Magnetic symmetry based deﬁnition of the chirality in the magnetically ordered media

8/6/2019 Magnetic symmetry based definition of the chirality in the magnetically ordered media

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Magnetic symmetry based definition of the chirality in

the magnetically ordered media

B. M. Tanygin

Kyiv Taras Shevchenko National University, Radiophysics Faculty, Glushkov av.4g, Kyiv,

Ukraine, MSP 01601

Abstract

Different definitions of the chirality have been considered for their applica-tions to the micromagnetic structures. It was suggested to use magneticsymmetry based Barron chirality definition. The Barron chirality was ob-tained for all magnetic domain walls as the example. The symmetry basedclassification of the domain walls has been used for the chirality determina-tion.

Keywords: Enantiomorphism, Barron chirality, Kelvin chirality,magnetically ordered medium, domain wall symmetry

1. Introduction

Chiral magnets investigations progress is caused by their usage in spin-tronics [1]. The term chirality was defined first time by Lord Kelvin [2]. Itis obvious that this parameter is widely using in all branches of the appliedscience. In case of the magnetic symmetry, the problems of the definition ofchirality have been formulated in conception of the complete symmetry byI.S. Zheludev [3] and investigations of L.D. Barron [4]. The latter showedthat in case of the systems with motion (magnetically ordered media) the dif-ferent types of the enantiomorphism exist regarding time direction parity. So,time-invariant and time-noninvariant enantiomorphism should be consideredseparately [5]. The difference between Barron and Kelvin definition is ap-pearing of two sub-types of enantiomorphic objects. The time-noninvariant

Email address: [email protected] (B.M.Tanygin)

Preprint submitted to Physica B: Condensed Matter July 11, 2011

Please, cite original work as:

B.M. Tanygin,Magnetic symmetry based definition of the chirality in the magnetically ordered media,

Physica B: Physics of Condensed Matter 406 (2011), pp. 3423-3424

http://dx.doi.org/10.1016/j.physb.2011.06.012

e-mail: [email protected]: http://sites.google.com/site/btanygin/


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enantiomorphism was called as the false chirality in contrast to the time-

invariant one (true chirality) [6].Frustrated magnets have specific chirality definitions: vector [7] and scalar[8] chirality. These definitions do not coincide with Kelvin one. The vectorchirality definition is widely used for the cycloid noncolinear magnetic order-ing [1]. Rotation of the cycloid around two-fold symmetry axis transformsright cycloid into the left. So, such type of the magnetic ordering (chi-ral cycloidal ordering) is achiral in scope of the Kelvin definition. Thisambiguity was considered in [9].

Consequently, the application of the chirality definition to the micromag-netic structures introduces some kind of contradictions. Definition of thechirality in magnetically ordered media should be harmonized with geomet-

ric definition to use the same formal language in different branches of theapplied science. For example, stereochemistry and condensed matter physicscan be related in material synthesis. The problems with chirality definitioncan appear, for example, in case of the synthesis of the chiral material.

Thus, the suggestion of the unified definition of the chirality for magnet-ically ordered media is the purpose of this work.

2. Symmetry based enantiomorphism

The unified approach of any type of definition means using of some typeof the formal theory. In case of the enantiomorphism definition the au-thor suggestion is to use the magnetic symmetry systematization as basis forenantiomorphism determination. Let us choose such constructive exampleas enantiomorphism definition of the magnetic domain walls (DWs).

The DW has enantiomorphism if its magnetic point group Gk [10] doesnot contain symmetry transformations n, where n is positive integer, i.e.symmetry should not include spatial inversion combined with any kind ofrotations. If group Gk contains symmetry transformation n

then DW enan-tiomorphism is time-noninvariant. Otherwise, it is time-invariant. Symmetryclassification of the magnetic 180 DWs have been presented in [11]. Exten-sion of this classification to the any type of DW (including 0 DW [12]) have

been provided in [10].Complete set of magnetic DW classes contains 64 ones (1


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(2 DW) and 42 classes (k = 2, 6


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The important difference between 0 DW and other types of DW is ex-

istence of the Bloch DW without enantiomorphism. The correspondencebetween DW symmetry classes and types of magnetization distribution ispresented in [11] and [10].

3. Discussion

Magnetic symmetry class of the DW is a sub-group of magnetic symme-try class Gp of the crystal paramagnetic phase [10]. Influence of the crystalsample shape on the micromagnetic structure (for example via dipole-dipoleinteraction) and all macroscopic properties of the DW (Neumann principle)means that real paramagnetic phase class must include only elements pre-

sented in class GS of crystal sample shape (Curie principle). Let us introducesymmetry class of spatially restricted crystal in paramagnetic phase [14]:

Gp = G

p GS (1)

In case of the (hkl) film or plate the class of crystal sample shape is givenby: [hkl]/mmm1

. In arbitrary magnetically ordered medium sample theclass of the DW must satisfy the following requirement:

Gk

Gp GS Gp (2)

Also, class Gk depends on the domains magnetizations [11], DW plane ori-entation [10], magnetization distribution [11] and electrical polarization [13]in the DW volume. All these factors are taking into account automaticallyin the symmetry based analysis [11].

If Gp has time-invariant enantiomorphism then all DWs of this materialhave the same enantiomorphism. If Gp has time-noninvariant enantiomor-phism then DWs can have either time-invariant or time-noninvariant enan-tiomorphism. If Gp is non-enantiomorphic then DW can have any kind ofenantiomorphism. The DW with time-invariant enantiomorphism can appearin any magnetic material.

The whole discussed theory can be generalized to the weak ferromagnetics

with non-collinear magnetic ordering as well as to the antiferromagneticsusing algorithms specified in [15] and [11] respectively. Also, presented heresymmetry classification automatically covers the structural changes causedby the DW motion [11].

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Presented here symmetry based classification of the magnetic DW enan-

tiomorphism is connected with different types of magnetoelectric effects asit was shown in: [16], [9] and [17].

4. Conclusions

Thus, the symmetry based classification is an approach to the formaldetermination of the Kelvin or Barron enantiomorphism. It was shown forall magnetic domain walls as example. All 64 magnetic point groups of theDWs have been distributed between time-invariant enantiomorphic, time-noninvariant enantiomorphic and non-enantiomorphic types.

5. Acknowledgements

I would like to express my sincere gratitude to my supervisor Prof. V.F.Kovalenko for his outstanding guidance, to my wife D.M. Tanygina for spellchecking, to Dr. A.P. Pyatakov for helpful discussion of flexomagnetoelectriceffects and to Dr. O.V. Tychko for valuable support in scientific methodology.

References

[1] M. Bode, M. Heide, K. von Bergmann et al., Nature 447 (2007) 190.

[2] Lord Kelvin, Baltimore Lectures on Molecular Dynamics and the WaveTheory of Light, in C.J. Clay and Sons (Eds.), Cambridge UniversityPress Warehouse, London 1904, Appendix H. 22, footnote p. 619.

[3] I.S. Zheludev, Soviet Physics Crystallography 5(1960)508.

[4] L. D. Barron, Nature 405 (2000) 895.

[5] L. D. Barron, New developments in molecular chirality, Kluwer Acad.Publishers, Dordrecht, 1991.

[6] L. D. Barron, True and false chirality and absolute asymmetric synthesis,

J. Am. Chem. Soc. 108 (1986) 5539.

[7] D. Grohol, K. Matan, J.-H. Cho et al., Nature Materials 4 (2005) 323.

[8] D. A. Rabson, S. A. Trugman, J. Phys.: Condens. Matter 7 (1995) 9005.

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[9] B. M. Tanygin, J. Magn. Magn. Mater 323 (2011) 616.

[10] B. M. Tanygin, O. V. Tychko, Phys. B: Condens. Matter 404 (2009)4018.

[11] V. G. Baryakhtar, V. A. Lvov, D. A. Yablonskiy, JETP 87 (1984) 1863.

[12] R. Vakhitov, A. Yumaguzin, J. Magn. Magn. Mater. 52 (2000) 215.

[13] V.G. Baryakhtar, V.A. Lvov, D.A. Yablonskiy, Theory of electric po-larization of domain boundaries in magnetically ordered crystals, in: A.M. Prokhorov, A. S. Prokhorov (Eds.), Problems in solid-state physics,Chapter 2, Mir Publishers, Moscow, 1984, pp. 56-80.

[14] B. M. Tanygin, O. V. Tychko, Acta Physic. Pol. A. 117 (2010) 214.

[15] V.G. Baryakhtar, E.V. Gomonaj, V.A. Lvov, Preprint/Inst. for Theor.Phys. ITP-93-66E, Kiev, 1993.

[16] B. M. Tanygin, IOP Conf. Ser.: Mater. Sci. Eng. 15 (2010) 012073.

[17] B. M. Tanygin, J. Magn. Magn. Mater 323 (2011) 1899.

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Magnetic symmetry based deﬁnition of the chirality in the magnetically ordered media