SUPPLEENTARY INFORATION DOI: 1.138/NAT3459 agnetic nano-oscillator driven by pure spin current Vladislav E. Demidov 1*, Sergei Urazhdin, Henning Ulrichs 1, Vasyl Tiberevich 3, Andrei Slavin 3, Dietmar Baither 4, Guido Schmitz 4 & Sergej O. Demoritov 1 1 Institute for Applied Physics and Center for Nonlinear Science, University of uenster, 48149 uenster, Germany Department of Physics, Emory University, Atlanta, GA 33, USA 3 Department of Physics, Oaland University, Rochester, I 4839, USA 4 Institute of aterial Physics, University of uenster, 48149 uenster, Germany *e-mail: demidov@uni-muenster.de Conditions for the onset of spin current-driven auto-oscillation in a multimode system. To get a qualitative insight into the effects of spin current on the spin wave dynamics, consider the Landau-Lifshitz-Gilbert equation with an additional Slonczewsi term [1,] describing the torque erted on the magnetization by the spin current [3,4]: d d σi = γ Heff + α G + [ ] dt dt s. (1) Here, =, γ is the gyromagnetic ratio, α G is the Gilbert damping constant, H eff is the sum of the ternal static magnetic field H and the effective fields caused by the interactions within the spin system of the ferromagnet, s is a unit vector in the direction of the polarization of spin current created by the electric current I, and σ is a coefficient describing the efficiency of the spin current generation by the spin Hall effect (see [5,6] for details). To analyze the dynamical processes, we press the magnetization as the sum of the static and NATURE ATERIALS www.nature.com/naturematerials 1 1 acmillan Publishers Limited. All rights reserved.
SUPPLEENTARY INFORATION DOI: 1.138/NAT3439 ~ the dynamic parts: = zˆ+ m = zˆ+ m xˆ + m y. ˆ The dynamic part of the magnetization z z x y vector m ~ describing all the cited dynamical modes is perpendicular to the static part z. ˆ z Since the total length of the magnetization vector is conserved, = + ( m ). ~ z By substituting the pression for the total magnetization into Eq. (1), one obtains the equation for its dynamic component m ~. The first term in Eq. (1) determines the frequencies of the modes, whereas the second (Gilbert) term determines their damping. Since the Gilbert term plicitly contains the time derivative d/dt, the corresponding damping rate is proportional to the mode frequency ω ( = α G ω). If s is parallel to the z-axis, which is the case σi ~ in our periments, the third term in Eq. (1) can be simplified to z m (see Eqs. (9) and (1) in [6]). It has the same form as viscous friction, and therefore also contributes to the damping. However, in contrast to usual systems where the viscous coefficient has a well defined sign to guarantee the stability of the ground state, here its sign can be reversed by changing the sign of the electric current I. For the appropriate sign of I, the term causes an effective negative damping z = σ I, which counteracts the natural Gilbert damping and amplifies the dynamic magnetization. Note that, since the spin current term does not contain spatial or temporal derivatives of the magnetization, the contribution of the negative damping is independent of the frequency and the wave vector of the dynamic modes. The frequency separation between the dynamic modes of a magnetic system generally increases with increasing size of the system. For a magnetic film with a relatively large area, such that the separation between the modes is smaller than their linewidth, single-mode approximation for auto oscillation induced by spin current is inadequate. In this case, the interaction between modes has a qualitative effect on the current-induced phenomena, as shown by the following analysis. NATURE ATERIALS www.nature.com/naturematerials 1 acmillan Publishers Limited. All rights reserved.
DOI: 1.138/NAT3459 SUPPLEENTARY INFORATION Below the auto-oscillation onset, thermal fluctuations of each mode are enhanced by the spin torque produced by the spin current. Introducing the dimensionless amplitude ~ = / c m for a spin-wave mode with a given wave vector, and the corresponding dimensionless power p = c, the total dimensionless power in all the modes is = p p. current As the total power is increased, the effective negative damping induced by the spin z = σi = σi(1 p) () is decreased, thus increasing the auto-oscillation onset current. To determine the dependence of the total power on spin current, we start with the effect of spin torque on the individual mode characterized by the wavevector where p = η (3) = (4) + α G D is the mode damping, η = γ BT /( V S ω ) is the mode power in thermal equilibrium, and D ω λ is the spin wave dispersion coefficient. Here, V is the volume of the film, and λ is the change length. To determine the total power, we for simplicity neglect the -dependence in the numerator of Eq. (3) and consider only the contribution of the low-frequency region < = 1/λ, since high-frequency modes are practically unaffected by the spin torque due to their strong damping. For a large film, we use a continuous-spectrum approximation. The total power p is then γbt d γ BT p = d S α ω λ + Δ 8πdλ ω G S αgω ln Δ, (5) NATURE ATERIALS www.nature.com/naturematerials 3 1 acmillan Publishers Limited. All rights reserved.
SUPPLEENTARY INFORATION DOI: 1.138/NAT3439 where Δ =, and we use approximations = α G ω /, Δ <<. According to Eq. (5), the condition Δ = for the onset of auto-oscillations cannot be achieved at any finite p. In other words, the effect of spin current is self limiting, since the current-induced increase of the total power results in a decreasing effect of spin current on the negative damping. In agreement with this analysis, measurements of the effect of spin current in micrometer- and millimeter-scale devices driven by uniformly distributed spin currents [7,8] showed an increase of the amplitude of magnetic oscillations to substantial levels, but their spectrum remained broad and did not hibit a transition to the coherent auto-oscillation regime. Qualitatively different behaviors are pected when a localized spin current is applied to an tended magnetic film. In this case, spin waves can escape from the active area, creating a new radiation channel for mode damping. Therefore, an additional term describing radiation losses needs to be added into the pression for the total effective damping (Eq. (4)): Here G + / = + α ω ( λ ) v L. (6) v = D is the group velocity of spin waves with the wavevector and L is the size of the spin-current localization area. With the radiation losses, the total power p is γbt d p =. (7) d α ω λ + D / L + Δ At the onset ( Δ = ) it taes the value S G p c γ BT d γbt α ω λ G = ln α ω λ + π λ ω L d D L d. S G / 4 S D One can see that, due to the presence of radiation losses, p c remains finite at Δ =, implying that in this case, the onset of auto-oscillations can be achieved. oreover, for practically important device geometries with the size of the active area of the order of several hundreds of nanometers, the estimated value of p c is relatively small (.1 -.1). As a consequence, 4 NATURE ATERIALS www.nature.com/naturematerials 1 acmillan Publishers Limited. All rights reserved.
DOI: 1.138/NAT3459 SUPPLEENTARY INFORATION the nonlinear increase of the onset current due to the multimode citation, given by Eq. (), does not ceed 1%. References 1. Slonczewsi, J. C. Current driven citation of magnetic multilayers. J. agn. agn. ater. 159, L1-L7 (1996).. Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353-9358 (1996). 3. Liu, L., oriyama, T., Ralph, D. C., & Buhrman, R. A. Spin-torque ferromagnetic resonance induced by the spin Hall effect. Phys. Rev. Lett. 16, 3661 (11). 4. Wang, Z., Sun, Y., Wu,., Tiberevich, V., & Slavin, A. Control of spin waves in a thin film ferromagnetic insulator through interfacial spin scattering. Phys. Rev. Lett. 17, 1466 (11). 5. Slavin, A.N. & Tiberevich, V.S. Nonlinear auto-oscillator theory of microwave generation by spin-polarized current. IEEE Trans. ag. 45, 1875-1918 (9). 6. Slavin, A.N. & Kabos, P. Approximate theory of microwave generation in a current-driven magnetic nanocontact magnetized in an arbitrary direction. IEEE Trans. ag. 41, 164-173 (5). 7. Kajiwara, Y. et al. Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. Nature 464, 6 66 (1). 8. Demidov, V. E. et al. Control of magnetic fluctuations by spin current. Phys. Rev. Lett. 17, 174 (11). NATURE ATERIALS www.nature.com/naturematerials 5 1 acmillan Publishers Limited. All rights reserved.