ЖЭТФ, 2021, том 159, вып. 4, стр. 594-597

ANALOGY BETWEEN THE MAGNETIC DIPOLE MOMENT AT

THE SURFACE OF A MAGNETOELECTRIC AND THE ELECTRIC

CHARGE AT THE SURFACE OF A FERROELECTRIC

N. A. Spaldin^*

Department of Materials, ETH Zurich

CH-8093, Zürich, Switzerland

Received November 29, 2020,

revised version December 21, 2020

Accepted for publication December 21, 2020

Contribution for the JETP special issue in honor of I. E. Dzyaloshinskii’s 90th birthday

∫

DOI: 10.31857/S0044451021040027

Energy = - μ(r) · H(r) d³r =

∫

μ(r) · H(0) d³r -

In a linear magnetoelectric material, an applied

∫

electric field induces a magnetization linearly propor-

- r_iμ_j(r)∂_iH_j(0)d³r - . . .

(1)

tional to the field strength, and an applied magnetic

field induces a corresponding linear electric polariza-

tion. The first mention of the phenomenon, to our

Here, the expansion in powers of the field gradients is

knowledge, is in the original 1958 edition of the clas-

calculated at some arbitrary reference point r = 0, and

sic Electrodynamics of Continuous Media by Landau

i, j are Cartesian directions with summation over re-

and Lifshitz [1], with the brief statement that an effect

peated indices implied. The usual magnetic dipole mo-

∫

resulting from a linear relation between the magnetic

ment, m =

μ(r)d³r appears in the first term of the

and electric fields in a substance is possible in princi-

expansion of Eq. (1); the M_ij tensor appears in the

ple. Soon after, Dzyaloshinskii proved using symmetry

second term. When appropriately normalized by the

arguments that the behavior should occur in chromia,

volume in the case of bulk, periodic systems we will

Cr₂O₃ [2]. This was then the material of choice for the

call it the magnetoelectric multipolization, by analogy

first experimental demonstration of the linear magne-

with the magnetization or polarization. It provides a

toelectric effect by Astrov [3].

bulk, thermodynamic quantity associated with “mag-

netoelectricness”, complementing the usual definition

of magnetoelectricity as a response function.

A symmetry requirement for the existence of a lin-

One scenario in which this thermodynamic aspect

ear magnetoelectric response is that both time-reversal,

manifests, which was pointed out by Dzyaloshinskii in

T, and space-inversion, P, symmetries are broken. This

1992 [7], is in the power-law decay of the external mag-

condition is the same as that for a non-zero magneto-

∫

netic field around an antiferromagnetic material with a

electric multipole tensor, M_ij =

r_iμ_j(r) d³r, which

net non-zero magnetoelectric multipolization. Power-

is the second order coefficient in the multipole expan-

law behavior is fundamentally different from the ex-

sion of the energy of a spatially varying magnetization,

ponential field decay expected around a conventional

μ(r), in a spatially varying magnetic field, H(r) [4-6]:

centro- or time-reversal symmetric antiferromagnet [7].

In the particular case of the prototypical magnetoelec-

tric Cr₂O₃, which is uniaxial and has non-zero magne-

toelectric multipolization below its Néel temperature,

Dzyaloshinskii showed that the external field should

^* E-mail: nicola.spaldin@mat.ethz.ch

have the angular form of a magnetic quadrupole. As in

594

ЖЭТФ, том 159, вып. 4, 2021

Analogy between the magnetic dipole moment. . .

Fig. 1. a) Surface charge associated with ferroelectric polarization, P. b) Surface magnetic dipole moment associated with

magnetoelectric multipolization, Mzz, which can be represented as the sum of a magnetoelectric monopole and z² quadrupole.

The - signs, + signs and small black arrows on the surfaces indicate negative charge, positive charge and magnetic dipole

moments. The ferroelectric has negative charge on its lower surface and positive charge on its upper surface; the magnetoelectric

has positive magnetic dipole moments (pointing outwards from the sample) on both its upper and lower surfaces

the case of the original magnetoelectric response predic-

be conveniently described in terms of the bulk magne-

tion, this was subsequently confirmed by Astrov [8, 9],

toelectric multipolization that is analogous to the fer-

although the measured field strength was smaller in

roelectric polarization. We define the intrinsic surface

magnitude than predicted. The intrinsic bulk nature

magnetization to be this surface magnetic dipole mo-

of the measured external field dependence was sub-

ment per unit area, and provide a convenient recipe

sequently questioned, however, when it was pointed

for extracting it for any surface plane, from knowledge

out that any antiferromagnet can in principle have a

of the bulk magnetic order. We demonstrate the pro-

surface magnetization that, depending on the sample

cedure for the prototypical magnetoelectric material,

shape and choice of surface termination, could give rise

Cr₂O₃, in which Dzyaloshinskii first identified the lin-

to a magnetic field [10]. The discussion was further

ear magnetoelectric effect, and compare the value of the

enriched by recent theoretical demonstrations that cer-

intrinsic surface magnetization to recent experimental

tain surfaces of a magnetoelectric antiferromagnet will

measurements. Finally, we show that the description is

always have a surface magnetization [11] and associ-

also relevant for non-magnetoelectric antiferromagnets,

ated external magnetic field [12] as a consequence of

allowing a classification into one of two types with fun-

the bulk magnetoelectric multipolization.

damentally different surface magnetic properties: the

trivial case, in which the allowed magnetoelectric mul-

In this paper, we revisit Dzyaloshinskii’s pioneer-

tipolization values contains zero, and non-trivial anti-

ing work on the linear magnetoelectric effect in light

ferromagnets whose magnetoelectric multipolization is

of the modern theory of ferroelectric polarization, and

non-zero, in spite of their not being magnetoelectric.

approach the description of the surface magnetism of

magnetoelectric antiferromagnets by making a corre-

From a quick glance at the units of electric polar-

spondence with the surfaces of ferroelectrics. We show

ization, which are dipole moment per unit volume or

that the surface magnetic dipole moment associated

equivalently charge per unit area, it is clear that a sur-

with magnetoelectric materials is analogous to the

face perpendicular to the polarization direction in a

bound surface charge in ferroelectrics, in that it can

ferroelectric material carries a bound charge per unit

595

N. A. Spaldin

ЖЭТФ, том 159, вып. 4, 2021

electric polarization [14, 15]. For a particular choice of

surface plane orientation and atomic termination, we

identify the unit cell that tiles the semi-infinite slab;

an example for the (0001) surface of Cr₂O₃ is shown in

Fig. 2. We then calculate the magnetoelectric multi-

pole of that unit cell, and normalize it to the unit cell

volume; by analogy with the ferroelectric case we call

this M^bulk. For the illustrated surface, domain and

unit cell of Cr₂O₃, only M^bulkzz is non-zero, and it has

the value -2.35μ_B/nm² (taking the atomic positions

and lattice parameters from Ref. [16]). The surface

magnetic dipole per unit area, which we define to be

the intrinsic surface magnetization, can then be read

off directly from the i, j components of the M^bulk ten-

sor, with the first index, i, indicating the x, y or z

Fig. 2. (Color online) A semi-infinite slab of Cr2O3 with a

orientation of the surface magnetic dipole moments at

(0001) surface, shown projected down the y axis. Cr and O

the surface plane normal to the second index, j. For

ions are shown in blue and red respectively, and the arrows in-

the case of Cr₂O₃ both (0001) surfaces shown have a

dicate the directions of the local magnetic moments on the Cr

surface magnetization of 2.35μ_B/nm² pointing into the

ions. The symbol . . . (black dots) indicate continuation of the

sample.

structure. The black rectangle shows a choice of hexagonal

In the full manuscript, the procedure is also ap-

unit cell, which, in combination with the numbered Cr ions,

plied to calculation of the interfacial magnetism in het-

can be periodically repeated to tile the slab

erostructures of Fe₂O₃/Cr₂O₃, and to model non-mag-

netoelectric systems.

area equal to the value of the polarization, with the sign

Summary and outlook. In summary, we re-

of the surface charge given by the direction of polariza-

viewed the phenomenology of magnetoelectric multi-

tion, as shown in Fig. 1a. (For a rigorous derivation

polization in bulk, periodic solids, and provided an

see Ref. [13].)

analogy with various aspects of the ferroelectric po-

While the ferroelectric polarization has units of

larization. We showed that the analogy provides a par-

charge per unit area, the magnetoelectric multipoliza-

ticularly convenient picture of the surface magnetiza-

tion, or magnetoelectric multipole per unit volume, has

tion that is associated with magnetoelectric materials

units of magnetic dipole moment per unit area. There-

[10, 11], and we provided the following straightforward

fore, by analogy with the ferroelectric case, the surface

recipe to extract it from the bulk magnetoelectric mul-

of a magnetoelectric should have a magnetic dipole mo-

tipolization for a given surface plane:

ment per unit area, whose size and orientation depends

1) for the surface plane and chemistry of interest,

on the bulk magnetoelectric multipolization. We refer

identify the unit cell and ionic basis that tiles a semi-

to this as the intrinsic surface magnetization, since it re-

infinite slab of the system;

sults from a bulk property of the material; it is this sur-

2) calculate the components of the bulk magneto-

face magnetization that was discussed in Ref. [11]. In

electric multipolization, M^bulkij, using this unit cell and

Fig. 1b we illustrate the analogy with ferroelectricity for

basis of ions, and normalizing it to the unit cell volume;

the case of a uniaxial magnetoelectric such as Cr₂O₃ in

3) the non-zero components of M^bulkij that have a

which the M_zz component of the magnetoelectric mul-

contribution normal to the surface plane then give di-

tipolization tensor (which can be decomposed into the

rectly the size and orientation of the intrinsic surface

magnetoelectric monopolar and z² quadrupolar contri-

magnetization.

butions shown) is non-zero. The M_zz component re-

We argued that such an intrinsic surface magnetiza-

sults in a z-oriented magnetic moment pointing away

tion is possible even at the surface or interface of a

from the sample on the (001) and (001) surfaces in this

non-magnetoelectric material, and distinguished two

example.

cases: In non-magnetoelectric materials whose multi-

Our procedure for extracting the surface magnetiza-

polization lattice contains zero it is always possible to

tion of a semi-infinite slab of an antiferromagnet from

choose a stoichiometric termination with zero magnetic

its bulk magnetoelectric multipolization follows that

moment for any choice of surface plane, although this

for determining the surface charge from the bulk ferro-

might not necessarily be the lowest energy termina-

596

ЖЭТФ, том 159, вып. 4, 2021

Analogy between the magnetic dipole moment. . .

tion. In non-magnetoelectric materials whose multi-

I. E. Dzyaloshinskii, Sov. Phys. JETP 10, 628 (1960).

polization lattice contains the half-multipolization in-

D. N. Astrov, Sov. Phys. JETP 11, 708 (1960).

crement, in contrast, surface planes exist for which an

intrinsic magnetic moment can not be avoided for sto-

C. Ederer and N. A. Spaldin, Phys. Rev. B 76, 214404

ichiometric terminations.

(2007).

We mentioned some phenomena for which these

N. A. Spaldin, M. Fiebig, and M. Mostovoy, J. Phys.

concepts might be relevant and which could provide

Condens. Matter 20, 434203 (2008).

interesting directions for future work. In particular,

the intrinsic surface magnetization arising from the

N. A. Spaldin, M. Fechner, E. Bousquet, A. Balatsky,

magnetoelectric multipolization could have implica-

and L. Nordström, Phys. Rev. B 88, 094429 (2013).

tions for the relative stability of antiferromagnetic

I. Dzyaloshinskii, Sol. St. Commun. 82, 579 (1992).

surfaces and interfaces, the formation of antiferromag-

netic domains, and the mechanism of exchange-bias

D. N. Astrov and N. B. Ermakov, JETP Lett. 59,

coupling. Finally, we suggested some experiments that

297 (1994).

could be used to verify or disprove our proposals, and

D. N. Astrov, N. B. Ermakov, A. S. Borovik-Roma-

we hope, in the spirit of Igor Dzyaloshinskii, that this

nov, E. G. Kolevatov, and V. I. Nizhankovskii, JETP

manuscript motivates future experimental work in

Lett. 63, 745 (1996).

these directions.

10.

A. F. Andreev, JETP Lett. 63, 758 (1996).

Funding. This work was supported by the Körber

Foundation, the European Research Council (ERC) un-

11.

K. D. Belashchenko, Phys. Rev. Lett. 105, 147204

der the European Union’s Horizon 2020 research and

(2010).

innovation programme grant agreement No 810451 and

by the ETH Zurich.

12.

Z. Jiang, D. West, and S. Zhang, Phys. Rev. B 102,

174411 (2020).

Acknowledgments. Thanks to Igor Dzyaloshin-

skii for the many inspiring discussions and fruitful

13.

R. D. King-Smith and D. Vanderbilt, Phys. Rev.

collaborations, and to Sayantika Bhowal, Christian

B 47, R1651 (1993).

Degen, Manfred Fiebig, Pietro Gambardella, Kane

Shenton, Tara Tošić and Xanthe Verbeek for helpful

14.

D. Vanderbilt and R. D. King-Smith, Phys. Rev.

comments on the manuscript.

B 48, 4442 (1993).

The full text of this paper is published in the English

15.

M. Stengel, Phys. Rev. B 84, 205432 (2011).

version of JETP.

16.

M. H. Dehn, J. K. Shenton, S. Holenstein,

Q. N. Meier, D. J. Arsenau, D. L. Cortie, B. Hitti,

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