Spin-Orbit Coupling Effects

Spin-Orbit Coupling Effects

Gemeinschaft - Mitglied der Helmholtz der Mitglied Spin-Orbit Coupling Effects and their modeling with the FLEUR code 24. September 2019| Gustav Bihlmayer Peter Grünberg Institut and Institute for Advanced Simulation Forschungszentrum Jülich, Germany Overview Ø basics • the Dirac equation • Pauli equation and spin-orbit coupling Ø relativistic effects in non-magnetic solids • bulk: Rashba and Dresselhaus effect • topological insulators OeNB Ø magnetic systems • Dzyaloshinskii-Moriya interaction • magnetic anisotropy images: Wikipedia/ images: 24. September 2019 Folie 2 Schrödinger type DFT Hamiltonian classical Hamiltonian image: OeNB image: continuity equation quantum mechanical Hamiltionian and interpretation of wavefunction spin enters (ad-hoc) as quantum number 24. September 2019 Folie 3 Non-collinear DFT calculation Noncollinear magnetism:120exampleo Néel state obtained from Schrödinger-type Hamiltonian: struct. energy moment (a) 5.4 meV 0 �9 (b) 5.4 meV 0 �9 (c) 0.0 meV 0.0042 � 9 (a) (b) (c) (d) (d) < 1.3 meV 0.0039 �9 Sandratskii & Kübler, PRL 76, 4963 (1996) Mn Sn 3 These configurations are indistighuishable without SOC! 24. September 2019 Folie 4 9. October 2008 NanoFerronics-2008 – p. 5 Relativistic extension by P.A.M.Dirac classical Hamiltonian Dirac’s Ansatz: image: Wikipedia image: 24. September 2019 Folie 5 2D- and 3D- Dirac equation 2D solution with Pauli spin matrices: Hˆ = c~ p~ + mc2σ · z 0 ~ 3D solution with 4x4 mat.: ~↵ = ~ 0 ✓ ◆ 2 Hˆ = βmc ~c~↵ p~ − · Bi (Z=83) bi-spinor wavefunction: In the FLEUR code we consider both components in the construction of the density. 24. September 2019 Folie 6 3D- Dirac equation Dirac equation with scalar (V) and vector potential (A): bi-spinor wavefunction: non-relativistic limit: 24. September 2019 Folie 7 Schrödinger and Pauli equation Usually, we ignore the vector potential in the Schrödinger equation: but: approximation to Dirac equation keeping terms up to 1/c2: mass- velocity term direct implementation in DarwinDFT Hamiltonian possiblespin (approximate-orbit (E+eVmagnetic)2 term ), SOC & magnetic field termterm couple the two spin channelscoupling field int. 24. September 2019 Folie 8 scalar relativistic calculations: block-diagonal equation in spin: 2 p~ e~ E + eV (~r) Bz(~r)σz + − 2m − 2mc 1 2 e~ (E + eV (~r)) + i E~ (~r) p~ =0 2mc2 (2mc)2 · ◆ with spin-dependent wave-function: 24. September 2019 Folie 9 relativistic effects in Ag and Au density of states (DOS): from www.oenb.at 24. September 2019 Folie 10 Spin-orbit coupling interaction with an (internal) magnetic field: similar to: with Thomas factor in a central potential (atom): note that the spin and the orbital momentum (L) couple antiparallel! 24. September 2019 Folie 11 Spin-orbit coupling effects in non-magnetic solids 24. September 2019 Folie 12 A typical semiconductor: Ge without SOC (l_soc=F) with SOC (l_soc=T) empty band heavy-hole band light-hole band split-off band 24. September 2019 Folie 13 Some symmetry considerations: Ge, Γ-point: Ø three p-orbitals, one split-off by SOC (atomic behavior) Ø all bands are doubly (spin) degenerate (Kramers pairs) Time reversal (TR) symmetry: ✏(~k, )=✏( ~k, ) " − # Inversion (I) symmetry: ✏(~k)=✏( ~k) − TR + I symmetry: ✏(~k, )=✏(~k, ) " # inversion center 24. September 2019 Folie 14 Broken I symmetry: Dresselhaus effect in presence of SOC: ✏ ( ~ k, ) = ✏ ( ~k, ) i.e. k-dependent spin splitting (here: α k3) " 6 # Dresselhaus Hamiltonian (kp-theory, e.g. in (111) direction): zincblende structure 24. September 2019 Folie 15 Broken I symmetry at a crystal surface Free electron gas in electric field: Suppose � = � �⃗ > and momentum confined in (x,y) plane: this describes electrons at a surface or an interface (e.g. doped layer between two semiconductors) 24. September 2019 Folie 16 Example: coinage metal surfaces (111) surface states www.almaden.ibm.cpm from from DFT calculations and SP-ARPES agree very well [experiment: Reinert et al., PRB 63, 115415 (2001)] 24. September 2019 Folie 17 Spin orientation in the Rashba effect good comparison between calculated and measured αR spin-dependent splittings require careful k-point sampling (±k) ! 24. September 2019 Folie 18 Sb2Te3 (0001) surface surface without SOC and with SOC: new states surface state with Rashba splitting 24. September 2019 Folie 19 The quantum spin Hall effect (QSHE) properties: Ø valence- and conduction band connected by edge states Ø spin-polarization of states is Rashba-like Ø one conduction channel per spin in the gap Kane & Mele, PRL 95, 226801 (2005) Ø topologically protected edge transport 24. September 2019 Folie 20 Band inversion in graphene (25μeV) Bandstructure around K (K’): mass inversion between K and K’: spin split edge state connecting K and K’ DFT calculation with SOC 24. September 2019 Folie 21 Band inversion II-VI semiconductors focus on Γ6 and Γ8: gap inv. gap responsible for band-inversion: Darwin-term of Pauli-equation 24. September 2019 Folie 22 . Spin-orbit effects in magnetic systems 24. September 2019 Folie 23 Magnetic interactions Interactions between two spins: on-site inter-site scalar traceless sym. scalar traceless sym. antisymmetric Stoner magnetic Heisenberg (pseudo)-dipolar Dzyaloshinskii magnet. anisotropy interaction interaction Moriya int. non-relativistic effects [T. Moriya, Phys. Rev. 120, 91 (1960)] 24. September 2019 Folie 24 Ansiotropic exchange ferromagnets: exchange >> Rashba splitting here: exchange interaction ≈ spin-orbit strength, e.g. two magnetic adatoms (SA,SB) on a heavy substrate with conduction electron σ: D. A. Smith, J. Magn. Magn. Mater. 1, 214 (1976) 24. September 2019 Folie 25 Ansiotropic exchange: E.g. 2 magnetic adatoms (Fe) on a heavy substrate (W) RKKY-type interaction: A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980). Dzyaloshinskii-Moriya (DM) term, anisotropic exchange interaction. 24. September 2019 Folie 26 Dzyaloshinskii-Moriya interaction: distinguish clockwise – counterclockwise rotations without SOC with SOC 24. September 2019 Folie 27 Simple 1D example: two domain walls topological index, winding number 24. September 2019 Folie 28 Simple 1D example: two domain walls topologically trivial structure 24. September 2019 Folie 29 Domain walls in 2 Fe / W(110) Due to the Dzyaloshinskii- This domain wall Moriya interaction this (Neél- does not exist! type) domain-wall is stabilized Theory: M. Heide, G. Bihlmayer and S. Blügel, Phys. Rev. B, 140403(R) (2008) 24. September 2019 Folie 30 Exchange interactions: 2-spin terms: ~ ~ H = SiJ ijSj i<j X relativistic on-site terms: i=j: trace(J)=Stoner I effects symmetric, traceless part: magnetic anisotropy intersite terms: trace(J)=Heisenberg-type exchange symmetric, traceless part: quasi-dipolar exchange antisymmetric part: Dzyaloshinskii-Moriya interaction H = J S~ S~ + D~ S~ S~ + S~T S~ ij i · j ij · i ⇥ j i Ki i i<j i X h ⇣ ⌘i X specify magnetization direction giving polar angles in the input! <soc theta=“0.00” phi=“0.00” l_soc=T ….> 24. September 2019 Folie 31 Magnetic Anisotropy: magnetization direction dependence of free energy of a cubic crystal: K F (Mˆ )=K + 1 (3 4 cos 2✓ + cos 4✓)(1 cos 4φ) + 8(1 cos 4✓) + K 0 64 { − − − } 2 4 Uniaxial system: F (Mˆ )=K0 + K1 sin ✓ + K2sin ✓ 24. September 2019 Folie 32 Effect of SOC on Fe band structure 1 2 2 dx -y dxy 0 dyz/dzx (eV) F 2 -1 dz E - dxy -2 dyz/dzx 10 k z d 2 2 H3 x -y N1 -3 R 8 Γ Γ H3 H H3 H F2 F1 6 Different band gaps open in formerly equivalent directions P (eV) F (dependingF on 4 the magnetization direction) D E - 2 G H k E. Młyńczak et al., Phys. Rev. X 6, 041048 (2016) N y k x 0 -2 24. September 2019 Folie 33 -4 H N P N Magneto-crystalline anisotropy (MCA): 2nd order perturbation theory: for a specific direction, �@, the matrix elements are: Hˆ L~ S~ ' L~ eˆ ' i| SOC| j / i| · | j / i| · | j D E D E D E Le < zx | < yz | < xy | < x2-y2 | < 3z2-r2 | | zx > 0 -iez iex -iey i√3ey | yz > iez 0 -iey -iex -i√3ex | xy > -iex iey 0 2iez 0 2 2 | x -y > iey iex -2iez 0 0 2 2 | 3z -r > -i√3ey i√3ex 0 0 0 24. September 2019 Folie 34 MCA of a molecular magnet: dimer model: HOMO level determines easy axis ΔL = Lz-Lr = 0.19 μB ΔE = Ez-Er = -13.7 meV N. Atodiresei et al., Phys. Rev. Lett. 100, 117207 (2008) 24. September 2019 Folie 35 A solid example: Magnetite Early compass: images: Wikipedia images: 24. September 2019 Folie 36 Magnetite: larger MCA with Co doping 2+ 2- 3+ 2- 4 -3 Fe3O4: (Fe O )(Fe2 O3 ) K1= -210 (J/m ) 2+ 2- 3+ 2- 6 -3 CoFe2O4: (Co O )(Fe2 O3 ) K1≈10 (J/m ) "e g " eg m= 2 1 d 0 -1 t2g -2 "t 2g " atomic spin-orbit splitting spherical cubic trigonal Co2+ Co2+ Co2+ 24. September 2019 Folie 37 Single ion anisotropy: 5 4 -3 2 1 7 Gd: K1= -1.210 K2= +8.010 (J/m ) conf: 6s 5d 4f 7 6 -3 2 1 8 Tb: K1= -5.710 K2= -4.610 (J/m ) conf: 6s 5d 4f 24. September 2019 Folie 38 Other relativistic effects in magnetism: Ø spin – other orbit coupling: H = C S~ L~ i,j i · j i,j X Ø spin – spin coupling (magnetic dipolar interaction between spin moments at the same ion) Ø quadrupole – quadrupole interaction (electrostatic interaction between electron clouds) Breit correction: captures relativistic 2-particle effects (dipole-dipole int.) e2 e2 (~↵ ~r )(~↵ ~r ) E + Hˆ + Hˆ + = ~↵ ~↵ + 1 · 1 2 · 2 1 2 r 2r 1 · 2 r2 ✓ 12 ◆ 12 12 not captured in our DFT formalism! 24. September 2019 Folie 39 Summary: relativistic effects: Ø single particle Dirac equation (can be studied with l_soc=“t”) Ø scalar relativistic effects (d-band position Au, Ag) Ø spin-orbit effects Ø T & S inversion symmetry (p1/2-p3/2 splitting) Ø T inversion symmetry (Rashba & Dresselhaus effect) Ø no T inversion symmetry (magneto-crystalline anisotropy) Ø no T & S (anisotropic exchange, Dzyaloshinskii-Moryia interaction) Ø topological effects Ø k-space: topological insulators Ø real space: magnetic skyrmions Ø two particle effects (Breit correction, dipole-dipole interaction) 24.

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