The relation between the radii and the densities of magnetic skyrmions
AAverage radius of skyrmions in the skyrmion phase
Yu-Jiao Bo ∗ , Wen-Wen Li † , Yu-Chen Guo ‡ , and Ji-Chong Yang § Department of Physics, Liaoning Normal University, Dalian 116029, ChinaFebruary 23, 2021
Abstract
Compared with the traditional magnetic bubble, skyrmion has smaller size, better stability and is con-sidered as a very promising candidate for future memory devices. It has been noticed that, the sizes ofskyrmions in the skyrmion phase are different from that of the isolated skyrmions, and the former needsmore exploration. In this paper, we use the lattice simulation of Landau-Lifshitz-Gilbert equation to studythe sizes of skyrmions in the skyrmion phase. We find that the shrinks of skyrmions in the skyrmion phasecan be attributed to the density. The radius as a function of different material parameters, the strength ofexternal magnetic field and the density is obtained. With this function, the skyrmion radius can be easilypredicted, which is helpful for the future study of skyrmion memory devices.
Keywords: radius of a skyrmion, shape of a skyrmion
PACS:
1. Introduction
Skyrmion is a topological soliton originally proposed to describe the baryons [1]. It was observed for the firsttime in 2D magnetic systems [2–5] involving Dzyaloshinskii-Moriya interactions (DMI) [6, 7], known as magneticskyrmions. Compared with the traditional magnetic bubble, the skyrmion is smaller, topologically protectedand needs lower power to manipulate, therefore, it has been proposed that the skyrmion is a promising candidatefor high density, high stability, high speed, high storage and low energy consumption memory devices [8–10].As a result, the magnetic skyrmions have draw a lot of attention and are studied intensively recently [10–12].A prerequisite for the use of skyrmions in devices is the knowledge of the relationship between the size ofa skyrmion and the external parameters such as exchange strength, DMI strength and the strength of externalmagnetic field. Such a relationship can be investigated by solving the Euler-Lagrange equation of a skyrmion,for example by using an ansatz [8], or by using a harmonic oscillation expansion [13], or by an asymptoticmatching [14]. It has been noticed that the radius of a skyrmion in the skyrmion phase is much smaller thanthat of an isolated skyrmion [13]. However, while the analytical results have been obtained for the isolatedskyrmions, up to our knowledge, the dependence of the radius of a skyrmion in the skyrmion phase on the ∗ E-mail:[email protected] † E-mail:[email protected] ‡ E-mail:[email protected] § Corresponding author. E-mail:[email protected] a r X i v : . [ c ond - m a t . s t r- e l ] F e b xternal parameters are poorly understood at a quantitative level. In this paper, we study this problem byusing the lattice simulation of Landau-Lifshitz-Gilbert (LLG) equation [15, 16]. We find that the radii in theskyrmion phase are smaller in the skyrmion phase mainly because of the density. A relationship between theradius and external parameters and density is obtained.The rest of paper is organized as the following. Our lattice simulation is discussed in Sec. 2.. The numericalresults on the average radius of the skyrmions in the skyrmion phase are presented Sec. 3.. A summary is madein Sec. 4..
2. Lattice simulation
The lattice simulation is based on the LLG equation, denoting n r as the local magnetic momentum at site r , the LLG can be written as [15–18] 𝑑𝑑𝑡 n r = − B eff ( r ) × n r − 𝛼 n r × 𝑑𝑑𝑡 n r , (1)where n r is the local magnetic moment, 𝛼 is the Gilbert damping constant and the effective magnetic field B eff is B eff ( r ) = − 𝛿𝐻𝛿 n r , (2)with the discretized version of Hamiltonian defined as [19, 20] 𝐻 = ∑︁ r ,𝑖 = 𝑥,𝑦 [ − 𝐽 ( r ) n r + 𝛿 𝑖 − 𝐷 ( r ) n r + 𝛿 𝑖 × e 𝑖 − B ] · n r , (3)where 𝐽 is the local ferromagnetic exchange strength, 𝐷 is the local strength of DMI, B is the strength ofexternal magnetic field and 𝛿 𝑖 refers to each neighbour. On a square lattice, one has 𝛿 𝑖 = e 𝑖 , therefore B eff ( r ) = ∑︁ 𝑖 = 𝑥,𝑦 [ 𝐽 ( r ) n r + 𝛿 𝑖 + 𝐽 ( r − 𝛿 𝑖 ) n r − 𝛿 𝑖 ]+ ∑︁ 𝑖 = 𝑥,𝑦 [ 𝐷 ( r ) n r + 𝛿 𝑖 × e 𝑖 − 𝐷 ( r − 𝛿 𝑖 ) n r − 𝛿 𝑖 × e 𝑖 ] + B ( r ) . (4)The simulation was carried out on GPU [15] which has a great advantage over CPU because of the abilityof parallel computing of the GPU. Eq. (1) is numerically integrated by using the fourth-order Runge-Kuttamethod.
3. Numerical results
We run the simulation on a 512 ×
512 square lattice. In the simulation, we use dimensionless homogeneous 𝐽 , 𝐷 and 𝐵 . 𝐽 = 1 is used as the definition of the energy unit [20–22], the results are presented with 𝐷/𝐽 and
𝐵/𝐽 . In the previous works, the Gilbert constant was chose to be 𝛼 = 0 .
01 to 1 [16, 18, 21–30]. In this paper,we use 𝛼 = 0 .
04 which is in the region of commonly used 𝛼 . The time step is denoted as ∆ 𝑡 . We use ∆ 𝑡 = 0 . steps starting with a randomized initialstate. The average radius of the skyrmions is denoted as 𝑟 , which is measured by using the method introducedin Ref. [13]. Denoting the The radius of a skyrmion is related to the number of sites in an isoheight contour of 𝑧 component of local magnetic moment 𝑛 𝑧 . The isoheight contour in use is 𝑛 𝑧 = ℎ with threshold ℎ = 0 . .1. Average radius of the skyrmions in the skyrmion phase To investigate the relationship between 𝑟 , 𝐷/𝐽 and
𝐵/𝐽 , we simulate with
𝐷/𝐽 in the range of 0 . .
6, and with growing
𝐵/𝐽 for each fixed
𝐷/𝐽 . We calculate 𝑟 for each 𝐷/𝐽, 𝐵/𝐽 when the configuration isin the skyrmion phase. Fig. 1 shows the configurations at
𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 . 𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 . 𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 .
04 and
𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 .
05 as examples. The phase diagram is shown in Fig. 2.The relationship between 𝑟 , 𝐷/𝐽 and
𝐵/𝐽 are fitted by a rational function, the result is 𝑟 ( 𝐽, 𝐷, 𝐵 ) = − . 𝐷 + 682 . 𝐷𝐵 + 273 . 𝐷𝐽 − . 𝐵 − . 𝐵𝐽 − . 𝐽 . 𝐷𝐽 + 51 . 𝐵𝐽 − . 𝐽 . (5) 𝑟 at different 𝐷/𝐽 and
𝐵/𝐽 and fitted 𝑟 ( 𝐽, 𝐷, 𝐵 ) (i.e. Eq. (5)) are shown in Fig. 3. One can see that therational function is consistent with the numerical results.
When the skyrmions are huddled together, the interactions between the skyrmions should not be neglected.We speculate that the interaction is the reason why the skyrmions are smaller in the skyrmion phase. Toinvestigate the relationship between density and the average radius of skyrmions, we define the density in termsof the area that is inside all skyrmions divided by the total area, which is denoted as 𝜌 . For simplicity, weuse the configurations at 𝐷/𝐽 = 0 . 𝐵/𝐽 = 0 . , . , . 𝜌 .Taking configuration 𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 . 𝑓 ( 𝜌 ) = 𝑟/𝑟 LO , where 𝑟 LO is themeasured isolated skyrmion radius. The function 𝑓 ( 𝜌 ) is shown in Fig. 5. We find that 𝑓 can be well fitted bya trilinear function, the result is 𝑓 ( 𝜌 ) = 1 − . 𝜌 . (6)Both the measured 𝑟/𝑟 LO and the fitted 𝑓 ( 𝜌 ) are shown in Fig. 5With Eq. (6) we can improve Eq. (5) to include the effect of density. 𝑟 LO can be obtained by solving 𝜃 LO ( 𝑟 ) = 𝜋 exp( − 𝑤 ( 𝐵/𝐷 ) 𝑟 /
2) = cos − ( ℎ ) where 𝑤 = 0 . 𝑟 ( 𝐵, 𝐷, 𝜌 ) = 𝑟 LO × 𝑓 ( 𝜌 ) = √︃ 𝑤 log (︂ 𝜋 cos − ( ℎ ) )︂ × 𝐷𝐵 × 𝑓 ( 𝜌 ) . (7)When ℎ = 0 . √︀ 𝜋/ cos − ( ℎ )) /𝑤 ≈ . -1 -0.50 0.5 1 (a) 𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (b)
𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (c)
𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (d)
𝐷/𝐽 = 0 . , 𝐵/𝐽 = 0 . Figure 1: The configurations in the skyrmion phase. The heat map represents the magnitude of 𝑛 𝑧 , the smallarrows represent ( 𝑛 𝑥 , 𝑛 𝑦 ). 4 .2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 D/J B / J Helical phaseSkyrmion phaseFerromagnetic phase
Figure 2: The phase diagram obtained by lattice simulation of LLG.Figure 3: 𝑟 at different 𝐷/𝐽 and
𝐵/𝐽 (marked as ‘+’) and the fitted 𝑟 ( 𝐽, 𝐷, 𝐵 ), i.e. Eq. (5).5 -1 -0.50 0.5 1 (a) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (b) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (c) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (d) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (e) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (f) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (g) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (h) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (i) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (j) 𝜌 = 0 .
50 100 150 200 250 300 350 400 450 50050100150200250300350400450500 -1 -0.50 0.5 1 (k) 𝜌 = 0 . Figure 4: The configurations corresponding to different 𝜌 .6 f () fitted f( )B/J=0.1B/J=0.15B/J=0.2 Figure 5: The measured 𝑟/𝑟 LO compared with the fitted 𝑓 ( 𝜌 ).Figure 6: 𝑟 calculated with Eq. (7) (marked as ‘+’) compared with Eq. (5) (the curved surface).7 r (r) LO (r/f( ))measured (r) Figure 7: The shape of a skyrmion in the skyrmion phase
Since the size of skyrmion is different for those in the skyrmion phase and the isolated skyrmions, a questionarises naturally. Are their shapes the same? The configuration of a skyrmion can be parameterized as n ( 𝑟, 𝜑, 𝑧 ) = sin[ 𝜃 ( 𝑟 )] e 𝜑 + cos[ 𝜃 ( 𝑟 )] e 𝑧 . (8)The function 𝜃 ( 𝑟 ) is often used to describe the shape of a skyrmion [13]. Choosing the configuration at 𝐷/𝐽 = 0 . 𝐵/𝐽 = 0 .
1, we measure 𝜃 ( 𝑟 ) with 𝜃 in the range arccos(0 . ≤ 𝜃 ≤ arccos( − . 𝜃 LO ( 𝑟 ) with 𝑟 rescaled according to Eq. (6), i.e. 𝜃 ′ ( 𝑟, 𝜌 ) = 𝜃 LO ( 𝑟/𝑓 ( 𝜌 )) in Fig. 7.As shown in Fig. 7, the shape of a skyrmion in the skyrmion phase is similar to the shape of an isolatedskyrmion with 𝑟 rescaled. This result implies that the skyrmions in the skyrmion phase can be seen as beingexperiencing an effective magnetic strength 𝐵 ′ = 𝐵/𝑓 ( 𝜌 ) when affected by other skyrmions. In the lattice simulation, we use dimensionless parameters. The numerical results can be matched tothe real material by using the rescaling introduced in Ref. [16, 31]. The rescaling factor is denoted as 𝑠 and 𝑠 = ( 𝐷/𝐽 ) 𝜆/ (2 𝜋 √ 𝑎 ) where 𝜆 is helical wavelength and 𝑎 is the lattice spacing. The helical wavelength of realmaterials can be found in Ref. [11]. For example, if we take 𝜆 ≈
60 nm, 𝑎 = 0 . 𝐷/𝐽 = 0 .
4, then 𝑠 ≈ .
75. Then 𝑟 = 6 . 𝑟 = 6 . × 𝑠 × 𝑎 ≈ .
97 nm. Meanwhile the time unit is rescaled as 𝑡 ′ = 𝑠 𝐽 (cid:126) /𝐽 ′ , where 𝐽 is the dimensionless exchange strength and 𝐽 ′ is the exchange strength of a real material.If we choose 𝐽 ′ ≈ 𝑡 ′ ≈ .
01 ns. The time step in the simulation is ∆ 𝑡 = 0 . 𝑡 ′ ≈ . Summary One of the reasons that the skyrmion is proposed as a candidate for the future memory devices is becausethe size of a skyrmion is small. The radius of a skyrmion in the skyrmion phase is an important issue which islack of exploration. In this paper we use the lattice simulation of LLG equation to study the average radius ofskyrmions in the skyrmion phase.The average radii at different
𝐷/𝐽 and
𝐵/𝐽 are measured. We confirm that the average radius of skyrmionsin the skyrmion phase is smaller than the radius of an isolated skyrmion. The relationship between the averageradius and the external parameters including exchange strength, DMI strength and the strength of externalmagnetic field are obtained. We also investigate the relationship between the average radius of skyrmions withthe density. The average radius 𝑟 can be presented as a function of isolated skyrmion radius 𝑟 LO ( 𝐵/𝐽, 𝐷/𝐽 ) anddensity 𝜌 , which is 𝑟 ( 𝐵, 𝐷, 𝜌 ) given in Eq. (7). It can be shown that with the decrease of density, the averageradius of skyrmion approaches that of an isolated skyrmion. With this relation, we improve our empiricalformula to include the effect of density. By using our empirical formula, the skyrmion radius for differentmaterials can be easily predicted. We also find that, the shapes of the skyrmions are not sensitive to thedensity, which implies that the interactions between skyrmions can be seen as an effective magnetic strength.
Acknowledgment
This work was partially supported by the National Natural Science Foundation of China under Grants No.12047570 and the Natural Science Foundation of the Liaoning Scientific Committee No. 2019-BS-154.
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