doi: 10.1209/0295-5075/109/17002
A Monte Carlo study of the anisotropy effects on the spin state evolution in ultrathin helimagnetnanorings
Guo Tian1, Xiao Song1, Peilian Li1, Minghui Qin1, Min Zeng1, Xingsen Gao1(a) and Junming Liu2(b)
1 Institute for Advanced Materials, South China Normal University - Guangzhou 510006, China
2 Laboratory of Solid State Microstructures, Nanjing University - Nanjing 210093, China received 23 September 2014; accepted in final form 16 December 2014
published online 14 January 2015
PACS 75.70.Kw– Domain structure (including magnetic bubbles and vortices)
PACS 75.30.Kz– Magnetic phase boundaries (including classical and quantum magnetic transitions, metamagnetism, etc.)
PACS 75.10.Pq– Spin chain models
Abstract– In this work, the magnetic states in ultrathin helimagnet nanorings were investigated using a Monte Carlo simulation, based on a Heisenberg model involving the short-range exchange coupling, Dzyaloshinsky-Moriya interaction, long-range dipolar interaction, and perpendicular anisotropy. The competition of these interactions leads to the stabilization of various spin states at zero magnetic field,e.g. necklace-like skyrmion chain state, twin-vortex with two coaxial vortices carrying two opposite charities, depending on geometric and anisotropy parameters. A vortex–to–
skyrmion-chain transformation triggered by varying the anisotropy was also revealed. Nanorings with different anisotropy also exhibit completely different magnetization reversal hysteresis loop against magnetic field. Finally, a spin state diagram based on anisotropy and magnetic field was constructed. The present work suggests a novel opportunity to manipulate the spin configurations in nanomagnets, which may find applications in high-density data storage or spintronic devices.
Copyright cEPLA, 2015
Introduction. – Recently, a kind of chiral-lattice heli- magnets have attracted widespread interests, due to their accommodation of unique non-trivial topological spin tex- tures such as magnetic skyrmion carrying a vortex-like spin-swirling spin structure [1–4]. The skyrmion lattice have been experimentally detected in a certain mate- rial family of metallic alloys which have B20 structure with a cubic chiral lattice space group P213, including MnSi [5], Fe1−xCoxSi [6], and FeGe [7], as well as the insulating magnet Cu2OSeO3 [8,9]. Due to the noncen- trosymmetric environment of the chiral lattice, an asym- metric Si ×Sj term called Dzyaloshinsky-Moriya (DM) interaction arises [10,11]. The DM interaction tends to stabilize screw-like spin textures, such as helical and nano- size skyrmion structures. These spin textures can be ma- nipulated by magnetic field, or created or annihilated by spin-polarized electric current [12,13], which are promising for high-density data storage or spintronic devices [14,15].
Moreover, due to the reversal DM effects, there may exist magnetic-field–induced ferroelectric polarization in certain
(a)E-mail:[email protected]
(b)E-mail:[email protected]
insulating helimagnets,e.g. Cu2OSeO3 [9], opening new possibilities for magnetoelectric devices.
It was also reported that skyrmions are very sensitive to geometric confinement. For instance, in bulk helimag- nets the skyrmion states usually occur in very narrow tem- perature and magnetic field windows [3], while they can be broadened to a much wider region in two-dimensional films [16]. In a monolayered Fe or PdFe film on an Ir substrate, an unexpected atomic scale skyrmion-like spin texture can also be observed thanks to the interfacial in- teractions [13,17–19]. It was also demonstrated that in nanolayered chiral magnets, the surface twist instability becomes significant, leading to some novel phenomena such as quantized helicodal states or twist rotation across the layer thickness [20–22]. In lateral confined nanodots, the surface twist from the circular nanodot edges allow for the isolation of single skyrmions, and confinement of quantized topological charges [23,24]. However, there still very few studies on helimagnetic nanorings, in which size confinement can be more significant [25].
On the other hand, it is well known that magne- toanisotropy also plays an essential role in conventional nanomagnets [4,26]. For instant, it greatly affects the
Guo Tianet al.
coercive fields, spin configurations, as well as the Curie temperatures of magnetic systems. In chiral magnets, it can lead to elliptically distorted skyrmion strings, and extend the stabilization region in the phase dia- gram [16,27,28]. In nanolayers, the anisotropy can be exerted by epitaxial strains [16]. In recent years, there have also been numerous reports that in certain magne- toelectric (ME) heterostructures, the magentoanisotropy can also be tuned by an electric field [17,29–33]. These het- erostructures allow for a novel way of using electric fields to modulate their magnetic states by anisotropy engineer- ing, which is promising for spintronic or magnetoelectric device applications.
In this work, we will explore the nanoscale spin states in helimagnetic nanorings by means of a Monte Carlo simu- lation. We have observed various unique spin states, such as vortex, twin-vortex with opposite chiralities, as well as skyrmion chains. A vortex–to–skyrmion-chain transi- tion triggered by the uniaxial anisotropy has also been revealed. This opens an effective way to manipulate the spin states in helimagnetic nanostructured systems.
Modeling and simulation method. – The object of the present study is an ultrathin nanoring on a simple cubic lattice with free boundary conditions. A generally accepted quasi–two-dimensional Heisengberg Hamiltonian for the ultrathin nanostructures of chiral magnets with Dzyaloshinsky-Moriya (DM) interaction and perpendicu- lar anisotropy is adopted, which is written as [25,34]
H =−J
i<j
Si·Sj−
i,j
KRij ·(Si×Sj)−Kz
i
(Si·ei)2
+d
i<j
Si·Sj
|Rij|3 −3(Si·Rij)(Sj·Rij)
|Rij|5
−H·
i
Si.
(1) In sequence, the five terms denote the ferromagnetic ex- change coupling with the exchange constant J, the DM interaction with constant|Kij =K| pointing to the vec- tor along site i and j, the perpendicular anisotropy en- ergy with constantKzwith the easy axis aligned along the z-axis, the dipolar interaction between the blocks with the dipolar strength constantd, and the Zeeman energy with external field, respectively. The ratio K/J with K = 1 was chosen to yield the spiral propagation wavelengths of Ts= 10/√
2 lattice constants.
A high-temperature annealing metropolis algorithm is employed to obtain the equilibrium spin configura- tions [35]. For this, the lattice is initialized from a random spin configuration corresponding to a paramagnetic state at sufficiently high temperature (T). The system is cooled down gradually until it reaches a very lowT. For the MC process, it is almost impossible to reach the temperature of the ground state (i.e., T = 0), hence we takeT = 0.01 as an approximation for the ground state, which is assumed to be low enough for the present system [35].
In addition, we use a ladder protocol [36] to approach to the equilibrium states at a series of discrete temperatures, i.e.,Tis varied following the linear protocolT(t) =T(0)± rt, whereT(0) is the initialT,ris a constant variance ratio ofT,tis measured in MC steps. At each temperature, the system is allowed to relax towards equilibrium for the first 1.5×105MC steps, and thermal averages are calculated over the subsequent 105steps. For simplification, we use a cut-off radiusRcut= 8 to limit their interactions within the neighbors ofRij < Rcut, as an approximation for the long-distance interaction [37].
Spin configurations for nanodisk and nanoring. – In magnetic nanoelements, the geometric parameter plays a vital role. Therefore, we first explore the effect of the ring width on the spin states of nanorings, as shown in fig. 1. In confined system, dipolar interaction (shape anisotropy) has become very crucial, which has been ex- perimentally observed in epitaxial FeGa(111) films [16].
Based on the spin state diagram in ref. [25], we choose d = 0.25 with a modulated dipolar interaction and use two different anisotropyKz (0.3 and 0.8). In this work, we fix the radius of the outer ringR1 = 9, and vary the inner ring radius ofR2 (with the unit of lattice space dis- tance).
For the smallerKz= 0.3, the spin textures of the nan- odisk (fig. 1(a),R1= 9, R2= 0) can be considered as the superposition of a perpendicular bubble domain state with a planar twin-vortex state in which two flux closure vor- tices carrying opposite chiralities coexist. If one removes the disk core forming a wide nanoring (R1 = 9, R2 = 3), the spin state evolves into four separated skyrmion-like states due to geometric confinement. With further re- ducing the ring width below 5 (R1 = 9, R2 = 5 or 3), the domain pattern transforms into a normal flux closure magnetic vortex, superimposed with bubble-like domain on vertical direction.
For the larger anisotropy (Kz= 0.8), we can see signifi- cant change in domain pattern for both the nanodisk and nanoring. For the nanodisk, the spin configuration shows a stripe helix structure. When removing the disk core (ring width = 7), the stripe domain evolves into a bro- ken twin-vortex state. More interestingly, at a modulated ring width of 5, the nanoring transfers into a well-ordered skyrmion chain sate, which is like a pearl necklace, hence we called it skyrmion chain state. If the ring width fur- ther reduces to 3, the skyrmion chain state breaks into an irregular vortex state.
The above spin evolution clearly indicates that the spin configurations can be greatly tailored by geometric and anisotropy conditions. With the involving of DM interac- tion, a large variety of the spin states can occur, which are rarely observed in conventional nanomagnets in par- ticular the unique twin-vortex and skyrmion chains. It is also interesting that at the same ring width we found completely different spin states,e.g. vortex structure and skyrmion chain, owing to the effect of anisotropy. It has
Fig. 1: (Color online) Spin state configurations for the nanodisks and nanorings with different ring width. The parameters (R1, R2) denote the outer disk radiusR1, and the inner ring radiusR2, for (a) d= 0.25,Kz = 0.3; (b)d = 0.25,Kz = 0.8.
The arrows denotes the in-planexy-component of local spin moments (Sxy), while the color scale indicates their perpendicular components (Sz).
also been mentioned that, in small helimagnetic nanos- tructures, a specific twist instability becomes very signif- icant due to the long circular edges [20–23]. To test the effects of surface twist, we examined the spin states evolu- tion without involving the dipolar interaction, precluding the shape anisotropy effect which is also very sensitive to the ring shape. We found that for both large and small Kz values, the spin texture shows helix stripe states in nanodisks, which evolves to skymion chains in nanorings, indicating that the surface twist favors the formation of skyrmion chain states.
Effects of perpendicular anisotropy. – To examine the effects of anisotropy, we study the evolution of spin states as a function of perpendicular anisotropy for various ring shapes. We begin from a zero-anisotropy state (vortex state), and sweep (or ramp) up and down the anisotropy following a ladder protocol. At each anisotropy value, we first performed the process for 1.5×105MC steps to reach the equilibrium state, and subsequently record the spin configuration. The relative local chiralityXr at lattice r is calculated by
Xr=Sr·
Sr+x×Sr+y
+Sr·
Sr−x×Sr−y
. (2)
Figure 2 shows the parameter Xr as a function ofKz
for a fixed d. At initial stage, Xr is rather stable at around ∼ 2.0 corresponding to a vortex state (or twin- vortex state). With increase of Kz, there occurs a quick rise of Xr to a maximum value of above 10, correspond- ing to the occurrence of a multiple-vortex–like skyrmion chain, as demonstrated in the spin configuration images.
Between these two states, there are also some transition states, like waved vortex states, belonging to the nucle- ation stage of the skyrmion. If we further increase Kz, Xr decreases smoothly again, in which case the skyrmion
Fig. 2: (Color online) Total chiralityXr as a function of Kz
for various ring shapes (d = 0.25), which shows a spin state transformation from vortex to skyrmion chain state.
chain domain still maintains but most of the spins are perpendicular aligned forming stripe-like domains, hence we may call it stripe-type skyrmion. All the nanorings with different shape show a similar trend. The above observations indicate that the skyrmion states are very sensitive to anisotropy, which makes it possible to manip- ulate the skyrmions by tailoring the magnetocrystalline anisotropy. This can be achieved by epitaxial strain or electric field.
The above transitions are also accompanied with changes in energy (fig. 3). Upon the transition from vor- tex to skyrmion chain state, both the DM and anisotropy energies drop down, while both exchange and dipolar en- ergy rise sharply. As a result, we are able to see a appar- ent variation in total energy at the transformation region.
At a relatively lower Kz, the exchange energy and the
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Fig. 3: (Color online) Average energies per unit cell as a func- tion ofKz: (a) the total energy per unit cell. The inset shows the energy data near the transition area. (b) The average ex- change, anisotropy, DM, and dipolar energies per unit cell.
dipolar energy are dominant, which favors the formation of vortex structures, whereas the anisotropy energy and DM energy become significant whenKzincreases, leading to the occurrence of helix or skyrmion chain structures.
This indicates that the magnetic structural transition can be driven by the anisotropy and DM energy. From the spin state evolution for different ring shapes, we can see that the surface twist is an essential driving force for the formation of skyrmion chain states.
Magnetization reversal behaviors. – We also ex- plored the magnetization reversal process as a function of magnetic field. Here we chose a fixed ring size (R1= 9, R2 = 5, d = 0.25) but with two different Kz
values (0.3 and 0.8), corresponding to a twin-vortex do- main pattern and a skyrmion chain pattern at zero field, respectively. Figure 4(a), (b) shows the magnetization hysteresis loops for the nanoring with the smallerKz(0.3), along both the out-of-plane (Hz) and in-plane (Hx) direc- tions, respectively. At the remnant state (in fig. 4(a)), the nanoring shows a twin-vortex spin state along the in-plane direction, which is superimposed with an out-of-plane bubble domain state. As the Hz sweeps from −1.8 to 1.8, firstly the magnetization steadily decreases. When
Hz arrives at ∼0.5, a sharp change from twin-vortex to reversed twin-vortex occurs. In between these two states, there is also a transition state such as the skyrmion-chain–
like spin state appearing in a narrow range of the magnetic field. The two remanent twin-vortex states carry reverse in-plane vortex chiralities and out-of-plane net magneti- zation, which may be useful in data storage. Figure 4(b) shows the magnetic reversal hysteresis loop along the in- plane orientation. It can be found that when the magnetic fieldHx sweeps from−0.4 to 0.4, the spin state evolution follows the sequences of onion-vortex-onion transforma- tion [38], which corresponds to the two sharp platform stages in the magnetization curve. This is a typical char- acteristic of vortex-onion magnetization reversal in con- ventional magnetic nanoring.
Figure 4(c), (d) shows the magnetic reversal process for the nanoring with the largerKz (0.8), using the same ge- ometric parameters as that of fig. 4(a), (b). The out-of- plane magnetic field Hz shows a skyrmion chain state at zero magnetic field (fig. 4(c)). AsHzramps up from−1 to 1, the spin state firstly gradually evolves into a twin-vortex state, and then to a skyrmion chain state, and subse- quently changes to a reverse twin-vortex state. The whole process is very smooth and completely reversible, with- out any sharp transformation and detectable hysteresis in magnetization curve. For the in-plane direction (fig. 4(d)), the spin state first evolves to an onion state, then be- comes a distorted skyrmion chain, and finally transforms to a reversed onion state, which is also very smooth and reversible like that along the out-of-plane direction.
Phase diagram. – To give an overview for the spin state evolution, a low-temperature phase diagram (fig. 5) as a function of perpendicular anisotropy and external magnetic field (Hz) along thez-direction was also plotted.
The temperature is set at T = 0.01, which is considered to be low enough to approximate the ground state. The whole diagram is mainly composed of three regions: (I) vortex state, (II) twin-vortex state, (III) skyrmion chain state. As shown in fig. 5, the skyrmion chain state occurs in the range of relatively largeKzand smallHz, while the planar vortex state exist in the range of relatively smallKz
and smallH. For relatively largeH, the domain pattern transforms into a twin-vortex state, which can be consid- ered as a combination of a perpendicular bubble state with a twin-vortex state. With this diagram, we are able to lo- cate a certain spin state by carefully choosing a specific magnetic field and anisotropy value. It is also worth men- tioning that for a certain anisotropy and magnetic field, there also exist some metastable states. Therefore, we should pay attention when locating a specific spin state, and more studies are still needed to comprehensively un- derstand the spin state evolution behaviors. It is also noted that for devices application, the spin state stabil- ity is rather crucial. Therefore, we tested the stability for both twin-vortex and skyrmion chain states by running the simulation beginning from random states using their
Fig. 4: (Color online) The magnetization reversal hysteresis loops and spin state evolution for the nanoring with a smaller Kz of 0.3 ((a), (b)), under the external magnetic field along the out-of-plane direction Hz (a) and in-plane directionHx (b), respectively; with a largerKz of 0.8 ((c), (d)), under the external magnetic field along the out-of-plane directionHz (c) and in-plane direction Hx (d), respectively.
Fig. 5: (Color online) Phase diagram of magnetic states for the helimagnet nanorings, as a function of perpendicular anisotropy (Kz) and the external magnetic field (Hz) along thez-direction at a low temperatureT = 0.01.
individual stabilization parameters. It turns out that each state can be reproduced from a random state after ∼105 MCS and then it is stable up to 106MCS, which indicates that it is most likely that the equilibrium states are stable long enough for certain applications.
Conclusions. – In summary, we explore the effects of anisotropy and magnetic field on the spin states in ultra- thin helimagnet nanorings based on Monte Carlo simu- lations. It was found that the spin state transformation
from a vortex state to a necklace-like skyrmion chain state can be triggered by magnetoanisotropy. It was revealed that the vortex state is favored by dipolar interaction and exchange coupling, while the skyrmion state is more fa- vored by DM interaction in particular the surface twist effect, as well as magentocrystalline anisotropy. We also investigate the magnetization reversal process for the two different anisotropy states, which shows a completely dif- ferent magnetization reversal process. Finally, a the spin state phase diagram based on anisotropy andH-field was constructed, which provides a guideline to design certain spin states. These suggest a new method to manipulate the spin state of nanostructures by external magnetic field or electric controllable anisotropy.
∗ ∗ ∗
The authors would like to thank the Natural Science Foundation of China (Grant Nos. 51031004, 51272078, 51332007), the State Key Program for Basic Researches of China (Grant No. 2015CB921202), the Project for Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2014), the International Sci- ence & Technology Cooperation Platform Program of Guangzhou (No. 2014J4500016), and the Program for Changjiang Scholars and Innovative Research Team in University of China (Grant No. IRT1243) for financial assistance.
Guo Tianet al.
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