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Interaction of two skyrmions stabilized by the ferromagnetic exchange, Dzyaloshinskii-Moriya interaction (DMI), and external magnetic field has been studied numerically on a 2D lattice of size large compared to the separation, $d$, between the skyrmions. We show that two skyrmions of the same chirality (determined by the symmetry of the crystal) repel. In accordance with earlier analytical results, their long-range pair interaction falls out with the separation as $\exp(-d/\delta_H)$, where $\delta_H$ is the magnetic screening length, independent of the DMI. The prefactor in this expression depends on the DMI that drives the repulsion. The latter results in the spiral motion of the two skyrmions around each other, with the separation between them growing logarithmically with time. When two skyrmions of the total topological charge $Q = 2$ are pushed close to each other, the discreteness of the atomic lattice makes them collapse into one skyrmion of charge $Q = 1$ below a critical separation. Experiment is proposed that would allow one to measure the interaction between two skyrmions by holding them in positions with two magnetic tips. Our findings should be of value for designing topologically protected magnetic memory based upon skyrmions.References: D Capic, DA Garanin and EM Chudnovsky, Skyrmion-skyrmion interaction in a magnetic film, Journal of Physics: Condensed Matter, https://doi.org/10.1088/1361-648X/ab9bc8 (2020). (Accepted Manuscript)

MMM 2020

Skyrmion-skyrmion interaction in a magnetic film

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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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).

Thoeretical Study on Anomalous Spin-Hall and Magnetic Spin-Hall Effect Induced by Anisotropic Spin-flip Scattering on Ferromagnets

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.

Hidden (staggered) spin hall conductivity

We present experimental evidence for coherent long-distance transport of angular momentum inside a non-magnetic dielectric via the coupling to circularly polarized sound waves that exceeds previous benchmarks set by magnon diffusion by orders of magnitude. Most of the spintronics devices rely on the delocalized electrons in metals to carry the spin information, which induces loss by Joule heating. It turns out that insulating magnetic materials, such as yttrium iron garnet (YIG), also allow the spin to propagate between localized magnetic moments via propagating spin-waves which transmit information from an atomic site to the other without any Joule effect. In principle, the induced angular momentum can also pass through a non-magnetic insulators by coupling to circular vibrations of the crystal lattice called chiral phonons. But the extent to which these phonons can mediate spin currents in a circuit remains to be demonstrated. In a recent experiment [1], a collaboration lead by researchers at Spintec used a microwave field to excite a current of chiral phonons through a half-millimeter thick layer of non-magnetic gadolinium gallium garnet sandwiched between two YIG films. By showing interference effects between the two magnetic layers, they demonstrated that circularly polarized acoustic phonons allow coherent transmission of spin information over millimeter distances. This work has since inspired theoretical work [2] showing the interest of using this discovery with an electric current which is more practical for powering spintronic devices.References: [1] K. An, A. N. Litvinenko, R. Kohno, et al. Phys. Rev. B 101, 060407(R) – Published 21 February 2020; [2] Rückriegel, A. & Duine, R. A. Physical Review Letters, American Physical Society (APS), 2020, 124

Coherent long-range transfer of angular momentum between magnon Kittel modes by circularly polarized phonons

Conversion of heat into spin current has recently drawn a lot of attention due to its potential usage on heat recovery via the spin Seebeck effect (SSE) [1] as well as fundamental interests including generation of spin super currents [2] and Bose-Einstein condensation of magnons [3]. The underlying mechansim for SSE is understood as a combination of interfacial and bulk (intrinsic) contriubtion. The interfacial SSE is driven by the tempearture drop at the interface between metal and magnetic layer while the bulk SSE is driven by diffusive thermal magnons. In the nonlocal spin transport measurements, the long range exponential decay of thermally generated spin signal was explained solely by the intrinsic SSE [4] while there are several studies showing that there is another decay mechanism in subhundred nanometer scale [5-7]. Here, we study the spatial decay of thermally excited magnon currents inside a thin magnetic insulator by focusing on the short-range behavior in the nonlocal geometry. We compare SSE signal on the same device before and after adding a nonmagnetic Al capping layer. The Al capping alters the thermal profile significantly near the heat source, which results in that the nonequilibrium thermal magnon profile deviates from an exponential decay and shows two sign reversals within 1 μm from the heat source. From simulated temperature profiles, we find the vertical temperature gradient also reverses twice within the same length scale in the Al deposited case. The correlation suggests that thermal magnons behave locally at this length scale. Using a phenomenological spin transport model, we calculate the spatial profile of nonequilibrium spin density. The observed short-range behavior can be accounted if the relavant decay length is on the order of subhundred nanometers. Our result shows the existance of the shorter decay length in the nonlocal magnon transport experiments. This suggests that the long-range SSE signal in previous measurements should be attributed to magnons with lower energy.References: [1] Boona, S. R., Myers, R. C., & Heremans, J. P. Energy & Environmental Science, 7(3), 885-910 (2014) [2] Bozhko, D. A., Serga, A. A., Clausen, P. et al., Nature Physics, 12(11), 1057-1062 (2016) [3] Tserkovnyak, Y., Bender, S. A., Duine, R. A. et al., Physical Review B, 93(10), 100402 (2016) [4] Cornelissen, L. J., Liu, J., Duine, R. A. et al., Nature Physics, 11(12), 1022-1026 (2015) [5] Kehlberger, A., Ritzmann, U., Hinzke, D. et al., Physical review letters, 115(9), 096602 (2015) [6] Prakash, A., Flebus, B., Brangham, J. et al., Physical Review B, 97(2), 020408 (2018) [7] Jamison, J. S., Yang, Z., Giles, B. L. et al., Physical Review B, 100(13), 134402 (2019)

Short-range Study of Thermal Magnon Transport in Magnetic Garnets

Usually negligible thermoelectric effects in superconductors can be increased using the simultaneous effects of spin splitting and spin filtering. These superconductor-ferromagnetic hybrids have been predicted to have thermal electric efficiencies (zT ~ 1.8) far exceeding any other cryogenic thermoelectric materials. Confirmation of these predictions requires controlled S-F interfaces, the generation of temperature gradients, and precise local cryogenic thermometry. S-F heterostructures have been fabricated using a combination of lithography, thermal evaporation, and magnetron sputtering, providing precise control over interfaces, materials, and geometries. Heaters and resistive thermometers, patterned via focused ion beam assisted platinum deposition have been calibrated and optimized for thermal transport measurements at cryogenic temperatures. Finally, the Seebeck coefficient of S-F heterostructures has been measured in the presence of a magnetic field. This measurement framework and measurement set the stage for additional studies of the effects of dimensionality, ferromagnet polarization, and interface quality.References: P. Machon, M. Eschrig, and W. Belzig, “Nonlocal thermoelectric effects and nonlocal onsager relations in a three-terminal proximity-coupled superconductor-ferromagnet device,” Phys. Rev. Lett., vol. 110, p. 47002, 2013. P. Machon, M. Eschrig, and W. Belzig, “Giant thermoelectric effects in a proximity couple superconductor- ferromagnet device,” New J. Phys., vol. 16, p. 073002, 2014.

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