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Inertia effect should be considered in ferromagnet magnetization dynamics on a sub picosecond-to-femtosecond-time scale. The inertia effect can be described by the inertial Landau-Lifshitz-Gilbert equation. This paper mainly introduces some theoretical and experimental developments of ultrafast ferromagnetic resonance, magnetization reversal and inertial spin dynamics. These results will be helpful in better understanding the basic mechanism of ultrafast demagnetization and magnetization reversal, and deepen the understanding of the microscopic mechanism of magnetic inertia. In the end, the development trend of future experimental and theoretical research are also presented.
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Keywords:
- inertia effect /
- ferromagnetic resonance /
- magnetization reversal
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图 1 磁化强度动力学示意图[40] (
$ {\boldsymbol{M}} $ 在$ {{\boldsymbol{H}}^{{\text{eff}}}} $ 周围的进动用蓝色实虚线表示, 章动用红色曲线表示)Figure 1. Schematic illustration of magnetization dynamics[40] (The precessional motion of
$ {\boldsymbol{M}} $ around $ {{\boldsymbol{H}}^{{\text{eff}}}} $ is depicted by the blue solid dashed curve, and the nutationis shown by the red curve).
图 2 磁化强度分量的时间演化
${M_x}/{M_{\text{S}}} = {\sin}\theta {\cos}\varphi $ (黑线),${M_y}/{M_{\text{S}}} = {\sin}\theta {\cos}\varphi $ (红线)和${M_z}/{M_{\text{S}}} = {\cos}\theta $ (绿线), 对于H = 2 T和${h_ \bot } = 0\;{\text{T}}$ 和任意的初始条件值θ0 = 30°,${\varphi _0} = {0^{\circ}}$ ,${\dot \theta _0} = 0\;{\text{rad/s}}$ 和${\dot \varphi _0} = - 2\sqrt 3 \times {10^{14}}\;{\text{rad/s}}$ (a) 短时间动力学$t < 10\tau $ 显示了由惯性项引起的章动振荡; (b) 长时间动力学$t \gg \tau $ 显示了固定应用场周围的进动振荡[55]Figure 2. Time evolution of the magnetization components
${M_x}/{M_{\text{S}}} = {\sin}\theta {\cos}\varphi $ (black line),${M_y}/{M_{\text{S}}} = {\sin}\theta {\cos}\varphi $ (red line), and${M_z}/{M_{\text{S}}} = {\cos}\theta $ (green line) for H = 2 T and${h_ \bot } = 0 \;{\text{T}}$ , and for arbitrary initial conditions${\theta _0} = {30^{\circ}}$ ,${\varphi _0} = {0^{\circ}}$ ,${\dot \theta _0} = 0\;{\text{rad/s}}$ , and${\dot \varphi _0} = - 2\sqrt 3 \times {10^{14}}\;{\text{rad/s}}$ : (a) Short time dynamics$t < 10\tau $ showing the nutation oscillations due to the inertial term; (b) long time dynamics$t \gg \tau $ showing the precession oscillations around the fixed applied field [55].
图 3 横向磁化强度率
${\chi _ \bot }\left( \omega \right)$ 相对于振荡场脉冲$\omega $ 的共振曲线, 观察到两个共振峰, 即低频的铁磁共振和高频的章动共振. 插图为横向磁化率${\chi _ \bot }$ 的计算实例, 对于ω = 2 × 1011 rad/s 得到$\left\langle {{M_ \bot }} \right\rangle = {\chi _ \bot }{h_ \bot }$ [55]Figure 3. Resonance curve of the transverse susceptibility
${\chi _ \bot }\left( \omega \right)$ with respect to the oscillating field pulsation$\omega $ . Two resonance peaks are observed: the ferromagnetic resonance at low frequency and the nutation resonance at high frequency. Inset: Example of the calculation of the transverse susceptibility${\chi _ \bot }$ such that$\left\langle {{M_ \bot }} \right\rangle = {\chi _ \bot }{h_ \bot }$ obtained for$\omega = 2 \times {10^{11}}\;{\text{rad/s}}$ [55].
图 4 惯性弛豫时间
$\eta = 0\;{\text{s}}$ 和$\eta = {10^{ - 13}}{\text{s}}$ 下的自旋泵浦直流电流, 使用的参数是${M_0} = 2{\mu _{\text{B}}}$ ,$K = {10^{ - 23}}{\text{J}}$ ,$\gamma = $ $ 1.76 \times {10^{11}}\;{{\text{T}}^{ - 1}} \cdot {{\text{s}}^{ - 1}}$ ,$\alpha = 0.05$ ,${H_0} = 1\; {\text{T}}$ ,$\left| h \right| = {10^{ - 3}}\;{\text{T}}$ ,$g_{\text{r}}^{ \uparrow \downarrow } = {10^{19}}{{\text{m}}^{ - 2}}$ [58]Figure 4. The calculated spin pumping dc current for inertial relaxation times
$\eta = 0\;{\text{s}}$ and$\eta = {10^{ - 13}}{\text{s}}$ . The used parameters are${M_0} = 2{\mu _{\text{B}}}$ ,$K = {10^{ - 23}}{\text{J}}$ ,$\gamma = 1.76 \times {10^{11}}\; {{\text{T}}^{ - 1}} \cdot $ $ {{\text{s}}^{ - 1}}$ ,$\alpha = 0.05$ ,${H_0} = 1\;{\text{T}}$ ,$\left| h \right| = {10^{ - 3}}\; {\text{T}}$ ,$g_{\text{r}}^{ \uparrow \downarrow } = {10^{19}}{{\text{m}}^{ - 2}}$ [58].
图 5 对于在章动共振到进动共振处的铁磁体, 自旋电流的比值对惯性弛豫时间
$\eta $ , 使用的参数是M0 = 2μB,$\gamma = 1.76 \times {10^{11}}\;{{\text{T}}^{ - 1}} \cdot {{\text{s}}^{ - 1}}$ ,$\alpha = 0.05$ ,$K = {10^{ - 23}}\;{\text{J}}$ 和${H_0} = $ $ 1\;{\text{T}}$ [58]Figure 5. Ratio of spin current for ferromagnets at the nutation resonance to the precession resonance vs. inertial relaxation time
$\eta $ , the used parameters are${M_0} = 2{\mu_{\rm{B}}}$ ,$\gamma = 1.76 \times {10^{11}}\;{{\text{T}}^{ - 1}} \cdot {{\text{s}}^{ - 1}}$ ,$\alpha = 0.05$ ,$K = {10^{ - 23}}\;{\text{J}}$ , and${H_0} = 1\; {\text{T}}$ [58].
图 6 (a) 正文中描述的数值模拟的球坐标; (b)考虑了薄膜体系的几何结构, 易磁化强度轴沿y方向, 平面内难磁化强度轴沿x方向; (c)通过数值求解ILLG方程得到的不同磁脉冲振幅和 FWHM的磁化强度状态图, 侧边图为主图选定点上的磁化强度进动轨迹, 颜色条显示了模拟结束时磁化强度矢量的y分量, 磁化强度总是从正y方向开始, 即平行于易磁化强度轴排列[63]
Figure 6. (a) Spherical coordinates used for the numerical simulations described in the main text; (b) geometry of the thin film system considered, the easy magnetization axis lies along the y direction, and the in-plane hard magnetization axis is along the x direction; (c) main plot is the magnetization state diagram for different magnetic pulse amplitude and FWHM obtained by numerically solving the ILLG equation, side plots are magnetization precession trajectories in selected points of the diagram. The color bar shows the y component of the magnetization vector at the end of the simulation. The magnetization starts always from the positive y direction, i.e., aligned parallel to the easy magnetization axis[63].
图 7 使用(a) LLG和(b) ILLG方程计算不同脉冲幅度和FWHM宽度的磁化强度翻转时间, 所有轴和振幅均为对数刻度; (c) 沿图(a)和(b)中黑色虚线所示的对角线切割(即垂直于恒定脉冲能量线)的翻转时间. 垂直虚线两次模拟的弹道翻转区域的边界[63]
Figure 7. Magnetization switching times calculated for different pulse amplitude and FWHM width using (a) the LLG and (b) the ILLG equations. All axes and amplitudes are in logarithmic scale; (c) switching time along the diagonal line cuts (i.e., perpendicular to the lines of constant pulse energy) shown by the black-dashed lines in panels (a) and (b). The dashed vertical lines indicate the boundaries of the ballistic switching region for the two simulations[63].
图 8 对于一些选定模拟参数, 动力学、势、耗散和沉积能量项的时间演变. 对于2 T振幅和1 ps FWHM的外加磁场的不同能量项的 (a) LLG 和 (b) ILLG 动力学, 即在进动翻转区域中; 对于8 T, 2.1 ps 磁场脉冲的(c) LLG 和 (d) ILLG 能量动力学, 即在弹道翻转区域[63]
Figure 8. Temporal evolution of the kinetic, potential, dissipated, and deposited energy terms for a few selected simulation parameters: (a) LLG and (b) ILLG dynamics of the different energy terms for an applied magnetic field of 2 T amplitude and 1 ps FWHM, i.e., in the precessional switching region; (c) LLG and (d) ILLG energy dynamics for 8 T, 2.1 ps magnetic field pulse, i.e., in the ballistic switching region[63].
图 9 对于±2 kG的平面外施加场, 典型的TR-MOKE对 (a)线性偏振(LP)光泵浦和(c)右(开圆)和左(全正方形)CP光泵浦的响应; (b) 通过平均图(a)中的曲线获得的对LP泵浦的真实磁化强度响应; (d) 通过对图(c)中相应的右CP曲线和左CP曲线求平均值获得的对右CP泵送(空心圆)和左CP 泵送的真实磁化强度响应; (b)和(d)中的实线使用(15)式拟合数据. 插图: 实验示意图; 倾斜磁化强度与表面法线成角度
$\varPhi $ ; CP泵浦光子携带整个量子的角动量$ \pm \hbar $ ; 探测脉冲对Mz敏感[70]Figure 9. Typical TR-MOKE response to (a) linearly polarized (LP) light pumping and (c) right (open circles) and left (full squares) CP light pumping, for an out of plane applied field of ±2 kG; (b) genuine magnetization response to LP pumping obtained by averaging the curves in panel (a); (d) genuine magnetization response to right (open circles) and left (full squares) CP pumping obtained by averaging the corresponding right and left CP curves in panel (c); the solid lines in panels (b) and (d) are fits to the data using Eq. (15). Inset: Schematic representation of the experiment; the canted magnetization forms an angle
$\varPhi $ with the normal to the surface; CP pump photons carry a whole quantum of angular momentum$ \pm \hbar $ ; probe pulses are sensitive to Mz[70].
图 12 实心符号: 在fcc, bcc和hcp钴薄膜上的时间分辨克尔旋转测量. 虚线: 泵浦 THz磁场HTHz的积分. 插图为t > 1.7 ps主要数据的放大图. 为了清晰起见, 数据被垂直移动. 连续线是用(16)式得到的最佳拟合[74]
Figure 12. Solid symbols: time-resolved Kerr rotation measurements on fcc, bcc, and hcp cobalt thin films. Dashed line: integral of the pump THz magnetic field HTHz. Inset: enlarged main panel data for t > 1.7 ps. The data are shifted vertically for clarity. The continuous lines are the best fits obtained using Eq. (16) [74].
图 13 (a) 符号, 对于不同最大振幅的THz磁场值, 在t > 1.7 ps时的时间分辨克尔信号, 为了清晰起见, 数据被垂直移动. 实线, 使用(16)式得到的最佳拟合. (b)符号, 提取振荡振幅B作为THz磁场和相应的标准差的函数. 虚线, 与施加零偏移量的数据进行线性拟合[74]
Figure 13. (a) Symbols, time-resolved Kerr signal at t > 1.7 ps for THz magnetic field values of different maximum amplitude. The data are vertically shifted for clarity. Solid lines, best fit obtained using Eq. (16). (b) Symbols, extracted oscillation amplitude B as a function of THz magnetic field and corresponding standard deviation. Dashed line, linear fit to the data with imposed zero offset[74].
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