- Email: [email protected]
Magnetic structures of 2D and 3D nanoparticles
Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetic structures of 2D and 3D nanoparticles J.-C.S. Levy Matériaux et Phenomenes Quantiques, University Paris Diderot, 75013 Paris, France
art ic l e i nf o
a b s t r a c t
Article history: Received 25 November 2013 Received in revised form 18 June 2014 Accepted 10 July 2014 Available online 1 August 2014
The minimization of exchange interactions and dipolar interactions in 2D and 3D nanoparticles is obtained from a powerful variational approach of the local spin Hamiltonian and leads to a different set of equations which correspond to different levels of screening of the long range dipolar interactions. These equations are shown to introduce topological defects which are analyzed on the basis of elementary spin clusters. Four basic topological defects are deduced for 2D nanoparticles, as observed in magnetic samples and simulations and 10 basic topological defects are deduced for 3D nanoparticles. These singularities induce complex variations of magnetization around them and non-linear properties. & 2014 Elsevier B.V. All rights reserved.
Keywords: Magnetic structure Magnetic nanoparticle Magnetic domain
1. Introduction Since the very beginning of magnetic domain observations in magnetic materials of various natures, sizes and shapes [1], there has been a strong experimental evidence for intrinsic walls, linear singularities and topological defects in the magnetic structure of materials. More specifically magnetic thin films and ultra-thin films [2] showed labyrinthine and vortex structures as well as other topological defects. More recently experimental results on the magnetic structure of nanofilms were helped by theoretical simulations used to deal with micromagnetism such as OOMMF method [3], Monte-Carlo computations [4] and Langevin dynamics [5] and these approaches showed localized topological defects and their consequences on dynamical properties [6]. 3D magnetic nanoparticles of interest for catalytic applications and for transport properties like printing or biological applications as guided vectors were recently shown to also exhibit a complex 3D domain structure at their external surfaces [7]. There are several difficulties in 3D analyses of magnetic nanoparticles, since mainly external surfaces are observed without data on internal structure. Moreover there is a lack in 3D completely reliable simulations up to now. Recent works explore local magnetic structures in nickel nanocylinders [8] as well as in antiferromagnets [9]. So the natural aim of this paper is to introduce a new theoretical analysis of the magnetic ground state of 2D and 3D nanoparticles by means of a variational approach and so to deduce the 3D observable topological defects. The obvious origin of complex magnetic structures is the long ranged dipole–dipole interaction competing with local exchange and anisotropy [10]. Real samples are finite, so all dipolar contributions
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmmm.2014.07.010 0304-8853/& 2014 Elsevier B.V. All rights reserved.
such as local demagnetizing field and dynamical matrix elements are far from being uniformly spread over the sample and the long range nature of dipolar interactions seriously increases this spreading [11]. Non-uniformity breaks the hopes of a magnetic sample translational invariance as assumed from [12] and followers about magnetic structures. As a result, the obvious finiteness of samples induces localization in the spin wave spectra as observed in numerical computations [11] and as a later consequence, it induces observed static topological defects [1] resulting from localized soft modes. So there is a real need for a theoretical approach of nanoparticle magnetic structure and especially for 3D nanoparticles. Here we are looking for a variational treatment from a generalized Landau's local version of the spin Hamiltonian. The long ranged dipole–dipole interaction is translated into a local interaction by means of Taylor expansion of the spin field introducing spin derivative fields as previously done about 2D samples [10], with lattice sums or more exactly sample sums with symmetry properties. This approach explains the occurrence of non-linear properties [6], which are efficient out of domains, i.e. at walls and defects as observed [6]. The zero order approximation of Taylor expansion is linked with the concept of demagnetizing field while the second order approximation of the spin Hamiltonian gives the main set of equations on magnetic structures deduced from minimization. These equations show a higher level of complexity for 3D samples than for 2D ones. Of course higher order equations correspond to higher levels of screening of the long ranged interaction and can also be deduced as in the twodimensional case [10]. The search for topological defects in 3D samples requires a local analysis within a spherical frame, i.e. for instance here a basic cluster. For 2D samples, just four basic topological defects when considering neighboring sites only: vortex, two antivortices and a strongly asymmetric one, while for 3D samples, 10 basic
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
Since variations for every site are independent, the variation basis is complete.
topological defects occur. For 2D samples as well as for 3D ones complex defects involving several basic defects occur also, so there are many complex topological defects. Harmonic magnetic excitations, i.e. spin waves, are defined in that context of topological defects [10] and can be observed by magnetic resonance [14]. From the local spin Hamiltonian a Landau like variational treatment enables us to obtain the basic equations which are satisfied by topological defects. These topological defects are then deduced from simple geometrical criteria.
3. Basic results At level 0 of the Taylor expansion of the dipolar interaction, omitting the spin anisotropy term, the effect of demagnetizing field only occurs as induced by the anisotropic character of dipolar interaction. The set of optimization equations reads, within the assumption of uniform sample sums:
2. The spin Hamiltonians
ðB CÞSz Sy ¼ 0
Here we consider the exchange interaction between spins located on neighboring states
ðC AÞSx Sz ¼ 0 ðA BÞSy Sx ¼ 0
! ! 1 H exc ¼ ∑J ij Si n Sj 2
! Sj¼ ∑
p;q;r
pþqþr
! ∂ S p!q!r! ∂xp ∂yq ∂zr i
x2 y2 z2 A ¼ L000;11 ∑ 5 B ¼ L000;22 ∑ 5 C ¼ L000;33 ∑ 5 r r r
It leads to a strictly local interaction in terms of the square of the gradient of the spin field, a standard result. The anisotropy interaction is local and so does not require any translation in terms of local field. The dipolar field induces the well known dipole– dipole interaction: ! ! ! ! ! ! ð Si n r ij Þð Si n r ij Þ Si n Si 3 ∑ 3 r 5ij i a j r ij iaj
ð3Þ
∂ Sx
A 3D
ð4Þ xpij yqij zrij 3=2 p!q!r! x2ij þ y2ij þ z2ij
Lp;q;r;α;β;i ¼ ∑ j
r α;ij r β;ij xpij yqij zrij 5=2 p!q!r! x2ij þ y2ij þ z2ij
ð5Þ
For infinite samples many of these lattice sums would diverge. Here sample sums remain finite and are often assumed to have a weak local variation, for the sake of simplicity. In that case, a continuous calculation by means of 3D integration will be introduced for the estimation of these sums as integrals. The principle of our variational approach consists of introducing a local arbitrary small deviation of a spin orientation on site k of weak amplitude C: ! Si ¼ Si;0 þ Cδði kÞSk;0 4 n ð6Þ ! Here n is an arbitrary unit vector, δðiÞ is the Dirac delta function and Eq. (6) ensures the spin amplitude conservation. 0
2
2
2
ðA 3DÞkx ðB 3GÞky ðC 3JÞkz þ λ B B 6Ek k detB x y @ 6Fkx kz
B 3G
C 3J
6E
6F
6Ekx ky
1
6Fkx kz 2
ð8Þ
Here the coefficients A, B and C are issued from isotropic sample sums I while coefficients D,E,..., L come from anisotropic sample sums L, and λ is a free parameter. Two other equations with similar coefficients are derived according to circular permutations. These equations complete the set of second order equations. It must be noticed that while lattice sums depend on lattice geometry and lattice parameter, sample sums and related coefficients depend also on sample shape as shown in Eq. (70 ). Thus Eq. (8) defines a large class of spin fields submitted to second order partial derivative equations. Introducing a common Fourier transform for the spin field with wavevectorðkx ; ky ; kz Þ, the set of equations leads to the characteristic condition on wavevectors: This set of properties of 3D magnetic structures is quite more complex than the set obtained for a 2D sample [10] which can be deduced from the more general 3D case from Eqs. (8) and (9). The characteristic Eq. (9) of order 6 means the occurrence of static wavy like deformations in the ground state. The next point to consider consists of deriving the topological defects which correspond to Eq. (8), within a spherical frame. Since these equations are submitted to very large changes
where isotropic and anisotropic sample sums are respectively
j
B ∂x C B ∂2 Sx C B ∂y2 C B C B ∂2 Sx C B ∂z2 C C λB B ∂2 Sy C ¼ 0 B C B ∂x∂y C B 2 C B ∂ Sz C @ ∂x∂z A Sx 2
Again using Taylor expansion of the spin field dipolar interaction reads as a local one in terms of all spin derivative fields when introducing lattice sums, i.e. more exactly sample sums I p;q;r;i and J p;q;r;α;β;i : ∂p þ q þ r ! ∂p þ q þ r H d ¼ ∑ I p;q;r;i Si n p q r S i 3 ∑ Lp;q;r;α;β;i Sα;i p q r Sβi ∂x ∂y ∂z ∂x ∂y ∂z p;q;r;i p;q;r;α;β;i
I p;q;r;i ¼ ∑
ð70 Þ
Here only order of magnitude is considered, omitting tedious numerical factors. In other words the classical shape effect of the demagnetizing field is found: for a flat sample in the xy plane, magnetization is in plane within this continuous model. Of course the detailed calculation can be achieved in a general 3D case, at the expanse of complexity according to the sample shape. At level 2 of the Taylor expansion of the dipolar Hamiltonian, always considering sample parameters as uniform over the sample, a set of three equations over the spin field components and their derivatives is obtained. The first equation of this set reads as a matrix product: 0 2 1
ð2Þ
Hd ¼ ∑
ð7Þ
With
ð1Þ
With infinite Taylor expansion of the spin field introducing the spin field derivatives: xpij yqij zrij
3
2
2
ðA 3EÞkx ðB 3HÞky ðC 3KÞkz þ λ
6Ikz ky
6Ikz ky
ðA 3FÞkx ðB 3IÞky ðC 3JlÞkz þ λ
2
2
2
C C C¼0 A
ð9Þ
4
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
according to sample shape, many topological defects must occur among samples. So it is more efficient to consider possible basic topological defects on the smaller cluster with full geometry, i.e. a complete set of neighbors. These clusters must satisfy flux closure properties associated with dipolar interactions, i.e. with surface spins parallel to sample surface, and this only constraint already defines several topological defects. There are three reasons of interest for such elementary clusters. First such small clusters can occur in nature. Secondly by a kind of renormalization process this calculation also enables us to consider larger samples by means of scaling. And the main point lies on the fact that in simulations as well as in observations, rather large uniform domains of magnetization appear and are split by rather thin walls [2,4,5]. So the renormalization process invoked before is well justified in 2D clusters with five sites in a simple square lattice and in 3D clusters with seven sites in a simple cubic lattice. The so defined 2D cluster can be compared to a square dot and similarly the 3D cluster is comparable to a cubic dot.
4. Topological defects in 2D and 3D samples In an elementary basic 2D cluster there are four sites around a central site. Assuming that spins are parallel to the border, i.e. to the four sides of this square, the only free parameter is the relative sense of these four spins. So the four spins can have the same sense as deduced by rotation and there are two possible chiralities. This defines a vortex V as shown in Fig. 1 and observed [15]. Then there is the case where every successive spins have opposite senses. This defines antivortex AV1 as also shown in Fig. 1, a kind of double flower where single flower state were already proposed about experimental observation [15]. There is also the case of antivortex AV2. Here two pairs of successive spins have the same sense while the two pairs have opposite senses as shown in Fig. 1. AV2 has been also called a “leaf” state according to observation [15]. Finally there are cases 1-3 where three successive spins share the same sense while the last one has opposite sense as observed in simulations [5]. As a matter of fact all these basic structures are observed, at least as metastable structures in simulations for a square dot [5]. Since these topological defects exhibit strong singularities, i.e. strong gradients of spin orientation they induce at the defect core a spin escape in the third dimension characterized by so called “Mexican hats” cores of vortices. Of course in real extended structures several topological defects can be observed playing together within the same sample. In this case the continuity of magnetic domains induces the existence of complementary topological defects as found both experimentally and in higher order terms of this energy minimization. In 3D the third dimension basically duplicate these structures since the two spins in the third direction can be either parallel or antiparallel. But in the last case 3-1 there is a basic dissymmetry between x and y axes. So finally 10 basic topological defects appear in 3D nanoparticles as shown from the principles exhibited in Fig. 2a and b. The two 3D versions of a 2D vortex are shown in Fig. 2. They exhibit a strong singularity so escapes in the third dimension occur in their core central part. On the left hand side the vortex
Fig. 2. (a) The two 3D topological defects issued from a 2D vortex. In red on line and in gray on paper the two spins lying on the third axis (z) are shown elongated for clarity, in antiparallel position and in parallel position respectively. (b) The four 3D topological defects issued from a 2D 1-3 defect. In red on line and in gray on paper the two spins lying on the third axis are shown elongated for clarity, in antiparallel position on the left part and in parallel position on the right part. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
line shows a S shape with a basic undulation while on the right hand side the vortex suffers a strong bending. Of course similar pairs of conjugated magnetic structures are deduced for antivortex 1 and antivortex 2. All these intrinsic defect structures exhibit strong variations of spin orientation within a restricted space so they must be rearranged in a complex way in order to minimize these discrete orientation variations in order to reduce extra exchange energy. Fig. 2b shows the four 3D basic magnetic structures issued from 2D 1–3 structure, always within the same scope, i.e. without exchange energy minimization. Once more in Fig. 2b defect lines deduced from xy plane either are distorted in a S shape or are bent. Quite generally these 10 3D topological defects exhibit strong local variations of the spin direction and twists. This property induces a strong competition between exchange and dipolar interactions which leads to an escape in other dimensions as observed for 2D samples and ultrathin films with Mexican hat shape. So finally realistic twisted topological defects differ a little bit from the basic shapes enumerated in Fig. 2 and deserve careful experimental and numerical investigations. Such high gradient spin configurations induce strong non-linear properties [10] and thus can be useful.
5. Conclusions
Fig. 1. The four basic topological defects V, AV1, AV2 and 1-3 in 2D samples.
This analysis of nanoparticle magnetic structures based upon a powerful variational treatment of a local Hamiltonian leads to several set of equations which result from successive screening levels of dipolar interaction. The first level expresses the global demagnetizing field. Further levels correspond to successive
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
differential equations on the spin field, harmonic and anharmonic too. The analysis of topological defects for 3D samples takes advantage of the previous work on 2D nanoparticles and uses an approach for clusters mixing both local and global constraints as early did about cluster structures and symmetry introducing what became quasicrystalline order [16]. The local constraint consists here in parting in a few sites, i.e. domains at a realistic size, while the global constraint consists in defining the spin orientation in such domains as parallel to external faces in order to satisfy dipolar energy minimization. This method enables us to introduce four basic topological defects for 2D magnetic nanoparticles, and 10 basic topological defects for 3D magnetic nanoparticles. These last 3D defects exhibit a strong twist of the spin orientation so they are rearranged in order to minimize exchange energy. Such configurations induce non-linear properties which can be used for magnonic applications. Acknowledgments It is a pleasure for the author to thank Dr. Philippe Depondt from INSP and Dr. Slawomir Mamica from UAM Poznan for so many fruitful discussions.
5
References [1] C. Guillaud, R. Vautier, Colloque National du Magnétisme Pierre Weiss, CNRS, Paris (1958) 25. [2 C. Stamm, et al., Science 282 (1998) 449. [3] M. Donahue, D. Porter, NIST (2004). [4] E.Y. Vedmedenko, et al., Surf. Sci. 402–404 (1998) 391; E.Y. Vedmedenko, et al., Phys. Rev. B 59 (1999) 3329. [5] Ph. Depondt, et al., Phys. Lett. A 376 (2012) 628; Ph. Depondt, et al., Phys. Lett. A 376 (2012) 3411. [6] H. Schultheis, et al., Phys. Rev. B 86 (2012) 054414; B. Obry, et al., Appl. Phys. Lett. 101 (2012) 192406. [7] I. Manke, et al., Nat. Commun. 1 (2010) 125. [8] N. Bizier, et al., Nano Lett. 13 (2013) 2053. [9] O. Shpryko, et al., Nature 447 (2007) 68; O. Shpryko, et al., Phys. Rev. Lett. 98 (2007) 117206. [10] J.-C.S. Levy, Phys. Rev. B 63 (2001) 104409; J.-C.S. Levy, in: J.-C.S. Levy (Ed.), Nanostructures and Their Magnetic Properties, Research Signpost, Trivandrum, Kerala, 2009, p. 117. [11] H. Puszkarski, et al., Phys. Rev. B 71 (2005) 014421; H. Puszkarski, et al., J. Appl. Phys. 101 (2007) 024326. [12] J.M. Luttinger, L. Tisza, Phys. Rev. 70 (1946) 954. [14] K. Vogt, et al., Phys. Rev. B 84 (2011) 174401. [15] R.P. Cowburn, et al., J. Appl. Phys. 87 (2000) 7082. [16] J.-C.S. Levy, D. Mercier, J. Appl. Phys. 53 (1982) 7709; D. Mercier, J.-C.S. Levy, Phys. Rev. B 27 (1983) 1293; S. Benakli, et al., in: J.-C.S. Levy (Ed.), Nanostructures and Their Magnetic Properties, 37, Research Signpost, Trivandrum, Kerala, 2009.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Magnetic structures of 2D and 3D nanoparticles J.-C.S. Levy Matériaux et Phenomenes Quantiques, University Paris Diderot, 75013 Paris, France
art ic l e i nf o
a b s t r a c t
Article history: Received 25 November 2013 Received in revised form 18 June 2014 Accepted 10 July 2014 Available online 1 August 2014
The minimization of exchange interactions and dipolar interactions in 2D and 3D nanoparticles is obtained from a powerful variational approach of the local spin Hamiltonian and leads to a different set of equations which correspond to different levels of screening of the long range dipolar interactions. These equations are shown to introduce topological defects which are analyzed on the basis of elementary spin clusters. Four basic topological defects are deduced for 2D nanoparticles, as observed in magnetic samples and simulations and 10 basic topological defects are deduced for 3D nanoparticles. These singularities induce complex variations of magnetization around them and non-linear properties. & 2014 Elsevier B.V. All rights reserved.
Keywords: Magnetic structure Magnetic nanoparticle Magnetic domain
1. Introduction Since the very beginning of magnetic domain observations in magnetic materials of various natures, sizes and shapes [1], there has been a strong experimental evidence for intrinsic walls, linear singularities and topological defects in the magnetic structure of materials. More specifically magnetic thin films and ultra-thin films [2] showed labyrinthine and vortex structures as well as other topological defects. More recently experimental results on the magnetic structure of nanofilms were helped by theoretical simulations used to deal with micromagnetism such as OOMMF method [3], Monte-Carlo computations [4] and Langevin dynamics [5] and these approaches showed localized topological defects and their consequences on dynamical properties [6]. 3D magnetic nanoparticles of interest for catalytic applications and for transport properties like printing or biological applications as guided vectors were recently shown to also exhibit a complex 3D domain structure at their external surfaces [7]. There are several difficulties in 3D analyses of magnetic nanoparticles, since mainly external surfaces are observed without data on internal structure. Moreover there is a lack in 3D completely reliable simulations up to now. Recent works explore local magnetic structures in nickel nanocylinders [8] as well as in antiferromagnets [9]. So the natural aim of this paper is to introduce a new theoretical analysis of the magnetic ground state of 2D and 3D nanoparticles by means of a variational approach and so to deduce the 3D observable topological defects. The obvious origin of complex magnetic structures is the long ranged dipole–dipole interaction competing with local exchange and anisotropy [10]. Real samples are finite, so all dipolar contributions
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmmm.2014.07.010 0304-8853/& 2014 Elsevier B.V. All rights reserved.
such as local demagnetizing field and dynamical matrix elements are far from being uniformly spread over the sample and the long range nature of dipolar interactions seriously increases this spreading [11]. Non-uniformity breaks the hopes of a magnetic sample translational invariance as assumed from [12] and followers about magnetic structures. As a result, the obvious finiteness of samples induces localization in the spin wave spectra as observed in numerical computations [11] and as a later consequence, it induces observed static topological defects [1] resulting from localized soft modes. So there is a real need for a theoretical approach of nanoparticle magnetic structure and especially for 3D nanoparticles. Here we are looking for a variational treatment from a generalized Landau's local version of the spin Hamiltonian. The long ranged dipole–dipole interaction is translated into a local interaction by means of Taylor expansion of the spin field introducing spin derivative fields as previously done about 2D samples [10], with lattice sums or more exactly sample sums with symmetry properties. This approach explains the occurrence of non-linear properties [6], which are efficient out of domains, i.e. at walls and defects as observed [6]. The zero order approximation of Taylor expansion is linked with the concept of demagnetizing field while the second order approximation of the spin Hamiltonian gives the main set of equations on magnetic structures deduced from minimization. These equations show a higher level of complexity for 3D samples than for 2D ones. Of course higher order equations correspond to higher levels of screening of the long ranged interaction and can also be deduced as in the twodimensional case [10]. The search for topological defects in 3D samples requires a local analysis within a spherical frame, i.e. for instance here a basic cluster. For 2D samples, just four basic topological defects when considering neighboring sites only: vortex, two antivortices and a strongly asymmetric one, while for 3D samples, 10 basic
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
Since variations for every site are independent, the variation basis is complete.
topological defects occur. For 2D samples as well as for 3D ones complex defects involving several basic defects occur also, so there are many complex topological defects. Harmonic magnetic excitations, i.e. spin waves, are defined in that context of topological defects [10] and can be observed by magnetic resonance [14]. From the local spin Hamiltonian a Landau like variational treatment enables us to obtain the basic equations which are satisfied by topological defects. These topological defects are then deduced from simple geometrical criteria.
3. Basic results At level 0 of the Taylor expansion of the dipolar interaction, omitting the spin anisotropy term, the effect of demagnetizing field only occurs as induced by the anisotropic character of dipolar interaction. The set of optimization equations reads, within the assumption of uniform sample sums:
2. The spin Hamiltonians
ðB CÞSz Sy ¼ 0
Here we consider the exchange interaction between spins located on neighboring states
ðC AÞSx Sz ¼ 0 ðA BÞSy Sx ¼ 0
! ! 1 H exc ¼ ∑J ij Si n Sj 2
! Sj¼ ∑
p;q;r
pþqþr
! ∂ S p!q!r! ∂xp ∂yq ∂zr i
x2 y2 z2 A ¼ L000;11 ∑ 5 B ¼ L000;22 ∑ 5 C ¼ L000;33 ∑ 5 r r r
It leads to a strictly local interaction in terms of the square of the gradient of the spin field, a standard result. The anisotropy interaction is local and so does not require any translation in terms of local field. The dipolar field induces the well known dipole– dipole interaction: ! ! ! ! ! ! ð Si n r ij Þð Si n r ij Þ Si n Si 3 ∑ 3 r 5ij i a j r ij iaj
ð3Þ
∂ Sx
A 3D
ð4Þ xpij yqij zrij 3=2 p!q!r! x2ij þ y2ij þ z2ij
Lp;q;r;α;β;i ¼ ∑ j
r α;ij r β;ij xpij yqij zrij 5=2 p!q!r! x2ij þ y2ij þ z2ij
ð5Þ
For infinite samples many of these lattice sums would diverge. Here sample sums remain finite and are often assumed to have a weak local variation, for the sake of simplicity. In that case, a continuous calculation by means of 3D integration will be introduced for the estimation of these sums as integrals. The principle of our variational approach consists of introducing a local arbitrary small deviation of a spin orientation on site k of weak amplitude C: ! Si ¼ Si;0 þ Cδði kÞSk;0 4 n ð6Þ ! Here n is an arbitrary unit vector, δðiÞ is the Dirac delta function and Eq. (6) ensures the spin amplitude conservation. 0
2
2
2
ðA 3DÞkx ðB 3GÞky ðC 3JÞkz þ λ B B 6Ek k detB x y @ 6Fkx kz
B 3G
C 3J
6E
6F
6Ekx ky
1
6Fkx kz 2
ð8Þ
Here the coefficients A, B and C are issued from isotropic sample sums I while coefficients D,E,..., L come from anisotropic sample sums L, and λ is a free parameter. Two other equations with similar coefficients are derived according to circular permutations. These equations complete the set of second order equations. It must be noticed that while lattice sums depend on lattice geometry and lattice parameter, sample sums and related coefficients depend also on sample shape as shown in Eq. (70 ). Thus Eq. (8) defines a large class of spin fields submitted to second order partial derivative equations. Introducing a common Fourier transform for the spin field with wavevectorðkx ; ky ; kz Þ, the set of equations leads to the characteristic condition on wavevectors: This set of properties of 3D magnetic structures is quite more complex than the set obtained for a 2D sample [10] which can be deduced from the more general 3D case from Eqs. (8) and (9). The characteristic Eq. (9) of order 6 means the occurrence of static wavy like deformations in the ground state. The next point to consider consists of deriving the topological defects which correspond to Eq. (8), within a spherical frame. Since these equations are submitted to very large changes
where isotropic and anisotropic sample sums are respectively
j
B ∂x C B ∂2 Sx C B ∂y2 C B C B ∂2 Sx C B ∂z2 C C λB B ∂2 Sy C ¼ 0 B C B ∂x∂y C B 2 C B ∂ Sz C @ ∂x∂z A Sx 2
Again using Taylor expansion of the spin field dipolar interaction reads as a local one in terms of all spin derivative fields when introducing lattice sums, i.e. more exactly sample sums I p;q;r;i and J p;q;r;α;β;i : ∂p þ q þ r ! ∂p þ q þ r H d ¼ ∑ I p;q;r;i Si n p q r S i 3 ∑ Lp;q;r;α;β;i Sα;i p q r Sβi ∂x ∂y ∂z ∂x ∂y ∂z p;q;r;i p;q;r;α;β;i
I p;q;r;i ¼ ∑
ð70 Þ
Here only order of magnitude is considered, omitting tedious numerical factors. In other words the classical shape effect of the demagnetizing field is found: for a flat sample in the xy plane, magnetization is in plane within this continuous model. Of course the detailed calculation can be achieved in a general 3D case, at the expanse of complexity according to the sample shape. At level 2 of the Taylor expansion of the dipolar Hamiltonian, always considering sample parameters as uniform over the sample, a set of three equations over the spin field components and their derivatives is obtained. The first equation of this set reads as a matrix product: 0 2 1
ð2Þ
Hd ¼ ∑
ð7Þ
With
ð1Þ
With infinite Taylor expansion of the spin field introducing the spin field derivatives: xpij yqij zrij
3
2
2
ðA 3EÞkx ðB 3HÞky ðC 3KÞkz þ λ
6Ikz ky
6Ikz ky
ðA 3FÞkx ðB 3IÞky ðC 3JlÞkz þ λ
2
2
2
C C C¼0 A
ð9Þ
4
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
according to sample shape, many topological defects must occur among samples. So it is more efficient to consider possible basic topological defects on the smaller cluster with full geometry, i.e. a complete set of neighbors. These clusters must satisfy flux closure properties associated with dipolar interactions, i.e. with surface spins parallel to sample surface, and this only constraint already defines several topological defects. There are three reasons of interest for such elementary clusters. First such small clusters can occur in nature. Secondly by a kind of renormalization process this calculation also enables us to consider larger samples by means of scaling. And the main point lies on the fact that in simulations as well as in observations, rather large uniform domains of magnetization appear and are split by rather thin walls [2,4,5]. So the renormalization process invoked before is well justified in 2D clusters with five sites in a simple square lattice and in 3D clusters with seven sites in a simple cubic lattice. The so defined 2D cluster can be compared to a square dot and similarly the 3D cluster is comparable to a cubic dot.
4. Topological defects in 2D and 3D samples In an elementary basic 2D cluster there are four sites around a central site. Assuming that spins are parallel to the border, i.e. to the four sides of this square, the only free parameter is the relative sense of these four spins. So the four spins can have the same sense as deduced by rotation and there are two possible chiralities. This defines a vortex V as shown in Fig. 1 and observed [15]. Then there is the case where every successive spins have opposite senses. This defines antivortex AV1 as also shown in Fig. 1, a kind of double flower where single flower state were already proposed about experimental observation [15]. There is also the case of antivortex AV2. Here two pairs of successive spins have the same sense while the two pairs have opposite senses as shown in Fig. 1. AV2 has been also called a “leaf” state according to observation [15]. Finally there are cases 1-3 where three successive spins share the same sense while the last one has opposite sense as observed in simulations [5]. As a matter of fact all these basic structures are observed, at least as metastable structures in simulations for a square dot [5]. Since these topological defects exhibit strong singularities, i.e. strong gradients of spin orientation they induce at the defect core a spin escape in the third dimension characterized by so called “Mexican hats” cores of vortices. Of course in real extended structures several topological defects can be observed playing together within the same sample. In this case the continuity of magnetic domains induces the existence of complementary topological defects as found both experimentally and in higher order terms of this energy minimization. In 3D the third dimension basically duplicate these structures since the two spins in the third direction can be either parallel or antiparallel. But in the last case 3-1 there is a basic dissymmetry between x and y axes. So finally 10 basic topological defects appear in 3D nanoparticles as shown from the principles exhibited in Fig. 2a and b. The two 3D versions of a 2D vortex are shown in Fig. 2. They exhibit a strong singularity so escapes in the third dimension occur in their core central part. On the left hand side the vortex
Fig. 2. (a) The two 3D topological defects issued from a 2D vortex. In red on line and in gray on paper the two spins lying on the third axis (z) are shown elongated for clarity, in antiparallel position and in parallel position respectively. (b) The four 3D topological defects issued from a 2D 1-3 defect. In red on line and in gray on paper the two spins lying on the third axis are shown elongated for clarity, in antiparallel position on the left part and in parallel position on the right part. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
line shows a S shape with a basic undulation while on the right hand side the vortex suffers a strong bending. Of course similar pairs of conjugated magnetic structures are deduced for antivortex 1 and antivortex 2. All these intrinsic defect structures exhibit strong variations of spin orientation within a restricted space so they must be rearranged in a complex way in order to minimize these discrete orientation variations in order to reduce extra exchange energy. Fig. 2b shows the four 3D basic magnetic structures issued from 2D 1–3 structure, always within the same scope, i.e. without exchange energy minimization. Once more in Fig. 2b defect lines deduced from xy plane either are distorted in a S shape or are bent. Quite generally these 10 3D topological defects exhibit strong local variations of the spin direction and twists. This property induces a strong competition between exchange and dipolar interactions which leads to an escape in other dimensions as observed for 2D samples and ultrathin films with Mexican hat shape. So finally realistic twisted topological defects differ a little bit from the basic shapes enumerated in Fig. 2 and deserve careful experimental and numerical investigations. Such high gradient spin configurations induce strong non-linear properties [10] and thus can be useful.
5. Conclusions
Fig. 1. The four basic topological defects V, AV1, AV2 and 1-3 in 2D samples.
This analysis of nanoparticle magnetic structures based upon a powerful variational treatment of a local Hamiltonian leads to several set of equations which result from successive screening levels of dipolar interaction. The first level expresses the global demagnetizing field. Further levels correspond to successive
J.-C.S. Levy / Journal of Magnetism and Magnetic Materials 373 (2015) 2–5
differential equations on the spin field, harmonic and anharmonic too. The analysis of topological defects for 3D samples takes advantage of the previous work on 2D nanoparticles and uses an approach for clusters mixing both local and global constraints as early did about cluster structures and symmetry introducing what became quasicrystalline order [16]. The local constraint consists here in parting in a few sites, i.e. domains at a realistic size, while the global constraint consists in defining the spin orientation in such domains as parallel to external faces in order to satisfy dipolar energy minimization. This method enables us to introduce four basic topological defects for 2D magnetic nanoparticles, and 10 basic topological defects for 3D magnetic nanoparticles. These last 3D defects exhibit a strong twist of the spin orientation so they are rearranged in order to minimize exchange energy. Such configurations induce non-linear properties which can be used for magnonic applications. Acknowledgments It is a pleasure for the author to thank Dr. Philippe Depondt from INSP and Dr. Slawomir Mamica from UAM Poznan for so many fruitful discussions.
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