Topological states and quantum effects in magnonics

Wang Zhen-Yu; Li Zhi-Xiong; Yuan Huai-Yang; Zhang Zhi-Zhi; Cao Yun-Shan; Yan Peng

doi:10.7498/aps.72.20221997

Abstract

In recent years, with the rapid development of the emerging technologies including the internet of things, cloud computing, big data, and artificial intelligence, higher computing capability is required. Traditional semiconductor devices are confronting huge challenges brought by device miniaturization, energy consumption, heat dissipation, etc. Moore’s law which succeeds in guiding downscaling and upgrading of microelectronics is nearing its end. A new information carrier, instead of electrons, is required urgently for information transmission and processing. Spin waves are collectively excited waves in ordered magnets, and the quantized quasi particle is referred to as magnon. The propagation of magnons does not involve electron motion and produces no Joule heating, which can solve the increasing significant issues of heating dissipation in electronic devices. Thus, magnon-based devices have important application prospects in low-power information storage and computing. In this review, we first introduce the recent advances in the excitation, propagation, manipulation, detection of spin waves and magnon-based devices. Then, we mainly discuss the researches of our group. This part is described from four aspects: 1) Chiral magnonics, including the chiral propagarion of magnetostatic spin waves, Dzyaloshinskii-Moriya interaction(DMI)-induced nonreciprocity of spin waves, spin-wave propagation at chiral interface, magnonic Goos-Hänchen effect, spin-wave lens, and magnonic Stern-Gerlach effect; 2) nonlinear magnonics, including three-magnon processes induced by DMI and noncollinear magnetic textures, skyrmion-induced magnonic frequency comb, twisted magnon frequency comb, and Penrose superradiance; 3) topological magnonics, including magnon Hall effect, magnonic topological insulator, magnonic topological semimetal, topological edge states and high-order corner states of magnetic solitons arranged in different crystal lattices; 4) quantum magnonics, including quantum states of magnon, magnon-based hybrid quantum systems, and cavity magnonics. Finally, the future development and prospect of magnonics are analyzed and discussed.

Keywords:

Authors and contacts

1.
State Key Laboratory of Electronic Thin Films and Integrated Devices, School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
2.
Institute for Theoretical Physics, Utrecht University, Utrecht 3584CC, The Netherlands

Corresponding author: Yan Peng, yan@uestc.edu.cn

Funds: Project supported by the National Key R&D Program, China (Grant No. 2022YFA1402802), the National Natural Science Foundation of China (Grant Nos. 12074057, 12204089, 11604041, 11704060, 11904048), and the China Postdoctoral Science Foundation (Grant Nos. 2019M653063, 2019M663461, 2020M673180)

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Cited By

图 1 磁子学领域的研究框架及其分支^{[11,16,25,35,55,72,73,88,198,229]}

Figure 1. Framework and its branches in the field of magnonics^{[11,16,25,35,55,72,73,88,198,229]}

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图 2 (a)静磁表面自旋波的非互易传播示意图^[8]; (b)静磁表面自旋波非互易传播对缺陷的鲁棒性^[80]

Figure 2. (a) Schematic illustration^[8]; (b) robustness character of nonreciprocal propagation of magnetostatic surface spin waves^[80]

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图 3 (a)右手和线性极化微波场驱动下的自旋波振幅; (b)左手微波场驱动下自旋波的振幅; (c)微波场频率为5.1和6.7 GHz时, 自旋波的非互易传播^[81]

Figure 3. (a) Spin-wave amplitudes under the right-handed and linearly polarized microwave fields; (b) amplitudes of spin waves driven by left-handed polarized microwave field; (c) nonreciprocal propagation of spin waves at two field frequencies 5.1 and 6.7 GHz^[81]

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图 4 (a)二维磁性薄膜示意图, $ {\boldsymbol{m}} $为磁矩单位矢量, 与$+\hat{{\boldsymbol{z}}}$轴之间的夹角为θ, $ {\boldsymbol{k}} $为自旋波波矢, 与$+\hat{{\boldsymbol{x}}}$轴之间的夹角为$ \phi_k $; (b)波长相同的自旋波沿垂直于磁矩方向传播时存在的频率差^[97]; (c)同频自旋波沿垂直于磁矩方向反向传播时的波长^[88]; (d)自旋波在波矢空间的等频曲线^[93]; (e)波矢平行于磁矩时, 自旋波的非共线传播^[93]; (f)在纳米带中传播的自旋波波前倾斜(群速平行于磁矩方向)^[90]

Figure 4. (a) Schematic illustration of an ultrathin film, $ {\boldsymbol{m}} $ is the unit magnetization vector having an angle θ with the $+\hat{{\boldsymbol{z}}}$ axis, $ {\boldsymbol{k}} $ is the wavevector of spin wave making an angle $ \phi_k $ with the $+\hat{{\boldsymbol{x}}}$ axis; (b) frequency difference of spin waves with opposite wave vectors perpendicular to the magnetization^[97]; (c) wavelength of spin waves with the same frequency and opposite $ {\boldsymbol{k}} $ perpendicular to the magnetization^[88]; (d) isofrequency curve of spin waves in the wave vector space^[93]; (e) non-collinear propagation of spin waves with opposite wave vectors parallel with the magnetization^[93]; (f) spin-wave canting for spin waves propagation in the nanostripe (the group velocity is parallel with the magnetization)^[90]

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图 5 (a)自旋波在DMI 界面传播所遵循的斯奈尔定律示意图. 不同角度入射的自旋波在DMI 界面处的传播　(b) $ \theta_\mathrm{i}=-60^{\circ} $; (c) $ \theta_\mathrm{i}=-18^{\circ} $; (d) $ \theta_\mathrm{i}=0^{\circ} $; (e) $ \theta_\mathrm{i}=18^{\circ} $; (f) $ \theta_\mathrm{i}=60^{\circ} $^[93]

Figure 5. (a) Schematic of the generalized Snell's law for the spin-wave scattering at a heterochiral interface. Spin-wave propagation through the DMI interface under different incident angles: (b) $ \theta_\mathrm{i}=-60^{\circ} $; (c) $ \theta_\mathrm{i}=-18^{\circ} $; (d) $ \theta_\mathrm{i}=0^{\circ} $; (e) $ \theta_\mathrm{i}=18^{\circ} $; (f) $ \theta_\mathrm{i}=60^{\circ} $^[93]

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图 6 (a)自旋波发生全反射时的强度分布, $ \varDelta_\mathrm{r} $为GH位移; (b) GH位移随入射角和DMI强度变化的相图; (c)固定DMI强度$D=3.0 \;\mathrm{mJ/m^2}$, GH位移随入射角度的变化; (d)固定入射角度为$ \theta_\mathrm{i}=-70^{\circ} $时, GH位移随DMI强度的变化^[94]

Figure 6. (a) Intensity map of spin waves reflected from the DMI interface, $ \varDelta_\mathrm{r} $ is the GH shift; (b) phase diagram of the GH shift in dependence on the incident angle and DMI strength; (c) GH shift as a function of the incident angle for $D=3.0\; \mathrm{mJ/m^2}$; (d) dependence of the GH shift on the DMI constant for $ \theta_\mathrm{i}=-70^{\circ} $^[94]

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图 7 (a)自旋波在半圆形界面散射所遵循的广义斯奈尔定律的示意图; (b)自旋波焦点的理论计算模型图; (c)自旋波离轴聚焦的微磁模拟结果; (d)焦点坐标随自旋波频率的变化^[128]

Figure 7. (a) Schematic plot of the generalized Snell’s law for the spin-wave scattering at a semicircle interface; (b) theoretical model of the focal-point coordinates calculation; (c) micromagnetic simulation results of the off-axis focusing of spin waves; (d) focal-point coordinates as a function of the spin-wave frequency^[128]

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图 8 (a)等磁程原理示意图; (b)利用椭圆界面构造磁子透镜聚焦自旋波的微磁模拟结果; (c)自旋波聚焦产生斯格明子的过程^[37]

Figure 8. (a) Schematic of the identical magnonic path length principle; (b) micromagnetic simulation of the spin-wave focusing by the magnonic lens constructed by an elliptical interface; (c) the process of the skyrmion generated by the spin-wave focusing^[37]

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图 9 (a)电子斯特恩盖拉赫效应的示意图; (b)磁子斯特恩盖拉赫效应的示意图; (c)一束线性极化的自旋波经过DMI界面被分为两束极化相反(左手和右手)的自旋波; (d)和(e)分别为线性极化自旋波经过半圆形异手性界面传播的理论和微磁模拟结果; (f)实验上利用自旋波双聚焦产生自旋流和探测的示意图^[136]

Figure 9. (a) Schematic illustration of the electronic Stern-Gerlach effect; (b) schematic illustration of the magnonic Stern-Gerlach effect; (c) a linearly-polarized spin-wave beam propagates through a DMI interface and is divided into two spin-wave beams with opposite polarizations; (d) analytical and (e) micromagnetic simulation results of the bi-focusing of spin waves propagating through a semi-circle DMI interface; (f) schematic of the spin-current generation by the bi-focusing of spin waves and detection^[136].

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图 10 (a)三磁子融合, (b)三磁子分裂, 以及(c)四磁子散射过程示意图

Figure 10. Schematic of (a) three-magnon confluence, (b) three-magnon splitting, and (c) four-magnon scattering process

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图 11 (a) DMI诱导的三磁子过程^[93]; (b)磁畴壁诱导的三磁子过程^[32]

Figure 11. Schematic illustration of three-magnon processes induced by (a) the DMI^[93] and (b) domain wall^[32]

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图 12 (a)磁子-斯格明子非线性散射产生自旋波频率梳的示意图; (b)微磁学模拟验证结果^[35]

Figure 12. (a) Schematic of nonlinear magnon-skyrmion scattering induced spin-wave frequency comb; (b) micromagnetic simulation results^[35].

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图 13 (a)涡旋自旋波与旋进涡核之间发生非线性散射产生涡旋自旋波频率梳的示意图以及微磁模拟验证结果; (b)涡旋自旋波频率梳的模式分布^[36]

Figure 13. (a) Schematic of twisted magnon frequency comb induced by nonlinear scattering between twisted spin waves and gyrating vortex core, and the verification of micromagnetic simulation results; (b) mode profiles of twisted magnon frequency comb^[36]

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图 14 涡旋自旋波发生彭罗斯超辐射效应时频率梳模式的振幅以及模式分布^[36]

Figure 14. Amplitude and mode profile of twisted magnon frequency comb with and without Penrose superradiance effect^[36]

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图 15 不同类型的拓扑绝缘体示意图　(a)一阶拓扑绝缘体及不同维度下系统的边界态; (b)二阶拓扑绝缘体及对应边界态(角态和铰链态); (c)三阶拓扑绝缘体及对应边界态(角态)^[171]

Figure 15. Schematic plot for different types of TIs: (a) The first-order TI and edge states in different dimensions; (b) the second-order TI and edge states (corner state and hinge states); (c) the third-order TI and edge states (corner states)^[171]

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图 16 (a) 烧绿石绝缘铁磁体$ {\rm{Lu}}_{2}{\rm{V}}_{2}{\rm{O}}_{7} $的晶体结构示意图; (b)磁子霍尔效应: 纵向的温度梯度导致横向热磁子流; (c)不同温度下, 热霍尔电导随磁场的变化^[176]

Figure 16. (a) Crystal structure of pyrochlore ferromagnet $ {\rm{Lu}}_{2}{\rm{V}}_{2}{\rm{O}}_{7} $; (b) magnon Hall effect: the longitudinal temperature gradient leads to the transverse thermal magnon current; (c) magnetic field dependence of the thermal Hall conductivity for various temperatures^[176]

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图 17 (a)磁子波包的自转产生的磁子边界流; (b)沿边界传播的磁子; (c)处于平衡态的边界磁子流; (d)温度梯度的施加会导致有限热霍尔电流的产生^[182]

Figure 17. (a) Self-rotation of a magnon wave packet with a magnon edge current; (b) magnon near the boundary; (c) magnon edge current in equilibrium; (d) a finite thermal Hall current emerges when temperature gradient is applied^[182]

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图 18 (a)半无限大kagome晶格结构; (b)系统的相图及Chern数; (c)—(f)不同拓扑非平凡相(图(b)中红点所示)所对应的能带结构^[187]

Figure 18. (a) Semi-infinite kagome lattice; (b) topological phase diagram of the system with different Chern numbers; (c)–(f) band structures for different topologically nontrivial phases as marked with red dots in panel (b)^[187]

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图 19 (a)堆叠蜂巢型铁磁体示意图; (b)第一体布里渊区和表面布里渊区; (c), (d)外尔磁子的能带结构, 红色和蓝色小球分别表示手性为$ +1 $和–1的外尔点; (e), (f)贝里曲率的空间分布; (g), (h)有限大系统的能带结构^[194]

Figure 19. (a) Schematic diagram of stacked honeycomb ferromagnets; (b) the first bulk Brillouin zone and the first surface Brillouin zone of the system; (c), (d) band structures of Weyl magnons, the Weyl nodes of chirality $ \pm1 $ are marked by red and blue dots, respectively; (e), (f) corresponding Berry curvatures of the magnon bands; (g), (h) band structures of finite system^[194]

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图 20 (a)自旋波传播示意图, 黑色箭头表示磁矩的方向, 黄色箭头表示自旋波的传播方向; (b)自旋波二极管; (c)自旋波分束器; (d)自旋波干涉仪示意图^[198]

Figure 20. (a) Schematic illustration of SW propagation, the black arrows denote the direction of magnetization, yellow arrows represent the propagation direction of SW; (b) illustrations of SW diode; (c) SW beam splitters; (d) SW interferometers^[198]

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图 21 (a)磁泡, (b)涡旋, (c)布洛赫型斯格明子, (d)奈尔型斯格明子, (e)反涡旋, (f)反斯格明子, (g)奈尔型畴壁, (h)涡旋型畴壁和(i)布洛赫型畴壁微磁结构示意图^[171]

Figure 21. Micromagnetic structures of (a) magnetic bubble, (b) vortex, (c) Bloch-type skyrmion, (d) Néel-type skyrmion, (e) antivortex, (f) antiskyrmion, (g) Néel-type, (h) vortex-type, and (i) Bloch-type domain walls^[171]

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图 22 (a)被周期性缺口钉扎的涡旋, 斯格明子和畴壁赛道示意图; (b)归一化磁矩分量沿畴壁赛道中心的分布; (c)无限大畴壁赛道的能带结构; (d) Zak相随比值$ d_{1}/d_{2} $的变化; (e)对不同的$ d_{1}/d_{2} $, 有限大畴壁赛道的能谱; (f)边界态对应畴壁振荡强度的分布^[213]

Figure 22. (a) Illustration of the vortex, skyrmion, and DW racetrack with periodic pinnings; (b) components of normalized magnetization along the center of DW racetrack; (c) band structure of an infinite DW racetrack; (d) dependence of the Zak phase on the ratio $ d_{1}/d_{2} $; (e) spectrum of a finite DW racetrack for different $ d_{1}/d_{2} $; (f) DW-oscillation amplitude for edge state^[213]

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图 23 (a)布洛赫型斯格明子组成的蜂巢阵列示意图; (b)整个系统的共振频谱; (c)半无限大系统的能带结构; 当频率f = 12.62 (d) 和16.65 GHz (e) 时边界态的传播图像^[209]

Figure 23. (a) Illustration of the honeycomb lattice with Bloch skyrmions; (b) resonant spectrum of the whole system; (c) band structure of the semi-infinite system; (d) snapshot of the propagation of edge states with frequency (d) f = 12.62, (e) 16.65 GHz^[209]

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图 24 (a)涡旋组成的呼吸型kagome晶格; (b)耦合常数$I_{{/ /}}$和$ I_{\perp} $随距离d的变化; (c)系统的相图; (d)系统处于高阶拓扑相时, 涡旋晶格的本征频率; (e)不同模式对应涡旋振荡的分布^[177]

Figure 24. (a) Illustration of the breathing kagome lattice of vortices; (b) dependence of the coupling strength $I_{{/ /}}$ and $ I_{\perp} $ on the vortex–vortex distance d; (c) phase diagram of the system; (d) eigenfrequencies of kagome vortex lattice for higher-order topological phase; (e) patial distribution of vortex gyrations for different states^[177]

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图 25 (a)涡旋组成的堆叠型蜂巢阵列; (b)一层结构的放大图; (c)系统的第一布里渊区; (d)系统的相图; (e)系统处于WSM2 相时, 对应半无限大晶格的能带结构^[202]

Figure 25. (a) Illustration of stacked honeycomb lattice composed of vortices; (b) zoomed in details of one layer; (c) the first Brillouin zone of the crystal; (d) phase diagram of the system; (e) band structures of semi-infinite system for WSM2 phase^[202]

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图 26 量子磁子学研究范畴, 包括磁子量子态以及磁子与比特系统、光学系统和声学系统的耦合^[229]

Figure 26. Framework of quantum magnonics that includes the quantum states of magnons and its coupling with qubit platforms, photonic platforms, and phononic platforms^[229]

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图 27 (a)磁子反聚束态示意图; (b)磁性小球里产生磁子反聚束态的思路示意^[231]; (c)磁子的单模压缩态示意图; (d)双子晶格反铁磁里磁子的双模压缩态示意图^[237]

Figure 27. (a) Scheme of magnon antibunching and (b) its generation in magnetic sphere^[231]; Scheme of (c) single-mode and (d) two-mode squeezed states of magnons^[237].

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图 28 基于磁子的混合量子体系示意　(a)磁体-微波腔^[68]; (b)磁体-量子比特^[246]; (c)磁体-声子复合系统^[247]

Figure 28. Hybrid magnonic platforms: (a) Magnet-microwave cavity^[68]; (b) magnet-qubit^[246]; (c) magnet-phonon systems^[247]

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图 29 第一性原理散射理论　(a) 一维散射模型; (b) 铁磁共振态与微波腔耦合的透射波谱, 在共振中心处, 两个耦合模的频率差为$ g_\text{eff} $; (c) 耦合强度随磁体厚度平方根的变化, 在厚度较小的时有良好的线性关系; 如考虑自旋交换相互作用, 高阶自旋波与微波腔的强耦合能够显示在透射波谱中 (d) 磁体厚度$ 1\; \text{μm}$, (e) 磁体厚度$ 5\; \text{μm} $; (f) 耦合强度随着自旋波阶数的增加而降低; (g) 自旋波与微波腔的耦合强度与磁体厚度的平方根均呈现线性关系, 当厚度增加时, 铁磁共振模($ p=1 $)的耦合强度较高阶模增加更多^[259]

Figure 29. First principle scattering theory: (a) 1-dimensional scattering model; (b) transmission spectrum of coupled microwave cavity and ferromagnet, the frequency difference between the hybrid modes at the resonant center is $ 2 g_\text{eff} $; (c) coupling strength is increasing with square root of ferromagnetic thickness, the linearity between them conforms well when the thickness is small; if we consider the exchange interaction, the standing spin wave modes are also coupled strongly with the microwave cavity, the transmission spectra are plotted when (d) $ d=1\; \text{μm}$ and (e) $d=5\; \text{μm}$; (f) coupling strength decreases with the increasing order of spin waves; (g) coupling strength for each spin wave is linearly increasing with the thickness however with different slopes. The FMR mode ($ p=1 $) has the largest slope^[259].

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图 30 (a) PT对称性微波腔磁子系统; 调制参数分别为 (b) $ \varDelta=0 $ 和 (c) $ \varDelta=-0.3 $ 时, 系统本征频率随耗散-增益参数P的变化, 微波腔的频率设置为$ \omega_c/g=5 $; (d) 系统PT对称性相图; 不同参数下的散射频谱$ \varDelta=0 $ (e) $ P=0.5 $, (f) $ P=\sqrt{2} $, 以及 (g) $ P=2 $; (h) 空微波腔的共振频谱; (i) 腔磁光极化子共振峰的半宽随着耗散-增益因子的变化; (j) 三阶EP点附近的磁性灵敏度^[269]

Figure 30. (a) PT symmetric cavity magnon polariton system. The eigenvalues varies with the loss-gain parameter P when the detuning (b) $ \varDelta=0 $ and (c) $ \varDelta=-0.3 $, with the solid and dashed curves respectively representing the real and imaginary part of eigenfrequencies. The cavity frequency is set as $ \omega_c/g=5 $. (d) PT-symmetric phase transition diagram; transmission spectrum for different gain-loss parameters: (e) $ P=0.5 $, (f) $ P=\sqrt{2} $, and (g) $ P=2 $. The right panel in (e)–(g) shows the zero-detuning spectrum. (h) Transmission spectrum of a bare cavity. (i) Half-linewidth of CMP modes as a function of the gain-loss parameter P at the zero detuning point. (j) Sensitivity at $ P=P_\mathrm{EP3} $, symbols denote numerical results and the blue curve represents the analytical formula (42)^[269].

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图 31 (a)光学腔磁子系统; (b)蓝色调制时, 入射光波频率大于光学腔共振频率, 释放出一个磁子和一个频率较低的光子; (c)红色调制时, 入射光波频率小于光学腔共振频率, 吸收出一个磁子和一个频率较高的光子; (d)光学诱导吉尔伯特系数和(e)磁场随调制因子的变化^[272]

Figure 31. (a) Schematic illustration of a macrospin $ {\boldsymbol{S}} $ interacting with three orthogonally propagating circularly-polarized lasers (red beams) in an optical cavity; off-resonant coupling between the driving laser ($ \omega_{\text{las}} $) and the cavity photon ($ \omega_{\text{cav}} $) mediated by magnons ($ \omega_{\text{m}}\ll\omega_{\text{cav}} $) in the (b) blue and (c) red detuning regimes; (d) optically induced magnetic gain and (e) induced magnetic field vs. the optical detuning parameter η^[272]

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