Antiferromagnetic Textures

Transcription

1 Antiferromagnetic Textures This image cannot currently be displayed. ULRICH K. RÖSSLE IFW DRESDEN SPICE Workshop Antiferromagnetic Spintronics /09/2016 Schloss Waldthausen

2 Magnetic Textures Topologically non-trivial texture in XY-ferromagnet or antiferromagnet SPICE Workshop Antiferromagnetic Spintronics /09/2016 Schloss Waldthausen

3 Magnetic 2D skyrmion Magnetic Textures Topologically non-trivial texture in XY-ferromagnet or antiferromagnet SPICE Workshop Antiferromagnetic Spintronics /09/2016 Schloss Waldthausen

4 Overview Order parameters in magnetism Problem of multidimensional static solitons Topology Continuum theory Ø Landau theory - Landau Ginzburg functionals Ø Classical Skyrme model Ø Dzyaloshinskii models Ø Skyrmions in chiral magnets Ø Problem of thermal phase transitions Ø Beyond simple magnetism 4

5 Vector order-parameters in 3D ferromagnetic (axial) vector) transformation under inversion antiferromagnetic staggered vector? polar vector/ gradients ( x y y )

6 Vector order-parameters in 3D ferromagnetic (axial) vector) transformation under inversion antiferromagnetic staggered vector or polar vector/ gradients ( x y y ) structure-symmetric or antisymmetric magnetic modes (E.A. Turov)

7 Carpets should be continuous the case for continuum theory 7

8 Constructing continuum models Pedestrian path I Heisenberg model: spins on a primitive orthorhombic lattice W= - å i å x=a,b,c J i+x S i S i+x Spatially slowly varying polarization use Taylor expansion s (r=r i+x ) = s (r=r i ) + å n (1/n!) (x Ñ) n s (r=r i ) Continuum approximation to energy W= - å i å x=a,b,c J i+x { s i s i + ( s i (x Ñ) s i ) + ( s i (x Ñ) 2 s i )/2 + } 8

9 W= - å i å x=a,b,c J i+x { s i s i + ( s i (x Ñ) s i ) + ( s i (x Ñ) 2 s i )/2 + } drop linear gradient term bc inversion symmetry integrate quadratic gradient term by parts (surface term!) Replace sum by integration with energy density continuum form of spin exchange energy only : w = a 0 s 2 + å x=a,b,c A x ( x s x s ) lattice anisotropy acts on inhomogeneous magnetization distributions e.g. directionality of domain walls. Compare with standard version for an isotropic lattice w = a 0 s 2 + A (Ñ s ) 2 9

10 Continuum model without symmetry Pedestrian path II Generalized Heisenberg model: spins on a primitive lattice Bilinear coupling terms W= - å i å x=a,b,c S i A i+x S i+x Arbitrary 3x3 matrices A i+x A = Tr (A) E + (A T + A )/2 - (A T - A )/2 symmetric axial vector J G D D W= - å i å x=a,b,c [ J i+x S i S i+x + S i G i+x S i+x - D i+x (S i S i+x ) ] 10

11 Generalized model contains Isotropic (in spin-space) exchange Exchange anisotropy Dzyloshinskii-Moriya couplings W= - å i å x=a,b,c [ J i+x S i S i+x + S i G i+x S i+x - D i+x (S i S i+x ) ] Continuum approximation same path w = s K i s + å x=a,b,c A x ( x s x s ) + å B ijkl ( i s k j s l ) + å x=a,b,c d kl (x) ( s l x s k - s k x s l ) Lifshitz invariants how many are here? 11

12 Symmetries Magnetically ordered materials form of the magnetic free energy constrained by symmetries frame indifference Crystalline Spin-space rotation Time inversion symmetries rigid body motion space group pm reference T(3) SO(3) Ä G Ä SO(3) Ä T Lab-frame coupled by SOC Spin Orbit Coupling 12

13 Landau theory again take the high road Ø Magnetic order breaks some symmetries Ø Microscopic nature of order generally unknown Ø Concept of order parameter (OP) Ø All possible free energy terms for the ordered state known Neel state of a two sublattice crystalline antiferromagnet (spin only) axial vector with 1 (point) group element broken time inversion OP is a staggered vector L pm afm 13

14 Neel state of a two sublattice crystalline antiferromagnet (spin only) axial vector with (point group) element broken time inversion OP is a staggered vector L manifold of ordered states is a sphere S 2 pm afm 120-degree Neel state of three-sublattice antiferromagnets OP is a Dreibein a noncoplanar arrangement of vectors internal degrees of freedom chirality can be broken manifold of ordered states corrresponds to rigid body rotations SO(3) 14

15 Spin-exchange structures Ø all possible types of magnetic orders in crystals are known well, in principle must be long-range ordered ferro- ferrimagnetic compensated antiferromagnet collinear & non-collinear & non-coplanar ferro-, ferri-, antiferro helicoidal hidden paramagnetic... Ø primary magnetic order parameters Ø weak anisotropies / SOC may distort these OPs Marchenko & Andreev UFN 1980, Izyumov / use color crystallographic groups ISOTROPY 15

16 Vistas of some skyrmionic states

17 What Skyrmions? A Skyrmion (in generalized sense) is a multidimensional, static, topological soliton a particle-like solution of a non-linear field equation Here: localized spin-states in certain 2 dimensional non-centrosymmetric magnets

18 Skyrmions Pions in a SU(2) field theory with the Skyrme term Hedgehog adult Skyrmion : (xyz) plane

19 Chiral Skyrmions How they look : Skyrmion filaments Profile Energy density Phase portrait Radial excitation mode

20 Localized states Skyrmion in a model for a ferromagnetic film 2 2 æ mx m m z y m ö z wm = A( Ñm) - Kmz +Q( z) ç mz - mx + mz -my è x x y y ø with surface-induced Dzyaloshinskii-Moriya interactions? A.N.Bogdanov + UKR, PRL 87 (2001)

21 Bogdanov-Hubert phase : a Skyrmion lattice Non-centrosymmetric magnetic crystals phenomenological free energy with Lifshitz invariants Stable localized multidimensional states Formation of condensed Skyrmionic phases Thermodynamically stable Skyrmion lattices in uniaxial magnets A.N. Bogdanov, D.A. Yablonskii, Sov. Phys. JETP 68 (1989) 101 A.N. Bogdanov, A. Hubert, JMMM 138 (1994) 255 Stability of Skyrmionic textures in cubic helimagnets A.N. Bogdanov, UKR, C. Pfleiderer Physica B 359 (2006) 1162 UKR, A.Bogdanov, C. Pfleiderer, Nature 442 (2006) 797

22 Non-collinear magnetism Effect of spin-orbit coupling Direct exchange vs. antisymmetric exchange? J (S 1 S 2 ) D ( S 1 x S 2 ) l m 1 m m 2 net magnetization m = ( m 1 + m 2 ) / 2 staggered (antiferromagnetic) vector l = ( m 1 - m 2 ) / 2 Weak ferromagnetism Néel 1953, Borovik-Romanov & Orlova 1956 Phenomenological theory: antisymmetric exchange, Dzyaloshinskii 1957 Microscopic explanation: antisymmetric contribution from superexchange, Moriya 1960

23 Radial chiral skyrmions in antiferromagnets 23

24 Homogeneous and modulated states in (certain) acentric antiferromagnets 24

25 Homogeneous states l staggered vector l m = 0, l >> m m net magnetization 25

26 Modulated states Simultaneous occurrence of weak-ferromagnetism and modulated states or localized Skyrmion tubes Ba 2 CuGe 2 O 7 class D 2d K 2 V 3 O 8 class C 4v A.Bogdanov, UKR, M.Wolf, K.-H.Müller, PRB 66 (2002)

27 How? Formation of skyrmionic matter

28 Basic Theory The Problem spatially localized stable objects in continuum theories u(x) continuous field Derrick-Hobart theorem energy functionals of type ò W 1D solitonic solutions, e.g. kinks for sine-gordon equation 1 2 [ 2 Ñ ( ) + ( ( ))] d ux Vux dx do not have nontrivial particle-like static solutions (solitons) for d > 1

29 Dzaloshinskii modes : Lifshitz invariants Phenomenological free energies including terms linear in spatial field gradients u/ x Dzyaloshinskii 1964 D F= ò[ f2( u) + f1( u) + f0( u)] d x f µ f ( u) µ ( u) 1, f ( u) µ ( u) ( u) ( u) 2, 2 f ( u) 1 = anti-symmetric Lifshitz-invariants

30 Dzaloshinskii modes : Lifshitz invariants Phenomenological free energies including terms linear in spatial field gradients u/ x Dzyaloshinskii 1964 D F= ò[ f2( u) + f1( u) + f0( u)] d x f µ f ( u) µ ( u) 1, f ( u) µ ( u) ( u) ( u) 2, 2 f ( u) 1 = anti-symmetric Lifshitz-invariants ( m x x m z - m z x m x )

31 D=1 D =1 kinks, walls W F = +W +W l (1) (1) 2 (1) (1) ( l) 1 0 l D = 2 D=2 vortices, baby-skyrmions l F ( l) =W +W l +W l (2) (2) (2) 1 (2) l D=3 D = 3 A.N. Boganov, JETP Lett. 62 (1995) 249 hedgehogs, spherulites F ( l) =W l+w l +W l (3) (3) (3) 2 (3) l F.N. Rybakov (2011)

32 1 st set of conclusions Ø Static multidimensional localized states are uncommon & require special mechanism to become stable Prohibition of such states in the standard continuum model of ordered states (Landau-Ginzburg free energy functionals)!