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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!
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