France



It has been shown that a rapid cooling of Yttrium-Iron-Garnet (YIG)/Pt nanostructures preheated by a DC electric current pulse sent through the Pt layer leads to an overpopulation of a magnon gas and to the subsequent Bose-Einstein condensation (BEC) of magnons [1]. Hereby, the application of a sufficiently-long DC heating pulse was used to generate a high population of magnons being in equilibrium with the phononic system. The subsequent rapid decrease in the temperature resulted in the break of the equilibrium, in the over-population of magnons over the whole magnon spectrum, and, in the following, in a redistribution of magnons from higher to lower energies. Finally, if the elevated temperature was high enough, and the cooling process fast enough, the chemical potential reached the minimal energy of the system and the BEC was triggered [1].&lt;br/&gt;&lt;br/&gt;In these experiments, a Pt or Al layer was used to heat the YIG nanostructure. For a Pt heater, in addition to the Ohmic heating effects, one expects the generation of a spin-polarized current transverse to the YIG/Pt interface due to the spin Hall effect (SHE) [2]. The spin current, consequently, is known to act on the magnetization dynamics in the YIG via spin-transfer torque (STT). Whereas in our previous studies the induced disequilibrium was found to be mainly given by the temperature increase and subsequent rapid cooling, we observe a significant contribution of the spin Hall effect (SHE) in a new set of samples. The larger effect of the SHE-STT was reached by a modification of the structures (see Fig. 1) with respect to those in the original study [1], i.e. by a decrease in the YIG thickness, and via using a sputtered Pt layer instead of an epitaxially grown Pt layer studies to enhance the spin Hall angle [3].&lt;br/&gt;&lt;br/&gt;It was found that SHE-induced STT enhances the BEC formation at the bottom of the magnon spectrum – in the minimum frequency of the fundamental spin-wave mode, and excites a nonlinear magnon mode – the spin-wave bullet, located below the linear magnon spectrum [4]. We investigated the dynamics of these magnon states employing Brillouin light scattering (BLS) spectroscopy. A sketch of the experimental setup and SEM images of the investigated structure is depicted in Fig. 1. The application of a 50-ns-long DC-pulse to a 7-nm- thin Pt layer on top of a 34-nm-thin and 2-μm-wide YIG strip results in the SHE-STT. At sufficiently large amplitudes of the DC-pulse applied, the bullet mode is formed during the DC pulse at a frequency around 1.8 GHz below the frequency of the fundamental mode, see Fig. 2b. The pronounced magnon density peak in the left panel in Fig. 2c (solid orange line), measured at the time just before the current pulse is switched off, clearly demonstrates the bullet mode formation. The SHE-STT nature of the mode is confirmed by the variation of the magnetisation orientation with respect to the current direction. The consequent switching of the DC pulse off results in the rapid cooling-induced BEC of magnons at the frequency around 6 GHz - see Fig. 2c. The results of the control experiment with no SHE-STT contribution under periodically changing polarity of the applied current (Fig. 2d) are in agreement with the original findings presented in Ref. [1].&lt;br/&gt;&lt;br/&gt;The interplay between the magnon condensate and the SHE-STT bullet mode, after the current pulse was gone, attracts special attention. Initially, in the rapid cooling phase of the process, thermally-induced excess magnons populate both the bullet mode and the condensate, see Fig. 2a. Further, both modes decay slower than expected from their lifteime indicating that they are still pumped from the gas. Finally, about 30 ns after the DC pulse is turned off, a qualitatively new process begins, resulting in the redistribution of magnons from the bullet mode to the condensate. This phenomenon manifests itself in an elevation of the condensate density accompanied by an increased decay of the bullet mode, and apparently occurs when the chemical potential of the magnon gas [1] decreases below the bullet mode frequency. The results suggest a path towards the application of macroscopic quantum states in spintronic devices.

**References:**

[1] M. Schneider, T. Brächer, D. Breitbach, V. Lauer, P. Pirro, D. A. Bozhko, H. Y. Musiienko- Shmarova, B. Heinz, Q. Wang, T. Meyer, F. Heussner, S. Keller, E. T. Papaioannou, B. Lägel, T. Löber, C. Dubs, A. N. Slavin, V. S. Tiberkevich, A. A. Serga, B. Hillebrands, and A. V. Chumak, Bose-Einstein condensation of quasiparticles by rapid cooling, Nature Nanotechnology 15, 457 (2020).&lt;br/&gt;[2] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016).&lt;br/&gt;[3] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Tuning the spin hall effect of pt from the moderately dirty to the superclean regime, Phys. Rev. B 94, 060412 (2016).&lt;br/&gt;[4] A. Slavin and V. Tiberkevich, Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet, Phys. Rev. Lett. 95, 237201 (2005).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/60e5026f5c65e8d34354a4da266411ef.jpg)
Colored SEM images of the investigated structures and sketch of the experimental setup. The structures consists of a 2-μm-wide and 34-nm-thick YIG strip. A 3-μm-long Platinum-heater (7 nm) on top is contacted by Ti/Au leads. The magnon intensity below the Pt-covered region is measured using time-resolved BLS spectroscopy. The shape of the Ti/Au contact pads attached to the Pt injector results in a DC current parallel to the short axis of the strip. Qualitatively the same results were obtained for another set of samples with a DC current flowing parallel to the short axis of the strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/534ab70ea9e9adb9ae113bed737dcfc7.jpg)
(a) Integrated BLS intensity from (b) as a function of the time within the two indicated frequency ranges. (b) BLS intensity color-coded as a function of time and the BLS frequency. During the pulse (0 ns &lt; &lt;i&gt;t&lt;/i&gt; &lt; 50 ns and &lt;i&gt;U&lt;/i&gt; = 2.0 V) the formation of a bullet mode is visible. The inset shows a sketch of the structure. (c) BLS intensity as a function of the BLS frequency at three different time steps, indicated by the dashed lines in (b). The black dashed lines depict the thermal BLS spectrum without a DC pulse present. (d) Integrated BLS intensity color coded as a function of time and the BLS frequency. In contrast to (b), the current polarity is alternating during the pulse, effectively switching off the contribution of the SHE-STT while kepping the heating constant.

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 The interplay between the rapid cooling-induced BEC and SHE-STT-driven bullet mode

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It has been shown that a rapid cooling of Yttrium-Iron-Garnet (YIG)/Pt nanostructures preheated by a DC electric current pulse sent through the Pt layer leads to an overpopulation of a magnon gas and to the subsequent Bose-Einstein condensation (BEC) of magnons [1]. Hereby, the application of a sufficiently-long DC heating pulse was used to generate a high population of magnons being in equilibrium with the phononic system. The subsequent rapid decrease in the temperature resulted in the break of the equilibrium, in the over-population of magnons over the whole magnon spectrum, and, in the following, in a redistribution of magnons from higher to lower energies. Finally, if the elevated temperature was high enough, and the cooling process fast enough, the chemical potential reached the minimal energy of the system and the BEC was triggered [1]. In these experiments, a Pt or Al layer was used to heat the YIG nanostructure. For a Pt heater, in addition to the Ohmic heating effects, one expects the generation of a spin-polarized current transverse to the YIG/Pt interface due to the spin Hall effect (SHE) [2]. The spin current, consequently, is known to act on the magnetization dynamics in the YIG via spin-transfer torque (STT). Whereas in our previous studies the induced disequilibrium was found to be mainly given by the temperature increase and subsequent rapid cooling, we observe a significant contribution of the spin Hall effect (SHE) in a new set of samples. The larger effect of the SHE-STT was reached by a modification of the structures (see Fig. 1) with respect to those in the original study [1], i.e. by a decrease in the YIG thickness, and via using a sputtered Pt layer instead of an epitaxially grown Pt layer studies to enhance the spin Hall angle [3]. It was found that SHE-induced STT enhances the BEC formation at the bottom of the magnon spectrum – in the minimum frequency of the fundamental spin-wave mode, and excites a nonlinear magnon mode – the spin-wave bullet, located below the linear magnon spectrum [4]. We investigated the dynamics of these magnon states employing Brillouin light scattering (BLS) spectroscopy. A sketch of the experimental setup and SEM images of the investigated structure is depicted in Fig. 1. The application of a 50-ns-long DC-pulse to a 7-nm- thin Pt layer on top of a 34-nm-thin and 2-μm-wide YIG strip results in the SHE-STT. At sufficiently large amplitudes of the DC-pulse applied, the bullet mode is formed during the DC pulse at a frequency around 1.8 GHz below the frequency of the fundamental mode, see Fig. 2b. The pronounced magnon density peak in the left panel in Fig. 2c (solid orange line), measured at the time just before the current pulse is switched off, clearly demonstrates the bullet mode formation. The SHE-STT nature of the mode is confirmed by the variation of the magnetisation orientation with respect to the current direction. The consequent switching of the DC pulse off results in the rapid cooling-induced BEC of magnons at the frequency around 6 GHz - see Fig. 2c. The results of the control experiment with no SHE-STT contribution under periodically changing polarity of the applied current (Fig. 2d) are in agreement with the original findings presented in Ref. [1]. The interplay between the magnon condensate and the SHE-STT bullet mode, after the current pulse was gone, attracts special attention. Initially, in the rapid cooling phase of the process, thermally-induced excess magnons populate both the bullet mode and the condensate, see Fig. 2a. Further, both modes decay slower than expected from their lifteime indicating that they are still pumped from the gas. Finally, about 30 ns after the DC pulse is turned off, a qualitatively new process begins, resulting in the redistribution of magnons from the bullet mode to the condensate. This phenomenon manifests itself in an elevation of the condensate density accompanied by an increased decay of the bullet mode, and apparently occurs when the chemical potential of the magnon gas [1] decreases below the bullet mode frequency. The results suggest a path towards the application of macroscopic quantum states in spintronic devices.

**References:**

[1] M. Schneider, T. Brächer, D. Breitbach, V. Lauer, P. Pirro, D. A. Bozhko, H. Y. Musiienko- Shmarova, B. Heinz, Q. Wang, T. Meyer, F. Heussner, S. Keller, E. T. Papaioannou, B. Lägel, T. Löber, C. Dubs, A. N. Slavin, V. S. Tiberkevich, A. A. Serga, B. Hillebrands, and A. V. Chumak, Bose-Einstein condensation of quasiparticles by rapid cooling, Nature Nanotechnology 15, 457 (2020). [2] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016). [3] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Tuning the spin hall effect of pt from the moderately dirty to the superclean regime, Phys. Rev. B 94, 060412 (2016). [4] A. Slavin and V. Tiberkevich, Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet, Phys. Rev. Lett. 95, 237201 (2005).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/60e5026f5c65e8d34354a4da266411ef.jpg)
Colored SEM images of the investigated structures and sketch of the experimental setup. The structures consists of a 2-μm-wide and 34-nm-thick YIG strip. A 3-μm-long Platinum-heater (7 nm) on top is contacted by Ti/Au leads. The magnon intensity below the Pt-covered region is measured using time-resolved BLS spectroscopy. The shape of the Ti/Au contact pads attached to the Pt injector results in a DC current parallel to the short axis of the strip. Qualitatively the same results were obtained for another set of samples with a DC current flowing parallel to the short axis of the strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/534ab70ea9e9adb9ae113bed737dcfc7.jpg)
(a) Integrated BLS intensity from (b) as a function of the time within the two indicated frequency ranges. (b) BLS intensity color-coded as a function of time and the BLS frequency. During the pulse (0 ns < t < 50 ns and U = 2.0 V) the formation of a bullet mode is visible. The inset shows a sketch of the structure. (c) BLS intensity as a function of the BLS frequency at three different time steps, indicated by the dashed lines in (b). The black dashed lines depict the thermal BLS spectrum without a DC pulse present. (d) Integrated BLS intensity color coded as a function of time and the BLS frequency. In contrast to (b), the current polarity is alternating during the pulse, effectively switching off the contribution of the SHE-STT while kepping the heating constant.

technical paper

Magnetic materials are at the hearts of billions of high frequency communication devices and on-chip integration of magnetic microwave devices is of paramount importance for efficient utilization of the huge electromagnetic (EM) spectrum beyond 5G. Magnonics – a sub-field of spintronics – employs magnetic waves and oscillations in magnetic thin films and patterned nanostructures for processing GHz to sub-THz EM signal [1-2]. Magnetic ferrites are conventionally used in such magnetic microwave devices. However, due to the need for external bias magnetic field and incompatibility with complementary metal oxide semiconductor (CMOS) process, ferrites are not suitable for miniaturized microwave components. On the other hand, metallic magnetic thin films and nanostructures have great potential for efficient engineering of microwaves. Although, the need for bias field remains a bottleneck. We have been working on solutions where the requirement of the bias-field is absent. We have successfully demonstrated several strategies for achieving such device architecture by utilizing self-biased nanomagnets with reprogrammable remanent states which are associated with distinct magnetization dynamics [2-11]. To illustrate reconfigurable microwave properties, we have shown in Fig 1 as an example, a pair of dipolarly coupled rhomboid nanomagnets (RNMs). Two different remanent states (↑↑ and ↑↓) are obtained by utilizing a simple field initialization (HI) scheme: applying a saturating field along short and long axes of the RNMs with subsequent removal of the field (Fig. 1(a)). Due to different distributions of the stray fields for the two remanent states, the effective fields (Fig. 1(b)) are different which manifest in the shift of the microwave spectra for the two remanent states. Measurement of microwave properties from a single isolated nanomagnetic pair was obtained by using micro-focused Brillouin light scattering (micro-BLS) experiment (Fig. 1(c)) and the micro-BLS spectra are shown in Fig. 1(d). This strategy for reconfigurable magnonic responses is expanded to 2D arrays of the RNMs for different magnetic configurations: antiferromagnetic-type and ferrimagnetic-type artificial magnonic crystals. The maximum amount of the shift in the microwave responses is achieved by using multilayer isolated as well as dipolar-coupled networks of RNMs. Beyond, RNM shape, we have also demonstrated reconfigurable microwave operation based on other strategic shapes with multiple remanent states, such as arrow-, C-, L- and S-type nanomagnets and zigzag nanowires. Furthermore, we emphasize that such self-biased nanomagnets are also suitable for magnonic device applications where the propagation and gating of magnons – quanta for spin waves – are of primary interest. Conventionally, a large bias magnetic field is used in order to satisfy an orthogonal orientation between spin wave propagation and magnetization direction for efficient transmission characteristics (known as surface geometry or Damon-Eshbach mode). The need of such bias field hinders device integration on-chip for magnonic waveguides. Therefore, we have proposed and demonstrated a self-biased magnonic waveguide based on dipolar-coupled but physically separated chain of RNMs (Fig. 2). A prototype of such magnonic devices is shown in Fig. 2(a) where a microwave antenna (input) excites the spin waves and the spin waves are detected using micro-BLS technique with a focused laser spot (output). In order to gate the spin wave propagation, we have flipped a RNM in the waveguide and it is marked as ‘gate’ in the scanning electron microscopy (SEM) image (Fig. 2(b)). Such waveguide design enables one to have two different remanent states: ferromagnetically ordered (FO) and FO with a defect (FO*) when field initialized (same scheme as mentioned above) along the width and the length of the waveguide, respectively. FO and FO* remanent magnetic states are shown in Fig. 2(b) using magnetic force microscopy (MFM) images. Gating of the spin wave propagation has been shown by plotting 2D spatial profiles of the spin wave intensities using micro-BLS technique which clearly indicate gated magnon flow for the FO* state in comparison to the FO state. Thus, one can also achieve binary gating of output signal amplitude using such self-biased nanomagnetic waveguide which is remarkable as it also eliminates the need for bias field. In this presentation, we would like to emphasize the need for bias-free reconfigurable magnonics along with recent developments and future roadmap.

**References:**

[1] A. Haldar, C. Tian and A. O. Adeyeye, Sci. Adv. 3, e1700638 (2017). [2] A. Haldar, D. Kumar and A. O. Adeyeye, Nat. Nanotech. 11, 437 (2016). [3] A. Haldar and A. O. Adeyeye, ACS Nano. 10, 1690 (2016). [4] A. Haldar and A. O. Adeyeye, Appl. Phys. Lett. 108, 192404 (2016). [5] A. Haldar and A. O. Adeyeye, Appl. Phys. Lett. 108, 022405 (2016). [6] A. Haldar and A. O. Adeyeye, J. Appl. Phys. 123, 243901 (2018). [7] K. Begari and A. Haldar, J. Phys. D: Appl. Phys. 51, 275004 (2018). [8] K. Begari and A. Haldar, J. Phys. D: Appl. Phys. 52, 335003 (2019). [9] K. Begari and A. Haldar, J. Phys. D: Appl. Phys. 53, 455005 (2020). [10] A. Haldar and A.O. Adeyeye, Appl. Phys. Lett. 116, 162403 (2020). [11] A. Haldar and A. O. Adeyeye, J Appl. Phys. 128, 240902 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d84bb5df54b8e122952d84e6e7237690.jpg)
Figure 1: (a) Reprogrammable remanent states (X- and Y-state) of a RNM network. (b) Stray field distributions showing difference in the effective field for the two remanent states. (c) Measurement scheme from a single RNM network using micro-BLS technique. (d) Reconfigurable microwave responses for the two different remanent states.

![](https://assets.underline.io/uploads/markdown_image/1/image/408cd838fb91485f6c26689a83c74bc1.jpg)
Figure 2: (a) Propagating spin wave measurement scheme from a RNM chain using micro-BLS technique. (b) Logic device based on dipolar coupled RNM chain is shown in the SEM image. MFM images show the two different magnetic states (FO and FO*). Logic operation is demonstrated by using 2D spatial maps of the spin wave propagation for the FO and FO* states.

 Reconfigurable magnonics using self-biased reprogrammable nanomagnetic structures INVITED

Magnonics where magnons, the associated quasiparticles of spin waves are used to perform advanced logical or analogue functions represents a promising paradigm for new means of information processing for beyond complementary metal oxide semiconductor (CMOS) based architectures. Furthermore, in view of the emerging nano-magnonics which can eventually lead to a large-scale integration of magnonic devices new approaches for a compact, reconfigurable operation of magnonic device are required. Although such a reconfigurability is key to any efficient application such as magnonic filters, it has hitherto only theoretically [1] but not experimentally been shown on a sub-micron scale. On the other hand, multiferroics inherently offer compact means to implement voltage-control and reconfigurability in such microscopic magnonic systems [2]. In this work, we combine multiferroics with ferromagnets by employing a fully epitaxial heterostructure to achieve such voltage-controlled and reconfigurable magnonic system. Frequency filtering is most often based on magnonic crystals (MC) where a periodic modulation of the spin wave’s energy landscape results in frequency bandgaps in the magnon spectrum [3,4]. Here, we employ a thin-film multiferroic bismuth ferrite BiFeO3 (BFO)- lanthanum strontium manganite La2/3Sr1/3MnO3 (LSMO) heterostructure with a low Gilbert damping ~6x10-3 for the LSMO layer. Instead of modulating the sample shape, we create a MC by imprinting a periodic remnant electrical polarization in the multiferroic BFO layer which modulates the effective magnetic field in the ferromagnetic layer. We characterize the spin wave propagation spectra in this artificial, voltage-induced magnonic crystal and demonstrate the occurrence of a robust magnonic bandgap with &gt;20 dB rejection, i.e. filtering. The imprinted ferroelectric domains can be reconfigured repeatedly and are long-lived. Our results open a new path for the transduction and configurable filtering of spin waves within a single magnonic device. In general, bridging between between the scientific fields of functional oxides and magnonics, it opens new perspectives for the development of beyond CMOS based technologies [5].

**References:**

[1] Q. Wang , A. V. Chumak, L. Jin et al., Phys. Rev. B. 95, 134433 (2017) [2] N. Spaldin and R. Ramesh, Nat. Mat. 18, 203-212 (2019) [3] P. Frey , A.A. Nikitin, D. Bozhko et al., Commun Phys 3, 17 (2020) [4] A. V. Chumak, A. A. Serga and B. Hillebrands, J. Phys. D: Appl. Phys. 50, 244001 (2017) [5] H. Merbouche, I. Boventer, V. Haspot et al., submitted

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/243806faa5791ce2503db35d16d621e6.jpg)
Schematics of the experimental setup shown for the periodic ferroelectric domain state for the BFO layer.

![](https://assets.underline.io/uploads/markdown_image/1/image/4e0581eb0df6af24b4bcc64b618bde47.jpg)
Evidence for the bandgap for the BFO/LSMO heterostructure with periodic domains in the BFO in the amplitude representation in logarithmic scale. The data is normalized by the mean amplitude. At the bandgap frequency, the amplitude is dropping by >20 dB. A reconfiguration of the periodic domain state to the uniform down state erases the gap and reconstitutes the initial uniform state's response. Pn,m label the different domain states where n is the cycle number (1 cycle : down (virgin)-> up-> periodic-> ) and m the direction of the ferroelectric polarization.

 Voltage-Controlled Magnonic Crystal at the Sub-Micron Scale

In the last decades, the investigation of microwave spin waves (SW) in thin magnetic films has been an attractive field of research due to their very short wavelength reaching down to nanometers at GHz frequencies. SW also have a rich dispersion relation, that depends on their propagation direction with respect to a magnetic field. Hence, SW -based devices are promising candidates for microwave information processing, and eventually for overcoming the limitations encountered with CMOS-devices such as power consumption. In magnonic devices, a key challenge is to achieve long propagation distances of the spin waves. This requires to work with magnetic materials with small magnetic damping such as Yttrium Iron Garnet (YIG). (YIG), has by far the lowest magnetic damping which allows the spin waves propagation to be spread over millimeter-scale distances making it a reference material for spin-wave dynamics studies. Kajiwara and al. [1] confirmed that an electric current injection in a YIG film can be converted into spin-waves. In fact, when placing a non-magnetic metal with a high spin orbit coupling such as platinum on top of a magnetic insulator such as YIG, flow of charge current in the metal leads to its conversion into a pure spin current (via the Spin Hall Effect), which is then injected in the magnetic insulator where it adds up as an additional torque on the YIG’s magnetization. The above-mentioned mechanism of STT generation originating from pure spin-current obtained through the spin-orbit interaction of a heavy metal is called spin orbit torque (SOT). Recent studies [2], [3] with Platinum stripes placed on top of a 20 nm thick YIG waveguide (Pt/YIG) showed that SOT could generate a full compensation of the damping, leading to auto-oscillations of the magnetization above a critical injected current density. When excited with a microwave field, spin waves propagation length was increased by a factor of 10 in the bilayer. However, the possibility to achieve an amplification of propagating spin-waves was not observed yet, due to the onset of nonlinear dissipative processes above the critical current. More recently, materials having perpendicular magnetic anisotropy have demonstrated a full damping compensation along with SW emission, such behavior is accomplished by substituting Bismuth in YIG [4]. Here, we report on a study of SW propagation driven by SOT in a Pt/Bi-YIG waveguide where SW are detected using micro-focused Brillouin Light Scattering spectroscopy (BLS). The 20 nm thick and 21 µm long Bi-YIG waveguide is covered with 7 nm of Platinum. The experiments show the lossless propagation of a pulse of spin-waves excited by an rf-field generated by an antenna placed on top of the waveguide, under zero or positive damping during the course of 200 ns. Here, we show the full scale micromagnetic simulations done to emulate this experiment. The result shows a verification of the lossless propagation of SWs as seen in the experiment. By including a realistic Gilbert damping parameter and the Slonczewski torque to the Bi-YIG structure to compensate the magnetic losses, the results reproduce not only the steady state dynamics but also transitional dynamics when the rf-excitation is turned on and off at the ns time-scale. In summary, the inclusion of STT in the micromagnetic simulations showed an accordance with the experimental results, which verifies the ability to obtain a lossless propagation of SWs in the Bi-YIG waveguide and their non-suppression even after a hundred of nanoseconds after cutting off the excitation field.

**References:**

[1] Y. Kajiwara et al., “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature, vol. 464, no. 7286, Art. no. 7286, Mar. 2010 [2] M. Collet et al., “Generation of coherent spin-wave modes in yttrium iron garnet microdiscs by spin–orbit torque,” Nat. Commun., vol. 7, no. 1, Art. no. 1, Jan. 2016 [3] M. Evelt et al., “High-efficiency control of spin-wave propagation in ultra-thin yttrium iron garnet by the spin-orbit torque,” Appl. Phys. Lett., vol. 108, no. 17, p. 172406, Apr. 2016 [4] L. Soumah et al., “Ultra-low damping insulating magnetic thin films get perpendicular,” Nat. Commun., vol. 9, no. 1, Art. no. 1, Aug. 2018

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2a26b6d8f9040dc1bef07c68aad030d7.jpg)
Fig.1: Two dimensional colormap showing the spin waves intensity in the simulated Bi-YIG waveguide after 80 ns from turning the rf-excitation on.

![](https://assets.underline.io/uploads/markdown_image/1/image/ea7d4882c8bc05ce1b677bad5a334f51.jpg)
Fig.2: Two dimensional colormap showing a lossless propagation of the SW intensity under SOT amplification, 100 ns after emission. The extracted group velocity is 55 m/s. The intensity of z-component of the magnetization is identical to that of Fig.1 recorded at a timestamp of 80 ns.

 Micromagnetic Simulations of Spin Waves propagation by SOT in a Bi-YIG waveguide

Conversion of heat into spin current has recently drawn a lot of attention due to its potential usage on heat recovery via the spin Seebeck effect (SSE) [1] as well as fundamental interests including generation of spin super currents [2] and Bose-Einstein condensation of magnons [3]. The underlying mechansim for SSE is understood as a combination of interfacial and bulk (intrinsic) contriubtion. The interfacial SSE is driven by the tempearture drop at the interface between metal and magnetic layer while the bulk SSE is driven by diffusive thermal magnons. In the nonlocal spin transport measurements, the long range exponential decay of thermally generated spin signal was explained solely by the intrinsic SSE [4] while there are several studies showing that there is another decay mechanism in subhundred nanometer scale [5-7]. Here, we study the spatial decay of thermally excited magnon currents inside a thin magnetic insulator by focusing on the short-range behavior in the nonlocal geometry. We compare SSE signal (Δ1,2) on the same device before and after adding a nonmagnetic Al capping layer. The Al capping alters the thermal profile significantly near the heat source, which results in that the nonequilibrium thermal magnon profile deviates from an exponential decay and shows two sign reversals within 1 μm from the heat source. From simulated temperature profiles, we find the vertical temperature gradient also reverses twice within the same length scale in the Al deposited case. The correlation suggests that thermal magnons behave locally at this length scale. Using a phenomenological spin transport model, we calculate the spatial profile of nonequilibrium spin density. The observed short-range behavior can be accounted if the relavant decay length is on the order of subhundred nanometers. Our result shows the existance of the shorter decay length in the nonlocal magnon transport experiments. This suggests that the long-range SSE signal in previous measurements should be attributed to magnons with lower energy.

**References:**

[1] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy & Environmental Science, 7, 885 (2014). [2] D. A. Bozhko, A. A. Serga, P. Clausen, et. al., Nature Physics, 12, 1057 (2016). [3] Y. Tserkovnyak,S. A. Bender, R. A. Duine, et. al., Physical Review B, 93, 100402. (2016). [4] L. J. Cornelissen, J. Liu, R. A. Duine, et. al., Nature Physics, 11, 1022 (2015). [5] A. Kehlberger, U. Ritzmann, D. Hinzke, et. al., Physical Review Letters, 115, 096602. (2015). [6] A. Prakash, B. Flebus, J. Brangham, et. al., Physical Review B, 97, 020408. (2018). [7] J. S. Jamison, Z. Yang, B. L. Giles, et. al., Physical Review B, 100, 134402. (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/71fcbac32cc3cda92f6d18b1228e05ae.jpg)
Fig.1: Comparison of the experimentally measured local and non-local SSE signals (Δ1,2) at 1 and 2 mA injection and the numerically calculated chemical potential profiles with different input values of decay length λ for YIG/Pt/Si3N4 (case A) on a symmetric log scale. The insets show the schematic of the sample structure (bottom left) and temperature profile (top right).

![](https://assets.underline.io/uploads/markdown_image/1/image/898460b298a266d784aeaeab1dbbca16.jpg)
Fig.2: Same plots as Fig.1 for YIG/Pt/Si3N4/Al (case B). The result with λ = 0.1 µm reproduces the measured double crossing as shown, revealing the short range thermal magnon diffusion. The second crossing does not appear anymore when increasing λ to 0.5 or 1 µm. The insets show the schematic of the sample structure (bottom left) and temperature profile (top right).

 Short-range Thermal Magnon Diffusion in Magnetic Garnet

Spin waves are collective excitations of a spin system, where the magnetic moments precess with a spatially periodic phase shift that determines the wavelength [1]. From a technological point of view, the use of spin waves as signal carriers in future spintronic logic and memory devices has the advantages of a potentially lower power consumption and improved miniaturization prospects compared to the present charge-based CMOS technology [2]. For achieving nanoscale integration, topological spin textures such as magnetic vortex cores can be driven by alternating magnetic fields to excite coherent spin waves with short, sub-micrometer wavelengths, as shown recently for both synthetic ferrimagnetic systems and single ferromagnetic layers [3, 4]. In this contribution, we will demonstrate the versatility of this excitation mechanism by showing that the application of a static magnetic bias field makes the vortex core a manipulable spin-wave source that can be continuously translated in space without altering the resulting spin-wave dispersion relation [5]. We employ time-resolved scanning transmission x-ray microscopy (TR-STXM) to directly image vortex-core spin textures as well as the emission and propagation of spin waves from them. By applying a static in-plane magnetic bias field, we can continuously displace the vortex core from its equilibrium position in the center of a structured element. For a synthetic ferrimagnet (Co/Ru/Ni81Fe19), it is possible to shift the core relatively close to the edge while maintaining a radial, isotropic spin-wave emission pattern (see Figure 1). The wavelengths of the excited spin waves are not noticeably changed by the applied magnetic bias field, but remain tunable by the driving frequency in the same way as without bias field. For a single Ni81Fe19 layer with a thickness of the order of 100 nm, we observe a similar behavior for relatively small magnetic bias fields but when the vortex core approaches the edge of the disk, it also expands (see Figure 2). This expansion can be seen as a transformation from a zero-dimensional point source to a one-dimensional curved object where its shape and extent is controlled by the magnitude of the applied magnetic bias field. Such an expansion leads to a directional emission of the spin waves that propagate away from the core in mainly two opposite directions rather than radially as observed for non-expanded cores. For certain combinations of magnetic bias fields and driving frequencies, focusing effects appear. Such effects provide promising means for a controlled and directional propagation of spin waves without the need for additional patterning or waveguides.

**References:**

[1] A.G. Gurevich and G.A. Melkov, Magnetization Oscillations and Waves. Vol. 1, CRC Press (1996) [2] A.V. Chumak et al., Nat. Phys. Vol.11, p.453-461 (2015) [3] S. Wintz et al. Nat. Nanotechnol. Vol.11, p.948-953 (2016) [4] G. Dieterle et al., Phys. Rev. Lett. Vol.122, p.117202 (2019) [5] S. Mayr et al., Nano Lett. accepted (2021)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1ab356e578b624eb1b59b9aeeb5b4f0a.jpg)
Fig. 1: TR-STXM snapshots of a synthetic ferrimagnet, taken with out-of-plane magnetic sensitivity at the Fe L3 edge at a magnetic bias field of 5 mT. (a) Absolute absorption image showing the black vortex core shifted from the center of the structure (green circle). (b) Normalized contrast displaying the radial spin-wave emission from the core for a 1 GHz alternating field excitation.

![](https://assets.underline.io/uploads/markdown_image/1/image/2d254de94235b1183f441f680c4fed5a.jpg)
Fig. 2: TR-STXM images with out-of-plane sensitivity displaying the response of the vortex in a Ni81Fe19 layer to a magnetic bias field of 22.5 mT. (a) Absolute absorption image showing the expansion of the vortex core (encircled in green). (b) Normalized image displaying only the temporal changes with respect to the average magnetization, showing the spin waves emitted from the expanded vortex core in two directions for an excitation frequency of 8.6 GHz.

 Spin-wave emission from vortex cores under static magnetic bias fields

Synthetic antiferromagnets (SyAFs) are an excellent platform to explore novel spintronic and magnonic concepts with coupled magnetic moments[1,2]. Unlike a homogeneously magnetised single-layer ferromagnet, coupled magnetic layers are able to offer rich magnetic states where the competition between interlayer exchange coupling, external-field-induced Zeeman interaction as well as other magnetic anisotropy terms plays a role. This can expand into their dynamic regimes as the coupled moment nature inherently provides two eigenmodes, acoustic and optical modes[3-7], where time-dependent components of two coupled moments are oscillating in-phase (acoustic) and out-of-phase (optical)in a spin-flop (canted) regime. When we excite spin-waves by oscillating magnetic fields, the spatial symmetry/profile of microwave excitation determines which spin-wave modes are excited. In the simplest case, a uniform distribution of microwave excitation fields across a magnet can excite the uniform spin-wave mode (wavevector k = 0) as well as higher-order spin-wave resonance modes (k &gt; 0) with odd index numbers since the spatial profile of the microwave excitation and spin-wave mode amplitude (with phase) matches to each other in terms of symmetry. This in turn suggests that it should be possible to control the selection rules of spin-wave excitations by designing the spatial profile of excitation fields. Accessing hidden spin-wave states as well as tailoring spin-wave excitation efficiency by this approach has not been much explored in the past. We report in this work the current-induced-torque excitation of acoustic and optical modes in Ta/NiFe/Ru/NiFe/Ta synthetic antiferromagnet stacks grown on SiO2/Si substrates(Fig 1(a)). The two Ta layers serve as spin torque sources with the opposite polarisations both in spin currents and Oersted fields acting on their adjacent NiFe layers. This can create the odd symmetry of spatial spin torque distribution across the growth direction, allowing us to observe different spinwave excitation efficiency from synthetic antiferromagnets excited by homogeneous torques. We analyse the torque symmetry by in-plane angular dependence of symmetric and anti-symmetric lineshape amplitudes for their resonance (as shown in Fig 2(a-b)) and confirm that the parallel (perpendicular) pumping nature for the acoustic (optical) modes in our devices, which is in stark difference from the modes excited by spatially homogeneous torques. The terms parallel (perpendicular) pumping here signify that the torque (hence spin-wave excitation efficiency) is maximised when the oscillating fields and dc magnetic field are collinear(perpendicular) to each other respectively. This is evident from Fig 2(c-d) that strongly suggests the torque symmetry for the acoustic(optical) mode excitation being of form sin Φ(cos Φ) . We also generalize macrospin model for this particular spin-torque excitation geometry, which excellently supports our experimental observation. Our results offer capability of controlling spin-wave excitations by local spin-torque sources and we can explore further spin-wave control schemes based on this concept. We envisage that the control of spin-wave excitations in STT nanodevices will be useful for future spintronic and magnonic nano-devices.

**References:**

[1] R. A. Duine, K.-J. Lee, S. S. P. Parkin, and M. D. Stiles, Nat. Phys. 14, 217 (2018). [2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453461 (2015). [3] F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952). [4] J. J. Krebs, P. Lubitz, A. Chaiken, and G. A. Prinz, J. Appl. Phys. 67, 5920 (1990). [5] S. M. Rezende, A. Azevedo, and R. L. Rodrguez-Surez, J. Appl. Phys. 126, 151101 (2019). [6] A. Sud, Y. Koike, S. Iihama, C. Zollitsch, S. Mizukami, H. Kurebayashi, arXiv:2012.12999 . [7] A. Sud, C. W. Zollitsch, A. Kamimaki, T. Dion, S. Khan, S. Iihama, S. Mizukami, and H. Kurebayashi, Phys. Rev. B 102, 100403(R) (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b406e148e73969fbd07f012bcc683fa7.jpg)
FIG. 1: (a) A schematic of the sample geometry used for STT-FMR measurements in our study. (b) V obtained at different frequencies for acoustic and optical modes in our device for Φ = 55ο (the values for some frequencies have been scaled to show them properly). (c) A 2D colorplot of V as a function of applied field and frequency, measured for Φ = 55ο . (d) Numerical results calculated for the experimental conditions as in (c).

![](https://assets.underline.io/uploads/markdown_image/1/image/aaeaf82e3f75f0759e29b390a7923bff.jpg)
FIG. 2: (a-b) Angular dependence of symmetric (Vsym) and antisymmetric (Vasy) components of voltage for (a) acoustic mode measured at 8 GHz and (b) optical mode measured at 8.5 GHz. The solid curves are obtained using theoretical model. (c-d) The symmetry of torques obtained by dividing the Voltage by (c) sin 2Φ for acoustic mode and (d) cos 2Φ for optical mode. The dominant sin Φ (cos Φ) dependence of torques confirms the parallel(perpendicular) pumping configuration.

 Parity-controlled spin-wave excitations in synthetic antiferromagnets

Spin wave (SW) based computing i.e. magnonics relies on propagating SW as information carriers. Being quasiparticles, magnons (the SW quanta) have a fine lifetime characterizing the exponential decay of the magnons population. Consequently, up to now, all magnonic devices operate within a short time-window and on short propagation distances. Finding a SW amplification paradigm is hence a prerequisite for the development of magnonics as a credible CMOS alternative. Two schemes are currently envisioned for SW amplification: Parametric pumping1 and spin-orbit-torque amplification (SOT)2. Here we will discuss SOT amplification in BiYIG/Pt bilayer3 (Fig.1). Using micro Brillouin light scattering spectroscopy (µ-BLS) on a 500 nm wide Pt/BiYIG waveguides, we take advantage of the strong BLS signal of BiYIG (nearly two orders of magnitude the one of YIG3) to extensively study the influence of SOT on coherent propagating magnon modes. By passing current in the Pt layer, the spin accumulation resulting from the spin-Hall effect induces a positive torque that compensates the Gilbert damping of BiYIG when the magnetic field orientation is perpendicular to the waveguide axis. At subcritical values, the current has a linear effect that lowers the effective damping, thus significantly increasing the decay length of the propagating magnons, similar to what was observed previously in a YIG/Pt waveguide4. However, by increasing the excitation frequency of these propagating magnons for a constant external field, we observe a gradual but strong decrease of the modulation efficiency of their lifetime in our BiYIG/Pt waveguide. Once the current in the Pt layer reaches a threshold value, the vanishing effective damping allows for the onset of the auto-oscillation regime5 where the thermal magnons get incoherently amplified and the magnetization auto-oscillates in the main uniform mode. However, coherent propagating magnons strongly interact with the auto-oscillations6, preventing the observation of their propagation at continuous supercritical current. To circumvent these non-linear interactions, we study the auto-oscillation transient regime for the first time by performing time-resolved µ-BLS measurements. When the SOT is switched on, the uniform auto-oscillation grows exponentially, in agreement with the Slavin & Tiberkevitch non-linear auto-oscillator model7 in the linear regime limit. The onset of non-linearities only occurs when the auto-oscillation reaches its stable orbital, after few hundreds of nanoseconds typically. Using a rf and DC pulse scheme, we propagate rf-excited magnons before the onset of non-linearities at supercritical currents and observe the amplification of propagating magnons (Fig. 2-a). Similarly to what was observed in the continuous regime at subcritical currents, the modulation efficiency of the propagating magnons lifetime strongly decreases with frequency (Fig. 2-b). This experiment sheds new light on the influence of SOT on propagating magnons and pave the way toward a magnonic spin-wave amplifier using DC current.

**References:**

1 T. Brächer, P. Pirro, and B. Hillebrands, Phys. Rep. 699, 1 (2017). 2 V.E. Demidov, S. Urazhdin, A. Anane, V. Cros, and S.O. Demokritov, J. Appl. Phys. 127, 170901 (2020). 3 L. Soumah, N. Beaulieu, J. Ben Youssef, and A. Anane, Nat. Commun. 9, 3355 (2018). 4 M. Evelt, V.E. Demidov, and A. Anane, Appl. Phys. Lett. 108, 172406 (2016). 5 M. Collet, V. Cros, A. Anane, G. de Loubens, and O. Klein, Nat. Commun. 7, 10377 (2016). 6 B. Divinskiy, S. Urazhdin, S.O. Demokritov, and V.E. Demidov, Nat. Commun. 10, 5211 (2019). 7 A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7c591b0f58bbf93bc6d6866a53fdf516.jpg)
Fig. 1 : Schematic representation of the 0.5µm-wide BiYIG/Pt waveguide. The DC current is applied in the Pt along the waveguide axis and a 300nm-wide strip rf-antenna is used to excite propagating magnons. The µ-BLS laser spot is focused from the back of the sample.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8aea6e40370d823a1c720f70ddf0448.jpg)
Fig. 2 : Spin-wave intensity as a function of the distance to the antenna. The experimental curves (solid lines) are linearly fitted (dotted lines) in log scale for 4 frequencies (only 2 shown here) 4.925 GHz (a), 4.95 GHz, 4.975 GHz and 5.0 GHz (b) and 4 DC voltages corresponding to one subcritical current (1.7V) and three supercritical currents (2.2, 2.3 and 2.4V). The FMR is constant during the pulse with value 4.885 ± 0.01 GHz. 
The positive slope at supercritical currents at 4.925 GHz indicates an amplification of the propagating spin-waves (negative effective damping) due to SOT. The damping compensation efficiency gradually decreases as the frequency is increased, and the spin-waves are no longer amplified at 5.0 GHz for the applied voltages.

 Insight on the effect of SOT on propagating magnons: Spin-wave amplification and frequency dependent efficiency

Artificial neural networks (NN) running on traditional computing hardware tend to be relatively slow and power-hungry. One possible solution to this issue is to create hardware with neurons that spike in a manner similar to biological neurons. For example, Intel [1] and IBM [2] have recently fabricated neuromorphic chips based on spiking NN. Still, neurons in these chips are fabricated with multiple silicon transistors, which limits power efficiency of such an approach. The development of neuromorphic chips would benefit from a nanoscale, single element synthetic neuron that behaves in a manner that closely resembles a biological neuron. Such a synthetic neuron can be fabricated with nanometer dimensions from an antiferromagnetic (AFM) material mounted on platinum [3]. These “AFM neurons” are spin Hall oscillators driven by a sub-threshold spin current that can produce ultra-short (~ 5 ps) spikes in response to a weak external stimulus. In traditional spiking NN, synapses that connect artificial neurons are formed by complex active circuits. A much simpler approach would be to use passive memristors as artificial synapses [4]. However, memristors create symmetric coupling between neurons, which allows a spike produced by one neuron to travel to all neighboring neurons, both forwards and backwards. Unfortunately, this bi-directional signal propagation makes use of standard machine learning algorithms impossible. Therefore, it would be interesting to design NN in which unidirectional spike propagation is possible even with symmetrically coupled neurons. Here, we demonstrate that the unidirectional signal propagation is realizable in symmetrically-coupled NNs based on AFM neurons. This regime is possible due to an effective inertia that originates from exchange coupling between the two magnetic sublattices [3] and creates a relatively long refractory period - an interval of time after a neuron fires when it is unable to fire again. We show, that it is possible to adjust synapse weights such that each neuron is still in its refractory period when it receives a backward propagating signal from its neighbors. Then, a simple chain of neurons receiving an input at one end carry the signal to the opposite end without any reflections. However, for such a simple chain of AFM neurons, an input can occur at either end, and thus spikes would be able to travel in both directions. It would be beneficial to create an ‘isolator’ neuromorphic circuit, which would allow the propagation of spikes in only one direction. Figure 1 depicts the schematic of a possible ‘isolator’ circuit composed of AFM neurons. This figure uses circles to represent AFM neurons. The isolator is composed of neurons 4, 5, and 6. In the figure, double arrows represent symmetrical synapses with arrow width proportional to the coupling strength. This isolator allows spikes to be carried “forwards” from neuron 1 to 9, but stops signals carried “backwards” from neuron 9 to 1, as illustrated by the results of numerical simulations shown in Fig. 2. The isolator functions as follows. When the spike is initiated in neuron 1 (Fig. 2(a)), it travels along the chain to neuron 4. Then, the spike generated by neuron 4 travels to both neuron 5 and neuron 6. Neuron 5 also generates a spike, which is carried to neuron 6. The synaptic weights connecting to neuron 6 are small enough that a single input is not sufficient to induce a spike in the AFM neuron. However, when both spikes combine at neuron 6, they provide enough stimulus for neuron 6 to fire. The spike can then continue to propagate from neuron 6 to 9. When traveling backwards (Fig. 2(b)), a spike begins at neuron 9 and travels unidirectionally to neuron 6. Neuron 6 will generate a spike, but the synaptic weights from neuron 6 to 4/5 are insufficient to induce an additional spike. This, in effect, prevents the backward travel of spikes through the isolator. Using the proposed isolator NN, it is possible to design a number of other neuromorphic circuits supporting unidirectional information propagation, for example, neuromorphic combiners and unidirectional memory rings. The creation of such circuits with symmetric synapses shows hope for AFM neurons to be used to create more realistic spiking NN. The delay caused by the exchange inertia in the AFM neurons plays a key role in any neuromorphic circuit constructed with these neurons. These physically realistic neurons will increase the possibility for more functional spiking neural network hardware.

**References:**

[1] M. Davies et al., Loihi: A Neuromorphic Manycore Processor with On-Chip Learning, in IEEE Micro, vol. 38, no. 1, pp. 82-99, January/February 2018, doi: 10.1109/MM.2018.112130359. [2] M. V. DeBole et al., TrueNorth: Accelerating From Zero to 64 Million Neurons in 10 Years, in Computer, vol. 52, no. 5, pp. 20-29, May 2019, doi: 10.1109/MC.2019.2903009. [3] R. Khymyn, I. Lisenkov, J. Voorheis, O. Sulymenko, O. Prokopenko, V. Tiberkevich, J. Akerman, and A. Slavin, Ultra-fast artificial neuron: generation of picosecond-duration spikes in a current-driven antiferromagnetic auto-oscillator, Scientific Reports 8, 15727 (2018). [4] Sung Hyun Jo, Ting Chang, Idongesit Ebong, Bhavitavya B. Bhadviya, Pinaki Mazumder, and Wei Lu Nanoscale Memristor Device as Synapse in Neuromorphic System, Nano Lett. 2010, 10, 4, 1297–1301

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/eb70901938b9e7ce0e61457aefc96cde.jpg)
Fig 1: Chain of spiking neurons, configured as an isolator circuit. Gray circles represent neurons, and black arrows represent coupling between neurons. Thick black arrows represent symmetric synaptic weight tuned such that a spike cannot flow backwards. Thin black double arrows represent symmetric synaptic weight weak coupling.

![](https://assets.underline.io/uploads/markdown_image/1/image/7460602315b4c8ef8afb70fa1a01a721.jpg)
 
Fig 2: Results of simulation of AFM neurons configured as an isolator circuit. (a) Initial spike is in neuron 1. The spike travels through the chain, to neuron 9. (b) The initial spike is at neuron 9. The spike travels to neuron 6, then is unable to continue.

 Antiferromagnetic Artificial Neural Networks with Symmetric Coupling

The mutual synchronization of spin-torque nano-oscillators (STNOs) arranged in a large array improves the individual properties of each STNO, increasing the output-power and reducing the linewidth of the emitted signal. These improvements are required to implement STNOs in wireless communication schemes [1] and furthermore open applications for different oscillator based neuromorphic computing schemes [2]. Many theoretical and experimental studies considered single STNO devices [3], while experimentally the synchronization of a large number of STNOs remains a challenging task [4]. This is because the synchronized state and the phase-locking depend on the geometry, device resistance, input signals, coupling mechanism, among others [5]. As a first step to guide experiments on the synchronization behavior of STNO arrays, we have undertaken a theoretical study, combining numerical and analytical approaches based on the spin-wave formalism [3]. The STNO devices considered here are of circular cross-section (100nm in diameter) with a perpendicular polarizer and an in-plane magnetized free layer under a strong out-of-plane field. This field is larger than the saturation field. The free layer magnetization is orientated out-of-plane leading to an out-of-plane precession mode that is advantageous to induce strong dipolar coupling between STNOs, which is the coupling mechanism considered here. We derived analytically the coupled equations of amplitudes and phases where the latter are similar to the Kuramoto-phase equation, but limited to a finite number of devices. Solving for the stationary states, this model provides analytical expressions for the locking-range, the stationary amplitudes, and the frequency of oscillations of the synchronized state. These solutions depend on the array geometry, number N of STNOs in the array, separation between STNOs, distribution of the DC current, as well as the different magnetic material parameters, and the external magnetic field. In this work, we investigate the solution of these equations for two specific array geometries: a straight line and a ring structure (N=3,6 and 10), shown in figure 1b. Our results show that the locking-range depends on the separation among the STNOs (because this scales the coupling strength), but it is independent of the angle among them. Furthermore, we show that the locking-range is larger when the STNOs are placed in a ring structure (or triangle for N=3) as compared to a straight-line structure. Finally, the most interesting result is that for the ring structure, we identify two dynamic modes that represent a standing wave for the STNO phases along the ring structure (observed for N&gt;5). These two dynamics modes are shown in Fig. 2 and are called here (a) the in-phase mode, with identical phases for all STNOs, and (b) the out-of-phase mode, where the phase difference between two adjacent STNOs is 2π/N. We study how these states and their existence range are modified upon variation of the current density through the different STNOs (identical and non-identical currents), the STNO separation, and the number of STNOs in the ring. The analytical results are in good agreement to the numerical solutions of the Landau-Lifshitz-Gilbert (LLG) equation, as long as the separation among the STNOs is above a certain critical value. For smaller separations, the dipolar interactions become too strong and non-linear effects would require to go beyond the first-order approximations used to solve the amplitude and phase equations. The results reported here represent a first step to validate the phase equations and to define their limitations, and can be used to address now the Kuramoto equations to find solutions for large scale structures (N very large). We acknowledge financial support in Chile from FONDECYT 1200867, and Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia AFB 180001. D. M.-A. acknowledges Postdoctorado FONDECYT 2018, 3180416. M. A. Castro acknowledges Conicyt-PCHA/Doctorado Nacional/2017-21171016. The project was supported in part by ERC Advanced Grant MAGICAL No. 669204

**References:**

[1] T. Chen, et al., Proceedings of the IEEE., 104, 1919-1945(2016). [2] J. Grollier, et al., Nat. Electron. 3, 360 (2020). [3] A. Slavin and V., Tiberkevich, IEEE Trans. Magn., 45,1875(2009). [4] S. Tsunegi, et al., Scientific Reports., 8, 13475(2018). [5] R. Lebrun, et al., Nat. Commun. 8, 15825 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/67e3f596fc435dd303beec991462b132.jpg)
Fig.1: (a) Schematics of an STNO device and (b) two configurations for a coupled STNO array.

![](https://assets.underline.io/uploads/markdown_image/1/image/fdc26dc3ca23f437e028887120de5d0d.jpg)
Fig. 2: Schematics of the two dynamic modes obtained for the ring structure. The snapshots show the instantaneous distribution of the x- and y- components of the magnetization in three consecutive time steps t1 < t2 < t3, for (a) the in-phase and (b) the out-of-phase modes.

 Mutual synchronization of an array of spin-torque oscillators with perpendicular polarizer

Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI), which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7-nm-thick YIG film by measuring the nonreciprocal spin-wave propagation in terms of frequency, amplitude, and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e., the film normal direction, applied field, and spin-wave wave vector. By measuring the asymmetric group velocities, we extract a DMI constant of 16 μJ/m2, which we independently confirm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG films is of key importance for functional chiral magnonics as ultralow spin-wave damping can be achieved.

**References:**

[1]. H. C. Wang, J. L. Chen, T. Liu, J. Y. Zhang, K. Baumgaertl, C. Y. Guo, Y. H. Li, C. P. Liu, P. Che, S. Tu, S. Liu, P. Gao, X. F. Han, D. P. Yu, M. Z. Wu, D. Grundler, and H. M. Yu*, Chiral Spin-Wave Velocities Induced by All-Garnet Interfacial Dzyaloshinskii-Moriya Interaction in Ultrathin Yttrium Iron Garnet Films, Phys. Rev. Lett. 124, 027203 (2020).

**Images:**

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