Spin-orbit interaction at interfaces: from Rashba states to chiral spin textures

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1 Spin-orbit interaction at interfaces: from Rashba states to chiral spin textures Stefan Blügel Peter Grünberg Institut and Institute for Advanced Simulation Spetses Spin-Orbit Coupling in Surface or Interface States 08. June 2015

2 Spin-Orbit Coupling v spin-orbit coupling has fascinating realizations and ramifications in solids Examples: o Rashba Effect o Dresselhaus Effect o Topological & Chern Insulator o Magnetic Anisotropy o Dzyaloshinskii-Moriya Interaction o Chiral magnets and skyrmions o Spin-Orbit Torque o Anomalous & Spin Hall Effect & Edelstein Effect o Quantum & Topological Spin Hall Effect o Spin-Relaxation (Elliot-Yafet, Dyakonov-Perel) June 11, 2015 Folie 2

Spin-Orbit Coupling v spintronics spin-orbitronics Examples: o Rashba Effect o Dresselhaus Effect o Topological & Chern Insulator o Magnetic Anisotropy o Dzyaloshinskii-Moriya Interaction o Chiral

3 Spin-Orbit Coupling v spintronics spin-orbitronics Examples: o Rashba Effect o Dresselhaus Effect o Topological & Chern Insulator o Magnetic Anisotropy o Dzyaloshinskii-Moriya Interaction o Chiral magnets and skyrmions o Spin-Orbit Torque o Anomalous & Spin Hall Effect & Edelstein Effect o Quantum & Topological Spin Hall Effect o Spin-Relaxation (Elliot-Yafet, Dyakonov-Perel) June 11, 2015 Folie 3

4 Outline v Aim: Space inversion asymmetry + Spin-orbit coupling + Lack of time inversion symmetry = chiral spin textures at metal surfaces à topologically protected spin-structures June 11, 2015 Folie 4

Mitglied der Helmholtz Gemeinschaft Thanks!

Gustav Bihlmayer Juba Bouaziz Samir Lounis Eugene Chulkov Pedro

Matthias Bode Ø Oliver Rader Ø Thomas Michely Ø Carlo Carbone Ø Roland

5 Mitglied der Helmholtz Gemeinschaft Thanks! Bernd Zimmermann Vasile Caciuc Nicolae Atodiresei Phivos Mavropoulos Gustav Bihlmayer Juba Bouaziz Samir Lounis Eugene Chulkov Pedro Echenique STM groups Frank Freimuth Yuriy Mokrousov PES groups Ø Matthias Bode Ø Oliver Rader Ø Thomas Michely Ø Carlo Carbone Ø Roland Wiesendanger SOCSIS, Spetses, June Ø Philip Hofmann Bertrand Dupé Stefan Heinze June 11, 2015 Folie 5

6 Tiny tiny SOC brings us to Spetses degenerate states k (r) =a k (r) k (r) =a k (r) e ik r e ik r E(k) ν k June 11, 2015 Folie 6

7 Tiny tiny SOC brings us to Spetses degenerate states Usually: k (r) = a k (r) + b k (r) e ik r a 2 1 k (r) = a k(r) b k(r) e +ik r b 2 1 b 2 : Ellio1- Yafet parameter H SOC = (r) L S E F E(k) n n Δ soc Perturba<on theory: b 2 nk = X 0 h nk (L S) "# 0 n0 k i 2 (E n 0 nk E n0 k) Fermi surface hot- spots k June 11, 2015 Folie 7

8 Fermi surface hot spots for Anisotropy of b 2 (ŝ) bcc W (Z=74) hcp Os (Z=76) details matter! B. Zimmermann et al., PRL 109, (2012) June 11, 2015 Folie 8

9 Tiny tiny SOC brings us to Spetses degenerate states Usually: k (r) = a k (r) + b k (r) e ik r a 2 1 k (r) = a k(r) b k(r) e +ik r b 2 1 b 2 : Ellio1- Yafet parameter Additional fun: Lifting degeneracies by breaking time and space inversion symmetry! June 11, 2015 Folie 9

10 Spin-Orbit Coupling: Space Inversion Symmetry Elemental solids (Cu, Si, Al.) For a given band ν the following two states have the same energy Proof: time reversal space inversion June 11, 2015 Folie 10 q.e.d.

11 Space inversion symmetry broken Time reversal + space inversion symmetry: (GaAs, InSb, interfaces, surfaces,...) Time reversal only!, Effective spin-orbit ( magnetic ) field Ω: Time reversal symmetry: I. Zˇuti ć, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). June 11, 2015 Folie 11

12 The functional forms of Ω(k): Symmetry Analysis J. Fabian June 11, 2015 Folie 12

13 Axis of spin-rotation Spin Splitting at constant E (Energy) k y <110> <110> <110> <110> k y <001> <001> k x k x Rashba type Inversion Asymmetry by surface, interface, heterostructure Dresselhaus type Inversion Asymmetry in crystal structure June 11, 2015 Folie 13

14 1 st evidence: Au(111) Surface Theory PES Experiment surface states E F bulk states Lashell et al., PRL 77, 3419 (1996) Reinert et al., PRB 63, (2001) June 11, 2015 Folie 14

15 Semi-metal Bi(111): Theory Without Spin-Orbit Interaction With Spin-Orbit Interaction Yu.M. Koroteev, G. Bihlmayer, J.E. Gayone, E.V. Chulkov, S. Blügel, P.M. Echenique, and PH. Hofmann, PRL 93, (2004) see Poster Aguilera June 11, 2015 Folie 15

11, 013035 (2009) (GGA), all relaxed with SOC position of surface state at Γ in

16 La(0001) and Lu(0001) Surface 11L dhcp La(0001) 12L hcp Lu(0001) O. Krupin et al. New J. Phys. 11, (2009) (GGA), all relaxed with SOC position of surface state at Γ in good agreement with experiment no detectable Rashba spin-orbit splitting for these surface states Z(La)=57; Z(Lu)=71; Z(Au)=79 June 11, 2015 Folie 16

17 La & Lu(0001) surface states: charge densities Lu: SS M point Lu: SS Γ La: SS Γ Bandstructure Lu(0001) Without SOC O. Krupin et al. New J. Phys. 11, (2009) Rashba-Spin-Orbit Splitting around point! (at M-point 0 due to symmetry) M June 11, 2015 Folie 17

18 La & Lu(0001) surface states: charge densities Bandstructure Lu(0001) With SOC O. Krupin et al. New J. Phys. 11, (2009) June 11, 2015 Folie 18

19 Magnetic Surface: Gd(0001) Surface state as for La(0001) or Lu(0001) But: Spin-splitted by exchange field (acts as Zeeman) June 11, 2015 Folie 19

20 Rashba-Effect at magnetic surfaces Surface electrons moving in a E field: 1 H = p 2 + α Rσ ( p E ) 2m H R Dispersion relation: 2 ( ) 2 ( ε k = ± α σ ) ( ) k R k E 2m Magnetic Surfaces: Dispersion relation: Exchange splitting 1 ± IM 2 2 (, ) 2 1 ( ε α ( k = ± ± ) M k IM RM k 2m 2 E) June 11, 2015 Folie 20

21 Gd(0001) Surface State Dispersion Exp. hν=36 ev T=80 K Theory Small Rashba-Spin-Orbit splitting at Γ-point (majority spin) Exp: O. Krupin, S. Gorovikov, J.E. Pietro, K. Döbrich, G. Kaindl, and K. Starke PRB 71, (2005) June 11, 2015 Folie 21

22 O on Gd(0001): Surface States Gd-(S) O Gd-(S) Gd-(S-1) June 11, 2015 Folie 22

23 Electrons at Surfaces E k -k June 11, 2015 Folie 23

24 Where does the SOC come from? Example: Rashba effect E k -k Spin-Orbit Strength (Z) Asymmetry of Ψ Orbital (s,p.d ) June 11, 2015 Folie 24

25 Where does the SOC come from? Example: Rashba effect Spin-Orbit Strength (Z) Atom Asymmetry of Ψ Orbital (s,p.d ) V (r) r 1 2mc 2 r V r L S Muffin-Tin Sphere June 11, 2015 Folie 25

26 Where does the SOC come from? Example: Rashba effect Atom V (r) r 1 2mc 2 r V r L S Rashba splitting (% of full value) Rashba splitting as function of radius of the sphere in which SOC is taken into account for Au(111) Sphere radius [a.u.] sphere radius (a.u.) 0.99 Muffin-Tin Sphere June 11, 2015 Folie 26

27 SOC+Inversion-asymmetry Example: Rashba effect Contribution of Individual layer to Rashba splitting Atom V (r) r 1 2mc 2 r V r L S 0 S-0 S-1 S-2 S-3 S-4 S-5 layer (S = surface) S S-1 S-2 S-3 S-4 S-5 Layer (S=Surface) 0% contribution of individual layer contribution of the idividual layers sphere radius (a.u.) 20% June 11, 2015 Folie 27 Rashba splitting (% of total) Rashba splitting (% of full value) 20 40% Au(111) 60 Rashba splitting as function of radius of the sphere in which SOC is taken into account for Au(111) 60% Au(111) 100

28 Rashba-Hamiltonian Electrons moving in a E field: 2D electron gas (in xy-plane) ψ ( ) ( k, r ) = 1 Ω ei k r ( ) > E(k) = 2 2m k 2 June 11, 2015 Folie 28

29 Mitglied der Helmholtz Gemeinschaft STS : Fe on Cu(111) Ill*tlsYOlsl.BlOsssl B 'S- -60 HI% \8~'17 <~ Distance (A) ' 1. (A) Constant otan( A x , current Aimage Fig. Fe adatomi (B) Solid line: average of three cross section Crommie etal, Science 218 ) onof the Fe adatom. the Cu(111) surface (V= 0.02 volt, I the center of the Fe adatom image in (A). Dashed line: na). Thefapparent height through the adatom is -0.9 A. The concentric rings surrounding the Fe adatomr 1 to the Cross section (the datawas fit only up to 18 A from th are standing waves due to the scattering of surface state electrons with of the adatom). = 1.0 sharp drop in di/dv at the surface state band June 11, 2015 Folie 29 shows the dil/dv spectrum measured with for imaging, the Fe adatoms did not

30 Rashba-Hamiltonian Electrons moving in a E field: ψ 2D electron gas (in xy-plane) 1 ik r ( k, r ) = e ( ) > ( ) Ω E Δ R Momentum k Δk x June 11, 2015 Folie 30

$Mitglied der Helmholtz Gemeinschaft STS : Fe on Cu(111) Ill*tlsYOlsl.BlOsssl.81.111. B Where 'S- is the second length scale due to spin orbit interaction -60 HI% \8~'17 <~ -- -40-20?$

31 Mitglied der Helmholtz Gemeinschaft STS : Fe on Cu(111) Ill*tlsYOlsl.BlOsssl B Where 'S- is the second length scale due to spin orbit interaction -60 HI% \8~'17 <~ ? 0 20 Distance (A) ' 1. (A) Constant otan( A x , current Aimage Fig. Fe adatomi (B) Solid line: average of three cross section Crommie etal, Science 218 ) onof the Fe adatom. the Cu(111) surface (V= 0.02 volt, I the center of the Fe adatom image in (A). Dashed line: na). Thefapparent height through the adatom is -0.9 A. The concentric rings surrounding the Fe adatomr 1 to the Cross section (the datawas fit only up to 18 A from th are standing waves due to the scattering of surface state electrons with of the adatom). = 1.0 sharp drop in di/dv at the surface state band June 11, 2015 Folie 31 shows the dil/dv spectrum measured with for imaging, the Fe adatoms did not

32 Standing waves at defect scattering E E B k k -k k -B Incoming states + reflected states June 11, 2015 Folie 32

33 Standing waves at defect scattering Fermi Surface Au(111) Scattering States E Γ M K E Γ June 11, 2015 Folie 33

34 Semi-metal Bi(110): Theory J.I. Pascual, G. Bihlmayer, Yu.M. Koroteev, H.-P. Rust, G. Ceballos, M. Hansmann, K. Horn, E.V. Chulkov, S. Blügel, P.M. Echenique, and Ph. Hofmann, PRL 93, (2004) June 11, 2015 Folie 34

35 Spin-dependent Interference: Bi(110) STM Scan Fourier TransF. J.I. Pascual, G. Bihlmayer, Yu.M. Koroteev, H.-P. Rust, G. Ceballos, M. Hansmann, K. Horn, E.V. Chulkov, S. Blügel, P.M. Echenique, and Ph. Hofmann, PRL 93, (2004) June 11, 2015 Folie 35

Scattering theory for magnetic adatom " Rashba

function G D G 0 = e iβ G ND " Scattering Amplitude

36 Scattering theory for magnetic adatom " Rashba Hamiltonian H = 1 2m * P2 + α R (σ x P y σ y P x ) " Dyson eq.: G = G 0 + G 0 tg 0 Surface " Surface Green function G D G 0 = e iβ G ND " Scattering Amplitude t = 1 ( m * e2iδ 1) e iβ G ND G D t = t 0! t 0! t t LDOS = 1 π ImG M = 1 π ImTr σ G June 11, 2015 Folie 36

37 Application: Adatoms on Au(111) surface LDOS(r) = 1 π Im ( (G G + G G )(t + t ) ) D D ND ND M z (r) = 1 π Im ( (G G G G )(t t ) ) D D ND ND STM M r (r) = 2 π Im ( G (t t )G ) D ND M z (r) 1 ( r k 1 cos(2k 1 r) + k 2 cos(2k 2 r) ) Surface June 11, 2015 Folie 37

38 Application: Adatoms on Au(111) surface LDOS(r) = 1 π Im ( (G G + G G )(t + t ) ) D D ND ND M z (r) = 1 π Im ( (G G G G )(t t ) ) D D ND ND STM M r (r) = 2 π Im ( G (t t )G ) D ND M r (r) 1 ( r k 1 sin(2k 1 r) k 2 sin(2k 2 r) ) Surface June 11, 2015 Folie 38

39 Mitglied der Helmholtz Gemeinschaft Three-dimensional Spin texture E=EF=410 mev Lounis, Bringer, Blügel, PRL108, (2012) June 11, 2015 Folie 39

40 Skyrmionic spin texture E=E F =410 mev high Result at E F Radius~70Å low June 11, 2015 Folie 40

41 Mitglied der Helmholtz Gemeinschaft Skyrmionic spin texture Bogdanov, Hubert, JMMM 195, 182 (1999) Rößler et al. Nature 442, 797 (2006) Mühlbauer et al. Science 323, 929 (2008) high Radius~70Å low June 11, 2015 Folie 41

Rößler et al. Nature 442, 797 (2006) Mühlbauer et al.

42 Mitglied der Helmholtz Gemeinschaft Complex Skyrmionic spin texture Bogdanov, Hubert, JMMM 195, 182 (1999) Rößler et al. Nature 442, 797 (2006) Mühlbauer et al. Science 323, 929 (2008) high Radius~70Å low June 11, 2015 Folie 42

43 Asymptotic behavior Distance from the adatom (Å) First nodes: wave length ~ 20 Å cos(2k F R) and λ = 2π = λ F 2k F 2 ~ 40 2 node Phase switch: ~ 60 Å cos(2k so R) λ and λ = so 2 ~ 120 Å 2 Å June 11, 2015 Folie 43

44 Interference of two skyrmionic waves M total = M k1 + M k Distance from the adatom (Å) M k1 k 1 ( cos(2k 1 r), sin(2k 1 r),0) k 1 = k + k so k 2 = k k so M k2 k 2 ( cos(2k 2 R),sin(2k 2 R),0) June 11, 2015 Folie 44

45 Scattering at two adatoms (Heisenberg) A. Fert & P. M. Levy, PRL 44, 1538 (1980). June 11, 2015 Folie 45

46 Rashba scattering at two adatoms Structure inversion asymmetric magnetism (SIA) E B k (Heisenberg) A. Fert & P. M. Levy, PRL 44, 1538 (1980). June 11, 2015 Folie 46

47 Dzyaloshinskii-Moriya Interaction (DMI): Structure inversion asymmetric magnetism (SIA) E B k (Heisenberg + Dzyaloshinskii-Moriya) June 11, 2015 Folie 47

48 Dzyaloshinskii-Moriya Interaction: Structure inversion asymmetric magnetism (SIA) E B k (Heisenberg) (Heisenberg A. Fert & P. M. + Dzyaloshinskii-Moriya) Levy, PRL 44, 1538 (1980). Example: Au(111) D y 12 (R) = ~ 2 apple sin(2kso R) sin(2k F R) m 2 + k so cos(2k so R)SI(2k F R) R 2R J. Bouaziz, M. d. Santos Dias, A. Ziane,, S. Blügel, S. Lounis June 11, 2015 Folie 48

Example: Au(111) D y ~ 2 h sin(2kf R) sin(2k so R) ij = m 2 R

49 Dzyaloshinskii-Moriya Interaction: Structure inversion asymmetric magnetism (SIA) D y k so k so E B k R x (Heisenberg) (Heisenberg A. Fert & P. M. + Dzyaloshinskii-Moriya) Levy, PRL 44, 1538 (1980). Example: Au(111) D y ~ 2 h sin(2kf R) sin(2k so R) ij = m 2 R 2R i + k so cos(2k so R) SI(2k F R) J. Bouaziz, M. d. Santos Dias, A. Ziane,, S. Blügel, S. Lounis June 11, 2015 Folie 49

50 DMI: Long range interaction DMI due to Rashba effect i R ij j Au(111) 1 1 Au(111) Reinert et al., PRB 63, (2001) Interaction D y ij [mev] R ij [Å] June 11, 2015 Folie 50

51 DMI: Long range interaction DMI due to Rashba effect 1 1 Au(111) Interaction D y ij [mev] R ij [Å] Reinert et al., PRB 63, (2001) D y ij = ~ 2 m 2 R h sin(2kf R) sin(2k so R) 2R June 11, 2015 Folie 51 Bouaziz, Blügel, Lounis i + k so cos(2k so R) SI(2k F R)

52 Real Fermi surface bcc W (Z=74) SOCSIS, Bernd Zimmermann Spetses, June June 11, 2015 Folie 52

53 Exchange and DMI in Pd/Fe/Ir(111) Top view of Fe Fe J 12 = 16.4 mev Ir Pd D 12 = -1.1 mev Bauer, Kiselev, Crum, Schweflinghaus, Bouhassoune, Bouaziz, Lounis, Blügel June 11, 2015 Folie 53

54 Exchange and DMI in 2Pd/Fe/Ir(111) Top view of Fe Fe Pd J 12 = 12.7 mev Ir D 12 = -1.4meV Bauer, Kiselev, Crum, Schweflinghaus, Bouhassoune, Bouaziz, Lounis, Blügel June 11, 2015 Folie 54

55 Dzyaloshinskii-Moriya Interaction o Well-known interaction in many TM oxides, chalcogenides.. Bulk inversion asymmetric (BIA) systems (non-centrosymmetric crystals): e.g. α-fe 2 O 3, MnCO 3, CrF 3. Antiferromagnets show weak ferromagnetism Anisotropic Superexchange Interaction + SOC o Metals : most metals bcc, fcc, hcp.. exceptions: B20 compounds, Spin glasses A. Fert & P. M. Levy, PRL 44, 1538 (1980). June 11, 2015 Folie 55

56 Bulk inversion asymmetric magnets E(c) FM/AFM L> R> c Dzyaloshinskii-Moriya Interaction June 11, 2015 Folie 56

57 Spin textures at surfaces skyrmions magnetism spin-orbit coupling inversion asymmetry chiral spin spirals Dzyaloshinskii-Moriya interaction Nature 465, 901 (2010) Nat. Phys. 7, 713 (2011) chiral domain walls magnon dispersion Nature 447, 190 (2007) PRL 88, (2002) PRB 78, (2008) PRL 102, (2009) PRL 104, (2010) June 11, 2015 Folie 57

Film Substrate with large SOC Dzyaloshinskii-Moriya Interaction

58 Spin-Orbit Interaction + Structure Inversion Asymmetry Break of inversion symmetry P(z) P(-z) Magn. Film Substrate with large SOC Dzyaloshinskii-Moriya Interaction But how large?? Which direction (sign)?? What about the metallic systems?? June 11, 2015 Folie 58

Skyrmions for Spintronics Chiral magnetism in thin films, but not too thin (min 3 layers) Try find small but not too small skyrmions 3-5 nm Above room temperature and zero magnetic field Fit to the

59 Skyrmions for Spintronics Chiral magnetism in thin films, but not too thin (min 3 layers) Try find small but not too small skyrmions 3-5 nm Above room temperature and zero magnetic field Fit to the field of spintronics: injection, transport, detection, manipulation at reasonable fields and currents Metallic magnetism Albert Fert, Vincent Cross and João Sampaio, Nature Nanotechnology 8, 152 (2013) June 11, 2015 Folie 59

Role of Dzyaloshinskii-Moriya Interaction in 1 + 2 Dimensions Micromagnetic energy functional: Z E(m) = A rm 2 + D (rm m)+m K m B m ê z dr 2 R 2 Exchange Dzyaloshinskii-M Anisotropy Ext.

60 Role of Dzyaloshinskii-Moriya Interaction in Dimensions Micromagnetic energy functional: Z E(m) = A rm 2 + D (rm m)+m K m B m ê z dr 2 R 2 Exchange Dzyaloshinskii-M Anisotropy Ext. Field Stretching transformation: m(r)! m (r) =m( r) 2D: Skyrmion Z apple A E( )= 2 rm 2 + D (rm m)+m K m B m ê 2 z dr 2 R Z 2 E( )= A rm 2 + D (rm m)+ 2 m K m 2 B m ê z dr 2 SOCSIS, R 2 Spetses, June June 11, 2015 Folie 60

61 Role of Dzyaloshinskii-Moriya Interaction in Dimensions Micromagnetic energy functional: Z E(m) = A rm 2 + D (rm m)+m K m B m ê z dr 2 R 2 Exchange Dzyaloshinskii-M Anisotropy Ext. Field Stretching transformation: m(r)! m (r) =m( r) 1D: Domain Wall Z apple A E( )= rm 2 + D (rm m)+ m K m B m ê z dr R Z E( )= A rm 2 + D (rm m)+ 2 m K m 2 B m ê z dr 2 SOCSIS, R 2 Spetses, June June 11, 2015 Folie 61

Continuum theory of chiral magnetic skyrmions Theoretical prediction: T.

Phys. JETP 68, 101 (1989). A.N. Bogdanov, A.

z dr 2 R 2 Exchange Dzyaloshinskii-M Anisotropy Ext.

62 Continuum theory of chiral magnetic skyrmions Theoretical prediction: T. Skyrme: Proc. Roy. Soc. A 260, 127 (1961) A.N. Bogdanov & D. A.Yablonski: Sov. Phys. JETP 68, 101 (1989). A.N. Bogdanov, A. Hubert: JMMM 138, 255 (1994) U. Rössler, A.N. Bogdanov, C. Pfleiderer, Nature 442, 797 (2006). Micromagnetic energy functional: Z E(m) = A rm 2 + D (rm m)+m K m B m ê z dr 2 R 2 Exchange Dzyaloshinskii-M Anisotropy Ext. Field min E{m} of continuous vector field m with unit sphere S 2 = {m 2 R 3 : m =1} è Topological concepts Smooth Mapping R 2! S 2 June 11, 2015 Folie 62

63 Conclusion Ø Many things are unknown and ready to discover Ø E.g. Manipulation and control of topological solitons by external fields Ø Topological, anomalous and thermal Hall effects in static skyrmion lattices Ø Spin orbit torque and damping in chiral magnets Ø Large Skyrmions we can describe in Berry phase physics and materials parameters for micromagnetic models Ø Small skyrmions we may treat completely from abinitio Ø Medium scale skyrmions not-adiabatic but too large of DFT. June 11, 2015 Folie 63

Continuum theory of chiral magnetic skyrmions E(m) = Z Mathematical analysis predicts: R 2 A rm 2 + D (rm m)+m K m B m ê z dr 2 Ø Dzyaloshinskii-Moriya Interaction (DMI) à new valley in topologically

64 Continuum theory of chiral magnetic skyrmions E(m) = Z Mathematical analysis predicts: R 2 A rm 2 + D (rm m)+m K m B m ê z dr 2 Ø Dzyaloshinskii-Moriya Interaction (DMI) à new valley in topologically non-trivial sector of energy landscape Ø Sectors are separated by an energy barrier (topological protection) Ø Theorem: For D 0, for each (D, A) there is a B-field, B 0.8 D 2 /A, at which an isolated chiral skyrmions stabilizes Ø Diameter: R S A/D Ø Critical T : T c D 2 /A Rößler, Leonov, and Bogdanov, J. Phys.: Conf. Series 303, (2011) Melcher Proc. R. Soc. A 470, 0394 (2014) June 11, 2015 Folie 64

65 Thanks! June 11, 2015 Folie 65

Skyrmions à la carte

Skyrmions à la carte This project has received funding from the European Union's Horizon 2020 research and innovation programme FET under grant agreement No 665095 Bertrand Dupé Institute of Theoretical Physics and Astrophysics,