Skyrmion à la carte. Bertrand Dupé. Skyrmion à la carte Bertrand Dupé. Institute of Physics, Johannes Gutenberg University of Mainz, Germany

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1 Skyrmion à la carte Bertrand Dupé Institute of Physics, Johannes Gutenberg University of Mainz, Germany 1

2 Acknowledgement Charles Paillard Markus Hoffmann Stephan von Malottki Stefan Heinze Sebastian Meyer Pavel Bessarab Gustav Bihlmayer Jairo Sinova Joo-Von Kim Stefan Blügel Melanie Dupé Ulrike Ritzmann Marie Böttcher 2

3 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 3

4 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 4

5 Skyrmions in bulk magnetic materials Micromagnetic model prediction Existence of magnetic skyrmions on a micrometer length scale A. N. Bogdanov & D. A.Yablonskii, Sov. Phys. JETP 68, 101 (1989). Experimental discovery Skyrmion race-track S. Mühlbauer et al Science 323, 915 (2009). X. Z. Yu et al Nature 465, 901 (2010). A. Neubauer et al PRL 102, (2009). M. Lee et al PRL 102, (2009). S. Parkin et al Science 320, 190 (2008). F. Jonietz et al Science 330, 1648 (2010). A. Fert et al Nature Nanotech. 8, 152 (2013). C. Moreau-Luchaire et al Nature Nanotech. 11, 444 (2016). W. Jiang et al Science 349, 283 (2015). 5

6 Skyrmions and topological charge Skyrmion number (topological charge) of a vector field n(x,y): Ferromagnet (S=0): topologically trivial state Skyrmion (S=+1) Antiskyrmion (S= 1) 6

7 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 7

8 Versatility of magnetism in Fe ultra-thin films 200 Fe/Ir(111): nano-skyrmion lattice FM 0 AFM EAFM-FM (mev/fe) Tc Ru Rh Pd -200 Re Os Ir Pt B. Hardrat et al PRB 79, (2009). Fe/Re(0001) Néel state S. Ouazi et al Sur. sci. 630, 280 (2014). S. Ouazi et al PRL 112, (2014). S. Heinze et al Nat. Phys. 7, 713 (2011). Pd/Fe/Ir(111) spin spiral ground state N. Romming et al Science 341, 636 (2013). B. Dupé et al Nature Comm. 5, 4030 (2014). 8

9 First-principles based spin Hamiltonian Density-functional theory (DFT) using the FLEUR code: energy of non-collinear magnetic structures energies of spiral spin-density waves Spin spirals Γ 1 st BZ M K AFM Néel state Energy dispersion!!!(!) =!!!,!!!!!!!!!!! FZ Jülich FM state 9

10 Example of DMI calculation Density-functional theory (DFT) using the FLEUR code: energy of non-collinear magnetic structures energies of spiral spin-density waves with and without spin-orbit coupling Spin spirals DM interaction!!" =!!". (!!!! )!!!!! FZ Jülich 10

11 First-principles based Hamiltonian Spin Hamiltonian solved by Monte-Carlo & spin dynamics!! =!!".!!.!! Magnetic exchange energy!!!!!". (!!!! ) Dzyaloshinskii-Moriya energy!!!!!.! +!!.!! 2 Zeeman energy Magnetocrystalline anisotropy energy interaction constants calculated from DFT No dipole-dipole interaction included 11

12 Stability diagram of Pd/Fe/Ir(111) B Monte-Carlo Simulation B. Dupé et al Nature Comm. 5, 4030 (2014). N. Romming et al Science (2013). 12

13 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 13

14 Frustration of exchange interaction: J eff Spin spiral ground state hcp fcc J 1 (mev) J eff (mev) A=2.0±0.4 pj.m -1 from N. Romming et al. PRL 114, (2015) J eff : approximation of spin stiffness only close to q=0 14

15 Frustration of exchange interaction: J eff PRL 114, (2015) Pd Fe Ir(111) B. Dupé et al Nature Comm. 7, (2016).

16 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 16

17 Calculation of the energy barrier: GNEB Geodesic nudged elastic band: GNEB Collapse of a skyrmion: first neighbor approximarion P. Bessarab et al Computer Physics Communications 196, 335 (2015). S. Rohart et al Phys. Rev. B 93, (2016).

18 Stability diagram of Pd/Fe/Ir(111) T= 0K J eff has very little effects on the stability diagram S. von Malottki et al submitted Arxiv

19 Radius depence with magnetic field J eff has very little effects on the stability diagram and very little effects on skyrmion properties with magnetic field at low temperature S. von Malottki et al submitted Arxiv

20 Energy barrier with frustrated exchange Jeff can not discribe excited states for a spin spiral ground state stabilized by magnetic exchange S. von Malottki et al submitted Arxiv

21 Frustrated exchange stabilizes antiskyrmions Frustration of exchange interaction can stabilize higher order topologically protected magnetic states S. von Malottki et al submitted Arxiv

22 Outline I. Introduction II. Methods q Spin spiral calculations applied on Pd/Fe/Ir(111) q DMI calculations applied on Pd/Fe/Ir(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics III. Magnetic exchange frustration and its consequences q Energy barriers for the collapse of skyrmion and antiskyrmion q Temperature dependence of skyrmion and antiskyrmion densities IV. Conclusion 22

23 Previous works Creation of skyrmion/antiskyrmion density pairs Pd/Fe/Ir(111) (c) (a) 7 χ/ T =0 6 χ/ B =0 fluctuation freezing FP B (T) 5 4 FP SkL PM 3 2 SkL 1 SS SkL SS FD T (K) (b) M. Lau et al PRB 39, 7212 (1989). L. Rosza et al PRB 93, (2016). Integrated skyrmion density is no order parameter MnSi 0.5 (c) s c 0.4 h B/J 0.3 conical 0.2 skyrmion lattice 0.1 helical T/J S. Bürhandt et al PRB 88, (2013). B. Dupé et al New Journ. Phys 18, (2016). 23

24 Temperature dependence of order parameters Finite temperature E - E SS (mev / spin) Stability diagram T=0K Spin spiral Skyrmion lattice (SkX) Ferromagnetic (FM) state Magnetic field (T) E (mev) M (arb. units) B (T) 0 T 1 T 2 T 3 T a c Temperature (K) ordered 5 T c b disordered d Temperature (K) C (arb. units) χ M (arb.units) Q (arb. units) Temperature (K) e 24

25 Phase diagram of Pd/Fe/Ir(111) Parallel tempering Q± (arb. units) Magnetic field (T) a Temperature (K) FM SkX Spin spiral = 16 = 14 T B = 1 T Q±,1 Q + Q = 12 Intermediate region = 10 Temperature (K) PM Temperature (K) = 8 = 6 = 4 = 2 T Q±, T c T Q+,1 T Q+,2 T Q-,1 T Q-,2 M. Böttcher et al submitted arxiv b B s c χ of Q± (arb. u.) (c) 8 B (T) FP FP SkL SkL Metropolis 2 χ/ T 2 =0 2 χ/ B 2 =0 fluctuation freezing FD 1 SS SkL SS T (K) PM L. Rosza et al PRB 93, (2016). 25

26 Skyrmion and anti-skyrmion density Q± (arb. units) Magnetic field (T) a Temperature (K) FM SkX Spin spiral = 16 = 14 T B = 1 T Q±,1 Q + Q = 12 Intermediate region = 10 Temperature (K) Temperature (K) = 8 = 6 = 4 = 2 T Q±,2 PM b T c T Q+,1 T Q+,2 T Q-,1 T Q-,2 B s c χ of Q± (arb. u.) I (arb. units) Magnetic field (T) T = T Q±,1 Q + = 10 Q- = 10 Q + = 17 Q- = 17 Q + = 32 T Q- = 32 Q + = 56 Q- = 56 Q- Q Magnetic field (T) Q + = 100 Q- = 100 Temperatur (K) Q + = 178 Q- = 178 a b Both skyrmion and antiskyrmion densities increase with temperature M. Böttcher et al submitted arxiv

27 Visualization of Sk and ask density Magnetic field (T) Q + = 10 Q- = 10 Q + = 17 Q- = 17 Q + = 32 T Q- = 32 Q + = 56 Q- = Q + = 100 Q- = 100 Temperatur (K) Q + = 178 Q- = 178 b M. Böttcher et al submitted arxiv

28 Conclusion 28

29 Parallel tempering Monte Carlo Parallel tempering!!!!" Parallel tempering allows: To overcome local minima with temperature To calculate thermodynamical quantities over a large volume of the phase space 29