Geometry and Spin Transport in Skyrmion Magnets

Transcription

1 Geometry and Spin Transport in Skyrmion Magnets Benjamin Brown University of Warwick Conference of Astronomy and Physics Students, University College London, 2017

2 What is a Magnetic Skyrmion? What is a Magnetic Skyrmion? Tony Hilton Royle Skyrme.

3 What is a Magnetic Skyrmion? What is a Magnetic Skyrmion? A skyrmion, i.e. a whirling magnetic texture. Tony Hilton Royle Skyrme.

4 What is a Magnetic Skyrmion? A skyrmion, i.e. a whirling magnetic texture.

5 What is a Magnetic Skyrmion? A skyrmion, i.e. a whirling magnetic texture. A Skyrmion is a smooth field configuration defined by a topologically non-trivial, surjective mapping from a base manifold M into the order parameter space O = S n, trivial on the surface of M and characterised by a finite integer-valued topological charge.

6 What is a Magnetic Skyrmion? Topological stability Wrapping a skyrmion around S 2. Skyrmion can be wrapped about the sphere W times. Call W the winding number. W is a homotopy invariant does not change under continuous deformation.

7 What is a Magnetic Skyrmion? Coordinate space M = R 2 to parameter space O = S 2. Skyrmion with W = 1.

8 What is a Magnetic Skyrmion? Coordinate space M = R 2 to parameter space O = S 2. Skyrmion with W = 1. Parameter space O = S 2 given by local magnetisation ˆM: (r, φ) ˆM(r) = ( 2λr r 2 + λ 2 cos(wφ), 2λr r 2 + λ 2 sin(wφ), r 2 λ 2 ) r 2 + λ 2 λ is the skyrmion radius. ˆM = 1.

9 Hamiltonian for an Electron in a Magnetic Texture Hamiltonian for an Electron in a Magnetic Texture [ p 2 i t ψ = 2m 1 + Jgµ ] B 2 σ ˆM(r) ψ ψ = [ψ, ψ ] T, 2-component spinor. σ ˆM spin-magnetisation coupling term.

10 Hamiltonian for an Electron in a Magnetic Texture [ p 2 i t ψ = 2m 1 + Jgµ ] B 2 σ ˆM(r) ψ n e Unitary rotation ψ U(r) ψ, M U(r) = n(r) σ, n(r) = ˆM(r) + ê z ˆM(r) + ê z. Premultiply by U (r); 2 nd term becomes U (r)σ ˆM(r)U(r) = σ z.

11 Hamiltonian for an Electron in a Magnetic Texture [ p 2 i t ψ = 2m 1 + Jgµ ] B 2 σ ˆM(r) ψ Unitary rotation ψ U(r) ψ, U(r) = n(r) σ, n(r) ˆM(r) + ê z. Premultiply by U (r); 2 nd term becomes U (r)σ ˆMU(r) = σ z. Electron coupling to the skyrmion texture.

12 Emergent Electromagnetic Fields The Hamiltonian [ p 2 i t ψ = 2m 1 + Jgµ ] B 2 σ ˆM(r) ψ

13 Emergent Electromagnetic Fields The Hamiltonian [ p 2 i t ψ = 2m 1 + Jgµ ] B 2 σ ˆM(r) ψ then becomes i t ψ = [ ] i U t U + 1 2m (1p i U U) 2 + Jgµ B 2 σ z ψ Single domain Stoner-Wolfarth model with minimal coupling substitution: p p i U U; t t + i U t U.

14 Emergent Electromagnetic Fields i t ψ = [ i U t U + 1 2m (1p i U U) 2 + Jgµ B 2 σ z ] ψ p p i U U; t t + i U t U. Introduce potentials/gauge fields V e = (i /q e )U t U, A e = (i /q e )U U,

15 Emergent Electromagnetic Fields i t ψ = [ i U t U + 1 2m (1p i U U) 2 + Jgµ B 2 σ z ] ψ p p i U U; t t + i U t U. Introduce potentials/gauge fields V e = (i /q e )U t U, A e = (i /q e )U U, p p q e A e ; t t + q e V e. [ ] i t ψ = q e V e + 1 2m (1p qe A e ) 2 + Jgµ B 2 σ z ψ V e and A e are analogous to electromagnetic potentials with charge q e.

16 Emergent Electromagnetic Fields V e = (i /q e )U t U; A e = (i /q e )U U. Recall: U(r) = n(r) σ, A e = ( /q e )σ (n n) V e = ( /q e )σ (n t n) Emergent electric and magnetic fields: E e = V e t A e = (2 /q e )σ ( n t n) B e = A e = ( /q e )σ ( n n)

17 Time-Independent Magnetisation Time-Independent Magnetisation E e = (2 /q e )σ ( n t n); B e = ( /q e )σ ( n n)

18 Time-Independent Magnetisation Time-Independent Magnetisation E e = (2 /q e )σ ( n t n); B e = ( /q e )σ ( n n) With the fields become n(r) = ˆM(r) + ê z ˆM(r) + ê z E e = 0, [ ] B e = 2W λ2 r λe iφ q e r(r 2 + λ 2 ) 2 λe iφ ê r z.

19 Time-Independent Magnetisation [ ] B e = 2W λ2 r λe iφ q e r(r 2 + λ 2 ) 2 λe iφ ê r z,

20 Time-Independent Magnetisation [ ] B e = 2W λ2 r λe iφ q e r(r 2 + λ 2 ) 2 λe iφ ê r z, Adiabatic approximation, project onto spin-axis: B e σ = σ z B e σ z = 2W λ2 q e r(r 2 + λ 2 ) 2 [ ] r 0 ê 0 r z.

21 Time-Independent Magnetisation [ ] B e = 2W λ2 r λe iφ q e r(r 2 + λ 2 ) 2 λe iφ ê r z, Adiabatic approximation, project onto spin-axis: B e σ = σ z B e σ z = 2W λ2 q e r(r 2 + λ 2 ) 2 i.e. electron experiences a magnetic field B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz. For λ = 100Å, B e σ 3.3 T, so quite significant! [ ] r 0 ê 0 r z.

22 Time-Independent Magnetisation Fun fact: magnetic flux from B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz is quantised!

23 Time-Independent Magnetisation Fun fact: magnetic flux from is quantised! B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz Integrate over the skyrmion unit-cell: c 1 = 1 B e σ ds = 4π skyrmion

24 Time-Independent Magnetisation Fun fact: magnetic flux from is quantised! B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz Integrate over the skyrmion unit-cell: c 1 = 1 ˆ λ B e σ ds = W λ2 r dr dθ 4π skyrmion q e 0 (r 2 + λ 2 ) 2 =

25 Time-Independent Magnetisation Fun fact: magnetic flux from is quantised! B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz Integrate over the skyrmion unit-cell: c 1 = 1 ˆ λ B e σ ds = W λ2 r dr dθ 4π skyrmion q e 0 (r 2 + λ 2 ) 2 = W /2qe ;

26 Time-Independent Magnetisation Fun fact: magnetic flux from is quantised! B e σ = 2W λ2 q e (r 2 + λ 2 ) 2 êz Integrate over the skyrmion unit-cell: c 1 = 1 ˆ λ B e σ ds = W λ2 r dr dθ 4π skyrmion q e 0 (r 2 + λ 2 ) 2 = W /2qe ; So magnetic flux comes in integer multiples of /2q e. c 1 is often called the 1 st Chern number of the system another topological invariant.

27 Time-Dependent Magnetisation Time-Dependent Magnetisation Apply magnetic field B = (0, 0, B). The dissipationless Landau-Lifschitz-Gilbert equation: ˆM(r, t) = t ˆM = αb ˆM ( 2λr 2λr r 2 cos Φ(φ, t), + λ2 r 2 + λ 2 sin Φ(φ, t), r 2 λ 2 ) r 2 + λ 2, with Φ(φ, t) = Wφ αbt.

28 Time-Dependent Magnetisation U(r, t) = σ n(r, t) inherits this time-dependence. Now V e = ( /q e )σ (n t n) is non-zero. = E e is non-zero: E e = V e t A e = (2 /q e )σ ( n t n) [ ] = 2αWB λ2 r λe iφ q e (r 2 + λ 2 ) 2 λe iφ ê r r

29 Time-Dependent Magnetisation E e = [ 2αWB λ2 q e (r 2 + λ 2 ) 2 r λe i Φ λe i Φ r ] ê r Project onto spin-axis again: E e σ = σ z E e σ z = ± 2αWB λ2 r q e (r 2 + λ 2 ) 2 êr, Faraday s law: E e σ = v d B e σ. Applying a current causes the skyrmion to drift need current density of about j 10 6 C/m 2. Domain walls require j C/m 2.

30 Time-Dependent Magnetisation Fin