Abstract
Ultrafast control of magnets using femtosecond light pulses attracts interest regarding applications and fundamental physics of magnetism. Antiferromagnets are promising materials with magnon frequencies extending into the terahertz range. Visible or nearinfrared light interacts mainly with the electronic orbital angular momentum. In many magnets, however, in particular with irongroup ions, the orbital momentum is almost quenched by the crystal field. Thus, the interaction of magnons with light is hampered, because it is only mediated by weak unquenching of the orbital momentum by spin–orbit interactions. Here we report alloptical excitation of magnons with frequencies up to 9 THz in antiferromagnetic CoO with an unquenched orbital momentum. In CoO, magnon modes are coupled oscillations of spin and orbital momenta with comparable amplitudes. We demonstrate excitations of magnon modes by directly coupling light with electronic orbital angular momentum, providing possibilities to develop magnetooptical devices operating at several terahertz with high outputtoinput ratio.
Introduction
The inverse effect of the magnetooptical Faraday effect, specifically light acting on magnetic systems, was first attempted by Faraday in 1845^{1} and verified more than 100 years later^{2, 3}. Nowadays, the various inverse magnetooptical effects (the inverse Faraday effect (IFE) or the inverse Cotton–Mouton effect (ICME))^{4} are used for nonthermal optical excitation of spin oscillations with frequencies in the terahertz range in transparent antiferromagnets^{5,6,7}. The effects enable the control of spin dynamics using optical polarization. In contrast to a recent realization of a nontrivial spin evolution for opaque metallic ferrimagnets^{8, 9}, nonthermal excitation does not lead to intense heating of the sample. This feature attracts particular attention for possible applications of antiferromagnets in magnetic recording and magnetooptical devices, e.g., terahertz radiation sources^{10, 11} and optomagnonic read–write transfer^{12}.
The coupled spin–orbit dynamics of cobalt oxide CoO has been studied for half a century. For CoO, quenching of the orbital angular momentum of a free Co^{2+} ion (L _{free} = 3) by the cubic crystal field is only partial, resulting in an effective angular momentum of L = 1 (orbital triplet), which should be treated as an additional degree of freedom in the magnetic subsystem of CoO. Being subject to a lowsymmetry crystal field, the ions with partial unquenching increase magnetic anisotropy (see ref. ^{13} and Supplementary Note 1); its magnitude is comparable to those of the effective spin–orbit and exchange interactions. All these specific features lead to magnon modes originating from spin and orbital degrees of freedom and the magnon frequencies are much higher than those for standard antiferromagnets containing transitionmetal ions^{14}. The magnon modes of CoO have been investigated using Raman scattering^{15}, infrared absorption^{13, 16}, infrared reflection^{17} and inelastic neutron scattering^{18, 19}, but their interpretation and theoretical description are still being debated. Further, the coupling of femtosecond laser pulses and unquenched orbital angular momentum has never been explored.
In the following, we report highly efficient nonthermal coherent excitation of magnon modes in CoO using femtosecond laser pulses at frequencies up to 9 THz. Symmetry analysis and a theoretical model calculation confirm the excitation of these modes, which consist of coupled dynamics that have comparable amplitudes of the oscillations of spin and orbital angular momenta.
Results
Analysis of magnon modes in CoO
Below the Néel temperature T _{N} = 292 K, CoO exhibits an antiferromagnetic order with the antiparallel spin orientations of Co^{2+} ions belonging to two sublattices (labeled with 1 and 2), S _{1} and S _{2} (effective S = 3/2). The unquenched part of the orbital angular momenta (effective L = 1) of Co^{2+} ions, L _{1} and L _{2} are expected to be antiparallel to the corresponding spin momenta for any sublattice because of the spin–orbit interaction. For the crystallographic and magnetic structures (Fig. 1a), CoO exhibits a lowsymmetry monoclinic ground state (crystal point group 2/m) with spin directions inclined from the crystalline axis [001] by an angle ρ. The value sin ρ = \(\sqrt {2{{\rm /}}51} \) is commonly accepted now^{20}. We choose a coordinate system with the zaxis along this direction and \({\hat{\bf y}}[\bar 110]\).
In CoO, the coupled spin–orbit dynamics provides a complex combination of magnon modes with different symmetries and frequencies. We introduce convenient combinations of the variables, total spin angular momentum m _{ S } = S _{1} + S _{2}, total orbital angular momentum m _{ L } = L _{1} + L _{2} (in the ground state m _{ S } = m _{ L } = 0), and spin and orbital antiferromagnetic vectors, N _{ S } = S _{1} − S _{2} and N _{ L } = L _{1} − L _{2}, respectively. The antiferromagnetic vectors can be present through their values in the ground state \({N_{S,L}}{\hat{\bf z}}\) and the small deviations from the ground state, \({{\bf{n}}_{S,L}} \bot {\hat{\bf z}}\). Our theoretical analysis (see Supplementary Note 1 and 2) suggests that four transverse magnon modes should be observed, which are classified into two modes with different symmetries (Γ_{1} and Γ_{2}). These magnon modes (Fig. 1b) belong to two symmetry classes: (1) Γ_{1}(S) and Γ_{1}(L)modes with nonzero (n _{ S })_{ x }, (n _{ L })_{ x }, (m _{ S })_{ y } and (m _{ L })_{ y }; and (2) Γ_{2}(S) and Γ_{2}(L) modes with nonzero (n _{ S })_{ y }, (n _{ L })_{ y }, (m _{ S })_{ x } and (m _{ L })_{ x } ^{14}. Here, (S) and (L) denote spin and orbitaldominated modes, respectively. They should exhibit different polarization dependence in the magnetooptical experiments.
Experimental geometries
To demonstrate the coherent excitation of magnons in CoO and to investigate these complex dynamics and symmetries of the magnons, we performed timeresolved pump–probe experiments. Here, optical pulses excite the magnons through the IFE and the ICME. In particular, we carefully chose the crystalline orientation and the optical polarizations to distinguish magnon modes with different symmetries. In the transverse and longitudinal geometries (TG and LG, respectively), the pump beam propagates along the [100] and [001] directions, which are nearly perpendicular and parallel to the z axis (Fig. 1c, d, respectively). The electric field of the pump light has the form \({{E_i}(t) = {{\rm Re}}\left[ {{{\cal E}_i}}(t){{\rm e}^{i\omega t}} \right]}\). The timedependent complex amplitude of the electric field \({{\cal E}_i}(t)\) takes the form \({{\cal E}_{[001]}}(t) \equiv {{\cal E}_0}(t){{\rm cos}}\,\theta \), \({{\cal E}_{\left[ {0\bar 10} \right]}}(t) \equiv {{\cal E}_0}(t){{\rm sin}}\,\theta {{\rm e}^{i\psi }}\) in the TG, and \({{\cal E}_{[100]}}(t) \equiv {{\cal E}_0}(t){{\rm cos}}\,\theta \), \({{\cal E}_{[010]}}(t) \equiv {{\cal E}_0}(t){{\rm sin}}\,\theta {{\rm e}^{i\psi }}\) in the LG, where 0° ≤ θ < 180°, −90° ≤ ψ ≤ 90°. For linearly polarized light, ψ = 0 with angle θ determining the azimuth of the polarization; for circularly polarized light, θ = 45° and ψ = ∓90° determine the two different helicities σ^{±}. For both geometries, pump pulses were circularly polarized (σ^{±}) or linearly polarized with different values of θ. For experimental details, see Methods.
Magnon excitation in the TG
Figure 2a expresses the change Δf in probe polarization, see Methods for definition, as a function of delay time t in the TG. Figure 2b gives the Fouriertransformed amplitude spectra of the oscillations for θ = 94° in Fig. 2a. Excitations with frequencies of 4.4, 6.6 and 8.9 THz are clearly observed. These modes have been attributed to magnetic excitations^{15}.
The temporal evolutions of the probe polarization observed in the TG were fitted by a superposition of damped oscillations \(\Delta f(t) \equiv \mathop {\sum}\nolimits_{j = 1}^3 {{F_j}{{\rm e}^{  {\alpha _j}{\Omega _j}t}}{{\rm sin}}\left( {{\Omega _j}t + {\vartheta _j}} \right)} \) with three frequencies Ω_{ j }/2π = 4.4, 6.6 and 8.9 THz, and damping constants α _{ j } = 0.011 ± 0.001, 0.004 ± 0.002 and 0.009 ± 0.003 for j = 1, 2 and 3, respectively. Here, F _{ j } is defined as the signed amplitude that may take negative values. Figure 2c, d shows the signed amplitude F _{ j } for linear and circular polarizations of pump, respectively. From Fig. 2c, the dependence of the pump polarization is almost proportional to (1 − cos 2θ) for the 4.4THz mode, cos 2θ for the 8.9THz mode and (cos 2θ + const.) for the 6.6THz mode, where const. is neither 0 nor −1. Surprisingly, for a circularly polarized pump beam, the amplitudes were almost independent of helicity σ^{±} and nearly equal to that for a linearly polarized pump beam with θ = 45°, 135°.
To explain this complicated picture, let us consider the phenomenological theory (see Supplementary Note 3 for detail). The (inverse) magnetooptical effects result from the dependence of the permittivity on the magnetic order parameter^{21}. Namely,
where k _{ ijk } and g _{ ijkl } determine the Faraday effect (FE) and the Cotton–Mouton effect (CME) (both direct and inverse), respectively. Here, k _{ ijk } = −k _{ jik }, g _{ ijkl } = g _{ jikl } = g _{ ijlk }, the structures of these tensors are determined by the magnetic symmetry group of the crystal^{22, 23}. It is noteworthy that the subscript “l” is not related to the orbital angular momentum, but a lower index for a tensor “ijkl”. For materials such as CoO with partially unquenched orbital angular momentum, the independent dynamics of the spin and orbital angular momenta should be considered. Formally, the independent contributions of these momenta, m _{ S }, n _{ S } or m _{ L }, n _{ L } can be written down in the form given in Eq. (1). The symmetric properties of the oscillations of the corresponding spin and orbital vectors are indeed the same, as well as the symmetric properties of the tensors k _{ ijk } and g _{ ijkl }. Although both the orbital and spin angular momenta contribute to the magnetism in CoO, the magnetooptical effect is dominated by the orbital angular momentum because the optical selection rule for the electricdipole interaction only allows changes in orbital angular momentum. A change in spin can be only allowed via spin–orbit interactions. In many other systems, however, the orbital angular momentum is quenched and cannot contribute to the magnetic properties of media, and thus magnetooptical effect is governed by the indirect coupling between spin and light^{24}. To simplify the expressions, we omit the index “L” from vectors m _{ L }, n _{ L } in the following. The effective energy of the magnetooptical interaction with the use of Eq. (1) for TG can be written as:
where \(I(t) = {\cal E}(t){{\cal E}^*}(t)\) for a given polarization of light. Here, Ks and Gs are related to the FE and CME (both direct and inverse), respectively, and are defined in Supplementary Eqs. (20)–(28).
When a material has high symmetry, such as cubic symmetry, the parameters G _{1}, G _{2}, G _{3} and G _{4} are zero and the amplitude of the magnetic oscillation induced by the ICME gives sin 2θ cos ψ dependence in Eq. (2)^{12, 25,26,27}. This implies that its sign changes when the pump azimuthal angle θ is changed from 45° to 135° for linearly polarized light (i.e., ψ = 0°), and that it vanishes for circularly polarized light (ψ = ±90°). Until very recently, magnetic excitations with only this particular polarization dependence were considered to originate from the ICME. However, the magnon modes experimentally observed in the TG never exhibit standard sin 2θ cosψ dependence. In a crystal with a reduced symmetry like in CoO, other Grelated terms, G _{1}–G _{4}, can also appear in Eq. (2). These terms may lead to magnetic oscillations induced by the ICME that are independent of helicity (ψ) and proportional to cos 2θ or (1 − cos 2θ). It is noteworthy that the (1 − cos 2θ) contribution from ICME does not vanish even for circularly polarized light with cos 2θ = 0 and the amplitude for circular polarization should be equal to that for linear polarization with θ = 45° and 135°, as mentioned above. This relationship enables other mechanisms to be excluded, e.g., thermal excitation. This helicityindependent excitation of the magnon modes by circularly polarized light is a nonstandard manifestation of the ICME and is a characteristic of materials with low symmetry^{28}.
Magnon excitation in the LG
Figure 3a shows Δf as a function of t in the LG. The amplitude of oscillations in Δf reaches large values, as high as 0.02. The outputtoinput ratio, which is defined as the amplitude of oscillation in Δf normalized by the pump fluence and pump–probe spectral weight is two orders of magnitudes higher than that observed in NiO^{28} (see Supplementary Note 4). It is noteworthy that Δf was measured under nonresonance conditions for both CoO and NiO. This demonstrates the high efficiency of the optical excitation for magnets with unquenched orbital angular momentum. Figure 3b presents the Fouriertransformed amplitude spectrum of the oscillations for θ = 0° in Fig. 3a. The temperature dependence of the spectral magnitude further confirms that the 4.4THz spectral peak is of magnetic origin (see Supplementary Fig. 1). Aside from the 4.4THz mode, a small peak at 8.9 THz was found. Figure 3c, d show the signed amplitude F _{ j } for linear and circular pump polarizations, respectively. F _{ i } are almost proportional to cos 2θ for the 4.4THz and nearly constant for the 8.9THz modes. In contrast to the TG, the sign of F for the 4.4THz mode changes by reversing the pump helicity, clearly identifying the IFE^{28,29,30,31,32}.
Discussion
We now discuss the symmetry of the magnon modes. Our analysis (see Supplementary Note 3) reveals that the 4.4THz mode is the Γ_{2} mode, which is excited with the (1 − cos 2θ)dependent ICME in the TG (Supplementary Eq. (31)), with the cos 2θdependent ICME (Supplementary Eq. (35)) and with the helicitydependent IFE (Supplementary Eq. (36)) in the LG. The polarization dependences of the 6.6THz mode in the TG is neither cos 2θ nor (1 − cos 2θ) (Supplementary Eq. (29)) and hence this mode can be interpreted as a Γ_{1} mode. If the highfrequency signal is caused by an excitation of one pure mode, either Γ_{1} mode or Γ_{2} mode, its θdependence should follow one of the dependencies observed for lowfrequency modes. From the experimental data, this is not evident for the TG. Thus, we suggest that the signal observed at 8.9 THz is a superposition of signals of two excited modes of different symmetries but similar frequencies. Our experimental results indicate that the lowerfrequency (4.4 and 6.6 THz) modes with different symmetries have significantly different frequencies, whereas the higherfrequency (8.9 THz) modes with different symmetries are degenerate. This finding contrasts with previous studies in which the lowfrequency Γ_{1} and Γ_{2} modes were claimed to be almost degenerate^{13, 15}, probably because the samples were not confirmed to be a single domain. Our spontaneous Raman scattering measurements on a single domain support our conclusion (see Supplementary Fig. 2).
Previous theoretical studies on CoO spin–orbit dynamics were based on the quasiuniaxial models, where the magnetic anisotropy originates mainly from the crystal field having the form \(  C\left( {L_{z,1}^2 + L_{z,2}^2} \right)\) ^{13,14,15,16,17,18,19}. Such models obviously lead to almostdegenerate doublets of the modes with Γ_{1} and Γ_{2} symmetries, which is not consistent with our observations. To describe our experimental results, we propose a quantum mechanical model with a biaxial crystal field for states with orbital angular momentum L = 1 (see Supplementary Note 1 for details). For transverse oscillations, four modes of coupled spin–orbit dynamics are found analytically and the parameters of the Hamiltonian are determined. The pair of lowerfrequency modes appears to be spin dominated, whereas the almostdegenerate pair of higherfrequency modes is orbital dominated. It is noteworthy that the different degrees of lifting the degeneracy of the lowerfrequency and higherfrequency modes require the notion of competing spin and orbital contributions to the anisotropy. See Supplementary Notes 1, 2 for more details.
Here we discuss the reason why the 4.4THz mode was not excited by the IFE in the TG. In principle, this mode can be excited by the IFE because of the term K _{2} in Supplementary Eq. (32), as well as the ICME from term G _{4} in Supplementary Eq. (31). The reason why this mode was not excited by the IFE is the following. The 4.4THz mode is a Γ_{2}(S) mode, where the trajectory of the magnetization vector is an ellipse elongated along the y axis. Excitation by the IFE initializes as a kick on the magnetization along the y direction, whereas that by the ICME is along the x direction, which corresponds to the short axis of the ellipse. Therefore, the ICME dominates the excitation efficiency and the helicitydependent excitation via the IFE was not observed.
To summarize, we demonstrated that femtosecond laser pulses can efficiently excite magnons consisting of the spin and unquenched orbital angular momentum in CoO. The study of coupled spin–orbital dynamics is of great interest in the promising area of optomagnonics. First, it enables the realization of faster spin dynamics, in particular excitation of magnons with higher frequencies than those mediated by pure spin dynamics. Second, the materials with a large fraction of orbital angular momentum oscillations are expected to exhibit higher efficiency in their magnetooptical effects (either inverse or direct), which leads to higher amplitudes of both excited magnons and of the probe signal. Indeed, we found that CoO produces quite a high magnetooptical signal from the excited magnons even for frequencies of order 10 THz.
Methods
Sample
The samples were CoO (001) biaxial single crystals^{33} grown by the floating zone method. Magnetostriction leads to contraction of the cubic unit cell along the 〈100〉 direction and gives rise to three types of T domains^{34}. From the TG and the LGconfigured pump–probe measurements, two types of T domains were classified from observations of differences in birefringence in the crossNicol configuration using a polarization microscope. In the TG, its optic axes were 4° out of alignment with the [010] and [001] directions, and \(\Delta n \simeq 0.02\) for 633 nm at 5 K; in the LG, the crystal had axes in the [110] and \([1\bar 10]\) directions, and \(\Delta n \simeq 0.001\). The values agree with those taken from the literature^{33, 34}. The thicknesses of the samples were 70 μm for the TG and 50 μm for the LG.
Pump–probe experimental setup
The samples were cooled at 5 K in a cryostat in the absence of an external magnetic field. A Ti: sapphire regenerative laser amplifier (SpectraPhysics, Spitfire Pro) was used as the fundamental light source producing a central wavelength of 800 nm, a pulse duration of 50 fs and a repetition frequency of 1 kHz. With an optical parametric amplifier, a part of this light was converted to a wavelength of 1,500 nm and used as pump pulses. The pump wavelength was chosen to avoid the real (d–d) excitation^{35}. The pump–pulse fluence was 130 mJ cm^{−2}. The rest of the light passed through a delay line before entering the sample as timedelayed probe pulses. To obtain a maximal signal, in the TG, probe pulses were linearly polarized with ϕ = 26.5° from [001] that enabled the simultaneous detection of the FE, CME and linear dichroism, whereas in the LG the probe pulses were linearly polarized at ϕ = 45° (see Supplementary Fig. 3). The change in the probe polarization was obtained by measuring the balance of the two linearly polarized components from the transmitted probe light; these orthogonal components, I _{1} and I _{2} have the same magnitude. I _{1} and I _{2} were ±45° from the reference angle when injected into the sample. The transmitted probe pulse was divided into two orthogonally polarized pulses by a Wollaston prism, each pulse was detected using a Si photodiode. We calculated f = (I _{1} − I _{2})/(I _{1} + I _{2}) and regarded Δf(t) as a modulation of the probe polarization^{26}.
Data availability
Data that support the findings of this study are available from the corresponding author on request.
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Acknowledgements
We thank H. Tamaru and K. Tomiyasu for valuable discussions, and K. Tsuchida for technical assistance. T. Satoh was supported by JSTPRESTO, JSPS KAKENHI (numbers JP15H05454 and JP26103004), JSPS CoretoCore Program (A. Advanced Research Networks) and Kyushu University Shortterm International Research Visitation Program. T.H. was supported by JSPS Postdoctoral Fellowship for Research Abroad and the European Research Council (Consolidator Grant NearFieldAtto). B.A.I. was supported by JSPS Invitation Fellowship Programs for Research in Japan and the National Academy of Sciences of Ukraine (No. 1/16N). V.I.B. was supported financially by the Russian Foundation for Basic Research (No. 160200069 a) and the Increase Competitiveness Program of NUST “MISiS” (Act 211 of the Russian Federation, contract number 02.A03.21.001).
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T. Satoh conceived the project. R.I. and T. Satoh carried out the pump–probe experiment. Y.F., A.K. and T. Satoh measured spontaneous Raman spectra. B.A.I. and V.I.B. performed the model calculation. B.A.I., V.I.B., R.I., T.H. and T. Satoh analyzed the data. H.U. provided the sample. T. Shimura and K.K. supervised the project. T. Satoh, R.I., T.H., B.A.I. and V.I.B. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Satoh, T., Iida, R., Higuchi, T. et al. Excitation of coupled spin–orbit dynamics in cobalt oxide by femtosecond laser pulses. Nat Commun 8, 638 (2017). https://doi.org/10.1038/s41467017006162
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