Spin-Wave Doppler Shift by Magnon Drag in Magnetic Insulators

Doppler shift of quasiparticle dispersion induced by charge currents is responsible for the critical supercurrents in superconductors and spin-wave instabilities by spin-transfer torque in metallic ferromagnets. Here we predict an analogous effect in thin films of magnetic insulators. A coherent spin current, excited by stripline microwaves, or a thermal magnon current, driven by a temperature gradient, can induce a Doppler shift that tilts the magnon dispersion in the spin-current direction. Around a critical driving strength, that is characterized by a spin-wave instability in the self-consistent mean-field treatment, the pumped magnon current reaches a maximum accompanied by a strong breaking of chiral pumping. The backaction of magnon currents on magnetic orders is therefore important for realizing large spin currents in low-dimensional magnonic devices.

Realizing large spin current is an important pursuit in spintronics. In metallic ferromagnets, electric currents excite magnetization dynamics by the spin-transfer torque [27,28]. The charge current induces a Doppler shift, i.e., a tilt of the spin-wave dispersion of a homogeneous magnetization in momentum space, which could trigger a spin-wave instability [29][30][31] and modulate the magnetic ground state [32]. In superconductors the critical supercurrent is caused by the Doppler shift of Bogoliubov quasiparticles. These obviously do not apply to magnetic insulators that cannot carry an electric charge current. However, magnetic insulators are also conduits for (magnonic) spin currents. By solving the self-consistent problem of a magnon gas interacting with its own current, we find that magnon spin currents can also affect the magnon dispersion via a Doppler effect.
In this Letter, we formulate the dynamics of long- thin magnetic films driven by pure magnon current. A long stripline along theẑ-direction is illustrated to pump the magnon current (the green thick arrow) in YIG films of thickness s that causes the tilt of magnon dispersion, as shown by the red thick arrow and parabolic bands. The in-plane magnetization is saturated with a relative angle ϕ to the stripline direction.
wavelength coherent magnons of thin YIG films in the presence of large magnon currents that are pumped by stripline microwaves, as depicted in Fig. 1, or injected thermally or electrically via Pt contacts. We predict an interaction-induced drag effect, in the form of a magnonic Doppler effect, on the magnon velocities, that tilts the spin-wave dispersion. The associated physics differs strongly from the magnon-drag by phonon [33] or electron [29][30][31]34] currents by its nonlinearity origin. Its phenomenology is intriguingly similar to an interfacial Dzyaloshinskii-Moriya interaction (DMI) [35][36][37]. The polarization-momentum locked AC magnetic field emitted by a microwave stripline [38][39][40] coherently populates magnon states at one side of the stripline through the chiral pumping effect, with a unidirectional magnon current that simultaneously modifies the magnon dispersion. Incoherent magnon accumulations injected thermally or electrically propagate diffusely and generate a magnon current proportional to their density gradient. At a critical value of any of these, the magnon excitation energy touches zero at a particular momentum, indicating an instability of the ground state with constant magnetization. Around this critical driving, we predict a maximum of the magnonic spin current accompanied with a strong breaking of the chirality in magnetization excitation by striplines. Our work points out the importance of the backaction of magnon currents on magnetic orders in the realization of large spin currents in magnetic insulators, providing a different scenario to the nonlinearity by populations.
Model and formalism.-We consider an in-plane magnetized YIG film with thickness s = O(10) nm, with surface normal oriented along thex-direction. An in-plane static magnetic field H app is applied at an angle ϕ to the striplineẑ-direction (Fig. 1). The Hamiltonian of the magnetic order readŝ where µ 0 is the vacuum permeability, α ex is the exchange stiffness, and M is the magnetization. We disregard anisotropies [41,42] because the crystal ones are small in YIG, while the dipolar ones are strongly suppressed in the thin film limit [43,44]. The exchange length in YIG is λ ex = 2π √ α ex = 109 nm since α ex = 3 × 10 −16 m 2 [45,46]. The magnetization dynamics then obeys a Landau-Lifshitz-Gilbert (LLG) equation where α G is the Gilbert damping constant and −γ is the electron gyromagnetic ratio. In the absence of external torques and damping the magnetization carries a magnetization current densitỹ that satisfies the continuity equation dM/dt + ∇ ·j = 0 [47]. The LLG phenomenology contains all of the nonlinearities that can be captured by interacting magnons to some extent. The Holstein-Primakoff transformation expresses the magnetization dynamics by bosonic magnon opera- where the spin opera-torsŜ = −M/ (γ ). The leading terms in the expansion of the square roots leads to a complete set of harmonic oscillators, that we use to expand the full problem. The eigenmodes normal to the film plane depend on the boundary conditions, that become free for thin films [48]. The magnon operators in position space can then be expanded in perpendicular standing spin waves (PSSWs) with index l [43,44] Substituting these expressions into the Holstein-Primakoff expansion, the Hamiltonian can be written asĤ =Ĥ L +Ĥ NL + · · · , whereĤ L describes the noninteracting magnon gas andĤ NL is the leading nonlinear term that introduces interactions between the magnons. At sufficiently low magnon densitieŝ contains two types of magnon-number conserving interactions derived in the Supplemental Material [49]. The potentials are governed by magnon-mode overlap integrals with dimensionless form factors When l 1 = 0, the scattering potentials obey selection rules U 0l2l3l4 ∝ l 3 l 4 (δ l2+l3,l4 + δ l2+l4,l3 − δ l3+l4,l2 ) and V 0l2l3l4 ∝ (δ l2+l3,l4 + δ l2+l4,l3 + δ l2+l3+l4,0 + δ l3+l4,l2 ). In the strictly two-dimensional limit, U 0000 = 0 vanishes, but The divergence for vanishing film thickness is an artifact of the continuum approximation that breaks down when s approaches unit cell dimensions. We are interested in the effect of a magnon current on a low-frequency coherent excitation, i.e., at excitation frequency ω/(2π) 1 GHz, which allows us to set l 1 = 0. Using the above selection rules of the scattering potentials and energy conservation, we prove in the Supplemental Material [49] that the incoherent scattering of these low-energy magnons by those in all other bands is marginally small. The leading nonlinearities in the coherent magnon states thus reduce to a self-consistent meanfield problem [50,51], in which the interaction renormalizes the energy dispersion but does not affect magnon dephasing and lifetime. The coherent magnon amplitude in the lowest band obeys a Heisenberg equation of motion that is augmented by the Gilbert damping [49], where · · · represents an ensemble average, is the linear-momentum magnon current density in subband l with contributions from both coherent and incoherent magnons, and P ex is a microwaves excitation source that will be specified below. The (locally) uniform magnon current hence engages the gradient (or momentum) of the magnon amplitude ∇ ρ Ψ 0 and tilts the magnon dispersion, which is an interaction-induced drag effect [33,34]. This drag depends only on the magnitude of the current, not on the coherent versus incoherent nature of the carriers. The magnon linear-momentum current density [Eq. (7)] is proportional to the magnon number current densityJ l defined by the continuity equation and the Heisenberg equation of motion for the non-interacting magnon Hamiltonian, since the exchange magnons have a constant mass /(2ω M α ex ). The former is also a spin current since in the absence of anisotropies the magnons carry angular momentum . With magnon density operatorρ l m (ρ) = Ψ † l (ρ)Ψ l (ρ) leading to J l (ρ) = (2ω M α ex / )J l (ρ), which is consistent with Eq. (3) since 1/(γ ) dxj(x, ρ) →J l (ρ) when l = 0 to linear order in the magnon operator.
Magnonic Doppler effect.-The microwaves emitted by a long stripline on top of a thin magnetic film launch a coherent magnon current normal to it. We consider a metallic wire of rectangular cross section 0 < x < d and −w/2 < y < w/2 ( Fig. 1) with an AC current density I of frequency ω s . The microwaves are uniform over the film thickness when s d. The Fourier component k y of the Oersted magnetic field in the thin film below the stripline (x → −s/2) reads [38][39][40][52][53][54], with F(d, w) = (2/k 2 y ) sin (k y w/2) 1 − e −|ky|d determined by stripline dimensions. Here we used |k y | ω s /c because the velocity of light c is much larger than that of the magnons. The magnetic field H y (k y , ω s ) = isgn(k y )H x (k y , ω s ) is right and left circularly polarized for positive and negative k y , respectively, so polarization and momentum are locked. This field couples to the magnons of the lowest PSSW band up to wave numbers k y ∼ π/w by the Zeeman interaction with coupling constant g = µ 0 γ M s s/2, so the excitation source P ex = g(H x (k y , t) − i cos ϕH y (k y , t)) in Eq. (6). The in-plane magnetization angle ϕ can be rotated by an applied DC magnetic field to tune the magnitude and direction of the pumped magnon current. When ϕ = 0, the stripline magnetic field launches a magnon current with k y > 0 into half space (see below). Thereby the excited magnon current J y (y > 0) = J y exp(−y/δ) decays exponentially with distance from the source on the scale of the decay length δ (ω s ) ∼ 2/ Im κ y ∼ (α ex ω M )(ω s − µ 0 γH app )/(α G ω s ), i.e. the root of (ω s − µ 0 γH app − ω M α ex κ 2 y ) 2 + (α G ω s ) 2 = 0. On the other hand, the amplitude Ψ 0 (ρ) oscillates rapidly with wavelength (1/|κ y | δ). Near the stripline, the magnon current in the lowest band obeys the integral equation, obtained from Eq. (6), with Doppler-shifted magnon frequencỹ which can be solved iteratively or graphically. A magnon current can also be carried by the incoherent thermal magnons driven by a magnon chemical potential or temperature gradient. These can be created either by the spin-Hall effect in, or Ohmic heating of, currentbiased Pt contacts [13,14]. We can estimate the steadystate magnon current under a temperature gradient ∇T by the linearized Boltzmann equation in the relaxationtime approximation [23,50,55], assuming that the drag term in the collision integral is small, where v k,l = (1/ )∂E l (k)/∂k = 2ω M α ex k is the magnon group velocity, f k,l = Ψ † l (k)Ψ l (k) is the magnon distribution in the l-band, f (0) is the equilibrium Planck distribution at temperature T , τ k,l ≈ /(α G E k,l ) is the magnon relaxation time [23]. Assuming a uniform ∇T = E yŷ applied along theŷdirection, the magnon momentum current [Eq. (7)] which also causes a Doppler shift [Eq. (11)] of the coherent magnon amplitude. Figure 2(a) illustrates the pumped magnon current J y as a function of the applied electric current density I with frequency ω s /(2π) ≈ 0.93 GHz across the stripline of width w = 150 nm and thickness d = 80 nm [21,54]  shows the coherently pumped magnon current Jy as a function of the applied electric current density I in the stripline from numerical LLG calculations ("LLG"), non-interacting spin-wave theory ("SW-free"), and spin-wave theory including the drag effect ("SW-drag"). The tilt of magnon dispersion at high excitation is illustrated in (b). We illustrate the chirality of the spin-current excitation for I < Ic [(c)] and I > Ic [(d)], respectively. The parameters used in the calculations are given in the text and figures. thickness s = 10 nm, the applied static magnetic field µ 0 H app = 10 mT that drives out domain walls [21,46], µ 0 M s = 0.18 T, and α G = 10 −4 . Magnons of wavelength 2w are resonantly excited and carry a current with decay length δ ≈ 333 µm. Here we compare the analytical solutions with the numerically exact solution of the LLG equation, which predicts a maximum spin-wave current for a stripline current I c ≈ 5 × 10 7 A/cm 2 . The noninteracting spin-wave theory (SW-free) fails already for small I, which emphasizes the importance of nonlinearities. When including the drag effect, the spin-wave theory Eq. (10) J y saturates at a current I ∼ I c , but returns to the non-interacting values at larger currents. I c is determined by the onset of a spin-wave instability that is characterized by negative magnon excitation energy (see below), which causes the discontinuous change of the spin current calculated by the self-consistent mean-field theory. When I > I c , the lowest-order nonlinearity of the Holstein-Primakoff expansion and thereby the mean-field theory may break down. The Doppler shift of the spinwave dispersion illustrated in Fig. 2(b) holds only for I < I c . When I I c , we observe that the chirality of the magnon excitation is strongly reduced, indicating that the backscattering of magnons becomes strong, as illustrated by Figs. 2(c) and (d), which is partly responsible for the suppression of spin current at strong driving.
A maximum value of the spin current in ultrathin YIG films was observed in a transistor in which a DC-current biased Pt gate injects magnons into the conducting channel [15]. Here we find in the coherent excitation regime that the magnon nonlinearity naturally limits the spin current to a maximum. The tilt of dispersion causes chiral velocities of spin waves of the same energy that should be observable by changes in the microwave transmission [21,46], nitrogen-vacancy center magnetometry [54,56], and Brillouin light scattering [57]. The dispersion tilts into the opposite direction when the magnetization direction is reversed (ϕ = π) and vanishes when perpendicular to the stripline (ϕ = π/2), i.e., it follows the current direction governed by the chirality of the stripline magnetic field. The basic features agree with recently reported experiments in YIG thin films of thickness s = 7 nm [21] that were interpreted in terms of the DMI by spinorbit interaction. A significant DMI has been reported for the interface between GGG and rare earth iron garnets [58], but has not yet been confirmed by other studies on YIG|GGG, to the best of our knowledge.
Magnetization chirality breaking.-We trace a close connection between the broken magnetization chirality and spin-wave instability, within the mean-field theory. Negative excitation energies of quasiparticles at finite momenta imply an instability of the ground state [29,30,32]. According to Eq. (11) a critical magnon current J (c) y generated by incoherent pumping can cause negative magnon excitation energiesẼ 0 (k) < 0 at the momentum k (c) y = 4V 0 J y /( 2 ω M α ex ), when the magnon current is larger than a critical value With the above YIG parameters, the critical magnon current J (c) y ≈ 10 −7 kg/(m · s). This value can be reached by incoherent spin injection with a critical temperature gradient E (c) y ≈ 4 K/µm when T = 300 K [Eq. (13)]. By stripline microwave excitation the critical spin current density is predicted when I ∼ I c by the drag theory. However, according to the LLG calculations in Fig. 2(a) nonlinearities might prohibit reaching this critical value.
Around the spin-wave instability driving strength I c the magnetization at one side of the stripline reaches its maximum with a rapid increase of the magnetization at the other side. Figure 3 shows the suppression of chirality under strong excitation. The nonequilibrium magnetization for y > 0 is maximal around I c , at which magnons accumulate also at y < 0. The chirality is strongly broken at a stronger driving with nearly equal excited magnetizations at the two sides of the stripline. We note the analogy with a magnetic domain wall centered at the stripline [32]. Under high excitation the injected power propagates in both directions, similar to the electric or thermal injection of an incoherent magnon accumulation. Chirality of magnetization is thereby a measure of magnon backscattering, which should be protected topologically for magnetostatic surface magnons [59]. Discussion.-Large spin currents can realize efficient spin transport that needs strong driving of the magnetic orders. We formulated the response of a ferromagnet to strong driving and found a Doppler shift of magnon dispersion by a magnon-interaction-driven drag effect. It requires a stripline current density ∼ 2 × 10 7 A/cm 2 in one or ∼ (2/N ) × 10 7 A/cm 2 in N striplines over a total width that should be small compared to the magnon propagation length, i.e., many micrometers. The chiral velocities here are controllable by the stripline chirality, which was observed at a YIG|GGG interface [21]. The nonmonotonic dependence of the spin current response to the microwave power might be related to the nonmonotonicity in nonlocal spin wave transport as a function of spin injection by a Pt gate [14,15]. Our study helps to realize large magnon spin currents and understand their effects on the magnetic orders in magnetic insulators.
This work is financially supported by DFG Emmy Noether program (SE 2558/2-1) as well as JSPS KAK-ENHI Grant Nos. 26103006. C.W. is supported by the National Natural Science Foundation of China under Grant No. 11704061. We thank Xiang-Yang Wei, Hanchen Wang, Haiming Yu, and Mehrdad Elyasi for valuable discussions.