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Magnetic interactions Ingrid Mertig Martin-Luther-Universitt Halle-Wittenberg Cargse 13.10.2017 Outline Introduction Interactions Models STONER model HEISENBERG model Cargse 13.10.2017 Introduction

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SLIDE 1

Magnetic interactions

Cargèse 13.10.2017

Ingrid Mertig Martin-Luther-Universität Halle-Wittenberg

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SLIDE 2
  • Introduction
  • Interactions
  • Models
  • STONER model
  • HEISENBERG model

Cargèse 13.10.2017

Outline

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SLIDE 3

Introduction

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SLIDE 4

Cargèse 13.10.2017

Quantum mechanical description of solids

R r ˆ H = ˆ TI + ˆ Te + VII + Vee + +VeI

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SLIDE 5

Cargèse 13.10.2017

Quantum mechanical description of solids

ˆ H = ˆ TI + ˆ Te + VII + Vee + +VeI ˆ H(R) = ˆ Te(R) + Vee(R) + +VeI(R)

Adiabatic approximation Electrons: Ions:

ˆ H = ˆ TI + VII + E(R) R = {R1, R2, R3, ...}

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SLIDE 6

Cargèse 13.10.2017

Solution of the electron problem

ˆ H(R) = ˆ Te(R) + Vee(R) + +VeI(R) ˆ H(R)Φ(r, R) = E(R)Φ(r, R)

Many-electron Schrödinger equation:

R = {R1, R2, R3, ...} r = {r1, r2, r3, ...}

Electron coordinates: Fixed ion coordinates:

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SLIDE 7

Cargèse 13.10.2017

Solution of the electron problem

  • Free electrons
  • Hartree approximation
  • Hartree-Fock approximation
  • Density functional theory

ˆ H(R)Φ(r, R) = E(R)Φ(r, R)

Many-electron Schrödinger equation: One-electron Schrödinger equation:

ˆ Hϕ(r) = (− ~2 2m ∂2 ∂r2 + V (r))ϕ(r) = εϕ(r)

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SLIDE 8

Magnetic interactions

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SLIDE 9

Cargèse 13.10.2017

Interactions

There is no elementary magnetic interaction! Dipol-dipol interaction between magnetic moments:

EDD(R) = 1 R3 (M1 · M2 − 3(M1 · ˆ R)(M2 · ˆ R))

M ∼ 1µB

µB = e~ 2mc

EDD ∼ 10−5eV

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SLIDE 10

Hartree-Fock approximation

Exchange interaction caused by Pauli principle: Ansatz for the wave function: Hartee-Fock energy:

ΦHF (r1...ri...rN) = 1 √ N! det |ϕαi(ri)| EHF [ϕα] =

N

X

i

Z d3rϕ∗

αi(r) ˆ

H(r)ϕαi(r) +1 2 X

i6=j

Z d3rd3r0 ✏2 |r − r0|['⇤

αi(r)'αi(r)'⇤ αj(r0)'αj(r0)

−ϕ⇤

αj(r)ϕαi(r)ϕ⇤ αi(r0)ϕαj(r0)]

Exchange of two electrons!

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SLIDE 11

Cargèse 13.10.2017

Limits of magnetic phenomena

Electrons in isolated atoms: Mostly magnetic, Hund‘s rule Electrons in an ideal Fermi gas: Mostly non-magnetic

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SLIDE 12

Localisation of the electrons

Atomic orbitals: localised Bloch waves: delocalised

Cargèse 13.10.2017

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SLIDE 13

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

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SLIDE 14

Cargèse 13.10.2017

Interatomic exchange

Direct exchange: Itinerant exchange: magnetism of delocalised electrons Indirect exchange: Superexchange:

Mn++ O- - Mn++

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SLIDE 15

Models

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SLIDE 16

Cargèse 13.10.2017

Models

ISING Magnetic insulators: EuO, EuS, MnO, … HEISENBERG HUBBARD Magnetic metals: Fe, Co, Ni, … ˆ H = − X

ij

Iij si · sj ˆ H = − X

ij

Jijsisj ˆ H = X

ijσ

tija+

iσajσ + 1

2U X

iσ

niσni−σ

niσ = a+

iσaiσ

σ = ±1 2 si = ±1

Mean field approximation WEISS STONER < ˆ A ˆ B >= ˆ A < ˆ B > + < ˆ A > ˆ B− < ˆ A >< ˆ B >

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SLIDE 17

STONER model

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SLIDE 18

Cargèse 13.10.2017

STONER model

One-electron Schrödinger equation for spin-dependent potential: Charge density: Magnetization density:

m(r) = n+(r) − n−(r) = X

m

|ϕ+

m(r)|2 −

X

m

|ϕ−

m(r)|2

n(r) = n+(r) + n−(r) = X

m

|ϕ+

m(r)|2 +

X

m

|ϕ−

m(r)|2

(− ~2 2m ∂2 ∂r2 + V ±(r))ϕ±

m(r) = ε± mϕ± m(r)

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SLIDE 19

Magnetization density and magnetization

M = Z

VZ

d3r m(r) m(r) = n+(r) − n−(r) = X

m

|ϕ+

m(r)|2 −

X

m

|ϕ−

m(r)|2

Local magnetic moment per unit cell M

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SLIDE 20

Cargèse 13.10.2017

STONER model

One-electron Schrödinger equation for spin-dependent potential:

V ±(r) = V (r) ⌥ 1 2IM

Spin-dependent potential:

M = Z

VZ

d3r m(r) (− ~2 2m ∂2 ∂r2 + V ±(r))ϕ±

m(r) = ε± mϕ± m(r)

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SLIDE 21

Cargèse 13.10.2017

Spin-polarized band structure

Wave function unchanged by spin polarization, constant potential: Splitting of the eigenvalues:

ϕ±

m(r) = ϕm(r)

ε±

m = εm ⌥ 1

2IM

k

ε± ε

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SLIDE 22

Spin-polarized density of states

D±(E) = D0(E ± 1 2IM)

D+ D− E

Majority electrons Minority electrons

EF

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SLIDE 23

Cargèse 13.10.2017

STONER model

Number of electrons: Magnetic moment: N = Z EF dE{D0(E + IM/2) + D0(E − IM/2)} M = Z EF dE{D0(E + IM/2) − D0(E − IM/2)} Fixed: To be determined: F(M) = Z EF (M) dE{D0(E + IM/2) − D0(E − IM/2)} N, D0(E) EF , M Self-consistent solution

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SLIDE 24

Cargèse 13.10.2017

STONER model

Properties of F(M):

  • .
  • bzw.
  • and
  • monotonically increasing

F(±∞) = ±M∞ −M∞ ≤ F(M) ≤ M∞ F(0) = 0 EF (−M) = EF (M) F 0(M) ≥ 1 D+ D− E EF D+ D− E EF +M∞ −M∞ F(−M) = −F(M)

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SLIDE 25

Cargèse 13.10.2017

STONER model

F(M) = Z EF (M) dE{D0(E + IM/2) − D0(E − IM/2)}

dF dM = Z EF (M) dE[ d dM {D0(E + IM/2) − D0(E − IM/2)} F 0(M) = Z EF (M) dE[{D0(E + IM/2) + D0(E − IM/2)} +{D0(E + IM/2) − D0(E − IM/2)}dEF dM ] +{D0(E + IM/2) − D0(E − IM/2)}dEF dM ]

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SLIDE 26

Cargèse 13.10.2017

STONER model

dN = dN dEF dEF + dN dM dM = 0

Calculation of from

dEF dM dN = 0

N = Z EF dE{D0(E + IM/2) + D0(E − IM/2)}

0 = (D+

0 + D− 0 )dEF + I

2(D+

0 − D− 0 )dM

dEF dM = I 2 (D+

0 − D− 0 )

(D+

0 + D− 0 )

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SLIDE 27

Cargèse 13.10.2017

STONER model

F 0(M) = Z EF (M) dE[{D0(E + IM/2) + D0(E − IM/2)} +{D0(E + IM/2) − D0(E − IM/2)}dEF dM ] dEF dM = I 2 (D+

0 − D− 0 )

(D+

0 + D− 0 )

F 0(M) = I 2(D+

0 + D 0 ){1 − (D+ 0 − D 0 )2

(D+

0 + D 0 )2 } ≥ 0

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SLIDE 28

Cargèse 13.10.2017

STONER model

M F(M)

Paramagnetic solution:

  • trivial solution M=0
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SLIDE 29

Cargèse 13.10.2017

STONER model

Ferromagnetic solution:

  • trivial solution M=0
  • two solutions with spontaneous magnetization

MS ±

M F(M) B +MS

  • MS

STONER criterion: F 0(0) = ID0(EF ) > 1

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SLIDE 30

Cargèse 13.10.2017

STONER model

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

D0(EF ) [eV −1] I [eV ] ID0(EF ) STONER criterion: F 0(0) = ID0(EF ) > 1 M [µB/atom]

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SLIDE 31

Cargèse 13.10.2017

Density of states for bulk ferromagnets

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SLIDE 32

HEISENBERG model

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SLIDE 33

Cargèse 13.10.2017

Magnons and second quantization

Dispersion relation of spin waves in ferromagnets: (only one basis atom) basis atoms lead to magnon branches

.

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SLIDE 34

Cargèse 13.10.2017

Magnons in second quantization

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

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SLIDE 35

Cargèse 13.10.2017

Magnons in second quantization

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

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SLIDE 36

Topological states and magnetism

Cargèse 14.10.2017

Ingrid Mertig Martin-Luther-Universität Halle-Wittenberg

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SLIDE 37
  • Introduction
  • Topological electron states
  • The quantum Hall effects
  • The topological Hall effect
  • Summary

Cargèse 14.10.2017

Outline

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SLIDE 38

What is a Berry phase?

Cargèse 14.10.2017

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SLIDE 39

Schrödinger equation and adiabatic evolution

Cargèse 14.10.2017

  • M. V. Berry, Proc. R. Soc. A 392, 1802 (1984)

C

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SLIDE 40

Berry phase: Berry connection: Berry curvature:

  • M. V. Berry, Proc. R. Soc. A 392, 1802 (1984)

What is a Berry curvature?

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SLIDE 41

8 8.5 9 9.5 10 U G Z F G L Energy (eV)

H(k) Ω(k)

Berry curvature of Bloch states

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SLIDE 42

Change of momentum: Change of position:

M.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010 (1996)

Lorentz force Anomalous velocity

Semiclassical equation of motion

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SLIDE 43

Transversal transport coefficients

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SLIDE 44

Cargèse 14.10.2017

      σxx σxy σyx σyy σzz      

Ohm’s law and conductivity tensor

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SLIDE 45

The Hall trio

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SLIDE 46

Hall effect Anomalous Hall effect Spin Hall effect 1879 1881 2004 Lorentz force spin-orbit interaction: Berry curvature

Nagaosa, Sinova et al., Rev. Mod. Phys. 82, 1539 (2010)

The Hall trio

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SLIDE 47

σ±

xy =

e2 ~(2π)3 X

n

Z

BZ

d3kfn(k)Ωn

z (k)

σ±

xy =

e2 ~(2π)3 X

n

Z EF dEΩn

z (E)

Anomalous Hall effect: Spin Hall effect:

σs

xy = σ+ xy − σ− xy

σxy = σ+

xy + σ− xy

Intrinsic anomalous and spin Hall conductivity

Cargèse 14.10.2017

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SLIDE 48

Intrinsic spin Hall conductivity

Pt: 2000 Au: 400 Guo et al., PRL 100, 096401 (2008); J. Appl. Phys. 105, 07C701 (2009)

σs

xy

σs

xy

Cargèse 14.10.2017

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SLIDE 49

Diabolic points

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SLIDE 50
  • J. von Neumann, E. Wigner, Phys. Z. 1928

kx kz k0 kz kx

Band crossing and diabolic points

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SLIDE 51

Point charge field: E±(r) = ±Q r − r0 |r − r0|3

P.A.M. Dirac, Phys. Rev. 1948

Magnetic monopole: B±(r) = ±g r − r0 |r − r0|3 Berry curvature monopole: Ω±(k) = ±g k − k0 |k − k0|3

Point charge and magnetic monopole

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SLIDE 52

P.A.M. Dirac, Phys. Rev. 1948

Monopole field: Dirac‘s quantization of the monopole field: gj = ±1 2 B(r) = ⇤ ⇥ A(r) + X

j

gj r rj |r rj|3 B±(r) = ±g r − r0 |r − r0|3 1 2π Z

V

dr ⇥ · B(r) = 1 2π Z

∂V

dσ n · B(r) = C C Z

Dirac quantization

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SLIDE 53

P.A.M. Dirac, Phys. Rev. 1948

Monopole field: Dirac‘s quantization of the monopole field: Ω±(k) = ±1 2 k − k0 |k − k0|3 1 2π Z

V

dk ⇥ · Ω(k) = 1 2π Z

∂V

dσ n · Ω(k) = C C Z gj = ±1 2 Ω(k) = ⇤ ⇥ A(k) + X

j

gj k kj |k kj|3

Berry curvature monopoles

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SLIDE 54
  • J. Henk, A. Ernst, S. V. Eremeev, E. V. Chulkov, I. V. Maznichenko, and I. M.
  • Phys. Rev. Lett. 108, 206801 (2012)
  • J. Henk, M. Flieger, I. V. Maznichenko, I. M., A. Ernst, S. V. Eremeev,

and E. V. Chulkov, Phys. Rev. Lett. 109, 076801 (2012)

  • P. Barone, T. Rauch, D. di Sante, J. Henk, I. M., and S. Picozzi, Phys. Rev. B

88, 045207 (2013)

  • T. Rauch, M. Flieger, J. Henk, and I. M., Phys. Rev. B 88, 245120 (2013)
  • T. Rauch, M. Flieger, J. Henk, I. M., and A. Ernst, Phys. Rev. Lett. 112, 016802

(2014)

  • T. Rauch, S. Achilles, J. henk, and I. M., Phys. Rev. Lett. 114, 236805 (2015)

Börge Göbel Tomáš Rauch Alexander Mook Jürgen Henk

Topological states

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SLIDE 55

kx ky

+

  • Intrinsic spin Hall conductivity
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SLIDE 56

+

  • kx

ky

+

  • Spin Hall effect of an insulator and Chern number
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SLIDE 57

Chern number

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SLIDE 58

+ +

  • SOC

+ +

  • +

+

  • CHERN insulator:

Gap in 2d WEYL semimetal: Gapless in 3d

Band inversion without TRS

Cargèse 14.10.2017

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SLIDE 59

SOC

+ +

  • Z2 TI:

gap in 2d and 3d Kramers degeneracy

+ +

DIRAC semimetal: no gap in 3d + crystal symmetry

+ +

  • Band inversion with TRS

Cargèse 14.10.2017

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SLIDE 60
  • B. A. Bernevig, T. L. Hughes, S. C. Zhang, Science 314, 1757 (2006)

Topological surface state of a Z2 TI

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SLIDE 61
  • T. Rauch, M. Flieger, J. Henk, and I. M., Phys. Rev. B 88, 245120 (2013)
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.0 0.1 0.2 0.3 0.4

Energy in eV

K Γ M

Bi2Te3 10

K Γ M

SnTe

K Γ M

Topological surface state in Bi2Te3

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SLIDE 62

M

S Oh Science 2013;340:153-154

The conductance is quantized!

The quantum Hall trio

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SLIDE 63

Topological Hall effect

  • B. Göbel, A. Mook, J. Henk, and I. M., Phys. Rev. B 95, 094413 (2017)
  • B. Göbel, A. Mook, J. Henk, and I. M., New Journ. Phys., accepted (2017)
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SLIDE 64

Cargèse 14.10.2017

VH V

Chiral spin texture

Experiment to measure the THE

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SLIDE 65

Cargèse 14.10.2017

  • M. Nagao et al., Experimental observation of multiple-q states for

the magnetic skyrmion lattice and skyrmion excitations under a zero magnetic field.

  • Phys. Rev. B 92, 140415 (2015)

Skyrmions

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SLIDE 66

Cargèse 14.10.2017

  • B. Göbel, A. Mook, J. Henk, and I.M., Phys. Rev. B 95, 094413 (2017)

Skyrmion lattice

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SLIDE 67

Cargèse 14.10.2017

ni

ˆ H = X

ij

tijc+

i cj − J

X

i

c+

i ˆ

σci · ni − µBˆ σ · B

Skyrmion – background spin texture

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SLIDE 68

Cargèse 14.10.2017

Electron bandstructure in background spin texture

ˆ H = X

ij

tijc+

i cj − J

X

i

c+

i ˆ

σci · ni − µBˆ σ · B

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SLIDE 69

Cargèse 14.10.2017

Electrons in the skyrmion field and THE

5 10

  • 902
  • 900
  • 898
  • 896

Conductivity σxy (σ0) Energy (t)

σ±

xy = e2

h 1 2π X

n

Z

BZ

d2kfn(k)Ωn

z (k)

Γ K M Γ

  • 902
  • 900
  • 898
  • 896

Wave vector Energy (t)

e2 h 2e2 h

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SLIDE 70

Börge Göbel, Alexander Mook, Jürgen Henk and Ingrid Mertig, Phys. Rev. B 95, 094413 (2017)

THE from Berry curvature of the electrons

σ±

xy = e2

h 1 2π X

n

Z

BZ

d2kfn(k)Ωn

z (k)

e2 h 2e2 h

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SLIDE 71

From THE to QHE

slide-72
SLIDE 72

Börge Göbel, Alexander Mook, Jürgen Henk and Ingrid Mertig, Phys. Rev. B 95, 094413 (2017)

Spin texture, skyrmion number and emergent field

Nsk = 1 4π Z

xy

d2r n(r) · [∂n(r) ∂x × ∂n(r) ∂y ]

Keita Hamamoto, Motohiko Ezawa, and Naoto Nagaosa, Phys. Rev. B 92, 115417 (2015)

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SLIDE 73

Free electrons in a triangular lattice

H = (p − qA)2 2m

Börge Göbel, Alexander Mook, Jürgen Henk and Ingrid Mertig, Phys. Rev. B 95, 094413 (2017)

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SLIDE 74

Comparison of THE and QHE

Cargèse 14.10.2017

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SLIDE 75

The conductance is quantized!

The quantum Hall trio and THE

Topological Hall

  • S. Oh, Science 340,153-154 (2013)
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SLIDE 76
  • Chern number – topological invariant
  • Berry curvature acts like a magnetic field and

causes anomalous veolcity!

  • Anomalous velocity is the origin of the transversal

transport coefficients: spin and anomalous Hall effect and quantum spin and anomalous Hall effect, as well as the topological Hall and quantum topological Hall effect!

Summary

Cargèse 14.10.2017