Multiferroic skyrmions

Transcription

1 Multiferroic skyrmions Maxim Mostovoy University of Groningen Zernike Institute for Advanced Materials ECRYS 2017 Cargese September 1, 2017

2 Collaborators RIKEN, U Tokyo Yoshi Tokura Naoto Nagaosa Xiuzhen Yu Shinichiro Seki Naoya Kanazawa Wataru Koshibae Aoyama Gakuin Uni Masahito Mochizuki Sungkyunkwan Uni Jung Hoon Haan UNSW Yaroslav Kharkov Oleg Sushkov Groningen Andrey Leonov Alessio Pozzi U of New Hampshire Jiadong Zang

3 Outline Skyrmions and skyrmion materials Physical meaning of topological charge Topological transport Skyrmion dynamics Skyrmions in frustrated magnets Electric field excitation of skyrmions

4 Skyrmion T. H. R. Skyrme, A Nonlinear field theory, Proc. Roy. Soc. London A 260, 127 (1961) Tony Skyrme ( ) π π + π + σ = S S 3 3 Topological charge = Baryon number

5 Skyrme model n =,,, ) n 2 = 1 ( 2 3 π π π σ 1

6 Nuclei as multi-q skyrmions 1 H 2 H 3 H, 3 He 4 He Q = 1 Q = 2 Q = 3 Q = 4 Q = 5 Q = 6 Q = 7 Q = 8 R.A. Battye& P.M. Sutcliffe, PRL (1997)

7 Lord Kelvin ( ) Atoms as vortices

8 Magnetic skyrmions

9 Magnetic skyrmion n 2 x + n 2 + n 2 y z = 1 S S 2 2

10 Topological charge Pontryagin number: Q 1 2 4π 1 4π ( n [ n n] ) = dω = integer # = d x x y n ( x, y + dy ) n ( x, y) n ( x + dx, y )

11 Skyrmions in chiral magnets A.N. Bogdanov& D.A. Yablonskii, Sov. Phys. JETP 95, 178 (1989) Mn Si x x Neutron scattering Si Mn Lorentz microscopy Mühlbauer et al, Science (2009) X. Yu et al. Nature (2010)

12 Non-centrosymmetric magnets I. Dzyaloshinskii, Sov. Phys. JETP 19, 960 (1964) P. Bak& M.H. Jensen, J. Phys. C: Solid State 13, L881 (1980) CubicB20 structurep32 1 MnSi, Fe 1-x Co x Si, FeGe Exchange energy f J = D 2 ( ) 2 n + n [ n] 2 O x S1 S2 DM energy [ ] r 12 = D S E DM 12 1 S2

13 Helical spiral state n = e cos( q x + ϕ ) + e sin( q x + ϕ) 1 2 e e e = e e q, q, f for for is the spin rotation axis J = q 2 D 2 D D > < 0 0 ( q e ) 3 e 3 q Spiral wave vector q = D / J Weak crystallization

14 q H Phase Diagram of MnSi Mühlbauer et al, Science 323, 915 (2009) q 3 q [111] λ =18 nm q1 q2 q H

15 Fluctuations M. Janoscheket al., PRB 87, (2013) Inverse correlation length S. A. Brazovskii, Sov. Phys. JETP 41, 85 (1975)

16 Skyrmion Crystal State

17 Stripes & magnetic bubbles X. Yu et al. PNAS 109, 8856 (2012)

18 Bubble arrays in ferromagnetic films

19 Stabilization of bubble array Free energy of Ising ferromagnet F d ( ( ) 2 c M + am + bm MH ) 2 4 = FMD 3 3 d x + Magnetodipolar energy favors non-uniform state F = 2 d f ( qd ) MD π M f ( x) f xm 4 = M M M M q 3 q1 q2 q3 q1 q2 H 0 T. Garel & S. Doniach, PRB 26, 325 (1982) q qd 2 ( qd ) 1, qd << 1 q 1 q + q + q = 0 2 M 3 1 q 2 M q 2 M q 3 M 0 e = 1 x 4th-order term stabilizes 3q-state in magnetic field x F MD linearlydecreaseswithq q 3 q1 q2

20 Order from disorder F = F + F MF fluct Monte Carlo F Mühlbauer et al, Science 323, 915 (2009) S. Buhrandt& L. Fritz, PRB (2013)

21 Rayleigh-Bénard convection

22 More skyrmion materials

23 B20-compounds Y. KITP

24 Universal phase diagram A. Bauer & C. Pfleiderer(2016)

25 Skyrmions in Cu 2 OSeO 3 P32 1 λ = 60 nm J-W. G. Bos et al PRB 78, (2008) S. Seki et al Science 336, 198 (2012)

26 Multiferroic behavior S. Seki et al Science 336, 198 (2012)

27 Rotation of SkX in electric field J. S. White, PRL (2014)

28 GaV 4 S 8 I. Kézsmárkiet al. Nat. Mater. 14, 1116 (2015) F 43m R3m at 42K

29 Lifshitz Invariant R3m Néel skyrmion ( n n n n ) + ( n n n n z x x x x z z y y y y z ) Cycloidal spiral

30 Room-temperature skyrmions Y. Tokunaga et al. Nature Communications (2015) Co 8 Zn 9 Mn 3

31 SkX in Co8Zn8Mn4

32 Metastable SkX with square lattice MnSi Co 8 Zn 8 Mn 4 Nakajima et al., Sci. Adv e K. Karubeet al Nat. Mater. 15, 1237 (2016)

33 Physical meaning of topological charge

34 Adiabatic motion of spin-polarized electrons Adiabatic projection Effective gauge potentials

35 Effective gauge fields Lorentz force acting on spin-polarized electron [ ] ( ) [ ] [ ] ( ) = = = = n n n a n n n k j ijk i i t i i t i i e c h e a c a e ε h h

36 Physical meaning of Skyrmion charge Topological charge: Q 1 = 2 d x x yn 4π ( n [ n ]) Q e hc 2 = d xhz Flux of internal magnetic field: 2 Φ = d xh = QΦ z 0 Φ 0 = hc e

37 Topological charge of vortex Vortex in superconductor iφ Ψ= Ψ e Winding number dx = φ π Q 2 1 Φ = d x A = Φ0 Q Φ 0 = hc 2e

38 Topological Hall Effect e A. Neubauer et al PRL 102, (2009) M. Lee et al PRL 102, (2009)

39 Emergent electric field Electric field of moving skyrmion Lorentz force 1 E = c [ V H] ( e) F = ) c [( v V H] J. Zang, MM et al PRL 107, (2011) T. Schultz et al Nat. Phys. 8, 301 (2012)

40 THE in MnGe λ = 3 nm N. Kanazawa et al. PRB 86, (2012) M = + (0,cosqx, sinqx) + ( sinqy,0,cosqy) (cosqz, sinqz,0) + (0,0, m z )

41 Lorentz TEM images of MnGe Y. Tonigakiet al. Nano Lett. 15, 5438 (2015)

42 Magnetic crystals B. Binzet al., PRL 96, (2006) J-H Park and J-H Han PRB 83, (2011)

43 Topological Magnon Hall Effect

44 Magnon effective gauge field δ n( x, t) = ψ e + ψ e e 1, e 2 n e 2 e 1 e 2 e 1 n n Effective gauge potential ai = ( e ) e 2 i 1

45 Topological Magnon Hall effect magnon For magnons, skyrmionis a flux of effective magnetic field Φ= ±2Φ 0

46 Magnon-skyrmion scattering J. Iwasaki et al. PRB 89, (2014)

47 Rotation of SkX M. Mochizuki et al, Nat. Mater. 13, 241 (2014)

48 Rotation of skyrmion crystal Cu 2 OSeO 3 MnSi

49 Heating by electron beam

50 Magnon-driven rotation of SkX

51 Room-temperature skyrmions in mulilayers C. Moreau-Luchaire et al Nat. Nano 11, 444 (2016) S. Woo et al Nat. Mater. 15, 501 (2016)

52 Interfacial DM interaction Lifshitzinvariant for broken m z M z x M x M x x M z + M z y M y M y y M z Cycloidal spiral z y x

53 Skyrmionics Nanometer size Topological protection Low critical currents Albert Fert et al Nature Nano (2013) C. Moreau-Luchaire et al Nature Nano 11, 444 (2016) S. Woo et al, Nature Materials (2016)

54 Magnetic bubble memory upgrade

55 Blowing magnetic bubbles W. Jiang, Science. 349, 283 (2015)

56 Skyrmion dynamics

57 Landau-Lifshitz-Gilbert equation Landau-Lifshitz-Gilbert equation: m & = γ m H + α m eff m& m = M M 0 S H eff = δ H δ M γ = gµ B h Dimensionless LLG equation: t t δh m & = m + α m δm m& M 0 γ

58 Thiele equations for Skyrmion ( ) ( ) ) (, t t R m x x m = Y X F G F G = Γ + = Γ Y X Y X & & & & α α Center-of-mass coordinates: ), ( Y X R = Q y x G π 4 = = x m m m = = Γ x x m m m m y y x x

59 Skyrmion in inhomogeneous magnetic field H z

60 Current-driven dynamics of spin textures

61 LLG equation in presence of electric current m& H = m + α m m& + β m ( j ) m m ( j )m spin-transfer torque

62 Spin transfer torque

63 LLG equation in presence of electric currents m& H = m + α m m& + β m ( j ) m m ( j )m β-term

64 Current-driven Skyrmion dynamics Thiele equations : α ΓX& GX& + α GY& ΓY& = = β Γj G j x x X& 2 2 G + αβ Γ = 2 2 G + ( αγ) j x j x & = ( β α ) ΓG Y 2 2 G + ( αγ) j x Skyrmion Hall Effect : V V y x ( α β ) Γ G

65 Relation between the THE and current-driven dynamics kr s >> 1 y F y e p = dxh, y z c ( x y) QΦ F = n v dy p = 0 y e x x c j x

66 Skyrmions in frustrated magnets J 2 > 0 J 1 < 0

67

68 Spiral ordering in SrFeO 3 T. Takeda JPSJ 33, 967 (1972); H. Oda JPSJ 42, 101 (1977); A. Lebon PRL 92, (2004) S. Ishiwata et al. PRB 84, (2011)

69 Skyrmions in frustrated triangular magnets T. Okubo et al. Phys. Rev. Lett. 108, (2012)

70 Frustratedtriangularmagnet withanisotropy,,

71 Continuum model E = d 2 x Spin stiffness ( 2 n n + n n 2hn Kn ) ρ µ i i ρ = ( J J ) µ = ( 9J J ) 1/ Skyrmions 2 1/ ρ > 0 ρ < 0 Ferromagnetic Lifshitz point 2 1 Non-collinear z J J Skyrmion crystal 2 1 z

72 Phase Diagram A. Leonov and MM, Nature Commun. 6, 8275 (2015) FM conical spiral SkX vertical spiral

73 Skyrmion crystal

74 Skyrmion charge, vorticity & helicity χ = 0 N. Nagaosa and Y. Tokura, Nature Nano 8, 899 (2013) Q = +1 χ = π χ = +π/ 2 χ = π/ 2 Q = -1

75 Arbitrary helicity and vorticity Θ = Θ( ρ ) Φ = vϕ + χ Q ( n ( 0) ( ) ) z n z = v 2

76 Skyrmion Hall effect W. Jiang at al. Nature Physics 13, 162 (2017)

77 Separation of skyrmions from A. Leonov& anti-skyrmions MM Nature Commun. 8, (2017)

78 Multi-Q skyrmion states in underfrustrated magnets Y. Kharkov, O.P. Sushkov & MM arxiv:

79 Skyrmions with large Q Y. Kharkov, O.P. Sushkov & MM arxive:

80 Q = -1 S z

81 Sz

82 Q = -2 S z

83 Q = -3 S z

84 Q = -4 S z

85 Q = -6 S z

86 Holographic multi-skyrmions 1 1 R. A. Battye& P.M. Sutcliffe, PRL 79, 363 (1997)

87 Topological stripe

88 Topological edge states A. Leonov& MM Nature Commun. 8, (2017)

89 Half-skyrmions

90 Skyrmion Bi-meron J 2

91 Bi-merons Spin configuration Topological density

92 Meron-antimeron crystal

93 Superfluid 3 He S = 1 L = 1 c k β ckα = d µ ( k )( ) σ σ µ i 2 αβ d ˆ ( k) d µ j k j µ = 3 He-A 3 He-B ˆ + ( e ie ) iφ d µ j = e d µ 1 j 2 = iφ d j e δ µ µ j j

94 Bi-meron U. Parts et al. PRL 75, 3320 (1975)

95 Topological stripe

96 Magnetoelectric properties of skyrmions in Mott insulators

97 Breaking of inversion symmetry by spin ordering Cycloidal spiral P Q e 3 InversionI: (x,y,z) (-x,-y,-z) Q Cycloidal spiral e3 P

98 Phase Diagram A. Leonov and MM, Nature Commun. 6, 8275 (2015) FM conical spiral SkX vertical spiral

99 Polarization at K = 0.04 flop skyrmion crystal 2q yz-spiral yz(q 1 ) x(q 2 )+x(q 3 ) 2q FM

100 Skyrmion electric dipole moment Magneto-electric coupling MM PRL 96, (2006) f me = g [ m( m m m] E ) ( ) Polarization induced by skyrmion P z = f E z = g dθ dr + v sin 2Θ 2r cos ((v 1) ϕ + χ ) Dipole moment for vorticity v = +1 ( χ) D(0) cos χ D z =

101 Magnetically-induced electric oscillations H x

102 Conclusions Non-trivial topology of magnetic defects gives rise to interesting physics Skyrmions, merons, monopoles, chiral bobbers, hopfions, Electric field control of skyrmions in magnetic insulators Topological memory