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Bose-Einstein condensate is a collective phenomenon which takes place when many bosons – particles having integer value of spin – experience transition to the same single quantum state. The famous example is formation of Cooper pairs of electrons, responsible for the superconducting properties of some materials at extremely low temperatures. Magnons – quasi-particles corresponding to spin waves propagating in magnetic media – are bosons as well. Hence, if we describe oscillation of magnetization as a set of magnons, the phenomenon like Bose-Einstein condensation should also exist in this case. Indeed, the Bose-Einstein condensation of magnons at room temperature has been observed under parametric pumping in thick ferrite films [1], and, recently, by rapid cooling in ultra- thin films [2].&lt;br/&gt;The stability of Bose-Einstein condensate of magnons (mBEC) is determined by the nonlinear interaction between magnons. This includes both interaction between magnons in the same spectral minimum (self-interaction), and the interaction between magnons belonging to two different spectral minima (cross-interaction). mBEC can be stable if the coefficient of nonlinear interaction is positive, i.e. if the interaction is repulsive [3]. It is well known that the self-interaction between magnons in relatively thick films is attractive [3]. However, mBEC has been experimentally proven to be stable [1, 2, 4]. It was shown in [5] that coefficient of nonlinear interaction in mBEC is negative, but rather small, so relatively strong repulsive interaction between magnons in different spectral minima may provide stability to mBEC. Recently, it has been also shown [6] that it is possible to separate mBEC in two parts, each containing magnons from &lt;i&gt;only one spectral minimum&lt;/i&gt;, which means that mBEC in each of the minima is stable, and this stability is independent of the strong cross-interaction of magnons from different minima.&lt;br/&gt;Thus, the theory developed in [5] cannot be used to explain stability of separated mBEC. Also, the theory [5] assumes a uniform profile of a magnon mode across the film thickness, and infinite in-plane sizes of the magnetic film sample. Both these simplifications are, generally, incorrect. Due to the long-range nature of the dipolar interaction, finite-size effects may have a significant impact on the nonlinear interactions of magnons [4]. Here, we present a simple, but rather accurate model of the nonlinear interactions in mBEC that takes into account the finite-size effects.&lt;br/&gt;Using Landau-Lifshits equation and vector Hamiltonian approach [7], we derived approximate analytical expressions for the linear dispersion characteristics and nonlinear interaction coefficients of magnons near the minima of the magnon spectrum (i.e. in the mBEC region).&lt;br/&gt;We showed that magnons undergoing mBEC can be described as a gas of weakly-interacting bosonic particles in an infinite potential well formed by a ferromagnetic film. The rest energy of the magnons in mBEC is determined by the dipolar interaction inside a film sample. This leads to a strong anisotropy of the effective mass of magnons in the magnetic film. In the direction along the applied in-plane magnetic field, the effective mass is &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt; = &lt;i&gt;hM&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt;/(32&lt;i&gt;π&lt;/i&gt;&lt;i&gt;γA&lt;/i&gt;) (where &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt; is the saturation magnetization, &lt;i&gt;h&lt;/i&gt; is the Planck constant, &lt;i&gt;γ&lt;/i&gt; is the gyromagnetic ratio, and &lt;i&gt;A&lt;/i&gt; is the exchange stiffness). For yttrium-iron garnet (YIG), this “longitudinal” mass is &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt; ≈ 1.5×10&lt;sup&gt;-30&lt;/sup&gt; kg ≈ 1.6 &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;e&lt;/sub&gt; (&lt;i&gt;m&lt;/i&gt;&lt;sub&gt;e&lt;/sub&gt; is the electron mass). In the directions perpendicular to the applied field magnons are described by the “perpendicular” mass &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;⊥&lt;/sub&gt; = (18&lt;i&gt;λ&lt;/i&gt;/&lt;i&gt;d&lt;/i&gt;)&lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt;, which depends on the ratio of the exchange length &lt;i&gt;λ &lt;/i&gt;= (2&lt;i&gt;A&lt;/i&gt;/&lt;i&gt;μ&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;s&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;1/2&lt;/sup&gt; to the thickness of the film &lt;i&gt;d&lt;/i&gt;. For typical experimental conditions &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;⊥&lt;/sub&gt; « &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt;; for example, &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;⊥&lt;/sub&gt; ≈ 0.06 &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt; in a YIG film of the thickness &lt;i&gt;d&lt;/i&gt; = 5 μm. The huge anisotropy of the effective mass leads to the anisotropy of the dynamical mBEC properties. In particular, the speed of the Bogolyubov waves [8] depends strongly on the direction of the wave propagation. Since the perpendicular mass m&lt;sub&gt;⊥&lt;/sub&gt; depends on the thickness of the film, it is possible to achieve the inverse mass relation &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;⊥&lt;/sub&gt; &amp;gt; &lt;i&gt;m&lt;/i&gt;&lt;sub&gt;∥&lt;/sub&gt; in ultra-thin YIG films with &lt;i&gt;d&lt;/i&gt; &amp;lt; 300 nm.&lt;br/&gt;Under the assumption of totally pinned spins, which is an accurate approximation in the mBEC region, we derived approximate analytical expressions for the nonlinear frequency shift due to the self- and cross- interactions in mBEC:&lt;br/&gt;Δ&lt;i&gt;f&lt;/i&gt; = (&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;self&lt;/sub&gt; + 2&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;cross&lt;/sub&gt;)&lt;i&gt;n&lt;/i&gt;;&lt;br/&gt;&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;self&lt;/sub&gt; = &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; (2&lt;i&gt;N&lt;/i&gt;&lt;sub&gt;d&lt;/sub&gt; – 3&lt;i&gt;πλ&lt;/i&gt;/&lt;i&gt;d&lt;/i&gt;√2);&lt;br/&gt;&lt;i&gt;F&lt;/i&gt;&lt;sub&gt;cross&lt;/sub&gt; = &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;.&lt;br/&gt;Here &lt;i&gt;F&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt; = &lt;i&gt;γ&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;i&gt;μ&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;&lt;i&gt;h&lt;/i&gt;/4&lt;i&gt;π&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; is the normalisation constant, &lt;i&gt;n&lt;/i&gt; is the magnon density, &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;d&lt;/sub&gt; is the component of the demagnetization tensor &lt;i&gt;N&lt;/i&gt; = diag(&lt;i&gt;N&lt;/i&gt;&lt;sub&gt;x&lt;/sub&gt;, &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;y&lt;/sub&gt;, &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;z&lt;/sub&gt;) corresponding to the direction along the applied in-plane magnetic field. These expressions show that the nonlinear cross-interaction stays large and positive independently of the confinement of the sample in the applied field direction, while the self-interaction coefficient changes its sign when &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;d&lt;/sub&gt; become greater than the critical value &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;cr&lt;/sub&gt;:&lt;br/&gt;&lt;i&gt;N&lt;/i&gt;&lt;sub&gt;cr&lt;/sub&gt; = (3&lt;i&gt;π&lt;/i&gt;/2√2)(&lt;i&gt;λ&lt;/i&gt;/&lt;i&gt;d&lt;/i&gt;).&lt;br/&gt;The result explains possibility to separate mBEC into two stable components corresponding to the different minima of the mBEC spectrum [4, 6].&lt;br/&gt;The developed model allows one to make simple and accurate predictions about the stability of the mBEC depending on the shape of the sample used, and to describe dynamics of fluctuations and waves of magnon density in the mBEC. Also, it makes possible to directly apply theoretical methods developed to describe condensates of atoms to study the mBEC dynamics, and, therefore, it might be very useful in further investigations of coherent magnonic phenomena.

**References:**

[1] S. O. Demokritov, V. E. Demidov, O. Dzyapko et al., &lt;i&gt;Nature&lt;/i&gt;, vol. 443, pp. 430-433, 2006.&lt;br/&gt;[2] M. Schneider, T. Brächer, D. Breitbach et al., &lt;i&gt;Nat. Nanotechnol.&lt;/i&gt;, vol. 15, pp. 457–461, 2020.&lt;br/&gt;[3] V. S. L’vov, Wave turbulence under parametric excitation, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1994).&lt;br/&gt;[4] I. V. Borisenko, B. Divinskiy, V. E. Demidov et al., &lt;i&gt;Nat. Commun.&lt;/i&gt;, vol. 11, no. 1691, 2020.&lt;br/&gt;[5] O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk et al., &lt;i&gt;Phys. Rev. B&lt;/i&gt;, vol. 96, no. 064438, 2017.&lt;br/&gt;[6] I. V. Borisenko, V. E. Demidov, V. L. Pokrovsky et al., &lt;i&gt;Sci. Rep.&lt;/i&gt;, vol. 10, no. 14881, 2020.&lt;br/&gt;[7] V. Tyberkevych, A. Slavin, P. Artemchuk, et al., arXiv:2011.13562 (2020).&lt;br/&gt;[8] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova et al., &lt;i&gt;Nat. Commun.&lt;/i&gt;, vol 10, no. 2460, 2019.

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INTERMAG 2021

 A Model for Description of Linear Properties and Stability of Bose-Einstein Condensate of Magnons at Room Temperature

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The great event is jointly organized by the Association for Computational Linguistics (ACL) and Asian Federation of Natural Language Processing (AFNLP).

Bose-Einstein condensate is a collective phenomenon which takes place when many bosons – particles having integer value of spin – experience transition to the same single quantum state. The famous example is formation of Cooper pairs of electrons, responsible for the superconducting properties of some materials at extremely low temperatures. Magnons – quasi-particles corresponding to spin waves propagating in magnetic media – are bosons as well. Hence, if we describe oscillation of magnetization as a set of magnons, the phenomenon like Bose-Einstein condensation should also exist in this case. Indeed, the Bose-Einstein condensation of magnons at room temperature has been observed under parametric pumping in thick ferrite films [1], and, recently, by rapid cooling in ultra- thin films [2]. The stability of Bose-Einstein condensate of magnons (mBEC) is determined by the nonlinear interaction between magnons. This includes both interaction between magnons in the same spectral minimum (self-interaction), and the interaction between magnons belonging to two different spectral minima (cross-interaction). mBEC can be stable if the coefficient of nonlinear interaction is positive, i.e. if the interaction is repulsive [3]. It is well known that the self-interaction between magnons in relatively thick films is attractive [3]. However, mBEC has been experimentally proven to be stable [1, 2, 4]. It was shown in [5] that coefficient of nonlinear interaction in mBEC is negative, but rather small, so relatively strong repulsive interaction between magnons in different spectral minima may provide stability to mBEC. Recently, it has been also shown [6] that it is possible to separate mBEC in two parts, each containing magnons from only one spectral minimum, which means that mBEC in each of the minima is stable, and this stability is independent of the strong cross-interaction of magnons from different minima. Thus, the theory developed in [5] cannot be used to explain stability of separated mBEC. Also, the theory [5] assumes a uniform profile of a magnon mode across the film thickness, and infinite in-plane sizes of the magnetic film sample. Both these simplifications are, generally, incorrect. Due to the long-range nature of the dipolar interaction, finite-size effects may have a significant impact on the nonlinear interactions of magnons [4]. Here, we present a simple, but rather accurate model of the nonlinear interactions in mBEC that takes into account the finite-size effects. Using Landau-Lifshits equation and vector Hamiltonian approach [7], we derived approximate analytical expressions for the linear dispersion characteristics and nonlinear interaction coefficients of magnons near the minima of the magnon spectrum (i.e. in the mBEC region). We showed that magnons undergoing mBEC can be described as a gas of weakly-interacting bosonic particles in an infinite potential well formed by a ferromagnetic film. The rest energy of the magnons in mBEC is determined by the dipolar interaction inside a film sample. This leads to a strong anisotropy of the effective mass of magnons in the magnetic film. In the direction along the applied in-plane magnetic field, the effective mass is m∥ = hMs/(32πγA) (where Ms is the saturation magnetization, h is the Planck constant, γ is the gyromagnetic ratio, and A is the exchange stiffness). For yttrium-iron garnet (YIG), this “longitudinal” mass is m∥ ≈ 1.5×10-30 kg ≈ 1.6 me (me is the electron mass). In the directions perpendicular to the applied field magnons are described by the “perpendicular” mass m⊥ = (18λ/d)m∥, which depends on the ratio of the exchange length λ = (2A/μ0Ms2)1/2 to the thickness of the film d. For typical experimental conditions m⊥ « m∥; for example, m⊥ ≈ 0.06 m∥ in a YIG film of the thickness d = 5 μm. The huge anisotropy of the effective mass leads to the anisotropy of the dynamical mBEC properties. In particular, the speed of the Bogolyubov waves [8] depends strongly on the direction of the wave propagation. Since the perpendicular mass m⊥ depends on the thickness of the film, it is possible to achieve the inverse mass relation m⊥ &gt; m∥ in ultra-thin YIG films with d &lt; 300 nm. Under the assumption of totally pinned spins, which is an accurate approximation in the mBEC region, we derived approximate analytical expressions for the nonlinear frequency shift due to the self- and cross- interactions in mBEC: Δf = (Fself + 2Fcross)n; Fself = F0 (2Nd – 3πλ/d√2); Fcross = F0. Here F0 = γ2μ0h/4π2 is the normalisation constant, n is the magnon density, Nd is the component of the demagnetization tensor N = diag(Nx, Ny, Nz) corresponding to the direction along the applied in-plane magnetic field. These expressions show that the nonlinear cross-interaction stays large and positive independently of the confinement of the sample in the applied field direction, while the self-interaction coefficient changes its sign when Nd become greater than the critical value Ncr: Ncr = (3π/2√2)(λ/d). The result explains possibility to separate mBEC into two stable components corresponding to the different minima of the mBEC spectrum [4, 6]. The developed model allows one to make simple and accurate predictions about the stability of the mBEC depending on the shape of the sample used, and to describe dynamics of fluctuations and waves of magnon density in the mBEC. Also, it makes possible to directly apply theoretical methods developed to describe condensates of atoms to study the mBEC dynamics, and, therefore, it might be very useful in further investigations of coherent magnonic phenomena.

**References:**

[1] S. O. Demokritov, V. E. Demidov, O. Dzyapko et al., Nature, vol. 443, pp. 430-433, 2006. [2] M. Schneider, T. Brächer, D. Breitbach et al., Nat. Nanotechnol., vol. 15, pp. 457–461, 2020. [3] V. S. L’vov, Wave turbulence under parametric excitation, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1994). [4] I. V. Borisenko, B. Divinskiy, V. E. Demidov et al., Nat. Commun., vol. 11, no. 1691, 2020. [5] O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk et al., Phys. Rev. B, vol. 96, no. 064438, 2017. [6] I. V. Borisenko, V. E. Demidov, V. L. Pokrovsky et al., Sci. Rep., vol. 10, no. 14881, 2020. [7] V. Tyberkevych, A. Slavin, P. Artemchuk, et al., arXiv:2011.13562 (2020). [8] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova et al., Nat. Commun., vol 10, no. 2460, 2019.

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The advent of materials with complex, topologically-stabilized magnetic textures, such as skyrmions, stemming from careful tuning of the Dzyaloshinskii–Moriya interaction, direct light-induced manipulation of spins, or geometric frustration [1], has opened up exciting avenues of research. Understanding the structure and dynamics of, and achieving control over, spin on the scale of the magnetic exchange interaction is both a question of fundamental physics and at the forefront of developing quantum technologies like spintronics and low-energy data transport and memory [2]. Because of the small size of these intricate nanoscale magnetic textures, few to hundreds of nm, and their low contrast, it is quite challenging to image them in 3D directly without prior knowledge of the sample or extensive simulation. They dynamics of such systems can also prove challenging to capture, as some magnetic processes occur on the timescale of twenty femtoseconds or less [3]. To fully understand the static and dynamic electronic, magnetic, and structural properties of complex samples we combine complementary nanoscale imaging and spectroscopy techniques: resonant x-ray magnetic circular dichroism (MCD) provides element-specific magnetic sensitivity; ptychography yields diffraction-limited images with phase contrast, ideal for low-contrast materials; and time-resolved extreme ultraviolet (EUV) spectroscopies can capture atto-to-nanosecond dynamics. By integrating soft x-ray MCD imaging with scalar ptycho-tomography [4, 5] we demonstrate a technique capable of directly imaging the 3D magnetic field within a Ni metalattice material, with no need for prior knowledge or simulation, that is applicable to a wide range of magnetic nanostructured samples.

**References:**

[1] J. E. Han and V. H. Crespi, Phys. Rev. Lett., Vol. 89, p. 197203 (2002) [2] P. Fischer, IEEE Trans. Magn., Vol. 51, p. 0800131 (2015) [3] W. You, P. Tengdin, C. Chen, et. al., Phys. Rev. Lett., Vol. 121, p. 77204 (2018) [4] C. Donnelly, M. Guizar-Sicairos, V. Scagnoli, et al., Nature, Vol. 547, p. 328 (2017) [5] A. Pryor, Y. Yang, A. Rana, et al., Scientific Reports, Vol. 7, p. 10409 (2017)

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 3D imaging of topologically stabilized spin textures in nanostructured materials – towards ultrafast imaging of spin transport

Antiferromagnets (AFMs) emerged as a versatile material science platform, which enabled numerous fundamental discoveries including the observation of monopole quasiparticles in frustrated systems and collective quantum effects, such as spin superfluidity and Bose–Einstein condensation of magnetic excitations [1]. Primary advantages of antiferromagnets are their terahertz operating frequencies, the absence of stray fields, magnetic field robustness, all of which result in numerous advantages including those in spintronics and spinorbitronics [2]. The key enabler of those applications is the presence of the Dzyaloshinskii-Moriya interaction (DMI). This, in turn, put stringent requirements on the magnetic symmetry of AFM, which should support weak ferromagnetism and chiral helimagnetism. This makes the portfolio of material systems available for these studies very limited that renders the progress in AFM-related fundamental and technological research to depend on time-consuming material screening and optimization of intrinsic chiral properties of AFMs. The field of curvilinear magnetism is well explored for ferromagnets, where magnetic responses are tailored by local curvatures [3]. By contrast, the topic of curvilinear AFMs is at its infancy [4-9]. The energy landscape of ring AFM and geometrically frustrated chains at non-zero temperature is characterized by a large number of metastable states including long-living noncollinear textures if anisotropy is strong enough [6]. In experiment, curvilinear AFMs are mainly represented by the molecular magnets [7] and metalized DNA molecules [8]. The shape anisotropy stemming from magnetostriction plays the major role in the ordering of the Neel vector in perovskite zig-zag stripes and nanodots determining the easy direction as the parallel or perpendicular to the boundary [9]. In this presentation, we will demonstrate that chiral responses of AFMs can be tailored by a geometrical curvature without the need to adjust material parameters. In a general case, an intrinsically achiral one-dimensional curvilinear AFM spin chain behaves as a chiral helimagnet with geometrically tunable DMI, orientation of the Neel vector and the helimagnetic phase transition, see Fig. 1 [4]. The helix-shaped spin chain possesses two ground states: the so-called homogeneous and periodic ones with respect to the motion along the chain. The energetically favorable state is determined by the direction of the geometry-driven DMI vector. In contrast to ferromagnets, there is no easy axis anisotropy competing with the geometry-driven one. Furthermore, the curvature-induced DMI results in the hybridization of spin wave modes. The low-frequency branch is gapless for straight chains and possesses the gap for any finite curvature. In addition, the DMI enables a geometrically-driven local minimum of the low frequency branch which increases for larger curvature and torsion, see Fig. 2. This opens exciting perspectives to study long-lived collective magnon states in AFMs. These findings position curvilinear 1D antiferromagnets as a novel platform for the realization of geometrically tunable chiral antiferromagnets for antiferromagnetic spinorbitronics and fundamental discoveries in the formation of coherent magnon condensates in the momentum space. The proposed description of vector fields living at curvilinear geometries can be applied for other systems with complex order parameters, such as ferroelectrics [10] or liquid crystals [11].

**References:**

[1] C. Castelnovo, R. Moessner, S. L. Sondhi, Nature, Vol. 451, p. 42 (2008); N. Nagaosa, Y. Tokura, Phys. Scr. Vol. T146, p. 014020 (2012); A. A. Zvyagin, Low Temp. Phys. Vol. 39, p. 901 (2013) [2] T. Jungwirth, J. Sinova, A. Manchon et al, Nat. Phys. Vol. 14, p. 200 (2018); V. Baltz, A. Manchon, M. Tsoi et al, Rev. Mod. Phys. Vol. 90, p. 015005 (2018) [3] R. Streubel, P. Fischer, F. Kronast et al, J. Phys. D: Appl. Phys. Vol. 49, p. 363001 (2016); A. Fernandez-Pacheko, R. Streubel, O. Fruchart et al, Nat. Comm. Vol. 8, p. 15756 (2016); D. D. Sheka, O. V. Pylypovskyi, P. Landeros et al., Comm. Phys. 3, 128 (2020) [4] O. V. Pylypovskyi, D. Y. Kononenko, K. V. Yershov et al, Nano Lett. Vol. 20, p. 8157 (2020) [5] K. K. Nanda, A. W. Addison, E. Sinn et al, Inorg. Chem. Vol 35, p. 5966 (1996); O. Cador, D. Gatteschi, R. Sessoli et al, Ang. Chem. Int. Ed. vol 43, p. 5196 (2004); M. Grzelczak, M. Correa-Duarte, V. Salgueirino-Maceira et al, Ang. Chem. Int. Ed. Vol. 46, p. 7026 (2007); T. Guidi, B. Gillon, S. A. Mason et al, Nat. Comm. Vol. 6, p. 7061 (2015) [6] S. Castillo-Sepulveda, R. A. Escobar, D. Altbir et al, Phys. Rev. B, Vol. 96, p. 024426 (2017) [7] S. J. Blundell, F. L. Pratt, J. Phys.: Cond. Matt. Vol. 16, p. R71 (2004) [8] K. Mizoguchi, S. Tanaka, M. Ojima et al, J. Phys. Soc. Jap. Vol. 76, p. 043801 (2007) [9] A. D. Bang, I. Hallsteinsen, R. V. Chopdekar et al, Appl. Phys. Lett. Vol. 115, p. 112403 (2019); M. S. Lee, P. Lyu, R. V. Chopdekar et al, J. Appl. Phys. Vol. 127, p. 203901 (2020) [10] M. Owczarek, K. A. Hujsak, D. P. Ferris et al, Nat. Comm. Vol. 7, p. 13108 (2016) [11] G. Napoli, L. Vergori, Phys. Rev. Lett. Vol. 108, p. 207803 (2012)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ae6c8e5f4d9443f9e0440eb83667f104.jpg)
Fig. 1. A general AFM spin chain and the helimagnetic phase transition of helix-shaped spin chain. The curvature-driven DMI and anisotropy modify the magnetic response as follows. The DMI is represented by vector d, while the equilibrium hard and easy axes for the Neel vectors are shown by e1 and e3, respectively. Adapted with permission from [4]. Copyright (2020) American Chemical Society.

![](https://assets.underline.io/uploads/markdown_image/1/image/0eef10e98fc40c90e95903e759da7ae1.jpg)
Fig. 2. The spin wave dispersion of the helix-shaped AFM spin chain and the respective depth δ of the local minimum at the dispersion curve. Adapted with permission from [4]. Copyright (2020) American Chemical Society.

 Curvilinear antiferromagnetism: Current state and perspectives INVITED

One current goal of spintronics is the development of a sustainable information technology based on the transport of pure spin currents. One promising way to do this is to use spin waves (SWs), the elementary excitations of magnetically ordered materials: by collective precession of magnetic moments, they can transport angular momentum. SW information processing also offers interesting alternatives to conventional microelectronics [1]. For this, yttrium iron garnet (YIG), an insulating ferrimagnet, is the ideal material because it has the lowest magnetic damping. Furthermore, thanks to the spin orbit torque (SOT), it is possible to control the relaxation time of SWs in YIG by an electric current injected into an adjacent platinum layer [2]. However, such metal/insulator hybrid devices presently have certain limitations, as a sudden drop in the amplitude of the main mode beyond a certain excitation threshold, which originates from nonlinear coupling with other SW modes [2]. Different strategies can be considered to overcome these issues. In nanopatterned samples the available SW modes become quantized, which severely limits the available nonlinear processes. As a result, nonlinear saturation thresholds are postponed, which allows to reach much higher amplitudes of single mode dynamics [3]. The perpendicular magnetic anisotropy (PMA) can also be used to control the sign of the nonlinear frequency shift. Recently, the growth of ultra-thin films of Bismuth doped YIG (BiYIG) with tunable PMA has been achieved while preserving a high dynamic quality [4]. It was shown that the emission characteristics of SWs emitted by SOT in thin BiYIG films where PMA compensates the in-plane shape anisotropy are greatly improved [5]. Here, we study the magnetization dynamics in individual nanodisks patterned from such a 30 nm thick BiYIG film, with diameters ranging from 1 µm down to 200 nm. The static magnetic field is applied perpendicularly to the sample plane. To excite the ferromagnetic resonance (FMR) of the BiYIG nanodisks, we use the spatially uniform in-plane microwave field produced by an antenna integrated on top. To detect it, we employ a magnetic resonance force microscope [6] in which a cobalt nanosphere grown at the tip of a soft cantilever [7] is dipolarly coupled to the longitudinal component of the magnetization of the nanodisk positioned underneath (Fig. 1). By studying the SW spectra of these BiYIG nanodisks in the linear regime, we find that their effective anisotropy field is weakly dependent of the diameter and negative, about -10 mT, which means that the PMA slightly overcompensates the shape anisotropy. The damping parameter is found to increase as the diameter is decreased and lies in the high 10-4 – low 10-3 range for all disks. We then perform measurement of the main resonance line, corresponding to the quasi-uniform mode, as a function of the microwave power injected into the antenna, which is varied by up to four orders of magnitude (excitation field ranging from 0.03 mT to 3 mT at 5 GHz). For all the disks, we observe a distortion of the resonance line at intermediate excitation power, corresponding to the onset of foldover. As expected, due to the negative effective anisotropy field, the resonance line shifts towards higher magnetic field as the precession angle increases, which is opposite to the behavior reported on undoped YIG nanodisks [3]. For the largest BiYIG disks, the dynamics witnessed at higher power is quite more complex and interesting, as shown in Fig. 2 for the 700 nm diameter nanodisk. We observe a rapid saturation of the peak amplitude, corresponding to average precession angles of the order of 10° to 20°, accompanied with a broadening of the line towards both lower and higher magnetic field. To understand this behavior, we perform extensive micromagnetic simulations. Using the magnetic parameters extracted in the linear regime and the nominal geometry of the nanodisks, these simulations reproduce the main characteristics of the measured FMR lines in the strongly nonlinear regime. They reveal that a dynamic instability is responsible of the observed behavior: as the amplitude of the quasi-uniform mode increases with the driving field, its precession profile gets more and more localized at the center of the disk, which eventually leads to the formation of a dynamic soliton similar to a wave bullet [8]. Moreover, this soliton is not stable over time, resulting in a rich temporal dynamics, which, depending on the exact driving conditions, can be quasi-periodic or intermittent. To experimentally access this complex temporal dynamics, we use the method introduced in [3], which consists in applying a second, much weaker excitation field of varying frequency, in addition to the main driving field at high power. The obtained frequency modulation spectra indeed reflect the existence of rich low-frequency temporal variations in the dynamics of the magnetization, as suggested by the simulations.

**References:**

[1] A. Chumak et al. Nat. Phys. 11, 453 (2015) [2] A. Hamadeh et al. Phys. Rev. Lett. 113, 197203 (2014); M. Evelt et al. Appl. Phys. Lett. 108, 172406 (2016) [3] Y. Li et al. Phys. Rev. X 9, 041036 (2019) [4] L. Soumah et al. Nat. Commun. 9, 3355 (2018) [5] M. Evelt et al. Phys. Rev. Appl. 10, 041002 (2018) [6] O. Klein et al. Phys. Rev. B 78, 144410 (2008) [7] S. Sangiao et al. Beilstein J. Nanotechnol. 8, 2106 (2017) [8] A. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201 (2005)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4929da18838a5e687ca4c091422c4fa2.jpg)
Fig.1 - Schematics of the experiment. The microwave field supplied by the antenna drives the magnetization of a BiYIG nanodisk into FMR, opening a precession angle around the perpendicularly applied field, which is detected by the cantilever of a magnetic resonance force microscope.

![](https://assets.underline.io/uploads/markdown_image/1/image/4ea71d26a653b41e10c95b4dc047a304.jpg)
Fig.2 - Main resonance line of the 700 nm diameter nanodisk recorded at 5 GHz as a function of the driving power, increased by step of 1 dB from -8 dBm to 10 dBm. The spectra have been vertically offset for clarity.

 Strongly nonlinear ferromagnetic resonance of Bi-doped YIG nanodisks

Spontaneous symmetry breaking is an ubiquitous ingredient in physics. It governs second order phase transitions whose statics are well described within the universal Ginzburg-Landau (GL) theory, as confirmed by a number of observations. However, the picture for the corresponding dynamics is less clear. Indeed, in the ordered phase, GL theory predicts two different low-energy excitations: the zero frequency Goldstone mode, corresponding to an oscillation of the phase of the order parameter, and the non-zero-frequency Higgs mode, corresponding to an oscillation of its amplitude.[1] In practice, the amplitude mode has remained very difficult to observe in condensed matter, either because it does not couple to the experimental probes or because it decays too fast into other types of excitations.[1] In this communication, we show that an archetypal micromagnetic texture, the so-called weak magnetic stripes,[2] constitutes an ideal platform to investigate such dynamics. Combining Brillouin light scattering with micromagnetic simulations and analytical modeling, we evidence spin wave features that can be directly identified to Goldstone/Higgs modes. The system is a 180nm thick amorphous CoFeB ferromagnetic magnetic film exhibiting a small perpendicular magnetic anisotropy K.[3] Under a strong enough in-plane magnetic field H, the film is homogeneously saturated. However, its spin-wave spectrum already displays a characteristic feature precursor of the nucleation of an inhomogeneous magnetic texture. This is shown in figure 1b, which displays a color plot of the spin-wave dispersion simulated in the Damon Eshbach configuration (wave-vector k⊥H). One recognizes a low frequency branch with a minimum around kc=21 rad/µm. Upon reducing the field, this minimum shifts to lower frequency and eventually reaches the horizontal axis (Fig. 1c), which is the behavior expected in the GL description: Upon approaching the critical field Hc, the curvature of the magnetic energy along a certain direction in the configuration space decreases to zero, which is associated to the softening of the corresponding normal mode. The mode profile at this point is shown in Fig. 1d: one recognizes an undulation of the vertical component of the magnetization in the middle of the film, together with quadrature undulations of the in-plane component of the magnetization on the top and bottom surfaces insuring magnetic flux closure. This pattern, and the critical fields and wavelength, correspond precisely to the weak magnetic stripes observed in this system. [3]. Indeed, upon further decreasing the field, the nucleation proceeds as a freeze of the soft spin-wave mode in a state with a given phase (or equivalently a given lateral position within the period), which constitutes the broken symmetry. Concomitantly, the characteristic cusp of the spin wave spectrum splits in two: One branch remains soft and a second one emerges with a gradual increase of its minimum frequency (Fig. 2a,b). Inspecting the spatial profile of these two modes (Fig. 2c,d) allows one to recognize them unambiguously as a Goldstone mode (rigid lateral oscillation of the stripe texture) and a Higgs mode (oscillation of the amplitude of the texture around its equilibrium value), respectively. We probe these modes experimentally using Brillouin light scattering onto thermally excited spin waves. The symbols in Fig.1b,c and Fig.2b show the position of the measured peaks. Although the limited wave-vector / frequency range accessible (&lt;20rad/µm, &gt;2GHz) does not allow to cover all the way down to the cusp, the agreement with the simulations is good. In particular, the measured dispersions display the very characteristic features predicted, namely the negative slopes (in the k⊥H configuration this can only happen due to the perpendicular anisotropy), the critical softening and the subsequent splitting in the broken symmetry phase. These essential features are visible around Hc and kc, which explains why they have been largely overlooked by the magnetism community, as previous dynamic studies of stripe domains had focused onto the k≈0 (ferromagnetic resonance) and/or deep supercritical (near remanence) regimes.[4,5] To support this identification, we shall present a Ginzburg Landau theory of weak stripe statics based on a stray-field-free ansatz [2,6] and extend it to the dynamics using the dipole-exchange spin wave formalism.[7] We shall also discuss the observability of these characteristic spin-wave modes for different measurements and compare them with Higgs/Goldstone modes occurring in other condensed-matter systems at completely different length/energy scales.[1,8]

**References:**

[1] D. Pekker and C.M. Varma, Annu. Rev. Condens. Matter Phys. 6, 269 (2015) [2] A. Hubert and R. Schäfer, Magnetic Domains, Springer (1998) [3] K. Ait Oukaci et al., Phys. Rev. Appl. 14, 024083 (2020) [4] N. Vukadinovic et al., Phys. Rev. Lett. 85, 2819 (2000) [5] C. Banerjee et al., Phys. Rev. B 96, 024421 (2017) [6] A. Hubert, Phys. Stat. Sol. 32, 519 (1969) [7] M. Grassi et al., Phys. Rev. Appl. 14, 024047 (2020) [8] Matsumoto M., J. Phys.: Conf. Ser. 592, 012123 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2f55dbdb13b46101fcd7656ce0220724.jpg)
Figure 1: Spin wave dispersions above Hc. (a) Geometry of the system. (b) Color plot: amplitude spectral density simulated for a 180nm thick CoFeB film in the Damon-Eshbach configuration (k⊥H) under a saturating in-plane field µ0H=30mT. Symbols: measured dispersion. (c) Idem at the critical field µ0Hc=14mT. (d) Vector plot of the dynamic magnetization distribution at the softening point [(kc,0) in panel c]. The first order magnetic anisotropy constant, saturation magnetization and exchange constant are K=36kJ/m3, µ0Ms=1.67T and A=16pJ/m, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/8a3afb52035833ea4a8fcfc84a4b58ea.jpg)
Figure 2: Spin-wave dispersion below Hc (a) Simulated amplitude spectral density at µ0H=12mT. (b) Idem at µ0H=7mT. The symbols show the measured dispersions. (c) Color plot of the vertical component of the dynamic magnetization for the Goldstone mode superimposed onto the nucleated weak stripe texture. (g) Same for the Higgs mode.

 Observation of Higgs and Goldstone spin-wave modes in weak magnetic stripes

The significance of phase noise lies in the fact that most if not all devices based on magnons employ the nonlinearity of the magnetic system in order to realize their functionalities. An important entity in such parametrically excited systems is the phase of the magnons, which plays a crucial role in governing the nonlinear physics and hence the physical properties of interest of such devices. It is due to this importance of phase that in the recent past, many devices have been proposed and/or demonstrated that directly employ the magnon phase for the realization of important applications. The analytical theory developed is supported using rigorous GPU-based micromagnetic simulations [1]. A 5.1 μm thick sample of yttrium iron garnet (YIG) with saturation magnetization, 145 emu/cm3 , and exchange constant, 3.77 x 10-7 erg/cm is used. Demagnetization-, exchange-, and thermal-fields are included in addition to the applied-fields. A primary microwave signal of 6 GHz is used at a bias-field of .200 Oe. In our previous study [2], we had provided an accurate theoretical prediction of the increase in the threshold-field when an additional secondary signal is introduced. The secondary signal modifies the phase-relationship between the magnons and the primary microwave signal. While the importance of the phase as a fundamental entity has increased, studies on the implications of the phase-noise are few. This is due primarily to the difficulties in accessing, manipulating, and analyzing the phase accurately. Previous studies of noise mainly consider thermal-noise and often make use of the macrospin approximation. In magnonics, noise studies exist largely only for spin-torque oscillators. In the radio frequency (RF) applications, noise is usually modeled at a circuit level [3], and a microscopic study of phase-noise remains to be done. Ours is a more general treatment of phase-noise implications and, consequently, applicable across all magnon-based nonlinear applications. We have used analytical techniques drawn from hydrodynamics to evaluate the effects of phase-noise. Moreover, the simulations performed are based on state-of-art GPUs, using software optimized to study the magnon based physics specifically. Such simulations would not have been computationally possible in the past. Hence, this is a timely study of an important entity. (1) At first, we introduce a Gaussian phase-noise of zero mean and standard-deviation,σ , to understand its impact on the nonlinearity threshold. For a temporal-noise, we find that the threshold-field scales as e(σ^2)/2, but the growth rate of the magnons that partly determines the transient-time scales as e-(σ^2)/2 (Fig. 1). We have demonstrated these results analytically as well as using micromagnetic simulations. The results are of particular importance for RF devices like Frequency Selective Limiters (FSLs), Signal to Noise-ratio Enhancers, etc., that are often employed in noisy environments and are therefore subjected to temporal phase-noise. As mentioned earlier, few studies of phase-noise at the magnonic scale exists for such conventional RF devices despite the nonlinear behavior being a crucial part of device-functionality. (2) We find that similar changes as above are seen in the threshold-field and the growth-rate when a spatial phase-noise of similar mean and standard-deviation is employed. As can be observed from Fig. 2, the simulation data-points are nearly identical to those obtained in the case of temporal noise. For magnonic applications [4], this points to the importance of process variations and microstructural quality during the production of magnetic samples that can give rise to spatial-noise conditions. (3) Compared with phase-noise, we find that frequency-noise leads to only trivial variations in the threshold-field. Frequency noise alters the amplitude of the spin-waves. Therefore, information when encoded into the spin-wave amplitude, is more likely to suffer distortion under frequency noise when compared with phase noise. Apart from lowering the signal to noise ratio, frequency noise increases the magnon frequency deviation. Some of the proposed magnonic logic schemes [5] employ a combination of amplitude-, phase- and frequency- encoding to realize beyond-Neumann architectures where the above considerations would be crucial. Likewise, these results also play an important role in conventional communication systems that employ a combination of modulation techniques. Our studies, thus, allow phase noise considerations to be included in the design and/or operation of nonlinear magnetic devices. These results are important for both the traditional and novel magnon-based applications since device performance characteristics often strongly depend on the threshold-field as well as the transient-time. This is evident, e.g., in the ongoing efforts to miniaturize nonlinear magnetic rf device -FSL- especially for autonomous automobile applications [6]. Also, our results are independent of the mode of pumping (parallel or perpendicular) and the magnetic-material (applies to insulators, e.g., YIG as well as metals, e.g., NiFe). The results, in general, could be used to understand the nonlinear properties of other parametric systems that involve three-particle processes.

**References:**

1. A. Venugopal, T. Qu, and R. H. Victora, IEEE Trans. Microwave Theory Tech., vol. 68, no. 2, pp. 602–610 (2020). 2. A. Venugopal, T. Qu, and R. H. Victora, Appl. Phys. Lett., vol. 117, no. 152404 (2020). 3. M. Shukla, M. Y. Koledintseva, M. Geiler, et al, "Adaptive Interference Mitigation Using Frequency-Selective Limiters over GPS Band for Automotive Applications," 2020 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMCSI), Reno, NV, USA, pp. 614-618 (2020). 4. A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys. D: Appl. Phys. vol. 43, no. 264002, (2010). 5. A. Khitun, M. Bao and K. L. Wang, J. Phys. D: Appl. Phys., vol. 43, no. 264005 (10pp), (2010). 6. A. O. Boryssenko, S. M. Gillette, M. Y. Koledintseva, IEEE Trans. Microwave Theory Tech., vol. 67, no. 12, pp. 4871-4880 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/874dd0937c495ae22693803fdb8ffab9.jpg)
Fig. 1. Comparison of simulated data and theoretical prediction of threshold-fields and the slope of growth rate w.r.t microwave field intensity.

![](https://assets.underline.io/uploads/markdown_image/1/image/af9ea0e88443cf351b0e314bebde8f05.jpg)
Fig. 2. Growth rate of magnons vs the primary rf field intensity for various phase-noise standard deviations. ‘T’ refers to temporal noise, ‘S’ to spatial noise.

 Phase Noise Considerations in Magnon based Parametric Excitations

We investigate the role of disorder on the various topological magnonic phases present in deformed honeycomb ferromagnets. To this end, we introduce a bosonic Bott index to characterize the topology of magnon spectra in finite, disordered systems. The consistency between the Bott index and Chern number is numerically established in the clean limit. We demonstrate that topologically protected magnon edge states are robust to moderate disorder and, as anticipated, localized in the strong regime. We predict a disorder-driven topological phase transition, a magnonic analog of the “topological Anderson insulator” in electronic systems, where the disorder is responsible for the emergence of the nontrivial topology. Combining the results for the Bott index and transport properties, we show that bulk-boundary correspondence holds for disordered topological magnons. Our results open the door for research on topological magnonics as well as other bosonic excitations in finite and disordered systems [1]. We start from a ferromagnetic honeycomb layer [Fig. 1(a), (c)]. The honeycombs can be distorted, controlled by the bond angle θ, as shown in Fig. 1. We consider nearest-neighbor (NN) ferromagnetic exchange energy, NN pseudodipolar interaction [2], and on-site anisotropy energy. For strong enough perpendicular easy-axis anisotropy, the out-of-plane ferromagnetic state is an equilibrium state. We then calculate the spin wave spectra upon this state using the standard methods. Because of the bosonic commutation relation, we result in a non-Hermitian generalized eigenvalue problem η H T=Tη E, where H is the Hermitian Hamiltonian, E is the diagonal matrix of eigenvalues, η is a metric due to the bosonic commutation relation, and T is the Bogoljiubov transformation matrix diagonalizing the Hamiltonian. We define a bosonic Bott index, which takes the same form as the electronic Bott index [3] but with a modified projector P=TηΓT† η, where Γ is a matrix electing certain states. We first consider the clean limit. Figure 1(b) and (d) show spin wave spectra for infinite system, 100-wide zigzag strip, and 40*40 finite flake (from left to right) for normal and squeezed honeycomb lattices, respectively. Gapped bulk spectra in infinite and periodic systems and gapless (crossing) edge states for an open strip can be observed, indicating a nontrivial topology. A topological transition happens at θ=π/2. When θ&gt;π/2 such as Fig. 1(a) (θ=2π/3), the upper (lower) band has Chern number +1 (-1), and the Bott indices for the finite flake give the same values as the Chern numbers. For θ&lt;π/2 such as Fig. 1(c) (θ=5π/12), the signs of Chern numbers as well as Bott indices flip. The system can also be tuned to be topologically trivial by staggered anisotropy. The Bott indices are still consistent with Chern numbers during this process. Then we add a non-correlated random anisotropy ranged [-W,W] onto each spin in the normal honeycomb lattice to study the effect of disorder. In this non-periodic system, the k-space Chern number cannot be directly used, and the Bott index becomes powerful. When the system is nontrivial in clean limit (Bott index is 1 for the upper band), for moderate disorder, the Bott index is still one, meaning that the system keeps nontrivial. When the disorder is strong enough, a topological transition occurs and the system becomes topologically trivial [Fig. 2(a)]. This phenomenon is consistent with the common wisdom that the topology is quite robust since very strong disorder is needed to break the topology. A more remarkable phenomenon occurs when the disorder affects a topologically trivial system [Fig. 2(b)]. For a trivial system in clean limit (the band structure of a strip near the gap in the clean limit is shown in the inset), there are no gapless edge states inside the bulk gap. Surprisingly, as the disorder strength increases, the Bott index increases from zero and reaches a plateau of B=1 and then drops to zero large disorder. This finding indicates that there exists a disorder-induced topological phase, similar to the topological Anderson insulator phase in electronic systems [4]. We further demonstrate the bulk-boundary correspondence in our topological magnon model by studying the transport properties. We calculate the total transmission of magnons at the mid-gap energy across a disordered sample sandwiched by two clean leads, and compare them with the Bott index results in Fig. 2(a) and (b). The agreement is quite good, demonstrating the fact that when the Bott index is nontrivial, there exists robust edge channel inside the bulk band gap, which is the common wisdom of bulk-boundary correspondence. The existence of edge states in strongly disordered magnets is further confirmed by the calculation of the real-space wave functions. Eigenstates whose energies are closest to the mid-gap energy for a certain disorder configuration are calculated, and the expectation values 〈Sx〉, 〈Sy〉 are plotted in Fig. 2(c) and (d), corresponding to the circled data points in (a) and (b), respectively.

**References:**

[1] X. S. Wang , A. Brataas, and R. E. Troncoso, Phys. Rev. Lett. 125, 217202 (2020). [2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [3] H. Huang and F. Liu, Phys. Rev. Lett. 121, 126401 (2018). [4] J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Phys. Rev. Lett. 102, 136806 (2009).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cca9b8afc32254bc08681bd6816a5560.png)
Fig. 1. (a) Schematic of a normal honeycomb magnet. (b) From left to right: the spin-wave spectra for infinite samples , zigzag strips of width 100 along rows, and 40*40 finite samples. (c) Schematic of a squeezed honeycomb magnet of θ=5π/12. (d) Spin-wave spectra for (c).

![](https://assets.underline.io/uploads/markdown_image/1/image/308b334f57cb35f1140a75f4540bcf6a.png)
Fig. 2. Comparison between the Bott index and the total transmission for (a) topologically nontrivial and (b) trivial systems. (c) and (d) plot 〈Sx〉, 〈Sy〉 of eigenstates. The size of the circles indicates the amplitude, and the color encodes the azimuthal angle.

 Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

Spin waves (SWs) are precessional magnetization disturbances propagating in magnetic media in the form of waves at microwave frequencies [1] that are the most commonly used frequencies range for wireless communication. Wavelengths of SWs are several orders of magnitude shorter than the lengths of electromagnetic waves of corresponding frequencies. The dynamics of SWs even in uniformly magnetized films is extraordinarily rich. It is easy to obtain effects difficult to be achived for other type of waves, such as nonreciprocity, excitation caustic-beam that are characteristic for hyperbolic media, easy external control over propagation (e.g., by inhomogeneous magnetic field) [2]. Non-linear phenomena are also easily available[1,3]. All these make SWs to be a promising candidate for information carriers, especially in the context of beyond-CMOS applications [4]. SWs may be confined to certain areas of the thin film, e.g., in a potential well created by an inhomogeneity of the static demagnetizing field in the vicinity of film’s edge. Such localized modes are called edge SWs (E-SWs). Usually, the frequencies of E-SWs are downshifted with respect to bulk-type SWs (B-SWs), i.e., SWs propagating far from the edge where the static effective field is practically homogenous. Here, we theoretically study by means of micromagnetic simulations [5] and analytical modelling the dynamics of the E-SWs localized at the edge of thin permalloy film and their interaction with B-SWs. Firstly, we focus on the utilization of the E-SWs to excite short B-SWs propagating in the form of plane waves. Secondly, we study the inelastic non-linear scattering of B-SW beam by the E-SWs. We consider propagation of the E-SW along the edge of 10 nm thick permalloy film that is uniformly magnetized by the external field of value 300 mT directed perpendicularly to the film’s edge (see Fig. 1a, where the system is presented along with exemplary result of a simulation). The E-SW is excited by the point-source-like microwave field of frequency 12 GHz located at the film’s edge. There are visible two prominent peaks in the simulated spectrum (Fig. 1b). The first one corresponds to the frequency of the microwave field (12 GHz) and propagation of the E-SWs, whereas the second peak has doubled frequency (24 GHz) and corresponds to the B-SWs propagating obliquely outwards the edge in the form of plane waves. It indicates excitation of the B-SWs by the propagating E-SWs in the process of second harmonic generation. Noteworthy, the wavelength of the B-SWs is over twice shorter than the wavelength of the E-SWs shown in Fig. 1a. The B-SW of wavelength equal to 78 nm is excited by the 285 nm long E-SW. For the same sample, we analytically and numerically analyze the interaction of an obliquely incident B-SW beam (at frequency f) with an E-SWs at frequency n. We find that due to the inelastic scattering the secondary B-SW beams with up- and down-shifted frequency (f-n, f+n) can be excited. In Fig. 2 an exemplary result of simulation for a B-SW beam incident at the films edge where E-SW propagates and the resulting secondary scattered beam with frequency (f+ν)=42 GHz are shown. Moreover, we observe angular shifts between the primary incident and the scattered beams with shifted frequencies and analyze the efficiency of the scattering process. Finally, we present an application of this effect for SW demultiplexing. Acknowledgments. The research was supported by the National Science Centre of Poland, project no. 2019/35/D/ST3/03729 (P.G. and M.K.); COST action under project CA17123 MAGNETOFON (I.L.); the Spanish MCIU grant PID2019-108075RB-C3-3 (K.G.). K.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science). The simulations were performed at the Poznan Supercomputing and Networking Center (Grant No. 398).

**References:**

 [1] A. Gurevich and G. Melkov, Magnetization oscillations and waves (CRC Press, Boca Raton, 1996). [2] A. Barman et al., J. Appl. Phys. 128, 170901 (2020). [3] A. V. Chumak et al., Nat. Phys. 11, 453 (2015). [4] B. Murmann, B. Hoefflinger, "IRDS—International Roadmap for Devices and Systems, Rebooting Computing, S3S" in NANO-CHIPS 2030 (Springer, Cham, 2020). [5] A. Vansteenkiste et al., AIP Advances 4, 107133 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6f369e1f64f9a7c41706068472bcb0f0.jpg)
Fig. 1. (a) System under consideration with the snapshot of the out-of-plane magnetization component (mz) in the steady state with E-SW excitation at 12 GHz. The source of E-SWs is located in the area indicated by the white triangle. (b) Simulated SW frequency spectrum.

![](https://assets.underline.io/uploads/markdown_image/1/image/3d4019bb873d5286156ad035fb24cd24.jpg)
Fig. 2. Scheme of the system under investigation with the result of micromagnetic simulation performed for B-SW beam at frequency f = 30 GHz incident under the angle 60° and scattered on an E-SW at ν =12 GHz. Red (at the edge), yellow and blue (beams) corresponds to presence of the SWs of frequencies ν, f, and f+ν, respectively. The white color denotes areas, where at least two frequencies are mixed.

 Inelastic scattering of spin wave beam at the edge localized spin waves and second harmonic generation of spin waves

Nowadays a lot of effort is put into the creation of a practical quantum computer. For this ambitious aim different research fields propose and investigate new materials and technologies. One of such promising systems, a magnon Bose-Einstein condensate (BEC)in magnetic crystals, may allow the creation of magnon-BEC-based qubit at room temperature. A prerequisite for such a qubit is the appearance of spontaneous coherence in magnon BEC. Also, the presence of coherence of the magnon Bose-Einstein condensate is essential for understanding such fundamental properties of it as the possibility of a dissipative magnon supercurrent [1], excitation of Josephson oscillations [2], and propagation of Bogoliubov waves [3]. In most of the previous studies in this field, the magnon BEC is investigated by means of Brillouin light scattering (BLS) spectroscopy [4,5] delivering information about the spectral density of a magnon gas. Unfortunately, due to the limited frequency resolution of the optical Fabry-Pérot interferometers used in BLS facilities, the conventional BLS technique does not allow one to prove the coherence of a magnon BEC directly. The insufficient frequency resolution makes it impossible to separate the relaxation dynamics of condensed and thermal magnons. Moreover, the possible outflow of the condensate from a spatially localized probing light spot complicates the interpretation of the obtained experimental results (see [6] and the corresponding discussion in [7]). For all these reasons only indirect indications of the coherency, obtained on the magnon BEC in magnetic crystals, could be provided yet. Up to this day, a decisive proof of the full coherence magnon BEC remained elusive. In our work, we report a first direct experimental observation of the evolution of magnon BEC towards full coherence, whose degree is limited only by the natural processes of the magnon relaxation into the crystal lattice. To achieve this, we use a novel concept that enhances the coupling of the parametrically-driven magnon system with the probing electromagnetic field. The obtained electric signal was investigated time-resolved by a spectrum analyzer to reach the required frequency resolution necessary for the proper display of the magnon BEC state. A pulsed electromagnetic parametric pumping was used to inject magnons, increase, thus the chemical potential of the magnon gas, and trigger the BEC formation process. It is found that the radiation spectrum picked up after the terminization of a pump pulse is narrowed by two orders of magnitude, compared to the broad-band spectrum obtained during the process of parametric pumping. The resulting spectrum width is only limited by the magnon relaxation frequency. We study this spectrum evolution in many details and identify the conditions for BEC creation. We believe that this direct demonstration of the coherence of the magnon BEC brings the implementation of the room temperature BEC-based computing closer. This research was funded by the European Research Council within the Advanced Grant No. 694709 “SuperMagnonics” and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Transregional Collaborative Research Center – TRR 173 – 268565370 “Spin+X” (project B01). The authors are grateful to G. A. Melkov and H. Yu. MusiienkoShmarova for the fruitful discussions.

**References:**

[1] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Supercurrent in a room-temperature Bose-Einstein magnon condensate, Nat. Phys. 12, 1027 (2016). [2] A. J. E. Kreil, A. Pomyalov, V. S. L’vov, H. Yu. Musiienko-Shmarova, G. A. Melkov, A. A. Serga, and B. Hillebrands, Josephson oscillations in a room-temperature Bose-Einstein magnon condensate, arXiv:1911.07802. [3] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova, A. A. Serga, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Bogoliubov waves and distant transport of magnon condensate at room temperature, Nat. Commun. 10, 2460 (2019). [4] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping, Nature 443, 430–433 (2006). [5] A. A. Serga, V. S. Tiberkevich, C. W. Sandweg, V. I. Vasyuchka, D. A. Bozhko, A. V. Chumak, T. Neumann, B. Obry, G. A. Melkov, A. N. Slavin, and B. Hillebrands, Bose-Einstein condensation in an ultra-hot gas of pumped magnons, Nat. Commun. 5, 3452 (2014). [6] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, Observation of spontaneous coherence in Bose-Einstein condensate of magnons, Phys. Rev. Lett. 100, 047205 (2008). [7] A. J. E. Kreil, H. Y. Musiienko-Shmarova, S. Eggert, A. A. Serga, B. Hillebrands, D. A. Bozhko, A. Pomyalov, and V. S. L’vov, Tunable space-time crystal in room-temperature magnetodielectrics, Phys. Rev. B 100, 020406(R) (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/72224fb139fe95447f22f54475f3939d.jpg)
Fig. 1 Sketch of the experimental setup. The sample holder section shows an open waveguide with an internally mounteddielectric resonator. A bulk YIG sample is placed in an opening of the resonator together with a large detection loop to detect the radiated signals from the whole sample volume.

![](https://assets.underline.io/uploads/markdown_image/1/image/84026c2fe3ed134957adcd953bd21b38.jpg)
Fig. 2 Radiated signal of the magnon BEC measured for different times during the free decay of the system. The signals demonstrate a lorentz shape with a FWHM bandwidth of approx. 1 MHz. In normalized view it is visible that the BEC signal demonstrates an unchanged singel-state profile for a long observation period.

 Evolution of room-temperature magnon gas toward coherent Bose-Einstein condensate

In thin-films-based experiments, spin waves (SWs) are often investigated in confined geometries or as plane waves in full films. There are exceptions, such as SW channeling along domain walls [1], which are appealing in terms of SW steering. However, this approach relies on the preparation of a specific magnetic texture and transferring the SW out of it, back into the domains, seems challenging. An alternative route towards the steering of strong, channeled SWs exists, namely caustic SW beams (CSWBs). Caustic phenomena in anisotropic wave propagation correspond to sharply-defined lines/surfaces with large intensities [2]. They have been investigated in a variety of fields [2], and can be expected in micromagnetism. In thin films, caustics can arise due to the marked anisotropy of the SW dispersion relation (DR), illustrated in Fig.1. From such a plot, the direction θv of the group velocity vg=▽kω can be determined from the local normal to the curve. It can be seen in Fig.1 that a portion of the red curve (at 8.5 GHz) is almost straight, leading to a stationary θv around a value θvc for this range of wavefront angles φ. Thus, an excitation of SWs with a broad wave vector spectrum would lead to strong emission in the direction θvc. The corresponding anisotropic enhancement defines a CSWB. Such beams have so far been studied notably upon scattering of a confined SW into a wide film [4,5,6]. In the present work, we tackle theoretically the emission of CSWBs in full, soft thin films; the latter could be stimulated by the tip of a microstrip antenna. We restrict ourselves to applied fields below ferromagnetic resonance (FMR). After dimensionless rewriting of the DR, only three free parameters are left: the ratio of dipolar-exchange length to film thickness η=lex/d, the normalized SW frequency ν=ω/(γ*Js) (Js: saturation induction), and the reduced applied field h=Ba/Js. We then investigate the properties of caustic points numerically and analytically, using suitable approximations. An experimental situation corresponds to fixed film thickness; we therefore record the caustic wavenumbers (WNs) kc*d, beam directions θvc and wavefront directions φc as a function of ν and h, at fixed η. As the example in Fig.2 shows, caustic points can exhibit a rich behaviour. Starting with the caustic WNs, one can see a low-frequency, low-field pocket in the dipolar-dominated regime. Above its sharp boundary, a relatively wide band appears, corresponding to kc*d varying slowly in the dipolar-exchange regime. A final transition brings kc*d into the exchange-dominated regime; the latter has an upper boundary above which no caustic point exists (blue region in Fig.2). The map of φc(h,ν) displays a similar aspect. In the bottom pocket, the caustic wavefront angle is almost constant, with a numerically found limit of about 54.7°. Our analytical approximation yields a value of arccos(1/√3)≈54.7°, in excellent agreement. The corresponding limit for θvc is numerically 109° and 103° from analytics. In the dipolar-exchange band, by contrast with kc*d, φc varies significantly and saturates at 90° in the exchange-dominated region. This saturation is in stark correlation to that of the caustic beam direction θvc, which also goes to 90°. All the above properties can be understood from the geometrical evolution of the DR curve when ν increases, as illustrated in Fig.1. The markers on the curves correspond to their caustic point (2.5 GHz omitted for clarity). At the bottom pocket's boundary, the tangent to the DR at kc*d passes close to the origin: hence, ∂kc/∂φc is large. In the exchange-dominated regime, the curve flattens near φ=π/2. This accelerates the increase of kc*d (cf. curves at 16 and 18.5 GHz) while bringing θvc and φc to π/2 at a certain threshold frequency. Beyond the latter, the curve rounds more and more: θv can no longer be stationary, thus, no caustic point exist. From this investigation, we can thus predict two very different types of CSWBs. At the low-kc pocket's upper boundary, due to the very straight and oblique DR curve, the CSWB should feature a broad wavelength spectrum. By contrast, at the upper boundary of the caustic band, the CSWB's wavelength spectrum should be much narrower. Indeed, the DR at kc*d is tangent to a circle k*d=cst. In addition, since this CSWB consists of two beams with overlapping directions, its amplitude should be correspondingly larger. This phenomenon has been studied theoretically by Kim et al. [7]. The first type of CSWB should be invisible in magneto-optical microscopy, but Brillouin light scattering should reveal the broadband SW beam. The second type of CSWB would require time-resolved Scanning Transmission X-ray Microscopy due to the higher kc.

**References:**

[1] V. Sluka, T. Schneider, R. A. Gallardo et al., Nature Nanotechnology, Vol.14, 4, pp.328-333 (2019) [2] Y. A. Kravtsov and Y. I. Orlov, "Caustic, Catastrophes and Wave Fields", Springer-Verlag 1993 [3] B. A. Kalinikos and A. N. Slavin, Journal of Physics C: Solid State Physics, Vol.19, 7013, pp.7013-7033 (1986) [4] V. E. Demidov, S. O. Demokritov, D. Birt et al., Physical Review B, Vol.80, 014429 (2009) [5] T. Schneider, A. A. Serga, A. V. Chumak et al., Physical Review Letters, Vol.104, 197203 (2010) [6] F. Heussner, A. A. Serga, T. Brächer et al., Applied Physics Letters, Vol.111, 122401 (2017) [7] J.-V. Kim, R. L. Stamps and R. E. Camley, Physical Review Letters, Vol.117, 197204 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/015b984df791782c0abb0d98462393a9.jpg)
Fig.1: Polar plot of the normalized SW wave number k*d (d: film thickness) as a function of the angle φ from the applied field to the wave vector. The DRs are calculated from the model by Kalinikos and Slavin [3], for an applied induction Ba of 20 mT, frequencies ω/(2π) of 2.5, 8.5, 11, 13.5, 16 and 18.5 GHz, a reduced gyromagnetic ratio γ/(2π)=29.3 GHz/T, saturation induction Js=μ0*Ms=0.92 T, film thickness 60 nm and dipolar-exchange length lex=6.2 nm. Owing to the DR's symmetries, the curves shown here generates the whole polar plots by line symmetry around the quadrant boundaries.

![](https://assets.underline.io/uploads/markdown_image/1/image/80218c582c1cedf81dd66f75f10ca21a.jpg)
Fig.2: Gray-level map of the normalized caustic wave number kc*d as a function of h and v, for η=0.10355. The yellow curve corresponds to FMR; the red filled region below is not considered here. In the top left corner's blue filled region, no caustic points exist.

 Investigation of Caustic Spin Wave Beams in Soft Thin Films

The magnetism in curved geometries is attracting attention because of several theoretical predictions with possible potential applications, such as a curvature induced effective Dzyaloshinskii-Moriya like interaction [1]. Within this setting, analogously to a geometric curvature, a curvature induced by a gauge field [2] is a fruitful aspect to consider in the interaction of the magnetization with an applied static electric field. In a quantum description of the motion of a magnetic moment, an electric field does not modify the energy but it rather modifies the linear momentum, therefore introducing an additional phase for the corresponding Schrödinger wave function, an effect which is known as the Aharonov-Casher’s effect [3]. The classical analog is the spin wave of a ferromagnet. It has been shown that a spin wave, which is carrying a magnetic moment, will acquire a phase when the system is subjected to an electric field [4]. This effect, even if rather small, can be experimentally observed [5]. It is therefore of interest to extend this idea to a general micromagnetic description of the problem. The aim of this research is to pose the problem of the micromagnetics under the effect of an electric field by using the so-called gauge invariance. Briefly the spin-waves, which are small oscillations, deviate from the uniform magnetization with respect to a local reference frame and around the ez direction. They interact with an applied electric field E Fig.(1) modifying the spin wave dispersion relation. Therefore, according to the so-called local exchange invariance [6], we propose a phase shift model, in the exchange spin waves regime, in order to possibly describe and interpret the data shown in [5]. More in details the local gauge symmetry is approximately a local invariance property (because of its thermodynamic origin), of the lagrangian density of the magnetization m(x, t) of a ferromagnet. It is known that it is invariant with respect to the SO(3) global symmetry [7]. In our case this symmetry reduces to a unitary representation of the SO(2) group, (i.e., U(1)). Due to the reason that this symmetry has to be local (i.e., to be position dependent) the invariance of the lagrangian density is broken unless introducing a gauge field able to solve the technical problem of the parallel transport of the magnetization vector. Therefore we show that a particular orientation of a reference frame has no physical relevance. In fact, gauge invariance should occur whenever one can define non-collinear, local reference frames. In our model, the orientation of the local non-coordinate reference frames cannot be uniquely defined because of the presence of an electric field. Summarizing in our invariant gauge formulation the reference frames’ orientation is unspecified with respect to a local phase which defines an invariance property of the exchange interaction terms in the lagrangian density of the magnetization m(x, t). The Lagrangian density of a ferromagnet, interacting with an electric field, is transformed into a new Lagrangian density upon a formal redefinition of the standard partial derivative. Moreover the transformation group, which expresses this invariance, imposes definite restrictions on the dispersion relation that at the first order in a Tylor expansion was previously found to be ω=ω(ky)-γLc-2∂ω/∂kyEx [4], with γL the gyromagnetic ratio and c the speed of light. Therefore our findings point to a more advanced theoretical interpretation with respect to previous discoveries [4] in order to drive experiments in manipulating spin waves and developing electrically tunable magnonic devices.

**References:**

[1] R. Streubel et al., J. Phys. D: Appl. Phys. 49, 363001 (2016). [2] S.G. Tan, et al., Physics Reports, Volume 882, Pages 1-36, (2020). [3] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319, (1984). [4] V. Basso and P. Ansalone, EPL, Vol. 130, Number 1, Article 17008, (2020). [5] X. Zhang, T.Liu, M.E. Flatté and H.X. Tang Phys. Rev. Lett. 113, (2014). [6] I.E. Dzyaloshinskii and G.E.Volovik, J.Physique, 39, 693, (1978). [7] W. F. Brown, Magnetoelastic interactions, pp. 26 and 79, (Vol. 9), Berlin: Springer, (1966).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/62bbb322a9e11ec137f86fd168238d14.jpg)
Fig. 1 The system under study is a two dimensional infinite ferromagnetic layer with primary magnetization along the direction ez , an external electric field is applied perpendicularly to both the magnetization and k spin-wave vector.

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