Reduced models for domain walls in soft ferromagnetic films Lukas - PowerPoint PPT Presentation

Reduced models for domain walls in soft ferromagnetic films Lukas Döring Conference on Nonlinearity, Transport, Physics, and Patterns Fields Institute, Toronto 06/10/14 Max Planck Institute for Mathematics in the Sciences

Modelling ferromagnetic thin films Ω ⊂ R 3 sample m : Ω → S 2 magnetization “Elementary magnets” � Unit-length vector field 1

Magnetization patterns in thin-film ferromagnets R. Schäfer J. Steiner, F. Otto Magnetization patterns in Permalloy films Numerical simulation of domain walls 2

Landau-Lifshitz (free) energy Observed patterns: Local minimizers m : Ω ⊂ R 3 → S 2 of � E ( m ) = d 2 |∇ m | 2 dx Exchange energy Ω � � ∇ · ( h str + 1 Ω m ) = 0 R 3 | h str | 2 dx + Stray-field energy ∇ × h str = 0 � 1 − ( e · m ) 2 dx Anisotropy energy for e ∈ S 2 , Q ≪ 1 + Q Ω � − 2 h ext · m dx Zeeman energy Ω Well-accepted Non-convex Non-local 3

Landau-Lifshitz (free) energy Observed patterns: Local minimizers m : Ω ⊂ R 3 → S 2 of � E ( m ) = d 2 |∇ m | 2 dx Exchange energy Ω � � ∇ · ( h str + 1 Ω m ) = 0 R 3 | h str | 2 dx + Stray-field energy ∇ × h str = 0 � 1 − ( e · m ) 2 dx Anisotropy energy for e ∈ S 2 , Q ≪ 1 + Q Ω � − 2 h ext · m dx Zeeman energy Ω Well-accepted Non-convex Non-local 3

Landau-Lifshitz (free) energy Observed patterns: Local minimizers m : Ω ⊂ R 3 → S 2 of � E ( m ) = d 2 |∇ m | 2 dx Exchange energy Ω � � ∇ · ( h str + 1 Ω m ) = 0 R 3 | h str | 2 dx + Stray-field energy ∇ × h str = 0 � 1 − ( e · m ) 2 dx Anisotropy energy for e ∈ S 2 , Q ≪ 1 + Q Ω � − 2 h ext · m dx Zeeman energy Ω Well-accepted Non-convex Non-local 3

Outline Single wall in infinitely extended film Periodic domain pattern with interacting wall tails

Wall patterns on cross-section of film Wall angle α Material anisotropy } Film “Easy axis” e thickness 2 t x 3 x 2 x 1 External magnetic field h ext 2 α m 2 Anisotropy Q and external field h ext e m 1 determine wall angle α . h ext Wall angle α and film thickness t determine wall type. 4

m 2 Three wall types − sin α 0 sin α Symmetric Néel wall m 3 m 1 Asymmetric Néel wall m 3 m 1 Asymmetric Bloch wall m 3 m 1 5

m 2 Three wall types − sin α 0 sin α Symmetric Néel wall m 3 m 1 Asymmetric Néel wall m 1 logarithmic tails m 3 m 1 ∼ t core Q ∼ d 2 Asymmetric Bloch wall t ln x 1 x 1 ≈ 1 m 3 m 2 m 1 m 1 5

m 2 Three wall types − sin α 0 sin α Symmetric Néel wall m 3 m 1 Asymmetric Néel wall m 3 m 1 Asymmetric Bloch wall m 3 m 1 Aim: Understand transitions between wall types for Q ≪ 1 5

m 2 Three wall types − sin α 0 sin α Symmetric Néel wall Wall types in Permalloy films m 3 Hubert, Schäfer: Magnetic Domains, Springer, 1998 m 1 Symmetric Néel Wall angle Asymmetric Néel Asymmetric Néel wall m 3 m 1 Cross-tie Asymmetric Bloch wall Asymm. Bloch Film thickness m 3 m 1 Aim: Understand transitions between wall types for Q ≪ 1 5

The critical regime: Optimal mix ... � t 2 ln − 1 t 2 if t 2 d 2 ≪ ln 1 d 2 Q , Q , Otto, ’02 min E 2D ( m ) ∼ if t 2 d 2 , d 2 ≫ ln 1 Q . m wall of angle π 2 t 2 d 2 = λ ln 1 What happens in critical regime: Q ? Optimal wall profile for angle α = asymm. “2 1 2 -d” core long-range 1-d tails m 1 = cos θ � � + � cos θ � cos θ − sin θ + sin θ 0 0 m 1 = cos α = = m m Optimal mix: θ α − θ Quantification of optimal mix difficult to access by brute-force numerics. ...of core and tails 6

Outline Single wall in infinitely extended film Periodic domain pattern with interacting wall tails

Expected behavior: Large domain width... x 2 H = 0 x 1 m 1 ¯ x 1 cos θ = H 0 ∼ t ∼ t core equilibrium core w x 3 x 1 ...similar to one-wall case 7

Expected behavior: Large domain width... x 2 H = 0 . 2 x 1 m 1 ¯ cos θ = H x 1 0 ∼ t ∼ t core equilibrium core w x 3 x 1 ...similar to one-wall case 7

Expected behavior: Large domain width... x 2 H = 0 . 4 x 1 m 1 ¯ cos θ H x 1 0 ∼ t ∼ t ∼ t ∼ t Q Q core equilibrium core tail tail w x 3 x 1 ...similar to one-wall case 7

Expected behavior: Small domain width... x 2 H = 0 x 1 m 1 ¯ x 1 cos θ = H 0 ∼ t ∼ t equilibrium core equilibrium core equilibrium w w w x 3 x 1 ...leads to coalescing tails 8

Expected behavior: Small domain width... x 2 H = 0 . 2 x 1 m 1 ¯ cos θ = H x 1 0 ∼ t ∼ t equilibrium core equilibrium core equilibrium w w w x 3 x 1 ...leads to coalescing tails 8

Expected behavior: Small domain width... x 2 H = 0 . 4 x 1 m 1 ¯ cos θ H x 1 0 � t � t � t ∼ t ∼ t Q Q Q core core tails tails tails w w w x 3 x 1 ...leads to coalescing tails 8

Strongly hysteretic transition between asym. walls... Elongated CoFeB elements 2 t = 120nm, Q = 1 . 55 · 10 − 3 , d = 3 . 86 ± 0 . 3nm. Origin of large jump in hard-axis magnetization? Domain width w x 1 x 2 ...due to interacting tails? C. Hengst, IFW Dresden 9

Just a few building blocks... m 1 ¯ cos θ cos α x 1 0 w tails w tails ∼ t ∼ t core core tail domain tail w x 3 x 1 E 2D = Exchange energy + Stray-field energy + Bulk energy � � � � 1 � 2 dx 1 ≈ d 2 |∇ m core | 2 dx + 2 t 2 � | d 2 m tails dx 1 | θ 1 � d ) 2 = λ ln 1 ( m tails − H ) 2 dx 1 , with ( t + 2 Qwt − Q . 1 Interesting regime: Qwt = κλ d 2 ; optimal w tails = w 2 . ...combined in an optimal way 10

Just a few building blocks... m 1 ¯ cos θ cos α x 1 0 w tails w tails ∼ t ∼ t core core tail domain tail w x 3 x 1 E 2D = Exchange energy + Stray-field energy + Bulk energy ≈ d 2 �� |∇ m core | 2 dx + 2 πλ ( cos θ − cos α ) 2 θ + 2 κλ ( cos α − H ) 2 � d ) 2 = λ ln 1 with ( t , Q . w κ Q ; optimal w tails = w Interesting regime: t = 2 . Q ln 1 ...combined in an optimal way 10

Just a few building blocks... m 1 ¯ cos θ cos α x 1 0 w tails w tails ∼ t ∼ t core core tail domain tail w x 3 x 1 =: E asym ( θ ) � �� � � � d − 2 min |∇ m | 2 dx m E 2D ( m ) ≈ min min θ m stray-field free wall of angle θ π ( cos α − H ) 2 �� ( cos θ − cos α ) 2 + κ � + 2 πλ min α d ) 2 = λ ln 1 as Q → 0, for ( t t Q , w = κ Q . Q ln 1 ...combined in an optimal way 10

Reduced model for the structure of domain walls Theorem ( κ = ∞ : D., Ignat, Otto; κ < ∞ : D.) There exist critical points m Q of E 2D , such that for Q → 0 , λ the relative film thickness, κ the relative domain width: � π + κ ( cos θ − H ) 2 � d − 2 E 2D ( m Q ) ≈ min κ E asym ( θ ) + 2 πλ θ ∈ [ 0 , π 2 ] and � π − m 1 , Q dx ≈ cos α opt = H + π + κ ( cos θ opt − H ) . domain ◮ Proof via Γ -conv. (minimize E 2D over periodic m ). ◮ Compactness requires “shifting argument” to ensure that { m Q } Q converges to a domain wall . 11

Stray-field free core: Néel and Bloch... �� � � � m : Ω → S 2 has wall angle θ, |∇ m | 2 dx E asym ( θ ) := min � with ∇· m ′ = 0 in Ω , m 3 = 0 on ∂ Ω � Ω 1 1 1 0.5 0.5 0.5 0 0 0 x 3 x 3 x 3 -0.5 -0.5 -0.5 deg = 0 -1 -1 -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 m 3 x 1 x 1 x 1 30 m 1 1 25 0.5 ? Asym. Bloch 0 x 3 20 -0.5 -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 E asym ( θ ) 15 x 1 deg = ± 1 1 10 0.5 0 x 3 Asym. Néel ≈ -0.5 5 4 π sin 2 θ + 148 35 π sin 4 θ ≈ -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 D., Ignat x 1 0 0 π /4 π /2 wall angle θ ...two topologically distinct wall types 12

Comparison of theory and experiments Co 40 Fe 40 B 20 films (lateral width 60 µ m) with parameters thickness/nm 102 153 212 Q / 10 − 3 1 . 36 0 . 93 1 . 16 µ 0 M s = 1 . 48T (measured in a single film of small thickness) d = 3 . 86nm (from Conca et al., J. Appl. Phys., 2013) For 2 t = 102nm: 0.6 0.6 w=7.5 µ m w=6.0 µ m w=8.2 µ m w=7.0 µ m w=9.0 µ m w=8.4 µ m 0.5 0.5 w=11.6 µ m w=13.6 µ m w=17.4 µ m 0.4 0.4 Magnetization Magnetization 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Reduced external field Reduced external field Experiments: C. Hengst 13

Comparison of theory and experiments Co 40 Fe 40 B 20 films (lateral width 60 µ m) with parameters thickness/nm 102 153 212 Q / 10 − 3 1 . 36 0 . 93 1 . 16 µ 0 M s = 1 . 48T (measured in a single film of small thickness) d = 3 . 86nm (from Conca et al., J. Appl. Phys., 2013) a) α = 0° 45° w 90° 30 µm stripe axis b) H a α H dem 20 µm Experiments: C. Hengst 13

Further questions x 1 Transversal (in)stability and x 2 path to cross-tie wall 1 0.5 Stability of asymmetric walls x 3 0 -0.5 1 -1 -2 -1.5 -1 -0.5 0 0.5 0.5 1 1.5 2 x 1 0 x 3 -0.5 -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 1 Comparison of critical θ wall angle α ∗ ≈ arccos ( 1 − 2 λ ) t 2 ( λ ≈ Q ) to experiments d 2 ln 1 α α ∗