Spin-orbit coupling effects on electrons, magnetic anisotropy, crystal field effects. Julie Staunton Physics Department, University of Warwick, U. K 1 / 3
Introduction Magnetism in condensed matter: ’exchange interactions’ between many interacting electrons. Electrons’ wavefunction, including their spins, must be antisymmetric , consistent with the Pauli Exclusion Principle. Generates spontaneous magnetisation in some materials. Insulators: H = − 1 � i , j J i , j s i · s j . 2 Spin-polarised electronic bands in metals. Nothing to link electron spins to any spatial direction -= ⇒ Spin-orbit coupling does this, H SO ∝ L · s . Role in magnetic easy axis and magnetic hardness, domain wall structure, transport properties for spintronics, e.g. Anomalous Hall Effect, Spin-Orbit Torque, topological materials, magnetic nanostructures,e.g. skyrmion structures, · · · 2 / 3
Spin-Orbit Coupling and Magnetic Anisotropy, U an . Free energy of magnet, large length scale, M = M ˆ n : (( ∇ M x ) 2 + ( ∇ M y ) 2 + ( ∇ M z ) 2 ) d r � F [ M ( r ) ] = A H app . · M ( r ) d r − 1 � � H ′ ( r ) · M ( r ) d r + � U an . ( M , ˆ n ) d r − 2 Fundamental property U an . links magnetisation direction to structure via spin-orbit coupling of electrons. Large U an . � magnetic hardness, permanent magnets. Small U an . � magnetic softness, high permeability. Large U an . � stability of magnetic information, � smaller magnetic particles, higher blocking temperatures T b ’s U an . is big for high Z materials. Has strong T , compositional and structural dependence. 3 / 3
Temperature and magnetocrystalline anisotropy - spin model illustration U an . = � l k l g l (ˆ n ), where k l are the magnetic anisotropy constants, and g l ’s are polynomials consistent with crystal point group symmetry. As T rises, the k l ’s decrease rapidly. Localised spin model for a uniaxial magnet, with H = − 1 n · s i ) 2 . � i , j J ij s i · s j − K � i (ˆ 2 k l ’s T dependence given by Zener, Akulov, Callen and Callen and others ( H.B.Callen and E.Callen, J.Phys.Chem.Solids, 27 , 1271, (1966) ). At low T, k l ( T ) k l (0) ≈ ( M ( T ) M (0) ) l ( l +1) / 2 , e.g. k 2 ≈ ( M ( T ) M (0) ) 3 At higher T, k l ( T ) k l (0) ≈ ( M ( T ) M (0) ) l , k 2 ≈ ( M ( T ) M (0) ) 2 4 / 3
Ab-initio calculations of K for L 1 0 FePt 1 0 Magnetic anisotropy energy (meV) MCA FePt M(T) FePt 0.8 −0.5 Magnetisation M(T) <−−model <−−−(0,0,1) 0.6 −1 0.3 0.4 0.2 0.1 −1.5 0.2 (1,0,0)−−> 0 860 900 940 0 −2 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Magnetisation squared Temperature T(K) ’Disordered local moment’ (DLM) Theory of magnetism to define ’spins’ in itinerant electron system, B. L. Gyorffy et al., J.Phys. F 15 , 1337, (1985); J. B. Staunton et al., Phys. Rev. Lett. 93 , 257204, (2004) . Shows anisotropy of exchange interactions (more later). 5 / 3
Magnetic anisotropy and crystal field effects Rare earth charge density from localised f-electrons, configuration determined by Hunds’ first and second Rules. e.g. Sm 4f 5 ↑ ↑ ↑ ↑ ↑ ◦ ◦ , S = 5 2 , L = 5 Spin-orbit coupling causes the f-electron charge distribution to be dragged around with spin as magnetisation direction is altered. There is an electrostatic energy cost from the surrounding charges, the crystal field, that generates an on-site magnetic anisotropy. 6 / 3
Magnetic anisotropy and domain wall widths Simple model of a Bloch domain wall M = (0 , sin θ ( x ) , cos θ ( x )), � L ∂ x ) 2 − K cos 2 ( θ ( x )) dx − L ( A ( ∂θ Free energy: F [ θ ( x )] = δ F Solution δθ ( x ) = 0 for θ ( − L ) = 0, θ ( − L ) = π , ( L → ∞ ). i.e. A d 2 θ dx 2 + K sin θ ( x ) cos θ ( x ) = 0 Non-linear sine-Gordon equation: � � tan( θ ( x ) − π K A 2 ) = sinh( x A ) and width of wall ∼ K . 7 / 3
K and blocking temperature in magnetic nanostructures S. Ouazi et al., Nature Commun.,12/2012; 3:1313 Blocking temperature, T b = K V k B 8 / 3
Materials Modelling and Density Functional Theory Computational materials modelling must describe ≈ 10 24 interacting electrons. Density functional theory (DFT) makes this problem tractable. It focusses on dependence of the energy of a material on electronic charge, ρ and magnetisation, M , densities, E [ ρ, M ]. Many interacting electrons described in terms of non-interacting electrons in effective fields (Kohn Sham). The effective fields have subtle exchange and correlation effects for nearly bound d- and f-electrons. Relativistic effects - spin-orbit coupling- lead to magnetic anisotropy, E [ ρ, M n 1 ] − E [ ρ, M n 2 ] or torque δ E δ n . 9 / 3
Fundamental origins - interacting electrons and relativistic effects Official starting point - Quantum Electrodynamics. Leads to Relativistic Density Functional Theory ( A.K.Rajagopal (1980),Adv.Chem.Phys. 41 ,59 ). E [ ρ, � M ] = � ( V ext ( r ) ρ ( r ) − µ B B ext ( r ) · M ( r )) d r T s [ ρ, M ] + E H [ ρ ] + E xc [ ρ, M ] + ρ ( r ) and M ( r ) given by solving single electron Kohn-Sham-Dirac equations self-consistently. ( − i � α · ∇ + β mc 2 + V eff ( r ) − µ B βσ · B eff ( r )) ψ λ ( r ) = ε λ ψ λ ( r ) ρ ( r ) = � occ . ψ † λ ( r ) ψ λ ( r ); M ( r ) = � occ . ψ † λ ( r ) βσψ λ ( r ) λ λ V eff ( r ) = V ext ( r ) + e 2 | r − r ′ | d r ′ + δ E xc ρ ( r ′ ) � δρ ( r ) ; 4 πǫ 0 B eff ( r ) = B ext ( r ) + δ E xc δ M ( r )( r ) 10 / 3
Origin of spin-orbit coupling Leading relativistic effects: ( − � 2 ∇ 2 2 m + V ( r )+ h mv + h D + h SO − µ B σ · B ( r )) φ λ ( r ) = ε λ φ λ ( r ) h mv , mass-velocity term, h D , Darwin term and � e h SO = − 4 m 2 c 2 ( E ( r ) × p ) · σ , E = −∇ V ( r ), factor 1 2 times smaller than term derived semi-classically (Thomas precession). Platinum bands Topological Insulator 11 / 3
Spin-orbit coupling and broken inversion symmetry When an inversion symmetry is broken there is a spin polarisation of the electronic states by SO coupling. Rashba E.I.Rashba and Y.A.Bychov,J.Phys.C 17 , 6039, (1984) and Dresselhaus G.Dresselhaus, Phys.Rev, 100 , 580, (1955) Effects. Electron confined in 2D ( x , y ,0 ≤ z ≥ d ) with external electric field 4 m 2 c 2 E ( σ x ∂ � e ∂ y − σ y ∂ E = (0 , 0 , E ) and h SO = − i ∂ x ) . Show that E ↑ ( ↓ ) = � 2 k 2 2 m + � 2 n 2 π 2 2 md 2 + ( − ) e � 2 Ek 2 m 2 c 2 and k , n � � ( − (+) k y + ik x ) k , n ( x , y , z ) ∝ e i ( k x x + k y y ) sin( n π z φ ↑ ( ↓ ) d ) 1 k ( − (+) k y − ik x ) Spin Field Effect Transistor (SFET) ( S.Datta and B.Das, App.Phys.Lett, 56 ,665,(1990); I.Zutic et al. Rev.Mod.Phys. 76 ,323,(2004) ). 12 / 3
Magnetic anisotropy again Time reversal invariance is broken when a magnetic term − µ B σ · B is added. With spin-orbit coupling and magnetic terms, electronic structure varies with direction of magnetisation, e.g. CoPt ( S.S.A.Razee et al., Phys.Rev. B 56 ,8082, (1997) ) Origin of anisotropy: ∆ E MAE = � occ . BZ ( ε n 1 λ ( k ) − ε n 2 � λ ) d k λ 13 / 3
Anisotropic magnetic interactions RKKY interaction between 2 magnetic impurities in free electron gas, (2 k F R 12 cos(2 k F R 12 ) − sin(2 k F R 12 )) H = V s 1 · s 2 . R 4 12 With spin-orbit coupling, interaction becomes anisotropic ( Staunton et al. JPCM, 1 , 5157, (1989) ), uniaxial anisotropy. H = H (( R 12 · s 1 )( R 12 · s 2 ) , ( R 12 . ( s 1 × s 2 )) 2 ). Break inversion symmetry by including third site ( A.Fert and A.M.Levy, Phys.Rev.Lett. 44 ,1538, (1980) ) and find unidirectional anisotropy, H = H (( R 1 × R 2 ) · ( s 1 × s 2 )) - Dzyaloshinskii-Moriya-type ( I.Dzyaloshinskii, J.Phys.Chem.Solids, 4 , 241, (1958); T.Moriya, Phys.Rev.Lett. 4 , 5 ,(1960) ) 14 / 3
Modelling magnetic nanostructures H = − 1 i , j ( J ij s i · s j + s i J s � ij s j + D ij · ( s 1 × s 2 )) + � i K i ( s i ) 2 Magnetic monolayer on f.c.c. (1 , 1 , 1) substrate, chiral magnetic structures. M. dos Santos Dias et al., Phys.Rev. B 83 , 054435, (2011) . 15 / 3
Summary F [ M ( r )], phenomenological free energy - sum of ’exchange’ + ’magnetic anisotropy’ + ’applied magnetic field interaction’ + ’magnetostatic’ energies. In materials formal underpinning from relativistic quantum electrodynamics → R-DFT. Magnetic anisotropy - hardness of permanent magnets, domain wall thicknesses, blocking temperatures. Crystal field origin of on-site (single ion) anisotropy. Spin-orbit coupling and spin polarisation of electrons as relativistic effects. Rashba, Dresselhaus effects. Broken spatial inversion symmetry. Disruption of time reversal invariance. Ab-initio DFT calculations used to explain/provide A , K parameters for modelling magnetic properties. 16 / 3
Summary Anisotropic magnetic interactions Dzyaloshinskii-Moriya interactions, Skyrmion lattices and data storage ( M.Beg et al. Sci. Reports 5 , 17137, (2015) ) 17 / 3
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