MP3 - Spin-transfer and spin-orbit torques, current topics in - PowerPoint PPT Presentation

ESM 2018 Krakow MP3 - Spin-transfer and spin-orbit torques, current topics in magnetisation dynamics Joo-Von Kim Centre for Nanoscience and Nanotechnology, Université Paris-Saclay 
 91120 Palaiseau, France joo-von.kim@c2n.upsaclay.fr

� 2 MP3: Spin-transfer and spin-orbit torques European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Brief review of concepts in spin-dependent transport 
 Spin-transfer torques (CPP , CIP) and spin-orbit torques 
 Slonczewski model, Zhang-Li model, spin Hall effect 
 E ff ects of current-driven torques on spin waves 
 Self-sustained oscillations, Doppler effect 
 E ff ect of current-driven torques on soliton dynamics 
 Domain wall propagation, vortex gyration

� 3 Magnetism affects transport: GMR European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Giant magnetoresistance (GMR): Electrical resistance of a metallic magnetic multilayer that depends on the relative orientation of the constituent layer magnetisations Current perpendicular-to-plane 
 CIP GMR CPP Current in-plane 
 CIP Antiferromagnetically coupled layers M Baibich et al, Phys Rev Lett 61 , 2472 (1988) G Binasch et al, Phys Rev B 39 , 4828 (1989) 2007 Nobel Prize in Physics

� 4 Two-channel model http://www.phys.ufl.edu/fermisurface/ European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV In metals, conduction processes occur at the Fermi surface Assume spin-up and spin-down electrons propagate independently (OK if spin- flip scattering is weak) Assign a resistance to each spin channel (Mott) In normal metals, spin-up and spin-down channels are equivalent Fermi surfaces of some nonmagnetic metals K Cu Al R ↑ = R ↓ 4s 3d 10 4s 3s 2 3p

� 5 Two-channel model http://www.phys.ufl.edu/fermisurface/ European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV In ferromagnetic metals, this degeneracy is lifted due to exchange splitting Spin-up and spin-down (majority/minority) resistances are di ff erent Fermi surfaces of some ferromagnetic metals bcc Fe hcp Co majority R ↑ ̸ = R ↓ minority 3d 6 4s 2 3d 7 4s 2

� 6 GMR with two-channel model European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Simple picture of giant magnetoresistance in terms of two-resistance model P AP R P ≠ R AP

� 7 ... But can transport affect magnetism? European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV sd model (Vonsovsky-Zener) : 
 Exchange interaction between local magnetisation ( M ) and conduction electron spin ( s ) E sd = − J sd M · s � F mobile 4s electrons J sd (conduction only) k ↑ k ↓ localised 3d electrons (magnetism only) Torques on the magnetisation can arise from this coupling

� 8 M D Stiles & A Zangwill, 
 Single electron at N/F interface Phys Rev B 66 , 014407 (2002) European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Exercise: Consider a free electron in the normal metal arriving at the normal metal (N)/ ferromagnet (F) interface. Solve 1D Schrödinger equation M quantisation axis | ψ i σ i z � F q y J sd f k ↑ k ↓ x N F h 2 k 2 h 2 ( k ↑ , ↓ F ) 2 � F = ¯ ∓ ∆ J sd � F = ¯ F 2 m 2 2 m 2 Because the bands in the ferromagnet are spin-split, there is a spin-dependent step potential at the interface k ↓ F < k ↑ F

� 9 Spin currents M D Stiles & A Zangwill, Phys Rev B 66 , 014407 (2002) European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Exercise: Calculate spin current through this interface. What is conserved? ↔ � i σ ( r ) ˆ s ⊗ ˆ Q ( r ) = Re ψ ∗ v ψ i σ ′ ( r ) i σσ ′ z y Conserved NOT conserved x M M Q in zx + Q ref zx = Q tr Q yx zx Q in ⊥ x + Q ref ⊥ x ̸ = Q tr ⊥ x Q xx longitudinal spin current transverse spin current From conservation of spin angular momentum, argue that missing transverse spin current is transferred to ferromagnet M  ∂ m � ∝ s ⊥ ∂ t STT

� 10 Spin-transfer torques European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Express transverse spin component in terms of vector products  ∂ m � s ⊥ ∝ ( m × s ) × m ∝ ( m × s ) × m ∂ t STT F2 N N M Typical realisations involve the CPP geometry where s is related to the magnetisation of a second (reference) layer Conduction electrons Co Cu Co N N N Nanopillars Nanocontacts <100 nm F1 F2 (<5 nm)

� 11 Slonczewski model of CPP torques European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Accounting for transport properties, obtain Slonczewski term for spin-transfer torques ∂ M ∂ t = − γ 0 M × H e ff + α M × ∂ M ∂ t + σ j e M × ( p × M ) M s Precession Damping Spin-transfer torque (Slonczewski) Current density j e p with spin σ = gµ B 1 polarisation P s dP N N N M 2 2 e j e efficiency factor d Current density matters, not currents. We did not observe STT before the advent of nanofabrication Need typical densities of 10 12 A/m 2 : 1 mA for 1000 nm 2 , 1 000 000 A for 1 mm 2 J C Slonczewski, J Magn Magn Mater 159 , L1 (1996)

� 12 Consequences on precessional dynamics European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Spin-transfer torques can reverse magnetisation reversal without magnetic fields F J Albert et al, Phys Rev Lett 89 , 226802 (2002) Nanopillar structure Cross-section 60 × 180 nm2 Basis of spin-torque magnetic random access memories STT-(M)RAM Everspin Samsung 28nm pMTJ STT-RAM

� 13 Current-in-plane (CIP) torques European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Spin-transfer torques also occur in continuous systems in which there are gradients in the magnetisation Important for micromagnetic states like domain walls, vortices Torques are governed by how well the conduction electron spin tracks the local magnetisation Like CPP case, spin transfer involves the absorption of transverse component of spin current Adiabatic Nonadiabatic M s e – Conduction electron spin Conduction electron spin precesses about sd field relaxes toward sd field

� 14 S Zhang & Z Li 
 Zhang-Li model of CIP torques Phys Rev Lett 93 , 127204 (2004) European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV In the drift-di ff usion limit (not detailed here), Zhang and Li derived ∂ M ∂ t = − γ 0 M × H e ff + α M × ∂ M ∂ t + T CIP M s T CIP = � b J M � � M � ( j e · � ) M � � c J M � ( j e · � ) M µ 0 M 2 µ 0 M s s adiabatic nonadiabatic P µ B ξ P µ B c J = b J = P: spin polarisation eM s (1 + ξ 2 ) eM s (1 + ξ 2 ) In this model, nonadiabaticity is a ratio between sd-exchange and spin flip time scales ξ = τ ex τ ex ∼ 10 − 15 s τ s f ∼ 10 − 12 s τ sf Many other theories have been proposed to describe this parameter

� 15 Re-interpreting Zhang-Li A Thiaville et al, Europhys Lett 69 , 990 (2005) European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV By recognising that the pre-factors in the CIP torques and the current density j e can be expressed in terms of an e ff ective spin-drift velocity u 1 j e = P � 1 u = Pg µ B j e [ u ] = m / s 2 e 2 e M s M s the equations of motion for the magnetisation M can be written as dt = � γ 0 M ⇥ H e ff + α dt � ( u · r ) M + β d M M ⇥ d M M ⇥ [( u · r ) M ] M s M s precession damping adiabatic nonadiabatic Rearranging into a more suggestive form: ✓ ∂ ✓ ∂ ◆ ◆ M = � γ 0 M ⇥ H e ff + α ∂ t + β ∂ t + u · r M ⇥ α u · r M M s Convective derivative

� 16 Convective derivatives European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV Consider time evolution of an element dV of a fluid Convective derivative D accounts for local variations and particle flow Dt = ∂ D ∂ t + ( u · r ) ρ ( t ) Particle density dV ∂ρ ( u · r ) ρ Time ∂ t ρ ( t + δ t ) flow velocity u

� 17 Analogy with fluid dynamics? ✓ ∂ ✓ ∂ European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV ◆ ◆ M = � γ 0 M ⇥ H e ff + α ∂ t + β ∂ t + u · r M ⇥ α u · r M M s This form can almost be obtained by replacing the time derivative of the usual Landau-Lifshitz equation ∂ M ∂ t = − γ 0 M × H e ff + α M × ∂ M ∂ t M s with the convective derivative ✓ ∂ ◆ ∂ ∂ t ! ∂ t + u · r It almost works except for the β / α term. u therefore represents the average drift velocity of the magnetisation (under applied currents), which for ferromagnetic metals makes some sense. No consensus (theoretically and experimentally) over the ratio β / α , which can vary between 0.1 and 10

� 18 Spin-orbit coupling European School on Magnetism 2018, Krakow – Magnetisation Processes (MP3) – Kim,JV In magnetic multilayered structures, metallic ferromagnets in contact with 5d transition metals (“heavy metals”) exhibit strong e ff ects due to spin-orbit coupling 3d ferromagnets 5d heavy metals