A theory on skyrmion size and profile

A magnetic skyrmion is a topological object consisting of an inner domain, an outer domain, and a wall that separates the two domains. The skyrmion size and wall width are two fundamental quantities of a skyrmion that depend sensitively on material parameters such as exchange energy, magnetic anisotropy, Dzyaloshinskii-Moriya interaction, and magnetic field. However, there is no quantitative understanding of the two quantities so far. Here, we present general expressions for the skyrmion size and wall width obtained from energy considerations. The two formulas agree almost perfectly with simulations and experiments for a wide range of parameters, including all existing materials that support skyrmions. Furthermore, it is found that skyrmion profiles agree very well with the Walker-like 360\textdegree{} domain wall formula.

Although much knowledge about magnetic skyrmions has been accumulated after intensive studies including skyrmion generation [8,10,17,18] and manipulation [14,[19][20][21], the dependence of skyrmion size (R) on material parameters such as exchange energy, magnetic anisotropy energy, and DMI strength is still poorly understood at a quantitative, or even qualitative level. A well-known formula of skyrmion size is R ∝ D/A, where D is the DMI strength and A is the exchange stiffness [14]. This formula cannot be correct because it does not capture the facts that the skyrmion size depends sensitively on the magnetic anisotropy K [16,22] and perpendicular external magnetic field B [7,23]. Another formula of R proposed in Ref. [24], which depends on D, K, and B, is incorrect because skyrmion size is sensitive to the exchange stiffness A. There are also other expressions for skyrmion size based on different ansatz about the skyrmion profile [25][26][27]. However, none of them agrees with experiments or micromagnetic simulations. Even worse, the physical pictures behind these expressions are either not clear or wrong. In this paper, we show that the skyrmion profiles agree well with Walker-like 360°domain wall formula. By minimizing the energy, we obtain the analytical expressions of the skyrmion size R and wall width w as functions of A, D, K, and B. Interestingly, R decreases with A while w is insensitive to A. In general, both A and w increases with D. These results agree perfectly with micromagnetic simulations and are consistent with experiments although they are against one's intuition.
We consider a two-dimensional (2D) ferromagnetic film in xy plane with an exchange constant A, an interfacial DMI coefficient D, a perpendicular easy-axis anisotropy K, and a perpendicular magnetic field B. The total energy E of the system consists of the exchange energy E ex , the DMI energy E DM , the anisotropy energy E an , and the Zeeman energy E Ze , where m is the unit vector of magnetization of a constant saturation magnetization M s and the integration is over the whole film. The energy reference is chosen in such a way that the energy of single domain state of m z = 1 is E = 0. The demagnetization energy is included in E an by using the effective anisotropy K = K u − µ 0 M 2 s /2 corrected by the shape anisotropy, where K u is the perpendicular magnetocrystalline anisotropy. It is convenient to use a polar coordinates so that a point r in the plane is denoted by r and φ. Magnetization at r is described by polar and azimuthal angles Θ(r, φ) and Φ(r, φ) so that m = (sin Θ cos Φ, sin Θ sin Φ, cos Θ). A skyrmion centered at r = 0 can be described [15] by, with boundary conditions of Θ(0) = 0(π) and Θ(∞) = π(0). v is the vorticity (v = 1 for a skyrmion and v = −1 for an antiskyrmion), and γ is a constant classifying type of skyrmions (γ = 0 or π for Néel skyrmions and γ = ±π/2 for Bloch skyrmions). A skyrmion consists of an inner domain, an outer domain, and a wall separating the two domains. We define the skyrmion size R as the radius of the m z = 0 contour. The wall width w is another fundamental skyrmion quantity often ignored in previous studies [24][25][26]. One can also define the skyrmion polarity as p = [m z (r = ∞) − m z (r = 0)]/2 so that p = 1 (−1), corresponds to spins in the inner and outer domains pointing respectively to the −z(+z) and +z(−z)-directions.
In terms of Θ, four energy terms for v = 1 are For a skyrmion of p = 1 as shown in Fig. 1, 2r rdr = ∞ 0 sin Θ cos Θdr < 0 is small because sin Θ cos Θ = 0 far from the skyrmion wall and sin Θ cos Θ changes its sign from negative near the inner domain to positive near the outer domain. Thus, to lower the total energy, one needs γ = 0(π) for D > 0 (< 0). This corresponds to a Néel skyrmion. Along a radial direction, the magnetization profile looks like a 360°Néel domain wall as illustrated in Fig. 1. This leads us to model a skyrmion profile by the Walker-like 360°domain wall solution [23,28,29], To test how good ansatz (4) is for a skyrmion, we use MuMax3 [30] to simulate various magnetic stable states in a magnetic disk of diameter 512 nm and thickness 0.4 nm. The mesh size of 1 nm × 1 nm × 0.4 nm is used in our simulations. A = 15 pJ/m, M s = 580 kA/m, and perpendicular easy-axis anisotropy K u = 0.8 MJ/m 3 [16] are used to mimic Co layer in Pt/Co/MgO system. The initial state is m z = 1 for r > 10 nm and m z = −1 for r ≤ 10 nm. The final stable state depends on the values of D and B. The lower left inset of Fig. 1 ○ is a stripe domains state for D = 5 mJ/m 2 and B = 0. The upper left plot of Fig. 1 shows the spatial distribution of m z of the skyrmion in 1 ○ along three radial directions, φ = 0 (crosses), 45 • (triangles), and 90 • (circles). All three sets of data are on the same smooth curve, showing m z is a function of r, but not φ. The curve can fit perfectly to Eq. (4) with R = 25.77 nm, w = 4.94 nm. We plotted also Φ(φ) at randomly picked spins from the simulated skyrmion. All numerical data (red circles) are perfectly on Φ = φ. These results not only confirm the validity of skyrmion expression of Eq. (2), but also suggest that m z (r) ≡ cos[Θ dw (r)] follows the Walker-like 360°DW profile (4).
The energy of a skyrmion can then be obtained from Eq. (3) by using the Walker-like 360°domain wall profile Θ dw (r). The total energy is, in general, a function of R and w [instead of a functional of Θ(r) and Φ(φ)] as where f i (x) (i = 1 ∼ 6) are non-elementary functions defined in the Supplemental Material [31]. The skyrmion size and wall width R and w are the values that minimize E(R, w).  5). Both micromagnetic simulations and analytical results clearly show that skyrmion can exist for D < 3.8 mJ/m 2 in the current case. Above the upper limit, the stable state is not a skyrmion, but stripe domains as shown in 3 ○ for D = 5 mJ/m 2 . E(R, w) of (5) has a minimum as long as |D| < 3.8 mJ/m 2 that indicates existence of skyrmion. However, micromagnetic simulation shows that skyrmion can only exist in the window of 1.2 mJ/m 2 < D < 3.8 mJ/m 2 when the skyrmion size is larger than 1 nm in the current case. Below 1.2 mJ/m 2 , the stable state is a single domain with all spins pointing up or down as shown in 2 ○ of Fig. 1. This discrepancy may be due to the discretization of continuous LLG equation in micromagnetic simulation. In principle, the mesh size should be much smaller than the skyrmion size. For a skyrmion of 1 nm, our mesh size is 0.01 nm. Due to the limited precision of the MuMax3 package, the mesh size cannot be further decreased. There exists also a minimal A of around 14 pJ/m as shown in Fig. 2(b) and a minimal K of around 0.56 MJ/m 3 as shown in Fig. 2(c), below which skyrmion does not exist, and the stable state is stripe domains as shown in 3 ○ of Fig. 1. The skyrmion size decreases with B, which is consistent with the experimental observations [7,11,23].
It is still unclear how R and w depend on A, D and K although Eq. (5) agrees almost perfectly with simulation results. Thus it is highly desirable to have a simple approximate expressions for R and w in terms of material parameters. The exchange and DMI energies come from the spatial magnetization variation rate. For a skyrmion, the magnetization variation rates in the radial and tangent directions scale respectively as 1/w and 1/R. The exchange energy is then proportional to skyrmion wall area of πRw multiplying the square of the magnetization variation rates 1/R 2 + 1/w 2 , i.e. E ex ∝ (R/w + w/R). For a Néel skyrmion, the magnetization variation rate along the tangent direction is perpendicular to m and does not contribute to the DMI energy. The DMI energy is then proportional to wall area (Rw) multiplying the magnetization variation rate along radial direction (1/w), i.e. E DM ∝ R. The anisotropy energy is mainly from the skyrmion wall area. Thus, E an ∝ Rw. The Zeeman energy of the skyrmion comes from the inner domain proportional to its area of π(R − cw) 2 , where c is a coefficient depending on the magnetization profile, and from the wall area proportional to its area of πRw. To obtain the proportional coefficients, one needs to find approximate expressions for f i (R/w) (i = 1, . . . , 6) in Eq. (5). In the case of R ≫ w (or x ≡ R/w ≫ 1), sinh(x) ≈ cosh(x) ≈ e x . Thus, function g(t, x) = [2 sinh 2 (x) cosh 2 (t)]/[sinh 2 (x) + sinh 2 (t)] 2 ≈ 2e 2(x−t) /[e 2(x−t) + 1] 2 is positive and significantly nonzero only near t = x, reflecting the fact that E ex , E DM , and E an are mainly from skyrmion wall region that is assumed to be very thin. Furthermore, the area bounded by g(t, x)-curve and t-axis is 1 so that g(t, x) ≈ δ(t − x) resembles the properties of a delta function.
We can evaluate f i 's under this approximation (See Supplemental Materials [31]). For example, f 1 (x) is The total energy is then Due to the specific form of the magnetization profile of Θ dw (r), Rw-term in E Ze vanishes and E Ze ≈ 4πµ 0 M s B R 2 2 + π 2 24 w 2 . The skyrmion size and wall width are then the values that make E(R, w) minimal, For B = 0, Eqs. (8) and (9) can be analytically solved.
The results are The dashed lines in Fig. 2(a)-(c) are the approximate formulas that compare quite well with simulation results too. For B = 0, it is difficult to analytically solve Eqs. (8) and (9), but their numerical solutions are easily obtained that are plotted as dashed lines in Fig. 2(d). In summary, our approximate formula agrees very well with the simulations for R ≫ w as expected from our approximation. For smaller skyrmions, the approximation is still not bad, and qualitatively gives correct parameter dependence. We can also determine the upper limit of D and lower limits of A, K, and B from the approximate formula. Since R must be real and finite, we have for B = 0. Note that these limits are consistent with the criteria of the existence of chiral domain walls [32,33].
We also compare our analytical results with micromagnetic simulations for PdFe/Ir [7,23,34], MnSi [22,36], and W/Co 20 Fe 60 B 20 /MgO [10]. The skyrmion sizes range from several nanometers to about 2 micrometers. All the comparisons give quite good agreement (See Supplemental Materials [31]). Our results show that skyrmion size increases with D, and decreases with A and K. Our results also show that not only DMI, but also magnetic anisotropy or perpendicular magnetic field is necessary for the formation of isolated skyrmions, which is consistent with all previous experiments and simulations [2-11, 14-16, 22, 23, 28].
It is natural to extend our approach to Bloch skyrmions in the systems with bulk inversion symmetry broken. The bulk DMI energy E DM = D m · (∇ × m)dS can be rewritten as where γ = π/2 gives minimal energy. Since all other discussions are the same as those for Néel skyrmions, the results about R and w are applicable for the Bloch skyrmions.
In conclusion, we found a single skyrmion can be well described by a 360°domain wall profile parametrized by two fundamental quantities, skyrmions size and wall width. Through the minimization of total energy with respect to skyrmion size and wall width, analytical formulas for skyrmion size and wall width as a function of exchange stiffness, anisotropy coefficient, DMI strength and external field are obtained. The formulas agree very well with simulations and experiments.

Derivation of energy expressions
To derive the functions f i (x) in the energy expression, we substitute Eq. (4) into Eq. (3). For the exchange energy, by defining x = R/w, t = r/w, we have Thus, we define where the coefficient I 1 is determined by For the DM energy, we have We define dt.
For f 3 (x), the function inside the integral sinh(x) cosh(t) sinh 2 (x)+sinh 2 (t) ≈ e (x−t) e 2(x−t) +1 is localize at x = t so that we have the approximation where I 3 is determined by The integrand in f 4 is 0 at r = 0, r = ∞ and r = R. Furthermore, it has opposite signs for r < R and r > R. So f 4 ∼ 0 after the integration.
For the anisotropy energy, we have Similar to the exchange energy and DM energy, the anisotropy energy is only non-zero near r = R, too. The approximate form of function f 5 is the same as f 1 , The Zeeman energy is non-zero for both the wall region and the inner domain. The function f 6 is Again, we replace sinh functions by exponential functions to obtain So we have f 6 (x) ≈ x 2 2 + π 2 24 .

Numerical verification of theoretical results for different materials
We compare the theoretical results with micromagnetic simulations. Similar to Fig. 2 in the main text, we compare the D-, A-, K-, and B-dependencies of skyrmion size r and skyrmion wall width w. The sample size ranges from 256 nm to 2048 nm, and the sample thickness is fixed to 0.1 nm. In each subfigure, one of D, A, K, and B is treated as a tuning parameter, and the other three parameters are fixed to the values listed in Table S1.