Dynamics of a skyrmion magnetic bubble. Oleg Tchernyshyov

Transcription

1 Dynamics of a skyrmion magnetic bubble Oleg Tchernyshyov ACMAC, University of Crete, 9 April 2013

2 Dynamics of a skyrmion magnetic bubble Oleg Tchernyshyov ACMAC, University of Crete, 9 April 2013

3 Outline Landau-Lifshitz, Thiele, and beyond motion of a magnetic bubble method of collective coordinates modes of a Bloch domain wall edge modes of a magnetic bubble I. Makhfudz, B. Krüger, and O. Tchernyshyov, Phys. Rev. Lett. 109, (2012)

4 Landau, Lifshitz & Gilbert ṁ = γh m + αm ṁ ṁ = γh m + α m (γh m) α 1 m(r) = M(r) M H(r) = δu[m(r)] δm(r) Nonlinear and nonlocal PDE

5 Thiele: forces ṁ = γh m + αm ṁ m = m(r R(t)), R(t) =vt steady motion Newton s 2nd law: G v + F Dv =0. F = U/ R conservative D ij = αm d 3 r i m j m γ viscous G i = M d 3 r ijk m j m k m γ gyro A.A.Thiele, Phys. Rev. Lett. 30, 230 (1973)

6 Caveat emptor m = m(r R(t)), R(t) =vt steady motion Even when the motion is not strictly steady state (such as motion driven by a thickness gradient), the instantaneous dissipation and gyroscopic forces may be used as first approximations. A.A.Thiele, Phys. Rev. Lett. 30, 230 (1973)

7 Beyond steady state? Vortex in a disk: G Ṙ kr DṘ =0. G =(0, 0, 2πtM/γ) G ṙ kr ṙ K. Yu. Guslienko et al., J. Appl. Phys. 91, 8037 (2002).

8 Beyond steady state? Vortex in a disk: G Ṙ kr DṘ =0. G =(0, 0, 2πtM/γ) G ṙ kr ṙ K. Yu. Guslienko et al., J. Appl. Phys. 91, 8037 (2002).

9 Beyond steady state? Vortex in a disk: G Ṙ kr DṘ =0. G =(0, 0, 2πtM/γ) G ṙ kr ṙ K. Yu. Guslienko et al., J. Appl. Phys. 91, 8037 (2002).

10 Rigid texture ṁ = γh m + αm ṁ m = m(r R(t)) arbitrary rigid motion Newton s 2nd law: G Ṙ + F DṘ =0. F = U/ R conservative D ij = αm d 3 r i m j m γ viscous G i = M d 3 r ijk m j m k m γ gyro

11 Magnetic skyrmion bubble G Ṙ kr DṘ =0. G =(0, 0, 4πtM/γ) Expect rotational motion with ω = k/g. C. Moutafis, S. Komineas, and J. A. C. Bland, PRB 79, (2009).

12 Simulated motion Expected Observed

13 Simulated motion Expected Observed

14 Actual COM trajectory

15 Actual COM trajectory

16 Actual COM trajectory A hypocycloid = superposition of two circular motions. The bubble s motion involves 2 soft modes instead of 1. What are they?

17 Phenomenology Thiele: G Ṙ kr DṘ =0. 1st-order ODE, 1 mode X, nm Y, nm Thiele + Newton: G Ṙ kr DṘ = m R 0.2 ΔX, nm ΔY, nm 2nd-order ODE, 2 modes time, ns

18 Phenomenology Thiele: G Ṙ kr DṘ =0. 1st-order ODE, 1 mode X, nm Y, nm Thiele + Newton: G Ṙ kr DṘ = m R 0.2 ΔX, nm ΔY, nm 2nd-order ODE, 2 modes Origin of inertia? time, ns

19 Systematic approach How do we deal with non-steady-state motion in a systematic way? The Landau-Lifshitz equation includes too many details about the dynamics, most of which are not interesting. If we could focus on a small number of key modes, perhaps the equations of motion would simplify.

20 Inspiration JOURNAL OF APPLIED PHYSICS VOLUME 90, NUMBER 1 Effective dynamics for ferromagnetic thin films Carlos J. García-Cervera a) Department of Mathematics, Princeton University, Princeton, New Jersey Weinan E b) Courant Institute of Mathematical Sciences, New York University, New York, New York and Department of Mathematics and PACM, Princeton University, Princeton, New Jersey Received 30 October 2000; accepted for publication 15 March 2001 In a ferromagnetic material, the dynamics of the relaxation process are affected by the presence of a strong shape or material anisotropy. In this article, we systematically explore this fact to derive the effective dynamical equation for a soft ferromagnetic thin film. We show that, as a consequence of the interplay between shape anisotropy and damping, the gyromagnetic term is effectively also a damping term for the in-plane components of the magnetization distribution. We validate our result through numerical simulation of the original Landau Lifshitz equation and our effective equation American Institute of Physics. DOI: /

21 Proc. R. Soc. A (2005) 461, doi: /rspa Published online 28 September 2004 Effective dynamics for ferromagnetic thin films: a rigorous justification By Robert V. Kohn and Valeriy V. Slastikov Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA (slastiko@cims.nyu.edu) Landau-Lifshitz equation m = (sin θ cos φ, sin θ sin φ, cos θ) Glauber s dynamics (A) m (cos φ, sin φ, 0) ṁ = δu[m(r)] δm(r) m + αm ṁ α φ = δu[φ(r)] δφ(r) θ(r),φ(r) φ(r)

22 Collective coordinates ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 F i = U/ ξ i D ij =(αm/γ) G ij =(M/γ) conservative force {ξ i } =(x, y) S. B. Choe et al., Science (2004). m m dv dissipation tensor ξ i ξ j m m m dv gyrotropic tensor ξ i ξ j O. A. Tretiakov et al., Phys. Rev. Lett. 100, (2008).

23 Collective coordinates m(t) =m({ξ i (t)}) F i = U/ ξ i ṁ = γh m + α m ṁ {ξ i (t)} a set of collective coordinates D ij =(αm/γ) G ij =(M/γ) G ij ξj + F i D ij ξj =0 conservative force m m dv dissipation tensor ξ i ξ j m m m dv gyrotropic tensor ξ i ξ j O. A. Tretiakov et al., Phys. Rev. Lett. 100, (2008). {ξ i } =(x, y) S. B. Choe et al., Science (2004). double twist required

24 Collective coordinates ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 F i = U/ ξ i conservative force {ξ i } =(x, y) S. B. Choe et al., Science (2004). L = 1 2 G ij ξ i ξ j U({ξ i }), R = 1 2 D ij ξ i ξj. Rayleigh function 1 1 S B = 2 G ij ξ i ξ j dt = 2 G ijξ j dξ i Berry-phase action D. J. Clarke et al., Phys. Rev. B (2008).

25 Collective coordinates ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 F i = U/ ξ i conservative force {ξ i } =(x, y) S. B. Choe et al., Science (2004). L = 1 2 G ij ξ i ξ j U({ξ i }), R = 1 2 D ij ξ i ξj. Rayleigh function 1 1 S B = 2 G ij ξ i ξ j dt = 2 G ijξ j dξ i D. J. Clarke et al., Phys. Rev. B (2008). Berry-phase action double twist required

26 Soft modes ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 Formally an exact description. Requires an infinite number of coordinates. Reasonable approach: focus on soft modes. slow motion: 0 inverse relaxation time mode Modes with τ 1 >T 1 are hard, can be ignored.

27 Soft modes ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 Formally an exact description. Requires an infinite number of coordinates. Reasonable approach: focus on soft modes. slow motion: T 1 0 inverse relaxation time mode Modes with τ 1 >T 1 are hard, can be ignored.

28 Soft modes ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 Formally an exact description. Requires an infinite number of coordinates. Reasonable approach: focus on soft modes. slow motion: T 1 0 inverse relaxation time mode Modes with τ 1 >T 1 are hard, can be ignored.

29 Soft modes ṁ = γh m + α m ṁ m(t) =m({ξ i (t)}) {ξ i (t)} a set of collective coordinates G ξj ij + F i D ξj ij =0 Formally an exact description. Requires an infinite number of coordinates. Reasonable approach: focus on soft modes. faster motion: T 1 0 inverse relaxation time mode Modes 0 and 1 are soft, the rest are hard.

30 Bloch domain wall in 1 dimension N. L. Schryer and L. R. Walker, J. App. Phys. 45, 5406 (1974). D. J. Clarke et al., Phys. Rev. B (2008).

31 Magnetization ˆm(x) = (sin θ cos φ, sin θ sin φ, cos θ). θ =0 θ = π dθ 2 + sin 2 θ dφ 2 K 0 = µ 0M 2 2 U[θ(x),φ(x)] = A dx exchange dx +K sin 2 θ + K 0 sin 2 θ cos 2 φ µ 0 HM cos θ. easy axis magnetostatic Zeeman

32 Bloch domain wall θ( ) =π θ(+ ) =0 Zero applied field: cos θ(x) = tanh x x 0, φ = ±π/2. λ x 0 location of the domain wall. λ = A/K width of the domain wall.

33 Soft modes x = x(t), φ = φ(t). U(x, φ) = HMx + k cos 2 φ. Potential energy. T = Gẋφ. Kinetic energy. Berry phase. R = bẋ 2 /2. Rayleigh s dissipation function. k = K 0 λ, G = 2M γ, b = 2αM γλ. λ = λ(φ) = A/(K + K 0 cos 2 φ). Equations of motion: G φ bẋ + HM =0. Gẋ + k sin 2φ =0.

34 Soft modes x = x(t), φ = φ(t). U(x, φ) = HMx + k cos 2 φ. Potential energy. T = Gẋφ. Kinetic energy. Berry phase. R = bẋ 2 /2. Rayleigh s dissipation function. k = K 0 λ, G = 2M γ, b = 2αM γλ. λ = λ(φ) = A/(K + K 0 cos 2 φ). Equations of motion (natural units): φ bẋ + h =0. ẋ + sin 2φ =0.

35 Steady-state motion in a low field: φ bẋ + h =0. ẋ + sin 2φ =0. Soft mode: x. Hard modes: φ, λ,... x = vt, v = h/b. sin 2φ = h/b, h b. The same result can be obtained directly from the Landau-Lifshitz equation in this regime.

36 Oscillatory motion in a high field: φ bẋ + h =0. ẋ + sin 2φ =0. Soft modes: x, φ. Hard modes: λ,... x = vt + cos (2ht)/2h, v = b/4h. φ = ht, h b. The Landau-Lifshitz equation is too complicated in this regime. 2 soft modes provide an excellent approximation. D. J. Clarke et al., Phys. Rev. B (2008).

37 Waves on the edge of a bubble r(φ) = r + m r m e imφ. CoPt (easy-axis) disk with two concentric domains Ground state m =1 m =2

38 Toy model: straight edge y ψ A Bloch domain wall is characterized by two fields: y(x), location of the line where Mz = 0,! ψ(x), the angle of in-plane (Mx,My) on that line. x The angle ψ and the displacement y are coupled: the in-plane (Mx,My) tends to be to the wall, + Berry-phase coupling between them.

39 Model dynamics y ψ L[y, ψ] = 0 dx gẏψ κ(ψ y ) 2 /2 U[y] x Gyrotropic coupling g between y and ψ (Walker). κ ties ψ to the wall direction.

40 Model dynamics y ψ x L[y, ψ] = 0 dx gẏψ κ(ψ y ) 2 /2 U[y] Integrate out field ψ: L[y] = 0 gẏ κ(ψ y )=0 Döring dx ρẏ 2 /2+gẏy U[y] ρ = g 2 /2κ

41 Model dynamics y ψ x Toy model: domain wall with (local) tension σ. U l [y] = L[y] = 0 0 σ dx 2 + dy 2 U l [0] + dx ρẏ 2 /2+gẏy σy 2 /2. ρÿ +2gẏ σy =0 0 dx σy 2 /2

42 Model dynamics y ψ x Toy model: domain wall with (local) tension σ. U l [y] = L[y] = 0 0 σ dx 2 + dy 2 U l [0] + dx ρẏ 2 /2+gẏy σy 2 /2. ρω 2 +2gωk + σk 2 =0 0 dx σy 2 /2 ω k

43 Dynamics of bubble edge Technical complications: circular shape and longrange stray dipolar field. Some parameters are easy to compute, others are not so easy. U l [r] = σ dr ω/2π, GHz analytical simulations U nl [r] σd 2 dr 1 dr 2 r 1 r 2 m

44 Eigenmodes m = 1 to 4

45 Bloch domain walls in CoPt and FePt thin films are chiral objects: left and right waves propagate on them with different speeds. The asymmetry can be traced to the breaking of the T and P symmetries in a ferromagnet with a domain wall. The soft and hard modes correspond to inplane magnetization and the direction of the wall oscillating in and out of phase. A bubble s inertia is generated by Doring s mechanism: a soft mode is canonically conjugate to a hard mode.