Temperature and Magnetic Field Dependence of the Internal and Lattice Structures of Skyrmions by Off-Axis Electron Holography

The internal and lattice structures of magnetic Skyrmions in $B20$-type FeGe are investigated using off-axis electron holography. The temperature, magnetic field, and angular dependence of the magnetic moments of individual Skyrmions are analyzed. The internal Skyrmion shape is found to vary with applied magnetic field. In contrast, the inter-Skyrmion distance remains almost unchanged in the lattice phase over the studied range of applied field. The amplitude of the local magnetic moment is found to vary with temperature, while the Skyrmion shape does not change significantly. Deviations from a circular to a hexagonal Skyrmion structure are observed in the lattice phase, in agreement with the results of micromagnetic simulations.

Figure 1

Dependence on temperature $T$ and applied magnetic field $B_{\mathrm{ext}}$ of the phase shift ($\varphi$) recorded from Skyrmions in a thin sample of FeGe using off-axis electron holography. (a)–(d) Phase shift recorded in a magnetic field of 100 mT at a temperature of (a) 95, (b) 160, (c) 200, and (d) 240 K. (e)–(h) Phase shift recorded at a temperature of 200 K in a magnetic field of (e) 100, (f) 200, (g) 300, and (h) 400 mT. The magnetic field was always applied perpendicular to the plane of the sample ($\simeq 120\mathrm{nm}$ thick). All scale bars are 100 nm.

Figure 2

Analysis of the $T$ and $B_{\mathrm{ext}}$ dependence of internal Skyrmion structure. (a) Schematic diagram of the magnetic configuration ($\mathit{m}$) of a Skyrmion, shown alongside the definition of the cylindrical coordinate system ($\rho$, $\phi$, $z$). ${\mathit{B}}_{\mathrm{ext}}$ and ${\mathit{k}}_{\mathrm{e}}$ are the external magnetic field and the wave vector of the incident electron beam, respectively. (b) Definition of spin rotation angle $\theta$. (c) $T$ dependence of the normalized phase distribution plotted as a function of distance ($\rho$) from the center of each Skyrmion. (d) $T$ dependence of $\theta$ plotted as a function of $\rho$. (e) and (f) show the $B_{\mathrm{ext}}$ dependence of the normalized phase distribution and $\theta$, respectively, plotted as a function of $\rho$. (g) Normalization factor $\mathrm{\Delta }\varphi$ for the phase curves shown in (c) plotted as a function of $T$. The right axis shows a rough estimation of $M_s$ in units of $M_0$. The dashed line is a theoretical curve for the critical exponent $\beta =0.336$ [23]. The vertical arrow indicates the magnetic transition temperature. (h) and (i) show the calculated $B_{\mathrm{ext}}$ dependence of the normalized phase distribution and $\theta$, respectively, plotted as a function of $\rho$.

Figure 3

Micromagnetic calculation of equilibrium SkLs for different $B_{\mathrm{ext}}$ values. (a) Simulated domain with dimensions $l_x$, $l_y$, $t=100\mathrm{nm}$. (b) Color map of the equilibrium distribution of magnetization averaged over the $z$ direction. Cartesian ($x$, $y$, $z$) and polar ($\rho$, $\phi$) coordinate systems used to describe the Skyrmion structure. (c) and (d) are the dependencies of the equilibrium period of the SkL and the average magnetization on $B_{\mathrm{ext}}$, respectively. The open circles correspond to the case when magnetostatic interactions are ignored; the demagnetizing fields inside the sample are assumed to be zero, $B_d=0$. Colored circles correspond to equilibrium states calculated for different spontaneous magnetization, where $M_0$ is the spontaneous magnetization at $T=0\mathrm{K}$.

Figure 4

Analysis of the SkL structure along the two typical directions of $\phi =0$ and $\pi /6$. (a)–(c) $\rho$ derivative of the phase shift $\varphi$ for $B_{\mathrm{ext}}$ values of (a) 100, (b) 200, and (c) 300 mT. (d)–(f) $\rho$ dependence of the calculated values of $m_{\phi }$ for $B_{\mathrm{ext}}$ of (d) 100, (e) 200, and (f) 300 mT.