Rotating vortex-antivortex dipoles in ferromagnets under spin-polarised current Stavros Komineas

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1 Rotating vortex-antivortex dipoles in ferromagnets under spin-polarised current Stavros Komineas Department of Applied Mathematics Archimedes Center for Modeling, Analysis & Computation University of Crete, Greece SIAM conference on Mathematical Aspects of Materials Science Philadelphia, 1 June 213

2 Injection of spin-polarized current through an aperture Fixed layer: in-plane magnetization. Free layer: elliptic element 25 nm 15 nm. Aperture to free layer: diameter 4 nm. G. Finocchio et al, PRB 28 Oscillations of the magnetization are measured ( 1 GHz). Simulations show: they are due to spontaneous generation of vortex-antivortex (VA) pair with opposite polarities, in rotation.

3 Spin-transfer nano-oscillators simple (quasi-linear) dependence with magnetic field not-simple behavior with current jumps: more than one oscillation modes Review in: Russek et al, Spin-Transfer Nano-Oscillators in Handbook of Nanophysics

4 A magnetic vortex 5 Vortex (S = 1) 5 Vortex (S = 1) Antivortex (S = 1) Vortex features S = ±1 the winding number (a topological invariant) λ = ±1 the vortex polarity

5 The skyrmion number N is a further topological invariant and it counts the number of times that the magnetization m covers the sphere m 2 = 1. For vortices: N = 1 n d 2 x = 1 4π 2 Sλ n = 1 2 ɛ µνm ( ν m µ m) :topological density Vortex (S = 1, N = ±1/2) 5 Antivortex (S = 1, N = ±1/2)

6 Vortex-antivortex dipole Vortex: S = 1, λ = 1 N = 1 2 Antivortex: S = 1, λ = 1 N = 1 2 Vortex pair N = 1. 6 Magnetization vector: m = (m 1, m 2, m 3 ) Vector plot: (m 1, m 2 ) [S. Komineas, PRL 27] Contour plot: m 3 Blue: m 3 < Red: m 3 >

7 The model: Spin-torque in the LL equation The polarized current-electrons exert a torque, modeled by an additional (Slonczewski) term in the Landau-Lifshitz (LL) equation: ṁ = m f + α m ṁ β m (m p) f := m m 3 ê 3 + h ext f : exchange + easy-plane anisotropy + external field α: damping constant Spin polarization βp = β(1,, ), β <, proportional to current density. External field h ext = (h ext,, ).

8 Rotating vortex dipole under an aperture 6 Spin-polarized current through 3 aperture of diameter 6l ex (dashed line). 3 Blue: polarity down Red: polarity up Simulation gives: Steady-state rotation. Note that ground state is: m = (1,, ).

9 Rotating vortex dipoles α =.2, β =.1, h ext =.4 6 Long VA pair 6 Short VA pair ω =.255, ω =.539

10 Rotating vortex dipoles α =.2, β =.2, h ext =.6 6 Wide VA pair ω =.82

11 Phase diagram h ext.6 short.4.2 long wide stars: long VA pairs circles: short VA pairs crosses: wide VA pairs Small regions of overlap Long VA pairs for h ext = No steady-states for β too small Empty region: VA pairs plus satellite pairs rotating (similar to [Berkov, Gorn, PRB 29])

12 The pure isotropic model...that is, we assume exchange only (and external field): f = m + h ext, h ext = (h ext,, ). Use the sterographic projection of m from the point m = (1,, ): Obtain equation of motion X = m 2 + im 3 1 m 1. (i α) Ẋ = 4 z z X + 8X 1 + X X zx z X (h ext iβ) X where z = x + iy the position on the complex plane, so X=X (z, z). Expect precession around m 1 due to field h ext = (h ext,, ).

13 Rotating solutions The simple form X = i z a, a : complex constant (N = 1) represents two merons: m 3 = ±1 at z = ±a [Gross, 1978], or a VA dipole. Solution of the eqn of motion for X (t = ) = X (z): ( z iβ X (z, t) = i a(t), a(t) = a hext exp i α VA dipole in steady rotation for αh ext + β = : X (z, t) = i z a e ihextt ) t. It represents rotation of the vortex positions ±a(t) = ±a e ihextt in the complex plane.

14 Virial relation (I) An exact virial relation (of Derrick type) can be derived, for steady-state rotation, involving the frequency (ω) of rotation. ( ω l + α 2 ) ( ɛ λν x λ x µ d µν d 2 x = E a + h ext µ 1 + β 2 ) x µ τ µ d 2 x, Frequency of rotation ω Angular momentum: l = 1 2 ρ 2 n d 2 x dva 2 (actually: l N dva 2 ) Anisotropy energy E a (E a π/2 for single vortex) Total in-plane magnetization: µ 1 = (1 m 1 ) d 2 x

15 Virial relation (II) An exact virial relation (of Derrick type) can be derived, for steady-state rotation, involving the frequency (ω) of rotation. A simplified form of the Derrick relation is ω =. ( ) Ea l + h µ 1 ext l Frequency of rotation ω Angular momentum: l = 1 2 ρ 2 n d 2 x dva 2 (actually: l N dva 2 ) Anisotropy energy E a (E a = π/2 for single vortex) Total in-plane magnetization: µ 1 = (1 m 1 ) d 2 x

16 Virial relation (III) An exact virial relation (of Derrick type) can be derived, for steady-state rotation, which involves the frequency (ω) of rotation. A simplified form of the Derrick relation is ω =. ( ) Ea l + h µ 1 ext l First term: rotation due to interaction between vortices Second term: rotation due to external field Both terms rely upon N (i.e., l ).

17 Asymptotics at large distances Away from the vortex pair, we have Bessel eqns, so: m 2 + im m 1 1 ρ e (iλ 1 λ 2 )ρ, ρ. λ 2 1: for h ext.1 VA pairs non-localized. λ 2 1: for β too small VA pairs collapse at ω max h ext (1 + h ext ), h ext.1. λ 1 : spiralling waves emanating from rotating pair.

18 6 A full rotation t= t= t=6 t=

19 Conclusions Vortices and antivortices can be generated by spin-polarized current. A vortex-antivortex dipole (with N = 1) is rotating. Three (at least) vortex-antivortex modes: well-separated vortex-antivortex or two-merons. In-plane field h ext = (h ext,, ), typically expected to induce magnetization precession (around x-axis), is actually giving rotation of a configuration with N = 1. Rotational motion is stabilized by the spin-polarized current. Rotation frequency can be tuned by current and external field. The magnetostatic field can be incorporated in the formalism (some simulations have been published).