IEEE Magnetics Summer School 2019

Simulation and theory on magnetism Part 2

Kyung-Jin Lee

Dept. of Mater. Sci. & Eng.

KU-KIST Graduate School of Converging Sci. & Technol.

Korea University, South Korea

Web: https://sites.google.com/site/kyungjinlee1/

Email: [email protected] Outline (Part 2)

1. Spin-orbit coupling origin

1. SOC origin

2. Spin Hall effect, Inverse spin Hall effect, Rashba effect

2. Rashbaness of FM/HM bilayers

1. Orbital moment

2. Signatures of Rashbaness in FM/HM bilayers

3. Interfacial magnetic properties

1. Interfacial or surface magnetocrystalline anisotropy energy

2. Interfacial Dzyaloshinskii-Moriya interaction & Chiral Damping

3. Angular dependence of spin-orbit torque (SOT) 1. Spin-orbit coupling origin Spin-orbit coupling

• The electron

– Spin (angular momentum) s (~ self-rotation of electron)

– Orbital angular momentum l (~ rotation of electron around nuclei)

• For each angular momentum, there is an associated magnetic moment,

µl and µs • Spin-orbit coupling (SOC) = Interaction between these two magnetic moments Orbital magnetic moment µl

• Electron orbit  current  magnetic moment

• µl = (current) X (area of the loop) = I A • Orbit period = T = 2 π/ω

• I = (charge)/(Time) = -e/T = -eω/(2π)

2 2 • µl = I A = -eω/(2π) X (π r ) = -e ω r /2

2 • Angular momentum |l| = |r x p| = m0vr = m0ωr

•  µl = -e/(2 m0) l

• Note1: µl and l are in opposite direction (e < 0) • Note2: If |l| = h/(2π) = hbar [ground state of Bohr model]

 µB = e hbar / (2 m0) : Bohr magneton = unit of magnetic moment Spin magnetic moment µs

• (electron) Spin = intrinsic angular momentum with quantum number s = ½

• Justified by Stern-Gerlach experiment + Dirac’s relativistic theory

•  µs = -g e/(2 m0) s • g = Landé g-factor = 2.002319 for an atom (usually larger than 2 in solid)

• Note1: g = 1 classically (Bohr model), g = 2 comes from Dirac’s relativistic quantum theory

• Small corrections from quantum electrodynamics Einstein-de-Haas effect vs Barnett effect

• Einstein-de-Hass  current pulse in solenoide  B field  sample M increases  sample rotates  M ~ angular momentum

• Barnett effect  inverse effect of Einstein-de-Hass Magnetic energy  SOC

• From “nucleus” point-of-view = electron rotates around me and generates a magnetic field

• From “electron” point-of-view = nucleus rotates around me and gerenrates a magnetic field = relativistic electron’s

orbital motion  Bl

• E = -µs·Bl  SOC • Note: Also an energy for “nucleus” (because nucleus has its own spin). But it is small and contributes to hyperfine structure SOC in Dirac equation

• 1/c expansion of Dirac eq.

• 4th term = SOC : σ = spin, V = potential

When an electron passes (p) through  U = σ ⋅(∇V ×p) the region having potential gradient (= 4m2c2 E field), its spin (σ) experiences an r dV ∇V = r dr effective magnetic field (~ E x p).  1 dV  1 dV U = σ ⋅(r ×p) = σ ⋅L 4m2c2 r dr 4m2c2 r dr Why SOC?

• Electron spin passing through E field feels an effective magnetic field == SOC

• Electron spin rotates around the magnetic field and then is aligned  Spin(-polarized) current

• Spintronics  Do something with spin current

• How to generate spin current = effective magnetic field

– True magnetic field (but require huge one)

– Passing through a magnet (exchange field >> 10 T)

– Use SOC (even without FM)  spin Hall effect

• What we can do with spin current

– Spin transfer torque  manipulation of local magnetization

– Spin current <-> Charge current conversion by SOC  Onsager reciprocity  Spin Hall effect and Inverse spin Hall effect Spin Hall effect (SHE)

• Charge current (Je) is converted to spin current (Js) by SOC ~ spin version of the Hall effect

VH = RH B× Je ⇔ J S = θSH σ× Je A simple picture of SHE

• Repulsive scatterer V(r) [dV/dr < 0] • Total scattering potential 1 dV V=+⋅ Vr() LS eff 2m22 c r dr

Spin Up

Spin Down Charge current  perpendicular Spin current due to L-S coupling

Spin Hall angle θSH = JS/JC

Courtesy of Prof. H.-W. Lee Experiment: SHE

• Y. K. Kato et al., Science 306, 1910 (2004) Inverse Spin Hall effect (ISHE)

• Spin current (Js) is converted to charge current (Je)

Je = θSH σ× J s

e e e e e e Spin Up

Spin current  perpendicular Charge current due to L-S coupling

Spin Hall angle θSH = JS/JC Experiment: ISHE

• E. Saitoh et al., APL 88, 182509 (2006)

Sensing the Spin Seebeck effect Anomalous Hall effect (AHE)

• Repulsive scatterer V(r) [dV/dr < 0] • Total scattering potential 1 dV V=+⋅ Vr() LS eff 2m22 c r dr

- - Spin Up ONLY

Longitudinal charge current  Transverse Hall voltage ~ SHE Three mechanisms of AHE

• Nagaosa et al. RMP ‘09

0 σ xy ∝ σ xx

1 σ xy ∝ σ xx

Note 1: “Intrinsic” and “side-jump” are not easy to be separated in experiment Note 2: “Skew” is dominant for very good metal. 2. “Rashbaness” of FM/HM bilayers

2-1. Orbital moment for RSOC

2-2. Signatures of Rashbaness in FM/HM bilayers Ferromagnet (FM)/Heavy metal (HM) bilayer

Interfacial spin-orbit coupling α H =R σ ⋅×pzˆ RSOC 

HM

Spin Hall Effect

• Effects of Interfacial (Rashba) SOC • Field-like spin torque: Manchon & Zhang, PRB 08; Obata & Tatara, PRB 08; Matos-Abiague & Rodriguez-Suarez, PRB 09 • Damping-like spin torque: Wang & Manchon, PRL 12; Kim, Seo, Ryu, KJL, HWL, PRB 12; Pesin & MacDonald, PRB 12; Kurebayashi et al., Nat. Nano. ‘14 • Experiment: Miron et al., Nat. Mat. 10, 11; Nature 2012 (Pt|Co|AlOx) A simple picture of Rashba effect

• Relativistic correction 1 e 1  HVso=−μ B ⋅Bveff =−σ  ⋅ − ×E =σ ⋅pr ×∇ () 22m c2 4mc22

• Structural inversion asymmetry in heterostructures

z y InAlAs x InGaAs

2DEG

Let p = kF xˆ αR H =σ ⋅×(pzˆ ) H = α k σ ⋅(xˆ × zˆ) = −σ ⋅α k yˆ R  R R F R F ER = −σ ⋅Beff ⇒ Beff ∝ α RkF yˆ

Rashba SOC generates spin density along y. RSOC strength I Courtesy of H.-W. Lee

• Ast et al., Phys. Rev. Lett. 98, 186807 (2007) • “Relativistic magnetic” field + Zeeman energy  =σ × ⋅∇ HVso 22 pr() “Mysteries” of RSOC strength4mc αR HR =(σ ×⋅pz) ˆ - “Relativistic magnetic field + Zeeman

2 1  α = ∇ energy” interpretation R Too smallV α 4 mc R

1 2 Work function difference ~ (αa ) - Correlation between atomic4 SOC0 andatomic α spaciR ng 1 ~α 2 ×⋅eVA ~10-4 eV ⋅A is missing 4

a0: Bohr radius α : Fine structure constant 1/137 RSOC strength II Courtesy of H.-W. Lee S. R. Park et al., PRL 107, 156803 (2011)

• Orbital angular momentum

S z y k JL x

If NOT Bloch state

Phase difference

+10o 170o 160°  slightly constructive (electron density = sin2(-20o)) 0o 180o 180°  destructive (electron density = 0) -10o o 190 200°  slightly constructive (electron density = sin2(+20o)) RSOC strength II Courtesy of H.-W. Lee S. R. Park et al., PRL 107, 156803 (2011)

• Orbital angular momentum

S z = π x = − 1 y =ππ = =x =1 k =0.1 π,Jm = − k0.05 , LkLxx 1 k = 0.10.1π,J ,m = 1 kL 0.1 , x2 1 high J k 2 x L z z

If Bloch state  additional phase of “k a”

Phase difference low y y +10o 175o 165°  electron density = sin2(-15o) = 0.067 0o 185o 185°  electron density = sin2(+5o) = 0.008 -10o∆ ∝195o× ⋅ E (L k) (inversion205°  electron symmetry density =breaking sin2(+25o) vector)= 0.18 when SOC is strong, S is aligned along L 

∝ λSOC (S × k)⋅(inversion symmetry breaking vector) αR : Magnitude estimation “Relativistic magnetic” field Orbital-induced electric dipole + Zeeman energy + Atomic spin-orbit coupling  =σ × ⋅∇ • ∆E ∝ (Electric dipole energy) HVso 22 pr() 4mc – Relevant length scale

αR • a0, ~ A HR =(σ ×⋅pz) ˆ  – λSOC ~ 0.1 – 1 eV 2 1  • αR ~ 0.1 – 1 eV⋅A αR = ∇V 4 mc – Similar to experimental values – Explains correlation with 1 2 Work function difference ~ (αa0 ) atomic number Z 4 atomic spacing 1 ~α 2 ×⋅eVA ~10-4 eV ⋅A 4

a0: Bohr radius α : Fine structure constant 1/137 Courtesy of H.-W. Lee Interfacial spin-orbit coupling (ISOC) Heavy Metal (HM) / Ferromagnet (FM) / Oxide trilayers

• Oxide/FM interface • FM/HM interface

• Gd/O; angle-resolved photoemission • Co/Pt; ab initio calculation (Park PRB

experiment (Krupin PRB 2005) 2013)

αR ~ 1 eV∙A splitting

αR ~ 0.5 eV∙A Rashba splitting Rashba Rashbaness of various Co/HM bilayers (1) Grytsyuk, …, KJL, and Manchon, PRB 93, 174421 (2016)

Top right = charge transfer from HM-d to Co-d Bottom right = induced orbital moment

All Co/HM bilayers show “Rashbaness” A. Orbital magnetization is a measure of “Rashbaness” Rashbaness of various Co/HM bilayers (2)

• In FM/HM bilayers, many experimental reports find the effective spin Hall angle

scales with exp(-tHM/λHM) where λHM is commonly identified as spin diffusion length of HM. • The reported spin diffusion length ~ 1 nm regardless of HM (Pt, Ta, W, etc) • The resistivity of Pt << resistivity of Ta or W  very different m.f.p. • Spin diffusion length ~ (m.f.p.)1/2

The “Rashbaness” A saturates in 2~3 monolayers of HM ~ 1 nm Damping-like SOT from Berry curvature Kurebyayshi et al. Nat Nano ‘13

• (Ga, Mn)As  no HM  no bulk spin Hall effect • Well-known band structure  Rashba SOC • Damping-like SOT ~ Field-like SOT  Both SOT components well described by Rashba model SOT engineering via oxygen manipulation Qiu, …, KJL, Yang, Nat Nano ‘15

Pt 4f XPS

• SiO thickness controls the oxidation of CoFeB layer  Change in the sign of damping-like SOT (note: Pt is not oxidized) • Close correlation with the orbital moment 3. Interfacial magnetic properties

3-1. Magnetocrystalline anisotropy energy

3-2. Chiral derivative  Interfacial DMI and Chiral damping

3-3. Angular dependence of SOT MagnetoCrystalline Anisotropy Energy (MCA)

• Bruno model (PRB 39, 865 (1989))  MCA ~ orbital magnetic moment anisotropy ⊥ // MCA ~ µL − µL

2k 2 H = + J σ ⋅m +α σ ⋅(k × z) & m = (m ,0,m ) 2m sd R x z 2 2 2 2 2 2 2 2 MCA  k  k  α R (k − k y mx )  ± =  +α × ≈   +α +  Ek J sd m R (k z)  J sd Rk ymx  2m 2m  2J sd  dk 2 E(m ) = (E + + E − ) ⇒ MCA = E(m =1)− E(m = 0) ≈ Cα 2 x ∫ (2π )2 k k x x R

Shi et al. PRL ‘07 e ˆ 2 OM μ = i ∑ ∇kunk [ε n (k) − H 0 (k)]∇kunk ~ α R M z 2 nk

Rashba model is able to capture the core physics of MCA. Interfacial magnetocrystalline anisotropy Kim, KJL, … PRB 94, 184402 (2016)

9 Free electron model (parabolic band) α . o R [eV A]  MCA vanishes once minority band 6 0.2 PMA 0.5 ) is occupied. 2 3 0.8 Tight-binding model 0 Jsd = 1 eV, me= m0 Square lattice -3 IMA IMA

MCA(mJ/m -6 Majority band Minority band -9 0.0 0.5 1.0 Electron Density (normalized)

• RSOC  Interfacial MCA • Band filling determines PMA / IMA Interfacial magnetocrystalline anisotropy

Tight-binding model First principles J = 1 eV, m = m sd e 0 Daalderop et al. PRL 68, 682 (1992). Triangular lattice Co/Ni, Co/Pd multilayers (fcc 111)

• Simple Rashba model captures the core effect of MCA Interfacial spin-orbit spin torques

“Magnetic interfaces (inversion asymmetry) + Spin-Orbit coupling”

Field-like torque Damping-like torque (Slonczewski-like)

2αR me α =ˆ × ˆ 2 R me Tf 2 bj ym T=−βb mˆ ××( ymˆ ˆ )  d 2 j

Adiabatic torque Nonadiabatic torque

Tmadia =bjx ∂ ˆ Tnon =−βbjx mmˆ ×∂ ˆ

Adiabatic + field-like torque Nonadiabatic + damping-like torque     Tmadia =bjx ∂ ˆ Tnon =−βbjx mmˆ ×∂ ˆ

 22ααRRmmee ∂≡∂+×=∂+×xxmˆ yˆ mˆ x mˆ ymˆ ˆ 22 Effects of Interfacial SOC (through 2D Rashba model Hamiltonian)

“Magnetic interfaces (inversion asymmetry) + Spin-Orbit coupling”

2D Rashba Hamiltonian

Unitary transformation

K.-W. Kim et al. PRL 111, 216601 (2013) Chiral derivative

 22ααRRmmee ∂≡∂+×=∂+×xxmˆ yˆ mˆ x mˆ ymˆ ˆ 22 “Chiral” derivative K.-W. Kim et al. PRL 111, 216601 (2013)

 22ααRRmmee ∂≡∂+×=∂+×xxmˆ yˆ mˆ x mˆ ymˆ ˆ 22

 ∂=xmˆ 0 z x θ y dθ 2α m = R e = precession rate of conduction dx 2 electron spins due to RSOC  No current-induced torque if magnetization precesses at the same rate as the conduction electrons would due to RSOC  RSOC induces the chirality in the spin torques  “Chiral” derivative captures RSOC effects Exchange “stiffness” energy

∂ˆ ⋅∂ ˆ A∫ dV xxmm Exchange “stiffness” interaction mediated by conduction electrons • Conduction electron mediation ∂ˆ ⋅∂ ˆ A∫ dV xxmm RSOC Continuum version of interfacial ∂ˆ ⋅∂ ˆ Dzyaloshinskii-Moriya interaction (DMI) A∫ dV xxmm 4α m =A dV ∂mˆ ⋅∂ mˆ + AR e dVymˆ ⋅ˆ ×∂ mˆ ∫∫xx 2 x 2 + O(αR ) A simple picture: why RSOC  DMI

• DMI = Current-INDEPENDENT magnetic energy

 Contributions from spins with +k + Cont. from spins with –k ≠ 0

• RSOC  effective B field in k x (interface normal)

Initial Final k > 0  B

Final Initial k < 0  B

Conduction electron spins are chiral  + s-d exchange Localized electron spins (m) tend to be chiral Spin motive force (SMF)

 Berger, PRB 1986 E± =±∑ xˆ it ∂ mˆ ×∂ i mmˆ ⋅ ˆ Volovik, JPC 1987 = 2e i xy, Barnes & Maekawa, PRL With RSOC 2007    E± =±∑ xˆ it ∂× mˆ ∂i mmˆ ⋅ˆ i= xy, 2e   2α m =±ˆ ∂ˆ ×∂ ˆ ⋅ ˆ ±ˆ ∂׈ R e ˆ ׈ ׈ ⋅ˆ ∑ xit m i mm ∑ xit m2 ( zxi) mm i= xy, 2e i= xy, 2e    α m = E ±R e zˆ ×∂ mˆ Kim, Moon, KJL, HWL, PRL 2012 ± e t Foros et al. PRB ’08 SMF Tserkovnyak & Wong, PRB ‘09 Zhang & Zhang, PRL 2009 m j  Feedback-induced modification of Landau-Lifshitz-Gilbert eq. STT Feedback effect on LLG equation (1)

• Feedback-induced damping torque ∂m Tm=×⋅δα feedback G ∂t

 22ααmm  ∂m µσ (δα ) =+ηηXXRReeεε +,, X =×=m Bc G ij ∑ ki  33ki  kj  kj  ki ∂ 2 k    xk i 2eMs Zhang & Zhang, PRL ’09 Kim, Moon, KJL, HWL, PRL 2012

Single domain Domain wall α + Cα 2 0 0    x   x   x   0 R  mˆ = cosφ sech , sinφ sech , tanh  ~ 2   λ   λ   λ  α =  0 α0 + Cα R 0    0 0 α ~ C  x  2  0  α 22 ≈ α0 − cosφ sech α R + O(α R ) λ  λ   Anisotropic enhanced Gilbert damping  Magnetic texture-dependent Chiral damping Chiral damping

zˆ . Interfacial Rashba effect ∝×α ˆ xˆ Effective magnetic field BRR( kz) yˆ

conduction spin

localized spin (m)

conduction spin

localized spin (m)

: Non-compensated chirality of conduction electron spin  Chirality of localized electron spin due to s-d exchange coupling (Dzyaroshinskii-Moriya interaction) : More frequent scattering  chiral damping 41 Spin-dependent magnetic field

 Volovik, JPC 1987 B± = zˆ ( ∂xy mˆ ×∂ mmˆ ) ⋅ ˆ 2e

With RSOC RSOC contribution to anomalous Hall effect    B± = zˆ (∂∂xy mˆ ×⋅ mmˆ ) ˆ (Bijl-Duine PRB ‘12) 2e 2 2  αR me 2αR me =zˆ ( ∂x mmˆ ×∂y ˆ )⋅mˆ ±∇××(zˆ mˆ ) ± mz zˆ 2e e e3

Topological Hall Effect (2nd Rashba term)/(1st Conv. Term) ~ 1 SdMF β becomes tensor, Spin spirals can be detected m anisotropic, and j texture-dependent STT Angular dependence of SOTs: Previous Experiments

SOT =τ f mˆ × yˆ +τ d mˆ ×(mˆ × yˆ)

Experiment: Garello et al. Nat. Nanotech. 2013

τ f τ d • Both τf and τd are angle-dependent.

Experiment: Qiu et al. Sci. Rep. 2013

τ d τ f Angular dependence of SOTs: Previous Theories

SOT =τ f mˆ × yˆ +τ d mˆ ×(mˆ × yˆ)

• Bulk SOC in HM (SHE: Haney et al. PRB ‘13)  No angular dependence (1st order of SH angle) • Interfacial SOC (Rashba: Pauyac et al. APL ‘13)  Weak Rashba regimes: both are almost constant

 Strong Rashba regimes: τf = constant, τd = angle-dependent

Berry phase contribution

1st principles: Pt/Co, Haney to τd (Kurebayashi et al. Nat. Nano. ‘14) Fermi surface distortion due to Rashba SOC

Ferromagnet αRkF/Jsd = 0.5

αR = 0 & Jsd ≠ 0 M = (0, 1, 0) M = (0, 1, 0)

Rashba non-magnet αRkF/Jsd = 0.5 α ≠ 0 & J = 0 R sd M = (1, 0, 0) SOT: Fermi surface contribution  τf

E-field M = Mz M = Mx

ky ky

kx kx

Circle & Concentric Ellipse & Non-concentric

τf(Mz) ≠ τf(Mx) SOT: Fermi sea contribution (Berry phase)  τd E-field

M = Mz M = Mx

ky ky

kx kx

Circle & Concentric Ellipse & Non-concentric

τd(Mz) ≠ τd(Mx) Field-like torque (τf): Fermi surface contribution Lee et al. PRB 91, 144401 (2015) z y mˆ = (cosφ sinθ, sinφ sinθ, cosθ ) J x EF = 10 eV, Jsd = 1 eV, m* = m0

Strong angular dependence near αRkF/Jsd = 1  FS distortion Damping-like torque (τd): Fermi sea contribution

z y mˆ = (cosφ sinθ, sinφ sinθ, cosθ ) J x EF = 10 eV, Jsd = 1 eV, m* = m0

Strong angular dependence near αRkF/Jsd = or > 1 Angle-dependent SOT: Switching and DW motion Seo-Won Lee and KJL, JKPS ‘15

∂mˆ ∂mˆ 2 = −γmˆ × H + α mˆ × + γτ mˆ × [(mˆ × yˆ ) + zˆ(mˆ ⋅ xˆ )(χ + χ (mˆ × zˆ) )] ∂t eff ∂t d 0 2 4 Angle-dependent part

Single domain switching

 Γ − h4 + 20h2 h6 + (h5 − 4h3 +12h) 8 + h2 +12h2 −16  τ sw = H  + χ , d 0 K ,eff  4 2 2   4 2 16 2 Γ − h + 20h 

DW speed  χ 3  λD τ 1+ 2 + χ  π 0 d 0  2 8 4  v = γ . DW 2 2  χ 3  D2α 2 + M 2τ 2 1+ 2 + χ  λ2 0 S d 0  2 8 4  Summary

1. Crucial role of SOC in magnetic properties and spin transport 2. Understanding of these SOC-related phenomena based on orbital moment 3. Rashbaness is always preserved for inversion symmetry broken systems 4. Orbital engineering is key to make SOC phenomena useful