France



A promising means for long-distance transport of angular momentum is spin superfluidity [1-4]. In this work, the unsteady nonlinear dynamics of spin transport in an easy-plane, thin-film, one-dimensional ferromagnetic channel is studied analytically. We formulate an initial-boundary value problem (IVBP) for the Landau-Lifshitz equation subject to spin injection into a static, uniformly magnetized nanowire. The long-time steady-state configuration in such channels with zero applied field has been identified as either a dissipative exchange flow (DEF) or a contact soliton DEF (CS-DEF) [5], depending on the strength of the injection. Here, we study the injection-induced highly nonlinear, transient magnetization dynamics that occur prior to the damping-dominated steady-state is reached and include a constant applied field. In particular, we focus on the 1 ns timescale where damping is weak, and the spin dynamics are dominated by spin injection, applied field, exchange interaction, and demagnetizing fields. In our analysis, we recast the Landau-Lifshitz equation into its dispersive hydrodynamic (DH) form [6], in terms of the longitudinal spin density and a magnetic analog of the fluid velocity that is proportional to the spin current. The spin injection is characterized by a nonzero fluid velocity at one end of the channel, hence can be viewed as a spin hydrodynamic “piston”. The DH form captures two important effects in the ferromagnetic system: nonlinearity and dispersion, which are inherited from anisotropy and exchange, respectively. These effects lead to the generation of unsteady nonlinear waves when large gradients occur. A rapid rise in the fluid velocity due to spin injection leads to large gradients that result in a new kind of spin shock wave that is an example of a dispersive shock wave (DSW). DSWs are expanding, highly oscillatory, nonlinear excitations that realize a coherent transition between two states in a variety of dispersive media [7]. Spin wave envelope DSWs have been observed in a Yttrium Iron Garnet feedback ring [8]. We provide a full analytical classification of nonlinear wave solutions to this IVBP in terms of the applied field and injection strength, using the Whitham modulation theory [9]. The solution components are rarefaction waves (RWs), DSWs, and solitons (see Fig. 1). Both RWs and DSWs have been identified as solutions to a pure initial value problem (the Riemann problem) in a two-component Bose-Einstein condensate [10], whose governing equations are equivalent to the DH form of the Landau-Lifshitz equation we study here. In contrast, we show that the injection boundary plays a prominent role in the dynamics. Depending on the interplay between the spin injection intensity and the applied magnetic field magnitude, different types and combinations of RWs, DSWs, and solitons arise. The generation of a soliton is a distinct feature due to the boundary and is identified with an important concept [10]: the magnetic sonic condition. Solitons only occur at the spin injection end and only in the supersonic injection regime. Our analytical solutions are confirmed by micromagnetic simulations.

**References:**

[1] E. Sonin, Spin currents and spin superfluidity, Advances in Physics Vol. 59, p.181 (2010).&lt;br/&gt;[2] S. Takei and Y. Tserkovnyak, Superfluid spin transport through easy-plane ferromagnetic insulators, Physical Review Letters Vol. 112, p.227201 (2014).&lt;br/&gt;[3] H. Skarsvåg, C. Holmqvist, and A. Brataas, Spin superfluidity and long-range transport in thin-film ferromagnets, Physical Review Letters Vol. 115, p.237201 (2015).&lt;br/&gt;[4] M. Evers and U. Nowak, Transport properties of spin superfluids: Comparing easy-plane ferromagnets and antiferromagnets, Physical Review B Vol. 101, p.184415 (2020).&lt;br/&gt;[5] E. Iacocca and M. A. Hoefer, Hydrodynamic description of long-distance spin transport through noncollinear magnetization states: Role of dispersion, nonlinearity, and damping, Physical Review B Vol. 99, p.184402 (2019).&lt;br/&gt;[6] B. Halperin and P. Hohenberg, Hydrodynamic theory of spin waves, Physical Review Vol. 188, p.898 (1969).&lt;br/&gt;[7] G. El and M. Hoefer, Dispersive shock waves and modulation theory, Physica D: Nonlinear Phenomena Vol. 333, p.11 (2016).&lt;br/&gt;[8] P. A. P. Janantha et al., Observation of self-cavitating envelope dispersive shock waves in Yttrium Iron Garnet thin films, Physical Review Letters Vol. 119, p.024101 (2017).&lt;br/&gt;[9] G. B. Whitham, Linear and nonlinear waves, Vol. 42 (John Wiley &amp; Sons, 2011).&lt;br/&gt;[10] S. K. Ivanov et al., Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate, Physical Review E Vol. 96, p.062202 (2017).&lt;br/&gt;[11] E. Iacocca, T. Silva, and M. A. Hoefer, Breaking of Galilean Invariance in the Hydrodynamic&lt;br/&gt;Formulation of Ferromagnetic Thin Films, Physical Review Letters Vol. 118, p.017203 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bc60f7f3bb99d10a60fec8c293fde70d.jpg)
Fig. 1: Example simulations of (top to bottom) the spin density &lt;i&gt;n&lt;/i&gt;, the magnetic fluid velocity &lt;i&gt;u&lt;/i&gt;, and an in-plane magnetization component &lt;i&gt;m&lt;sub&gt;x&lt;/sub&gt;&lt;/i&gt;, at &lt;i&gt;t&lt;/i&gt; = 1 ns in (left) the subsonic regime and (right) the supersonic regime.



INTERMAG 2021

 Spin-Injection-Generated Shock Waves and Solitons in a Ferromagnetic Nanowire

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MMM 2020

A promising means for long-distance transport of angular momentum is spin superfluidity [1-4]. In this work, the unsteady nonlinear dynamics of spin transport in an easy-plane, thin-film, one-dimensional ferromagnetic channel is studied analytically. We formulate an initial-boundary value problem (IVBP) for the Landau-Lifshitz equation subject to spin injection into a static, uniformly magnetized nanowire. The long-time steady-state configuration in such channels with zero applied field has been identified as either a dissipative exchange flow (DEF) or a contact soliton DEF (CS-DEF) [5], depending on the strength of the injection. Here, we study the injection-induced highly nonlinear, transient magnetization dynamics that occur prior to the damping-dominated steady-state is reached and include a constant applied field. In particular, we focus on the 1 ns timescale where damping is weak, and the spin dynamics are dominated by spin injection, applied field, exchange interaction, and demagnetizing fields. In our analysis, we recast the Landau-Lifshitz equation into its dispersive hydrodynamic (DH) form [6], in terms of the longitudinal spin density and a magnetic analog of the fluid velocity that is proportional to the spin current. The spin injection is characterized by a nonzero fluid velocity at one end of the channel, hence can be viewed as a spin hydrodynamic “piston”. The DH form captures two important effects in the ferromagnetic system: nonlinearity and dispersion, which are inherited from anisotropy and exchange, respectively. These effects lead to the generation of unsteady nonlinear waves when large gradients occur. A rapid rise in the fluid velocity due to spin injection leads to large gradients that result in a new kind of spin shock wave that is an example of a dispersive shock wave (DSW). DSWs are expanding, highly oscillatory, nonlinear excitations that realize a coherent transition between two states in a variety of dispersive media [7]. Spin wave envelope DSWs have been observed in a Yttrium Iron Garnet feedback ring [8]. We provide a full analytical classification of nonlinear wave solutions to this IVBP in terms of the applied field and injection strength, using the Whitham modulation theory [9]. The solution components are rarefaction waves (RWs), DSWs, and solitons (see Fig. 1). Both RWs and DSWs have been identified as solutions to a pure initial value problem (the Riemann problem) in a two-component Bose-Einstein condensate [10], whose governing equations are equivalent to the DH form of the Landau-Lifshitz equation we study here. In contrast, we show that the injection boundary plays a prominent role in the dynamics. Depending on the interplay between the spin injection intensity and the applied magnetic field magnitude, different types and combinations of RWs, DSWs, and solitons arise. The generation of a soliton is a distinct feature due to the boundary and is identified with an important concept [10]: the magnetic sonic condition. Solitons only occur at the spin injection end and only in the supersonic injection regime. Our analytical solutions are confirmed by micromagnetic simulations.

**References:**

[1] E. Sonin, Spin currents and spin superfluidity, Advances in Physics Vol. 59, p.181 (2010). [2] S. Takei and Y. Tserkovnyak, Superfluid spin transport through easy-plane ferromagnetic insulators, Physical Review Letters Vol. 112, p.227201 (2014). [3] H. Skarsvåg, C. Holmqvist, and A. Brataas, Spin superfluidity and long-range transport in thin-film ferromagnets, Physical Review Letters Vol. 115, p.237201 (2015). [4] M. Evers and U. Nowak, Transport properties of spin superfluids: Comparing easy-plane ferromagnets and antiferromagnets, Physical Review B Vol. 101, p.184415 (2020). [5] E. Iacocca and M. A. Hoefer, Hydrodynamic description of long-distance spin transport through noncollinear magnetization states: Role of dispersion, nonlinearity, and damping, Physical Review B Vol. 99, p.184402 (2019). [6] B. Halperin and P. Hohenberg, Hydrodynamic theory of spin waves, Physical Review Vol. 188, p.898 (1969). [7] G. El and M. Hoefer, Dispersive shock waves and modulation theory, Physica D: Nonlinear Phenomena Vol. 333, p.11 (2016). [8] P. A. P. Janantha et al., Observation of self-cavitating envelope dispersive shock waves in Yttrium Iron Garnet thin films, Physical Review Letters Vol. 119, p.024101 (2017). [9] G. B. Whitham, Linear and nonlinear waves, Vol. 42 (John Wiley & Sons, 2011). [10] S. K. Ivanov et al., Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate, Physical Review E Vol. 96, p.062202 (2017). [11] E. Iacocca, T. Silva, and M. A. Hoefer, Breaking of Galilean Invariance in the Hydrodynamic Formulation of Ferromagnetic Thin Films, Physical Review Letters Vol. 118, p.017203 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bc60f7f3bb99d10a60fec8c293fde70d.jpg)
Fig. 1: Example simulations of (top to bottom) the spin density n, the magnetic fluid velocity u, and an in-plane magnetization component mx, at t = 1 ns in (left) the subsonic regime and (right) the supersonic regime.

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Introduction: Spin current driven by charge current is one of the key technologies for spin-based memory device. Among them, spin current generated transverse to the charge current has been attracting a great deal of attentions. This is because this spin current generates spin-orbit torques (SOTs) to the adjacent ferromagnet, which could be a promising writing method with high endurance and fast operation. The underlying physics of this charge-to-spin conversion is a spin-orbit coupling related phenomena such as spin Hall effect in non-magnetic materials [3]. Recently, it has been revealed that not only non-magnetic materials but also ferromagnetic materials (FMs) can generate transverse spin current [4-8]. Moreover, FM has a great potential as a spin source for SOT because of the breaking of time reversal symmetry due to the magnetization. This leads to an additional magnetization-dependent transverse spin current generation which can be used as field-free SOT switching [5]. Although the existence of the charge-to-spin conversion phenomena has been mainly studied on in-plane magnetized ferromagnets [5,6], there are few reports on the perpendicularly magnetized case [7]. Here, we focused on the perpendicularly magnetized ferromagnet as a spin current source and quantitatively investigated the charge-to-spin conversion phenomena [8]. Experimental Results: We prepared the following tri-layer base structure; buffer-layer / PML / Cu (3.0 nm) / Fe75B25 (1.3 nm)/ MgO (see Fig. 1a). In this structure, in-plane magnetized highly resistive Fe75B25 layer acts as a detecting layer, and perpendicularly magnetized layer (PML) acts as a spin source material. The PML used here is a Co/Ni multilayer, which is a typical structure for obtaining strong perpendicular magnetic anisotropy. Charge-to-spin conversion in PML is quantitatively investigated by measuring the spin-torque ferromagnetic resonance of the Fe75B25 layer with DC bias current application [8]. We found that the resonance linewidth W is linearly modulated by DC current injection indicating spin injection from PML (Fig. 1b). The efficiency of W modulation significantly changes with the magnetization reversal of PML (+MPML and -MPML state respectively), indicating the presence of magnetization-dependent charge-to-spin conversion. The spin current generated from PML is further investigated by angular dependence (Fig. 2a). From the angular dependence, we revealed that two types of spin current with different spin polarization are generated from PML as shown in Fig. 2b, and they are polarized along je×z and m×(je×z) direction respectively (m stands for magnetization of PML) [8]. Furthermore, we investigated the reciprocal process of the charge-to-spin conversion, i.e., spin-to-charge conversion, using a spin-pumping method to inject pure spin current to PML [9]. As a result, we revealed that the spin pumping-induced electromotive force is well explained by the inverse process of charge-to-spin conversion in PML. Conclusion: We reveal that two charge-to-spin conversion mechanisms with different spin polarization coexist in single ferromagnetic material. In addition to this, we observed an inverse process of the charge-to-spin conversion via spin pumping method. Our findings contribute to the deeper understanding of the spin current generation in ferromagnetic materials and will promote the development of SOT-based applications. Acknowledgement: This work was partly supported by JSPS KAKENHI (Grant No. 19J01643), JST CREST (Grant No. JPMJCR18T3), Japan. Part of this work was conducted at the AIST Nano-Processing Facility, supported by the Nanotechnology Platform Program of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

**References:**

[1] A. Manchon et.al, Rev. Mod. Phys., Vol. 91, 035004 (2019) [2] L. Liu et.al, Science, Vol. 336, 555 (2012) [3] J. Sinova, Rev. Mod. Phys., Vol. 87, 1213 (2015) [4] A. Davidson et.al, Phys. Lett. A, Vol. 384, 126228 (2020) [5] S. Iihama et.al, Nat. Electro., Vol. 1, 120 (2018) [6] S. C. Baek et.al, Nat. Mater., Vol. 17, 509 (2018) [7] A. Humphries et.al, Nat. Comm., Vol. 8, 911 (2017) [8] Y. Hibino et.al, Phys. Rev. B, Vol. 101, 174441 (2020) [9] K. Ando et.al, J. Appl. Phys., Vol. 109, 103913 (2011).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9b680dc7b89b6216d97f73dceb88b6e2.jpg)
Fig. 1: (a) Schematic of tri-layer based structure (left) and measurement setup of the spin-torque ferromagnetic resonance (right). (b) Resonance linewidth W as a function of applied DC bias current IDC under +MPML (left) and -MPML (right) respectively. The inset schematic shows the magnetization configuration of each measurement.

![](https://assets.underline.io/uploads/markdown_image/1/image/dc47615b64008028cec24209cf049b74.jpg)
Fig. 2: (a) Angular dependence of efficiency of W modulation via IDC. Upper (downer) triangle shows the measurement under +MPML (-MPML) state, and solid curve shows the best fitting curve. (b) Schematic of two components of spin current flowing towards out-of-plane direction (+z direction) under +MPML and -MPML state (left and right respectively) obtained from the fitting.

 Charge-to-spin Conversion in Perpendicularly Magnetized Ferromagnetic Materials

An efficient electrical switching of thin film metallic nanomagnets has been demonstrated by using ferromagnet/heavy metal thin film heterostructures [1][2]. The spin-Hall effect in the heavy metal layer and/or the Rashba-Edelstein effect at the interface have been used to control the nanomagnets efficiently by electrical means. On the other hand, optically induced switching of nanomagnets might be useful for optically-switchable magnetic memories, promising candidate for future fast and low power consumption information processing system using photonic integrated circuits [3]. An increase of optical spin generation efficiency in the materials is crucial for such photo-spintronics devices. Recently, optical helicity induced torque on magnetization in metallic thin films [Fig. 1], termed as optical spin torque, has been reported. It was demonstrated that the optical spin torque is induced by the inverse Faraday effect in ferromagnetic metals and the optical orientation effect in nonmagnetic heavy metal layers [4][5]. Both the inverse Faraday effect and the optical orientation effect can be considered as bulk effects in the ferromagnetic layers and the heavy metal layers, respectively. However, interfacial nature of the optical spin generation such as the optical Rashba-Edelstein effect [6] has not been addressed so far. In this study, optical spin torque vectors in the ferromagnet/heavy metal heterostructure with different thicknesses and symmetry of stacking structures were investigated to gain insight into the interfacial nature of optical spin generation [7]. Thin film samples were fabricated by using magnetron sputtering technique. The stacking structure is as follows, Si/SiO2 sub./ MgO(10) / Fe50Co50 (dFeCo) / Pt (dPt) (thickness is in nm). FeCo (Pt) thickness dFeCo (dPt) was varied from 1 (2) nm to 4 (10) nm. To study the effect of symmetries for the stacking structure, samples with following stacking structure were also made, Si/SiO2 sub./ Pt(3) / Co(2) / Pt(3), Si/SiO2 sub./ Pt(3) / Co(2) / Ta(2), Si/SiO2 sub./ Co(2) / Pt(3). The optically induced spin was detected by using all-optical time-resolved magneto-optical Kerr effect (TRMOKE). The wavelength and pulse duration were 800 nm and 120 fs, respectively. The pump beam is irradiated on the film at an angle of 10 deg. measured from film normal. Polar MOKE signals were detected to measure normal component of magnetization. In-plane magnetic field up to 2 T was applied. Right circularly polarized (RCP) and left circularly polarized (LCP) pump pulses were irradiated on the ferromagnet/heavy metal heterostructure films. A change in the phase of magnetization precession was observed when the pump laser pulses with different optical helicities were irradiated, indicating the magnetization precession was excited by the optical spin torque. Figure 2 shows optical helicity induced TRMOKE signals where signals are obtained by taking differences between the signals with RCP and LCP pump pulses. Large peak at around delay zero is due to the so-called specular inverse Faraday effect, which is used to define the time when the pump pulse arrives at the film surface. Oscillation signals were fitted by the sinusoidal decayed function, as shown with broken curves in Fig. 2 bottom. The optical helicity induced TRMOKE signals were decomposed into sine and cosine components, as shown with red and blue solid curves in Fig. 2, respectively. This phase analysis was used to evaluate optical spin torque vectors in the heterostructures. The cosine signal indicates that magnetization precession is excited by out-of-plane optical spin torques, which can be explained by the optical orientation effect in the heavy metal Pt layer. In addition, remarkable sine signals was observed. The in-plane field-like optical spin torque evaluated from the sine signal was found to be increased with decreasing FeCo thickness. This increase cannot be explained by the inverse Faraday effect in the ferromagnetic FeCo layer [7]. A possible explanation is that the remarkable in-plane field-like optical spin torque is attributed to the interface spin generated by the optical Rashba-Edelstein effect. Interfacial in-plane torques can be induced by exchange coupling between generated spins and magnetization, which is analogous to the electrical manipulation of magnetization via the Rashiba-Edelstein effect. To validate the optical spin torque induced by the optical Rashba-Edelstein effect, we measured the optical spin torque with changing stacking structure symmetry. It was found that the in-plane optical spin torque was enhanced when Co was sandwiched by SiO2 and Pt layer, i.e., asymmetric heterostructure [7]. This result supports the optical spin torque induced by the interfacial Rashba spin-orbit coupling owing to the structural inversion symmetry breaking. In conclusion, we studied optical spin torque vectors in the ferromagnet/heavy metal heterostructure. By changing the thickness and the stacking structure symmetry of the heterostructure, we found new type of the interface optical spin generation, optical Rashba-Edelstein effect. This work will lead to efficient optical manipulation of thin film nanomagnets.

**References:**

[1] I. M. Miron, K. Garello, G. Gaudin et al., Nature, Vol. 475, p. 189 (2011) [2] L. Liu, C. -F. Pai, Y. Li et al., Science, Vol. 336, p. 555 (2012) [3] H. Becker, C. J. Krückel, D. V. Thourhout et al., IEEE J. Sel. Top. Quant., Vol. 26, p. 8300408 (2020) [4] G. -M. Choi, A. Schleife, D. G. Cahill, Nat. Commun. Vol. 8, p. 15085 (2017) [5] G. -M. Choi, J. H. Oh, D. -K. Lee et al., Nat. Commun. Vol. 11, p. 1482 (2020) [6] V. M. Edelstein, Phys. Rev. Lett. Vol. 80, p. 5766 (1998) [7] S. Iihama, K. Ishibashi, S. Mizukami, Nanophotonics, 20200571 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/622c5ff6e97b66c2b5430e3384c20386.jpg)
Schematic illustration of the optical spin torque in a ferromagnet/heavy metal heterostructure.

![](https://assets.underline.io/uploads/markdown_image/1/image/466eff3367e9340d6519f6562b628362.jpg)
Optical helicity induced TRMOKE signal obtained by taking differences between signals with RCP and LCP pump pulses for a FeCo(2) / Pt(3) thin film. The optical helicity induced signals at around t = 0 [top] and magnetization precession due to the optical spin torque [bottom].

 Interface optical spin generation in a ferromagnet/heavy metal heterostructure



The relaxation of a spin current into a ferromagnetic material occurs through different processes depending on the spin currents’ polarization orientation. For a polarization collinear with the material magnetization, the relaxation occurs through diffusive processes controlled by the spin diffusion length [1]. When the polarization is transverse with the magnetization, the relaxation is driven by ballistic processes due to band structure mismatch and non-coherent spin precession [2]. Despite being a key ingredient for the understanding of phenomena like Spin Transfer Torque, this anisotropy has never been directly observed so far. In this presentation, we report the observation of the spin relaxation anisotropy on a single device by using a spin absorption method in Lateral Spin Valves (LSVs) [3]. Using a magnetic field, we control the relative orientation between the spin current polarization and the magnetization of a nanodisk-shaped ferromagnetic absorber (fig. 1). We observed that the efficiency of the spin current absorption increases when this polarization and the absorber magnetization have transverse directions. We will present a device that allows the study of spin transport in a LSV with non-collinear ferromagnetic component. By demonstrating a good control over its magnetic states, we will show that we indeed observed the spin current relaxation anisotropy in a single device. We carried out an analysis by introducing bulk characteristic lengths of the transverse spin relaxation, and the spin mixing conductance describing transverse interfacial spin relaxation processes. This data analysis points out the role of both interfacial and bulk contribution in the non-collinear spin current relaxation, and allows the electrical measurement of the spin mixing conductance at 3d ferromagnets-Cu interfaces [3].

**References:**

[1] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993) [2] C. Petitjean et al., Phys. Rev. Lett. 109, 117204 (2012) [3] G. Zahnd et al., Phys. Rev. B 98, 175514 (2018) [3] M. Cosset-Cheneau et al, arXiv:2007.14058 (2020), accepted in Physical Review Letters

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/26188d6aedb9ad2a57f1c2367cd5d7bf.jpg)
Drawing and SEM picture of the device used for the absorption anisotropy measurement. The nanodisk absorbs the spin current that flows between the ferromagnetic electrodes. Its isotropic shape allows to change its magnetization orientation by using a weak external magnetic field.



 Measurement of the Spin Absorption Anisotropy in Lateral Spin Valves

Spin currents generated from thermal gradients in magnetic tunnel junctions (MTJs) via the tunnel magneto-Seebeck effect (TMS) [1] or in ferromagnetic insulators via the spin Seebeck effect (SSE) [2,3] have high potential for future nano-electronic devices [4]. The TMS effect can be enhanced by proper choice of barrier [5] and electrode [6] materials. In addition, recent reports highlight the enhancement of thermally induced spin currents by the insertion of antiferromagnetic insulating layers [7] and assign this effect to the improved spin transport properties of the antiferromagnetic oxide layer. However, quantitative determination and comparison of the TMS or SSE coefficients require accurate knowledge of the temperature drop across the insulating tunnel barrier or antiferromagnetic insulating layer. As a promising alternative, the detected thermovoltages are related to the heat flux that passes the insulating material. Unknown properties of the thermal connection between sample and setup make the temperature difference a less reliable quantity compared to the heat flux detection [8-10]. The key property here is the thermal conductivity, which determines both temperature drop and heat flux, and which is needed, e.g., to compare SSE coefficients scaled to the heat flux with literature values. However, the thermal conductivity of oxide materials is experimentally difficult to access. In my talk, I will introduce two spin caloric devices that include an oxide interlayer, and discuss the role of the thermal conductivity of these layers. In particular, these will be the oxide tunnel barrier of an MTJ in a TMS device and the inserted antiferromagnetic oxide layer in an SSE stack (see Fig. 1). Additionally, I will present several experimental approaches to obtain quantitative values for the thermal conductivity in insulators utilizing laser-induced TMS in combination with finite element modeling [11] as well as time-domain thermoreflectance (see Fig. 2) and time-resolved magnetooptic Kerr effect thermometry [12]. Here, we extract values of the thermal conductivity for MgAl2O4 and MgO barrier materials in half-MTJ stacks with Co-Fe-B electrode. The results are in nice agreement with theoretical predictions for ultra-thin oxide barriers [13]. Furthermore, we experimentally found that the thermal conductivity does not significantly change when an antiferromagnetic insulating layer, such as NiO, is inserted into SSE devices. Therefore, the temperature profile of the SSE stack does not change as well and cannot explain the enhanced thermal spin current due to the insertion of an antiferromagnetic oxide layer. Thus, we confirm the enhanced spin conductivity induced by the insertion of an antiferromagnetic insulating layer.

**References:**

[1] T. Kuschel et al., J. Phys. D: Appl. Phys. 52, 133001 (2019) [2] K. Uchida et al., Appl. Phys. Lett. 97, 172505 (2010) [3] J. Kimling, TK et al., Phys. Rev. Lett. 118, 057201 (2017) [4] U. Martens, TK et al., Commun. Phys. 1, 65 (2018) [5] T. Huebner, TK et al., Phys. Rev. B 96, 214435 (2017) [6] A. Boehnke, TK et al., Nat. Commun. 8, 1626 (2017) [7] W. Lin et al., Phys. Rev. Lett. 116, 186601 (2016) [8] A. Sola, TK et al., Sci. Rep. 7, 46752 (2017) [9] P. Bougiatioti, TK et al., Phys. Rev. Lett. 119, 227205 (2017) [10] A. Rastogi, TK et al., Phys. Rev. Appl. 14, 014014 (2020) [11] T. Huebner, TK et al., J. Phys. D: Appl. Phys. 51, 224006 (2018) [12] H. Jang, TK et al., Phys. Rev. Appl. 13, 024007 (2020) [13] J. Zhang et al., Phys. Rev. Lett. 115, 037203 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3e03416a3d998c8d7f49347f5d0986e8.jpg)
Investigated SSE systems which contain antiferromagnetic NiO interlayers of various thicknesses on top of Fe3O4. The heater generates the temperature gradient that induces the spin current detected by the orthogonal inverse spin Hall effect voltage in Pt. The obtained electric field V/D is related to the temperature difference T1-T2 and to the heat flux which are detected by thermocouples and by a heat flux sensor, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/d9e05b68fd41757831c9ac4fcbdb3c01.jpg)
Time-domain thermoreflectance measurements (open symbols) on the half-MTJ samples with Co-Fe-B electrode and MgO as well as MgAl2O4 barrier material. In addition, the reference sample with just the seed layer Ta has been studied. The thick Ru layer absorbs the pump pulse and passes on the heat towards the half MTJ. This reduces the reflected intensity of the probe beam on a 100-1000 ps time scale. The solid lines are fits based on a model that includes the Ru layer, the substrate and an intermediate thermal conductance containing all intermediate layers and interfaces. Figure taken from Ref. [10].

 Impact of thermal conductivity in insulators on thermally induced spin currents INVITED

Current-induced spin-orbit torque enables highly efficient manipulation of magnetization for spintronic applications. The anti-damping torque is responsible for efficient magnetization manipulation when the torque is collinear with the magnetization leading to a directly change of the effective magnetic damping. But as the anti-damping torque is restricted to lie along an in-plane direction, it is efficient for manipulating only samples with magnetic anisotropy along the in-plane y axis not that along the out-of-plane (z axis) or collinear with the current (x axis). To efficiently and deterministically drive, for example, perpendicularly-magnetized devices that are preferred for high-density memories, an out-of-plane anti-damping torque is required. Such an out-of-plane spin-torque can originate from spin-orbit scattering from ferromagnetic interfaces(1–3), or can arise at the interface in systems with reduced symmetry, such as in bilayers of the non-magnetic transition-metal dichalcogenide and ferromagnetic metals(4); however these effects based on interface or heterostructure engineering have not been demonstrated strong enough for practical anti-damping switching. Here we demonstrate an alternative strategy to achieve unconventional spin-orbit torques, based on long-range non-collinear magnetic order within the bulk of the spin-source layer(5). In particular, we use the spin-Hall effect in epitaxial thin films of Mn3GaN, a metallic antiferromagnet that has a 120° triangular spin texture, which reduces the symmetry sufficiently to allow spin current generation with different spin polarization directions to generate unconventional spin-torques. In heterostructures of epitaxial Mn3GaN/Cu/permalloy, we observe not only the out-of-plane anti-damping torque, but also the anti-damping torque corresponding to a Dresselhaus-like spin polarization, besides the conventional Rahsba-like symmetry. When the non-collinear spin texture is eliminated by heating above the Néel temperature of Mn3GaN (345 K), the unconventional spin-torques go to zero. Such a control of the spin polarizations is coincident with our symmetry analysis and theory calculation upon the magnetic space groups across the Néel transition. Our results demonstrate an approach based on spin-structure design for controlling spin-orbit torque, enabling high efficient manipulation of magnetization for antiferromagnetic spintronics. The presenter thanks C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, K. Song, S. Y. Choi, L. Guo, R. D. Johnson, P. Manuel, R. V. Chopdekar, I. Hallsteinsen, T. Tybell, P. J. Ryan, J. W. Kim, Y. S. Choi, P. G. Radaelli, D. C. Ralph, E. Y. Tsymbal, M. S. Rzchowski, C. B. Eom for their contributions in this research.

**References:**

1. V. P. Amin, M. D. Stiles, Spin transport at interfaces with spin-orbit coupling: Phenomenology. Phys. Rev. B. 94, 104420 (2016). 2. S. H. C. Baek, V. P. Amin, Y. W. Oh, G. Go, S. J. Lee, G. H. Lee, K. J. Kim, M. D. Stiles, B. G. Park, K. J. Lee, Spin currents and spin-orbit torques in ferromagnetic trilayers. Nat. Mater. 17, 509–513 (2018). 3. A. M. Humphries, T. Wang, E. R. J. Edwards, S. R. Allen, J. M. Shaw, H. T. Nembach, J. Q. Xiao, T. J. Silva, X. Fan, Observation of spin-orbit effects with spin rotation symmetry. Nat. Commun. 8, 911 (2017). 4. D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, D. C. Ralph, Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers. Nat. Phys. 13, 300–305 (2016). 5. T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, K. Song, S. Y. Choi, L. Guo, R. D. Johnson, P. Manuel, R. V. Chopdekar, I. Hallsteinsen, T. Tybell, P. J. Ryan, J. W. Kim, Y. Choi, P. G. Radaelli, D. C. Ralph, E. Y. Tsymbal, M. S. Rzchowski, C. B. Eom, Controlling spin current polarization through non-collinear antiferromagnetism. Nat. Commun. 11, 4671 (2020).

**Images:**

 Controlling spin current polarization through non-collinear antiferromagnetism INVITED

Spintronics has been expanding its fields to cover not only electronics and magnetism but also photonics, caloritronics and mechanics in recent years. Among them, photoexcitation using circularly polarised light has been commonly used to generate a spin polarised electrical current mainly in a semiconductor [1]. However, the linkage between a spin current and a propagating surface plasmon polariton (SPP) has not been well investigated theoretically until recently [2]. SPPs are generated in a PHz regime in a non-magnetic metal, which can generate a magnetisation gradient and a resulting spin current. In this study, we have experimentally measured the SPP generated spin current using an inverse spin Hall effect and confirmed it agrees with the above theoretical prediction. A high vacuum sputtering system (PlasmaQuest, HiTUS) was used to deposit 20 nm thick Ag films on a thermally oxidised Si substrate. The films were also patterned into a Hall bar with the width of 1.2 mm and the length of 11 mm. By placing a prism (ThorLab, N-BK7) on the surface of the Hall bar, the Kretschmann arrangement was achieved for SPP introduction by a semiconductor laser (Thorlabs, LDM635 with the wavelength of 635 nm and the power of 0.4 mW). As shown in Fig. 1, a linear polariser and a half-wave plate (Thorlabs, SM05 mounted zero-order 633 nm) were used to generate both s- and p-polarisations. The spot size of the incident beam was then controlled using a pair of objective and a convex lens. The measured laser spot size was 1.73 mmf. The incident angle q of the beam to the prism was controlled by a mirror. The generated spin current was detected using an inverse spin Hall effect as schematically shown in Fig. 3. The source meter (Keithley, 2400) was used to detect the spin Hall voltage by connecting electrical contacts at the diagonal corners of the continuous film and the ends of the Hall bars. The incident angle q was adjusted to show the largest Hall voltage. The measurements were carried out with 600~1,000 repetitions with 0,1 ms interval, providing an average value with a standard deviation. A source current was set to be 0.0 and 0.1 mA for the Hall voltage and resistance measurements, respectively. The continuous films were measured to demonstrate the spin current generation by SPP as shown in Fig. 1a. Ag, Pt and W films were used for the inverse spin Hall measurement with and without a prism. In order to subtract any parasitic effects, e.g., spin caloritronic and photovoltaic effects induced by the laser introduction, a spin current generated by the electro-motive force Iemf was first estimated by subtracting the current generated by the electro-motive force with and without the prism. The spin current generated by SPP ISPP was then estimated by taking the difference between those generated by s- and p-polarisations since only the p-polarised light generates SPP. By solving the wave-number matching condition between the laser and SPP, the plasma frequency of Ag is estimated to be 2.18 PHz. ISPP is estimated to be (2.128 ± 0.281) and (6.650 ± 0.169) µA for the Ag continuous film and Hall bar, respectively, confirming the SPP-induced spin currents. Such a new linkage between spintronics and photonics and lead to a new coupler to convert optical data signals into spin currents for processing and storage and vice versa.

**References:**

[1] A. Hirohata, "Optically Induced and Detected Spin Current," in Spin Current S. Maekawa, S. O. Valenzuela, E. Saitoh and T. Kimura (Eds.) (Oxford University Press, Oxford, UK, 2012) p. 49-64. [2] D. Oue and M. Matsuo, New J. Phys. 22, 033040 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3bcba07ccbb8f6e4d8383f907974c798.jpg)
Fig. 1 Schematic diagram of the spin current generation by SPP.

 Spin Current Induced by a Surface Plasmon Polariton

Recent demonstration of efficient transport and manipulation of spin information by magnon currents has opened exciting prospects for processing information in devices. Magnon currents can be excited in magnetic insulators by applying charge currents in an adjacent metal layer. Here, by implementing a non-local device scheme, we study the magnon diffusion length (MDL) for electrically and thermally excited magnon currents in Y3Fe5O12 (YIG) and Tm3Fe5O12 (TmIG) [1,2]. In contrast to earlier reports, our temperature and thickness-dependence studies reveal that the MDL depends on the way the magnon currents are generated, evidencing that magnons of different energies are excited (sub-thermal and thermal for electrically- and thermally-driven magnon currents, respectively) [1]. At room temperature, the MDL in YIG is ~9 μm for thermally excited magnons, which is almost twice the value extracted from those excited electrically. This difference gradually decreases as temperature decreases, which is consistent with the expected convergence of the excited magnon distributions at low temperatures. Moreover, we demonstrate that the MDL of thermally induced magnons in YIG is the same regardless of the film thickness and growth conditions, evidencing the robustness of the measurement method to reliably extract the intrinsic MDL. We find the MDL of perpendicularly magnetized TmIG films to be ~300 nm [2]. The shorter diffusion length of TmIG compared to YIG is attributed to the larger Gilbert damping of TmIG (α~0.01) and the vertical confinement of the magnon modes for ultrathin films. Besides, we investigate the magnetic field dependence of the MDL and analyse it in terms of a modified magnon diffusion equation considering a linear with field enhancement of the magnon damping. We also demonstrate that the non-local thermal signals are dominated by diverse thermoelectric effects of magnetic and spin origin occurring at the detector electrode, and provide a guide on how to disentangle thermally-induced diffusive magnon transport signals from those originated by thermoelectric effects. Finally, by employing a third gate electrode, we demonstrate current-induced modulation of the magnon conductivity in TmIG. The possibility of combining magnon transport in perpendicularly magnetized layers with other device functionalities such as current-induced switching and wall motion [3] open prospects for novel spintronic devices concepts.

**References:**

[1] J. M. Gomez-Perez, S. Vélez, L. E. Hueso, and F. Casanova, Phys. Rev. B 101, 184420 (2020). [2] J. Gao, S. Vélez et al., submitted. [3] S. Vélez et al., Nature Communications 10, 4750 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7be3bce9c7a40293a2abf6a2591dfbff.jpg)
FIG. 1. (a), (b) Schematic representation of the creation, transport, and detection of electrically-induced and thermally-induced magnon currents in nonlocal devices. Is represent the spin current generated by I, s the vector spin, Ie (Ith) and Ie (Ith) the spin and charge currents generated at the detector, λe (λth) the magnon diffusion length, and ▽T the thermal gradient. (c) Scanning electron microscopy image of a representative non-local device with d = 6 μm.

![](https://assets.underline.io/uploads/markdown_image/1/image/484103aeedfce274088e8a4b34751eec.jpg)
FIG. 2. Temperature dependence of the MDL for electrically- and thermally-driven magnon currents at two different magnetic fields.

 Injection, Transport, Detection, and Modulation of Magnon Spin Currents in Magnetic Insulators

In the field of spintronics, pure spin currents are discussed for spin transport and spin-based information processing at low dissipation levels. Here, magnetically ordered insulators (MOIs) take an exceptional role, as magnon transport can be investigated in an isolated fashion. In addition, its properties can be measured by all-electrical means combining heavy metals (HM) and MOIs in heterostructures. In nanopatterned devices, this allows to excite, control, and detect incoherent magnons and investigate their transport properties. In our experiments, we deposit three electrically isolated platinum (Pt) strips on top of an 11 nm thick yttrium iron garnet (YIG) thin film to realize three-terminal devices [1], as shown in Fig. 1. Driving a charge current through the first Pt strip acting as the injector, a pure spin accumulation is created in the YIG via the spin Hall effect (SHE) and Joule heating. The excited magnons diffuse and can be detected at the detector Pt strip via the inverse SHE. A constant dc charge current Imod applied to the modulator strip placed between the two outer Pt strips allows to manipulate the magnon transport from injector to detector via a SHE induced spin accumulation and Joule heating effects [1]. The magnon diffusion between injector and detector can be modeled as magnon conductivity, which changes in a nonlinear fashion with increasing modulator current and exhibits a threshold behavior. Above threshold, the enhanced magnon conductivity is attributed to a zero effective damping state via SHE induced damping-like spin-orbit torque underneath the modulator [2]. Here, we quantitatively compare two measurement schemes: (i) a dc-technique and (ii) an ac-technique. We demonstrate that both schemes allow to investigate the magnon transport in three-terminal devices and can efficiently distinguish between electrically and thermally injected magnons [3]. In the first case, we apply a varying dc charge current to the injector and measure the detector voltage Vdc, utilizing the current reversal method to distinguish between the electrically and thermally injected magnons. For the ac-technique, an ac-current is applied to the injector, while the detector voltage Vac is measured via lock-in detection. We develop a model to compare the SHE contributions resulting in the measured values of Vdc and Vac. Both are expected to give identical values if we focus on the linear regime. We experimentally corroborate this model by performing an angle-dependent measurement, where we find a quantitative agreement of the dc and ac voltages, as depicted in Fig. 2(a) and 2(b), respectively. A more detailed comparison of the extracted detector signal amplitudes Adc and Aac of the SHE induced magnons shows that both schemes yield identical results in the low modulator charge current regime. However, we find clear differences above a certain threshold current, as shown in Fig. 2(c), where the ratio Aac/Adc exhibits a clear deviation from 1 for Imod &gt; 0.55 mA. This indicates contributions of higher orders to the detector voltage originating from the injector current. Our findings shed new light onto nonlinear effects on the magnon conductance, contributing to the understanding of the manipulation of magnon currents using a three-terminal device, which is crucial for implementing a logic based on incoherent magnons. We gratefully acknowledge financial support by the German Research Foundation via Germany’s Excellence Strategy (Grant No. EXC-2111-390814868) and project AL 2110/2-1.

**References:**

[1] L. J. Cornelissen et al., Physical Review Letters 120, 097702 (2018). [2] T. Wimmer et al., Physical Review Letters 123, 257201 (2019). [3] J. Gückelhorn et al., Applied Physics Letters 117, 182401 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ac31d13b82c713ff614f865f5515b607.jpg)
Fig. 1 Sketch of the sample configuration and the experimental external magnetic field geometry.

![](https://assets.underline.io/uploads/markdown_image/1/image/1bf844b707812981e29cefb260a0e89b.jpg)
Fig. 2 Detector signals (b) Vdc and (c) Vac of the SHE induced magnons plotted as a function of the magnetic field orientation with constant magnitude µ0H = 50 mT for positive modulator currents Imod at 280 K. The detector signals induced by SHE magnon transport are in perfect agreement. (c) Ratio of the extracted amplitudes Adc and Aac of the SHE injected magnons for µ0H = 60 mT. For large modulator current values, the ratio clearly deviates from 1.

 Magnon transport in three-terminal YIG/Pt nanostructures studied by dc and ac detection techniques

Artificial intelligence (AI) is gaining acceptance in a variety of applications. In order to develop an energy-efficient architecture for AI, neuromorphic computing, which draws inspiration from the human brain, is being investigated. This architecture is a complex interconnection of numerous synthetic neurons and synapses, which are the processing and memory units, respectively. To realize neuromorphic architecture, one needs to fabricate artificial neurons as well as synapses. Several charge and spin-based material systems are being studied towards developing neuromorphic architecture. However, magnetic domain wall (DW) devices are one of the most energy-efficient contenders1. Besides, utilizing spin-orbit torque to drive DWs provides more degrees of freedom in achieving higher energy efficiency. Therefore, in the present study, we are studying spin-orbit torque for driving the DWs in the proposed novel material engineering to further reduce energy consumption. In SOT driven DW motion, the power can further be reduced either by increasing spin Hall angle or by reducing pinning field while keeping spin Hall angle at optimized values. Since W in β phase results in the highest spin Hall angle (θSH)2, we decided to study β-W-based spin Hall layer. However, β-W is obtained by depositing the W films at low power and high Ar gas pressure (HP). These depositions conditions result in granular W films, which in turn results in (a) more defects, (b) higher critical current, (c) larger power, and (d) lower TMR2,3. To resolve these, we have proposed the concept of the dual W layer, where a low-pressure (LP) W, which is expected to be smoother (lesser grains), is deposited between β-W and ferromagnetic layers. The studied stack involves dual W spin Hall layer (SHL) (6 nm)/ CoFeB (1 nm)/ MgO (~1 nm)/ Ru (2 nm). The studied stack is schematically illustrated in figure 1 (a). Firstly, we characterized the thin film samples and observed that the HP3LP3 (this means SOT layer consists of high pressure deposited W (3 nm)/ low pressure deposited W (3 nm)) dual-layer sample exhibits the best structural, magnetic, and electrical characteristics. Therefore, we fabricated DW devices from the HP3P3 sample. Moreover, we prepared DW devices from single LP6 and HP6 spin Hall layers for comparison purposes. The details of the measurements at the thin film level will be presented in detail during the conference. Subsequently, we studied the DW dynamics in our samples using Kerr microscopy. First, we saturated the DW devices with a large magnetic field. Then, we applied a reversed magnetic field to insert DW in the DW devices. Once the DW is inserted into the devices, we applied current pulses of various amplitudes and observed the DW motion. When current pulses were applied the DW moves in a direction opposite to the current (please refer to figure 1 (b-d). Moreover, as the amplitude of the current decreases the DW velocity decreases. Most importantly, the DW moves at the current density as small as 107 A/m2, as shown in figure 1 (d). The detailed measurements revealed that the DW can be moved to a minimum current density of 106 A/m2. Note, a small OOP magnetic field of -1 mT, which is approximately one-third of the coercivity of the devices, was applied during these measurements. On contrary, we observed that devices with a single spin Hall layer do not show the DW motion. Further harmonic Hall measurements and pinning field measurements showed that ultra-low pinning field (figure 2) in addition to the optimum value of spin Hall angle results in DW motion at such low currents in our dual-layer devices. We have also compared the energy consumption in our devices with respect to the current state-of-art and found a significant reduction in energy consumption in our devices4-6. In addition, we have proposed the design of DW-based neuron and synaptic devices. Moreover, we have proposed and studied the synaptic functions in meander wire, in which two neighboring segments meet at a small offset of “d”. We also demonstrated the working stochastic, which is for d= 15%, and non-stochastic, which is for d=50%, type synapses in our dual-layer meander devices. In conclusion, the observation of such energy-efficient DW motion is a significant contribution to the field of neuromorphic computing. The results in detail will be presented during the conference.

**References:**

P. Chenyun & A. Naeemi. IEEE J. Explor. Solid-State Comput. Devices Circuits 3, 101-110 (2017). C. Zhang, S. Fukami, K. Watanabe et al. Applied Physics Letters 109, 192405 (2016). K. Demasius, T. Phung, W. Zhang et al. Nature communications 7, 10644 (2016). S. DuttaGupta, S. Fukami, C. Zhang et al., Nature Physics, 12, 333-336 (2016). K. Fritz, L. Neumann, and M. Meinert, arXiv:1910.13837 (2019). K. Cai, Z. Zhu, J. Lee et al., Nature Electronics, 3, 37-42 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/448b927e09ca7d07c2f688e4d05410f2.jpg)
(a) Schematic illustration of the studied dual-layer stack. DW motion in our dual-layer devices at a current density of (b) 1011 A/m2, (c) 109 A/m2, and 107 A/m2.

![](https://assets.underline.io/uploads/markdown_image/1/image/8e024781e04b07a4dfb7cbde9e6afd60.jpg)
The DW velocity vs out-of-plane (OOP) magnetic field curves. A linear fitting of these curves provides information about the pinning field of the studied samples.

 Ultra-low Power Domain Wall Device for Spin-based Neuromorphic Computing

Rare earth iron garnet (REIG) thin films have attracted a great deal of attention for spintronic applications over the past few years. Properties of REIG|Pt heterostructures have also been extensively studied. However, effects of nonmagnetic ion substitution on iron sublattice of REIG thin films have not been investigated so far. We report growth, structural, magnetic and spintronic properties of epitaxial scandium substituted terbium iron garnet (TbScIG) thin films on gadolinium gallium garnet (GGG, (111) orientation) substrate. Reciprocal space mapping shows that films are lattice matched to substrate without strain relaxation up to thickness of 66 nm. All the deposited films with varying scandium content have been found to have perpendicular magnetic anisotropy (PMA) due to magnetoelastic anisotropy contribution arising from lattice strain. Their compositions have been determined using wavelength dispersive spectroscopy (WDS) to be Tb2.8Fe5.2O12, Tb2.8Sc0.3Fe4.9O12, Tb2.9Sc0.5Fe4.6O12 and Tb2.8Sc0.8Fe4.4O12. Thickness of these films have been determined to vary within 63-73 nm by fitting Laue fringes observed in symmetric high-resolution x-ray diffraction (HRXRD) scans around the GGG (444) peak. Compensation temperatures of TbScIG thin films have been found to decrease with increasing scandium content which is in agreement with larger Sc3+ ions occupying octahedral sublattice as reported in past for bulk yttrium iron garnet. Increasing scandium content has been found to be accompanied by increase in room temperature saturation magnetization from 37 emu/cm3 for Tb2.8Fe5.2O12 to 59 emu/cm3 for Tb2.8Sc0.8Fe4.4O12. Anomalous Hall effect (AHE) like spin-Hall magnetoresistance (SMR) measurements have been performed to determine the effect of scandium content on spin mixing conductance of TbScIG|Pt interface. Ferromagnetic resonance (FMR) measurements have been performed to determine the Gilbert damping parameter of these thin films.

**References:**

1. Rosenberg, E. R. et al. Phys. Rev. Mater. 2, 094405 (2018). 2. Chen, Y.-T. et al. Phys. Rev. B 87, 144411 (2013). 3. Geller, S. J. Appl. Phys. 37, 1408–1415 (1966). 4. Espinosa, G. P. J. Chem. Phys. 37, 2344–2347 (1962). 5. Geller, S. & Gilleo, M. A. J. Phys. Chem. Solids 3, 30–36 (1957).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6622f0b7058f2890a716a1b0e7583354.jpg)
Fig. 1 High-resolution XRD scans of TbScIG/GGG thin films with varying Sc content (vertically offset for clarity with film and substrate peaks labeled F and S respectively)

![](https://assets.underline.io/uploads/markdown_image/1/image/a0dcfa0d077c1af474271af74f5d7897.jpg)
Fig. 2 Magnetic moment versus temperature for TbScIG/GGG thin films with varying Sc content. A typical error bar is attached.

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