Coherent terahertz spin-wave emission associated with ferrimagnetic domain wall dynamics

Se Hyeok Oh; Se Kwon Kim; Dong Kyu Lee; Gyungchoon Go; Kab Jin Kim; Teruo Ono; Yaroslav Tserkovnyak; Kyung Jin Lee

doi:10.1103/PhysRevB.96.100407

Coherent terahertz spin-wave emission associated with ferrimagnetic domain wall dynamics

Se Hyeok Oh, Se Kwon Kim, Dong Kyu Lee, Gyungchoon Go, Kab Jin Kim, Teruo Ono, Yaroslav Tserkovnyak, Kyung Jin Lee

Department of Materials Science and Engineering

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35 Citations (Scopus)

Abstract

We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum.

Original language	English
Article number	100407
Journal	Physical Review B
Volume	96
Issue number	10
DOIs	https://doi.org/10.1103/PhysRevB.96.100407
Publication status	Published - 2017 Sep 28

ASJC Scopus subject areas

Electronic, Optical and Magnetic Materials
Condensed Matter Physics

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10.1103/PhysRevB.96.100407

Cite this

@article{2687b2e9832340c9923c92aeadaeb760,

title = "Coherent terahertz spin-wave emission associated with ferrimagnetic domain wall dynamics",

abstract = "We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum.",

author = "Oh, {Se Hyeok} and Kim, {Se Kwon} and Lee, {Dong Kyu} and Gyungchoon Go and Kim, {Kab Jin} and Teruo Ono and Yaroslav Tserkovnyak and Lee, {Kyung Jin}",

note = "Funding Information: Oh Se-Hyeok 1 Kim Se Kwon 2 Lee Dong-Kyu 3 Go Gyungchoon 3 Kim Kab-Jin 4,5 Ono Teruo 5 Tserkovnyak Yaroslav 2 Lee Kyung-Jin 1,3,6 * Department of Nano-Semiconductor and Engineering, 1 Korea University , Seoul 02841, Korea Department of Physics and Astronomy, 2 University of California , Los Angeles, California 90095, USA Department of Materials Science and Engineering, 3 Korea University , Seoul 02841, Korea Department of Physics, 4 Korea Advanced Institute of Science and Technology , Daejeon 34141, Korea Institute for Chemical Research, 5 Kyoto University , Kyoto 611-0011, Japan KU-KIST Graduate School of Converging Science and Technology, 6 Korea University , Seoul 02841, Korea * kj_lee@korea.ac.kr September 2017 28 September 2017 96 10 100407 4 May 2017 {\textcopyright}2017 American Physical Society 2017 American Physical Society We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum. National Research Foundation of Korea http://dx.doi.org/10.13039/501100003725 NRF http://sws.geonames.org/1835841/ 2015M3D1A1070465 2017R1A2B2006119 2017R1C1B2009686 2016R1A6A3A1193588 Ministry of Science, ICT and Future Planning http://dx.doi.org/10.13039/501100003621 MSIP http://sws.geonames.org/1835841/ 17-BT-02 Army Research Office http://dx.doi.org/10.13039/100000183 ARO http://sws.geonames.org/6252001/ http://sws.geonames.org/4482348/ W911NF-14-1-0016 Japan Society for the Promotion of Science http://dx.doi.org/10.13039/501100001691 JSPS http://sws.geonames.org/1861060/ 15H05702 26103002 In modern communications, information is carried by electromagnetic waves whose frequency is limited to ≈ 0.1 terahertz (THz), the frequency of oscillating circuits based on high-speed transistors [1] . On the other hand, semiconductor lasers generate coherent light with a frequency > 30 THz [2] . The terahertz gap refers to the fact that no relevant technology exists in the frequency range between these two limits ( 0.1 – 30 THz). Therefore, it is of critical importance to find relevant physical phenomena that fill in the terahertz gap. In this respect, antiferromagnets whose resonance frequencies are in the THz range [3,4] are of interest [5,6] . It has been reported that coherent THz magnons or spin waves are generated in antiferromagnets, driven by a laser [7,8] or an electrical current [9,10] . However, THz spin-wave excitations by a dc magnetic field are in principle not possible for antiferromagnets as their magnetic moments are compensated on an atomic scale. In this Rapid Communication, we theoretically show that the generation of coherent THz spin waves can be achieved by field-driven domain wall (DW) motion in ferrimagnet/heavy metal bilayers in which the interfacial Dzyaloshinskii-Moriya interaction (DMI) is present. As far as the terahertz gap is concerned, this dc field-driven scheme could be beneficial as it allows a low-power operation by avoiding laser-induced or current-induced heating. It is also fundamentally interesting as THz spin-wave emission is caused by a relativisticlike dynamics of a ferrimagnetic DW. Relativistic kinematics refers to kinematics compatible with the theory of relativity [11] , whose key ingredient is the Lorentz invariance with a limiting velocity c , the speed of light. When the dispersion of a wave satisfies the Lorentz invariance, a quasiparticle corresponding to the wave follows an analogous relativistic kinematics by replacing the speed of light with the maximum group velocity of the wave. When the velocity of the quasiparticle approaches the maximum group velocity, it undergoes the Lorentz contraction and its speed saturates to the limiting velocity. An example of such quasiparticles is an antiferromagnetic DW [10,12,13] . When the DW velocity approaches the maximum spin-wave group velocity, the DW width shrinks by emitting spin waves [10] . Similarly, the dynamics of a ferrimagnetic DW is also expected to exhibit the features of relativistic kinematics provided that the net magnetization and DMI, which break the Lorentz invariance of the system, are sufficiently ineffective. Here, we show that such a relativisticlike DW dynamics is achievable for a class of ferrimagnets, rare earth (RE) and transition metal (TM) compounds, in which the spin moments are antiferromagnetically coupled. As RE and TM elements have different Land{\'e}- g factors [14] , RE-TM ferrimagnets have two distinct temperatures: the magnetic moment compensation point T M where the net magnetic moment vanishes, and the angular momentum compensation point T A where the net angular momentum vanishes. See Fig. 1(a) for an illustration of spin configurations in a ferrimagnet. For RE-TM ferrimagnets, resonance [15,16] , switching [17–21] , domain wall motion [22–24] , and skyrmion (or bubble domain) motion [25–27] near these compensation points have been explored experimentally and theoretically. In particular, an experimental observation of fast field-driven DW motion at T A in GdFeCo single-layered ferrimagnets was recently reported [23] . This observation reveals two distinguishing features of RE-TM ferrimagnets at T A . One is that the spin dynamics is antiferromagnetic and thus fast because of zero net angular momentum at T A . The other is that this fast antiferromagnetic dynamics is achieved by a field because the net magnetic moment is nonzero at T A and thus couples with a magnetic field. We begin by deriving the equations of motion for a ferrimagnetic DW based on the collective coordinate approach [28] . The dynamics of a general collinear ferrimagnet at sufficiently low temperatures can be described by the following Lagrangian density [27,29] , (1) L = ρ n{\. } 2 / 2 − δ s a [ n ] · n{\. } − U , where n is the unit vector along the collinear order, ρ parametrizes the inertia of the dynamics, δ s is the spin density in the direction n , a [ n ] is the vector potential generated by a magnetic monopole of unit charge satisfying ∇ n × a = n , and U is the potential-energy density. Here, the first term is the spin Berry phase associated with the staggered spin density, which thus appears in the Lagrangian for collinear antiferromagnets; the second term is the Berry phase associated with the net spin density δ s , which is used to describe the dynamics of uncompensated spins in ferrimagnets. We consider the following potential-energy density, (2) U = A ( ∇ n ) 2 / 2 − K ( n · z{\^ } ) 2 / 2 + κ ( n · x{\^ } ) 2 / 2 + D y{\^ } · ( n × ∂ x n ) / 2 − h · n . Here, the first term is the exchange energy with A > 0 ; the second term is the easy-axis anisotropy along the z axis with K > 0 ; the third term is the weaker DW hard-axis anisotropy along the x axis with κ > 0 ; the fourth term is the interfacial DMI; the last term is the Zeeman coupling with h = M net H , where M net is the net magnetization in the direction n . The dissipation can be accounted for by introducing (the spatial density of) the Rayleigh dissipation function, R = s α n{\. } 2 / 2 . Here, s α is a phenomenological parameter quantifying the energy and spin loss due to the magnetic dynamics. For example, in the ferromagnetic regime, i.e., away from T A , it can be considered as the product of the effective Gilbert damping constant and the net spin density. The low-energy dynamics of a DW can be described by the two collective coordinates, the position X ( t ) and the azimuthal angle ϕ ( t ) . We consider the Walker ansatz [30] for the DW profile, n ( x , t ) = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) , where θ = 2 tan − 1 { exp [ ( x − X ) / λ ] } and λ = A / K is the DW width. The equations of motion can be derived from Eqs. (1) and (2) in conjunction with the Rayleigh dissipation function, (3) M X{\" } + G ϕ{\. } + M X{\. } / τ = F , (4) I ϕ{\" } − G X{\. } + I ϕ{\. } / τ = − κ{\~ } sin ϕ cos ϕ + D{\~ } sin ϕ , where M = 2 ρ A / λ is the mass, I = 2 ρ A λ is the moment of inertia, G = 2 δ s A is the gyrotropic coefficient, τ = ρ / s α is the relaxation time, F = 2 h A is the force exerted by an external field, κ{\~ } = 2 κ λ A , D{\~ } = π D A / 2 , and A is the cross-sectional area of the DW. From Eq. (3) , we obtain the steady-state solution of the DW velocity, (5) V DW = M net λ s α H , where H is the external field applied along the z axis. In this steady state, the DW moves at a constant velocity V DW with a constant angle ϕ . When the field becomes sufficiently strong such that V DW exceeds a certain threshold V max , the DW begins to precess, engendering the phenomenon known as the Walker breakdown [31,32] . The Walker breakdown field H WB can be obtained from Eq. (4) by (6) H WB = V max s α M net λ . In the absence of DMI ( D = 0 ), the threshold velocity is given by V max = κ{\~ } / 2 G and thus H WB = κ{\~ } s α / 2 G M net λ . When DMI is much stronger than the DW anisotropy in the y direction, i.e., | D{\~ } | ≫ κ{\~ } , then | V max | = | D{\~ } | / G . In this strong DMI limit, the Walker breakdown field is given by (7) H WB = | D{\~ } | G s α M net λ = π | D | 4 δ s s α M net λ . We note that H WB is inversely proportional to G and thus to the net spin density δ s . As a result, the Walker breakdown is absent at T A where the net spin density vanishes, δ s = 0 . This suppression of the Walker breakdown at T A can be understood as a result of decoupling of the DW position X and the angle ϕ at T A [23] . It is worthwhile comparing Eq. (7) to the Walker breakdown field for ferromagnetic DWs in the strong DMI limit [33] , H WB,FM = α π D FM / 2 M FM λ FM , which can be obtained from Eq. (7) by taking the ferromagnetic limit. From this comparison, one finds that in the vicinity of T A , H WB for ferrimagnetic DWs is much larger than that for ferromagnetic DWs because δ s ≈ 0 and | M net | ≪ M FM . Moreover, this very large H WB for ferrimagnetic DWs suggests that V DW can reach the maximum group velocity of the spin wave more easily without experiencing a Walker breakdown, and thus ferrimagnetic DWs can generate THz spin waves in a wide range of net angular momentum δ s . Finally, the time-averaged velocity V ¯ for a one period far above the Walker breakdown is given as (8) V ¯ = M net λ s α + δ s 2 / s α H . To verify these theoretical predictions on the DW velocity and THz spin-wave emission, we perform atomistic model calculations [10,34] for two-sublattice ferrimagnets, which correspond to RE-TM compounds. Two sublattices possess the magnetization M 1 and M 2 , which are coupled by the antiferromagnetic exchange. The spin densities are given by s 1 = M 1 / γ 1 and s 2 = M 2 / γ 2 , where γ i = g i μ B / ℏ is the gyromagnetic ratio of the lattice i , μ B is the Bohr magneton, and g i is the Land{\'e}- g factor. The parameters in the above descriptions for general ferrimagnets are given by δ s = s 1 − s 2 , M net = M 1 − M 2 , and s α = α 1 s 1 + α 2 s 2 , where α i is the Gilbert damping constant for the lattice i . The one-dimensional discrete Hamiltonian that we use for numerical calculations is given by (9) H = A sim ∑ i S i · S i + 1 − K sim ∑ i ( S i · z{\^ } ) 2 + κ sim ∑ i ( S i · x{\^ } ) 2 + D sim ∑ i y{\^ } · ( S i × S i + 1 ) − g i μ B μ 0 ∑ i H · S i , where S i is the normalized spin moment vector at lattice site i [i.e., an even (odd) i corresponds to a RE (TM) atomic site], A sim , K sim , κ sim , and D sim denote the exchange, easy-axis anisotropy, DW hard-axis anisotropy, and DMI constants, respectively, and H is the external field. We numerically solve the atomistic Landau-Lifshitz-Gilbert equation, (10) ∂ S i ∂ t = − γ i S i × H eff , i + α i S i × ∂ S i ∂ t , where H eff , i = − 1 M i ∂ H ∂ S i is the effective field. We use the following simulation parameters, A sim = 7.5 meV, K sim = 0.4 meV, κ sim = 0.2 μ eV , damping constant α TM = α RE = 0.002 , lattice constant d = 0.4 nm, and Land{\'e} g -factors g TM = 2.2 for TM, and g RE = 2 for the RE element [14] . The relations for the material parameters between Eqs. (2) and (9) are A = 4 A sim / d , K = 2 K sim / d 3 , κ = 2 κ sim / d 3 , and D = 8 D sim / d 2 . Figure 1(b) shows the assumed temperature-dependent change in the magnetic moment M i and corresponding δ s . For simplicity, we assume other parameters are invariant with temperature. 10.1103/PhysRevB.96.100407.f1 1 FIG. 1. (a) A schematic illustration of a ferrimagnet in which neighboring spins are coupled antiferromagnetically. (b) The assumed magnetic moments of TM (red) and RE (blue) elements as a function of the temperature T . Black symbols represent the net magnetic moment ( = M TM − M RE ), and dark yellow symbols represent the net angular momentum δ s . Zero δ s corresponds to the angular momentum compensation temperature T A (purple). Figure 2(a) shows V DW for D = 0 as a function of H . Below H WB , V DW increases linearly with H , in agreement with Eq. (5) (solid lines). For H > H WB , the Walker breakdown occurs except for T = T A at which V DW keeps increasing because of the absence of the Walker breakdown. Figure 2(b) shows H WB as a function of δ s at various DMIs. Two features are worth mentioning. First, H WB diverges at T A (i.e., δ s = 0 ). Second, H WB for a finite DMI becomes much larger than that for D = 0 , in agreement with Eq. (6) (solid lines). Figure 2(c) shows V DW in the high-field regime for D = 0 . The numerically obtained values of V DW at T A (green symbols) deviate from Eq. (8) (a green solid line), which predicts a linear increase in V DW with H . This deviation is a manifestation of the relativisticlike dynamics of ferrimagnetic DWs, as we will explain below in detail. For D sim = 0.25 meV, however, nonlinearity appears in all tested ranges of T [or δ s ; Fig. 2(d) ]. It means that the relativisticlike dynamics occurs in a wide range of T , resulting from the largely enhanced H WB near T A . 10.1103/PhysRevB.96.100407.f2 2 FIG. 2. (a) Domain wall velocity as a function of the external field H in the low-field regime ( μ 0 H < 10 mT). Symbols indicate the calculation results and solid lines indicate Eq. (5) . (b) Walker breakdown field H WB as a function of net angular momentum δ s at various DMI constants D sim . Domain wall velocity in the high-field regime for (c) D sim = 0 and (d) D sim = 0.25 meV. Solid straight lines in (c) indicate Eq. (8) . The red solid line in (c) and solid curved lines in (d) represent Eq. (14) . Horizontal dashed lines in (c) and (d) represent v g , max [Eq. (12) ]. For the cases showing nonlinearity, we observe spin-wave emission from the DW [Fig. 3(a) ]. The spin-wave frequency is in the THz range [Fig. 3(b) ]. To elaborate the nonlinearity of V DW on H and associated THz spin-wave emission, we derive the spin-wave dispersion for ferrimagnets on top of a uniform ground state, n = z{\^ } . Starting from the Lagrangian density Eqs. (1) and (2) , we derive the equation of motion by using the spin-wave ansatz n x + i n y = δ n exp [ i ( ω t − k x ) ] , where δ n ( ≪ 1 ) is the amplitude of the spin wave, and k is the wave number. Following mathematical procedures, we finally get the dispersion of the spin wave given by (11) ω ± = ± δ s + δ s 2 + 4 ρ ( A k 2 + K − h ) 2 ρ , where the upper (lower) sign corresponds to the right-handed (left-handed) circular mode. The spin-wave dispersion is reduced to that of ferromagnets ω = A k 2 + K − h in the limit ρ → 0 , and that of antiferromagnets ω 2 = ( A k 2 + K − h ) / ρ in the limit δ s → 0 . We note that the DMI does not contribute to spin-wave dispersion as it is effective only for the y component of magnetization, which is negligible for perpendicularly magnetized ferrimagnets. From the dispersion [Eq. (11) ], we obtain the maximum spin-wave group velocity v g , max as (12) v g , max = A / d s , where s = ( s 1 + s 2 ) / 2 . We note that v g , max is indicated by horizontal dashed lines in Figs. 2(c) and 2(d) . With v g , max , the nonlinear dependence of V DW on H is readily interpreted based on the relativisticlike kinematics of a DW, similar to the dynamics of an antiferromagnetic DW [10] : v g , max acts as the speed of light [13] and the DW width shrinks as V DW approaches v g , max via the Lorentz contraction. The Lorentz contraction of DW is described by (13) λ DW = λ eq 1 − ( V DW / v g , max ) 2 , where λ eq is the equilibrium DW width. The inertial DW mass also varies with the Lorentz factor 1 / 1 − ( V DW / v g , max ) 2 , i.e., M = 2 ρ A / λ DW . Figure 3(c) shows that the DW width decreases as V DW approaches v g , max while the inertial mass increases with V DW . With the Lorentz contraction, we modify Eq. (5) relativistically as (14) V DW = v g , max 1 − ( λ DW / λ eq ) 2 , which is represented by a red solid line in Fig. 2(c) and solid lines in Fig. 2(d) . Excellent agreement between the numerically obtained V DW and Eq. (14) confirms the relativistic dynamics of ferrimagnetic DWs. 10.1103/PhysRevB.96.100407.f3 3 FIG. 3. (a) Configuration of domain wall and spin waves for the staggered vector n . The inset shows the overall shape of the domain wall. (b) The frequency of emitted spin waves as a function of H with finite δ s . (c) Domain wall width and mass as a function of V DW / v g , max at T A . The orange solid line indicates Eq. (13) . (d) Domain wall velocity as a function of the current density J at several temperatures. Symbols represent calculation results and solid lines represent Eq. (15) . Dashed lines represent relativistic modified solutions, Eq. (14) . We also show that spin-orbit torque drives the fast relativistic motion of ferrimagnetic DWs. We suppose that the charge current flows along the x direction in the adjoining heavy metal layer, where spin accumulation resulting in spin-orbit torques is manifested at the interface. We consider a dampinglike torque only for simplicity. Using the collective coordinate approach [10] , we obtain the steady-state solution of ferrimagnetic DW velocity as (15) V DLT = λ DW π s B{\~ } DLT 2 α , where B{\~ } DLT = ℏ θ SH J / 2 e t z s 1 s 2 is the effective field corresponding to the dampinglike torque, θ SH is the effective spin Hall angle of the ferrimagnet/heavy metal bilayer, J is the current density, e is the electron charge, and t z is the ferrimagnet thickness. A numerical simulation including the dampinglike torque for D sim = 0.25 meV [Fig. 3(d) ] shows that spin-orbit torque combined with the DMI effect is highly efficient for ferrimagnetic DW motion and relativisticlike dynamics occurs for all tested ranges of T (or δ s ). This spin-orbit-torque-driven ferrimagnetic DW motion also accompanies THz spin-wave emission (not shown). Finally, we discuss the origin of spin-wave emission from ferrimagnetic DWs. Two mechanisms have been proposed, a Cherenkov-like process [35] and internal DW structure distortion [36] . The former occurs when the DW velocity matches the spin-wave phase velocity. In antiferromagnets or ferrimagnets near T A , the phase velocity is always higher than the group velocity, as one finds from the dispersion [Eq. (11) ]. Therefore, this Cherenkov process is irrelevant to our case. For ferrimagnetic DWs, instead, the spin waves can be emitted by releasing the DW energy enhanced through the Lorentz contraction as in the case of antiferromagnetic DWs [10] . In conclusion, we have shown field-driven THz spin-wave emission for ferrimagnetic DWs, which is not possible for ferromagnetic or antiferromagnetic DWs. In ferrimagnet/heavy metal bilayers in which the interfacial DMI arises naturally, field-driven THz spin-wave emission can be observed in a wide range of T (or δ s ), thereby largely enhancing the experimental accessibility to our prediction. Moreover, an in-plane current can also excite THz spin waves in a wide range of T (or δ s ) through the combined effect between spin-orbit torque and DMI. Our finding suggests that ferrimagnetic DWs are potentially useful for high-speed and high-frequency spintronics devices. K.-J.L. was supported by the National Research Foundation of Korea (NRF) (NRF-2015M3D1A1070465, NRF-2017R1A2B2006119) and by the DGIST R&D Program of the Ministry of Science, ICT and Future Planning (17-BT-02). S.K.K. and Y.T. were supported by the Army Research Office under Contract No. W911NF-14-1-0016. T.O. was supported by JSPS KAKENHI Grants No. 15H05702 and No. 26103002. K.J.K. was supported by NRF-2017R1C1B2009686. G.G. was supported by NRF-2016R1A6A3A1193588. S.-H.O. and S.K.K. contributed equally to this work. Publisher Copyright: {\textcopyright} 2017 American Physical Society.",

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doi = "10.1103/PhysRevB.96.100407",

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TY - JOUR

T1 - Coherent terahertz spin-wave emission associated with ferrimagnetic domain wall dynamics

AU - Oh, Se Hyeok

AU - Kim, Se Kwon

AU - Lee, Dong Kyu

AU - Go, Gyungchoon

AU - Kim, Kab Jin

AU - Ono, Teruo

AU - Tserkovnyak, Yaroslav

AU - Lee, Kyung Jin

N1 - Funding Information: Oh Se-Hyeok 1 Kim Se Kwon 2 Lee Dong-Kyu 3 Go Gyungchoon 3 Kim Kab-Jin 4,5 Ono Teruo 5 Tserkovnyak Yaroslav 2 Lee Kyung-Jin 1,3,6 * Department of Nano-Semiconductor and Engineering, 1 Korea University , Seoul 02841, Korea Department of Physics and Astronomy, 2 University of California , Los Angeles, California 90095, USA Department of Materials Science and Engineering, 3 Korea University , Seoul 02841, Korea Department of Physics, 4 Korea Advanced Institute of Science and Technology , Daejeon 34141, Korea Institute for Chemical Research, 5 Kyoto University , Kyoto 611-0011, Japan KU-KIST Graduate School of Converging Science and Technology, 6 Korea University , Seoul 02841, Korea * kj_lee@korea.ac.kr September 2017 28 September 2017 96 10 100407 4 May 2017 ©2017 American Physical Society 2017 American Physical Society We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum. National Research Foundation of Korea http://dx.doi.org/10.13039/501100003725 NRF http://sws.geonames.org/1835841/ 2015M3D1A1070465 2017R1A2B2006119 2017R1C1B2009686 2016R1A6A3A1193588 Ministry of Science, ICT and Future Planning http://dx.doi.org/10.13039/501100003621 MSIP http://sws.geonames.org/1835841/ 17-BT-02 Army Research Office http://dx.doi.org/10.13039/100000183 ARO http://sws.geonames.org/6252001/ http://sws.geonames.org/4482348/ W911NF-14-1-0016 Japan Society for the Promotion of Science http://dx.doi.org/10.13039/501100001691 JSPS http://sws.geonames.org/1861060/ 15H05702 26103002 In modern communications, information is carried by electromagnetic waves whose frequency is limited to ≈ 0.1 terahertz (THz), the frequency of oscillating circuits based on high-speed transistors [1] . On the other hand, semiconductor lasers generate coherent light with a frequency > 30 THz [2] . The terahertz gap refers to the fact that no relevant technology exists in the frequency range between these two limits ( 0.1 – 30 THz). Therefore, it is of critical importance to find relevant physical phenomena that fill in the terahertz gap. In this respect, antiferromagnets whose resonance frequencies are in the THz range [3,4] are of interest [5,6] . It has been reported that coherent THz magnons or spin waves are generated in antiferromagnets, driven by a laser [7,8] or an electrical current [9,10] . However, THz spin-wave excitations by a dc magnetic field are in principle not possible for antiferromagnets as their magnetic moments are compensated on an atomic scale. In this Rapid Communication, we theoretically show that the generation of coherent THz spin waves can be achieved by field-driven domain wall (DW) motion in ferrimagnet/heavy metal bilayers in which the interfacial Dzyaloshinskii-Moriya interaction (DMI) is present. As far as the terahertz gap is concerned, this dc field-driven scheme could be beneficial as it allows a low-power operation by avoiding laser-induced or current-induced heating. It is also fundamentally interesting as THz spin-wave emission is caused by a relativisticlike dynamics of a ferrimagnetic DW. Relativistic kinematics refers to kinematics compatible with the theory of relativity [11] , whose key ingredient is the Lorentz invariance with a limiting velocity c , the speed of light. When the dispersion of a wave satisfies the Lorentz invariance, a quasiparticle corresponding to the wave follows an analogous relativistic kinematics by replacing the speed of light with the maximum group velocity of the wave. When the velocity of the quasiparticle approaches the maximum group velocity, it undergoes the Lorentz contraction and its speed saturates to the limiting velocity. An example of such quasiparticles is an antiferromagnetic DW [10,12,13] . When the DW velocity approaches the maximum spin-wave group velocity, the DW width shrinks by emitting spin waves [10] . Similarly, the dynamics of a ferrimagnetic DW is also expected to exhibit the features of relativistic kinematics provided that the net magnetization and DMI, which break the Lorentz invariance of the system, are sufficiently ineffective. Here, we show that such a relativisticlike DW dynamics is achievable for a class of ferrimagnets, rare earth (RE) and transition metal (TM) compounds, in which the spin moments are antiferromagnetically coupled. As RE and TM elements have different Landé- g factors [14] , RE-TM ferrimagnets have two distinct temperatures: the magnetic moment compensation point T M where the net magnetic moment vanishes, and the angular momentum compensation point T A where the net angular momentum vanishes. See Fig. 1(a) for an illustration of spin configurations in a ferrimagnet. For RE-TM ferrimagnets, resonance [15,16] , switching [17–21] , domain wall motion [22–24] , and skyrmion (or bubble domain) motion [25–27] near these compensation points have been explored experimentally and theoretically. In particular, an experimental observation of fast field-driven DW motion at T A in GdFeCo single-layered ferrimagnets was recently reported [23] . This observation reveals two distinguishing features of RE-TM ferrimagnets at T A . One is that the spin dynamics is antiferromagnetic and thus fast because of zero net angular momentum at T A . The other is that this fast antiferromagnetic dynamics is achieved by a field because the net magnetic moment is nonzero at T A and thus couples with a magnetic field. We begin by deriving the equations of motion for a ferrimagnetic DW based on the collective coordinate approach [28] . The dynamics of a general collinear ferrimagnet at sufficiently low temperatures can be described by the following Lagrangian density [27,29] , (1) L = ρ n ̇ 2 / 2 − δ s a [ n ] · n ̇ − U , where n is the unit vector along the collinear order, ρ parametrizes the inertia of the dynamics, δ s is the spin density in the direction n , a [ n ] is the vector potential generated by a magnetic monopole of unit charge satisfying ∇ n × a = n , and U is the potential-energy density. Here, the first term is the spin Berry phase associated with the staggered spin density, which thus appears in the Lagrangian for collinear antiferromagnets; the second term is the Berry phase associated with the net spin density δ s , which is used to describe the dynamics of uncompensated spins in ferrimagnets. We consider the following potential-energy density, (2) U = A ( ∇ n ) 2 / 2 − K ( n · z ̂ ) 2 / 2 + κ ( n · x ̂ ) 2 / 2 + D y ̂ · ( n × ∂ x n ) / 2 − h · n . Here, the first term is the exchange energy with A > 0 ; the second term is the easy-axis anisotropy along the z axis with K > 0 ; the third term is the weaker DW hard-axis anisotropy along the x axis with κ > 0 ; the fourth term is the interfacial DMI; the last term is the Zeeman coupling with h = M net H , where M net is the net magnetization in the direction n . The dissipation can be accounted for by introducing (the spatial density of) the Rayleigh dissipation function, R = s α n ̇ 2 / 2 . Here, s α is a phenomenological parameter quantifying the energy and spin loss due to the magnetic dynamics. For example, in the ferromagnetic regime, i.e., away from T A , it can be considered as the product of the effective Gilbert damping constant and the net spin density. The low-energy dynamics of a DW can be described by the two collective coordinates, the position X ( t ) and the azimuthal angle ϕ ( t ) . We consider the Walker ansatz [30] for the DW profile, n ( x , t ) = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) , where θ = 2 tan − 1 { exp [ ( x − X ) / λ ] } and λ = A / K is the DW width. The equations of motion can be derived from Eqs. (1) and (2) in conjunction with the Rayleigh dissipation function, (3) M X ̈ + G ϕ ̇ + M X ̇ / τ = F , (4) I ϕ ̈ − G X ̇ + I ϕ ̇ / τ = − κ ̃ sin ϕ cos ϕ + D ̃ sin ϕ , where M = 2 ρ A / λ is the mass, I = 2 ρ A λ is the moment of inertia, G = 2 δ s A is the gyrotropic coefficient, τ = ρ / s α is the relaxation time, F = 2 h A is the force exerted by an external field, κ ̃ = 2 κ λ A , D ̃ = π D A / 2 , and A is the cross-sectional area of the DW. From Eq. (3) , we obtain the steady-state solution of the DW velocity, (5) V DW = M net λ s α H , where H is the external field applied along the z axis. In this steady state, the DW moves at a constant velocity V DW with a constant angle ϕ . When the field becomes sufficiently strong such that V DW exceeds a certain threshold V max , the DW begins to precess, engendering the phenomenon known as the Walker breakdown [31,32] . The Walker breakdown field H WB can be obtained from Eq. (4) by (6) H WB = V max s α M net λ . In the absence of DMI ( D = 0 ), the threshold velocity is given by V max = κ ̃ / 2 G and thus H WB = κ ̃ s α / 2 G M net λ . When DMI is much stronger than the DW anisotropy in the y direction, i.e., | D ̃ | ≫ κ ̃ , then | V max | = | D ̃ | / G . In this strong DMI limit, the Walker breakdown field is given by (7) H WB = | D ̃ | G s α M net λ = π | D | 4 δ s s α M net λ . We note that H WB is inversely proportional to G and thus to the net spin density δ s . As a result, the Walker breakdown is absent at T A where the net spin density vanishes, δ s = 0 . This suppression of the Walker breakdown at T A can be understood as a result of decoupling of the DW position X and the angle ϕ at T A [23] . It is worthwhile comparing Eq. (7) to the Walker breakdown field for ferromagnetic DWs in the strong DMI limit [33] , H WB,FM = α π D FM / 2 M FM λ FM , which can be obtained from Eq. (7) by taking the ferromagnetic limit. From this comparison, one finds that in the vicinity of T A , H WB for ferrimagnetic DWs is much larger than that for ferromagnetic DWs because δ s ≈ 0 and | M net | ≪ M FM . Moreover, this very large H WB for ferrimagnetic DWs suggests that V DW can reach the maximum group velocity of the spin wave more easily without experiencing a Walker breakdown, and thus ferrimagnetic DWs can generate THz spin waves in a wide range of net angular momentum δ s . Finally, the time-averaged velocity V ¯ for a one period far above the Walker breakdown is given as (8) V ¯ = M net λ s α + δ s 2 / s α H . To verify these theoretical predictions on the DW velocity and THz spin-wave emission, we perform atomistic model calculations [10,34] for two-sublattice ferrimagnets, which correspond to RE-TM compounds. Two sublattices possess the magnetization M 1 and M 2 , which are coupled by the antiferromagnetic exchange. The spin densities are given by s 1 = M 1 / γ 1 and s 2 = M 2 / γ 2 , where γ i = g i μ B / ℏ is the gyromagnetic ratio of the lattice i , μ B is the Bohr magneton, and g i is the Landé- g factor. The parameters in the above descriptions for general ferrimagnets are given by δ s = s 1 − s 2 , M net = M 1 − M 2 , and s α = α 1 s 1 + α 2 s 2 , where α i is the Gilbert damping constant for the lattice i . The one-dimensional discrete Hamiltonian that we use for numerical calculations is given by (9) H = A sim ∑ i S i · S i + 1 − K sim ∑ i ( S i · z ̂ ) 2 + κ sim ∑ i ( S i · x ̂ ) 2 + D sim ∑ i y ̂ · ( S i × S i + 1 ) − g i μ B μ 0 ∑ i H · S i , where S i is the normalized spin moment vector at lattice site i [i.e., an even (odd) i corresponds to a RE (TM) atomic site], A sim , K sim , κ sim , and D sim denote the exchange, easy-axis anisotropy, DW hard-axis anisotropy, and DMI constants, respectively, and H is the external field. We numerically solve the atomistic Landau-Lifshitz-Gilbert equation, (10) ∂ S i ∂ t = − γ i S i × H eff , i + α i S i × ∂ S i ∂ t , where H eff , i = − 1 M i ∂ H ∂ S i is the effective field. We use the following simulation parameters, A sim = 7.5 meV, K sim = 0.4 meV, κ sim = 0.2 μ eV , damping constant α TM = α RE = 0.002 , lattice constant d = 0.4 nm, and Landé g -factors g TM = 2.2 for TM, and g RE = 2 for the RE element [14] . The relations for the material parameters between Eqs. (2) and (9) are A = 4 A sim / d , K = 2 K sim / d 3 , κ = 2 κ sim / d 3 , and D = 8 D sim / d 2 . Figure 1(b) shows the assumed temperature-dependent change in the magnetic moment M i and corresponding δ s . For simplicity, we assume other parameters are invariant with temperature. 10.1103/PhysRevB.96.100407.f1 1 FIG. 1. (a) A schematic illustration of a ferrimagnet in which neighboring spins are coupled antiferromagnetically. (b) The assumed magnetic moments of TM (red) and RE (blue) elements as a function of the temperature T . Black symbols represent the net magnetic moment ( = M TM − M RE ), and dark yellow symbols represent the net angular momentum δ s . Zero δ s corresponds to the angular momentum compensation temperature T A (purple). Figure 2(a) shows V DW for D = 0 as a function of H . Below H WB , V DW increases linearly with H , in agreement with Eq. (5) (solid lines). For H > H WB , the Walker breakdown occurs except for T = T A at which V DW keeps increasing because of the absence of the Walker breakdown. Figure 2(b) shows H WB as a function of δ s at various DMIs. Two features are worth mentioning. First, H WB diverges at T A (i.e., δ s = 0 ). Second, H WB for a finite DMI becomes much larger than that for D = 0 , in agreement with Eq. (6) (solid lines). Figure 2(c) shows V DW in the high-field regime for D = 0 . The numerically obtained values of V DW at T A (green symbols) deviate from Eq. (8) (a green solid line), which predicts a linear increase in V DW with H . This deviation is a manifestation of the relativisticlike dynamics of ferrimagnetic DWs, as we will explain below in detail. For D sim = 0.25 meV, however, nonlinearity appears in all tested ranges of T [or δ s ; Fig. 2(d) ]. It means that the relativisticlike dynamics occurs in a wide range of T , resulting from the largely enhanced H WB near T A . 10.1103/PhysRevB.96.100407.f2 2 FIG. 2. (a) Domain wall velocity as a function of the external field H in the low-field regime ( μ 0 H < 10 mT). Symbols indicate the calculation results and solid lines indicate Eq. (5) . (b) Walker breakdown field H WB as a function of net angular momentum δ s at various DMI constants D sim . Domain wall velocity in the high-field regime for (c) D sim = 0 and (d) D sim = 0.25 meV. Solid straight lines in (c) indicate Eq. (8) . The red solid line in (c) and solid curved lines in (d) represent Eq. (14) . Horizontal dashed lines in (c) and (d) represent v g , max [Eq. (12) ]. For the cases showing nonlinearity, we observe spin-wave emission from the DW [Fig. 3(a) ]. The spin-wave frequency is in the THz range [Fig. 3(b) ]. To elaborate the nonlinearity of V DW on H and associated THz spin-wave emission, we derive the spin-wave dispersion for ferrimagnets on top of a uniform ground state, n = z ̂ . Starting from the Lagrangian density Eqs. (1) and (2) , we derive the equation of motion by using the spin-wave ansatz n x + i n y = δ n exp [ i ( ω t − k x ) ] , where δ n ( ≪ 1 ) is the amplitude of the spin wave, and k is the wave number. Following mathematical procedures, we finally get the dispersion of the spin wave given by (11) ω ± = ± δ s + δ s 2 + 4 ρ ( A k 2 + K − h ) 2 ρ , where the upper (lower) sign corresponds to the right-handed (left-handed) circular mode. The spin-wave dispersion is reduced to that of ferromagnets ω = A k 2 + K − h in the limit ρ → 0 , and that of antiferromagnets ω 2 = ( A k 2 + K − h ) / ρ in the limit δ s → 0 . We note that the DMI does not contribute to spin-wave dispersion as it is effective only for the y component of magnetization, which is negligible for perpendicularly magnetized ferrimagnets. From the dispersion [Eq. (11) ], we obtain the maximum spin-wave group velocity v g , max as (12) v g , max = A / d s , where s = ( s 1 + s 2 ) / 2 . We note that v g , max is indicated by horizontal dashed lines in Figs. 2(c) and 2(d) . With v g , max , the nonlinear dependence of V DW on H is readily interpreted based on the relativisticlike kinematics of a DW, similar to the dynamics of an antiferromagnetic DW [10] : v g , max acts as the speed of light [13] and the DW width shrinks as V DW approaches v g , max via the Lorentz contraction. The Lorentz contraction of DW is described by (13) λ DW = λ eq 1 − ( V DW / v g , max ) 2 , where λ eq is the equilibrium DW width. The inertial DW mass also varies with the Lorentz factor 1 / 1 − ( V DW / v g , max ) 2 , i.e., M = 2 ρ A / λ DW . Figure 3(c) shows that the DW width decreases as V DW approaches v g , max while the inertial mass increases with V DW . With the Lorentz contraction, we modify Eq. (5) relativistically as (14) V DW = v g , max 1 − ( λ DW / λ eq ) 2 , which is represented by a red solid line in Fig. 2(c) and solid lines in Fig. 2(d) . Excellent agreement between the numerically obtained V DW and Eq. (14) confirms the relativistic dynamics of ferrimagnetic DWs. 10.1103/PhysRevB.96.100407.f3 3 FIG. 3. (a) Configuration of domain wall and spin waves for the staggered vector n . The inset shows the overall shape of the domain wall. (b) The frequency of emitted spin waves as a function of H with finite δ s . (c) Domain wall width and mass as a function of V DW / v g , max at T A . The orange solid line indicates Eq. (13) . (d) Domain wall velocity as a function of the current density J at several temperatures. Symbols represent calculation results and solid lines represent Eq. (15) . Dashed lines represent relativistic modified solutions, Eq. (14) . We also show that spin-orbit torque drives the fast relativistic motion of ferrimagnetic DWs. We suppose that the charge current flows along the x direction in the adjoining heavy metal layer, where spin accumulation resulting in spin-orbit torques is manifested at the interface. We consider a dampinglike torque only for simplicity. Using the collective coordinate approach [10] , we obtain the steady-state solution of ferrimagnetic DW velocity as (15) V DLT = λ DW π s B ̃ DLT 2 α , where B ̃ DLT = ℏ θ SH J / 2 e t z s 1 s 2 is the effective field corresponding to the dampinglike torque, θ SH is the effective spin Hall angle of the ferrimagnet/heavy metal bilayer, J is the current density, e is the electron charge, and t z is the ferrimagnet thickness. A numerical simulation including the dampinglike torque for D sim = 0.25 meV [Fig. 3(d) ] shows that spin-orbit torque combined with the DMI effect is highly efficient for ferrimagnetic DW motion and relativisticlike dynamics occurs for all tested ranges of T (or δ s ). This spin-orbit-torque-driven ferrimagnetic DW motion also accompanies THz spin-wave emission (not shown). Finally, we discuss the origin of spin-wave emission from ferrimagnetic DWs. Two mechanisms have been proposed, a Cherenkov-like process [35] and internal DW structure distortion [36] . The former occurs when the DW velocity matches the spin-wave phase velocity. In antiferromagnets or ferrimagnets near T A , the phase velocity is always higher than the group velocity, as one finds from the dispersion [Eq. (11) ]. Therefore, this Cherenkov process is irrelevant to our case. For ferrimagnetic DWs, instead, the spin waves can be emitted by releasing the DW energy enhanced through the Lorentz contraction as in the case of antiferromagnetic DWs [10] . In conclusion, we have shown field-driven THz spin-wave emission for ferrimagnetic DWs, which is not possible for ferromagnetic or antiferromagnetic DWs. In ferrimagnet/heavy metal bilayers in which the interfacial DMI arises naturally, field-driven THz spin-wave emission can be observed in a wide range of T (or δ s ), thereby largely enhancing the experimental accessibility to our prediction. Moreover, an in-plane current can also excite THz spin waves in a wide range of T (or δ s ) through the combined effect between spin-orbit torque and DMI. Our finding suggests that ferrimagnetic DWs are potentially useful for high-speed and high-frequency spintronics devices. K.-J.L. was supported by the National Research Foundation of Korea (NRF) (NRF-2015M3D1A1070465, NRF-2017R1A2B2006119) and by the DGIST R&D Program of the Ministry of Science, ICT and Future Planning (17-BT-02). S.K.K. and Y.T. were supported by the Army Research Office under Contract No. W911NF-14-1-0016. T.O. was supported by JSPS KAKENHI Grants No. 15H05702 and No. 26103002. K.J.K. was supported by NRF-2017R1C1B2009686. G.G. was supported by NRF-2016R1A6A3A1193588. S.-H.O. and S.K.K. contributed equally to this work. Publisher Copyright: © 2017 American Physical Society.

PY - 2017/9/28

Y1 - 2017/9/28

N2 - We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum.

AB - We theoretically study the dynamics of ferrimagnetic domain walls in the presence of a Dzyaloshinskii-Moriya interaction. We find that an application of a dc magnetic field can induce terahertz spin-wave emission by driving ferrimagnetic domain walls, which is not possible for ferromagnetic or antiferromagnetic domain walls. The Dzyaloshinskii-Moriya interaction is shown to facilitate terahertz spin-wave emission in a wide range of the net angular momentum by increasing the Walker breakdown field. Moreover, we show that spin-orbit torque combined with the Dzyaloshinskii-Moriya interaction also drives fast ferrimagnetic domain wall motion by emitting terahertz spin waves in a wide range of the net angular momentum.

UR - http://www.scopus.com/inward/record.url?scp=85050364037&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.96.100407

DO - 10.1103/PhysRevB.96.100407

M3 - Article

AN - SCOPUS:85050364037

VL - 96

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 10

M1 - 100407

ER -

Coherent terahertz spin-wave emission associated with ferrimagnetic domain wall dynamics

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