Magnetism and Magnetic Switching Robert Stamps SUPA-School of Physics and Astronomy University of Glasgow
A story from modern magnetism: The Incredible Shrinking Disk Instead of this: (1980)
A story from modern magnetism: The Incredible Shrinking Disk We have this: (10 3 greater capacity)
Size and Sensitivity Nobel Prize 2007 The physics of GMR sensors allow 10 to 100 times sensitivity over previous sensors. Greater sensitivity means ability to read data at higher densities.
Devices for Tomorrow Magnetic RAM Racetrack Memory Nanomagnetic Logic U. Twente U. Sheffield IBM Almaden Reversal involves precessional magnetic rotation, domain wall formation and domain wall motion
Spin Torque Transfer & Spin Currents current of unpolarized electrons Ferromagnet 1 Ferromagnet 2 Spin accumulation and torque transfer Optical analogy: polarising filter measurement filter incident beam of unpolarized light
Magnonics Control GHz properties with patterned magnetic films: Spinwave based logic:
Outline: Starting points: Magnetic moment and exchange - Electrostatics and exchange energy - Magnetic ordering and mechanisms for exchange Magnetisation dynamics - Torque equations and effective fields - Precessional dynamics - Switching Spin waves - Correlated precession
Starting Point 1: Magnetic Moment Spin and orbital angular momentum: ħ J =ħ(s+l) µ=γ(ħ J ) Gyromagnetic ratio γ : γ= g µ B ħ Energy and precession: U= µ B Γ= d dt L=µ B L=L o +l e iω L t l B Larmor precession ω L = µ B ħ B
Concept: Exchange Energy Pauli exclusion separates like spins: Can be energetically favourable: suppose parallel alignment results in a small change in average separation. Then if: r a 0.3 nm r p 0.31nm r p e 2 4.8 ev r a e 2 4.75eV r p r a E E =0.05 ev (580 K) equivalent field: E E µ B =870T
Example: Direct Exchange Electrons on two neighbouring atoms: H= p 2 1 2m + p 2 2 2m + e2 R a R b e2 r 1 r a e2 r 2 r b + e2 r 1 r 2 core-core intra-atomic inter-atomic Assume know solutions for isolated orbitals with energy E on atoms a and b. Solve two electron problem with combination of product orbitals Ψ I =ϕ a (r 1 )ϕ b (r 2 ) ψ=c I Ψ I +c II Ψ II Ψ II =ϕ a (r 2 )ϕ b (r 1 )
Example: Direct Exchange Solution involves several overlap integrals: 2( 2 V= d r 1 d r 2 Ψ I, II e 1 + 1 1 1 1 R ab r 12 r ab r 1a r 2b) U= d r 1 d r 2 Ψ I * Ψ II e 2( 1 R ab + 1 r 12 1 r ab 1 r 1a 1 r 2b) l= d rϕ a * (r)ϕ b (r) Two electron wavefunctions: space symmetric space anti-symmetric c I =c II c I = c II E + =2 E+ V +U E 1+l 2 =2 E+ V U 1 l 2
Example: Direct Exchange For Pauli exclusion require ψ s ψ a (space symmetric)(spin antisymmetric) (space antisymmetric)(spin symmetric) ψ s 1 2 ( ) ψ a ( ) 1 ( + ) ( ) 2 spin 0 spin -1, 0, 1 Energy difference determines whether spins prefer parallel or antiparallel alignment: J=E E + U l 2 V Heitler & London (1927)
Examples of Magnetic Ordering Ferromagnetic: Antiferromagnetic: Ferrimagnetic: Helical:
Summary I Bohr and Pauli Study Angular Momentum Angular momentum & magnetic moment: Defines energy, torque and precession Exchange: electrostatic repulsion & Pauli exclusion: Determines long range order and phase transitions
Starting Point 2: Phenomenology Relevant energy scales (P. W. Anderson, 1953): 1 10 ev Atomic Coulomb integrals Hund's rule exchange energy Electronic band widths Energy/state at ε f spin waves ordering 0.1 1.0 ev Exchange energy splitting 10-2 10-1 ev Spin-orbit coupling 10-4 ev Magnetic spin-spin coupling Interaction of a spin with 10 kg field 10-6 10-5 ev Hyperfine electron-nuclear coupling
Dynamics & Effective Field Magnetic parameters describe energy & torques: E= m(r,t) H eff (r,t) t m r = m r H eff H eff H eff =H a +K(m(r))+A 2 m(r) h dip applied field anisotropy exchange
Magnetisation & Exchange Parameters Basic idea: define magnetisation density M(r)=gµ B j δ(r r j ) σ j Exchange energy must be compatible with symmetry of local atomic environment: E ex = α k l C kl m α (r) r k Example: isotropic medium m α (r) r l E ex =A[( m x ) 2 +( m y ) 2 +( m z ) 2 ]
Magnetic Anisotropy Parameter Local atomic environment affects spin orientation: [Kittel, Introduction to Solid State] Spin orbit interaction and crystal field effects Anisotropy energy and symmetries: Uniaxial: E ani (m z )=E ani ( m z ) E ani = K u (1) m z 2 K u (2) m z 4 +... Cubic: E ani (m x, m y, m z )=E ani ( m x,m y,m z ), etc. E ani =K 4 (m x 2 m y 2 +m x 2 m z 2 +m y 2 m z 2 )+...
h dip (r ij )= 1 2 g 2 µ B 2 Dipolar Interactions Magnetic moments are point dipoles: ij( m(r i) m(r j ) 3 [r ij m(r i )][r ij m(r j )] 3 r ij r ij 5 ) wondermagnet.com θ All moments interact throughout sample. Sample shape contributes to anisotropy. For an ellipsoid: E ani = M 2 V 2µ o (N x sin 2 θ cos 2 ϕ+n y sin 2 θsin 2 ϕ+n z cos 2 θ) Easy direction Hard direction
Dissipation Damping: additional torques www.medical.siemens.com www.ptb.de Gilbert damping: Rayleigh dissipation form t m(r)=γ m(r) H m(r) eff α m(r) t
Magnetic Switching
Fast Reversal of Single Particles Switching involves precession + dissipation: Experiment: Apply pulse, drive reversal H bias h pulse Thirion, et al., Nat. Materials 2003
Reversal: Nonlinearities Bauer, Hillebrands, Stamps, Phys. Rev. B 2000
Sensitivity To Pulse Orientation pulse pulse
Bauer, Hillebrands, Stamps, Phys. Rev. B 2000 Reversal: Nonlinear Dynamics Anisotropy creates windows for precessional reversal Pulse: 0.25 ns Ellipsoidal particle Polar plot: field pulse orientation and strength Bright = switched Dark = not switched
Thermal Activation: H<H c Climbing to the top: fluctuations Energy transfer between spin system and heat bath Torque equation of motion: thermal fluctuation field t m(r)=γ m(r) H eff α m(r) m(r) +h t f random thermal driving torque
Single Particle Switching: Stoner- Wohlfarth Model Approximate reversal as pure relaxation: H θ E=V H M cos K sin 2 =V K [ H M 2K ]2 ɛ Rate depends on activation energy and attempt frequency = 1 = f o exp / k B T [Kramers, 1940; Langer, 1969; Büttiker, 1981]
Switching in Elements Attempt frequency ~ energy landscape curvatures
Spin Waves
Fluctuations in Magnetic Density Energy to reverse one spin in chain: 2 J H = J ij S i S j ij Superposition of ways to flip one spin: n = 1 = + + +K Spin wave excitation (boson)
Classical Picture: Correlated Precession Ground state magnetic orderings: 1) Magnetic moments 2) Exchange coupling Excitations: Precession dynamics H eff slide courtesy J-V Kim
Dispersion: Spinwaves Contribution from exchange: m x, m y ~exp i t k r H ex ~A 2 m r H ex ~ A k 2 m r 2 2 = H a A k 2 H a 2 K M s A k 2 resonance at k=0 field + anisotropy k
Dispersion: Dipole Exchange Modes Dipolar: long range nonlocal term Dynamics: compete with exchange at long wavelengths h dip (r)= g(r r ')m(r ')d r ' Shape anisotropies: create effective K terms
Thermal Reduction of Magnetisation M s (0) M s (T) k n k T 3/2 Linear spin waves Large amplitude fluctuations [Weiss & Forrer]
Summary II 'Macroscopic' models of magnetic configurations & dynamics Effective fields: magnetic parameters Reversal processes: Thermal activation and precession Spin waves: Fluctuations of magnetisation density walyou.co m
The End
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