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The intrinsic bulk spin Hall effect plays an important role in spintronics applications, such as spin-orbit torque-based memory. The bulk space group symmetry ultimately determines the form of spin current conductivity tensor under an applied electric field. The high symmetry normally present in heavy metals, such as Pt, only allows the electric-field-induced transverse spin current flow with spin polarization perpendicular to both electric field and spin current flow directions. Recent studies of spin Hall effects in semi-metallic transition metal dichalcogenides demonstrate the existence of additional spin conductivity tensor components when the crystal symmetry is reduced. In this paper we present an explicit first-principles method to compute the site-resolved spin current conductivity, which may be different from the bulk form. Because the local point group symmetry of individual atom is lower compared to the global space group symmetry, the unconventional spin current conductivity tensor component is staggered in the real-space, and hidden in the direct bulk spin Hall calculations. Our layer-resolved numerical study of 1T&#39;-\ch{WTe2} shows a uniform conventional spin Hall conductivity tensor component $\sigma^y_{xz}$ and a staggered unconventional component $\sigma^z_{xz}$. The magnitude of these two components, around 100 and 20 $\hbar/2e~(\rm \Omega\cdot cm)^{-1}$, are comparable to the spin-orbit torque exerted on adjacent ferromagnets. We present another example of orthorhombic PbTe in which both uniform and staggered spin current conductivity are one order larger. The position-dependent intrinsic spin current conductivity can broaden the applications of spin Hall effect.

MMM 2020

Hidden (staggered) spin hall conductivity

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MMM 2022

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The intrinsic bulk spin Hall effect plays an important role in spintronics applications, such as spin-orbit torque-based memory. The bulk space group symmetry ultimately determines the form of spin current conductivity tensor under an applied electric field. The high symmetry normally present in heavy metals, such as Pt, only allows the electric-field-induced transverse spin current flow with spin polarization perpendicular to both electric field and spin current flow directions. Recent studies of spin Hall effects in semi-metallic transition metal dichalcogenides demonstrate the existence of additional spin conductivity tensor components when the crystal symmetry is reduced. In this paper we present an explicit first-principles method to compute the site-resolved spin current conductivity, which may be different from the bulk form. Because the local point group symmetry of individual atom is lower compared to the global space group symmetry, the unconventional spin current conductivity tensor component is staggered in the real-space, and hidden in the direct bulk spin Hall calculations. Our layer-resolved numerical study of 1T'-\ch{WTe2} shows a uniform conventional spin Hall conductivity tensor component $\sigma^y_{xz}$ and a staggered unconventional component $\sigma^z_{xz}$. The magnitude of these two components, around 100 and 20 $\hbar/2e~(\rm \Omega\cdot cm)^{-1}$, are comparable to the spin-orbit torque exerted on adjacent ferromagnets. We present another example of orthorhombic PbTe in which both uniform and staggered spin current conductivity are one order larger. The position-dependent intrinsic spin current conductivity can broaden the applications of spin Hall effect.

technical paper

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The electrical detection of ferromagnetic resonance through the spin rectification (SR) effect has inspired great interest to investigate the magnetization dynamics in the last two decades. Furthermore, a comprehensive understanding of SR is essential to study the spin pumping and spin Hall effects. Recently, the single crystalline Co0.5Fe0.5 (001) alloy film has attracted attention due to the giant damping anisotropy and current-orientation effect of anisotropic magnetoresistance [Phys. Rev. Lett. 122, 117203 (2019)]. In this experimental work, we studied the SR dependence on the magnetization and current orientations in the single crystalline Co0.5Fe0.5 (001) alloy film. We present the experimental evidence that the magnetization-orientation dependence of the SR signal in Co0.5Fe0.5 strongly differs from those reported in the polycrystalline films. The strong anisotropy ratio of the damping constant up to 520% is determined, which can induce the strong anisotropy of the resonant susceptibility driven by the microwave field. By considering the current-orientation dependent anisotropic magnetoresistance and the anisotropic susceptibility in single crystalline Co0.5Fe0.5, we explain such unusual angular dependent SR signal by the SR theory. Our measurements show the amplitude of the SR signal could have 13 times difference for the devices along the <110> and <100> directions, which introduces the crystalline symmetry as a new approach to engineer the microwave characteristics in microwave spintronics.

Role of crystalline and damping anisotropy to the angular dependences of spin rectification effect in single crystal CoFe film

Hybrid magnonic systems have recently attracted wide attentions due to due to rich potential applications in coherent information processing. One particularly interesting platform is the thin-film-based magnon-magnon hybrid system, which has been recently explored in magnetic bilayers [1-4]. Because magnon excitations are confined within the magnetic media, it is convenient to build up more compact hybrid platforms in micron scale compared with millimeter-scale microwave circuits. Furthermore, the abundant spintronic properties, such as spin-torque manipulation and spin pumping, can be used to control and engineer the hybrid dynamics especially for magnetic thin-film devices. In this work, we demonstrate electrical detection of strongly coupled magnon-magnon hybrid system in yttrium iron garnet/permalloy (YIG/Py) bilayer devices. Direct microwave current injection through the conductive Py layer drives the hybrid dynamics consisting of the uniform mode of Py and the first standing spin wave (n=1) mode of YIG, which are strongly coupled via interfacial exchange coupling. The hybrid dynamics is detected via the spin rectification signal from Py, which provides phase resolution of the coupled dynamics. For the Py-dominated mode we measure a constant phase evolution. For the YIG-dominated mode a π phase shift is measured across the avoided crossing, which agrees with theoretical prediction. In addition, the characterization of nonlocal devices consisting two Py devices on a YIG base device shows that the two Py devices are coupled by inductive coupling, allowing spin rectification measurements of the nonlocal Py device. Our results provide a new device platform for exploring hybrid magnonic dynamics and probing their phases, which are crucial for implementing coherent information processing with magnon excitations. Sample preparation and measurements were supported by the US Department of Energy, Basic Energy Sciences, Materials Science and Engineering Division. Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.References: [1] S. Klingler, et al., Phys. Rev. Lett. 120, 127201 (2018). [2] J. Chen, et al., Phys. Rev. Lett. 120, 217202 (2018). [3] H. Qin, et al., Sci. Rep. 8, 5755 (2018). [4] Y. Li, et al., Phys. Rev. Lett. 124, 117202 (2020)

Phase-resolved electrical detection of strongly coupled magnon-magnon hybrid device

Reducing energy dissipation while increasing speed in computation and memory is a long-standing challenge for spintronics research [1]. In the last 20 years, femtosecond lasers have emerged as a tool to control the magnetization in specific magnetic materials at the picosecond timescale [2]. However, the use of ultra-fast optics in integrated circuits and memories would require a major paradigm shift. An ultrafast electrical control of the magnetization [3] is far preferable for integrated systems. Here we demonstrate reliable and deterministic control of the out-of-plane magnetization of a 1 nm-thick Co layer with single 6 ps-wide electrical pulses that induce spin orbit torques on the magnetization. We can monitor the ultrafast magnetization dynamics due to the spin orbit torques with sub-picosecond resolution, thus far accessible only by numerical simulations. Due to the short duration of our pulses, we enter a counter-intuitive regime of switching where heat dissipation assists the reversal. Moreover, we estimate a low energy cost to switch the magnetization, below 50 pJ for our micrometer sized device. These experiments show that spintronic phenomena can be exploited on picosecond time-scales for full magnetic control and should launch a new regime of ultrafast spin torque studies and applications.References: [1] Åkerman, J. Toward a universal memory. Science. 308, 508–510 (2005). [2] Kirilyuk, A., Kimel, A. V & Rasing, T. Ultrafast optical manipulation of magnetic order. Rev. Mod. Phys. 82, (2010). [3] Garello, K. et al. Ultrafast magnetization switching by spin-orbit torques. Appl. Phys. Lett. 105, 1–12 (2014).

Picosecond Spin Orbit Torque Switching

Vortex-based magnetic tunnel junctions (MTJs) are a rich dynamic system with a variety of different resonant modes, spanning a wide frequency spectrum from hundreds of MHz to tens of GHz [1, 2]. The diverse spectral properties of vortex-based MTJs makes them potential candidates for high frequency signal detectors [3, 4], with a dc rectification ‘spin-diode’ effect resulting when one of the resonant modes are excited with an rf current passing across the MTJ. In this study, we investigate the spin-wave modes present in a vortex-based MTJ, by analysis of the spin-diode rectification effect and by comparison with Fourier analysis of micromagnetic simulations. We demonstrate that the nature of the spin wave mode is strongly related to the position of the vortex core, varying from a pure azimuthal mode when the core is located at the centre of the MTJ, to a more standing wave edge mode when the core is strongly displaced to the edge. Furthermore, we demonstrate that the dominant excitation mechanism also changes depending upon core position, with the azimuthal mode being more sensitive to the field-like spin torque effect, and with the edge mode being more susceptible to the radial Oersted field. In addition to analysing the nature and the excitation scheme of these modes, we also show how their resonant excitation can lead to the modification of the critical current of the steady state excitation of the gyrotropic mode. The ability to control the gyrotropic mode via excitation of the higher order mode gives rise to a frequency down converter which can demodulate a signal with a carrier signal at 7.7 GHz with a modulation rate of 1 MHz. By harnessing the higher order spin wave modes present in vortex-based MTJs, new radio-frequency technologies such as signal detection and demodulation, which target GHz frequency bands can be envisaged. VC and PB acknowledge the ANR project “SPINNET” ANR-18-CE24-0012.References: [1] K. Y. Guslienko, A. N. Slavin, V. Tiberkevich, and S. K. Kim, “Dynamic origin of azimuthal modes splitting in vortex-state magnetic dots,” Phys. Rev. Lett., 101, 24, 1–4, 2008. [2] M. Kammerer et al., “Magnetic vortex core reversal by excitation of spin waves,” Nat. Commun., 2, 1, 279, 2011. [3] - A. S Jenkins et al., “Wideband High-Resolution Frequency-to-Resistance Converter Based on Nonhomogeneous Magnetic-State Transitions”, Phys. Rev. Applied 13, 014046, (2020) [4] - A. S Jenkins et al., “Spin-torque resonant expulsion of the vortex core for an efficient radiofrequency detection scheme”, Nature Nanotechnology volume 11, 360–364, (2016)

Controlling the gyrotropic critical current by resonant excitation of the higher order spin wave modes in vortex-based magnetic tunnel junctions

Recently, the Co8Zn8Mn4 alloy was found to host the skyrmion crystal phase at room temperature [1,2]. Understanding the GHz spin dynamics and magnonic eigenmodes of this phase is key for developing room temperature applications in spintronics and magnonics. Here, we report the broadband microwave absorption spectroscopy measurement of bulk Co8Zn8Mn4 crystal, mounted onto a coplanar waveguide (CPW). We have determined the resonance frequencies of the eigen-excitations of the various magnetic phases in Co8Zn8Mn4 - at room temperature - including the helical, skyrmion crystal, conical and fully polarized magnetic order (Fig. 1 and Fig. 2). In particular, we found that the clockwise, counter-clockwise and breathing modes of the skyrmion crystal magnetic configuration resides between ~0.8 and 2.5(2) GHz. The frequencies of the modes depend on the orientation of the applied field with respect to the CPW and the long axis of the crystal. We account for the frequency variations by means of magnetic anisotropies. Furthermore, we augment our results with single crystal x-ray diffraction measurements and temperature-dependent magnetic susceptometry. We thank S. Watanabe, P. R. Baral and I. Zukovic for experimental support. The research was financially supported by the SNSF (grant no. 171003 - sinergia project Nanoskyrmionics) and the EPSRC (grant no. EP/M020517/1).References: 1. Y. Tokunaga, X. Z. Yu, J. S. White, et al. Nature Communication, Vol. 6 p. 7638 (2015) 2. K. Karube, J. S. White, N. Reynolds, et al. Nature Materials, Vol. 15, p. 4752 (2016)

Magnetization Dynamics of Helical and Skyrmion Phases in Bulk Co8Zn8Mn4 at Room Temperature

The finding of skyrmions at room temperature has boosted the potentiality of skyrmionic devices. From the theoretical point of view, the temperature in micromagnetic systems has been taken into account either using stochastic spin dynamics, the stochastic Landau-Lifshitz equation, or the Landau-Lifshitz-Block equation. Also, the rigid model has been used to describe the skyrmion dynamics taking into account thermal effects, using the stochastic Thiele equation [1]. Here we present a deterministic method to study the skyrmion dynamics [2], an approach in which the temporal motion of skyrmions can be studied without relying on repetition of stochastic simulations, but on solving once the associated deterministic Fokker-Planck equation. This is a substantial advance, since by finding and solving a single partial differential equation, one could obtain all the relevant probabilistic information of the system. Indeed, within the rigid approximation, the probability density for the presence of skyrmions under any external potential can be found at any time and position, and as a function of the temperature. This allows one to evaluate the probability of presence/survival trapping/escaping ... of skyrmions in many practical situations, which is a key information to ensure the viability of applications. In particular, the probability of trapping/escaping of a skyrmion that encounters a pinning site, and the probability of survival of a skyrmion along a track are presented as a function of temperature.References: [1] D. Pinna, F. Abreu Araujo, J. Grollier, Phys. Rev. Applied., Vol. 9, (2018) [2] J. Castell-Queralt, L. González-Gómez, C. Navau. Phys. Rev. B 101, 140404 (R) (2020)

Deterministic Approach to Skyrmionic Dynamics at Nonzero Temperatures: Pinning Sites and Racetracks

Skyrmions in antiferromagnetic (AFM) materials with the Dzyaloshinskii-Moriya (DM) interaction are expected to exist for essentially the same reasons as in DM ferromagnets (FM). On the other hand, the dynamics of solitons in AFM is substantially different [1,2]. We give the fundamental formula and a corresponding argument to show that traveling solitary waves (such as skyrmions) are possible in an AFM in contrast to the dynamics observed in FM. The formula shows that the skyrmion number, known to be linked to the dynamics of topological solitons in FM, is, here, unrelated to the dynamical behavior. We calculate numerically the configuration of skyrmions in chiral antiferromagnets that are traveling as solitary waves. They are found to be elongated and they expand in size with the velocity (Fig. 1). The velocity takes values from zero up to a maximum value, which depends on a scaled DM parameter λ. The maximum skyrmion velocity is found analytically as a function of λ. The formula is based on a strong theoretical argument that links the breakdown of a traveling skyrmion with the DM parameter value λ=2/π for the transition from the uniform to the spiral state. The energy and the linear momentum of an AFM skyrmion lead to a proper definition of its mass. We give the details of the energy-momentum dispersion of traveling skyrmions and explore their particle-like character based on exact relations. The traveling skyrmion behaves as a Newtonian particle for small velocities, while it presents relativistic behavior for velocities close to the critical. It is notable that the energy-momentum formula differs from the standard relativistic one by a constant shift of the energy. The solitonic and particle-like behavior of skyrmions in AFM is in stark contrast to the dynamical behavior of their FM counterparts [3]. We acknowledge support by the project ThunderSKY, funded by HFRI and GSRT, under Grant No. 871.References: [1] E. G. Galkina and B. A. Ivanov, Low Temp. Phys. 44, 618 (2018) [2] H. Velkov, O. Gomonay, M. Beens, et al, New J. Phys. 18, 075016 (2016) [3] S. Komineas and N. Papanicolaou, SciPost Phys. 8, 086 (2020)

Traveling Skyrmions in Chiral Antiferromagnets

The bursty motion of magnetic domain walls (DWs) in disordered ferromagnets, known as Barkhausen noise, exhibits critical behavior in its depinning dynamics, giving rise to universal power laws [1]. For example, the average DW velocity V depends on the external magnetic field B as (B-Bd)θ for a depinning field Bd and a critical exponent θ [2], and the distribution of velocity burst sizes S goes like P(S) ~ S-τ for a critical exponent τ [3]. This is similar to many critical systems in statistical physics, ranging from earthquakes and geophysical flows [4] to dislocation avalanches in plastically deforming crystals [5]. However, current studies exploring the critical depinning dynamics of Barkhausen noise using micromagnetic modeling is severely limited by small system sizes [6]. We develop an effective model for the motion of domain walls in perpendicular magnetic anisotropy (PMA) thin films, by reducing the description of the magnet to a one-dimensional description of the DW, allowing access to large system sizes. For such Bloch-type DWs, the direction of in-plane magnetization is an important dynamical variable that leads to inertial effects in the DW dynamics, such as Walker Breakdown and the formation of Bloch lines (Fig. 1). We consider the critical depinning dynamics of such DWs in quenched disorder. For weak disorder strengths, the internal dynamics are suppressed, so the DW motion reduces to the quenched Edwards-Wilkinson (qEW) equation. With increasing disorder strength, the internal dynamics of the DW become increasingly important, leading to nucleation of Bloch lines. Using our effective model to access large system sizes, we characterize the critical exponents of the depinning problem, and explore the effect of internal dynamics and Bloch lines on the critical behavior.References: [1]: S. Zapperi et. al., Phys. Rev. B, Vol. 58, p. 6353 (1998) [2]: R. D. Pardo et. al., Phys. Rev. B, Vol. 95, p. 184434 (2017) [3]: D. Spasojević et. al., Phys. Rev. E, Vol. 54, p. 2531 (1996) [4]: J. P. Sethna, K. A. Dahmen and C. R. Myers, Nature, Vol. 410, p. 242 (2001) [5]: See e.g., P. Moretti et. al., Phys. Rev. B, Vol. 69, p. 214103 (2004) [6]: T. Herranen and L. Laurson, Phys. Rev. Lett, Vol. 122, p. 117205 (2019)

Critical depinning dynamics of domain walls with internal degrees of freedom

Spin transfer across the interface of magnetic insulator/heavy metal is dominated by the magnetic state at interface. Utilizing aberration corrected scanning transmission electron microscopy and electron energy loss spectroscopy, we demonstrate the existence of metallic Fe0 atoms at YIG/Pt interface. The observed spin pumping signal is linearly proportional to total Fe0 magnetic moments at YIG/Pt interface, revealing the critical role of the interfacial magnetic moments on spin transfer. In addition, PtFe alloying might contribute to a new channel of spin sink at YIG/Pt interface.References: Xingtao Jia, Kai Liu, Ke Xia and Gerrit E. W. Bauer, EPL, Vol.96, p.17005 (2011)

Interfacial modulation of the spin pumping in YIG/Pt

Efficient spin generation methods like spin Hall effect (SHE) are being explored for the realization of next-generation spintronics devices[1]. SHE has been studied mainly in non-magnetic materials. However, because of the recent discoveries of various novel and promising effects in magnetic materials, such as anomalous spin-Hall effect (ASHE)[2,3], magnetic spin-Hall effect (MSHE)[4,5], and the spin-current driven by anisotropic magnetoresistance (AMR)[2,6], the spin-currents in magnetic materials have attracted much attention both from fundamental and applied viewpoints. Although the ASHE was initially thought to be a spin current driven by the anomalous Hall effect (AHE), some experimental results show an unaccountable spin current contribution that cannot be explained by AHE alone, and this is still an open question[7]. The theoretical studies of MSHE have so far assumed spin-independent scattering and focused largely on antiferromagnets. In ferromagnetic materials, the spin-dependent scattering can play an important role, however. In this study, we investigate the spin-dependent scattering mechanisms of ASHE and MSHE by using a microscopic model that assumes s-d scattering in 3d ferromagnetic metal (FM). The electrons in FM are described within the framework of the impurity Anderson model with spin splitting and spin-orbit coupling for 3d orbitals[8]. The spin-conductivities are calculated by using Kubo formula. We obtained both contributions of ASHE and MSHE (Fig.1) and showed that these phenomena are caused by anisotropic spin-flip scattering (Fig. 2). In particular, we found this mechanism of ASHE is different from that of conventional extrinsic SHE such as skew-scattering and side-jump. In the conference, we will present a complete expression of the magnetization direction dependence of these phenomena. In addition, we will discuss the effect of the crystal-field splitting.References: [1] J. Sinova, S. O. Valenzuela, J. Wunderlich, et al., Rev. Mod. Phys., Vol. 87, p.1213 (2015), [2] T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev. Appl., Vol. 3, p.044001 (2015), [3] C. Qin, S. Chen, Y. Cai, et al., Phys. Rev. B, Vol. 96, p.134418 (2017), [4] J. Zelezny, Y. Zhang, C. Felser, and B. Yan, Phys. Rev. Lett., Vol. 119, p.187204 (2017), [5] M. Kimata, H. Chen, K. Kondou. et al., Nature, Vol 565, p.627 (2019), [6] C. Safranski, E. A. Montoya, and I. N. Krivorotov, Nature Nano., Vol. 14, p.27 (2019), [7] Y. Omori, E. Sagasta,Y. Niimi. et al., Phys. Rev. B, Vol. 99, p. 014403 (2019), [8] Y. Yahagi, D. Miura, and A. Sakuma, J. Phys. Soc. Jpn., Vol. 89, p.044714 (2020).

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