376
X. S. Wang and X. R. Wang
dependences of the skyrmion size R (left y-axis) and the skyrmion wall width w
(right y-axis), with other parameters fixed to the values for Co mentioned earlier. The
symbols are the micromagnetic simulation data [R is the size of m z = 0 contour and
w is the fit of the skyrmion profile to dw (r )]. The sample size used in the simulations
is large enough to avoid the boundary effects. Solid lines are numerical results from
minimising E of (14.23). The simulation results agree almost perfectly (except a
slight deviation in the D-dependence of w for smaller D) with our analytic results
of (14.23). Both micromagnetic simulations and analytical results clearly show that
the 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 Fig. 14.11e
for D = 5 mJ/m
2 . E(R, w) of (14.23) has a minimum as long as |D| < 3.8 mJ/m
2
that indicates existence of the skyrmion. However, micromagnetic simulations show
that the skyrmion can only exist in the window of 1.2 mJm
−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
Fig. 14.12d. This discrepancy is due to the discretization of continuous LLG equation
in micromagnetic simulations. There exists also a minimal A of around 14 pJ/m as
shown in Fig. 14.12b and a minimal K of around 0.56 MJm
−3 as shown in Fig. 14.12c,
below which the skyrmion does not exist, and the stable state is stripe domains as
shown in Fig. 14.11e. The skyrmion size decreases with B, which is consistent with
the experimental observations [70, 75, 88].
It is still unclear how R and w depend on A, D and K although (14.23) agrees
almost perfectly with the simulation results. Thus it is highly desirable to have 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 the skyrmion wall area
of π Rw multiplying the square of the magnetization variation rates 1/R
2
+ 1/w
2 ,
i.e. E ex ∝ (R/w + w/R). The integrand of the second term (tangential term) of E DM
is proportional to sin 2 due to the triple product of the interfacial DMI vector (along
the radial direction), the tangential variation of m (along the tangential direction), and
m itself. Because sin 2 is almost asymmetric about r = R (positive when r > R and
negative when r < R) and nonzero only near r = R, the second term in E DM is vanishingly small after integral over r . The DMI energy is then mainly contributed by the
first term (radial term), which is proportional to wall area (Rw) multiplying the magnetization variation rate along radial direction (1/w), i.e. E DM ∝ R. The isotropic
energy is mainly from the skyrmion wall area. Thus, E an ∝ Rw. The Zeeman energy
of the skyrmion comes from the skyrmion core 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 π Rw. To obtain the proportional coefficients, one needs
to find approximate expressions for f i (R/w) (i = 1, . . . , 6) in (14.23). 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 non-zero only near t = x, reflecting the fact that E ex , E DM , and E an are
mainly from the skyrmion wall region that is assumed to be very thin. Furthermore,
X. S. Wang and X. R. Wang
dependences of the skyrmion size R (left y-axis) and the skyrmion wall width w
(right y-axis), with other parameters fixed to the values for Co mentioned earlier. The
symbols are the micromagnetic simulation data [R is the size of m z = 0 contour and
w is the fit of the skyrmion profile to dw (r )]. The sample size used in the simulations
is large enough to avoid the boundary effects. Solid lines are numerical results from
minimising E of (14.23). The simulation results agree almost perfectly (except a
slight deviation in the D-dependence of w for smaller D) with our analytic results
of (14.23). Both micromagnetic simulations and analytical results clearly show that
the 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 Fig. 14.11e
for D = 5 mJ/m
2 . E(R, w) of (14.23) has a minimum as long as |D| < 3.8 mJ/m
2
that indicates existence of the skyrmion. However, micromagnetic simulations show
that the skyrmion can only exist in the window of 1.2 mJm
−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
Fig. 14.12d. This discrepancy is due to the discretization of continuous LLG equation
in micromagnetic simulations. There exists also a minimal A of around 14 pJ/m as
shown in Fig. 14.12b and a minimal K of around 0.56 MJm
−3 as shown in Fig. 14.12c,
below which the skyrmion does not exist, and the stable state is stripe domains as
shown in Fig. 14.11e. The skyrmion size decreases with B, which is consistent with
the experimental observations [70, 75, 88].
It is still unclear how R and w depend on A, D and K although (14.23) agrees
almost perfectly with the simulation results. Thus it is highly desirable to have 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 the skyrmion wall area
of π Rw multiplying the square of the magnetization variation rates 1/R
2
+ 1/w
2 ,
i.e. E ex ∝ (R/w + w/R). The integrand of the second term (tangential term) of E DM
is proportional to sin 2 due to the triple product of the interfacial DMI vector (along
the radial direction), the tangential variation of m (along the tangential direction), and
m itself. Because sin 2 is almost asymmetric about r = R (positive when r > R and
negative when r < R) and nonzero only near r = R, the second term in E DM is vanishingly small after integral over r . The DMI energy is then mainly contributed by the
first term (radial term), which is proportional to wall area (Rw) multiplying the magnetization variation rate along radial direction (1/w), i.e. E DM ∝ R. The isotropic
energy is mainly from the skyrmion wall area. Thus, E an ∝ Rw. The Zeeman energy
of the skyrmion comes from the skyrmion core 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 π Rw. To obtain the proportional coefficients, one needs
to find approximate expressions for f i (R/w) (i = 1, . . . , 6) in (14.23). 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 non-zero only near t = x, reflecting the fact that E ex , E DM , and E an are
mainly from the skyrmion wall region that is assumed to be very thin. Furthermore,