Skyrmions à la carte

Transcription

1 This project has received funding from the European Union's Horizon 2020 research and innovation programme FET under grant agreement No Bertrand Dupé Institute of Theoretical Physics and Astrophysics, University of Kiel, Germany Skyrmions à la carte 1

2 This project has received funding from the European Union's Horizon 2020 research and innovation programme FET under grant agreement No Bertrand Dupé Institute of Theoretical Physics and Astrophysics, University of Kiel, Germany Skyrmions à la carte 2

3 Acknowledgement Marie Böttcher Fabian Otte Charles Paillard Markus Hoffmann Stephan von Malottki Stefan Heinze Kirsten von Bergmann Christian Hannecken André Kubetzka Niklas Romming Roland Wiesendanger Gustav Bihlmayer Bernd Zimmermann Yuriy Mokrousov Stefan Blügel 3

4 Outline I. Method: First principles calculations q Spin spiral calculations applied on /Fe/(111) q DMI calculations applied on /Fe/(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics II. Fe based multilayers: exchange tuning q Advantage of exchange tuning q Tuning the exchange in ultra-thin films q Prediction of skyrmions in multilayers based on effective exchange q Effect of DMI and exchange frustration on an isolated skyrmion q Determination of T c in the case of spin spiral III. Co based system: the importance of the magnetic moments q Back to basics: Fe and Co freestanding monolayers q Application to Co/Pt(111) 4

5 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 Γ 1 st BZ M K AFM Néel state Spin spirals Energy dispersion!!!(!) =!!!,!!!!!!!!!!! FZ Jülich FM state 5

6 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 6

7 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 with and without spin-orbit coupling Spin spirals all interaction constants calculated from DFT FZ Jülich Spin Hamiltonian solved by Monte-Carlo & spin dynamics!! =!!".!!.!!!!!!!". (!!!! )!!! Exchange interaction Dzyaloshinskii-Moriya!!.! +!!.!! 2 Zeeman energy Magnetocrystalline anisotropy energy 7

8 Outline I. Method: First principles calculations q Spin spiral calculations applied on /Fe/(111) q DMI calculations applied on /Fe/(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics II. Fe based multilayers: exchange tuning q Advantage of exchange tuning q Tuning the exchange in ultra-thin films q Prediction of skyrmions in multilayers based on effective exchange q Effect of DMI and exchange frustration on an isolated skyrmion q Determination of T c in the case of spin spiral III. Co based system: the importance of the magnetic moments q Back to basics: Fe and Co freestanding monolayers q Application to Co/Pt(111) 8

9 Versatility of magnetism in Fe ultra-thin films Fe/(111): nano-skyrmion lattice /Fe/(111) spin spiral ground state S. Heinze et al Nat. Phys. 7, 713 (2011) N. Romming et al Science 341, 636 (2013) Fe/Re(0001) Néel state S. Ouazi et al Sur. sci. 630, 280 (2014) S. Ouazi et al PRL 112, (2014) E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM B. Hardrat et al PRB 79, (2009) 9

10 /Fe/(111): exchange stabilized skyrmions 120 Γ Κ Isolated skyrmion 100 fcc hcp 80 Energy (mev) /Fe/(111) The skyrmions are - Stable in fcc - Metastable in hcp Spin spiral vector q (2π/a) Non-collinear spin structures are stabilized by exchange in (fcc)/fe/(111) 10

11 Metastable states in (fcc)/fe/(111) STM tip Q=-2 Similar contrast of all spin structures for outof-plane magnetized tip Q=-1 Q=1 11

12 Metastable states in (fcc)/fe/(111) (a) STM tip B. Dupé et al New Journ. Phys. 18, (2016) (b) (c) In-phase rotation of the SP-STM contrast Anti-phase rotation of the SP-STM contrast 12

13 Non-Collinear magneto-resitance (NCMR) Change of magnetization gradient at the center of a skyrmion Change of the resistance at the center of a skyrmion a Skyrmion centre b Skyrmion centre di/d U (a.u.) Δd = 0.40 nm LDOS (a.u.) Δd = 0.27 nm Experiment, 2.5 T FM Theory, 2.5 T Sample bias (V) E E F (ev) c d 0.15 Exp. (a) TB ( b) Exp. TB ΔE (ev) FM B = 2.5 T B = 1.0 T Distance (nm) Distance (nm) N. Romming et al PRL 114, (2015) C. Hannecken et al Nature Nano. 10, 1039 (2015) D. Crum et al Nature Comm. 6, 8541 (2015) 13

14 Further tuning of the exchange: /Fe/(111) E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM B. Hardrat et al PRB 79, (2009) Very good agreement with experiments - λ fcc 1.1nm and λ hcp 1.5nm A. Romming et al in preparation Exchange energy (mev) J (mev/fe) λ (nm) shell number (fcc)/fe/(111) (hcp)/fe/(111) (hcp)/fe/(111) (fcc)/fe/(111) - No change with magnetic field: use the exchange bias to create isolated skyrmions A. Nandy et al PRL 116, (2016) G. Chen et al APL 106, (2015) q vector (2π/a) 14

15 Tuning magnetic states in ultra-thin films Ferromagnetic ground state Fe/(111): D= -0.1 mev Co/Pt(111): D=-1.8 mev 15

16 Tuning magnetic states in ultra-thin films Ferromagnetic ground state Fe/(111): D= -0.1 mev Co/Pt(111): D=-1.8 mev Spin spiral ground state single skyrmions in magnetic field /Fe/(111): D 1.2 mev 16

17 Tuning magnetic states in ultra-thin films Ferromagnetic ground state Fe/(111): D= -0.1 mev Co/Pt(111): D=-1.8 mev Spin spiral ground state single skyrmions in magnetic field /Fe/(111): D 1.2 mev Skyrmion lattice ground state stabilized by 4-spin term Fe/(111): D=1.7 mev 17

18 Tuning magnetic states in ultra-thin films Ferromagnetic ground state Fe/(111): D= -0.1 mev Co/Pt(111): D=-1.8 mev Spin spiral ground state single skyrmions in magnetic field Skyrmion lattice ground state stabilized by 4-spin term Fe/(111): D=1.7 mev Spin spiral ground state no skyrmions B. Dupé et al, Nature Communications 5, 4030 (2014) N. Romming et al in preparation 18

19 Skyrmions à la carte J eff : parabolic approximation of E(q) close to FM state (q=0)! =!!"" 3!! 2!!! Fe/(111) Co/Pt(111) J 1 (mev) J eff (mev)

20 Skyrmions à la carte Spin spiral ground state hcp fcc J 1 (mev) (mev) J eff (pj.m -1 ) 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 20

21 Skyrmions à la carte Spin spiral ground state hcp fcc J 1 (mev) (mev) J eff (pj.m -1 ) 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 21

22 Skyrmions à la carte Fe (111) B. Dupé et al Nature Comm. 7, (2016) 22

23 Skyrmions à la carte Fe (111) B. Dupé et al Nature Comm. 7, (2016) 23

24 Skyrmions à la carte E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM Fe (111) Tuning of the exchange Tuning of the DM Fe Fe B. Dupé et al Nature Comm. 7, (2016) 24

25 Skyrmions à la carte E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM Tuning of the exchange Tuning of the DM ( x 1-x ) Fe Fe B. Dupé et al Nature Comm. 7, (2016) 25

26 Skyrmions à la carte E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM Tuning of the exchange Tuning of the DM ( x 1-x ) Fe Fe B. Dupé et al Nature Comm. 7, (2016) 26

27 DM interaction in [//2Fe/2] n DM model (fit):!!" =!!". (!!!! )!!!!! [/ x 1-x /2Fe/2] n : D 1.3 mev for all x! /Fe/(111): D=1.2 mev Fe/(111): D=1.7 mev Fe/(111): D=-0.1 mev B. Dupé et al Nature Comm. 7, (2016) Energy (mev/f.u.) E DM (mev/f.u.) Left rotating spiral Right rotating spiral 27

28 MC and SD simulations of double magnetic layers!! =!!".!!.!!!!!!!"!.!!.!!!!!!!!". (!!!! )!!!!!.! 2 +!!.!! J ij J ij J ij B. Dupé et al Nature Comm. 7, (2016) 28

29 MC and SD simulations of double magnetic layers J ij J ij Fast rotating spin spiral ground state 29

30 MC and SD simulations of double magnetic layers J ij Ferromagnetic ground state J ij J ij 30

31 MC and SD simulations of double magnetic layers J ij Slow rotating spin spiral ground state J ij DM=1.3 mev J ij 31

32 Phase diagram of [//2Fe/2] n λ = 2.7 nm Profile of isolated skyrmion at B= 3.5 T r = 1.2 nm at B=3.5T B. Dupé et al Nature Comm. 7, (2016) 32

33 Energy contributions of an isolated skyrmion E (mev per Fe) B. Dupé et al Nature Comm. 7, (2016) Isolated skyrmion gains 32meV as compared to FM background at B=3.5T 33

34 Frustration between the bilayers J ij J ij DM=1.3 mev J ij 34

35 Layer dependent energy dispersion 80 Γ both Fe layers Κ J ij 0 J ij 0 J ij 0 60 Exchange energy (mev) J ij =0 J ij 0 J ij =0 0 J ij =0-20 J ij = q vector (2π/a) J ij 0 35

36 Effect of frustration of the exchange Energy (mev per unit cell) SS: λ=2.7 nm SkX: r = 1.7 nm at B = 2 T θ (in radian) Profile of an isolated skyrmion at B= 3.5 T distance from the center (nm) r = 1.6 nm at B=3.5T -0.8 SS SkX FM B (T) 36

37 Energy density with different intra-layer coupling AFM ground state FM ground state Exchange energy (mev per Fe) 2 (a) (b) θ (in radian) y (nm) x (nm) (c) Energy (mev per Fe) x (nm) (d) Total energy Isolated skyrmion gains 6.5 mev as compared with the FM background at B=3.5T distance from the center (nm) distance from the center (nm) 37

38 DM and Exchange frustration in the bilayers DM 1 =-0.2 mev J ij DM 2 =1.5 mev J ij J ij FM ground state AFM ground state 38

39 Energy density with different intra-layer coupling Isolated skyrmion gains 23.1 mev as compared with the FM background at B=3.5T (a) (b) θ (in radian) Profile of an isolated skyrmion at B= 3.5 T distance from the center (nm) r = 1.8 nm at B=3.5T x (nm) (c) (e) //2Fe/ y (nm) DM energy (mev/fe) (d) Total energy (mev/fe) Energy (mev/fe) (f) distance from the center (nm) Total energy density (mev/fe)

40 MC study including inter-bilayer exchange coupling J DM=1.3 mev J J J z B. Dupé et al Nature Comm. 7, (2016) 40

41 overcoming local minima in frustrated spin systems Parallel tempering J z!!!!!!!!!" Parallel tempering allows: To overcome local minima with temperature To calculate thermodynamical quantities over a large volume of the phase space B. Dupé et al Nature Comm. 7, (2016) 41

42 Toward room temperature skyrmions J z 460 temperatures of 20x20x6x2 spins starting from the spin spiral ground state at B=0T after temperature swaps with T auto = MC steps //2Fe/2 [//2Fe/2] n [2/2Fe/2] n χ (arb. units) J z = 0 mev J z = 1.75 mev J z = 5 mev J z = 10 mev J z = 15 mev T c (K) J z (mev) Increase of T c with J z Requires further optimization of the / and spacers Temperature (K) 42

43 skyrmions!!! Do it yourself. E AFM-FM (mev/fe) Tc Ru Re Os Pt FM AFM Tuning Tuning Fe Fe B. Dupé et al Nature Comm. 7, (2016) 43

44 Outline I. Method: First principles calculations q Spin spiral calculations applied on /Fe/(111) q DMI calculations applied on /Fe/(111) q Effective extended Heisenberg Hamiltonian for MC and spin dynamics II. Fe based multilayers: exchange tuning q Advantage of exchange tuning q Tuning the exchange in ultra-thin films q Prediction of skyrmions in multilayers based on effective exchange q Effect of DMI and exchange frustration on an isolated skyrmion q Determination of T c in the case of spin spiral III. Co based system: the importance of the magnetic moments q Back to basics: Fe and Co freestanding monolayers q Application to Co/Pt(111) 44

45 Conclusions 45