Magnetization Processes I (and II) C.H. Back Universität Regensburg Bert Koopmans / TU Eindhoven 1
Tentative Title: Magnetization processes in bulk materials and films, and control: H and E, current, strain, photons Overarching Question: How can we manipulate (switch) the magnetization? Magnetic field Electric field (magneto-electricity, voltage control) Stress/strain (magneto-striction, phonons) Spin polarized current (spin transfer torque) Light
Simple magnetization reversal prozesses (Stoner-Wohlfarth-model, buckling, curling) and hysteresis Effects of thermal agitation Magnetization reversal by domain wall motion Precessional magnetization reversal Magnetization reversal by spin polarized currents Electric field induced magnetization reversal Light induced magnetization reversal mostly Blundell! 3
Technological Aspects Short Introduction to Micromagnetics Stoner Wohlfarth model Temperature effects 4
Scaling 5 Mbyte Microdrive 000 15. Gbits/in 1 x 1 dia disk 1 Gbyte 70 kbits/s RAMAC 1956 kbits/in 50x 4 inch diameter disks >100 Mbits/s Source: Hitachi Global Storage 5
Time Scales for Magnetization Processes Stability owledge Kn 1 s Thermally activated magnetization processes (viscous regime, domain 1 ms nucleation, domain growth) 1 s 1 ns 1 ps 1 fs Precessional Regime (Precessional Switching) Ultrafast demagnetization processes (optical excitation) Landau-Lifshitz-Gilbert equation: M = M H eff + (M M) M 6
Technological Aspects Short Introduction to Micromagnetics Stoner Wohlfarth model Temperature effects 7
The Energy Landscape / The Total Effective Field Precessional Motion of a Single Spin (Quantum Mechanics) The Landau-Lifshitz Equation 8
The Total Effective Field Zeeman-Energy E z : E z 0M s H ext(r) m(r) dv where M( r,t) = M s m(r,t) ( m = 1 ) Exchange-Energy E ex : Attention: E ex A ( grad m(r)) dv Not the divergence (A: exchange constant) of a vector field! 9
Side note Isotropic exchange Dzyaloshinskii-Moriya Interaction (DMI) General exchange anisotropic anisotropic isotropic symmetric antisymmetric Usual DMI form rewrite Prerequisites: - Broken inversion symmetry - Spin-Orbit coupling 10
Anisotropy-Energy E an : E an(m) an dv E m K f(,, ) an 0 1 c i cos i with the anisotropy energy density an m 3 b cubic crystals: 1 a an K (m 1 x m y m x m z m y m z ) K (m x m y m z ) uniaxial anisotropy (x-axis = easy-axis): an - K u m x (K 0, K 1, K, K u : anisotropy constants) 11
Example: magneto-crystalline anisotropy of bulk Fe (bcc) H(A/ (A/m) Bozorth 1
Magneto-crystalline anisotropies For cubic crystals: For symmetry reasons only terms containing can appear, since,... 4 i i i i j i thi th t t ith t ll d ),, ( 1 0 f K E m c j i j i this means that terms with are not allowed ),, ( 1 0 f K E m c m For cubic crystals no axis is special, thus b 3 3 1 3 1 K K K E m a 1 It follows that:... 3 1 3 1 3 3 1 3 1 3 1 1 0 K K K K E m i i cos 13
Magneto-crystalline anisotropies in Fe, Ni und Co [001] bcc [001] fcc [0001] hcp [100] [010] [100] [010] 14
Magneto-crystalline anisotropies Expand the angular dependence of the energy: for cubic symmetry 15
Origin of the magneto-crystalline anisotropy In a crystal the electron orbits are tied tied to the lattice due to the crystal field. L is quenched but not completely Spin-orbit coupling mediates the energetic anisotropy with respect to the orientation of the magnetic moment m 3 1 free atom crystal Em K f,, ) 0 ( 1 cos i i Energy functional and angles. The functional dependence results from the symmetry, the strength from the SOC 16
For simplicity we almost always assume only the uniaxial anisotropy in the following - K m an u x 17
Magneto-striction and magneto-elastic coupling Mechanical strain/stress breaks the crystal symmetry and an additional contribution to the energy density results: E K sin... tension in %... C stress Inverse effect: The magnetization leads to a slight deformation of the sample. E 1 K sin C E 0 K sin C Length change: l / l 10 5 experimental verification and experimental verification and measurement by cantilever method! 18
Strayfield-Energy E d : 1 E d - 0 H (r) M(r) dv d the strayfield H d is given by div H (r) div (M) d d (from Maxwell s equations) because rot H d = 0 we can express H d via a skalar potential d : H d (r) grad d (r) where M s div m(r ) m(r ) n(r ) d (r) dv ds 4 r r rr 19
Simplification: We consider only the demagnetizing g field of an ellipsoid Shape anisotropy 0
Demagnetizing field for sphere, wire and thin film Energy (density): In general: Demagnetizing tensor D: Sphere Infinitely long wire Infinite thin film 1
shape anisotropy Thin film Thin wire Sphere
Demagnetizing i field for an infinite it thin film 3
Technological Aspects Short Introduction to Micromagnetics Stoner Wohlfarth model Temperature effects 4
Equilibrium of the magnetization in an applied field Consider only shape anisotropy and Zeeman energy Energy minimization with respect to the angle: E V cos s 0 M sin cos M 0 M s cos H H M M s Magnetization component M(H) M ( H ) M S cos H M increases linearly until the value s H sin H M(H) H M S Is reached. In general we obtain: H NM S 5
Uniform rotation: Stoner-Wohlfarth model (1949) Calculation of hysteresis loops: Ideal magnetizion loop of a magnetic particle with uniaxial anisotropy (e.g. ellipsoid with short axes a=b, and long axis c) For K U >0 the c-axis is the easy axis of the magnetization. Total energy for the determination of the minimum energy and thus the angle (,H,M,K( U U) ): K U M S Anisotropy energy: E K0 sin Aniso K U H Zeeman energy: Zeeman 0 E HMcos( ) Energy minimum E E 0 0 E K Usin cos 0HMSsin( ) 0 6
Stoner-Wohlfarth model Calculation of Hysteresis loops for We search for the magnetization along the direction of H K U E K Usin cos 0HMSsin( ) M S 0 K sincos HM cos or U 0 S cos 0 K sin M H U 0 S sin M K S U 0 H 3, H With the magnetization component M in the direction of H M ( H ) M S sin M(H) M S M S K U 0 H Saturation is reached for: MS 1 K U 0H H S KU 0MS 7
Stoner-Wohlfarth model Calculation of magnetization loop for M K U M S 1 M S H S K M U 0 S H H -1 For H perpendicular to the easy axis M grows linearly with H H S M(H) M S K M U 0 S M S K U 0H 8
Stoner-Wohlfarth model Calculation of magnetization loop for 0, E or K sin cos HM sin 0 sin 0 K cos U 0 S 0, MS U 0 U 0 S cos K M H H No minimum since 0 E This means that the magnetization jumps between the two values 0, M ( H ) M S 1,1 The magnetic field at which M jumps is H K M S U 1 K U 0H C 0MS 9
Stoner-Wohlfarth model Calculation of hysteresis loops: 0, M M S H C 3 H H C 1 1 H C H C KU M 0 H K U E/K H 0-1 Solutions are only 0, The magnetization jumps between M ( H ) M S ( ) -1 and 1 The coercive field is H C K M U 0 S 30
Stoner-Wohlfarth model: energy landscape Energy landscape for both extremes: #1 Steady rotation from 90 to -90 90 # Jump into the second minimum i via a saddle point 30 Blundell 31
Stoner-Wohlfarth Astroid Hysteresis loops for various angles Switching behavior For arbitrary angles one obtains a mixture of both cases. The coercive fields (H C ) for switching into the stable minimum for arbitrary directions show a minimum. The switching fields for coherent rotation can be summarized in the switching asteroid. 3
Stoner-Wohlfarth astroid Within the asteroid more than one minimum exists, outside only one! 33
Stoner-Wohlfarth model: experiment SQUID: Superconducting Quantum Interference Device Sensitive method for the measurement of single clusters ( nm Diameter, corresponding to 10-10 3 atoms) 3 nm Co Cluster, embedded in a Niobium film. Wernsdofer et al. PRL 00 Jamet et al. PRL 001 34
Technological Aspects Short Introduction to Micromagnetics Stoner Wohlfarth model Temperature effects (very briefly) 35
Stoner-Wohlfarth Astroid at various temperatures HB M BH M Thermal agitation effectivly reduces the energy barrier! Exponentially! 3 nm Co Cluster, Micro-SQUID Experiment Wernsdofer et al. PRL 00 36
Beyond S-W-Model: e.g. Curling Mode Curling Reversal Mode Hertel, Kirschner, Physica B (004) 37
Real magnetic materials 38
Mechanisms for magnetization reversal Nucleation and domain wall motion Domain creation (smaller particle size) Why is it easier to nucleate a domain wall from an edge of the sample? Increase of the local demagnetizing field by surface roughness (asperities) reversal of the magnetization becomes more easy (a similar process occurs at defects inside a sample!) Coherent rotation (see Stoner Wohlfarth) 39
Magnetic Hysteresis (micron sized element) / Domain rotation 40
Barkhausen jumps E cc cc 1 cc: Configuration Coordinate actual state or configuration of system 41
magnetization reversal as a function of size 4
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Magnetization Processes II C.H. Back Universität Regensburg 44
Stoner-Wohlfarth model What happens if a short magnetic field pulse is applied (let s say < 1 ns) M K U M S 1 M S H S K M U 0 S H H -1 45
Precessional magnetization reversal 46
Precessional motion of a single electron spin (Quantum Mechanics) The time evolution of an observable is given by its commutator with the Hamilton operator. For the spin operator this means: The Hamilton operator consists only of the Zeeman term z-component: whereby the last step makes use of the commutation rules for the spin operator: 47
Thus, one obtains: The same holds for the other components of S: We now want to extend this equation to the magnetization. In the macrospin model the magnetization is considered uniform and is given as the average of the spin magnetic moments (we ignore a contribution from the orbital momentum) : From here we arrive at the analog relation for the magnetization: with the gyromagnetic ratio and g =.003 the gyromagnetic splitting factor for a free electron. This is the first part of the Landau-Lifshitz equation. 48
Landau- Lifshitz- Equation: d M ( M H ) ( ) eff M M H eff dt M H eff tot M 0 : gyromagnetic ratio sets the time scale 49
The total energy E tot : E tot E z E ex E an E d 1 A (grad m) an (m) 0M s Hext m 0M s Hd m dv (m) tot The variation of ε tot (m) with respect to m will result in the effective field H eff that will excert a torque on the magnetization M : 1 A 1 H m grad (m) H H tot eff m an ext d M s m Ms Ms Exchange field H ex Anisotropy field H an 50
The energy landscape of confined magnetic systems (4 micron objects) 51
Response of a single magnetic moment to a static DC-field The damping para- meter governs how quickly the moments relax to the effective field direction, and in nature ~0.01for Fe for example. dm dt (MH eff ) 5
Example for a precessional switching experiment explained by the macrospin model The SLAC experiment N ~ 10 10 e - Parameters: -5 ps pulse duration up to 0 T amplitude In-plane and perpendicular p magnetized Films 53
Current Density Magnetic Field Profile Torque M x H H M micron micron micron 54
special case for dipolar stray field (shape anisotropy): homogeneously magnetized ellipsoid: H D ( N M e N M M 0 xx x x yy y y zz z z with N ii being the diagonal elements of the demagnetizing g tensor: e N e ) In this example: thin film: N xx = N yy = 0, N zz = 1 H D 0 M z e z 55
Response of a single magnetic moment (with perpendicular crystalline anisotropy, and shape anisotropy) to a short pulsed field (in the x-y plane) for precessional switching 56
Short field pulse (some picosecond) (much shorter than relaxation time) 57
Perpendicularly Magnetized Co/Pt Multilayer 0 M = 1.6 T s,eff In-plane magnetized 0 nm Co Film H = 168 ka/m, 0 M=17T 1.7 k s ent compone z-c y-component x-component Phys. Rev. Lett. 81, 351 (1998) Science 85, 864 (1999) 58
Experiment Macrospin Calculation Fairly good agreement! But: the effective damping is much too large: =0.0 and the inner structure cannot be explained Total internal field oversimplified? Non uniform excitation field! Excitations? 59
Simple example: two dipoles coupled by their own stray fields (Micromagnetic Simulations using M.Scheinfein s LLG code) 0 10ns =17.6 MHz/Oe =0.01 M s =1714 emu/cc E 0 time 0ns 60
Spatially and time resolved experiments Laser pulse Wave guide I Photoconductive switch GaAs Magnetic Element H S.-B. Choe et al Science 304, 40 (004) Magnetic Patterns 10 m Current Conducting wire Magnetic field 61
Time resolved (~1 ps resolution) scanning (~300nm resolution) Kerr microscopy polar Kerr geometry Probe pulse fixed delay polares Kerr-Signal Pulsed fs-laser Ti:Sapphire @ 80 MHz -doubler 400 nm 0.0 0.5 1.0 1.5.0 Time [ns] Lock-in Amplifier @ 1.5 khz Probe pulse Polarizer Polarization sensitive detector 800 nm 400 nm Chopper Pump pulse Objective lens Optical delay line Sample Waveguide Oscilloscope
Time resolved (~1 ps resolution) scanning (~300nm resolution) Kerr microscopy polar Kerr geometry Probe pulse fixed delay polares Kerr-Signal Pulsed fs-laser Ti:Sapphire @ 80 MHz -doubler 400 nm 0.0 0.5 1.0 1.5.0 Time [ns] Lock-in Amplifier @ 1.5 khz Probe pulse Polarizer Polarization sensitive detector 800 nm 400 nm Chopper Pump pulse Objective lens Optical delay line Sample Waveguide Oscilloscope
Time resolved (~1 ps resolution) scanning (~300nm resolution) Kerr microscopy polar Kerr geometry Probe pulse fixed delay polares Kerr-Signal Pulsed fs-laser Ti:Sapphire @ 80 MHz -doubler 400 nm 0.0 0.5 1.0 1.5.0 Time [ns] Lock-in Amplifier @ 1.5 khz Probe pulse Polarizer Polarization sensitive detector 800 nm 400 nm Chopper Pump pulse Objective lens Optical delay line Sample Waveguide Oscilloscope
Time resolved switching experiment Nature 418, 509 (00) 65
Nature 418, 509 (00) 66
Electric field control? Weisheit et al., Science 315, 349 (007) 67
Bi-axial anisotropy uniaxial anisotropy D. Chiba et al., Nature 455, 07318 (008) 68
M. Sawicki et al., Nature Physics 009 (DOI: 10.1038/NPHYS1455) 69
S. Miwa, Nature Communications (017) (DOI: 10.1038/ncomms15848) 70
Spin-polarized currents? Reminder giant magneto resistance (GMR) GMR: Effect of the magnetization distribution on the electric conductivity 71
Spin Transfer Torque: Action of the conduction electrons (for example 4s-like) on the local magnetization (for example 3d-like) Independent proposals by L. Berger and J. Slonczewski 7
73 Transfer of angular momentum between conduction electrons and the electrons which make up the local magnetization results in a torque on the local magnetization
First experimental demonstration 74
???????? M Quantum mechanics of spin: A cos = A + B B sin Quantum mechanical probabilities: 1 Pr A 1cos 1 Pr B 1cos An arbitrary spin state is a coherent superposition of up and down spins. 75
Absorption of transverse angular momentum 1 1 Magnetization 1 1 ˆx Stiles and Zangwill PRB 00 76
Reflection and transmisson coefficients are spin dependent Reflection and transmisson coefficients are complex leading to rotation (classical dephasing) Different wave vectors for spin up and spin down in FM lead to precession M. D. Stiles and J. Miltat, Topics in App. Phys. 101, 5 (006). 77
electron flow electron flow 78
Control of the magnetization by light? 79
Inverse Faraday effect: 80
Does it work in metallic systems? Here: ferrimagnet GdFeCo + dichroism? All-Optical Magnetic Recording with Circularly Polarized Light Stanciu et al., Phys Rev Lett 99, 047601 (007) 81
Manipulation of magnetization is possibe by many different methods: Magnetic field control Electric field control Strain control Spin polarized currents Light 8