this term, if to neglect weak antiferromagnetic
modes, can be written as follows:

$\Delta{\bf
g}=\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\beta}}}{\bf G}$ (99)

with the only nonzero, and generally speaking unequal, components $\beta_{zx}$
and $\beta_{xz}$ of the
$\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\beta}}}$ tensor.
Despite for main isotropic contribution the components of the
$\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\alpha}}}$ tensor are
expectedly significantly larger than the
$\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\beta}}}$ tensor
components, the relation $F\ll G$ which is typical for weak ferromagnets
points to a puzzling effect of a probably leading antiferromagnetic
contribution to the gyration vector.

It is generally accepted that the major (if not dominant) role in the
magnetooptic rotation of visible and ultraviolet light in rare-earth
orthoferrites is played by allowed electric-dipole O2$p$–Fe3$d$ charge
transfer ${}^{6}A_{1g}-{}^{6}T_{1u}$ transitions in octahedral complexes FeO6
[58, 118, 120], in particular, due to an orbital Zeeman splitting of the
excited ${}^{6}T_{1u}$ term with effective orbital moment $L$ = 1. In addition
to conventional orbital Zeeman $V_{Z}^{orb}=-\beta_{e}\sum_{m}({\bf
L}_{m}\cdot{\bf H}_{ext})$ and on-site spin-orbital
$V_{SO}=\lambda\sum_{m}({\bf L}_{m}\cdot{\bf S}_{m})$ couplings such a
splitting is caused by unconventional spin-other-orbit coupling (90). In all
the cases we deal with a real or effective orbital magnetic field. The
contribution of the isolated ${}^{6}T_{1u}$ term to the gyration vector can be
represented as follows [117, 118]:

${\bf g}=\left(\frac{n_{0}^{2}+2}{3}\right)^{2}\frac{2\pi
e^{2}f_{AT}}{m\omega_{0}}\frac{\partial
F(\omega,\omega_{0})}{\partial\omega_{0}}$ $(-N\beta_{e}{\bf
H}_{ext}+\lambda_{eff}\sum_{m}\langle{\bf
S}_{m}\rangle+\sum_{m>n}\lambda^{(0)}_{mn}\langle{\bf S}_{n}\rangle-$
$\sum_{m>n}[\boldsymbol{\lambda}_{mn}\times\langle{\bf
S}_{n}\rangle]+\sum_{m>n}\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\lambda}}}_{mn}\langle{\bf
S}_{n}\rangle)\,,$ (100)

where $N$ is the number of FeO6 clusters in the unit volume, $f_{AT}$ and
$\hbar\omega_{0}$ are the oscillator strength and the energy of the
${}^{6}A_{1g}-{}^{6}T_{1u}$ transition, respectively, $F(\omega,\omega_{0})$
is the dispersion factor. Here, the first and second terms determine
conventional "local" diamagnetic and isotropic ferromagnetic contributions,
respectively, while the three other terms derived from the spin-other-orbit
interaction determine unconventional "nonlocal" contribution, though the first
one gives a simple correction to the ferromagnetic term. However, the second
and third nonlocal terms provide novel antisymmetric and symmetric anisotropic
antiferromagnetic contributions to gyration vector, respectively. Their effect
was experimentally studied in orthoferrite YFeO3 [117]. An analysis of the
dependence $\Theta_{F}({\bf H}_{ext})$ made it possible to determine all the
contributions to the gyration vector ($\lambda$ = 0.6328 $\mu$m):

$\alpha_{zz}F_{z}=(0.95\pm 0.55)\cdot 10^{-3};\,\beta_{zx}G_{x}=(3.15\pm
0.55))\cdot 10^{-3};\,$ $\alpha_{xx}F_{x}=(0.2\pm 0.7)\cdot
10^{-3};\,\beta_{xz}G_{z}=(-2.1\pm 1.0))\cdot 10^{-3};\,$
$\gamma_{zz}\approx\gamma_{xx}=(-1.1\pm 2.8)\cdot 10^{-6}\,kOe^{-1}\,.$ (101)

Interestingly, rather large measurement errors allow for certain to determine
only the fact of a large if not a dominant antisymmetric antiferromagnetic
contribution related with antisymmetric spin-other-orbit coupling.

Existence of spontaneous spin-reorientational phase transitions
$\Gamma_{4}(F_{z}G_{x})\rightarrow\Gamma_{2}(F_{x}G_{z})$ in several rare-
earth orthoferrites does provide large opportunities to study anisotropy of
circular magnetooptics [58, 115, 116, 118]. Gan’shina et al. measured the
equatorial Kerr effect in TmFeO3 and HoFeO3 and have found the the gyration
vector anisotropy in a wide spectral range 1.5-4.5 eV [118]. The
magnetooptical spectra were nicely fitted within a microscopic model theory
based on the dominating contribution of the O2$p$–Fe3$d$ charge transfer
transitions and spin-other-orbit coupling in FeO${}_{6}^{9-}$ octahedra.
Studies have demonstrated a leading contribution of the antisymmetric spin-
other-orbit coupling and allowed to estimate effective orbital magnetic fields
in excited ${}^{6}T_{1u}$ states of the FeO${}_{6}^{9-}$ octahedra, $H_{L}\sim
100\,T$. These anomalously large fields can be naturally explained to be a
result of strong exchange interactions of the charge transfer ${}^{6}T_{1u}$
states with nearby octahedra that are determined by a direct $p$ \- $d$
exchange. Whereas the existence of the antiferromagnetic contribution to the
gyration vector is typical of a large number of multisublattice magnetic
materials, the antisymmetry of the tensor
$\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\beta}}}$ is a specific
feature of weak ferromagnets alone. In the case of rhombohedral weak
ferromagnets such as FeBO3, FeF3, or $\alpha$-Fe2O3, the tensor
$\stackrel{{\scriptstyle\leftrightarrow}}{{\boldsymbol{\beta}}}$, governing
the antiferromagnetic contribution to the Faraday effect is entirely due to
the antisymmetric contribution, in view of the requirements imposed by the
crystal symmetry. In crystals of this kind the appearance of the
antiferromagnetic contribution to the gyration vector is entirely due to
allowance for the antisymmetric spin-other-orbit coupling.

##  10 Antisymmetric exchange-relativistic spin-dependent electric
polarization

I. E. Dzyaloshinskii in 1959 theoretically predicted the existence of the
magnetoelectric effect (ME) in antiferromagnetic Cr2O3 [121], and a year
later, D. N. Astrov recorded the magnetization induced by an electric field
[122]. Since the prediction and discovery of the ME effect in Cr2O3, several
different mechanisms of magnetoelectric coupling have been proposed [123], but
a real breakthrough in this direction is associated with the discovery and
studying of multiferroics.

Currently two essentially different spin structures of net electric
polarization in crystals are considered: i) a bilinear nonrelativistic
symmetric spin coupling [124]

${\bf P}_{s}=\sum_{mn}{\bf\Pi}_{mn}^{s}({\bf S}_{m}\cdot{\bf S}_{n})\,$ (102)

or ii) a bilinear relativistic antisymmetric spin coupling[Katsura, Sergienko]

$\hat{\bf
P}_{a}=\sum_{m>n}{\bf\stackrel{{\scriptstyle\leftrightarrow}}{{\Pi}}}_{mn}[{\bf
S}_{m}\times{\bf S}_{n}])\,\,$ (103)

respectively. Strictly speaking,
${\bf\stackrel{{\scriptstyle\leftrightarrow}}{{\Pi}}}_{mn}$ is a rank-2
tensor, whose antisymmetric component yields the contribution to the electric
polarization ${\bf P}_{a}$ as follows

${\bf P}_{a}=\sum_{mn}\left[{\bf\Pi}_{mn}^{a}\times\left[{\bf S}_{m}\times{\bf
S}_{n}\right]\right]\,,$ (104)

The effective dipole moments ${\bf\Pi}_{mn}^{s,a}$ depend on the $m,n$ orbital
states and the $mn$ bonding geometry.

If the first term stems somehow or other from a spin isotropic Heisenberg
exchange interaction (see, e.g. Refs. [124, 125]), the second term does from
antisymmetric Dzyaloshinskii-Moriya (DM) coupling. Following Katsura et al.
[37] the electric dipole ${\bf P}_{a}$ is said to be induced by a spin current
mechanism, since the vector product $\left[{\bf S}_{m}\times{\bf
S}_{n}\right]$ is proportional to the bond spin current, where the
Dzyaloshinskii vector ${\bf d}_{mn}$ acts as its vector potential. Namely the
second, or "spin-current" term is frequently considered to be one of main
contributors to multiferroicity [126], however, at present there is no
reliable theoretical justifications and experimental evidences for its
dominating over conventional symmetric isotropic term [39, 127].

Microscopic quantum theory of ME effect has not yet been fully developed,
although several scenarios for particular materials have been proposed.
Katsura et al. [37] presented a mechanism of the giant ME effect theoretically
derived "in terms of a microscopic electronic model for noncollinear magnets".
The authors derived the expression for the electric dipole moment for the spin
pair as follows:

${\bf P}_{ij}=a\left[{\bf R}_{ij}\times\left[{\bf S}_{i}\times{\bf
S}_{j}\right]\right]\,,$ (105)

where ${\bf R}_{ij}$ denotes the vector connecting the two sites $i$ and $j$,
${\bf S}_{i,j}$ are spin moments, $a$ is an exchange-relativistic parameter.
However, the original "spin-current" model by Katsura et al. [37] seems to be
questionable as the authors proceed with an unrealistic scenario [127].

The spin-current model can explain direction of ferroelectric polarization for
Pbmn spin-cycloidal perovskite manganites, however, cannot explain the
polarization anisotropy in spin-spiral LiCu2O2 and LiVCuO4, the direction of
ferroelectric polarization for spin-cycloilal delafossites, such as AgFeO2 and
$\alpha$-NaFeO2, the absense of polarization in spin-spiral NaCu2O2. It cannot
explain an emergence of ferroelectricity associated with proper crew magnetic
ordering in several multiferroics, including CuFeO2, CuCrO2, AgCrO2, Cu3Nb2O8,
CaMn7O12, and RbFe(MoO4)2, because the propagation vector ${\bf
R}_{ij}\parallel[{\bf S}_{i}\times{\bf S}_{j}]$. In other words, for a most
part of multiferroics the spin-current model does not work.

Alternative mechanism of giant magnetoelectricity in the perovskite manganites
based on the antisymmetric DM type magnetoelastic coupling was proposed by
Sergienko and Dagotto [38]. The authors took into account strong dependence of
the Dzyaloshinskii vector on the superexchange bond angle and the displacement
of the intermediate ligand. However, here we meet with a "weak" contributor.
Indeed, the minimal value of $\gamma$ parameter ($\gamma=d{\bf D}/d{\bf R}$)
needed to explain experimental phase transition in multiferroic manganites is
two orders of magnitude larger than the reasonable microscopic estimations
[38].

Size of the macroscopic polarization P in nonmagnetic ferroelectrics computed
by modern ab-initio band structure methods agrees exceptionally well with the
ones observed experimentally. However, state of the art ab-initio computations
for different multiferroics: manganites HoMnO3, TbMn2O5, HoMn2O5, spin spiral
chain cuprates LiCuVO4 and LiCu2O2 yield data spread within one-two orders of
magnitude with absolutely ambiguous and unreasonable values of polarisation.
Indeed, the basic starting points of the current versions of such spin-
polarized approaches as the LSDA exclude any possibility to obtain a reliable
quantitative estimation of the spin-dependent electric polarization in
multiferroics. Basic drawback of the spin-polarized approaches is that these
start with a local density functional which implies presence of a large
fictious local one-electron spin-magnetic field. Magnitude of the field is
considered to be governed by the intra-atomic Hund exchange, while its
orientation does by the effective molecular, or inter-atomic exchange fields.
Despite the supposedly spin nature of the field it produces an unphysically
giant spin-dependent rearrangement of the charge density that cannot be
reproduced within any conventional technique operating with spin Hamiltonians.
In such a case the straightforward application of the LSDA scheme can lead to
an unphysical overestimation of the effects or even to qualitatively incorrect
results due to an unphysical effect of a breaking of spatial symmetry induced
by a spin configuration.

Overall, the LSDA approach seems to be more or less justified for a
semiquantitative description of exchange coupling effects for materials with a
classical Néel-like collinear magnetic order. However, it can lead to
erroneous results for systems and effects where the symmetry breaking and
quantum fluctuations are of a principal importance such as: i) noncollinear
spin configurations, in particular, in quantum s=1/2 magnets, ii) relativistic
effects, such as the symmetric spin anisotropy, antisymmetric DM coupling,
and, iii) spin-dependent electric polarization. Indeed, a correct treatment of
these high-order perturbation effects needs in a correct account both of local
symmetry and of quantum fluctuations (see, e.g., Ref. [31]).

A systematic standard microscopic theory of spin-dependent electric
polarization which implies the derivation of effective spin operators for
nonrelativistic and relativistic contributions to electric polarization of the
generic three-site two-hole cluster such as Cu1 \- O - Cu2 has been proposed
in Ref. [39, 127]. The authors made use of conventional well-known approaches
to account for the $p$ \- $d$ covalent effects, intra-atomic correlations,
crystal field, and spin-orbital coupling. Despite the description was focused
on a three-site Cu1 \- O - Cu2 two-hole system typical for cuprates with a
tetragonal Cu on-site symmetry and Cu3$d_{x2-y2}$ ground states, the
generalization of the results on the M1 \- O - M2 clusters in other 3$d$
oxides seems to be a trivial procedure.

The effective electric polarization differs for the singlet and triplet
pairing due to a respective singlet-triplet difference in the hybridization
amplitudes. Hence we may introduce an effective nonrelativistic exchange-
dipole spin operator

${\bf\hat{P}}_{12}={\bf\hat{P}}_{12}^{(0)}+{\bf\Pi}_{12}({\bf\hat{s}}_{1}\cdot{\bf\hat{s}}_{2})$
(106)

with an exchange-dipole moment

${\bf\Pi}_{12}=\langle{\bf P}\rangle_{S=1}-\langle{\bf P}\rangle_{S=0}\,$
(107)

and a spinless contribution
${\bf\hat{P}}_{12}^{(0)}=\frac{1}{4}\left(3\langle{\bf
P}\rangle_{S=1}+\langle{\bf P}\rangle_{S=0}\right)$. It is worth noting that
the net local electric polarization lies in the Cu1 \- O - Cu2 plane. As it
was shown in Ref. [39] (see, also Refs. [127]), in general, the exchange-
dipole moment can be written as a superposition of the longitudinal and
transversal contributions as follows:

${\bf\Pi}_{12}=p_{\parallel}{\bf R}_{12}+p_{\perp}{\boldsymbol{\rho}}_{12}\,,$
(108)

(${\bf R}_{12}={\bf R}_{1}-{\bf R}_{2}$, ${\boldsymbol{\rho}}_{12}=({\bf
R}_{1}+{\bf R}_{2})$), where $p_{\parallel}$ does not vanish only for a
specific crystallographic nonequivalence of the centers 1 and 2 when there is
no inversion center even for collinear Cu - O - Cu chain.

The spin-orbital coupling $V_{SO}$ for copper and oxygen ions drives the
singlet-triplet mixing which gives rise to a relativistic contribution to
electric polarization deduced from an effective spin operator, or an exchange-
relativistic-dipole moment which can be written as follows

${\bf\hat{P}}_{12}=-\frac{1}{J_{12}}{\bf\Pi}_{12}\left({\bf
D}_{12}\cdot[\hat{\bf s}_{1}\times\hat{\bf s}_{2}]\right)=$
$-\frac{d_{12}(\theta)}{2l^{2}J_{12}}(p_{\parallel}{\bf
R}_{12}+p_{\perp}{\boldsymbol{\rho}}_{12})\left([{\bf
R}_{12}\times{\boldsymbol{\rho}}_{12}]\cdot[\hat{\bf s}_{1}\times\hat{\bf
s}_{2}]\right)\,.$ (109)

In other words, the exchange-relativistic contribution to the dipole moment is
a superposition of the two (longitudinal and transversal) mutually orthogonal
contributions determined only by the superexchange Cu - O - Cu geometry, while
the spin-current factor does only modulate its value.

The DM type exchange-relativistic dipole moment (109) is believed to be a
dominant relativistic contribution to electric polarization in the Cu1 \- O -
Cu2 or similar clusters. It is worth noting that the exchange-dipole moment
operator (106) and exchange-relativistic-dipole moment operator (109) are
obvious counterparts of the Heisenberg symmetric exchange and Dzyaloshinskii-
Moriya antisymmetric exchange, respectively. Hence, the Moriya like relation
$|\Pi_{ij}|\sim\Delta g/g|{\bf\Pi}|$ seems to be a reasonable estimation for