Magnetic interactions Ingrid Mertig Martin-Luther-Universitt - PowerPoint PPT Presentation

Magnetic interactions Ingrid Mertig Martin-Luther-Universität Halle-Wittenberg Cargèse 13.10.2017

Outline • Introduction • Interactions • Models • STONER model • HEISENBERG model Cargèse 13.10.2017

Introduction

Quantum mechanical description of solids R r H = ˆ ˆ T I + ˆ T e + V II + V ee + + V eI Cargèse 13.10.2017

Quantum mechanical description of solids H = ˆ ˆ T I + ˆ T e + V II + V ee + + V eI Adiabatic approximation Electrons: H ( R ) = ˆ ˆ T e ( R ) + V ee ( R ) + + V eI ( R ) Ions: H = ˆ ˆ T I + V II + E ( R ) R = { R 1 , R 2 , R 3 , ... } Cargèse 13.10.2017

Solution of the electron problem H ( R ) = ˆ ˆ T e ( R ) + V ee ( R ) + + V eI ( R ) Many-electron Schrödinger equation: ˆ H ( R ) Φ ( r , R ) = E ( R ) Φ ( r , R ) Electron coordinates: r = { r 1 , r 2 , r 3 , ... } Fixed ion coordinates: R = { R 1 , R 2 , R 3 , ... } Cargèse 13.10.2017

Solution of the electron problem Many-electron Schrödinger equation: ˆ H ( R ) Φ ( r , R ) = E ( R ) Φ ( r , R ) • Free electrons • Hartree approximation • Hartree-Fock approximation • Density functional theory One-electron Schrödinger equation: H ϕ ( r ) = ( − ~ 2 ∂ 2 ˆ ∂ r 2 + V ( r )) ϕ ( r ) = εϕ ( r ) 2 m Cargèse 13.10.2017

Magnetic interactions

Interactions There is no elementary magnetic interaction! Dipol-dipol interaction between magnetic moments: E DD ( R ) = 1 R 3 ( M 1 · M 2 − 3( M 1 · ˆ R )( M 2 · ˆ R )) M ∼ 1 µ B e ~ E DD ∼ 10 − 5 eV µ B = 2 mc Cargèse 13.10.2017

Hartree-Fock approximation Exchange interaction caused by Pauli principle: Ansatz for the wave function: 1 Φ HF ( r 1 ... r i ... r N ) = det | ϕ α i ( r i ) | √ N ! Hartee-Fock energy: N Z α i ( r ) ˆ X d 3 r ϕ ∗ E HF [ ϕ α ] = H ( r ) ϕ α i ( r ) i ✏ 2 +1 Z X d 3 rd 3 r 0 | r − r 0 | [ ' ⇤ α i ( r ) ' α i ( r ) ' ⇤ α j ( r 0 ) ' α j ( r 0 ) 2 i 6 = j − ϕ ⇤ α j ( r ) ϕ α i ( r ) ϕ ⇤ α i ( r 0 ) ϕ α j ( r 0 )] Exchange of two electrons!

Limits of magnetic phenomena Electrons in isolated atoms: Mostly magnetic, Hund‘s rule Electrons in an ideal Fermi gas: Mostly non-magnetic Cargèse 13.10.2017

Localisation of the electrons Atomic orbitals: localised Bloch waves: delocalised Cargèse 13.10.2017

Localisation of the electrons Degree of electron localisation causes magnetism or not! • Simple metals and semiconductors: non-magnetic • Rare earth atoms: atomic magnetic moments • Transition metals and actinide: weakly localised electrons Cargèse 13.10.2017

Interatomic exchange Direct exchange: Indirect exchange: Superexchange: Mn ++ O - - Mn ++ Itinerant exchange: magnetism of delocalised electrons Cargèse 13.10.2017

Models

Models Magnetic insulators: Magnetic metals: EuO, EuS, MnO, … Fe, Co, Ni, … ISING HEISENBERG HUBBARD i σ a j σ + 1 X X X ˆ ˆ t ij a + H = 2 U n i σ n i − σ H = − I ij s i · s j ij ij σ i σ X ˆ n i σ = a + H = − J ij s i s j i σ a i σ σ = ± 1 ij s i = ± 1 2 Mean field approximation < ˆ A ˆ B > = ˆ A < ˆ B > + < ˆ A > ˆ B − < ˆ A >< ˆ B > WEISS STONER Cargèse 13.10.2017

STONER model

STONER model One-electron Schrödinger equation for spin-dependent potential: ( − ~ 2 ∂ 2 ∂ r 2 + V ± ( r )) ϕ ± m ( r ) = ε ± m ϕ ± m ( r ) 2 m Charge density: m ( r ) | 2 + X X n ( r ) = n + ( r ) + n − ( r ) = | ϕ + m ( r ) | 2 | ϕ − m m Magnetization density: X X m ( r ) = n + ( r ) − n − ( r ) = | ϕ + m ( r ) | 2 − m ( r ) | 2 | ϕ − m m Cargèse 13.10.2017

Magnetization density and magnetization X X m ( r ) = n + ( r ) − n − ( r ) = | ϕ + m ( r ) | 2 − m ( r ) | 2 | ϕ − m m Local magnetic moment per unit cell M Z d 3 r m ( r ) M = V Z

STONER model One-electron Schrödinger equation for spin-dependent potential: ( − ~ 2 ∂ 2 ∂ r 2 + V ± ( r )) ϕ ± m ( r ) = ε ± m ϕ ± m ( r ) 2 m Spin-dependent potential: V ± ( r ) = V ( r ) ⌥ 1 Z d 3 r m ( r ) M = 2 IM V Z Cargèse 13.10.2017

Spin-polarized band structure Wave function unchanged by spin polarization, constant potential: ϕ ± m ( r ) = ϕ m ( r ) Splitting of the eigenvalues: m = ε m ⌥ 1 ε ± 2 IM ε ± ε k Cargèse 13.10.2017

Spin-polarized density of states D + Majority electrons D ± ( E ) = D 0 ( E ± 1 2 IM ) E Minority electrons D − E F

STONER model Number of electrons: Z E F N = dE { D 0 ( E + IM/ 2) + D 0 ( E − IM/ 2) } Magnetic moment: Z E F M = dE { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } Fixed: To be determined: N, D 0 ( E ) E F , M Z E F ( M ) F ( M ) = dE { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } Self-consistent solution Cargèse 13.10.2017

STONER model Properties of F(M): • . F (0) = 0 • bzw. F ( − M ) = − F ( M ) E F ( − M ) = E F ( M ) • and − M ∞ ≤ F ( M ) ≤ M ∞ F ( ± ∞ ) = ± M ∞ • monotonically increasing F 0 ( M ) ≥ 1 + M ∞ − M ∞ D + D + E E D − D − E F E F Cargèse 13.10.2017

STONER model Z E F ( M ) F ( M ) = dE { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } Z E F ( M ) dF dE [ d dM = dM { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } + { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } dE F dM ] Z E F ( M ) F 0 ( M ) = dE [ { D 0 ( E + IM/ 2) + D 0 ( E − IM/ 2) } + { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } dE F dM ] Cargèse 13.10.2017

STONER model dE F Calculation of from dN = 0 dM dN = dN dE F + dN dM dM = 0 dE F Z E F N = dE { D 0 ( E + IM/ 2) + D 0 ( E − IM/ 2) } ( D + 0 ) dE F dM = I 0 − D − 0 ) dE F + I 0 = ( D + 2( D + 0 + D − 0 ) dM 0 − D − ( D + 2 0 + D − 0 ) Cargèse 13.10.2017

STONER model Z E F ( M ) F 0 ( M ) = dE [ { D 0 ( E + IM/ 2) + D 0 ( E − IM/ 2) } + { D 0 ( E + IM/ 2) − D 0 ( E − IM/ 2) } dE F dM ] ( D + 0 ) dE F dM = I 0 − D − ( D + 2 0 + D − 0 ) 0 ) { 1 − ( D + 0 ) 2 0 − D � F 0 ( M ) = I 2( D + 0 + D � 0 ) 2 } ≥ 0 ( D + 0 + D � Cargèse 13.10.2017

STONER model Paramagnetic solution: • trivial solution M=0 F(M) M Cargèse 13.10.2017

STONER model Ferromagnetic solution: • trivial solution M=0 • two solutions with spontaneous magnetization M S ± F(M) B M -M S +M S STONER criterion: F 0 (0) = ID 0 ( E F ) > 1 Cargèse 13.10.2017

STONER model STONER criterion: F 0 (0) = ID 0 ( E F ) > 1 D 0 ( E F ) [ eV − 1 ] M [ µ B /atom ] I [ eV ] ID 0 ( E F ) Na 0.23 1.82 0.41 Al 0.21 1.22 0.25 Cr 0.35 0.76 0.27 Mn 0.77 0.82 0.63 Fe 1.54 0.93 1.434 2.22 Co 1.72 0.99 1.70 1.71 Ni 2.02 1.01 2.04 0.61 Cu 0.14 0.73 0.11 Pd 1.14 0.68 0.78 Pt 0.79 0.63 0.5 Cargèse 13.10.2017

Density of states for bulk ferromagnets Cargèse 13.10.2017

HEISENBERG model

Magnons and second quantization Dispersion relation of spin waves in ferromagnets: (only one basis atom) . basis atoms lead to magnon branches Cargèse 13.10.2017

Magnons in second quantization Hamiltonian: lowers z component raises z component Bosonization: is the ground state (magnon vacuum); analyze small fluctuations Holstein-Primakoff Semiclassic approximation transformation (linear spin-wave theory) "cold large spins" creates a boson destroys a boson counts bosons Plug this into Hamiltonian and ignore non-bilinear terms. Cargèse 13.10.2017

Magnons in second quantization Free-boson Hamiltonian: creates a magnon Fourier transformation: destroys a magnon k -space Hamiltonian: energy with (only one basis atom) basis atoms lead to magnon branches . Cargèse 13.10.2017

Topological states and magnetism Ingrid Mertig Martin-Luther-Universität Halle-Wittenberg Cargèse 14.10.2017

Outline • Introduction • Topological electron states • The quantum Hall effects • The topological Hall effect • Summary Cargèse 14.10.2017

What is a Berry phase? Cargèse 14.10.2017

Schrödinger equation and adiabatic evolution C M. V. Berry, Proc. R. Soc. A 392 , 1802 (1984) Cargèse 14.10.2017

What is a Berry curvature? Berry phase: Berry connection: Berry curvature: M. V. Berry, Proc. R. Soc. A 392 , 1802 (1984)

Berry curvature of Bloch states 10 9.5 Energy (eV) 9 8.5 Ω ( k ) H ( k ) 8 U G Z F G L

Semiclassical equation of motion Change of momentum: Lorentz force Change of position: Anomalous velocity M.-C. Chang and Q. Niu, Phys. Rev. B 53 , 7010 (1996)

Transversal transport coefficients

Ohm’s law and conductivity tensor   0 σ xx σ xy     0 σ yx σ yy       0 0 σ zz Cargèse 14.10.2017

The Hall trio

The Hall trio Hall effect Anomalous Hall effect Spin Hall effect 1879 1881 2004 Lorentz force Berry curvature spin-orbit interaction: Nagaosa, Sinova et al., Rev. Mod. Phys. 82 , 1539 (2010)