The spin-torque magnetic diode (STMD) effect [1] is a quadratic rectification effect of the input microwave current IRF(t) in a magnetoresistive nano-junction, which is commonly observed in the traditional regime of operation of an STMD, when the magnetization of the “free” layer (FL) lies in-plane, and when the frequency fs of the current IRF(t) is close to the ferromagnetic resonance (FMR) frequency f0 of the junction. In this case, the induced resonance oscillations of the junction resistance R(t) can mix with the oscillations of the input microwave current IRF(t) , and produce an output DC voltage UDC = &lt;IRF(t)R(t)&gt; (where "&lt;&gt;" denotes time averaging) across the junction. It was demonstrated theoretically in [2] that in an STMD, biased by an out-of-plane static magnetic field, a different dynamical regime of the STMD operation characterized by large-angle out-of-plane magnetization precession can be realized. The excitation of the large-angle precession has a threshold character, and is possible only for input microwave currents exceeding a certain frequency-dependent critical value, and sufficiently low frequency. In this regime of the out-of-plane precession the output DC voltage of an STMD increases with the frequency of the input signal, but is virtually independent of its power. An STMD working in the regime of large-amplitude out-of-plane precession functions as a non-resonant threshold detector of low frequency microwave signals, due to the large nonlinear shift of its operating frequency. Therefore, this precession regime is particularly suitable for applications in the microwave energy harvesting. It was demonstrated experimentally in [3] that the out-of-plane magnetization precession regime in an STMD operation predicted in [2] can be realized without any bias magnetic field, if an STMD FL has a perpendicular magnetic anisotropy. In such a regime an STMD has the rectification properties that are independent of the input microwave frequency, and the performance of the STMD-based energy harvesting device is comparable to the performance of the state-of-the-art Schottky diode for low-power rectification. It was further shown in [3] that the developed bias-free STMD provides sufficient dc voltage to power a practical nanodevice − a black phosphorus photosensor. Here we present an analytical and numerical theory explaining the performance of a bias-free STMD with perpendicular magnetic anisotropy [4]. We show that in an STMD, having a magnetic FL with perpendicular magnetic anisotropy of the first and second order, and no external bias magnetic field, the out-of-plane regime of magnetization precession can be excited by sufficiently large (exceeding a certain threshold) RF signals with the frequencies smaller than 250 MHz . We also show that such a device can operate as a broadband energy harvester capable of converting incident RF power into a DC power with a conversion efficiency of ∼5%. The developed analytical theory of the bias-free STMD operation can be used for the optimization of high-efficiency RF detectors and energy harvesters based on STMDs. Our results are paving a way for use of spintronic detectors and harvesters as building blocks for self-powered nano-systems, such as implantable biomedical devices, wireless sensors, and portable electronics.

Signal Detector and RF Energy Harvester Based On A Spintronic Diode With Perpendicular Anisotropy

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

**Session chairs:** Riccardo Tomasello, Politecnico di Bari and Mario Carpentieri, Politecnico of Bari



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Signal Detector and RF Energy Harvester Based On A Spintronic Diode With Perpendicular Anisotropy

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

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