Peter I. Karpov and Sergei I. MukhinDept. of Theoretical Physics and Quantum Technologies, National University of Science and Technology “MISiS”

IntroductionTopological defects (for example, magnetic vortices and skyrmions) are extensively investigated in magneticmaterials, because they provide new possibilities for memory devices. Even more interesting is to considertopological defects in materials with magnetoelectric coupling (such as multiferroics) due to the possible controlover magnetic microstructures with the electric field, which is much easier and more reliable comparing withconventional magnetic read-and-write heads.

Multiferroics are materials that in some area of the phase diagram possess both polarization andmagnetization. According to classification of D. Khomskii [2] there are two types of multiferroics:

• In type-I multiferroics magnetic and electric orderings are largely independent. • in type-II multiferroics (“magnetoelectric multiferroics”) the magnetic order implies the electric order

The main ideaAbstractIn the present work [1] we consider a model of quasi-2D magnetoelectric material as the XY model for a spinsystem on a lattice with a local multiferroic-like interaction of the spin and electric polarization vectors. Wecalculate the contribution of magnetic (spin) vortex-antivortex pairs (which form electric dipoles) to the dielectricsusceptibility of the system. We show that in the approximation of non-interacting pairs as T →TBKT(Berezinskii-Kosterlitz-Thouless temperature) the dielectric susceptibility diverges.

arXiv: 1506.07856

We know that at any temperature belowTBKT there exists a certain amount ofthermally activated vortex-antivortex pairs(because the energy of the pair is finiteE ∼ log r/a). Since a vortex carries apositive charge and an antivortex carries anegative charge, a pair forms an electricdipole. The main goal of the work is tocalculate the dielectric susceptibility ofsuch dipole gas with a variable number ofdipoles, which is done in the next section.

−−−−−−−−−−−−−

+++++++++++++

−−++

++

−−++

−−

++

−−

• in type-II multiferroics (“magnetoelectric multiferroics”) the magnetic order implies the electric order (polarization) due to the interaction of the magnetic and the electric subsystems – in the present work we are dealing with this type of multiferroic.

To describe the interaction of the magnetic and electric subsystems, we use a phenomenological modelproposed by M. Mostovoy [3] to explain induced ferroelectricity in spiral magnets. From the symmetry arguments(time and spatial reversal shouldn’t change the energy) the magnetoelectric coupling energy density can bewritten as

where γ is the coupling constant. This phenomenological formula describes how inhomogeneous magnetizationcan create the electric polarization in magnetoelectric materials and type-II multiferroics.

Ordinary XY model

In the work we consider a thin film of almost identical layers, where magnetic moments are forced to lie in the filmplane due to the magnetostatic energy. So that we describe the magnetic subsystem by the classical 2D XY model.Let Mi = M0 (cos ϕi , sin ϕi ) be magnetic moment at the i-th site. Then the Hamiltonian of the system can bewritten as

where 〈i,j〉denotes summation over all the nearest neighbors and we consider ferromagnetic case when theexchange integral J > 0; the last equality represents the continuous limit ϕi → ϕ(x) where is thespin-wave stiffness and the constant term is neglected. Local energy minimum equation∇2ϕ = 0 possesses the vortex class of the solutions ϕ = nθ + ϕ0 , where θ is the polar angle and n ∈ ℤ is

called the topological charge or vorticity. And since the equation is linear, there exists a vortex-antivortex pairsolution that is superposition of single vortex and antivortex.

Topological defects play crucial role in the XY model: they are responsible for the famous Berezinskii-Kosterlitz- Conclusions

Calculation of the dielectric susceptibility

Consider a system of non-interacting vortex-antivortex quasi-2D dipole pairs of vortex lines (at temperatureT<TBKT ), which exist in the disk-shaped thin film with thickness h and radius R in the electric fieldE. The non-interacting approximation is reasonable since the dipole formation energy |μ|h ≫TBKT , therefore, the dipoleconcentration is very low (μ<0 is an analogue of the chemical potential per unit film thickness). For simplicity weconsider only the lowest-energy topological defects with n = ±1 topological charges. The part of energy (per initfilm thickness) of one dipole pair that depends on the distance between the vortex and antivortex cores (r) is

where . Then the grand partition function can be written as

Here we summate over the number of pairs np . Calculating the grand partition function (see [1] for the details)we can find the contribution of vortex-antivortex pairs to the dielectric susceptibility of the system in the linearresponse approximation E ≈ 0

where I0, I1 are the modified Bessel functions ofthe first kind and . This temperaturedependence is sketched on the right. As T →TBKT ≃ h qm

2/2 the susceptibility divergeslike |T –TBKT |-1. When T≪TBKT the susceptibility takes the activation exponential form

single vortex solutions vortex-antivortex pair

R – is of order of the system size

─ la ce const.

Topological defects play crucial role in the XY model: they are responsible for the famous Berezinskii-Kosterlitz-Thouless (BKT) phase transition [4,5]. Below a certain temperature (TBKT ) vortices and antivortices are boundedin pairs and when the temperature rises above TBKT the pairs dissociate.

The model: XY + magnetoelectric couplingTo describe the interaction of the magnetic and electric subsystems, we use the phenomenological modelproposed of M. Mostovoy [3]. We write our energy density of magnetoelectric material in electric field E as

where χe is the dielectric susceptibility in the absence of M, γ is the coupling constant, and |M|=M0 = const is the saturation magnetization. Minimization of the energy density with respect to P gives

This formula tells us that magnetic structure can be electrically polarized and possess an electric charge.Let , then the energy density can be rewritten as

In the absence of the electric field this gives us the XY model withthe effective spin-wave stiffness

In magnetic vortices, inhomogeneous magnetization induces the electricpolarization (due to (1), see picture on the right) and the vortex core acquires an electric charge with linear density

so that, the electric charge is proportional to the topological charge of thevortex (vortices become positively charged and antivortices become negatively charged). Also the edge of the sample acquires charge that is opposite to the vortex charge. This edge charge effectively shields the core charge and value of shielding depends on the shape of the sample. For disk-shaped sample the effective charge is exactly twice less than (2):

ConclusionsIn conclusion, in this work we investigated the properties of a magnetoelectric thin film with type-II multiferroic-like interaction of the electric and the magnetic subsystems. Magnetic vortices in such materials possess electriccharges and vortex-antivortex pairs form electric dipoles. Such dipole pairs have finite energies, so at anytemperature T <TBKT there exists a certain amount of thermally activated vortex-antivortex pairs. We calculatedthe contribution of vortex-antivortex pairs to the static dielectric susceptibility of the system at temperaturesT < TBKT in the approximation of non-interacting dipoles. This approximation is valid up to the temperatureswhich are not extremely close to TBKT (namely, it is valid outside of very narrow region near TBKT where

). In the low temperature limit, the dielectric susceptibility takes theactivation exponential form (4), which is consistent with the fact that at T → 0 the number of vortex pairs isproportional to . As T →TBKT formula (3) gives the diverging susceptibility. This reflects the process ofvortex-antivortex pairs unbinding and the phase transition. However, we expect that close to TBKT interaction ofdipole pairs becomes important and, therefore, at TBKT the susceptibility stays finite.

Literature[1] P.I. Karpov, S.I. Mukhin. Dielectric susceptibility of magnetoelectric thin films with vortex-antivortex dipole pairs. arXiv:1506.07856 (2015).[2] D.I. Khomskii. Classifying multiferroics: Mechanisms and effects. Physics 2, 20 (2009).[3] M.V. Mostovoy. Ferroelectricity in spiral magnets. Phys.Rev.Lett. 96, 067601 (2006).[4] J.M. Kosterlitz and D.J. Thouless. Ordering, metastability and phase transitions in two-dimensional systems. J.Phys. C 6, 1181 (1973).[5] V.L. Berezinskii. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Sov.Phys. JETP 32, 493 (1971).

Arrows show polarization distribution that is created by single vortex in disk-shaped sample. Vortex core becomes positively charged, edge of the sample – negatively charged.

AcknowledgmentsThe authors are grateful to I.S. Burmistrov, D.I. Khomskii, M.V. Mostovoy, and S.A. Brazovskii for helpfuldiscussions. Part of this work was carried out with the financial support of the Ministry of Education and Scienceof the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISIS” (No. K2-2014-015). PK also acknowledges the financial support of the Dynasty Foundation.