Encoding orbital angular momentum of light in magnets
Breaking the diffraction limit and focusing laser beams to subwavelength scale are becoming possible with the help of recent developments in plasmonics. Such subwavelength focusing bridges different length scales of laser beams and matter. Here we consider optical vortex, or laser beam carrying orbital angular momentum (OAM) and discuss potential subwavelength magnetic phenomena induced by such laser. On the basis of numerical calculations using Landau-Lifshitz-Gilbert equation, we propose two OAM-dependent phenomena induced by optical vortices, generation of radially anisotropic spin waves and generation of topological defects in chiral magnets. The former could lead to the transient topological Hall effect through the laser-induced scalar spin chirality, and the latter reduces the timescale of generating skyrmionic defects by several orders compared to other known means.
PhySH: Angular momentum of light, Ultrafast phenomena, Spin waves, Skyrmions, LLG
Introduction— Realization of ultrashort laser pulses with femto- to pico- second width now offers a powerful tool for the study of non-equilibrium, ultrafast phenomena in solids. Since the pioneering observation of ultrafast demagnetization in Nickel by Beaurepaire et al in 1996 Beaurepaire et al. (1996), such highly non-equilibrium, laser-induced physics is one of the most important research subjects in the field of magneto-optics Kirilyuk et al. (2010); Stanciu et al. (2007); Satoh et al. (2012); Finazzi et al. (2013); Caviglia et al. (2013); Mikhaylovskiy et al. (2015); Subkhangulov et al. (2016); Sato et al. (2014); Takayoshi et al. (2014a, b); Mentink et al. (2015); Pimenov et al. (2006); Mochizuki and Nagaosa (2010); Takahashi et al. (2011); Sato et al. (2016).
So far, most of ultrafast magneto-optical phenomena in solids are explored using laser beams with Gaussian spatial profile. In 1992, however, Allen et al. proposed Allen et al. (1992) a new type of laser beam, now called optical vortex Andrews and Babiker (2012). Optical vortex is a beam carrying orbital angular momentum (OAM), which can be transferred to physical systems as, for example, mechanical rotation of (semi-) classical particles Friese et al. (1998); He et al. (1995); Paterson et al. (2001); Shigematsu et al. (2016). In the past decades, many applications such as super-resolution microscope Bretschneider et al. (2007) and optical ablation Toyoda et al. (2012); Hamazaki et al. (2010) are developed. However, the use of optical vortex for controlling microscopic magnetic degrees of freedom of solids is almost unexplored.
The main difficulty in is in the mismatch between the timescales. Typically, the timescale of spin dynamics is in the Tera Hz (THz) or Giga Hz region, but the wavelength of optical vortex of such frequencies is too large for individual spins to “feel” its characteristic spatial profile. Using heating effect, as discussed in Ref. Fujita and Sato, 2017 is one option to resolve this issue, but here we take another approach: breaking the diffraction limit Maier and Atwater (2005); Hibbins et al. (2004); Huang and Zhu (2007); Gramotnev and Bozhevolnyi (2010) of THz beams to realize the subwavelencth optical vortices Heeres and Zwiller (2014). Stimulated by the enormous successes for the microwave and visible light, breaking the diffraction limit of THz beams is actively explored and the technologies are rapidly developing recently with the help of plasmonics. We can now design the spatial profile of THz beams at the subwavelength scale Morimoto et al. (2016) and actually focus the THz optical vortices Arikawa et al. (2017) using designed metallic structures such as an array of antennas. Although the technology is still primitive, the prospect for the deep subwavelength focusing of THz beams stimulates studies of ultrafast magnetic phenomena induced by such beams.
In this paper, with numerical calculations based on the Landau-Lifshitz-Gilbert (LLG) equation, we seek the ultrafast manipulation of magnets through Zeeman and magneto-electric (ME) coupling between spins and the subwavelength optical vortices. We find that optical vortex can be used to excite multipolar and spiral spin waves with OAM dependent wavefronts, which enable us to dynamically generate inhomogeneous spin texture and would induce the transient topological Hall effect Ye et al. (1999); Ohgushi et al. (2000); Taguchi et al. (2001); Shindou and Nagaosa (2001); Tatara and Kawamura (2002); Neubauer et al. (2009).
Moreover, for chiral ferromagnets with Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinsky (1958); Moriya (1960) we observe OAM-dependent generation of topological magnetic defects such as skyrmioniums Bogdanov and Hubert (1994); Finazzi et al. (2013) and skyrmions Bogdanov and Pfleiderer (2006); Mühlbauer et al. (2009); Fert et al. (2013); Seki and Mochizuki (2016), both of which are prospected as ingredients of future magnetic memory devices. We find that OAM of lasers can be transferred as the topological number, i.e. the number of skyrmions generated by the beam. We show that the spatial profile of the subwavelength optical vortex offers an ideal tool for creating skyrmionic defects and the timescale of their generation can be orders of magnitude shorter than other known schemes like heating Koshibae and Nagaosa (2014); Fujita and Sato (2017) and electric current pulses Yuan and Wang (2016); Zhang et al. (2016).
Optical vortex— Optical vortex, or Laguerre-Gaussian (LG) mode is a class of solutions of Maxwell’s equations in a vacuum under the paraxial approximation. The derivation of LG modes can be found in literatures Allen et al. (1992); Andrews and Babiker (2012). In the cylindrical coordinate , where is the radial coordinate and the azimuthal angle, the field configuration of LG modes propagating in the direction is given as at the focal plane (. Here is the polarization vector, for linearly polarized light and for circularly polarized light. The spatial profile is characterized by two integers, radial index and OAM :
where is the generalized Laguerre function.
The non-vanishing phase twist for requires the field to vanish at the topological singularity . The beam waist represents the size of optical vortex (see Fig. 1), and usually cannot be smaller than a half the wavelength because of the diffraction limit. However, as we mentioned, by using plasmonics techniques Maier and Atwater (2005); Hibbins et al. (2004); Huang and Zhu (2007); Gramotnev and Bozhevolnyi (2010); Heeres and Zwiller (2014); Morimoto et al. (2016), in principle we can take to be much smaller than the wavelength. In the rest of this paper, we consider the simplest optical vortex with the radial index . We note that LG modes of electric fields, which we consider in the supplementary material, have the same form as magnetic fields presented above, as far as we are discussing the beams propagating in a vacuum.
Model and numerical method— In this paper, we numerically investigate the laser-driven dynamics of (chiral) ferromagnets. Particularly, in the following, we focus on Zeeman coupling between spins and magnetic fields of optical vortices and study the spin dynamics in the framework of LLG equation. The effect of ME coupling, which can be important and useful in multiferroic materials Cheong and Mostovoy (2007); Tokura et al. (2014), is discussed in the supplementary material, where we observe qualitatively the same results obtained for Zeeman coupling presented below. Depending on the sign of , these magnetic field distributions rotate in either clockwise (CW) or counter-clockwise (CCW) way. As we discuss below, the spatially-inhomogeneous in-plane structure of optical vortices shown in Fig. 1 and its time-dependent rotation induce various characteristic phenomena.
Let us consider the situation where a square lattice classical (chiral) ferromagnet is placed at the focal plane of the optical vortex. The Hamiltonian we consider is
where is the lattice constant and is the unit vector along the -axis (). The vector represents the spin at the site , with its norm normalized to unity. We have the ferromagnetic Heisenberg interaction , DM interaction with DM vector on the bond 111With this choice of DM vectors, we obtain so-called Bloch-type skyrmions., and the static external magnetic field applied in the -direction aside from the optical vortex. The last term describes Zeeman coupling between spins and the optical vortex.
The Hamiltonian Eq. (S1) is a canonical model of (chiral) ferromagnets. Despite its simplicity, this model well describes some experimental results of the actual three dimensional (thin film) materials and is widely used for the study of their topological defects, i.e. skyrmions Bogdanov and Pfleiderer (2006); Mühlbauer et al. (2009); Fert et al. (2013); Seki and Mochizuki (2016) (see Ref. Seki and Mochizuki, 2016 for the review). With increasing the external field , the model shows the helical ordered phase, skyrmion lattice phase, and the ferromagnetic phase as observed in thin film materials Yu et al. (2010). In the supplementary material, we give a brief review of this model, the phase diagram and topological defects therein. Depending on the materials, the size of skyrmions can vary from nm to m Fert et al. (2013); Seki and Mochizuki (2016); Shibata et al. (2013).
The time coordinate is normalized by , which corresponds to ps for meV. The second term in the right hand side of Eq. (3) is the Gilbert damping term describing the dissipation with strength determined by the dimensionless parameter . We see that each spin precesses around the normalized effective field , and damps towards that. Hereafter, we take . Namely, in the following , , and are measured in the unit of , and the time is in the unit of .
In the THz region, heating caused by the laser absorption is small, so that we ignore the laser heating effect in the following. Actually, magnetic resonance experiments on magnets and multiferroic materials for THz light can be well explained by theory without taking the heating effect into account (for example, Refs Mukai et al., 2016; Mochizuki et al., 2010). In Ref. Mukai et al., 2016, for an antiferromagnetic dielectric HoFeO the temperature change caused by a short intense ( 1 Tesla) THz magnetic field pulse at the magnetic resonance is estimated to be about 1mK ( meV). This is orders of magnitude smaller than the energy scale of the direct coupling between spins and light.
For the numerical calculations, we use the fourth-order Runge-Kutta method with numerical time step of the calculation . We consider a system consisting of 150 times 150 sites with periodic boundary condition imposed in both and directions. In all the cases below, we assume a simple pulse of optical vortex:
where is the frequency, determines the strength of the magnetic field, and gives the beam duration. For meV, is about Tesla, about 3 ps, and about 1.2 THz. In the following, we propose two ultrafast applications of optical vortex, anisotropic spin wave excitations and generation of topological defects. In both cases, below we assume THz optical vortices with nanoscale beam waist , which is orders of magnitude smaller than the wavelength. As we show in the supplementary material and discuss lat er, even with much larger beam waists, qualitatively the same results can be observed by using proper materials.
Spin waves and magnetic resonance— First we apply optical vortex to ferromagnets with . In this case the sign of OAM is unimportant, but the spatial profile of optical vortices still leads to spin wave excitations with characteristic spatial distribution depending on the value of OAM. Here we only focus on linearly polarized waves with finite OAM. In the supplementary material, we give a discussion for circularly polarized optical vortices and Gaussian beams without OAM.
For high frequency beams satisfying , as shown in Fig. 2, we have multipolar spin waves (dipolar, quadrupolar, octapolar) depending on OAM of those beams. On the other hand, at the magnetic resonance (), the spin wave amplitudes become drastically larger, and the multipolar wavefronts connect with each other to be spiral-shaped as shown in Fig. 3. In both cases, the spin structure is modulated from the collinear ferromagnetic state in an inhomogeneous way. Therefore, if (and only if) the laser beam carries OAM, we can dynamically induce the scalar spin chirality ( are neighboring sites of the site ) as shown in the right-bottom panel of Fig. 3. The non-vanishing net chirality would lead to the topological Hall effect Ye et al. (1999); Ohgushi et al. (2000); Taguchi et al. (2001); Shindou and Nagaosa (2001); Tatara and Kawamura (2002); Neubauer et al. (2009) in itinerant magnets, but the quantitative analysis is beyond the scope of this paper and may be presented elsewhere.
Generation of topological defects— As we see in Fig. 1, the optical vortex with non-vanishing OAM has radially anisotropic field distribution. This induces either chiral or anti-chiral twist to the spin texture depending on the sign of the OAM. In chiral magnets with , such twisted nature of the perturbation should compete with the intrinsic chirality of the magnets determined by their DM interaction.
In the following calculations, we take left-handed optical vortices with polarization vector . The polarization dependence and the case for Gaussian beams without OAM are discussed in the supplementary material. For left-handed beams, the twist induced to the spin texture by them is CW (CCW) for negative (positive) OAM. Since our model Eq. (S1) stabilizes skyrmions with CW spin twisting, for negative OAM, the chiralities of light and magnets are in a sense consistent. Here we take the following parameters: , , , , , , , and 222We confirm that distinct behaviors between positive and negative OAM is not an artifact of fine-tuning by performing calculations with slightly different beam profiles.. With these beam parameters, the magnetic field Eq. (4) becomes a half-cycle pulse both for and as shown in Fig. 4, and the beam waist is comparable with the size of skyrmions. For meV, the frequency is about THz and the peak values of the fields are about Tesla. We assume the initial state at to be the meta-stable perfect ferromagnetic state in the skyrmion crystal phase of the model Eq. (S1). In this phase skyrmionic defects are once formed, stable against weak perturbations such as thermal fluctuations Bogdanov and Hubert (1994) (experimentally skyrmions in some materials can survive even at the room temperature). Hence, even if we take the heating effect ignored in this paper into account, the following results will hold at least qualitatively.
In Fig. 4, we present the OAM dependence of the dynamics of the laser-irradiated chiral magnet. As we noted, in the present setup, optical vortex twists the spin texture in a way consistent with the intrinsic chirality of the target. In combination with the topological singularity () where the field amplitude is zero, this results in the generation of skyrmioniums for . Due to the destructive effect of the frequent changes in the sign of magnetic fields around the topological singularity, for we do not obtain skyrmionium. Nevertheless, the OAM dependence of the outcome is clear: the number of skyrmions after the irradiation is given by . Therefore, we can encoded OAM of optical vortices into chiral magnets as their topological charge. Although the field strength assumed here is strong as THz beams, the field enhancement accompanied by the subwavelength focusing Gramotnev and Bozhevolnyi (2010) would resolve this issue.
Here we comment on the timescale of the process we discussed. As for the creation of skyrmionium, the timescale of its creation with the present scheme (with beam) turns out to be much shorter than other schemes, and is essentially unchanged even the size of skyrmions is varied (see the supplementary material). The scheme using heating with vortex beams Fujita and Sato (2017) requires beams with period O long and that using spin-polarized current Zhang et al. (2016) needs a pulse with duration of O 333They apply spin-polarized current of a few hundred pico seconds for the exchange coupling pJm, which would correspond to meV., while with our scheme its creation completes within O (see more details in the supplementary material). Behind the short timescale, there are two features of optical vortices: coherent coupling with spins and the -dependent spatial profile, which generate the desired twisted spin texture directly in the THz timescale.
Concluding remarks– In this paper, we proposed two ultrafast magnetic phenomena induced by subwavelength optical vortices. We found that OAM of optical vortex can be encoded in magnets in the form of anisotropic spin waves or topological defects.
We show that there appears the non-vanishing net spin chirality due to the anisotropic spin waves, which would lead to the laser-induced topological Hall effect. With regard to topological defects in chiral magnets our findings offer a scheme for the ultrafast generation of them. Unlike other known schemes, our method can generate multiple skyrmions at the same time in a controlled way.
Finally, we comment on the experimental feasibility of the proposed phenomena. First, the excitation of the anisotropic spin waves in Figs. 2 and 3 are rather easy, since they are long-wavelength phenomena by nature and do not actually require the subwavelength focusing. On the other hand, the feasibility of the generation of topological defects in Fig. 4 is more subtle. As we mentioned in the introductory part, the THz focusing is at the very early stage of its study, and at present the maximum focusing achieved experimentally is by a factor of three to four Arikawa et al. (2017), and it is indeed quite challenging to realize the “nanometre” scale focusing we assumed in Fig. 4. However, as discussed in the supplementary material, we can verity that the proper beam waist for the proposed phenomena simply scales with the intrinsic length scale of the target materials, the size of skyrmions. Therefore, by using materials with large skyrmions discovered recently Shibata et al. (2013); Jiang et al. (2015); Yu et al. (2016); Woo et al. (2016); Moreau-Luchaire et al. (2016); Boulle et al. (2016); Jiang et al. (2017), the requirement for the focusing factor can be drastically relaxed and we could realize the OAM encoding in Fig. 4.
Acknowledgement— We thank Koichiro Tanaka for useful comments. H. F. is supported by Advanced Leading Graduate Course for Photon Science (ALPS) of Japan Society for the Promotion of Science (JSPS) and JSPS KAKENHI Grant-in-Aid for JSPS Fellows Grant No. JP16J04752. M. S. was supported by Grant-in-Aid for Scientific Research on Innovative Area, âNano Spin Conversion Scienceâ (Grant No.17H05174), and JSPS KAKENHI (Grant No. JP17K05513 and No. JP15H02117). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.
- Beaurepaire et al. (1996) E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).
- Kirilyuk et al. (2010) A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010).
- Stanciu et al. (2007) C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 047601 (2007).
- Satoh et al. (2012) T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Nat. Photon. 6, 662 (2012).
- Finazzi et al. (2013) M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh, L. Duò, A. Kirilyuk, T. Rasing, and M. Ezawa, Phys. Rev. Lett. 110, 177205 (2013).
- Caviglia et al. (2013) A. D. Caviglia, M. Först, R. Scherwitzl, V. Khanna, H. Bromberger, R. Mankowsky, R. Singla, Y.-D. Chuang, W. S. Lee, O. Krupin, W. F. Schlotter, J. J. Turner, G. L. Dakovski, M. P. Minitti, J. Robinson, V. Scagnoli, S. B. Wilkins, S. A. Cavill, M. Gibert, S. Gariglio, P. Zubko, J.-M. Triscone, J. P. Hill, S. S. Dhesi, and A. Cavalleri, Phys. Rev. B 88, 220401 (2013).
- Mikhaylovskiy et al. (2015) R. V. Mikhaylovskiy, E. Hendry, A. Secchi, J. H. Mentink, M. Eckstein, A. Wu, R. V. Pisarev, V. V. Kruglyak, M. I. Katsnelson, T. Rasing, and A. V. Kimel, Nat. Commun. 6, 8190 (2015).
- Subkhangulov et al. (2016) R. R. Subkhangulov, R. V. Mikhaylovskiy, A. K. Zvezdin, V. V. Kruglyak, T. Rasing, and A. V. Kimel, Nat. Photon. 10, 111 (2016).
- Sato et al. (2014) M. Sato, Y. Sasaki, and T. Oka, arXiv:1404.2010 (2014).
- Takayoshi et al. (2014a) S. Takayoshi, H. Aoki, and T. Oka, Phys. Rev. B 90, 085150 (2014a).
- Takayoshi et al. (2014b) S. Takayoshi, M. Sato, and T. Oka, Phys. Rev. B 90, 214413 (2014b).
- Mentink et al. (2015) J. H. Mentink, K. Balzer, and M. Eckstein, Nat. Commun. 6, 6708 (2015).
- Pimenov et al. (2006) A. Pimenov, A. A. Mukhin, V. Y. Ivanov, V. D. Travkin, A. M. Balbashov, and A. Loidl, Nature Phys. 2, 97 (2006).
- Mochizuki and Nagaosa (2010) M. Mochizuki and N. Nagaosa, Phys. Rev. Lett. 105, 147202 (2010).
- Takahashi et al. (2011) Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, and Y. Tokura, Nature Phys. 8, 121 (2011).
- Sato et al. (2016) M. Sato, S. Takayoshi, and T. Oka, Phys. Rev. Lett. 117, 147202 (2016).
- Allen et al. (1992) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
- Andrews and Babiker (2012) D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University Press, 2012) cambridge Books Online.
- Friese et al. (1998) M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
- He et al. (1995) H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
- Paterson et al. (2001) L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
- Shigematsu et al. (2016) K. Shigematsu, K. Yamane, R. Morita, and Y. Toda, Phys. Rev. B 93, 045205 (2016).
- Bretschneider et al. (2007) S. Bretschneider, C. Eggeling, and S. W. Hell, Phys. Rev. Lett. 98, 218103 (2007).
- Toyoda et al. (2012) K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, Nano Lett. 12, 3645 (2012).
- Hamazaki et al. (2010) J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, Opt. Express 18, 2144 (2010).
- Fujita and Sato (2017) H. Fujita and M. Sato, Phys. Rev. B 95, 054421 (2017).
- Maier and Atwater (2005) S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 (2005).
- Hibbins et al. (2004) A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, Phys. Rev. Lett. 92, 143904 (2004).
- Huang and Zhu (2007) C.-P. Huang and Y.-Y. Zhu, Act. passive electron. compon. 2007, 13 (2007).
- Gramotnev and Bozhevolnyi (2010) D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photon. 4, 83 (2010).
- Heeres and Zwiller (2014) R. W. Heeres and V. Zwiller, Nano Lett. 14, 4598 (2014).
- Morimoto et al. (2016) S. Morimoto, T. Arikawa, F. Blanchard, K. Sakai, K. Sasaki, and K. Tanaka, in 2016 41st International Conference on Infrared, Millimeter, and Terahertz waves (IRMMW-THz) (Copenhagen, Denmark, 2016) pp. 1–3.
- Arikawa et al. (2017) T. Arikawa, S. Morimoto, and K. Tanaka, Opt. Express 25, 13728 (2017).
- Ye et al. (1999) J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar, and Z. Tešanović, Phys. Rev. Lett. 83, 3737 (1999).
- Ohgushi et al. (2000) K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, R6065 (2000).
- Taguchi et al. (2001) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573 (2001).
- Shindou and Nagaosa (2001) R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001).
- Tatara and Kawamura (2002) G. Tatara and H. Kawamura, Journal of the Physical Society of Japan 71, 2613 (2002).
- Neubauer et al. (2009) A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Phys. Rev. Lett. 102, 186602 (2009).
- Dzyaloshinsky (1958) I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).
- Moriya (1960) T. Moriya, Phys. Rev. 120, 91 (1960).
- Bogdanov and Hubert (1994) A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255 (1994).
- Bogdanov and Pfleiderer (2006) A. N. Bogdanov and C. Pfleiderer, Nature 442, 797 (2006).
- Mühlbauer et al. (2009) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Science 323, 915 (2009).
- Fert et al. (2013) A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013).
- Seki and Mochizuki (2016) S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials, 1st ed., SpringerBriefs in Physics (Springer International Publishing, Berlin, 2016).
- Koshibae and Nagaosa (2014) W. Koshibae and N. Nagaosa, Nat. Commun. 5, 5148 (2014).
- Yuan and Wang (2016) H. Y. Yuan and X. R. Wang, Sci. Rep. 6, 22638 (2016).
- Zhang et al. (2016) X. Zhang, J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, and M. Ezawa, Phys. Rev. B 94, 094420 (2016).
- Cheong and Mostovoy (2007) S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007).
- Tokura et al. (2014) Y. Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77, 076501 (2014).
- (52) With this choice of DM vectors, we obtain so-called Bloch-type skyrmions.
- Yu et al. (2010) X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).
- Shibata et al. (2013) K. Shibata, X. Z. Yu, T. Hara, D. Morikawa, N. Kanazawa, K. Kimoto, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Nanotechnol. 8, 723 (2013).
- Mukai et al. (2016) Y. Mukai, H. Hirori, T. Yamamoto, H. Kageyama, and K. Tanaka, New J. Phys. 18, 013045 (2016).
- Mochizuki et al. (2010) M. Mochizuki, N. Furukawa, and N. Nagaosa, Phys. Rev. Lett. 104, 177206 (2010).
- (57) We confirm that distinct behaviors between positive and negative OAM is not an artifact of fine-tuning by performing calculations with slightly different beam profiles.
- (58) They apply spin-polarized current of a few hundred pico seconds for the exchange coupling pJm, which would correspond to meV.
- Jiang et al. (2015) W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, Science 349, 283 (2015).
- Yu et al. (2016) G. Yu, P. Upadhyaya, X. Li, W. Li, S. K. Kim, Y. Fan, K. L. Wong, Y. Tserkovnyak, P. K. Amiri, and K. L. Wang, Nano Lett. 16, 1981 (2016).
- Woo et al. (2016) S. Woo, K. Litzius, B. Kruger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Klaui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016).
- Moreau-Luchaire et al. (2016) C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J. M. George, M. Weigand, J. Raabe, V. Cros, and A. Fert, Nat. Nanotechnol. 11, 444 (2016).
- Boulle et al. (2016) O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza Chaves, A. Locatelli, T. O. Menteş, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigné, A. Stashkevich, S. M. Chérif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Nat. Nanotechnol. 11, 449 (2016).
- Jiang et al. (2017) W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Benjamin Jungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L. Wang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis, Nature Phys. 13, 162 (2017).
Appendix A Supplementally Material for
“Encoding orbital angular momentum of light in magnets”
As in the main text, in this supplementary material we set , namely measure energy in the unit of and time in . For all the numerical calculations based on the Landau-Lifshiz-Gilbert (LLG) equation below, we use the forth-order Runge-Kutta method with timestep in spatially periodic systems. When meV, corresponds to 26 fs. The Gilbert damping constant is , same as in the main text.
Appendix A Two dimensional chiral magnets and skyrmions
In the main text, we used a canonical model of two dimensional chiral ferromagnets on a square lattice:
Here we shortly review the static properties of this model, particularly focusing on topological defects, skyrmions in that. The first term () describes the ferromagnetic Heisenberg interaction and the second term does the Zeeman coupling between magnetic moments and the external magnetic field . The third term, called the Dzyaloshinskii-Moriya (DM) term favors the twisted spin texture. These three terms compete with each other and result in the ground state phase diagram shown in Fig. S1.
There appear three distinct phases depending on the ratio (here ). Under the strong external magnetic field , the system is ferromagnetic and all the spins align in the positive -direction. On the other hand, if the magnetic field is very weak, ferromagnetic Heisenberg interaction and DM interaction turn the system into the helically ordered state. In between them, there exists the skyrmion crystal phase where skyrmions, localized magnetic defects with (clockwise) in-plane spin configuration, form a triangular lattice in the ground state. Skyrmions can also appear in the form of isolated magnetic defects in the ferromagnetic phase as metastable defects.
In the continuum space, Eq. (S2) is quantized to an integer and any smooth deformations to the spin texture cannot change . Since the spin texture of skyrmions give a non-vanishing , they are stable against continuous deformations. Although this quantization is imperfect in lattice systems, the topological nature of skyrmions makes them energetically meta-stable magnetic defects.
At the center of each skyrmion, spins point to the opposite direction with respect to the background spin structure. In other words, spins in skyrmions are -rotated from those in the outside. Then, we can easily generalize this into -rotation. If , spins in and outside that defects point to the same direction (-direction for example), but they are separated by a closed domain formed by spins with negative . As a whole, the defect with spin rotation, called skyrmionium, looks like a ring in terms of . In Fig. S2, we show the spin texture of a skyrmion ( vortex) and skyrmionium ( vortex).
Appendix B Use of beams with larger waist
In the main text, we assumed the deep subwavelength focusing of optical vortices. In this section, we show that this assumption is not quite essential for our results, i.e. we can reproduce the qualitatively the same results using optical vortices with much larger beam waist for proper parameters.
In Fig. S3, we show the anisotropic spin wave excitation induced by optical vortices with , which is four times larger than that assumed in the main text (the temporal profile of the beam is the same as in the main text Fig. 3). We see that the OAM dependence of the shape of the spin wavefronts is the same as in the main text. This is because there is no length scale in our model other than the lattice constant () (here we assume ). Consequently as long as the beam waist of the optical vortex is much larger than the lattice constant, the shape of wavefronts induced by the beam becomes at least qualitatively the same even for different beam waists.
As for the generation of topological defects, however, matching the length scale of the beam waist and that determined by the DM interaction, i.e. the skyrmion size, is important. Although it is quite challenging to achieve the deep focusing down to nanometer scale, as we mentioned in the main text, the proper beam waist of optical vortex will be more accessible values by using materials with large skyrmions.
As Fig. S4 shows, by using smaller and , or a model with larger skyrmions, indeed we can reproduce the results in Fig. 4 of the main text. Namely, the relationship between and the topological number after the irradiation for optical vortex in the main text holds for larger beam waist (here we assume , five times larger than that in the main text). The important point is that we could obtain Fig. S4 by using the optical vortex with the same temporal profile (a half-cycle pulse with the same frequency) as in Fig. 4 of the main text. That is, the proper frequency to cause this phenomenon, i.e. the OAM encoding, does not change even if we decrease , or use materials with larger skyrmions. Therefore, from Fig. S4 we can expect that the proper beam waist for THz optical vortex scales with the size of skrymions and the strong nano-focusing required to realize OAM encoding can be relaxed to a micrometer scale in reality.
Finally we discuss the timescale of the formation process. In Fig. S4 we show the snapshot for at , which are both five times larger and longer than those in the main text. For materials with larger skyrmions, it takes longer time for the laser-induced defects to be relaxed to their equilibrium states. However, we find that for , the topological structure itself develops at the very early stage of the equilibration process as shown in Fig. S5. Here we plot the “cummulative” skyrmion number
calculated for spins within the circle with radius (measured from the center of the beam spot) for and , . We fix and take . Since the skyrmionium is a bound state of a skyrmion and anti-skyrmion, its cummulative skyrmion number shows the bump structure as a function of . In Fig. S5, we see that the formation of the bump is qualitatively completed even right after the optical vortex pulse passes (we are assuming a pulse with , ) even when the beam waist (thus the size of skyrmion) is large. Therefore, the timescale of generating skyrmioniums with optical vortices is essentially unchanged even for materials with large skyrmions. For , on the other hand, as we mentioned in the main text, the formation of the laser-induced defects depends strongly on the spin dynamics governed by the DM interaction. As a result, we see that the timescale becomes longer for larger beam waist. In the figure, as an example we show the results for beams under which a pair of skyrmions (each has the skyrmion number ) are created, and the total topological number after the laser irradiation is .
With increasing the size of skyrmions (and skyrmioniums) in the steady state, their creation/annihilation take longer time under the conventional schemes since their timescale is determined by the magnitude of , which is smaller for large skyrmion materials. Our results show that at least for the creation (and possibly the annihilation) of skyrmioniums, optical vortices could resolve the timescale issue. Hence, as far as the topological structure is concerned, by using the subwavelength optical vortices, we can generate the large skyrmioniums in the THz timescale.
Appendix C Magneto-electric coupling
In insulating chiral magnets like CuOSeO, magneto-electric (ME) coupling allows us to control the magnetic texture with electric fields, instead of magnetic fields. Because the electric field of a laser beam is usually much larger than the magnetic field, in such multiferroic materials with the large ME coupling constant, the magnetization dynamics can be dominated by the coupling of magnets with the electric fields. The ME coupling we discuss here is described by the following Hamiltonian:
In multiferroic materials, the local electric polarization is a function of local spins. The electric field at the focal plane is with polarization vector and
Here is the LG mode of electric field with the radial index and OAM .
In the following, we discuss two major mechanisms of ME coupling in multiferroic magnets: Dzyaloshinskii-Moriya (DM) interaction induced coupling Katsura et al. (2005) and - hybridization induced one Arima (2007); Seki et al. (2012). The former mechanism gives the electric polarization proportional to the outer-product of two nearby spins: , where is the unit vector pointing from site to site . Finite polarization from this mechanism requires a canted magnetic structure and does not exist in the collinear magnetic states. Therefore, even if the coupling constant between spins and electric field is large, as long as we are considering of sufficiently smooth magnetic textures like in spin waves or skyrmions, ME coupling of this type becomes effectively weaken and Zeeman coupling discussed in the main text would dominate the response.
On the other hand, ME coupling from - hybridization could be important in our setup. The ME coupling of this kind is experimentally observed in CuOSeO, and described by the polarization vector Mochizuki (2016), if we take the -axis to be the -axis. For CuOSeO, experimentally we have Cm Seki et al. (2012); Mochizuki (2016). Contrary to the DM induced one, this ME coupling can exist in collinear magnetic phase and can dominate over the effect of Zeeman coupling. Substituting this polarization vector into the Hamiltonian and differentiate it by , we obtain the following effective magnetic field from the ME coupling, which enters in the LLG equation as
Then we notice that the in-plane components of the effective field are proportional to . Therefore, the torque which is to rotate spins from the initial state gets weaker and weaker as approaches to zero. As a result, the precession induced by ME coupling tends to confine spins inside - plane rather than to induce magnetization reversal.
Even though the precession itself is not completely suitable for the creation of topological defects, it still twists the spin texture and could generate topological defects with the help of DM and Heisenberg interaction. With numerical calculations based on LLG equation for the model:
we find that for right-handed optical vortices with , the ME coupling from - hybridization Eq. (S6) can indeed generate topological defects just as the Zeeman coupling with left-handed optical vortices. In Fig. S6, we show time evolution of spins for , , and under the electric field Eq. (S5) with , , , , , and . For meV and Cm, the electric field strength corresponds to MV cm. We see that the final states after the irradiation are qualitatively the same as those in the case of Zeeman coupling discussed in the main text. Namely, we can generate a skyrmionium and skyrmions depending on the OAM of optical vortices. For left-handed electric field, we do not observe any topological defects after the irradiation for all these OAMs.
We also observe multipolar spin wave excitations for linearly polarized electric fields with as shown in Fig. S7, just as those from Zeeman coupling discussed in the main text. Here we assume and . For meV, MVcm and THz. In this setup, even under the laser irradiation, , , and are preserved since the electric field field is rather weak. Therefore the effective field Eq. (S6) is almost equivalent to Zeeman coupling with a magnetic field , so it is natural to have qualitatively the same outcome. The situation is the same at the resonance and we have spiral-shaped spin waves with larger amplitudes.
Appendix D Gaussian beam
In the main text, we saw that linearly polarized optical vortices with non-vanishing OAM can induce spin dipolar-, quadrupolar-, and octapolar- radiations, and left-handed, circularly polarized optical vortices offer an efficient way of creating topological defects in chiral magnets. Here for comparison we show the result for , namely ordinary Gaussian beams without carrying OAM. As shown in Fig. S8, even for the linearly polarized Gaussian beam at the resonance, the spin wave is excited, but its spatial distribution is radially isotropic, contrary to the multipolar radiations from finite OAM lasers discussed in the main text. Since the Gaussian beam does not twist the spin texture non-uniformly unlike optical vortex, as shown in Fig. S9, we could not obtain any topological defects after the irradiation for left-handed beam even with the same beam strength used in the main text.
Appendix E Optical polarization dependence
Here we discuss the optical polarization dependence of the phenomena discussed in the main text, namely spin waves and creation of topological defects.
e.1 Anisotropic spin waves
In the spin wave part of the main text, we focused on the linearly polarized beams and discussed multipolar spin waves from high frequency beams and spiral spin waves from beams with the frequency at the magnetic resonance. Here we discuss spin wave excitations at the magnetic resonance by circularly polarized beams. We consider both left and right handed beams . For OAM , in Fig. S11, we show the -component of spins after the laser irradiation. For the left handed cases, we clearly see the spiral-shaped wavefronts with the wave amplitude larger than the linearly polarized case in the main text. For the right handed cases, however, the amplitude is much smaller. The spatial structure of the wavefronts is spiral but unclear from the figure due to the small amplitude of the spin wave. The spiral wave fronts can be observed for the left-handed off-resonant beams with , but the wave amplitude gets much smaller just as we saw for linearly polarized case in the main text.
e.2 Creation of topological defects
In the main text, we considered left-handed waves with to create magnetic defects. If we take , left-handed optical vortices with OAM and right-handed ones with OAM are equivalent. However, once we have finite , there appears clear difference. It turns out that right-handed optical vortex with is not suitable for the creation of topological defects as opposed to the left-handed one. We calculate the time evolution of spins for the right handed beams using the same parameters in the Fig. 4 of the main text: , , , , , , , , , and . As Fig. S12 shows, with these parameters, we cannot get any topological defects, contrary to the left-handed.
We can understand the difference between left and right handed beams by moving into the rotating frame with frequency as shown in Fig. S13. In this frame, the magnetic field of optical vortex is transformed into a superposition of the static in-plane (anisotropic) magnetic field and the rotation-induced static magnetic field in the -direction. The sign of the rotation-induced field depends on whether the original optical vortex is left or right handed. In our setup, rotation-induced field is in the -direction for left-handed optical vortex, which is the opposite to the uniform external magnetic field. Therefore, left-handed beams are more suitable for exciting spin waves and creating topological defects.
We also comment on the use of linearly polarized magnetic fields for the creation of defects. Since linearly polarized field with, for example, is a superposition of left and right-handed beams, as a tool for creating defects, it would be more suitable than right-handed beams. We calculated the time evolution for , , , , , , , , and and summarize the result in Fig. S14. Here the beam is still a half-cycle like but with the longer duration . With , that used in the circularly polarized cases in the main text, we could not obtain topological defects. We see from Fig. S14 that the linearly polarized laser indeed produces topological defects, though the systematic OAM dependence breaks down at .
- Katsura et al. (2005) H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).
- Arima (2007) T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007).
- Seki et al. (2012) S. Seki, S. Ishiwata, and Y. Tokura, Phys. Rev. B 86, 060403 (2012).
- Mochizuki (2016) M. Mochizuki, Adv. Electron. Mater. 2 (2016), 1500180.