arXiv:1008.3092v1 [cond-mat.mtrl-sci] 18 Aug 2010
Dzyaloshinskii–Moriya interaction: How to measure its sign in weak ferromagnetics?
Vladimir E. Dmitrienko∗, Elena N. Ovchinnikova†, Jun Kokubun‡, Kohtaro Ishida‡
A.V. Shubnikov Institute of Crystallography, 119333 Moscow, Russia
†Department of Physics, Moscow State University, Moscow, Russia
‡Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan
Three experimental techniques sensitive to the sign of the Dzyaloshinskii–Moriya interaction are
discussed: neutron diffraction, M¨ ossbauer γ-ray diffraction, and resonant x-ray scattering. Classical
examples of hematite (α-Fe2O3) and MnCO3 crystals are considered in detail.
PACS numbers: 61.05.C-, 61.05.F-, 76.80.+y
Weak ferromagnetism (WF) of antiferromagnetics is a
classical example of an initially small and controversial
physical problem that later produces a strong impact on
the general picture of magnetic phenomena. From the
very beginning, the modern theoretical consideration of
WF developed by Dzyaloshinskii and Moriya was based
on symmetry arguments, both phenomenological [1, 2]
and microscopic [3, 4]. It was shown that appropriate
crystal symmetry allows the following term in the
interaction of two antiferromagnetic sublattices S1 and
D · [S1× S2],
which favors to (usually small) canting angle between
S1 and S2; here D is a vector parameter of the
Dzyaloshinskii–Moriya interaction. Possible directions
of D were found  for different local symmetries.
Significant progress was recently achieved in ab initio
calculations of D (see  and references therein).
Moriya interaction is a very important ingredient of
magnetoelectric effects with possible applications to
The canted spin arrangement is just responsible for
WF. Both the magnitude of WF and the canting angle
are proportional to |D| and therefore it seems that the
sign of D is not important at all. According to Eq.
(1), the sign obviously depends on our choice which of
sublattices is 1 or 2 and therefore it is usually claimed
that the sign is conventional. From the phenomenological
point of view this is true because in macroscopic
theory the phase of antiferromagnetic arrangement is
not fixed relative to the crystal lattice. However at
the atomic level the phase can be fixed owing to the
Dzyaloshinskii–Moriya interaction and the sign of this
interaction is crucial for relation between the local crystal
structure and magnetic ordering. For instance, this sign
determines the handedness of spin helix in crystals with
the noncentrosymmetrical B20 structure [6, 7]. In this
paper we show how one can measure it in classical WF
crystals like α-Fe2O3or MnCO3.
Let us rewrite Eq. (1) in a more invariant form not
depending on any arbitrary choice of sublattices. If two
atoms with spins s1and s2are located at the points r1
and r2, then we can add the following scalar to the energy
of their interaction
where an antisymmetric tensor, Tjkm
characterizes interaction of spins s1 and s2 through
intermediate crystal matter. The properties (in particular
symmetry) of the intermediate matter determines the
properties of this tensor including symmetry restrictions
on its tensor components. It is well known also that
any third-rank antisymmetric tensor is equivalent to
a second-rank pseudo-tensor Anm: Tjkm = ǫjknAnm
where ǫjkna unitary antisymmetric pseudo-tensor (Anm
changes its sign under inversion). The relation between
D, Anmand Tjkmis given by
2ǫjknTjkm(r1− r2)m= Anm(r1− r2)m.
Using the well known symmetry restrictions on the
third-rank antisymmetric tensors  we can obtain from
Eq. (2) all the symmetry restrictions on D found in .
In particular, Anm = 0, Tjkm = 0 and Dm = 0 if the
points r1and r2are related by inversion symmetry (rule
1 from ). If there is an n-fold rotation axis (n ≥ 2)
along r1− r2then D is parallel to r1− r2(rule 5 from
However, there is an important principal difference
between D and Tjkm: tensor Tjkm (or pseudo-tensor
Anm) can be considered as a field on the lattice, it should
be invariant relative to all the symmetry operations of the
space group. In particular, it is determined by the same
parameters at all equivalent lattice points; of course, one
should take into account corresponding crystallographic
operations connecting those equivalent points: rotations
(changing orientations of the principal axes) and space
inversions (changing signs of all components of Tjkm).
On the contrary, the pattern of vector D on the
lattice cannot be obtained by pure crystallographic
operations and some additional consideration is needed
(see discussion of La2CuO4in [9, 10]).
We conclude this short introduction with a remark
that Eq. (2) cannot be used for quantitative description
of WF; modern first-principles theoretical considerations
are more appropriate . Nevertheless this expression can
be used for better understanding of symmetry aspects of
the problem and now we will show that this is really the
The appearance of the antisymmetric third-rank tensor
suggests an idea that there is some chiral effect behind
this. In fact, it is known for a long time  that
the Dzyaloshinskii–Moriya interaction can produce long-
period magnetic spiral structures in ferromagnetic and
antiferromagnetic crystals lacking inversion symmetry.
This effect was suggested for MnSi and other crystals
with B20 structure [6, 7] and it has been carefully proved
that the sign of the Dzyaloshinskii–Moriya interaction,
hence the sign of the spin helix, is determined by
the crystal handedness . More delicate situation
with chirality occurs in typical WF crystals with
R¯3c symmetry (α-Fe2O3, MnCO3, etc.) which are
At first let us consider classical WFs, carbonates
of transition metals, for instance MnCO3 . In its
primitive rhombohedral unit cell with the space group
R¯3c, there are two Mn atoms at crystallographically
equivalent inversion centers¯3, vertices (0,0,0) and body-
2). Atoms at the vertices and body-centers
have almost opposite magnetic moments lying in the
planes normal to the threefold axis. According to the
crystal symmetry the moments should not be exactly
opposite and in fact the moments are slightly canted
so that the resulting WF moment is along one of three
twofold axes. There are also two carbon atoms at points
The physical origin of WF in MnCO3 is a weak
relativistic interaction between spins in the lattice with
the R¯3c space group. But what is the structural origin
of this R¯3c symmetry? If we consider only Mn atoms,
the symmetry of lattice would be R¯3m and WF would
be impossible. This symmetry will not change if the
carbon atoms are taken into account. Only oxygen atoms
change R¯3m symmetry to R¯3c; thus their configuration
is crucial for the value (and sign) of the Dzyaloshinskii–
Moriya interaction and it is worthy of a more careful
In MnCO3, there are six hexagonal Mn layers per the
lattice period along the threefold axis, so that atoms of
the next layer is just under the centers of triangles formed
by atoms from the previous layer, and layer sequence is
ABCABC ..., like in the fcc lattice. These equidistant
layers has z-coordinates equal to z = 0,1
each layer all the spins are parallel and are lying in the
layer plane. The spin of neighboring layers are almost
opposite. Here and below the standard hexagonal setting
of the rhombohedral lattice is used .
Considering the first two layers one can see that
between them, at z =
12, there is a layer of oxygen and
carbon atoms; the point symmetry of this layer is 32,
just because of low symmetry of oxygen positions: oxygen
atoms are at the 18e positions with point symmetry
4) and (3
right-hand twistright-hand twistright-hand twistright-hand twist
z =1/6z =1/6z =1/6z =1/6
z =0z =0z =0 z =0
left-hand twistleft-hand twistleft-hand twistleft-hand twist
z =1/6z =1/6z =1/6z =1/6
z =1/3z =1/3z =1/3z =1/3
Рис. 1: Right-hand and left-hand twists of moments between
layers alternating along z axis. Triangles indicate the threefold
axis normal to the figure plane; all possible directions of
WF moments in MnCO3 (twofold axes) are shown by small
arrows. Big arrows are spin directions in neighboring layers
at different z levels for the case when external magnetic field
is applied in horizontal direction and DM is positive. A bold
small arrow indicates the direction of WF moment.
2 and with coordinates equivalent to (xO,0,1
xO≈ 0.27. Carbon atoms are at the 6a positions (0,0,1
with point symmetry 32.
It is very important that this intermediate layer is
noncentrosymmetric and therefore both vector D and
tensor Tjkmcan have some nonzero values. For pairwise
interaction of spins from different layers, tensor Tjkm
has symmetry 1, but being averaged over all pairs it
has of course the symmetry of the intermediate layer.
For symmetry 32, tensor Tjkm is determined by two
independent parameters, say Tyzxand Txyz, but only
the latter leads, according to Eq. (2), to a twist angle
between the Mn spins lying in the first and second layers.
The sign of this twist angle is just the sign of Txyz. This
twist violates the right-left symmetry and, in a figurative
sense, we can say that the intermediate C-O layer is
The next intermediate C-O layer is at z =
between the Mn layers at z =1
layer all the components of tensor Tjkmchange sign due
to inversion centers at z =1
6. Thus the "chirality" of this
layer is opposite to that of the first intermediate layer,
hence the small twist angle between spins is also opposite
and for the Mn layer at z =
exactly coincides with the spin orientation for the Mn
layer at z = 0 (see Fig. 1). And then this repeats from
layer to layer. We see that two alternating local twists,
left and right, between neighboring Mn layers result in
macroscopic canting angle between magnetic sublattices
in centrosymmetric WF crystals.
If we change the sign of xO so that x′
0.73, then the layer "chirality" changes to opposite, Txyz
change the sign and the twist angle also change the sign.
6and z =1
3, and for this
3the spin orientation
O= 1 − xO ≈
From pure crystallographic point of view both values,
xO and x′
O, are equivalent, they simply correspond to
the lattice origin shifted at a half-period, from (0,0,0) to
2). In the crystallographic databases, both values
of xOare cited for α-Fe2O3; nevertheless sometimes the
first-principle results for WF in α-Fe2O3do not indicate
which value of xO they really adopt (see for instance
). We see also that in R¯3c crystals the sign of the
Dzyaloshinskii–Moriya interaction changes to opposite at
one half-period, therefore the idea to measure this sign
by the M¨ ossbauer absorption  cannot be correct.
According to R¯3c symmetry there are six possible
orientations of the WF moments (along plus and
minus directions of three twofold axes; in Fig. 1,
Application of an external magnetic field along one of
these directions makes the corresponding ferromagnetic
domain energetically favorable. And if the orientation
Dzyaloshinskii–Moriya interaction fixes the phase of
antiferromagnetic sequence of moments in this domain
Now we are ready to consider the main item of this
paper: how to measure the sign of the Dzyaloshinskii–
Moriya interaction in WFs? First of all, a strong
enough magnetic field should be applied to obtain
the single domain state where the Dzyaloshinskii–
Moriya interaction 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, M¨ ossbauer γ-ray diffraction, and resonant x-
ray scattering. We will discuss now their advantages and
Because of the layered magnetic structure alternating
along z-axis, the reflections with strong magnetic
scattering correspond to the reciprocal lattice vectors
Hhkℓ with odd ℓ. At first let us consider scattering on
oxygen atoms; the expressions for the oxygen structure
amplitudes Fox(H) are similar for all three techniques
and looks like (ℓ = 2n + 1):
Fox(H) = 2Aox[cos2π(hxox+ ℓ/4) + cos2π(kxox+ ℓ/4)
+cos2π(hxox+ kxox− ℓ/4)
= 8Aox(−1)n+1sinπhxoxsinπkxoxsinπ(h + k)xox,(4)
scattering factor (for x-rays) or to the nuclear scattering
length (for neutrons); it is practically real because oxygen
absorption is very small for thermal neutrons or hard
x-rays. There is no contribution from carbon . Here
and below some factors (such as atomic and magnetic
formfactors, the Debye–Waller factor, etc.) are omitted
because corresponding expressions are well known and
is proportional to the oxygen atomic
implemented into routine computer programs used for
diffraction experiments. It is clear from this equation that
one should measure reflections with hk(h + k) ?= 0.
In the case of neutron diffraction, one can adopt
the standard technique using the polarization ratio
R, i.e. the ratio of reflection intensities for incoming
neutrons with spin σ parallel and antiparallel to the
direction of applied magnetic field. For ℓ = 2n + 1
this ratio is given by the following expression containing
interference between nuclear scattering by oxygen atoms
and magnetic scattering by Mn atoms
R(hkℓ) =|Fox(H) + σ · Q(H)|2
|Fox(H) − σ · Q(H)|2
=|Fox(H) − sDMAmagk(2h + k)|2
|Fox(H) + sDMAmagk(2h + k)|2,
where Q(H) is the magnetic structure amplitude for
reflection H, Q(H) ∝ MH− H(H · MH)/H2, MH
is the correspondent Fourier harmonic of the vector
field describing the electron-magnetization distribution,
sDMis the sign of the Dzyaloshinskii–Moriya interaction
between the first two layers of Mn atoms and Amag
includes all routine factors describing neutron magnetic
scattering. Calculating (6) from (5) we took into
account the geometry shown in Fig. 1 (i.e. σ is
directed horizontally, MH is directed vertically so that
σ · MH
=0 if small canting is neglected, etc.).
In particular, factor sDM appears just because the
phase of antiferromagnetic sequence is fixed by the
Dzyaloshinskii–Moriya interaction. All other factors in
Eq. (6) are more or less known and one can determine
sDM from rather rough measurements giving R(hkℓ) <
1 or R(hkℓ) > 1. Notice that there is an additional
condition for this measurements, 2h + k ?= 0, which
appears in Eq. (6) from H · MH
confusion we should notice that expression (4) obeys
threefold symmetry whereas magnetic scattering does
not, because of the external field applied perpendicular
to the threefold axis.
The technique of polarized neutron diffraction was
used for measurements of the sign of small angular
deviations of moments in MnF2  but in that case
the deviation is introduced by the single-spin anisotropy
. To the best of our knowledge, there were no
attempts to measure with this technique the sign of the
In hematite (α-Fe2O3), the situation is slightly more
complicated because iron atoms are in the positions
(0,0,zFe) with the point symmetry 3 and neighboring
Felayers are coupledeither
antiferromagnetically.Like in MnCO3, the R¯3c symmetry
is induced by oxygen atoms. The Fe layers coupled
feromagnetically are related by inversion and, according
to the Morya rules, there is no canting between them
(Tjkm = 0 for inversion centers). The antiferomagnetic
neighboring layers interact via an oxygen layer with
symmetry 32 and alternating right and left twists of their
?= 0. To avoid
moments lead to a macroscopic WF moment. Eq. (6)
R(hkℓ) =|Fox− sDMAmagk(2h + k)cos2πℓzFe|2
|Fox+ sDMAmagk(2h + k)cos2πℓzFe|2. (7)
The additional factor cos2πℓzFe (where zFe ≈ 0.355)
allows us to change the value and sign of magnetic
scattering just changing ℓ. A suitable reflection (210
rhombohedral, i.e. 2¯13 hexagonal) had been studied in
; however, as it was noted in that paper, the result
was controversial: in the great majority of observations,
R(2¯13) = 1/R(¯21¯3) whereas R(2¯13) = R(¯21¯3) would
be expected from the symmetry of hematite. Thus we
cant extract the sought sign of the Dzyaloshinskii–Moriya
interaction and more careful experiments are needed.
It should be again emphasized that our symmetry-
based arguments are only qualitative: the ab initio
calculations for α-Fe2O3 show that the torque induced
by neighboring antiferomagnetic layer is opposite to the
total torque . This confirms importance of suggested
direct measurements of the sign of the total torque.
The M¨ ossbauer diffraction can be used in a similar
way. In this case, there is no need to vary the photon
polarization and one can study intensity Ihkℓ(E) of
reflections as a function of photon energy E:
Ihkℓ(E) = |Fox(H) − sDMBmag(H,E)|2,
where Bmag(H,E) is the magnetic M¨ ossbauer structure
factor of the hkℓ reflections with ℓ = 2n+1,hk(h+k) ?= 0.
The function Bmag(H,E) is well known  and its
real part, which interferes with the first term in (8),
changes sign when E passes through resonances provided
by the hyperfine splitting of nuclear levels. This should
facilitate the observation of interference between the
two terms. For57Fe both terms in (8) may be of the
same order of magnitude. The M¨ ossbauer diffraction was
observed in many crystals including WF α-Fe2O3 
and FeBO3 (an analog of MnCO3) but these studies
were concentrated mainly on pure magnetic scattering
rather then on its interference with scattering on oxygen
atoms. Contrary to neutrons, the M¨ ossbauer diffraction
can be used for very thin layers but the number of
possible crystals is rather limited by the list of suitable
M¨ ossbauer isotopes.
measurements can be related with resonant x-ray
diffraction, i.e. diffraction near x-ray absorption edges.
It is sensitive both to structural and magnetic ordering
especially near L absorption edges (see recent surveys
[23–25]). However, in d magnetic metals, the K edge
is the only appropriate for diffraction, and for this
edge magnetic scattering is several orders of magnitude
smaller than conventional charge scattering by electrons.
Thus, for reflections of ℓ = 2n + 1,hk(h + k) ?= 0 type,
the intensity is given by eq. (8) with the second term
much smaller then the first one. Therefore the reliable
observation of interference between two terms will be
perhaps very difficult.
However, resonant x-ray scattering provides another
nontrivial way to measure sDM. The asymmetric oxygen
environment of transition metals induces some additional
anisotropy of their non-magnetic scattering amplitude,
so that just owing to this anisotropy the reflections with
ℓ = 2n + 1 can be excited even if hk(h + k) = 0. These
reflections do not exist out of the resonant region and
they are referred to as "forbidden reflections". There is no
direct contribution to forbidden reflections from oxygen
atoms, equation (4) gives zero, but the sign of the induced
anisotropy depends on asymmetrical arrangement of the
oxygen atoms and, correspondingly, the sign of the non-
magnetic structure amplitude of forbidden reflections
with ℓ = 2n + 1,hk(h + k) = 0 is proportional to
the sign of xO. For crystals with R¯3c symmetry these
reflections were first observed in α-Fe2O3 . Then it
was predicted that there should be some "chiral" dipole-
quadrupole contribution to these reflections  and
interference between different contributions (including
magnetic scattering) have been studied in detail for
α-Fe2O3 and Cr2O3 crystals . It was shown that
the azimuthal dependence of reflection intensity could
be strongly influenced by this interference (especially
for the weak 009 reflection, see Figs. 11 and 12 from
) and orientation of antiferromagnetic moment was
determined in α-Fe2O3 from the observed azimuthal
dependence (without external magnetic field). Exactly
the same measurements in orienting magnetic field would
allow us to determine the sign of the Dzyaloshinskii–
The only problem with the last method is that we
should rely on the sign of the x-ray anisotropy of iron
atoms calculated with rather sophisticated computer
codes. However it was proved experimentally for Ge 
that those codes (we used FDMNES ) are rather
reliable. It is worth noting that resonant x-ray diffraction
and M¨ ossbauer diffraction are element sensitive and
moreover the former can distinguish orbital and spin
contributions to magnetic moments.
In conclusion, we see that the experiments similar
to those needed for the sign measurements had been
already performed (some of them long time ago) for all
three considered techniques. Therefore we believe that
this paper will stimulate these measurements in different
types of the weak ferromagnetics.
This work is partly supported by Presidium of Russian
Academy of Sciences (program 27/21) and by the
Russian Foundation for Basic Research (project 10-02-
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