Advanced micromagnetics and atomistic simulations of magnets Richard - PowerPoint PPT Presentation

Advanced micromagnetics and atomistic simulations of magnets Richard F L Evans ESM 2018

Overview • Landau-Lifshitz-Bloch micromagnetics • Applications of atomistic spin dynamics • Simulations of ultrafast magnetisation processes ! −

Landau Lifshitz Bloch micromagnetics

Next generation micromagnetics: Landau Lifshitz Bloch equation • Conventional micromagnetics ubiquitous but does a poor job of thermodynamics of magnetic materials • Atomistic models in principle resolve this but horrendously computationally expensive • Landau Lifshitz-Bloch micromagnetics is an advanced micromagnetic approach which attempts to correctly simulate the intrinsic thermodynamic properties of magnets • Still only a partial solution - crystal structure, interfaces, surfaces, local defects, finite size e ff ects all still not really accessible to a micromagnetic model

Landau Lifshitz Bloch (LLB) equation • An additional dynamic term compared to the LLG equation m = γ [ m × H eff ] + | γ | α || ˙ m 2 ( m · H eff ) m − | γ | α ⊥ m 2 [m × [ m × ( H eff + η ⊥ )] ] + η || • Derived from the thermodynamic behaviour of a collection of classical spins by D. Garanin [1] • Longitudinal fluctuations (and damping) of the magnetization are now included in the dynamics, enabling simulations up to and above the Curie temperature • Also quantum flavours of the LLB [1] D. A. Garanin, Phys. Rev. B 55, 3050 (1997)

Longitudinal term in the Landau Lifshitz Bloch (LLB) equation • Longitudinal fluctuations of the + | γ | α || magnetization have their own dynamics m 2 ( m · H eff ) m • Di ff erent e ff ects below and above the Curie temperature, T c • The e ff ective magnetic field that constrains the magnetization length is given by ⎧ � � 1 − m 2 1 T � T c , m , ⎨ m 2 2 ˜ χ ∥ e H eff = H + H A + � T − T c m 2 � T c − 1 1 + 3 T � T c m , ⎩ ˜ 5 χ ∥

Energy terms in the Landau Lifshitz Bloch (LLB) equation • Conventional energy terms used in micromagnetics cause numerical issues for the LLB, as any “applied” magnetic field will cause the moment length to grow + | γ | α || m 2 ( m · H eff ) m • Therefore need to treat internal fields in a special way so that in thermal equilibrium , the net magnetic field is zero ⎧ 2 m 2 x + m 2 + ( m 2 − m 2 e ) y , T � T c , ⎪ F ⎨ χ ∥ m 2 2 ˜ χ ⊥ 8 ˜ e s V = � 2 m 2 x + m 2 M 0 � m 2 + 5 T c T − T c 3 y , T > T c . + ⎪ ⎩ 2 ˜ 20 ˜ T − T c 3 T c χ ⊥ χ ∥ Evans et al , Phys. Rev. B 85 , 014433 (2012)

Parameters for the LLB equation can be derived from mean field or atomistic/multiscale simulations 1.0 0.2 ˜ χ ⊥ χ ( T ) ˜ 0.8 χ � 0.15 0.6 χ [1/T] m e 0.1 0.4 ˜ m e ( T ) 0.05 0.2 0.0 0 T c 0 300 600 900 0 300 600 900 T [K] T [K] 2 . 5 10 − 11 x 2 10 − 11 x A [J/m] 1 . 5 10 − 11 x 1 10 − 11 x A ( T ) 5 10 − 12 x 0 T c 0 300 600 900 T [K] Kazantseva et al , Phys. Rev. B 77 , 184428 ︎ (2008)

Comparative dynamics for LLB and atomistic simulations 1 T = 300 K 0.8 T = 500 K 0.6 m z 0.4 T = 650 K 0.2 T = 800 K 0 0 1 2 3 4 5 t [ps] Kazantseva et al , Phys. Rev. B 77 , 184428 ︎ (2008)

Applications of atomistic spin dynamics

Atomistic spin dynamics and temperature dependent properties of Nd 2 Fe 14 B

Permanent magnetic materials Source: siemens.com

Structure at atomic and granular length scales determines overall material performance Acta Materialia 77 , 111-124 (2014)

Atomistic spin Hamiltonian for Nd 2 Fe 14 B H = H Nd + H Fe H Nd = − ∑ J NdFe S i · S δ i , δ E k , Nd − ∑ − µ Nd ∑ H app · S i i i i H Fe = − ∑ J Fe ( r ) S ν · S δ − ∑ J NdFe S ν · S j ν , j ν , δ E k , Fe − ∑ − µ Fe ∑ H app · S ν ν ν ν

Fe-Fe Exchange interactions J Fe ( r ) = J 0 + J r exp ( − r / r 0 ) 3.5 ab − initio data Fitted function 3.0 Exchange energy ( × 10 − 21 J) Scaled data 2.5 2.0 ab-initio data for BCC Fe from 1.5 M. Pajda et al, Phys. Rev. B 64 , 1.0 174402 (2001) 0.5 0.0 − 0.5 2 2.5 3 3.5 4 4.5 5 Interatomic spacing (Å)

Magnetic anisotropy energy Fe Nd a 3.0 Fe (0) 2.5 Fe ( T ) / k 2 E k , Nd = − κ Nd e 2 − κ Nd e 2.0 P P 4 2 4 i 1.5 k 2 1.0 b 30 Fe (kOe) e e 20 10 H k e 3 ( 3 S 2 0 2 = − 1 z − 1 ) P 0 100 200 300 400 500 600 Temperature (K) e 12 ( 35 S 4 z − 30 S 2 4 = − 1 z + 3 ) P 2 ( e T ) = f ( σ ( e k Fe T )) σ ) = 1 + κ ca f ( e r tanh ( r e σ ) J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991)

Simulated temperature dependent magnetization a 1.0 Simulation Fit 0.8 Magnetisation ( m / m s ) 0.6 0.4 0.2 0.0 0 100 200 300 400 500 600 Temperature (K) The temperature-dependent xpression m ( T ) = ( 1 � T / T c ) β (sho T = 631 82 K and exponent

Quantitative modelling of temperature dependent properties in ferromagnets Richard F L Evans , Unai Atxitia and Roy W Chantrell Evans et al , Phys. Rev. B 91 , 144425 (2015)

Classical spin model m(T) simulation

Real ferromagnets: Kuz’min equation 1 � � 0.8 m � � � � � 1 � s � 3 = 2 � � 1 � s � � p � 1 = 3 ; 0.6 m Fe 0.4 0.2 (b) 0 0 0.2 0.4 0.6 0.8 1 τ Real ferromagnets very | 1 v different from classical model 0.8 → problem! 0.6 m Co 0.4 0.2 (c) 0 0 0.2 0.4 0.6 0.8 1 M. D. Kuz’min, Phys. Rev. Lett. 94 , 107204 (2005) τ

Phenomenological temperature rescaling Classical model xpression m ( T ) = ( 1 � T / T c ) β fitted T = 631 82 K and exponent Assume m(T) well fitted by Curie-Bloch equation m ( τ ) = ( 1 − τ α ) β Classical model: α = 1 Real ferromagnets: α ≠ 1 � Simplest rescaling: 1 � τ = τ α

Spin temperature rescaling (STR) method Universe � T exp � α T sim m sim = 0.9 = Simulation T c T c T sim = 50 K m exp = 0.9 T exp = 300 K

Spin temperature rescaling (STR) method 1.0 α = 1 3 / 2 ~ Experimental Temperature τ 5 / 2 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Simulation Temperature τ

Classical spin model with Heisenberg exchange X H exc = � J i j S i · S j 20 nm i 6 = j 20 nm 20 nm ������� ����������� VAM PIR E vampire.york.ac.uk R F L Evans et al , J. Phys.: Condens. Matter 26 103202 (2014)

a ! b ! 1.0 1.0 Fit Fit Calculated Calculated Kuzmin Kuzmin 0.8 0.8 Magnetization ( m / m s ) Magnetization ( m / m s ) Corrected Corrected Co Fe 0.6 0.6 Relative error (%) Relative error (%) 3 1 0.4 0.4 2 0 1 0 0.2 0.2 -1 -1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 τ τ 0.0 0.0 0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 Temperature (K) Temperature (K) c ! d ! 1.0 1.0 Fit Fit Calculated Calculated Kuzmin Kuzmin 0.8 0.8 Magnetization ( m / m s ) Magnetization ( m / m s ) Corrected Corrected Ni Gd 0.6 0.6 Relative error (%) 1 Relative error (%) 1 0.4 0.4 0 0 -1 0.2 0.2 -1 -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 τ τ 0.0 0.0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 Temperature (K) Temperature (K)

Physical picture of temperature rescaling S S S Quantum Classical Rescaled

Back to NdFeB

Apply temperature rescaling to achieve exact agreement with experimental M s (T) b 1.0 Temperature rescaling Corrected simulation Classical fit Kuz’min fit 0.8 Magnetisation ( m / m s ) m ( τ ) = ( 1 − τ α ) β 0.6 0.4 � T exp � α T sim 0.2 = T c T c 0.0 0 100 200 300 400 500 600 Temperature (K) R. F. L. Evans et al , Phys. Rev. B 91 , 144425 (2015)

Temperature dependent magnetization with temperature rescaling 40 Nd 2 Fe 14 B Fe 35 fit Magnetisation ( µ B / f.u.) Nd 30 25 20 15 10 5 0 0 100 200 300 400 500 600 Temperature (K)

Spin-reorientation transition b 30 Model Experiment 25 Angle from c axis ( ° ) 20 𝜄 15 10 5 0 0 20 40 60 80 100 120 140 E k , Nd e e = − κ Nd 2 − κ Nd P P Temperature (K) 4 2 4 i

Anisotropy field calculation 40 35 Magnetic moment ( µ B / f.u.) 30 25 20 15 Simulation [001] 10 [110] 5 Experiment [001] [110] 0 0 5 10 15 20 25 Applied field strength (T)

Domain wall structures in Nd 2 Fe 14 B 10 nm

Domain wall profile 1.0 δ 001 δ 100 Reduced magnetization 0.5 0.0 δ 001 = 2.182 nm − 0.5 δ 100 = 2.471 nm − 1.0 − 4 − 2 0 2 4 Distance (nm) 1 nm