Stripe-vortex transitions in ultrathin magnetic nanostructures

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J. Appl. Phys. 113, 054312 (2013); https://doi.org/10.1063/1.4790483 113, 054312

Stripe-vortex transitions in ultrathin magnetic nanostructures

Cite as: J. Appl. Phys. 113, 054312 (2013); https://doi.org/10.1063/1.4790483

Submitted: 17 December 2012 . Accepted: 22 January 2013 . Published Online: 06 February 2013 J. P. Chen, Z. Q. Wang, J. J. Gong, M. H. Qin, M. Zeng, X. S. Gao, and J.-M. Liu

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Stripe-vortex transitions in ultrathin magnetic nanostructures

J. P. Chen,¹Z. Q. Wang,¹J. J. Gong,¹M. H. Qin,¹M. Zeng,¹X. S. Gao,^1,a)and J.-M. Liu^2,b)

1Institute for Advanced Materials, South China Normal University, Guangzhou 510006, China

2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

(Received 17 December 2012; accepted 22 January 2013; published online 6 February 2013) In this work, the magnetic states in ultrathin nanostructures are investigated using Monte Carlo simulation, based on a Heisenberg model involving the short-range exchange coupling, long-range dipole-dipole interaction, and perpendicular anisotropy. An intriguing thermally driven magnetic structural transition from perpendicular stripe domain to flux closure (planar vortex) state, accompanied by an apparent thermal hysteresis effect and typical characteristics of the first-order phase transition, is revealed. Furthermore, it is found that the transition can be remarkably modulated by perpendicular anisotropy. The present work suggests a promising approach to manipulate the spin configurations in nanomagnets by adjusting temperature and perpendicular anisotropy.V^C 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790483]

I. INTRODUCTION

In the past decades, there has been a surge of research interest in nanomagnets, owing to the advances in modern nanofabrications and microscopic techniques, as well as the growing demand for high density recording, sensors, and spintronic devices.¹In geometrically confined nanomagnetic systems, a great variety of magnetic topological defects such as bubble domains, vortices, solitons, and Skyrmions can frequently occur.² An interesting example is the magnetic vortex state, a curling magnetic structure occurring in ferro- magnetic (FM) structures like disks or rings.^3–6 In many cases, it possesses a vortex core with out-of-plane magnetization, as confirmed by experimental observations using magnetic force microscopy⁵ and spin-polarized scanning tunneling microscopy.⁶ Another example is more complex structure named Skyrmion, a vortex-like spin-swirling object. A Skyrmion state was originally proposed to describe a localized, particle-like excitation in the continuous fields with nonlinear interactions.⁷It was subsequently observed in many condensed-matter systems such as quantum Hall ferro- magnets⁸and cold atoms,⁹and more recently in chiral magnets MnSi, Fe_0.5Co_0.5Si, and FeGe (Refs. 10–13) and even multiferroic Cu₂OSeO₃.¹⁴

It is known that the spin structures in a magnetic system are governed by the competition among various fac- tors, including exchange energy, dipole-dipole interaction, magnetocrystalline anisotropy, and so on. For a thin nanostructure, the shape anisotropy originating from the dipole- dipole interaction always favors the in-plane magnetization, whereas magnetocrystalline anisotropy often promotes the out-of-plane magnetization. Therefore, the competition between the shape and the magnetocrystalline anisotropy can largely determine the micromagnetic structures of nanomagnets. This offers us a good opportunity to adjust the contributions of these effects to manipulate the mag-

netic pattern of the nanostructures, by varying parameters such as the system shape and/or size,¹⁵ temperature (T),^16,17 and external fields,¹⁸ which can be generally con- sidered as a kind of domain engineering for designing magnetic states.

In the past decades, many theoretical and experimental works have been devoted to control the micromagnetic structures in ultrathin films and multilayers. One of the most con- spicuous features in these systems is their spin configurations are sensitive dependent onT. For instance, a spin reorienta- tion transition from an out-of-plane to predominant in-plane orientation can occur in ultra-thin magnetic films as T varies.^16,17For magnetic nanostructures, most of earlier studies aimed at micromagnetic properties of ground state or low T state,^19,20 and seldom addressed their thermodynamic effects. In this sense, the thermodynamic effects over magnetic nanostructures deserve for exploration due to the poten- tial existence of fascinating properties in the thermodynamic evolution of the system.

On the other hand, magnetism in FM materials is tra- ditionally controlled directly by applying an external magnetic field. Recent studies²¹ revealed that the anisotropy can be tailored by applying an electric field without pres- ence of magnetic field. So far, electric field control of magnetic behavior was realized by the magnetoelectric (ME) coupling effect in multiferroics,²² which has poten- tial applications in electric field-controlled magnetic data storage. Hence, the ability to control the anisotropy with an applied electric field stimulates us to further explore the effects of anisotropy on the thermodynamic behaviors of nanomagnets.

Based on these considerations, we investigate the thermal evolution of the micromagnetic structures in ultrathin magnets with a specific geometry (an ultrathin rectangular structure) using Monte Carlo (MC) simulation.²³An intriguing transition from a planar vortex state (with an out-of-plane core) to a perpendicular stripe state was observed in this system, which can be understood as the result of the competition between internal energy and entropy. In addition, it is also

a)Electronic address: [email protected].

b)Electronic address: [email protected].

0021-8979/2013/113(5)/054312/6/$30.00 113, 054312-1 VC2013 American Institute of Physics

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found that the characteristics of these transitions (e.g., stripe- vortex transition temperature) are greatly depended on their perpendicular anisotropy.

The remainder of this paper is organized as follows:

In Sec. II, the model and the simulation method will be described. Section III is attributed to the simulation results and discussions. At last, the conclusion is presented in Sec.IV.

II. MODEL AND SIMULATION METHOD

The object of the present study is an ultrathin nanostructure comprising LLh spins on a simple cubic lattice with free boundary conditions. The MC simulation is conducted to track the evolution of the spin configuration as a function ofT. The Hamiltonian of the system we are interested in consists of nearest-neighbor FM exchange interactions, dipole-dipole interactions, and perpendicular anisotropy, as given below²⁰

H¼ JX

hi;ji

SiSjþDX

ði;jÞ

SiSj

jrijj³ 3ðSirijÞðSjrijÞ jrijj⁵

" #

Kz

X

i

ðSizÞ²; (1)

whereJ>0 defines the FM exchange constant,Dis the dipolar constant, andKz>0 determines the strength of the anisotropy along the easy axis (z-axis),Si¼(Six,Siy,Siz) denotes a classical Heisenberg spin with unit moment at site i (i.e., jSij ¼1), andjrijj jrirjjis the distance betweenith spin andjth spin.hi,jidenotes that the summation is restricted to the nearest-neighbor pairs. In the second term, the interaction is long-ranged and the summation is over all pairs in the lattice. For simplicity, we set the Boltzmann constant kB¼1, exchange constantJ¼1, and lattice constanta¼1.

The simulations are conducted following the Metropolis algorithm. To investigate the ground states, we use the simulated annealing technique.²⁴For this, the lattice is initialized from a random spin configuration corresponding to the paramagnetic state at sufficiently highT. The system is cooled down gradually until it reaches a very low T. For the MC process, it is almost impossible to reach the temperature of the ground state (i.e.,T¼0), hence we takeT¼0.01 as an approximation, which is assumed to be low enough for the present system.²⁴

In addition, to evaluate the magnetic structural transitions, we use a ladder protocol²⁵ to reach the equilibrium states at a series of discrete temperatures, i.e., temperatureT is varied following the linear protocolT(t)¼T(0)6rt, where T(0) is the initialT, ris a constant variance ratio ofT,tis measured in MC steps. To locate the accurate transition temperature for very abrupt magnetic transitions, a histogram method is applied.²⁶It is worthy of mention that most of our computing time is consumed on calculating the dipolar interactions over the whole lattice. Although there are also some approaches using a cut-off radiusrcutto limit their interactions within the neighbors ofrijrcut,²⁷such a cut-off radius may also bring about artificial effects.^20,24Therefore, we do

not conduct any cut-off to ensure the accuracy of our simulation.

The transitions in our MC simulation are monitored by evaluating several physical quantities. The specific heat is measured as the fluctuation of energy

CðTÞ hE²i hEi²

NkBT² ; (2)

whereEis the internal energy of the lattice andNis the lattice volume.

The out-of-plane magnetization hSz2iis used to characterize the extent of perpendicular orientation. For instance, hSz2i ¼1 means that all the spins are oriented out-of-plane, whilehSz2i ¼0 corresponding to the case that almost all spins are aligned in plane.

The degree of perfection of a striped domain in magnetic nanostructures can be quantified by an orientational order parameter¹⁷given by

Ohv¼nvnh

nvþnh

; (3)

where nh (nv) is the number of horizontal (vertical) nearest neighbor spins which are antiparallel. This order parameter describes the degree of ordering of the stripe domains. It is saturated atOhv¼1 for a fully striped domain phase.

The order parameter, named toroidal moment, is introduced to characterize the vortex texture in the present lattice, following earlier literature,²⁸it is defined as

G¼ 1 2N

X

i

riSi: (4)

III. SIMULATION RESULTS AND DISCUSSIONS A. Low temperature phase diagram

Fig.1shows the lowT phase diagram of a 20203 cubic lattice as a function of the perpendicular anisotropy

FIG. 1. Phase diagram for magnetic states of magnetic nanodots on a 20203 simple cubic lattice, as a function of dipolar constant (D) and perpendicular anisotropy (Kz) at low temperatureT¼0.01. Region I is an in- plane vortex state, region II is out-of-plane striped domain state, and region III is mixture of both states. Inserts show some typical spin configurations (planar component of the spins (Sixy) in region I and perpendicular component of the spins (Siz) in region II).

054312-2 Chenet al. J. Appl. Phys.113, 054312 (2013)

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(Kz) and the dipolar interaction (D) atT¼0.01. Two major magnetic states are observed: One is the perpendicular striped domain phase in which the majority of spins is aligned out-of-plane, forming a striped structure, while the other phase is a planar vortex texture in which the spins form a flux closure configuration on the xy plane. As shown in Fig.1, the planar vortex states exist in the range of relatively small Kz and large D (region I), while the perpendicular striped domain phase occurs in the range of large Kz and small D (region II). In between these two regions, there exists a mixture of vortex and striped domains (region III).

This indicates that the dipolar interaction favors the planar vortex domain, while the perpendicular anisotropy promotes the formation of striped domain.

B. Stripe-vortex transitions

To understand the evolution of magnetic state as a function ofT, a specific vortex ground state was chosen (at fixed Kz¼0.6 and D¼0.4, in region I) and the T-dependence of this state is simulated. Fig.2shows the snapshots of spin lattice obtained at a series of temperatures. To illustrate the three-dimensional magnetic moments, we use the color maps to demonstrate the perpendicular component of spins (Siz) and spin vectors (arrows) to describe thexy component of spins (Sixy

). It is noted that thexycomponent of spins for the consecutive three layers exhibits nearly identical configura-

tions, hence we only present the spin vectors for the middle layer.

As shown in Fig.2(a), at highT¼6.0, the spins are in a disordered (paramagnetic) state. With decreasing T, the lattice first evolves continuously into a so-called tetragonal phase [Fig. 2(b)] with a fourfold symmetry as proposed in the earlier study.²⁹ AsTdecreases further, a slightly defec- tive stripe pattern [Fig. 2(c)] emerges and then transforms into a regular stripe domain pattern [Fig. 2(d)] with straight domain borders at T as low as 0.15. Surprisingly, when T drops down slightly to 0.13, the stripe domains suddenly transform into a planar vortex state [Fig.2(e)]. At thisT, the vortex phase is a little deformed due to the thermal fluctua- tions. At even lower T¼0.01, we observe a perfect vortex pattern with an apparent out-of-plane vortex core, which is expected to be the ground state [Fig.2(f)]. When heating up, the vortex state can evolve back into the stripe domain pattern again, whereas the transitions corresponding to the heating and cooling processes occur at different temperatures, indicating the existence of a thermal hysteresis effect.

To further characterize the structural evolution, several structural order parameters as a function of T in the cooling and heating cycle are evaluated. We usehSz2ito quantify the extent of spin orientation [Fig.3(a), the cooling sequence]. As temperature decreases from highT,hSz2irises smoothly from 0.50 and reaches the peak value of0.81, which means that about 90% of spin components are aligned out-of-plane.

FIG. 2. Evolution of spin configurations of magnetic nanodots on a 20203 simple cubic lattice (withD¼0.4,Kz¼0.6), during cooling down process.

Snapshots for thexycomponent of spinsSixy

(left) and out-of-plane component of spinsSiz

(right) at different temperatures:T¼6.00 (a),T¼1.40 (b),T¼0.90 (c),T¼0.15 (d),T¼0.13 (e), andT¼0.01 (f). The spin vectors (arrows) and color maps are used to illustrate the spin components along in-plane and out-of- plane direction, respectively. The system was initialized from a random state, which was first relaxed toT¼6.00, and then cooled down using a ladder cooling protocol to a lowT(T¼0.01).

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During the evolution, the spin lattice first evolves into a tetragonal phase, and subsequently transforms into a stripe domain phase. Further decrease ofT leads to a sudden drop of hSz2i to nearly zero, manifesting that almost all spins flip abruptly from the out-of-plane stripe to in-plane vortex state.

It is noted that, at highT¼6.0 (corresponding to a paramagnetic phase),hSz2i is about 0.5. However, in a fully random state, spins are expected to orientate statistically in all direc- tions, yieldinghSx2i ¼ hSy2i ¼ hSz2i ¼1/3.²⁰ In our simulation, due to the existence of perpendicular anisotropy, the spins tend to align along thez-axis, leading to a considerably larger hSz2ithan that of randomly orientated state. The orientational order parameterOhvand toroidal momentG(Gx,Gy,Gz) as a function ofTduring the cooling sequence are plotted, respectively, in Fig.3(b). In between 0.15<T<1.2,Ohvgrows up gradually until it reaches the maximal value (1) atT¼0.15, which corresponds to the formation of perfect striped states.

Further decrease ofT leads to a suddenly drop down inOhv, accompanied with a sharp rise ofGz simultaneously, which associates with the formation of planar vortex state.

The transitions can be further seen in T-dependence of the specific heat as shown in Fig. 3(c). It shows a broad hump atT¼1.2 along with a sharp peak atT0.15. The for- mer corresponds to a transition from paramagnetic phase to stripe domain phase, while the latter is to a sudden transition to the vortex state. The inset shows the specific heat data near the stripe-vortex transition temperature, calculated by the histogram method,²⁶ which reveals that the transition occurs within an extremely narrow range of 0.001 K. The transition is also accompanied with a sudden change in energy [Fig.3(d)]. Upon the transition from stripe to vortex, the anisotropy energy increases abruptly, while both the dipolar and the exchange energy drop down sharply, leading to an overall reduction in total energy. This indicates that the transition is the result of competition between different components of internal energy. At a relatively higherT, anisotropy energy is dominant, which favors the formation of stripe domains, while the exchange energy and the dipolar energy become significant at low T, leading to the occur- rence of planar vortex structure. It is worth of mention that the specific heat is not always a reliable quantity to charac-

terize a phase transition. However, in our case, the locations of maximum specific heat are very close to the onset temperatures for magnetic structure change, and jumps in energy and order parameters, which can still give an important clue to the transition behaviors in our model.

To further examine the transition behavior, we study the thermal hysteresis effect by the cooling-heating cycle, as seen in Fig.4. It is interesting that a thermal hysteresis loop

FIG. 3. Plots of various physical quantities as a function of T: hSz2

i (a), toroidal moment G and orientational order parameterOhv(b), specific heatC(T) (c), and the average exchange, dipolar, anisotropic and total energy per spin (d).

FIG. 4. A strong thermal hysteresis effect is reflected by the distinct thermal hysteresis loops ofhSz2

i(a), the average energy per spin (b), and orientational order parameterOhv(c) around the region of stripe-vortex transition.

All the results are obtained from a cooling-heating procedure, with which the system is cooled down from highTto a very low temperatureT¼0.01, and then heated up with opposite process by adopting a ladder protocol.

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arises in the region of stripe-vortex transition, in which the transition temperature Tc for heating process is notably higher than that for cooling process. According to Ehren- fest’s proposition,³⁰there are three criteria identifying a first- order phase transition, including hysteresis, latent heat, and an abrupt change in order parameter. This allows us to con- clude that the above transition exhibits the characteristics of the first-order phase transition. Although the above results obtained by MC simulation show clear evidence of the existence of stripe-vortex transition in present system, the underlying mechanisms are still unclear and need further investigations.

C. Effect of perpendicular anisotropy

Recently, it was reported that in some thin film heterostructures (e.g., ultrathin Fe, Co based film heterostructures), magnetocrystalline anisotropy can be modified by electric field.²¹This stimulates us to further explore the effect of anisotropy on the magnetic states, which can be utilized as an effective tool for manipulating the magnetism of nanomagnets. In this work, we explore the magnetic state transition behaviors for various anisotropy value Kz during cooling down process. As shown in Fig.5, there are two types of magnetic structure transitions. For Kz<0.7, the transitions follow the sequences as paramagnetic-tetragonal-stripe-vortex. ForKz>0.7, only the paramagnetic-tetragonal-stripe transition occurs, without involving of planar vortex states. The detailed simulations produce aKz-T phase diagram, as illus- trated in Fig.6. It is interesting that the phase boundary which separates the region I and region II shows thatTcis almost lin- early dependent on Kz in the range of low anisotropy (Kz<0.7). ForKz>0.7, the tetragonal-stripe transition occurs at a nearly fixed temperature of around 1.1, which is similar to the results obtained from thin film systems.^17,25 The above observations indicate that the magnetic states are very sensi- tively to the change of anisotropy, which make it possible to

manipulate the magnetic structure transitions and tailor spin patterns by using external electric field to control the magnetocrystalline anisotropy.

D. Discussion

From our previous analysis, it is evident that various spin textures can be manipulated by tuning temperature and anisotropy. This may offer us opportunities to design specific domain structure or magnetic structural transition for spintronics or multiferroic applications. It is worth noting that due to the limitation of computational resource, our simulation is more suitable for small-sized systems. To extend to large-sized systems, a scaling technique^31,32can be integrated which allows extending the simulation of small lattice to much larger systems by using a proper scaling relation, which has been suc- cessfully applied in cylinders,³¹cones,³³nanowires,³⁴and so on. Besides that, it was also found that such anisotropy can be introduced by external electric fields, this provides the possi- bility of using external electric fields to control magnetic states, even design specific magnetic domain structures.

IV. CONCLUSIONS

In this work, a magnetic structural transition driven byT arising in an ultrathin FM nanostructure is observed. The transition sequences can be summarized as follows: paramagnetic

! tetragonal!perpendicular striped domain! planar sin- gle vortex (with an out-of-plane vortex core). During the stripe-vortex transition, the spins undergo a collective and abrupt change, bridging two completely different spin configurations. In addition, a strong thermal hysteresis effect, latent heat, combined with an abrupt change in theT-dependence of hSz2

i, toroidal moment G, and orientational order parameter Ohvoccur in this transition, exhibiting the features of the first- order phase transition. Furthermore, the characteristics of the transitions (e.g., vortex-stripe transition temperature, and type of transitions) are strongly depended on anisotropy. These offer new opportunities to tailor the magnetic states of nanostructures usingTas well as external fields.

ACKNOWLEDGMENTS

We thank Mr. Y.-L. Xie for stimulating discussions.

This work was supported by the Natural Science Foundation

FIG. 5. Evolution ofhSz2i(a), and orientational order parameterOhv(b) as a function ofTfor various anisotropy strength (Kz).

FIG. 6. Phase diagram for the spin states of magnetic nanodots on a 20203 simple cubic lattice withD¼0.4, as a function ofKz andT.

Regions I, II, and III are corresponding to the vortex spin configuration, striped domain, and tetragonal or paramagnetic state, respectively.

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of China (Grant Nos. 51072061, 51031004, and 51272078) and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.

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