ESM 2018 Krakow MP2 - Precessional dynamics, dissipation processes, elementary and soliton excitations Joo-Von Kim Centre for Nanoscience and Nanotechnology, Université Paris-Saclay 91120 Palaiseau, France joo-von.kim@c2n.upsaclay.fr
� 2 MP2: Precessional dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Overarching theme: Landau-Lifshitz equation Linear excitations – spin waves Dispersion relations, applications in information processing Dissipation processes Intrinsic and extrinsic contributions, Gilbert damping Dynamics of topological solitons Lagrangian formulation, domain wall motion, vortex gyration
� 3 Time scales European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Today’s lecture Ultrafast laser-induced thermalisation Domains in nanoparticle arrays Data storage Spin waves Y h s µs ns ps fs x U L c Domain wall creep Spin glass relaxation Conduction spin relaxation
� 4 Magnetisation dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV In MP1, we saw how magnetic moments couple to each other and to their environment (e.g., exchange, dipole-dipole interactions). But how do they evolve in time? Consider Heisenberg picture in quantum mechanics, i ~ d ⌦⇥ ⇤↵ dt h S ( t ) i = S , H Consider a single spin in an applied magnetic field H . The Zeeman Hamiltonian is H = − gµ 0 µ B S · H To see how this works, expand out the S x term: [ S x , H ] = − gµ 0 µ B [ S x , S x H x + S y H y + S z H z ] = − gµ 0 µ B ( H y [ S x , S y ] + H z [ S x , S z ])
� 5 Magnetisation dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV By applying the usual commutation rules for the spin operators H [ S x , S y ] = iS z [ S y , S z ] = iS x [ S z , S x ] = iS y we obtain d S /dt d M /dt [ S x , H ] = − gµ 0 µ B i ( H y S z − H z S y ) M S Combining with the other spin components, we find (a) d h S ( t ) i = gµ 0 µ B h S i ⇥ H dt ~ This describes the precession of a spin in a magnetic field. With the definition of the gyromagnetic constant g | µ B | γ = gq e 2 m = gµ B ~28 GHz/T < 0 γ 0 = µ 0 = − µ 0 γ � �
� 6 Magnetisation dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV By averaging over the spins in the Bloch equation, d h S ( t ) i H = gµ 0 µ B h S i ⇥ H dt ~ d M /dt we can express the torque equation for a general magnetisation field M as M d M dt = − γ 0 M × H (a) M = gµ B N h S i /V The micromagnetics approach allows for a classical description of the magnetisation dynamics by treating the magnetisation as a continuous field M subject to torques applied by magnetic fields H .
� 7 Precessional dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Generalise torque equation to any magnetic energy by replacing H with the e ff ective field H e ff H e ff d M dt = − γ 0 M × H e ff d M /dt where M δ E H e ff = − 1 δ M µ 0 (a) The energy density accounts for all relevant contributions to the magnetic Hamiltonian (see MP1) Magnetisation precesses about its local e ff ective field Note that this torque equation conserves the norm of the magnetisation vector and describes dynamics at constant energy d d dt k M k 2 = 0 dt ( M · H e ff ) = 0
� 8 Linear excitations - Spin waves European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Small amplitude (linear) excitations of magnetisation are described by spin waves Consider a chain of spins uniformly aligned along an applied field H 0 E = − N ( gµ B S ) H 0 ≡ E 0 H 0 What is the smallest excitation possible? One spin reversal . There are two ways to accomplish this: 1) Flip one spin along the chain E − E 0 = 2 J 2) Distribute the spin reversal by canting all spins E � E 0 = ~ ω ⌧ 2 J
� 9 Spin waves European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV λ ⃗ k Spin waves are elementary excitations of a magnetic system � � ( � k ) Quantised spin-wave: magnon ( cf phonons for elastic waves) It is more favourable energetically to distribute flipped spin over all lattice sites, rather than to have it localised to one lattice site. (NB. Such excitations do exist - Stoner excitations - and these are important at high energies)
� 10 Spin wave dispersion relations European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Consider a uniformly magnetised system along the positive z axis. Suppose there is an applied external field H 0 along the positive z direction: H 0 Let k m k = 1 M = M s m If we allow for spatial variations in m , we need to also include exchange, ( r m x ) 2 + · · · h i E = E Z + E ex = � µ 0 M s H 0 m z + A From this expression, we can derive an expression for the e ff ective field ∂ E 2 A 1 r 2 m ∂ m = H 0 ˆ H e ff = � z + µ 0 M s µ 0 M s
� 11 Linearising the equations of motion European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Study small amplitude fluctuations of the magnetisation by linearising the equations of motion Write the magnetisation in terms of static and dynamic components. Assume the ground state consists of uniform magnetic state along +z: m ( r , t ) = m 0 + δ m ( r , t ) = (0 , 0 , 1) + ( m x ( r , t ) , m y ( r , t ) , 0) static dynamic Similarly, decompose the e ff ective field into static and dynamic components: H e ff = H e ff , 0 + h e ff ( r , t ) Terms that Terms that depend on δ m depend on m 0
� 12 Linearising the equations of motion European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Rewrite the precession term in the Landau-Lifshitz equation in terms of static and dynamic parts, retain only linear terms in the dynamic components: d m dt = − γ 0 m × H e ff d m dt = − γ 0 ( δ m × H e ff , 0 + m 0 × h e ff ) dynamic dynamic Assume plane wave solutions for the dynamic part m x,y ( r , t ) = c 0 e i ( k · r − ω t ) Left-hand side of the torque equation becomes m x � d δ m δ m = ( m x , m y , 0) = − i ω m y dt dynamic magnetisation
� 13 Linearising the equations of motion European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV In a similar way, the terms on the right-hand side (RHS) of the equation become d m dt = − γ 0 ( δ m × H e ff , 0 + m 0 × h e ff ) ✓ 2 A m x � ◆ r 2 δ m − i ω δ m × H 0 ˆ z ⇥ ˆ z m y µ 0 M s which leads to the matrix equation � m x � � m x � � � � � 2 A 0 − ω k k 2 H 0 + = ω k = γ 0 − i ω m y 0 m y µ 0 M s ω k � i ω � � m x � − ω k = 0 i ω m y ω k
� 14 Spin wave dispersion relation European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Condition of vanishing determinant of the 2x2 matrix gives the dispersion relation for the spin waves: ω − ω 2 + ω 2 k = 0 ⇒ ω = ω k = γ 0 H 0 + Dk 2 where we have defined a spin-wave sti ff ness γ 0 H 0 D ≡ 2 γ A k M s Spin waves in ferromagnets are dispersive with a “band gap” due to applied and anisotropy fields ω k � = ∂ω ∂ k Other energy contributions will bring supplementary terms to the dispersion relation
� 15 Brillouin light scattering spectroscopy European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Probe spin wave spectra by scattering light o ff surfaces z Reflected photons give information about spin waves that Backscattering are created (Stokes) or annihilated (anti-Stokes) geometry y x Stokes Anti-Stokes ω I ω I + ω ω I − ω
� 16 Mode confinement in nanostructures European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Translational invariance is broken in nanostructured magnetic elements Boundary conditions determine the quantisation conditions ∂ ⃗ � M Boundary condition = κ ⃗ � M for magnetisation � ∂⃗ n � S Micromagnetics R D McMichael & M D Stiles, J Appl Phys 97 , 10J901 (2005) Circular Elliptical
� 17 Mode confinement in nanostructures Experiments European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Brillouin light scattering with nano-sized apertures and near-field imaging allows confined modes to be probed Microfocus BLS setup Edge modes in a ferromagnetic ellipse Incident & scattered light Cantilever w i t h h Microwave 700 Oe t i 350 Oe p current 200nm 200 nm 1570 Oe 1000 Oe Permalloy H ellipse Tip apex Microstrip with a line nano-size aperture J Jersch et al, Appl Phys Lett 97 , 152502 (2010)
� 18 Spin waves as probes of magnetic properties European School on Magnetism 2018, Krakow – Magnetisation Processes (MP2) – Kim,JV Example: Determine exchange constant A from frequencies of perpendicular standing spin waves (PSSW) ⌘ 2 � ⌘ 2 � ω H + ω e ff + γ 2 A ⇣ π p ω H + γ 2 A ⇣ π p ω 2 PSSW = M s d M s d C Bilzer et al, J Appl Phys 100 , 053903 (2008)
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