Basic Theory of Antiferromagnets I

September 26, 2016 Antiferromagnetic Spintronics

Waldhausen Schloss

Helen Gomonay Johannes Gutenberg Universität Mainz

Take-home message

2

• Antiferromagnets: Neel order parameter, distinguish between micro- and macro description!

• Two types of exchange, exchange enhancement

• Newtonian-like dynamics vs gyrotropic dynamics in FM

Application

• High frequencies • Zero magnetization • Magnetomechanical coupling • Combined with

semiconductors

New physics

• Variety of structures • Nontrivial dynamics • Spin-orbit coupling • Complicated, less studied

FM ⇒ AFM

3

Motivation

Outline

4

• Basics of antiferromagnetism: exchange interactions,Neel states, magnetic sublattices

• Phenomenological description, spin-flop transitions • Magneto elastic effects • Basics of dynamics: equation of motion

Outline

5

• Basics of antiferromagnetism: exchange interactions,Neel states, magnetic sublattices

• Phenomenological description, spin-flop transitions • Magneto elastic effects • Basics of dynamics: equation of motion

Hierarchy of atomic interactions

6

 

1µeV

1 meV

1 eV

energy, eV

 

 k k

AF exchange interactions

Co80Ni20 / Ru / Co80Ni20

Parkin et al., Phys.Rev. B 44, 7131 (1991) 7

• Superexchange (insulators) • RKKY (4-f metals) • Exchange in 3-d metals • Double exchange (transition metal oxides) • DMI (anisotropic exchange)

Quantum state vs Neel state

,

ˆ ˆˆjk j k

j k

H J=∑ S S

Quantum state, T=0

 

8

Quantum state vs Neel state

,

ˆ ˆˆjk j k

j k

H J=∑ S S

Neel state, T≠0

s1 s2

1 2= −S S

{s1, s2,…, sj, sj+1}

9

Different experimental technique

ARPES

XMLD

RIXS

Raman spectroscopy

ND

10

Bryan Gallagher: afternoon session

Sublattices

Sublattice magnetizations (Neel, 1948)

( ) ( )j

k k jgN τ

τ= +∑M r S r

r

τ•Physically small volumes •Macroscopic vectors •Field variables

11

and order parameters

NiO, IrMn

M2

M1

Symmetry relations: 2[110] : M1 ↔ M2

Order parameter (Neel vector):

Magnetization: MAF= M1+M2 ≈0

L ⊥ MAF ,⏐L⏐ ≈2Ms

M1

M2

MAF

L

12

M1

M2

L= M1—M2

Noncollinear structures

IrMn, Mn3NiN

M2

M3

M1

2[110] : M1 ↔ M2, M3 ↔ M3

3[111] : M1 → M2 → M3

Order parameters (Neel vectors):

Magnetization: MAF= M1+ M2 + M3 ≈0

M1

M3

M2

L1

L2

L1= M1+M2 - 2M3 L2= M1- M2

13

Variety of AFM structures

α-Mn

NiONiMn

Mn3NiN

 

14

Take-home messages

15

• AF = variety of exchange mechanisms • AF = metals,insulators and in between • AF = variety of structures • Macroscopic description = sublattice

magnetizations

Outline

16

• Basics of antiferromagnetism: exchange interactions,Neel states, magnetic sublattices

• Phenomenological description, spin-flop transitions • Magneto elastic effects • Basics of dynamics: equation of motion

Hierarchy of interactions

1 T

100 T

1000 T

H, T

intra NJ T⇔

inter 1/J χ⊥∝

s-f inter anisH J H∝

17

Free energy, Landau approach

18

Symmetry-based modeling

Low temperature T,H ⌧ Jinter

w =1

2JinterM

2AF �MAFH+

1

2A (rL)2

+1

2K(2)

anisL2z

� 1

4K(4)

anis

�L4x

+ L4y

�

MAF ⌧ L, L = 2Ms ⇡ const

Equilibrium state

19

0F wdVδ δ= =∫Principle of energy minimum:

Exchange approximation: excluding MAF

( )2inter

14 sJ M

= × ×M L H L

M1

H=0

M2

L

H

L M1

M2

MAF

wZeeman = � (L⇥H)2

8JinterMs

Spin-flop transition

20

w =H

kanis

8Ms

L2z

� H?anis

32Ms

�L4x

+ L4y

�

LLM1

M1M2 M2

Possibility for information coding!

Spin-flop transition

21

w =H

kanis

8Ms

L2z

�H2L2

y

8JinterMs

� H?anis

32Ms

�L4x

+ L4y

�

H Hsf =q2JinterH?

anis

L

HH

LMAF

MAF=0

Spin-flop transition

22

H

L MAF

H > Hsf =q2JinterH?

anis

Exchange enhancement

Take-home messages

23

• Macroscopic description: order parameters and magnetization vector

• Inter-sublattice exchange – exchange enhancement

• Static magnetic field – quadratic effects • Equivalent states– information coding • Spin-flop transition – information control

Outline

24

• Basics of antiferromagnetism: exchange interactions,Neel states, magnetic sublattices

• Phenomenological description, spin-flop transitions • Magneto elastic effects • Basics of dynamics: equation of motion

Hierarchy of atomic interactions

25

 

1µeV

1 meV

1 eV

energy, eV

 

 k k

Magnetoelastic interactions

Covalent bonds ⇒ spin-orbit coupling ⇒ mag.-el.

Spontaneous striction:

26

uspon

= � �me

c0L⌦ L

wel =1

2cjklmujlukm

wme = �jklmujlLkLm

Spontaneous striction

L

L

27

uspon

= �⇤

c0L2

uspon

=⇤

c0L2

uspon

= 0

uspon

= � �me

c0L⌦ L

Magnetoelastic anisotropy

•Additional 4-th order anisotropy

•Variation of potential barrier

28

uspon

= � �me

c0L⌦ L wme = �jklmujlLkLm

w =H

kanis

8Ms

L2z

�H?

anis

32Ms

+�2

c0

� �L4x

+ L4y

�

Example: domain walls

spon1u

spon2u

L1 L2

X

ψ

ϕLY x

y

29

�Ad2�

dz2+

Kanis +

�2

c0f(z)

�sin� cos� = 0

Elastic dipoles: long-range forces

1 23

1 2

ˆ ˆ( ) ( )Φ ∝

−

in in

ddu ur rr r2ˆ( )u r

1ˆ( )u r

30

uspon

= � �me

c0L⌦ L

Shape-induced effects

2shape 2 2destr x y

aW K L L

b⎛ ⎞= −⎜ ⎟⎝ ⎠

Inhomogeneous sample, destressing energypy

31

Magnetostriction and stress

T>TN T<TN

σin

32

Take-home message

33

• Magnetostriction = source of additional anisotropy

• Orientation domains = magnetoelastic • Shape anisotropy = magnetoelastic • Different effects in nano and macro-

samples

Outline

34

• Basics of antiferromagnetism: exchange interactions,Neel states, magnetic sublattices

• Phenomenological description, spin-flop transitions • Magneto elastic effects • Basics of dynamics: equation of motion

Spin Torques in antiferromagnet

35

dM1

dt= �M1 ⇥H1 + ⇤M1 ⇥ p1 ⇥M1

dM2

dt= �M2 ⇥H2 + ⇤M2 ⇥ p2 ⇥M2

Hierarchy of interactions

1 T

100 T

1000 T

H, T

intra NJ T⇔

inter 1/J χ⊥∝

s-f inter anisH J H∝

36

Exchange enhancement

37

𝜞1N

𝜞2N

𝜞ex

𝜞ex

M1

M2

BN1

BN2

Solid-like dynamics

38

inter, JHT <<

Equations of motion

39

Dynamic magnetisation

M ⌘ 1

Hex

L⇥ L = �L⇥HL

Magnetisation balance

M =1

Hex

L⇥ L+1

Hex

L⇥H⇥ L

Equation of motion

AF dynamics: Newton-like

FM dynamics: precession

40

L|{z}acc

= � �⇣H⇥ L+ 2H⇥ L

⌘

| {z }gyroforce

+ �2Jinter

HL| {z }potent.force

Take-home messages

41

• AF: dynamics magnetisation • AF: inertia due to exchange • Dynamics = balance equation for

magnetizations

Conclusions

42

• AF different from FM • Field effects => weak • Exchange interaction => important for dynamics • Strong magneto elastic effects

Thank you!

Surface vs bulk anisotropy

xy

44

Wsurf = Ksurf

Z

S(L · n)2dS

Wbulk = �1

4Kbulk

Z

V

�L4x

+ L4y

�dV

Shape-induced effects

( ) ( )anis 4 4 shape 2/x y xW V K L L K a b L⊥⎡ ⎤= + +⎣ ⎦

Homogeneous sample, shape-induced anisotropy

M

L

a=b

M

L

a>>b

45

Kbias > Kshape

⇣ab

⌘= 0 Kbias < Kshape

⇣ab

⌘

Magnetoelastic interactions

m-e jklm jk l mw u L Lλ=

Covalent bonds ⇒ spin-orbit coupling ⇒ mag.-el.

Spontaneous striction:

L

L

x

y y

x46

uspon

= � �me

c0L⌦ L

Basic Theory of Antiferromagnets II

September 26, 2016 Antiferromagnetic Spintronics

Waldhausen Schloss

Helen Gomonay Johannes Gutenberg Universität Mainz

Take-home message

2

• Exchange enhancement ⇒fast antiferromagnetic dynamics

• Antiferromagnetic states can be effectively manipulated by spin and charge current

Motivation

3

19th century physics

• All-electrical control and manipulation of AF states

• Information and data storage with AF

Outline

4

• Dynamics: spin-waves

• Dynamics: domain walls • Current-induced dynamics • Switching and STO

Outline

5

• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • What is beyond?

Hierarchy of interactions

1 T

100 T

1000 T

H, T

intra NJ T⇔

inter 1/J χ⊥∝

s-f inter anisH J H∝

6

Spin waves as “classical” excitations

7

L⇥ L = �2JinterL⇥HL

L = L(0) + lei(kr�!t)

x

y

z

!AFMR = �pJinterHanis

!sw =q!2AFMR + c2k2

L(0)kz, lky

M / i!

JinterL(0) ⇥ le�i!t kx

Circular polarised modes

8

M / i!l2

JinterM / � i!l2

Jinter

!± = ±�H + !AFMR

Magnetoelastic gap

Large sample: “frozen” lattice

soundds

τ ∝ d= 1 mm, τsound∝10-6 sec, d= 30 nm, τsound ∝10-11 sec

τmag∝10-12 sec

sponˆ

ˆ'

u constcλ

= − ⊗ =L L

( )

sponanis anis

sponAFMR inter anis

2

2

s

s

H H M u

J H M u

λ

ω γ λ

→ +

= +

9

Magnetoelastic waves

( ) ( )(0) ,i t i tje eω ω− −= + =kr krL L l u u

10

!21,2(k) = !2

AFMR + s2k2 ±q(!2

AFMR � s2k2)2+ 4�Jinter�2k2

Take-home messages

11

• Spin-wave spectra: many modes • Spin waves transfer magnetization • Exchange enhancement • Magneto elastic gap, size effects

Outline

12

• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO

Below Walker breakdown in FM

13

Velo

city

Field

B

x

N. L. Schryer, L. R. Walker, JAP, 1972µ

FM ⌘dv

FMsteady

dBZee=

�xDW

↵G

Anatomy of FM DW motion

14

𝜞an

𝜞ZM

BZ

⌧FM =1

�↵GHan

MFMDW =MsS

�2HanxDW

MFMDW

⌧FM=

↵GMsS

�xDW

Above Walker breakdown in FM

15

Velo

city

Field

No Walker breakdown in AFM

16

Velo

city

Field

I.V. Bar’yakhtar, B.A. Ivanov, Solid State Comm., 1980

Anatomy of AF DW motion

17

𝜞1N

𝜞2N

𝜞ex

𝜞ex

M1

M2

BN1

BN2

MAFDW

=MsS

�2Hex

xDW

MAFDW

⌧AF=

↵GMsS

�xDW

⌧AF

=1

�↵GHex

Dynamics of DW in AF and FM

18

Velo

city

Field

DW motion: “relativistic” dynamics

19

dPx

dt= �↵

G

�Hex

Px

+ Fx

P

x

/ �Z

@✓

@x

✓dx

Px

/ vp1� v2/c2

c2@2✓

@x2

� ¨✓ � �2Hex

Han

sin ✓ cos ✓ = ↵G�Hex

˙✓ + �2Hex

BNeel

sin ✓

Take-home messages

20

• No Walker breakdown • Small mass and exchange enhancement • Relativistic dynamics

Ulrich Rössler: today and tomorrow session

Outline

21

• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO

Hierarchy of interactions

1 T

100 T

1000 T

H, T

intra NJ T⇔

inter 1/J χ⊥∝

s-f inter anisH J H∝

22

Spintronic: sd-exchange in FM

Polarization:

Scattering:

Effective field:

mH δsdsd J= Hsd

23

⇧in

� ⇧out

/ f(Jsd

)�m⌦ je

Hsd = �JsdX

j

sj · Sj ) �Jsd�m ·M

�m / hsjikM

AF: sd-exchange?

Polarization:

Scattering:

Effective field:

mH δsdsd J= Hsd

24

⇧in

� ⇧out

/ f(Jsd

)�m⌦ je

Hsd = �JsdX

j

sj · Sj ) �Jsd�m ·MAF

�m / hsjikMAF ! 0

Equations of motion

25

Dynamic magnetisation

M ⌘ 1

Hex

L⇥ L = �L⇥HL

Magnetisation balance

M =1

Hex

L⇥ L+1

Hex

L⇥H⇥ L

Magnetization⇒ rotation

const=+ AFMmδ

26

MAF

�m ! MAF / L⇥ L ! L(t)

MAF

Spin transfer in AF and spin balance

( ) sinkˆˆAF +⋅Π−Π= NMoutindt

d

H.Gomonay, V.Loktev,2008

 

27

L� �2JinterHL =

✓�dI

dt+ JinterI�

◆s⇥ L� �↵GJinterL⇥ L

Outline

28

• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO

Dynamics in the dc spin current

29

II

L

anerint

AFMRAFcr H

JI γ

σ

α

γσ

ωγ==

s

L� �2JinterHL = JinterI�s⇥ L� �↵GJinterL⇥ L

s

Critical current

I

anerint

AFMRAFcr H

JI γ

σ

α

γσ

ωγ==

0 10 20 30 40 50

-2

0

2

4

6

8

10

t

-10

1

-10

1-1

0

1

lXlY

l Z

-10

1

-10

1-1

0

1

lXlY

l Z

0 10 20 30 40 50-0.5

0

0.5

1

t

l Z

epol

30

Critical current

FM

AFM, uniaxial

anGFMSTT HH α=

AFM, cubic

31

HAFSTT = ↵G

pHanJinter

HAFSTT =

r↵2GHanJinter +

⇣H

kan �H?

an

⌘2

FM vs AFM, possible dynamics near the critical current

32

Take-home messages

33

• Spin transfer = magnetisation = dynamics • Exchange enhancement of spin • Different dynamics of FM and AF

What’s beyond?

34

• Quantum fluctuations, quantum excitations • Large fields ~ intersublattice exchange • Ultrafast dynamics, magnetooptics • Small AF particles (mesoscales)

Welcome to AF spintronics!