Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system

Transcription

1 Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system Lihui Chai Department of Mathematics University of California, Santa Barbara Joint work with Carlos J. García-Cervera, and Xu Yang YRW 2016, Duke

2 Outline 1 Motivation and introduction 2 The Schrödinger-Poisson-Landau-Lifshitz-Gilber system 3 Existence of weak solutions 4 Semiclassical limit of SPLLG

3 Outline 1 Motivation and introduction 2 The Schrödinger-Poisson-Landau-Lifshitz-Gilber system 3 Existence of weak solutions 4 Semiclassical limit of SPLLG

4 Magnetic devices Magnetic recording devices and computer storages Spinvalues 1 Magnetic random access memory Domain walls 2 Racetrack memories 1 Science@Berkeley Lab: The Current Spin on Spintronics 2

5 Methodology for detecting the orientation Tunnel magnetoresistance 3 Julliere s model: Constant tunneling matrix Giant magnetoresistance 4 Albert Fert & Peter Grünberg: 2007 Nobel Prize in Physics

6 Methodology for rotating the orientation Spin transfer torque (STT) 5 Two layers of different thickness: different switching fields The thin film is switched, and the resistance measured 5

7 Micromagnetics: Landau-Lifshitz-Gilbert model Basic quantity of interest: m : Ω R 3 ; m = M s Landau-Lifshitz energy functional: ( ) m F[m] = φ dx + A Ω M s Ms 2 m 2 dx Ω µ 0 H s m dx µ 0 H e m dx 2 Ω Ω

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9 ( ) φ m : Anisotropy Energy: Penalizes deviations from the Ms easy directions. For uniaxial materials φ(m) = K u ( (m2 /M s ) 2 + (m 3 /M s ) 2).

10 ( ) φ m : Anisotropy Energy: Penalizes deviations from the Ms easy directions. For uniaxial materials φ(m) = K u ( (m2 /M s ) 2 + (m 3 /M s ) 2). A m 2 : Exchange energy: Penalizes spatial variations. Ms 2

11 ( ) φ m : Anisotropy Energy: Penalizes deviations from the Ms easy directions. For uniaxial materials φ(m) = K u ( (m2 /M s ) 2 + (m 3 /M s ) 2). A M 2 s m 2 : Exchange energy: Penalizes spatial variations. µ 0 H e m: External field (Zeeman) energy.

12 ( ) φ m : Anisotropy Energy: Penalizes deviations from the Ms easy directions. For uniaxial materials φ(m) = K u ( (m2 /M s ) 2 + (m 3 /M s ) 2). A m 2 : Exchange energy: Penalizes spatial variations. Ms 2 µ 0 H e m: External field (Zeeman) energy. µ 0 2 H s m: Stray field (self-induced) energy.

13 ( ) φ m : Anisotropy Energy: Penalizes deviations from the Ms easy directions. For uniaxial materials φ(m) = K u ( (m2 /M s ) 2 + (m 3 /M s ) 2). A m 2 : Exchange energy: Penalizes spatial variations. Ms 2 µ 0 H e m: External field (Zeeman) energy. µ 0 2 H s m: Stray field (self-induced) energy. The stray field, H s = u is obtained by solving the magnetostatic equation: u = div m, x Ω, u = 0, x Ω c, with jump boundary conditions [ ] u [u] Ω = 0, ν Ω = m ν.

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15 Landau-Lifshitz-Gilbert equation

16 Landau-Lifshitz-Gilbert equation m t = γ m H + αm m t, where H = δf δm and the second is Gilbert damping term.

17 Landau-Lifshitz-Gilbert equation m t = γ m H + αm m t, where H = δf δm and the second is Gilbert damping term. γ = T 1 s 1

18 Landau-Lifshitz-Gilbert equation m t = γ m H + αm m t, where H = δf δm and the second is Gilbert damping term. γ = T 1 s 1 α << 1: Damping coefficient

19 Landau-Lifshitz-Gilbert equation m t = γ m H + αm m t, where H = δf δm and the second is Gilbert damping term. γ = T 1 s 1 α << 1: Damping coefficient With the injection of spin current

20 Landau-Lifshitz-Gilbert equation m t = γ m H + αm m t, where H = δf δm and the second is Gilbert damping term. γ = T 1 s 1 α << 1: Damping coefficient With the injection of spin current m t = γ m (H + Js) + αm M t, where s is the spin procession and J is the coupling strength between spin and magnetization

21 How to describe the spin dynamics?

22 How to describe the spin dynamics? Quantum (Schrödinger equation in spinor form) i ψ ( ( ) = 2 µ B ω t 2m 2 x + V (x) )Î σ m(x, t) ψ. 2

23 How to describe the spin dynamics? Quantum (Schrödinger equation in spinor form) i ψ ( ( ) = 2 µ B ω t 2m 2 x + V (x) )Î σ m(x, t) ψ. 2 Kinetic (Boltzmann equation) t W (x, v, t) + v x W (x, v, t) e m E vw (x, v, t) i 2 [µ Bω σ m(x, t), W (x, v, t)] = W W τ 2 τ sf ( W Î 2 Tr C 2 W )

24 How to describe the spin dynamics? Quantum (Schrödinger equation in spinor form) i ψ ( ( ) = 2 µ B ω t 2m 2 x + V (x) )Î σ m(x, t) ψ. 2 Kinetic (Boltzmann equation) t W (x, v, t) + v x W (x, v, t) e m E vw (x, v, t) i 2 [µ Bω σ m(x, t), W (x, v, t)] = W W τ Hydrodynamic (Diffusion equation) s t = div J s 2D 0 (x) s λ 2 sf 2D 0 (x) s m λ 2, J 2 τ sf ( W Î 2 Tr C 2 W ) J s = βµ B e J n m 2D 0 (x) [ s ββ ( s m) m].

25 Outline 1 Motivation and introduction 2 The Schrödinger-Poisson-Landau-Lifshitz-Gilber system 3 Existence of weak solutions 4 Semiclassical limit of SPLLG

26 The SPLLG system

27 The SPLLG system The Schrödigner-Poisson (SP) equation iε t ψ ε j (x, t) = ε2 2 ψε j (x, t) + V ε ψ ε j (x, t) ε 2 mε σψ ε j (x, t), x V ε = ρ ε, ψ ε j (x, t = 0) = ϕε j (x) x R 3, t > 0, j N

28 The SPLLG system The Schrödigner-Poisson (SP) equation iε t ψ ε j (x, t) = ε2 2 ψε j (x, t) + V ε ψ ε j (x, t) ε 2 mε σψ ε j (x, t), x V ε = ρ ε, ψ ε j (x, t = 0) = ϕε j (x) x R 3, t > 0, j N The Landau-Lifshitz-Gilbert (LLG) equation t m ε = m ε H ε eff + αmε t m ε, m ε (x, t) = 1, x Ω, t > 0 m ε = 0 on Ω ν m ε (x, t = 0) = m 0

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30 The occupation numbers λ ε j > 0 satisfy that there exist C > 0 such that λ ε j + ε 2 λ ε j ϕε j 2 L 2 (R 3 ) + ε 3 (λ ε j )2 C j=1 j=1 j=1

31 The occupation numbers λ ε j > 0 satisfy that there exist C > 0 such that λ ε j + ε 2 λ ε j ϕε j 2 L 2 (R 3 ) + ε 3 (λ ε j )2 C j=1 j=1 Physical observables ρ ε (x, t) = j ε (x, t) = ε s ε (x, t) = λ ε j ψ ε j (x, t) 2, j=1 j=1 λ ε j Im ( ψ ε j (x, t) x ψ ε j (x, t)), j=1 j=1 λ ε j Tr C 2 ( σ ( ψ ε j (x, t)ψε j (x, t) ))

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33 The Landau-Lifshitz-Gilbert energy ( 1 F LL = 2 mε Hε s m ε ε ) 2 sε m ε dx, Ω

34 The Landau-Lifshitz-Gilbert energy ( 1 F LL = 2 mε Hε s m ε ε ) 2 sε m ε dx, Ω The the effective field H ε eff = δf LL δm ε = mε + H ε s + ε 2 sε

35 The Landau-Lifshitz-Gilbert energy ( 1 F LL = 2 mε Hε s m ε ε ) 2 sε m ε dx, Ω The the effective field H ε eff = δf LL δm ε = mε + H ε s + ε 2 sε The stray field H ε s(x) = with N (x) = 1 4πx. Ω N (x y) m ε (y) dy,

36 The Wigner-Poisson equation

37 The Wigner-Poisson equation The Wigner transformation W ε (x, v) = 1 (2π) 3 R 3 y j=1 λ ε j ψε j ( x + εy ) ( ψ ε j x εy ) e iv y dy. 2 2

38 The Wigner-Poisson equation The Wigner transformation W ε (x, v) = 1 (2π) 3 R 3 y j=1 λ ε j ψε j ( x + εy ) ( ψ ε j x εy ) e iv y dy. 2 2 The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i ) 2 Γε [m ε ] W ε = 0,

39 where the operator Θ ε is given by Θ ε [V ε ]W ε (x, v) 1 iε = 1 (2π) 3 [ ( V ε x εy ) ( V ε x + εy )] W ε (x, v ) 2 2 e i(v v ) y dy dv, and the operator Γ ε is given by Γ ε [m ε ]W ε (x, v) = 1 (2π) 3 [ M ε ( x εy 2 e i(v v ) y dy dv, where the matrix M ε = σ m ε. ) ( W ε (x, v ) W ε (x, v )M ε x + εy )] 2

40 Outline 1 Motivation and introduction 2 The Schrödinger-Poisson-Landau-Lifshitz-Gilber system 3 Existence of weak solutions 4 Semiclassical limit of SPLLG

41 Consider the case ε = 1.

42 Consider the case ε = 1. The Schrödinger equations i t ψ j = 1 2 ψ j + V ψ j 1 2 m σψ j, x K, j N, t > 0, ψ j (t = 0, x) = ϕ j (x), ψ j (t, x) = 0, x K x K

43 Consider the case ε = 1. The Schrödinger equations i t ψ j = 1 2 ψ j + V ψ j 1 2 m σψ j, x K, j N, t > 0, ψ j (t = 0, x) = ϕ j (x), ψ j (t, x) = 0, x K x K The Landau-Lifschitz-Gilbert equation t m = m H + αm t m, (x, t) Ω R +, m(t = 0, x) = m 0 (x), x Ω, ν m = 0, (x, t) Ω R +,

44 Consider the case ε = 1. The Schrödinger equations i t ψ j = 1 2 ψ j + V ψ j 1 2 m σψ j, x K, j N, t > 0, ψ j (t = 0, x) = ϕ j (x), ψ j (t, x) = 0, x K x K The Landau-Lifschitz-Gilbert equation t m = m H + αm t m, (x, t) Ω R +, m(t = 0, x) = m 0 (x), x Ω, ν m = 0, (x, t) Ω R +, The Poisson potential V = N ρ

45 Consider the case ε = 1. The Schrödinger equations i t ψ j = 1 2 ψ j + V ψ j 1 2 m σψ j, x K, j N, t > 0, ψ j (t = 0, x) = ϕ j (x), ψ j (t, x) = 0, x K x K The Landau-Lifschitz-Gilbert equation t m = m H + αm t m, (x, t) Ω R +, m(t = 0, x) = m 0 (x), x Ω, The Poisson potential V = N ρ ν m = 0, (x, t) Ω R +, The effective field H = m + H s s, and H s = ( N m),

46 Existence of weak solution

47 Existence of weak solution Denote Ψ = {ψ j } j N and Φ = {ϕ j } j N.

48 Existence of weak solution Denote Ψ = {ψ j } j N and Φ = {ϕ j } j N. Introduce the summation Hilbert space H r λ (K ) by defined Ψ 2 H r λ (K ) := j=1 λ j ψ j 2 H r (K ).

49 Existence of weak solution Denote Ψ = {ψ j } j N and Φ = {ϕ j } j N. Introduce the summation Hilbert space H r λ (K ) by defined Ψ 2 H r λ (K ) := j=1 λ j ψ j 2 H r (K ). Theorem:

50 Existence of weak solution Denote Ψ = {ψ j } j N and Φ = {ϕ j } j N. Introduce the summation Hilbert space H r λ (K ) by defined Ψ 2 H r λ (K ) := j=1 λ j ψ j 2 H r (K ). Theorem: Given any initial conditions with Φ H 1 λ (R3 ) and m 0 H 1 (Ω), m 0 = 1, a.e.. Then there exists Ψ L ([0, ), H 1 λ (R3 )) and m C([0, ), H 1 (Ω)), m = 1, a.e., such that the SPLLG system hold weakly.

51 Existence of weak solution Denote Ψ = {ψ j } j N and Φ = {ϕ j } j N. Introduce the summation Hilbert space Hλ r (K ) by defined Ψ 2 Hλ r (K ) := j=1 λ j ψ j 2 H r (K ). Theorem: Given any initial conditions with Φ Hλ 1 (R3 ) and m 0 H 1 (Ω), m 0 = 1, a.e.. Then there exists Ψ L ([0, ), Hλ 1 (R3 )) and m C([0, ), H 1 (Ω)), m = 1, a.e., such that the SPLLG system hold weakly. We prove this theorem first in a bounded domain K and then let K R 3.

52 Galerkin approximations in a bounded domain

53 Galerkin approximations in a bounded domain Assume K R 3 bounded.

54 Galerkin approximations in a bounded domain Assume K R 3 bounded. Construct solutions by Galerkin approximations:

55 Galerkin approximations in a bounded domain Assume K R 3 bounded. Construct solutions by Galerkin approximations: Let {θ n } n N be the normalized eigenfunctions of θ = µθ in K, θ K = 0. Let {ω n } n N be the normalized eigenfunctions of ω = µω in Ω, ν ω Ω = 0.

56 Galerkin approximations in a bounded domain Assume K R 3 bounded. Construct solutions by Galerkin approximations: Let {θ n } n N be the normalized eigenfunctions of θ = µθ in K, θ K = 0. Let {ω n } n N be the normalized eigenfunctions of ω = µω in Ω, ν ω Ω = 0. We seek the approximate solutions ψ j N (x, t) = N n=1 α jn(t)θ n (x), and m N (x, t) = N n=1 β n(t)ω n (x),

57 Galerkin approximations in a bounded domain Assume K R 3 bounded. Construct solutions by Galerkin approximations: Let {θ n } n N be the normalized eigenfunctions of θ = µθ in K, θ K = 0. Let {ω n } n N be the normalized eigenfunctions of ω = µω in Ω, ν ω Ω = 0. We seek the approximate solutions ψ j N (x, t) = N n=1 α jn(t)θ n (x), and m N (x, t) = N n=1 β n(t)ω n (x), and satisfy the initial conditions ψ jn (, 0) = Π K N ϕ j, and m N (, 0) = Π Ω N m 0, where Π K N and ΠΩ N is the orthogonal projections to Span{θ n } N n=1 and Span{ω n} N n=1, resp..

58 Galerkin solutions 6 F. Alouges & A. Soyeur 1992

59 Galerkin solutions Let ψ jn and m N be the solutions of 6 F. Alouges & A. Soyeur 1992

60 Galerkin solutions Let ψ jn and m N be the solutions of (i t ψ j N = 12 ψ j N + V N ψ j N 12 ) m N σψ j N, Π K N 6 F. Alouges & A. Soyeur 1992

61 Galerkin solutions Let ψ jn and m N be the solutions of Π K N (i t ψ j N = 12 ψ j N + V N ψ j N 12 ) m N σψ j N, ( α t m N = m N t m N H N + k( m N 2 ) 1)m N, 6 Π Ω N 6 F. Alouges & A. Soyeur 1992

62 Galerkin solutions Let ψ jn and m N be the solutions of Π K N (i t ψ j N = 12 ψ j N + V N ψ j N 12 ) m N σψ j N, ( α t m N = m N t m N H N + k( m N 2 ) 1)m N, 6 Π Ω N where H N = m N + ( H sn s N). 6 F. Alouges & A. Soyeur 1992

63 Galerkin solutions Let ψ jn and m N be the solutions of Π K N (i t ψ j N = 12 ψ j N + V N ψ j N 12 ) m N σψ j N, ( α t m N = m N t m N H N + k( m N 2 ) 1)m N, 6 Π Ω N where H N = m N + ( H sn s N). Estimates then are based the following conservation d λ j ψ dt jn 2 + d V N 2 K dt j=1 K + d m N 2 + d H sn 2 dt Ω dt R 3 + k d ( ) 2 m N α t m N 2 = d 2 dt dt Ω Ω Ω s N m N. 6 F. Alouges & A. Soyeur 1992

64 Limits as N Then we have the convergence w.r.t. corresponding topology:

65 Limits as N Then we have the convergence w.r.t. corresponding topology: Ψ N N Ψ k L (R +, Hλ 1 (K )), weak *,

66 Limits as N Then we have the convergence w.r.t. corresponding topology: Ψ N N Ψ k L (R +, Hλ 1 (K )), weak *, N t Ψ N t Ψ k L (R +, H 1 (K )), weak *, λ

67 Limits as N Then we have the convergence w.r.t. corresponding topology: Ψ N N Ψ k L (R +, Hλ 1 (K )), weak *, N t Ψ N t Ψ k L (R +, H 1 (K )), weak *, m N N m k L (R +, H 1 (Ω)), weak *, λ

68 Limits as N Then we have the convergence w.r.t. corresponding topology: Ψ N N Ψ k L (R +, Hλ 1 (K )), weak *, N t Ψ N t Ψ k L (R +, H 1 (K )), weak *, m N t m N N m k L (R +, H 1 (Ω)), weak *, N t m k L 2 (R +, L 2 (Ω)), weakly, λ

69 Limits as N Then we have the convergence w.r.t. corresponding topology: Ψ N N Ψ k L (R +, Hλ 1 (K )), weak *, N t Ψ N t Ψ k L (R +, H 1 (K )), weak *, m N t m N N m k L (R +, H 1 (Ω)), weak *, N t m k L 2 (R +, L 2 (Ω)), weakly, m N 2 1 N m k 2 1 L (R +, L 2 (Ω)), weak*, λ

70 Weak Solutions to the penalized problem Lemma For all χ H 1 ([0, T ] Ω) and η C([0, T ], H 1 (K )), it holds that T i t ψ k j η = 1 T 0 K 2 0 K 1 T 2 0 T T α t m k χ = 0 Ω + 0 T 0 Ω Ω T ψ k j η + V k ψ k j η 0 K K m k σψ j k η, (m k t m k H s k 12 sk ) χ k ( ) m k 2 1 m k χ + m k χ.

71 Weak Solutions to the SPLLG

72 Weak Solutions to the SPLLG

73 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, H 1 λ(k )) weak*,

74 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, λ

75 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, λ

76 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, t m k k t m L 2 (R +, L 2 (Ω)) weakly, λ

77 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, t m k k t m L 2 (R +, L 2 (Ω)) weakly, m k 2 1 k 0 L 2 ([0, T ] Ω) weakly and a.e., λ

78 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, t m k k t m L 2 (R +, L 2 (Ω)) weakly, m k 2 1 k 0 L 2 ([0, T ] Ω) weakly and a.e., Take χ = m k ξ with ξ C ([0, T ] Ω) in the penalized LLG equation and obtain λ

79 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, t m k k t m L 2 (R +, L 2 (Ω)) weakly, m k 2 1 k 0 L 2 ([0, T ] Ω) weakly and a.e., Take χ = m k ξ with ξ C ([0, T ] Ω) in the penalized LLG equation and obtain T T ( t m ξ = m (α t m H s 12 )) s ξ 0 Ω + 0 Ω T 0 Ω λ m m ξ.

80 Weak Solutions to the SPLLG Ψ k k Ψ L (R +, Hλ(K 1 )) weak*, t Ψ k k t Ψ L (R +, H 1 (K )) weak*, m k k m L (R +, H 1 (Ω)) weak*, t m k k t m L 2 (R +, L 2 (Ω)) weakly, m k 2 1 k 0 L 2 ([0, T ] Ω) weakly and a.e., Take χ = m k ξ with ξ C ([0, T ] Ω) in the penalized LLG equation and obtain T T ( t m ξ = m (α t m H s 12 )) s ξ 0 Ω + 0 Ω T 0 Ω λ m m ξ. Since m = 1 a.e., by a density argument, we also obtain the above equation holds for all ξ H 1 ([0, T ] Ω).

81 Weak solutions in whole space R F. Brezzi & P.A. Markowich 1991

82 Weak solutions in whole space R 3 7 For each R fixed, there exist weak solutions to the SPLLG in K = B(0, R). 7 F. Brezzi & P.A. Markowich 1991

83 Weak solutions in whole space R 3 7 For each R fixed, there exist weak solutions to the SPLLG in K = B(0, R). Conservation law and energy dissipation. 7 F. Brezzi & P.A. Markowich 1991

84 Weak solutions in whole space R 3 7 For each R fixed, there exist weak solutions to the SPLLG in K = B(0, R). Conservation law and energy dissipation. The energy estimates does not depend on the radius R. 7 F. Brezzi & P.A. Markowich 1991

85 Weak solutions in whole space R 3 7 For each R fixed, there exist weak solutions to the SPLLG in K = B(0, R). Conservation law and energy dissipation. The energy estimates does not depend on the radius R. There exit subsequences of solutions converge as R. 7 F. Brezzi & P.A. Markowich 1991

86 Weak solutions in whole space R 3 7 For each R fixed, there exist weak solutions to the SPLLG in K = B(0, R). Conservation law and energy dissipation. The energy estimates does not depend on the radius R. There exit subsequences of solutions converge as R. The limit satisfy the SPLLG weakly in R 3. 7 F. Brezzi & P.A. Markowich 1991

87 Outline 1 Motivation and introduction 2 The Schrödinger-Poisson-Landau-Lifshitz-Gilber system 3 Existence of weak solutions 4 Semiclassical limit of SPLLG

88 The WPLLG system in the semiclassical regime.

89 The WPLLG system in the semiclassical regime. The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i 2 Γε [m ε ] ) W ε = 0

90 The WPLLG system in the semiclassical regime. The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i 2 Γε [m ε ] ) W ε = 0 The Landau-Lifshitz-Gilbert (LLG) equation t m ε = m ε H ε eff + αmε t m ε

91 The WPLLG system in the semiclassical regime. The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i 2 Γε [m ε ] ) W ε = 0 The Landau-Lifshitz-Gilbert (LLG) equation t m ε = m ε H ε eff + αmε t m ε The Poisson potential V ε = N ρ ε

92 The WPLLG system in the semiclassical regime. The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i 2 Γε [m ε ] ) W ε = 0 The Landau-Lifshitz-Gilbert (LLG) equation t m ε = m ε H ε eff + αmε t m ε The Poisson potential V ε = N ρ ε The effective field H ε eff = m + Hε s sε and H ε s = ( N m ε )

93 The WPLLG system in the semiclassical regime. The Wigner equation t W ε + v x W ε ( Θ ε [V ε ] + i 2 Γε [m ε ] ) W ε = 0 The Landau-Lifshitz-Gilbert (LLG) equation t m ε = m ε H ε eff + αmε t m ε The Poisson potential V ε = N ρ ε The effective field H ε eff = m + Hε s sε and H ε s = ( N m ε ) The behavior of the solution (W ε, m ε ) in the semiclassical limit ε 0.

94 Main theorem: The semiclassical limit There exists a subsequence of solutions (W ε, m ε ) to the WPLLG system such that W ε ε 0 W in L ((0, T ); L 2 (R 3 x R 3 v)) weak m ε ε 0 m in L ((0, T ); H 1 (Ω)) weak and (W, m) is a weak solution of the following VPLLG system, t W = v x W + x V v W + i [ σ m, W ], 2 t m = m H eff + αm t m, V (x) = 1 W (y, v, t) dv dy, 4π x y R 3 x R 3 v H eff = m + H s, ( 1 H s (x) = 4π Ω m(y) x y ) dy.

95 Conservation quantities.

96 Conservation quantities. Conservation of the total mass. ρ ε (x, t) dx = R 3 x = R 3 x R 3 x R 3 v Tr C 2( W ε (x, v, t) ) dv dx ρ ε (x, 0) dx = λ ε j. j=1

97 Conservation quantities. Conservation of the total mass. ρ ε (x, t) dx = R 3 x = R 3 x R 3 x Conservation of the L 2 -norm of W ε. W ε (t) 2 L 2 (R 3 x R 3 v ) := R 3 v Tr C 2( W ε (x, v, t) ) dv dx ρ ε (x, 0) dx = R 3 x R 3 v λ ε j. j=1 Tr C 2{[ W ε (x, v, t) ] 2} dv dx = W ε I 2 L 2 (R 3 x R 3 v ) = 2 (4πε) 3 (λ ε j )2. j=1

98 Conservation quantities. Conservation of the total mass. ρ ε (x, t) dx = R 3 x = R 3 x R 3 x Conservation of the L 2 -norm of W ε. W ε (t) 2 L 2 (R 3 x R 3 v ) := R 3 v Tr C 2( W ε (x, v, t) ) dv dx ρ ε (x, 0) dx = R 3 x R 3 v λ ε j. j=1 Tr C 2{[ W ε (x, v, t) ] 2} dv dx = WI ε 2 L 2 (R 3 x R 3 v ) = 2 (4πε) 3 Conservation in the magnetization. m ε (x, t) = m 0 (x) 1; m ε (x, t) L 2(Ω) = m 0 (x) L 2(Ω) = Ω. (λ ε j )2. j=1

99 Energy dissipation.

100 Energy dissipation. Schrödinger-Poisson energy F SP = Ekin ε + 1 V ε 2 dx 2 R 3 x

101 Energy dissipation. Schrödinger-Poisson energy F SP = Ekin ε + 1 V ε 2 dx 2 R 3 x The Landau-Lifschitz ( energy 1 F LL = 2 mε Hε s 2 ε ) 2 sε m ε dx Ω

102 Energy dissipation. Schrödinger-Poisson energy F SP = Ekin ε + 1 V ε 2 dx 2 R 3 x The Landau-Lifschitz ( energy 1 F LL = Ω 2 mε Hε s 2 ε ) 2 sε m ε dx The energy dissipation d dt (F LL + F SP ) + α t m ε (x, t) 2 dx 0 R 3 x

103 Energy dissipation. Cont. Using the estimate s ε m ε dx Ω Ω s ε m ε dx R 3 x s ε dx C we get there exists a constant C independent of ε, such that 8 m ε (t) 2 dx + Ekin ε (t) + 1 t V ε (x, t) 2 dx + α t m 2 Ω 2 R 3 x 0 R 3 x C + F LL (0) + Ekin ε (0) + V ε (x, 0) 2 dx C. R 3 x 8 Brezzi & Markowich 1991, Arnold 1996

104 Boundedness From the conservation equations and the energy dissipation we get the following boundedness Ekin ε (t) = v 2 Tr C 2( W ε (x, v, t) ) dv dx C, R 3 x R 3 v V ε L ((0, ),L 6 (R 3 x )) + V ε L ((0, ),L 2 (R 3 x )) C. W ε L 2 (R 3 x R 3 v ) + mε (t) L 2 (Ω) + mε (t) L 2 (Ω) C, and t m ε L 2 ([0,T ],L 2 (R 3 )) C.

105 From the interpolation lemma 9 Lemma Let 1 p, q = (5p 3)/(3p 1), s = (5p 3)/(4p 2), and θ = 2p/(5p 3). Then C > 0 s.t. ρ ε L q C λ ε θ ( p ε 2 Ekin ε ) 1 θ, j ε L s C λ ε θ ( p ε 2 Ekin ε ) 1 θ, ( ) 1/p where λ ε p = j=1 λε j p under the assumption λ ε 2 C we get the estimats ρ ε L ((0, ),L q (R 3 x )) + sε L ((0, ),L q (R 3 x )) C, q [1, 6/5], j ε L ((0, ),L s (R 3 x )) + Jε s L ((0, ),L s (R 3 x )) C, s [1, 7/6]. 9 Arnold 1996

106 Convergence subsequences. W ε V ε ρ ε s ε m ε m ε H ε s V ε ε 0 W in L ((0, ); L 2 (R 3 x R 3 v)) weak, ε 0 ρ in L ((0, ); L q (R 3 x)) weak, q [1, 6/5] ε 0 s in L ((0, ); L q (R 3 x)) weak, q [1, 6/5] ε 0 m in L ((0, ); H 1 (Ω)) weak, ε 0 m in L 2 ([0, T ], L 2 (R 3 x)) strongly. ε 0 H in L ((0, ); L 2 (Ω)) weak, ε 0 V in L ((0, ); L 6 (R 3 x)) weak, ε 0 V in L ((0, ); L 2 (R 3 x)) weak.

107 Passing to the limit of the Wigner equation We study the weak formulation of the Wigner equation [ ( W ε ( t φ + v x φ) + Θ ε [V ε ] + i ) ] 2 Γε [m ε ] W ε φ = Markowich & Mauser 1993

108 Passing to the limit of the Wigner equation We study the weak formulation of the Wigner equation [ ( W ε ( t φ + v x φ) + Θ ε [V ε ] + i ) ] 2 Γε [m ε ] W ε φ = 0. lim ε 0 W ε ( t φ + v x φ) = W ( t φ + v x φ). 10 Markowich & Mauser 1993

109 Passing to the limit of the Wigner equation We study the weak formulation of the Wigner equation [ ( W ε ( t φ + v x φ) + Θ ε [V ε ] + i ) ] 2 Γε [m ε ] W ε φ = 0. lim W ε ( t φ + v x φ) = ε 0 10 lim ε 0 Θ ε [V ε ]W ε φ = W ( t φ + v x φ). W x V v φ 10 Markowich & Mauser 1993

110 Passing to the limit of the Wigner equation We study the weak formulation of the Wigner equation [ ( W ε ( t φ + v x φ) + Θ ε [V ε ] + i ) ] 2 Γε [m ε ] W ε φ = 0. lim W ε ( t φ + v x φ) = W ( t φ + v x φ). ε 0 10 lim Θ ε [V ε ]W ε φ = W x V v φ ε 0 lim ε 0 Γ ε [m ε ]W ε φ? = [m σ, W ]φ 10 Markowich & Mauser 1993

111 Recall that the operator Θ ε is given by Θ ε [V ε ]W ε (x, v) 1 iε = 1 (2π) 3 [ ( V ε x εy ) ( V ε x + εy )] W ε (x, v ) 2 2 e i(v v ) y dy dv, and the operator Γ ε is given by Γ ε [m ε ]W ε (x, v) = 1 (2π) 3 [ M ε ( x εy 2 e i(v v ) y dy dv, where the matrix M ε = σ m ε. ) ( W ε (x, v ) W ε (x, v )M ε x + εy )] 2

112 lim ε 0 Γ ε [m ε ]W ε

113 lim ε 0 Γ ε [m ε ]W ε If m ε H 1 (R 3 ), we can prove (by Taylor s theorem) Γ ε [m ε ]W ε φ = [m σ, W ]φ lim ε 0

114 lim ε 0 Γ ε [m ε ]W ε If m ε H 1 (R 3 ), we can prove (by Taylor s theorem) Γ ε [m ε ]W ε φ = [m σ, W ]φ lim ε 0 But m ε / H 1 (R 3 ), since m ε 1 in Ω, m ε 0 in Ω c

115 lim ε 0 Γ ε [m ε ]W ε If m ε H 1 (R 3 ), we can prove (by Taylor s theorem) Γ ε [m ε ]W ε φ = [m σ, W ]φ lim ε 0 But m ε / H 1 (R 3 ), since m ε 1 in Ω, m ε 0 in Ω c Let m ε,β = m ε x ϕ β, and then m ε = (m ε m ε,β ) + m ε,β, where ϕ β (x) = ϕ(x/β) and ϕ is a positive mollifier.

116 lim ε 0 Γ ε [m ε ]W ε If m ε H 1 (R 3 ), we can prove (by Taylor s theorem) Γ ε [m ε ]W ε φ = [m σ, W ]φ lim ε 0 But m ε / H 1 (R 3 ), since m ε 1 in Ω, m ε 0 in Ω c Let m ε,β = m ε x ϕ β, and then m ε = (m ε m ε,β ) + m ε,β, where ϕ β (x) = ϕ(x/β) and ϕ is a positive mollifier. (Γ ε [m ε ]W ε [M, W ] ) φ dx dv dt Γ ε [ m ε m ε,β] W ε φ dx dv dt (Γ + ε [ m ε,β] W ε [ M β, W ]) φ dx dv dt [M + β M, W ] φ dx dv dt, where M = σ m and M β = M x ϕ β.

117 lim ε 0 Γ ε [m ε ]W ε Cont.

118 lim ε 0 lim ε 0 Γ ε [m ε ]W ε Cont. For the second integral, since m ε,β H 1 (R 3 ) and m ε,β m x ϕ β strongly in L 2 ([0, T ] R 3 ), we have ( Γ ε [m ε,β ]W ε [M β, W ]) φ = 0.

119 lim ε 0 lim ε 0 Γ ε [m ε ]W ε Cont. For the second integral, since m ε,β H 1 (R 3 ) and m ε,β m x ϕ β strongly in L 2 ([0, T ] R 3 ), we have ( Γ ε [m ε,β ]W ε [M β, W ]) φ = 0. For the third integral we have [ ] M β M, W φ dx dv dt C β 0, as β 0.

120 lim ε 0 lim ε 0 Γ ε [m ε ]W ε Cont. For the second integral, since m ε,β H 1 (R 3 ) and m ε,β m x ϕ β strongly in L 2 ([0, T ] R 3 ), we have ( Γ ε [m ε,β ]W ε [M β, W ]) φ = 0. For the third integral we have [ ] M β M, W φ dx dv dt C β 0, as β 0. For the first integral, we use triangle inequality to get [ Γ ε m ε m ε,β] W ε φ dx dv dt C m ε m L 2 ([0,T ] R 3 x ) + C mβ m ε,β L 2 ([0,T ] R 3 x ) + C m m β L 2 ([0,T ] R 3 x ) C m ε m L 2 ([0,T ] R 3 x ) + C m mβ L 2 ([0,T ] R 3 x ),

121 lim ε 0 Γ ε [m ε ]W ε Cont..

122 lim ε 0 Γ ε [m ε ]W ε Cont.. Then lim ε 0 [ Γ ε m ε m ε,β] W ε φ dx dv dt C β.

123 lim ε 0 Γ ε [m ε ]W ε Cont.. Then lim ε 0 [ Γ ε m ε m ε,β] W ε φ dx dv dt C β. And Then (Γ lim ε [m ε ]W ε [M, W ] ) φ dx dv dt C β. ε 0

124 lim ε 0 Γ ε [m ε ]W ε Cont.. Then lim ε 0 [ Γ ε m ε m ε,β] W ε φ dx dv dt C β. And Then (Γ lim ε [m ε ]W ε [M, W ] ) φ dx dv dt C β. ε 0 But the left hand side of above inequality is independent of β, we then have (Γ lim ε [m ε ]W ε [M, W ] ) φ dx dv dt = 0. ε 0

125 Passing to the limit of the LLG equation The weak formulation of the LLG equation m ε t φ = m ε H ε eff φ α m ε t m ε φ.

126 Passing to the limit of the LLG equation The weak formulation of the LLG equation m ε t φ = m ε H ε eff φ α m ε t m ε φ. Since m ε m in L 2 ([0, T ], L 2 (Ω)) strongly, we have m ε t φ = m t φ. lim ε 0

127 Passing to the limit of the LLG equation The weak formulation of the LLG equation m ε t φ = m ε H ε eff φ α m ε t m ε φ. Since m ε m in L 2 ([0, T ], L 2 (Ω)) strongly, we have m ε t φ = m t φ. lim ε 0 Since t m ε t m ε in L 2 ([0, T ], L 2 (Ω)) weakly, we have m ε t m ε t φ = m t m t φ. lim ε 0

128 Passing to the limit of the LLG equation The weak formulation of the LLG equation m ε t φ = m ε H ε eff φ α m ε t m ε φ. Since m ε m in L 2 ([0, T ], L 2 (Ω)) strongly, we have m ε t φ = m t φ. lim ε 0 Since t m ε t m ε in L 2 ([0, T ], L 2 (Ω)) weakly, we have m ε t m ε t φ = m t m t φ. lim ε 0 m ε H ε eff φ dx dt = m ε m ε φ dx dt + m ε H ε sφ dx dt + ε m ε s ε φ dx dt. 2

129 Passing to the limit of the LLG equation

130 Passing to the limit of the LLG equation Thus we can take the limit lim m ε H ε eff φ dx dt = lim m ε m ε φ dx dt ε 0 ε 0 + lim m ε H ε ε 0 sφ dx dt ε + lim m ε s ε φ dx dt ε 0 2 = m m φ dx dt + m H s φ dx dt.

131 The limit of the WPLLG system (W, m) is a weak solution of the following VPLLG system, t W = v x W + x V v W + i [ σ m, W ], 2 t m = m H eff + αm t m, V (x) = 1 W (y, v, t) dv dy, 4π x y R 3 x R 3 v H eff = m + H s, ( 1 H s (x) = 4π Ω m(y) x y ) dy.

132 Summary

133 Summary Use the Schödinger-Poisson-Landau-Lifshitz-Gilbert system to model the spin-magnetization coupling.

134 Summary Use the Schödinger-Poisson-Landau-Lifshitz-Gilbert system to model the spin-magnetization coupling. Prove the existence of H 1 solutions.

135 Summary Use the Schödinger-Poisson-Landau-Lifshitz-Gilbert system to model the spin-magnetization coupling. Prove the existence of H 1 solutions. Use Wigner transformation to get the kinetic description.

136 Summary Use the Schödinger-Poisson-Landau-Lifshitz-Gilbert system to model the spin-magnetization coupling. Prove the existence of H 1 solutions. Use Wigner transformation to get the kinetic description. In the semiclassical limit, the spin-magnetization coupling dynamics can be described by a Vlasov-Poisson-Landau-Lifshitz system.

137 THANKS FOR YOUR ATTENTION!