Magnonics in skyrmion-hosting chiral magnetic materials

Transcription

1 Magnonics in skyrmion-hosting chiral magnetic materials TU Dresden

2 Collaboration theory: experimental groups: Johannes Waizner Peter Böni (München) Neutron scattering Achim Rosch (Köln) Dirk Grundler (Lausanne) Magnetic resonance Shinichiro Seki (Riken) Spinwave spectroscopy

3 Outline: Introduction to chiral magnets Spin-wave dynamics of the magnetic helix Spin-wave dynamics of the magnetic skyrmion Spin-wave dynamics of the magnetic skyrmion lattice

4 Introduction to chiral magnets

5 Chiral magnets non-centrosymmetric, cubic magnets: MnSi, FeGe, FexCo1-xSi, Cu2OSeO3, Bravais lattice: space group: simple cubic P213 (B20) chiral atomic crystal lattice D ~ M(r ~ M) Dzyaloshinskii-Moriya interaction crystal chirality inherited by magnetism chiral magnetic textures helices skyrmions

6 Phase diagram of chiral magnets example: MnSi B MnSi Skyrmion crystal S. Mühlbauer et al. Science (2009) [111] helix

7 Skyrmions Tony Skyrme (1961,1962) solutions of a non-linear field theory, model for baryons B=3 (isospin doublet 3 H/ 3 He) stereographic projection from sphere to plane: 2 (S 2 )=Z hedgehog topologically stable object with quantized winding number W = 1 4 d 2 r ˆM x ˆM y ˆM essential for skyrmion dynamics!

8 Phase diagram of MnSi con t. in a magnetic field under pressure unexplained mystery! slow magnetisation dynamics destroying the Fermi liquid? resistivity T 3/2 extended non-fermi liquid regime three (!) decades in temperature Pfleiderer, Julian, Lonzarich, Nature 2001 Ritz Pfleiderer, Nature 2013

9 This talk: Spin-wave dynamics of chiral magnetic textures 1d helix 2d skyrmion lattice wavelength: 18 nm in MnSi 70 nm in FeGe

10 Magnetization dynamics Landau-Lifshitz-Gilbert t ~ M = ~ M ~ Be +... precession in effective field damping, driving currents etc. wikipedia effective field is determined by the magnetic texture: ~B e = with the energy functional 1 M s F ˆM F = Z d~r V( ˆM)

11 Linear spin-wave theory small amplitude excitation of the equilibrium magnetisation texture ˆM eq ˆM = ˆM eq p ê + +ê with local orthogonal frame Landau-Lifshitz equation ê 1 (r) ê 2 (r) = ˆM eq (r) ê ± = 1 p 2 (ê 1 ± iê 2 t ~ M = ~ M ~ Be i~ t ~ = H ~ magnon spinor wave function ~ T =(, ) U(1) charge = spin angular momentum of magnon not conserved! due to spin-orbit coupling, texture and dipolar interactions Bogoliubov-deGennes 2 2 matrix Hamiltonian H

12 Spin-wave dynamics of the magnetic helix

13 Magnon excitations of the magnetic helix 2 /Q helix = 1d magnetic crystal magnon excitations should obey Bloch s theorem magnon band structure Kugler, MG et al PRL (2015)

14 Magnon excitations of the magnetic helix 2 /Q helix = 1d magnetic crystal q? z magnon excitations should obey Bloch s theorem magnon band structure magnon wave equation i~ t ~ = H ~ for spinor ~ =(, ) with the magnon Hamiltonian: H 0 = D apple1(q 2z) i2 z Qq? cos(qz)+ Q2 2 (1 x ) variant of the Mathieu equation particle in a one-dimensional periodic cosine potential Kugler, MG et al PRL (2015)

15 Magnon excitations of the magnetic helix 2 /Q helix = 1d magnetic crystal magnon excitations should obey Bloch s theorem q? q k magnon band structure q? transversal momentum tunes strength of periodic potential crossover from weak to tight-binding limit Q Q Q flat magnon bands q k (Q) q k (Q) q k (Q) Kugler, MG et al PRL (2015)

16 Magnon excitations of the magnetic helix 2 /Q helix = 1d magnetic crystal magnon excitations should obey Bloch s theorem q? q k magnon band structure Inelastic neutron scattering on MnSi: Q q k (Q) five magnon bands well-resolved Kugler, MG et al PRL (2015)

17 Non-reciprocal magnon dynamics magnon dispersion in the background of field-polarised state: E E E finite group velocity q 0 q 0 q without Dzyaloshinskii-Moriya interaction (DMI) with DMI: shifted parabola non-reciprocal dispersion E(q) 6= E( q) magnon emission and absorption at different energies

18 Magnon spectrum as a function of magnetic field magnetic state helix conical helix field-polarised 0 Hc2 H 3) magnon spectrum for q? =0 (2) (1) E (mev) 16 spin-flip channels 816 non spin-flip channel ) H int c2 ) µ 0 H int =0.00T q = 0 k h NSF _ SF SF+ (3) E (mev) (2) (1) (1') -0.5 (2') (3') q (k h ) 1 0 µ 0 H int =0.22T E (mev) (3) (2) (1) (1') (2') (3') q (k h ) µ 0 H int =0.48T E (gµ 0 µ B H c2 int ) E (mev) (3) (2) (1) (1') (2') (3') q (k h ) µ 0 H int =0.61 T E (gµ 0 µ B H c2 int ) (1) (3') q (k h ) q k (Q) q k (Q) q k (Q) q k (Q) reciprocal spectrum but non-reciprocal weight distribution non-reciprocal spectrum E (gµ 0 µ B H c2 int ) Weber, MG et al arxiv:

19 Inelastic neutron scattering on MnSi positive H 0 non-reciprocal magnon spectrum!(q k ) 6=!( q k ) (a) Set-up 3 TASP µ 0 H int > 0 (1) µ 0 H int = +260 mt q = +2.5 k h (2) (3) (e) µ 0 H int < 0 µ 0 H int = -260 mt q = +2.5 k h (1) (2) (3) H 0 negative H 1200 (b) µ 0 H int = +440 mt q = +2.5 k h (f) (1') µ 0 H int = -440 mt q = +2.5 k h 1000 (1') (2) (3) (1) (2) (3) Hc2 S(q,E) (a.u.) (c) (1') (1) µ 0 H int = +610 mt q = +2.5 k h (2) (3) (g) (1') µ 0 H int = -610 mt q = +2.5 k h (1) (2) (3) -Hc2 (1) 400 (d) µ 0 H int = +610 mt q = -2.5 k h (h) µ 0 H int = -610 mt q = -2.5 k h 200 (1) (1') H 0 (1') E (mev) (1) E (mev) quantitative agreement between theory and experiment (including instrumental resolution) Weber, MG et al arxiv:

20 Magnetic microwave resonances ac magnetic field two resonances: exciting magnons at zero momentum mean magnetisation oscillates clockwise counter-clockwise mev q k ( Demagnetization field splits the degeneracy: Schwarze, MG et al Nat Mater (2015)

21 Polarization dependence of the helix modes at zero field: helix possesses π-rotation symmetry a ˆQ

22 Polarization dependence of the helix modes at zero field: helix possesses π-rotation symmetry a ˆQ

23 Polarization dependence of the helix modes at zero field: helix possesses π-rotation symmetry a ˆQ b c a ˆQ

24 Polarization dependence of the helix modes at zero field: helix possesses π-rotation symmetry a ˆQ b c d +Q -Q a ˆQ +/- Q modes strictly linearly polarized at zero field! (for non-circular sample shape) similar to easy-plane antiferromagnets

25 Polarization dependence of the helix modes +/- Q modes strictly linearly polarized at zero field! allows to address each mode selectively I. Stasinopolous, MG, et al, Scientific Reports (2017)

26 Spin-wave dynamics of the magnetic skyrmion

27 Skyrmion in a field-polarised background static skyrmion-soliton solution B Bogdanov & Hubert (1994) spin-waves scatter off the skyrmion magnon scattering problem magnon Hamiltonian H = ~2 ( i z ~ r 1~a) 2 2M mag + 1V 0 + x V x scattering vector potential scattering potentials

28 Magnon-skyrmion bound states Magnon continuum breathing mode quadrupolar mode magnetic field sextupolar mode local elliptical instability global instability magnon-skyrmion bound states thermally excited subgap states! see also Lin, Batista & Saxena PRB (2014) Schütte & MG PRB (2014)

29 Emergent magnon Lorentz force adiabatic adjustment of local frame Berry phase vector scattering potential ~a = cos with quantised total flux Q sin ( sin, cos ) Z d 2 r(r ~a) = Z d 2 r ˆM(@ x ˆM) topological winding number magnon scatter off a localised emergent magnetic flux due to non-trivial topology of skyrmion emergent Lorentz force

30 Topological magnon skew scattering WKB wave function incoming magnon wave scattered to the right-hand side emergent Lorentz force leads to skew scattering! topological magnon Hall effect! see also Iwasaki, Beekman & Nagaosa PRB (2014) Mochizuki et al. Nat. Mat. (2014) Schütte & MG PRB (2014) Schroeter & MG LTP (2015)

31 Skew & rainbow scattering magnon differential cross section asymmetric & oscillations Different classical trajectories contribute and interfere! rainbow scattering! Schütte & MG PRB (2014) Schroeter & MG LTP (2015)

32 Skew & rainbow scattering magnon differential cross section asymmetric & oscillations Different classical trajectories contribute and interfere! rainbow scattering! Schütte & MG PRB (2014) Schroeter & MG LTP (2015)

33 How to drive skyrmions with magnon currents? incoming momentum magnon wave exerts a pressure on the skyrmion in which direction will it move? momentum conservation?

34 Driving skyrmions with magnon currents counting net flux of incoming & outgoing momentum outgoing momentum incoming momentum conservation law of linear momentum 4 S" 0µ j top µ µ T stat µ =0 space-time topological current j top µ = 1 8 µ ˆM(@ ˆM) (static part) energy-momentum tensor expanding conservation law up to quadratic order in and integrating over the space

35 Linear response approximation Thiele equation with a magnon force t ~ R(t) = ~ F Linear response: evaluate force F for skyrmion at rest Ṙ t ~ R ~F with after some algebra using optical theorem: ~F = J " k k(")?(") magnon force determined by transport scattering cross sections: J " momentum-transfer force skew scattering finite transversal force

36 Spin-wave dynamics of the magnetic skyrmion lattice

37 Magnon-band structure of skyrmion lattice skyrmion lattice magnon dispersion for in-plane momenta 2d magnetic Brillouin zone

38 Topological magnon-band structure non-trivial topology of skyrmions topological magnon band structure each skyrmion acts like a source of emergent magnetic flux Chern numbers emergent magnon electrodynamics emergent magnon Landau levels bands with finite Chern numbers topologically protected magnon edge states

39 Magnetic skyrmion resonances magnetic resonances at the Γ point clockwise (CW) breathing counterclockwise (CCW)

40 Magnetic skyrmion resonances field-dependence of the resonance frequencies clockwise (CW) breathing (a) 1.0 H / H c2 0.5 conical helical FM skyrmion lattice PM T / T c for a field sweep through the phase diagram counterclockwise (CCW) see also Mochizuki PRL (2012); Onose et al. PRL (2012)

41 Comparison experiment & theory different field sweeps normalized with Hc2(T) three different materials MnSi, Fe0.8Co0.2Si and Cu2OSeO3 with three different shapes (demagnization factors) excellent quantitative theoretical understanding T. Schwarze, et al. Nat Mater (2015)

42 Non-reciprocity of skyrmion-lattice magnons magnon dispersion for out-of-plane momenta skyrmion lattice: 4 H k k 2 Q Q k k non-reciprocity!(k k ) 6=!( k k )

43 Non-reciprocity of skyrmion-lattice magnons magnon dispersion for out-of-plane momenta skyrmion lattice: 4 H k k 2 CW breathing CCW Q Q k k non-reciprocity!(k k ) 6=!( k k ) most pronounced for the CCW mode! confirmed by spin-wave spectroscopy experiments by S. Seki!

44 Non-linear excitations

45 Skyrmion creation at the edge uncompensated Dzyaloshinskii-Moriya interaction at the edge: boundary condition: ˆn ~ r ˆM Qˆn ˆM =0 Rohart & Thiaville, PRB (2013) Meynell et al PRB (2014) twist of the magnetisation even in the field-polarised state attractive potential for spin waves: creation of skyrmions using condensation of edge magnon magnon localised to the edge Müller, Rosch, MG, New J. Phys. (2016)

46 Topological defects of helimagnetic order a ˆQ magnetic helix = lamellar structure similar to cholesteric liquid crystals or stripe domain patterns pitch ˆQ is a director ± vortices are possible = disclinations defects D disclinations combine to form a dislocation with Burgers vector B B

47 Skyrmion winding number of dislocations A B C D skyrmion number W = -1 skyrmion number W = -1/2 skyrmion embedded in a topologically trivial background dislocation (with B = λ) = meron general relation for skyrmion number of dislocation with Burgers vector B: W = 1 2 mod 2 B only dislocations with half-integer B contribute to topological Hall effect & emergent electrodynamics

48 Magnetic relaxation by climb motion of dislocations magnetic force microscopy: surface of FeGe slow power-law relaxation ~ 1000 sec! climb motion of dislocations 180 o phase shift A. Dussaux et al., Nat Comm 2016

49 Topological domain walls depending on relative angle: three types of domain walls a Q 1 α type I curvature wall Q 2 1 µm b type II α Q 2 Q 1 1 µm zig-zag disclination wall c Q 1 α type III Q 2 dislocation wall 1 µm MFM: surface of FeGe P. Schoenherr, MG, D. Meier, arxiv:

50 Summary: Spin-wave dynamics of the magnetic helix band structure resonances linear polarisation Spin-wave dynamics of the magnetic skyrmion internal modes skew scattering rainbow scattering Spin-wave dynamics of the magnetic skyrmion lattice 4 2 topological band structure resonances k k Q Q non-reciprocity REVIEW: MG, J. Waizner, D. Grundler, J. Phys. D: Appl. Phys. 50, (2017)