France



We theoretically investigated the four-fold symmetric term of the anisotropic magnetoresistance (AMR) effect in cubic ferromagnetic metals (FMs), which has recently been observed experimentally for certain metals, by using fourth-order perturbation theory with respect to the spin-orbit interaction (SOI). The AMR effect is a magnetoresistance effect specific to magnetic metals in which the charge conductivity changes depending on the direction of magnetization in FMs [1], and attracted much attention in the field of spintronics to apply for evaluation of half-metallicity [2] or spin-orbit torque generation [3].&lt;br/&gt;Usually, in experiments on bulk polycrystalline FMs, the longitudinal conductivity is proportional to the cos(2θ) where θ is the relative angle between the current direction and the magnetization direction, that is, the AMR effect has two-fold symmetry under the magnetization rotation. Theoretically, this two-fold AMR effect is caused by the SOIs and has been successfully explained by Campbell’s model, an s-d impurity scattering model with considering the second-order perturbation in terms of the SOIs [4].&lt;br/&gt;In single-crystal FMs such as Fe4N, however, the AMR measurements exhibited not only the conventional two-fold component but also the unconventional four-fold component proportional to cos(4θ) [5]. While this four-fold AMR effect is expected to stem from the symmetry of the cubic crystal, the microscopic origin is not so straightforward because the magnetization direction dependence of the conduction band is too small to explain this effect.&lt;br/&gt;Recently, a possible mechanism of the four-fold AMR effect is theoretically proposed by Kokado [6], wherein the s-d scattering coupled with the SOIs and “tetragonal” symmetric crystal fields is responsible for this effect. This theory is an extension of Campbell’s model in polycrystalline FMs to the single crystal FM model by taking into account the energy level splitting due to the tetragonal crystal fields on the d-states (Fig. 1). Since it is argued that the splitting among d-states is significant for the presence of a four-fold component, this theory requires the extra assumption of existing the tetragonal (uniaxial or planar) distortion to apply to the cubic crystals including the Fe4N. However, the experiment is confronted with the fact that the four-fold component appears even though the sample is almost cubic crystal and has little distortion. The microscopic origin is still controversial; therefore, it is worthwhile to explore a possible mechanism of the four-fold AMR effect in the cubic symmetric single-crystal FMs without tetragonal distortion.&lt;br/&gt;&lt;br/&gt;In this work, we show the presence of the four-fold AMR effect in cubic single crystal FMs from a microscopic viewpoint [7]. Our main finding is that the higher-order perturbation terms of SOI can also be responsible for the four-fold AMR effect in cubic FMs, instead of the second-order SOI with tetragonal distortion as mentioned in Kokado’s theory. Inspired by the Kokado model, we use the s-d impurity scattering model expressed by an impurity Anderson Hamiltonian with cubic crystal fields and SOI. The conductivity is represented on the basis of the Kubo–Greenwood formula. This approach enables us to consider the non-perturbative role of SOIs on the AMR effects.&lt;br/&gt;Firstly, to overlook a physical aspect of this effect, we carried out perturbative analysis with respect to the SOIs. The results indicate that the fourth-order terms of SOI can give rise to the four-fold AMR effect. Furthermore, we extract the two important roles of the SOI: first, to couple the electron’s momentum and magnetization, and second, to induce the orbital splitting among d-states instead of the tetragonal crystal field.&lt;br/&gt;Secondly, we performed numerical calculations to evaluate the non-perturbative behavior of the AMR effects. As the results, we successfully obtained both the two-fold and four-fold components of the AMR effect without assuming the tetragonal distortion, which is consistent with the perturbative analysis (Fig. 2). Other parameter dependencies will be presented at the conference.

**References:**

[1] T. McGuire and R. Potter, IEEE Trans. Magn., Vol. 11, p.1018 (1975).&lt;br/&gt;[2] S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J. Phys. Soc. Jpn., Vol. 81, p.024705 (2012).&lt;br/&gt;[3] T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev. Applied, Vol. 3, p.044001 (2015).&lt;br/&gt;[4] I. A. Campbell, A. Fert, and O. Jaoul, J. Phys. C, Vol. 3, p.S95 (1970).&lt;br/&gt;[5] M. Tsunoda, H. Takahashi, S. Kokado, Y. Komasaki, A. Sakuma, and M. Takahashi, Appl. Phys. Express, Vol. 3, p.113003 (2010).&lt;br/&gt;[6] S. Kokado and M. Tsunoda, J. Phys. Soc. Jpn., Vol. 84, p.094710 (2015).&lt;br/&gt;[7] Yuta Yahagi, Daisuke Miura, and Akimasa Sakuma, J. Phys. Soc. Jpn., Vol. 89, p.044714 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8e02bcab2e43d5b25b65902dc8186411.jpg)
Fig.1: Schematics of the energy level of the d-states under (a) a cubic symmetric and (b) a tetragonal symmetric crystal-field, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/04a12b54dea732a7b428e0597df7effd.jpg)
Fig.2: Two-fold (c2) and four-fold (c4) components of the AMR ratio.

INTERMAG 2021

 Theoretical Study on Four-fold Symmetric Anisotropic Magnetoresistance Effect in Cubic Single-crystal Ferromagnetic Model

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We theoretically investigated the four-fold symmetric term of the anisotropic magnetoresistance (AMR) effect in cubic ferromagnetic metals (FMs), which has recently been observed experimentally for certain metals, by using fourth-order perturbation theory with respect to the spin-orbit interaction (SOI). The AMR effect is a magnetoresistance effect specific to magnetic metals in which the charge conductivity changes depending on the direction of magnetization in FMs [1], and attracted much attention in the field of spintronics to apply for evaluation of half-metallicity [2] or spin-orbit torque generation [3]. Usually, in experiments on bulk polycrystalline FMs, the longitudinal conductivity is proportional to the cos(2θ) where θ is the relative angle between the current direction and the magnetization direction, that is, the AMR effect has two-fold symmetry under the magnetization rotation. Theoretically, this two-fold AMR effect is caused by the SOIs and has been successfully explained by Campbell’s model, an s-d impurity scattering model with considering the second-order perturbation in terms of the SOIs [4]. In single-crystal FMs such as Fe4N, however, the AMR measurements exhibited not only the conventional two-fold component but also the unconventional four-fold component proportional to cos(4θ) [5]. While this four-fold AMR effect is expected to stem from the symmetry of the cubic crystal, the microscopic origin is not so straightforward because the magnetization direction dependence of the conduction band is too small to explain this effect. Recently, a possible mechanism of the four-fold AMR effect is theoretically proposed by Kokado [6], wherein the s-d scattering coupled with the SOIs and “tetragonal” symmetric crystal fields is responsible for this effect. This theory is an extension of Campbell’s model in polycrystalline FMs to the single crystal FM model by taking into account the energy level splitting due to the tetragonal crystal fields on the d-states (Fig. 1). Since it is argued that the splitting among d-states is significant for the presence of a four-fold component, this theory requires the extra assumption of existing the tetragonal (uniaxial or planar) distortion to apply to the cubic crystals including the Fe4N. However, the experiment is confronted with the fact that the four-fold component appears even though the sample is almost cubic crystal and has little distortion. The microscopic origin is still controversial; therefore, it is worthwhile to explore a possible mechanism of the four-fold AMR effect in the cubic symmetric single-crystal FMs without tetragonal distortion. In this work, we show the presence of the four-fold AMR effect in cubic single crystal FMs from a microscopic viewpoint [7]. Our main finding is that the higher-order perturbation terms of SOI can also be responsible for the four-fold AMR effect in cubic FMs, instead of the second-order SOI with tetragonal distortion as mentioned in Kokado’s theory. Inspired by the Kokado model, we use the s-d impurity scattering model expressed by an impurity Anderson Hamiltonian with cubic crystal fields and SOI. The conductivity is represented on the basis of the Kubo–Greenwood formula. This approach enables us to consider the non-perturbative role of SOIs on the AMR effects. Firstly, to overlook a physical aspect of this effect, we carried out perturbative analysis with respect to the SOIs. The results indicate that the fourth-order terms of SOI can give rise to the four-fold AMR effect. Furthermore, we extract the two important roles of the SOI: first, to couple the electron’s momentum and magnetization, and second, to induce the orbital splitting among d-states instead of the tetragonal crystal field. Secondly, we performed numerical calculations to evaluate the non-perturbative behavior of the AMR effects. As the results, we successfully obtained both the two-fold and four-fold components of the AMR effect without assuming the tetragonal distortion, which is consistent with the perturbative analysis (Fig. 2). Other parameter dependencies will be presented at the conference.

**References:**

[1] T. McGuire and R. Potter, IEEE Trans. Magn., Vol. 11, p.1018 (1975). [2] S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J. Phys. Soc. Jpn., Vol. 81, p.024705 (2012). [3] T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev. Applied, Vol. 3, p.044001 (2015). [4] I. A. Campbell, A. Fert, and O. Jaoul, J. Phys. C, Vol. 3, p.S95 (1970). [5] M. Tsunoda, H. Takahashi, S. Kokado, Y. Komasaki, A. Sakuma, and M. Takahashi, Appl. Phys. Express, Vol. 3, p.113003 (2010). [6] S. Kokado and M. Tsunoda, J. Phys. Soc. Jpn., Vol. 84, p.094710 (2015). [7] Yuta Yahagi, Daisuke Miura, and Akimasa Sakuma, J. Phys. Soc. Jpn., Vol. 89, p.044714 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8e02bcab2e43d5b25b65902dc8186411.jpg)
Fig.1: Schematics of the energy level of the d-states under (a) a cubic symmetric and (b) a tetragonal symmetric crystal-field, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/04a12b54dea732a7b428e0597df7effd.jpg)
Fig.2: Two-fold (c2) and four-fold (c4) components of the AMR ratio.

technical paper

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

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

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.

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

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.

 Magnetic and Spintronic Properties of Scandium Substituted Terbium Iron Garnet Thin Films

The transverse electric field generated by anomalous Nernst effect (ANE) in a magnetic material is perpendicular to both the temperature gradient (▽T) and its magnetization (M), different from the parallel relationship between the generated electric field and ▽T emerges from Seebeck effect (SE). This allows different design principles for thermoelectric power generation and heat flux sensing devices, which could be advantageous in creating a simple and flexible device with better integratability and scalability.[1-3] However, the practical application of ANE is hampered by the fact that the thermopower of ANE (SANE) is still orders of magnitude smaller compared to that of SE (SSE), and there has been increasing studies in order to find magnetic materials with large SANE in recent years. SANE can be separated into two components as SANE = ρxxαxy − ρAHEαxx, where ρxx is the longitudinal resistivity, ρAHE is the anomalous Hall resistivity, and αxx and αxy are the diagonal and off-diagonal components of the Peltier tensor, respectively. The first term on the right hand side (ρxxαxy) is considered the intrinsic term of ANE, as αxy directly converts ▽T into a transverse electrical current. The L21-ordered Co2MnGa Heusler alloys have shown large values of αxy originated from their topological band structures, which lead to record high values of SANE.[4-7] The second term on the right hand side (− ρyxαxx) can be rewritten as – SSE × ρAHE / ρxx, indicating it originates from the anomalous Hall effect (AHE) of the longitudinal carrier flow induced by SE. Here, inspired by the second term, we propose a different approach for transverse thermoelectric generation. We considered a hybrid system comprising a thermoelectric material and a magnetic material. The electric field generated by SE of the thermoelectric material was converted to transverse thermoelectric generation by AHE of the magnetic material, which showed a similar symmetry to ANE, but surprisingly, more than one order of magnitude enhancement in thermopower. This is referred to as the Seebeck-driven transverse thermoelectric generation (STTG).[8] To model STTG and derive an expression for its transverse thermopower, we consider a system that consists of a thermoelectric material and a magnetic material electrically connected at both ends along the ▽T direction to form a closed circuit, as shown in Fig. 1. Both materials have a rectangular shape, with LTEx, LTEy, and LTEz (LMx, LMy, and LMz) respectively being the length along x, y, and z directions for the thermoelectric (magnetic) material. The magnetic material has the Seebeck coefficient SM, anomalous Nernst coefficient SANE, resistivity ρM, and anomalous Hall resistivity ρAHE. The thermoelectric material has the Seebeck coefficient STE, and resistivity ρTE. By solving the linear-response transport equations for the hybrid system, we derived the total transverse thermopower as Stoty = (EMy / –▽T) = SANE – ρAHE (STE – SM) / ((ρTE / r) + ρM), where EMy is the transverse electric field along the y direction in the magnetic material and r = (LMx / LTEx) × (LTEyLTEz / LMyLMz) is the size ratio between the thermoelectric and magnetic materials. In addition to ANE from the magnetic material, Stoty includes the contribution from the artificial hybridization of SE of the thermoelectric material through STE and AHE of the magnetic material through ρAHE (the second term on the right hand side), which is STTG. For the proof-of-concept demonstration of STTG, we deposited a 50-nm-thick L21-ordered Co2MnGa thin film on a single-crystalline MgO (100) substrate and used it as the magnetic material. The MgO substrate was then cut into a rectangular shape and bonded onto a rectangular-shaped 0.5-mm-thick Si substrate, which was used as the thermoelectric material. The Co2MnGa film and Si substrate were electrically connected to each other through Ohmic contacts and bonding wires near the ends to form the closed circuit. EMy in the Co2MnGa layer was measured while the magnetic field (H) was swept in the direction perpendicular to the sample plane. The same measurement was also carried out without the Co2MnGa-Si connection. Figure 2 shows the H dependence of EMy divided by ▽T for the samples using whether an n-type Si (STE = −1.3×103 µV K−1) or a p-type Si (STE = 1.3×103 µV K−1) as the thermoelectric material. Without the Co2MnGa-Si connection, only the ANE contribution from the Co2MnGa layer was measured, which showed SANE ~ 6.2 µV K−1, consistent with previous reports.[4-7] With the Co2MnGa-Si connection, however, the total transverse thermopower Stoty increased by one order of magnitude to 82.3 µV K−1 for using n-type Si; while Stoty changed sign to −41.0 µV K−1 for using p-type Si. The experimentally obtained Stoty is consistent with the predicted values made using the equation and the properties of the Co2MnGa and Si, which is 77.2 (−45.3) µV K−1 for Co2MnGa/n(p)-type-Si system. Our result indicates that STTG can make significant enhancement from ANE in the transverse thermopower by optimizing transport properties and dimensions of the materials, which would provide great flexibility for designing future thermoelectric power generation and heat flux sensing devices.

**References:**

[1] K. Uchida, H. Adachi, T. Kikkawa et al., Proc. IEEE, Vol. 104, p.1946 (2016). [2] Y. Sakuraba, Scr. Mater., Vol. 111, p.29 (2016). [3] W. Zhou, and Y. Sakuraba, Appl. Phys. Express, Vol. 13, p.043001 (2020). [4] A. Sakai, Y. P. Mizuta, A. A. Nugroho et al., Nat. Phys., Vol. 14, p.1119 (2018). [5] S. N. Guin, K. Manna, J. Noky et al., NPG Asia Mater., Vol. 11, p.16 (2019). [6] L. Xu, X. Li, L. Ding et al., Phys. Rev. B, Vol. 101, p.180404(R) (2020). [7] K. Sumida, Y. Sakuraba, K. Masuda et al., Commun. Mater., Vol. 1, p.89 (2020). [8] W. Zhou, K. Yamamoto, A. Miura et al., Nat. Mater. (2021), DOI: 10.1038/s41563-020-00884-2.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/af67d3057028ca8c7187a10cb62d178a.jpg)
Schematic illustration of the model system for Seebeck-driven transverse thermoelectric generation.

![](https://assets.underline.io/uploads/markdown_image/1/image/d8560fa996a7031109c92672c1f01103.jpg)
(a) H dependence of the transverse electric field EMy divided by ▽T in the Co2MnGa layer of the Co2MnGa/n-type-Si sample and (b) Co2MnGa/p-type-Si sample with (STTG + ANE) and without (ANE) the Co2MnGa-Si connection.

 Seebeck-driven Colossal Transverse Thermoelectric Generation

Spintronics combining with superconductor electronics is of exciting and challenging subject because of its intriguing and amazing properties. Spin-polarized current injected from the ferromagnetic metal can be carried by quasi-particles in a superconductor. The injected quasiparticles diffusively flow and gradually convert the Cooper pairs within a few micron length scale. Since a conventional Cooper pair consists of a pair of up- and down-spin electrons, the spin-polarized current is expected to take longer time to be transformed to the supercurrent. To investigate the relaxation processes of spin-polarized current in superconductor, we developed a Nb-based multi-terminal device with a ferromagnetic CoFeAl spin injector as schematically shown in Fig. 1. The injected quasiparticles relax with decaying electrochemical potentials. In the probe configuration shown in Fig. 1, the decreasing non-local quasiparticles electrochemical potentials can be detected as non-local resistances. Moreover, by changing the injector from Cu to CoFeAl, we can compare the relaxation lengths for non-polarized and polarized quasiparticles. Figure 2 shows non-local resistances measured with nonmagnetic injectors with changing the distance from 0 to 3000 nm. Non-local resistance depends mainly on a charge imbalance which are a nonequilibrium phenomenon that arises from the conversion of quasiparticle current in the normal metal to supercurrent in the superconductor [1]. From the electrodes-distance dependence of non-local resistances, we estimated the relaxation length of non-polarized quasiparticles is 950 ± 30nm, while that of polarized quasiparticles is 1240 ± 130nm. This difference can be explained by the spin diffusion length at the vicinity of superconducting transition temperature which is longer than that in a normal conducting state.

**References:**

[1] P. Cadden-Zimansky and V. Chandrasekhar, Phys. Rev. Lett. 97, 237003 (2006).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6a7ff821738fccf3134555c8773ca6f1.jpg)
Fig.1 Multi-electrode mesoscopic device

![](https://assets.underline.io/uploads/markdown_image/1/image/1a92e23000507f88fdf5c54608e252c3.jpg)
Fig. 2 Temperature dependence of non-local resistances with various distances between an injector and detectors (top), and the electrodes-distance dependence of the maximum values of non-local resistances (bottom) for (A) Non spin polarized quasiparticles and (B) Spin polarized quasiparticles.

 Relaxation Process of Spin-polarized Quasiparticles in a Superconducting Nb Wire

In spintronics, Cu is often considered as a passive material which cannot generate spin current by itself due to negligible spin-orbit coupling (SOC). However, recent experiments reported observation of gigantic current-induced spin-orbit torque in surface oxidized Cu/ferromagnet structures [1-5], whose magnitude is comparable to material systems with large SOC such as Co/Pt. These experimental results cannot be explained neither by spin Hall effect nor by Rashba-Edelstein effect since these mechanisms assume strong SOC. On the other hand, electric generation of the orbital angular momentum does not require SOC at all, which can be achieved via orbital Hall effect [6] or orbital Rashba-Edelstein effect [7]. Furthermore, according to a recent theory, injection of orbital angular momentum to a ferromagnet can generate torque on the magnetization, which is called orbital torque (OT) [8]. Thus, it has been proposed that a microscopic mechanism responsible for the torque generation in surface oxidized Cu/ferromagnet is OT, originating in the orbital Rashba effect (ORE) at oxidized region of Cu film. The OT mechanism qualitatively explains anomalous features observed in experiments such as strong suppression of the torque by intermixing of atoms at the interface [3], sign change of the torque for depending on ferromagnets [4], and gigantic enhancement of the torque efficiency by insertion of an ultrathin Pt layer [5]. However, to our best knowledge, there is no direct evidence of the ORE nor theoretical investigation of the electronic structure of Cu film under influence of surface oxidization. In this contribution, we present our recent theoretical results showing that surface oxidization of a Cu film leads to strong enhancement of the ORE [9], which is assessed from first-principles calculations based on density functional theory. We find that microscopic origin of the ORE is the resonant hybridization between O p-orbitals and Cu d-orbitals. We investigate various stacking geometries (fcc hollow, hcp hollow, bridge, on-top structures) and find that the ORE emerges in a robust manner. The ORE leads to a pronounced orbital texture at the Fermi surface (Fig. 1), where the magnitude of the orbital polarization reaches as much as 0.5 \hbar per state. Because of the orbital texture, application of an external electric field leads to a generation of orbital Hall current, which by far exceeds the magnitude of the spin current of heavy elements. This implies that the OT mechanism can be significant in surface oxidized Cu/ferromagnet structures. Our theoretical results also encourage experimental verification of the orbital texture, i.e. by angle-resolved photoesmission spectroscopy or momentum microscopy.

**References:**

[1] An et al. Spin-torque generator engineered by natural oxidization of Cu, Nat. Commun. 7, 13069 (2016). [2] Otano et al. Nonreciprocal Spin Current Generation in Surface-Oxidized Copper Films, Phys. Rev. Lett. 122, 217701 (2019). [3] Kim et al. Non-trivial charge-to-spin conversion in ferromagnetic metal/Cu/Al2O3 by orbital transport, Phys. Rev. B, in press. [4] Tazaki et al. Current-induced torque originating from orbital current, arXiv:2004.09165. [5] Ding et al. Harnessing Orbital-to-Spin Conversion of Interfacial Orbital Currents for Efficient Spin-Orbit Torques, Phys. Rev. Lett. 125, 177201 (2020). [6] D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic Spin and Orbital Hall Effects from Orbital Texture, Phys. Rev. Lett. 121, 086602 (2018). [7] Go et al. Toward surface orbitronics: giant orbital magnetism from the orbital Rashba effect at the surface of sp-metals, Sci. Rep. 7, 46742 (2017). [8] D. Go and H.-W. Lee, Orbital torque: Torque generation by orbital current injection, Phys. Rev. Res. 2, 013177 (2020). [9] Go et al. Orbital Rashba effect in surface oxidized Cu film, arXiv:2011.08601.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4a85b5b77611539a80e30b149ba500dc.jpg)
Orbital angular momentum texture of the surface oxidized Cu film at the Fermi surface. The colormap represents the magnitude and the arrow indicates direction of the angular momentum at each k-point.

Theoretical Study on Four-fold Symmetric Anisotropic Magnetoresistance Effect in Cubic Single-crystal Ferromagnetic Model

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