COMPUTER SIMULATION STUDIES OF SKYRMIONIC TEXTURES IN HELIMAGNETIC NANOSTRUCTURES Marijan Beg*, Hans Fangohr Faculty of Engineering and the Environment, University of Southampton , Southampton, United Kingdom * email : mb4e10@soton.ac.uk
OVERVIEW 1. Initial states (analytic model) 2. Equlibrium states in a nano disk 3. Ground state phase diagram 4. Robustness 5. Hysteretic behaviour (DMI anisotropy) 6. Reversal mechanism 7. Summary 2
SKYRMIONIC TEXTURES IN CONFINED HELIMAGNETIC NANOSTRUCTURES 3
MOTIVATION Yu et. al., Nature 465, 901-4 (2011) a b Magnetic skyrmions possess interesting • properties promising for the development of future data-storage and information processing devices. One of the main problems, obstructing the • Schematic Lorentz TEM development of skyrmion-based devices using helimagnetic materials, is their magnetic and thermal stability . In infinitely large thin film or bulk B20 • helimagnetic samples, skyrmion phase is stabilised in presence of an external field . The motivation for this work is to explore • the skyrmionic textures in finite size B20 helimagnetic nanostructures . Thin film phase diagram 4
SYSTEM UNDER STUDY Sample geometry is 10 nm thin film • disk with varying diameter Cubic B20 helimagnetic FeGe • M S = 3.84 x 10 5 Am -1 • A = 8.78 x 10 -12 Jm -1 • D = 1.58 x 10 -3 Jm -2 . • Helical period 4 π A / D = 70 nm • Sample geometry and Finite elements mesh maximum • sample skyrmion ground neighbouring node spacing smaller state than 3 nm. Finite size effects, stability, hysteretic behaviour, External field applied uniformly and • and reversal mechanism of skyrmionic textures perpendicular to the film in + z in nanostructures, direction . Marijan Beg, Dmitri Chernyshenko, Marc- Antonio Bisotti, Weiwei Wang, Maximilian zero temperature micromagnetic • Albert, Robert L. Stamps, Hans Fangohr, model arxiv:1312.7665 http://arxiv.org/abs/1312.7665 (2014) 5
MICROMAGNETIC MODEL - HAMILTONIAN AND DYNAMICS - FINMAG • Finite elements based simulator. • successor of Nmag, http://nmag.soton.ac.uk • HAMILTONIAN: • Z ⇥ A ( r m ) 2 + D m · ( r ⇥ m ) � µ 0 m · H + w d d 3 r ⇤ W = No anisotropy (isotropic helimagnetic B20 material). • Full 3D model - no assumption about translational invariance of • magnetisation in out-of-film direction which radically changes the skyrmion energetics [Rybakov et al., PRB 87 , 094424 (2013)]. 6
MAGNETISATION DYNAMICS Magnetisation dynamics is governed by the LLG EQUATION . • damping ∂ m ∂ t = γ ∗ m × H e ff + α m × ∂ m ∂ t precession H e ff H e ff H e ff = + = + H e ff H e ff H e ff 7
ENERGY LANDSCAPE initial state relaxed state 8
SIMULATION METHOD d and H are varied in steps: • ∆ d = 2 nm µ 0 ∆ H = 2 mT Gilbert damping • α = 1 System is relaxed from multiple initial • states by computing the magnetisation’s time development The relaxed state with the lowest energy is chosen as the ground state for • the phase space point ( d , H ). The scalar parameter S a is computed as: • � ◆� ✓ ∂ m ∂ x × ∂ m S a = 1 � d 3 r � � � m · � � 4 π ∂ x Phase diagram: S a = f ( d, H ) • 9
INITIAL CONFIGURATIONS 10
DEFINING SKYRMIONIC INITIAL STATES – ANALYTIC MODEL The chiral skyrmion profile is approximated in cylindrical coordinates: • Schematic Lorentz TEM a b y =0 1 m z Yu et. al., Nature 465, 901-4 (2011) m y m r = 0 0 m θ = sin( kr ) m x m z = − cos( kr ) x -1 � w 1 The effective field due to • ~ H e ff = − symmetric exchange and DMI � ~ µ 0 M s m 2 ( no external field, isotropic A r 2 ~ ~ ⇥ ⇤ H e ff = m � D ( r ⇥ ~ m ) µ 0 M S B20 material ): 11
ANALYTIC MODEL - ZERO TORQUE EQUATION - In equilibrium state, the torque is zero: • m × H e ff = 0 Computing the zero radial torque at r=R for assumed chiral skyrmion • profile results in condition: kA sin 2 ( kR ) − sin(2 kR ) g ( kR ) ≡ − D + 1 = 0 2 kR This equation has solution if : • P = D kA > 2 D > 2 3 kA ⇒ 3 Two scalar parameters are computed: • ✓ ∂ m ◆ Z ∂ x × ∂ m S = d x d y m · ∂ y Z � ◆� ✓ ∂ m ∂ x × ∂ m � � S a = � d x d y � m · � � ∂ y 12
m r = 0 m θ = sin( kr ) SOLUTION A m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 13
m r = 0 m θ = sin( kr ) SOLUTION B m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 14
m r = 0 m θ = sin( kr ) SOLUTION C m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 15
m r = 0 m θ = sin( kr ) SOLUTION D m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 16
m r = 0 m θ = sin( kr ) SOLUTION E m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 17
m r = 0 m θ = sin( kr ) SOLUTION F m z = − cos( kr ) 1.5 1.0 0.5 A B C D E F 0.0 g ( kR ) -0.5 P =2.0 -1.0 -1.5 -2.0 1.5 0.0 0.5 1.0 2.0 2.5 3.0 3.5 kR ( π ) 3.5 F z H =0 3.0 E y =0 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B A 1.0 0.5 D C 0.0 1 S -0.5 A E F B -1.0 mz ( x ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 0 x -1 18
ANALYTIC MODEL RESULTS 1.5 c C a A B P =0 P =2/3 1.0 P =0.5 y =0 0.5 P =1.0 A B C D E F 0.0 g ( kR ) P =1.5 1 -0.5 P =2.0 mz ( x ) 0 -1.0 P =2.5 x -1.5 -1 P = D -2.0 D E F kA 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) 3.5 b F z y =0 H =0 3.0 E 2.5 R D S ( kR ), S a ( kR ) C 2.0 S a 2R 1.5 B 1 A 1.0 0.5 mz ( x ) 0 D C 0.0 S -0.5 A E F B x -1.0 -1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 kR ( π ) Du, H., Ning, W., Tian, M., & Zhang, Y. (2013), Physical Review B, 87, 014401. Du, H., Ning, W., Tian, M., & Zhang, Y. (2013). EPL, 101(3), 37001. 19
SIMULATION RESULTS 20
EQUILIBRIUM CONFIGURATIONS 21
GROUND STATE PHASE DIAGRAM We select the state with the lowest energy • Two different ground states. • FeGe thin film disk phase diagram 22
d =80 nm INCOMPLETE μ 0 H =0.2 T SKYRMION (ISK) 1 mz ( x ) 0 x -1 - d /2 0 d /2 No complete spin rotation. • | S| < 1 • In literature also called “quasi- • ferromagnetic” or “vortex” state. 23 23
d =160 nm μ 0 H =0.3 T ISOLATED SKYRMION (SK) 1 mz ( x ) 0 x -1 - d /2 0 d /2 Complete spin rotation • present. Significant tilt of magnetisation • at the edge which reduces | S |. 24
ENERGIES OF METASTABLE STATES 1.2 iSk 1.0 3Sk z H 0.8 d /2 10 nm 2Sk μ 0 H (T) 0.6 d Sk 0.4 oSk 0.2 H3 H H2 0.0 40 60 80 100 120 140 160 180 d (nm) 25
ROBUSTNESS 1.0 0.5 z H =0 10 nm d /2 m z ( x ) 0.0 x - d /2 d /2 0 -0.5 d =120 nm d =140 nm d =160 nm d =100 nm d =80 nm d =180 nm -1.0 -90 -60 -30 0 30 60 90 Skyrmionic textures able to adapt • x (nm) 220 their size to accommodate the z m z z s -cos( kx ) 200 H size of a hosting nanostructure. s =2 π / k (nm) 180 d /2 iSk 10 nm x This provides the robustness of - d /2 0 d /2 • 160 technology built on skyrmions. d 140 120 Sk iSk and Sk have different core • 100 orientation . 80 40 60 80 100 120 140 160 180 d (nm) Du, H., Ning, W., Tian, M., & Zhang, Y. (2013), Physical Review B, 87, 014401. 26
POSSIBLE STABILISING MECHANISM Rybakov et al., PRB 87 , 094424 (2013) no demagnetisation 2D 27
POSSIBLE STABILISING MECHANISM 28
Recommend
More recommend
