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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

S2

D · [S1× S2],

(1)

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 [4] for different local symmetries.

Significant progress was recently achieved in ab initio

calculations of D (see [5] and references therein).

Besidesfundamentalinterest,

Moriya interaction is a very important ingredient of

magnetoelectric effects with possible applications to

spintronics.

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.

the Dzyaloshinskii–

∗dmitrien@crys.ras.ru

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

Tjkms1js2k(r1− r2)m,

(2)

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

=−Tkjm,

Dn=1

2ǫjknTjkm(r1− r2)m= Anm(r1− r2)m.

(3)

Using the well known symmetry restrictions on the

third-rank antisymmetric tensors [8] we can obtain from

Eq. (2) all the symmetry restrictions on D found in [4].

In particular, Anm = 0, Tjkm = 0 and Dm = 0 if the

points r1and r2are related by inversion symmetry (rule

1 from [4]). If there is an n-fold rotation axis (n ≥ 2)

along r1− r2then D is parallel to r1− r2(rule 5 from

[4]), etc.

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

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of WF; modern first-principles theoretical considerations

are more appropriate [5]. 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

case.

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 [11] 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 [12]. More delicate situation

with chirality occurs in typical WF crystals with

R¯3c symmetry (α-Fe2O3, MnCO3, etc.) which are

centrosymmetric.

At first let us consider classical WFs, carbonates

of transition metals, for instance MnCO3 [13]. 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-

centers (1

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

(1

4).

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

consideration.

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 [14].

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

2,1

2,1

4,1

4,1

4) and (3

4,3

4,3

6,1

3,1

2,2

3,5

6; in

1

right-hand twistright-hand twistright-hand twistright-hand twist

z =1/6z =1/6z =1/6z =1/6

z =0z =0z =0z =0

left-hand twistleft-hand twistleft-hand twistleft-hand twist

z =1/6 z =1/6z =1/6z =1/6

z =1/3z =1/3z =1/3 z =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[8], 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

chiral.

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.

4) where

4)

1

4, i.e.

6and z =1

3, and for this

1

3the spin orientation

O= 1 − xO ≈

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3

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

(0,0,1

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

[15]). 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 [16] 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,

theyare shownby short

Application of an external magnetic field along one of

these directions makes the corresponding ferromagnetic

domain energetically favorable. And if the orientation

of theferromagneticdomain

Dzyaloshinskii–Moriya interaction fixes the phase of

antiferromagnetic sequence of moments in this domain

[17].

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

disadvantages.

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):

arrows,like in [14]).

isfixed thenthe

Fox(H) = 2Aox[cos2π(hxox+ ℓ/4) + cos2π(kxox+ ℓ/4)

+cos2π(hxox+ kxox− ℓ/4)

= 8Aox(−1)n+1sinπhxoxsinπkxoxsinπ(h + k)xox,(4)

where Aox

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 [14]. 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,

(5)

(6)

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 [18] but in that case

the deviation is introduced by the single-spin anisotropy

[19]. To the best of our knowledge, there were no

attempts to measure with this technique the sign of the

Dzyaloshinskii–Moriya interaction.

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

ferromagnetically or

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moments lead to a macroscopic WF moment. Eq. (6)

transforms to

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

[17]; 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 [5]. 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,

(8)

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 [20] 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 [21]

and FeBO3[22] (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.

Anotherpromisingapproaches

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,

to the sign

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 [26]. Then it

was predicted that there should be some "chiral" dipole-

quadrupole contribution to these reflections [27] and

interference between different contributions (including

magnetic scattering) have been studied in detail for

α-Fe2O3 and Cr2O3 crystals [28]. 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

[28]) 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–

Moriya interaction.

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 [29]

that those codes (we used FDMNES [30]) 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-

00768). Discussions with V.I. Anisimov and E.I. Kats are

gratefully acknowledged.

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