磁性史科子(Magnetic skyrmion) 是種漩渦狀的結構,它被用來預測可以作為儲
存裝置。 塊狀材料如 MnSi,FeGe 和一些超薄磁性物質裡頭以觀察到有如此的結
構。 Skyrmion結構中異常的自傾斜旋被認為是 Dzyaloshinskii-Moriya interaction
(DMi) 所 造 成 , 但 是 此 種 現 象 在 磁 轉 向 過 程 , 即spin reorientation transition
(SRT)中也被觀察到,因此我們想證明就算沒有 DMi 也能產生 skyrmion。 在
SRT 附近的模擬中我們觀察到複雜的結構,其中有條紋狀、不規則塊狀和圓形如
skyrmion 般的形狀。 隨著磁場的增加,反向的磁區縮小減少,而其中有些封閉
條紋和圓形變成如 skyrmion 一般的漩渦。 在某些磁區牆的不連續處形成了被稱
為反漩渦的結構。 為了去確認這些粒子般的結構為何,我們用了稱之為 skymion
number 的方法來分析。 得到了想要的答案後便證明了即使沒有 DMi, 在 SRT 的
範圍內一樣可以找到 skyrmion。

Magnetic skyrmion, a quasiparticle-like spin texture, attracted lots of attention
due to its promising potential on magnetic memory device and spintronics.The
formation of skyrmion in ultrathin regime is usually attributed to the broken
symmetry induced Dzyaloshinskii-Moriya interaction (DMI), resulting in the spin
canting at the surface of ultrathin film.In the regime of spin reorientation transi-
tion (SRT)the metastable phase in magnetism might provide similar spin config-
urations. This suggests that it is possible to create skyrmion without the DMI.
In this thesis, we address this issue by the micromagnetic simulation taking ex-
change, magnetic anisotropy, demagnetization and Zeeman energies into consid-
erations. The system we explored is cobalt on gold (111). Skyrmion like particles
were found in the SRT range of cobalt. At the field between 5 mT to 11 mT,
skyrmions formed with high symmetry circular shape. The size of skyrmions
varied with field, from 216 nm in 5 mT to 48 nm in 11 mT. We also found that
vortexes can only form from the closed stripe domains, and anti-vortexes come
from somewhere the domain wall alignments are discontinuous. To confirm the
vortex as skyrmion, we computed the skyrmion number of the vortex.The num-
ber is close to a unity, suggesting the vortex texture is a skyrmion. This implies
that the skyrmion can exist without DMI in the SRT regime.

1 Introduction. . . . . . . . . . . . . . . . . . . . . . .1
1.1
Skyrmion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2
Computer Simulation . . . . . . . . . . . . . . . . . . . . . . .2

2 Basic Concept. . . . . . . . . . . . . . . . . . . . . . .4
2.1
Magnetic energy. . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1
Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2
Uniaxial Anisotropy . . . . . . . . . . . . . . . . . . . . . . .6
2.1.3
Demagnetization . . . . . . . . . . . . . . . . . . . . . . . . .6
2.1.4
Zeeman Energy . . . . . . . . . . . . . . . . . . . . . . . . . .8
2.2
Precession of Moments . . . . . . . . . . . . . . . . . . . . . .8
2.3
Spin Reorientation Transition . . . . . . . . . . . . . . . . . .8
2.4
Simulation Princple . . . . . . . . . . . . . . . . . . . . . . .9

3 Results and discussion. . . . . . . . . . . . . . . . . . . . . . .10
3.1
Parameters in Simulation . . . . . . . . . . . . . . . . . . . . 10
3.2
Configurations near the transition thickness . . . . . . . . . . 11
3.3
Evolution of skyrmion in external field . . . . . . . . . . . . .14
3.4
Particles Annihilation . . . . . . . . . . . . . . . . . . . . . 16
3.4.1
Anti-vortexes . . . . . . . . . . . . . . . . . . . . . . . . . .16
3.4.2
Vortex and Anti-vortex . . . . . . . . . . . . . . . . . . . . . 18
3.5
Skyrmion Number . . . . . . . . . . . . . . . . . . . . . . . . .21
3.6
Local Charge Density . . . . . . . . . . . . . . . . . . . . . . 23
3.7
Topological Invariant . . . . . . . . . . . . . . . . . . . . . .23
3.8
Field Dependent Skyrmion Size . . . . . . . . . . . . . . . . . .28
3.9
MATLAB function . . . . . . . . . . . . . . . . . . . . . . . . .29
3.9.1
avf2ovf . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
3.9.2
oommf2matlab . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.9.3
Skyrmion Number . . . . . . . . . . . . . . . . . . . . . . . . 35
3.9.4
Skyrmion Size . . . . . . . . . . . . . . . . . . . . . . . . . .35

4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .39

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