![Peter I. Karpov and Sergei I. Mukhin
Dept. of Theoretical Physics and Quantum Technologies, National University of Science and Technology “MISiS”
Introduction
Topological defects (for example, magnetic vortices and skyrmions) are extensively investigated in magnetic
materials, because they provide new possibilities for memory devices. Even more interesting is to consider
topological defects in materials with magnetoelectric coupling (such as multiferroics) due to the possible control
over magnetic microstructures with the electric field, which is much easier and more reliable comparing with
conventional magnetic read-and-write heads.
Multiferroics are materials that in some area of the phase diagram possess both polarization and
magnetization. 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 ideaAbstract
In the present work [1] we consider a model of quasi-2D magnetoelectric material as the XY model for a spin
system on a lattice with a local multiferroic-like interaction of the spin and electric polarization vectors. We
calculate the contribution of magnetic (spin) vortex-antivortex pairs (which form electric dipoles) to the dielectric
susceptibility 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 below
TBKT there exists a certain amount of
thermally activated vortex-antivortex pairs
(because the energy of the pair is finite
E ∼ log r/a). Since a vortex carries a
positive charge and an antivortex carries a
negative charge, a pair forms an electric
dipole. The main goal of the work is to
calculate the dielectric susceptibility of
such dipole gas with a variable number of
dipoles, 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 model
proposed 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 be
written as
where γ is the coupling constant. This phenomenological formula describes how inhomogeneous magnetization
can 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 film
plane 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 be
written as
where 〈i,j〉denotes summation over all the nearest neighbors and we consider ferromagnetic case when the
exchange integral J > 0; the last equality represents the continuous limit ϕi → ϕ(x) where is the
spin-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 pair
solution 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 temperature
T < TBKT ), which exist in the disk-shaped thin film with thickness h and radius R in the electric field E. The non-
interacting approximation is reasonable since the dipole formation energy |μ|h ≫TBKT , therefore, the dipole
concentration is very low (μ<0 is an analogue of the chemical potential per unit film thickness). For simplicity we
consider only the lowest-energy topological defects with n = ±1 topological charges. The part of energy (per init
film 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 linear
response approximation E ≈ 0
where I0, I1 are the modified Bessel functions of
the first kind and . This temperature
dependence is sketched on the right.
As T →TBKT ≃ h qm
2/2 the susceptibility diverges
like |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
systemsize
─ 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 bounded
in pairs and when the temperature rises above TBKT the pairs dissociate.
The model: XY + magnetoelectric coupling
To describe the interaction of the magnetic and electric subsystems, we use the phenomenological model
proposed 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 with
the effective spin-wave stiffness
In magnetic vortices, inhomogeneous magnetization induces the electric
polarization (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 the
vortex (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):
Conclusions
In 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 electric
charges and vortex-antivortex pairs form electric dipoles. Such dipole pairs have finite energies, so at any
temperature T <TBKT there exists a certain amount of thermally activated vortex-antivortex pairs. We calculated
the contribution of vortex-antivortex pairs to the static dielectric susceptibility of the system at temperatures
T < TBKT in the approximation of non-interacting dipoles. This approximation is valid up to the temperatures
which 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 the
activation exponential form (4), which is consistent with the fact that at T → 0 the number of vortex pairs is
proportional to . As T →TBKT formula (3) gives the diverging susceptibility. This reflects the process of
vortex-antivortex pairs unbinding and the phase transition. However, we expect that close to TBKT interaction of
dipole 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.
Acknowledgments
The authors are grateful to I.S. Burmistrov, D.I. Khomskii, M.V. Mostovoy, and S.A. Brazovskii for helpful
discussions. Part of this work was carried out with the financial support of the Ministry of Education and Science
of 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.](https://image.slidesharecdn.com/posterkarpov-150924143514-lva1-app6892/85/Poster-karpov-1-320.jpg)
магнитные вихри Poster karpov
- 1. Peter I. Karpov and Sergei I. Mukhin Dept. of Theoretical Physics and Quantum Technologies, National University of Science and Technology “MISiS” Introduction Topological defects (for example, magnetic vortices and skyrmions) are extensively investigated in magnetic materials, because they provide new possibilities for memory devices. Even more interesting is to consider topological defects in materials with magnetoelectric coupling (such as multiferroics) due to the possible control over magnetic microstructures with the electric field, which is much easier and more reliable comparing with conventional magnetic read-and-write heads. Multiferroics are materials that in some area of the phase diagram possess both polarization and magnetization. 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 ideaAbstract In the present work [1] we consider a model of quasi-2D magnetoelectric material as the XY model for a spin system on a lattice with a local multiferroic-like interaction of the spin and electric polarization vectors. We calculate the contribution of magnetic (spin) vortex-antivortex pairs (which form electric dipoles) to the dielectric susceptibility 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 below TBKT there exists a certain amount of thermally activated vortex-antivortex pairs (because the energy of the pair is finite E ∼ log r/a). Since a vortex carries a positive charge and an antivortex carries a negative charge, a pair forms an electric dipole. The main goal of the work is to calculate the dielectric susceptibility of such dipole gas with a variable number of dipoles, 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 model proposed 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 be written as where γ is the coupling constant. This phenomenological formula describes how inhomogeneous magnetization can 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 film plane 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 be written as where 〈i,j〉denotes summation over all the nearest neighbors and we consider ferromagnetic case when the exchange integral J > 0; the last equality represents the continuous limit ϕi → ϕ(x) where is the spin-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 pair solution 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 temperature T < TBKT ), which exist in the disk-shaped thin film with thickness h and radius R in the electric field E. The non- interacting approximation is reasonable since the dipole formation energy |μ|h ≫TBKT , therefore, the dipole concentration is very low (μ<0 is an analogue of the chemical potential per unit film thickness). For simplicity we consider only the lowest-energy topological defects with n = ±1 topological charges. The part of energy (per init film 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 linear response approximation E ≈ 0 where I0, I1 are the modified Bessel functions of the first kind and . This temperature dependence is sketched on the right. As T →TBKT ≃ h qm 2/2 the susceptibility diverges like |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 systemsize ─ 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 bounded in pairs and when the temperature rises above TBKT the pairs dissociate. The model: XY + magnetoelectric coupling To describe the interaction of the magnetic and electric subsystems, we use the phenomenological model proposed 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 with the effective spin-wave stiffness In magnetic vortices, inhomogeneous magnetization induces the electric polarization (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 the vortex (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): Conclusions In 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 electric charges and vortex-antivortex pairs form electric dipoles. Such dipole pairs have finite energies, so at any temperature T <TBKT there exists a certain amount of thermally activated vortex-antivortex pairs. We calculated the contribution of vortex-antivortex pairs to the static dielectric susceptibility of the system at temperatures T < TBKT in the approximation of non-interacting dipoles. This approximation is valid up to the temperatures which 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 the activation exponential form (4), which is consistent with the fact that at T → 0 the number of vortex pairs is proportional to . As T →TBKT formula (3) gives the diverging susceptibility. This reflects the process of vortex-antivortex pairs unbinding and the phase transition. However, we expect that close to TBKT interaction of dipole 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. Acknowledgments The authors are grateful to I.S. Burmistrov, D.I. Khomskii, M.V. Mostovoy, and S.A. Brazovskii for helpful discussions. Part of this work was carried out with the financial support of the Ministry of Education and Science of 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.