basic theory of antiferromagnets i - uni-mainz.de · basic theory of antiferromagnets i september...
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Basic Theory of Antiferromagnets I
September 26, 2016 Antiferromagnetic Spintronics
Waldhausen Schloss
Helen Gomonay Johannes Gutenberg Universität Mainz
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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
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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
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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
![Page 5: Basic Theory of Antiferromagnets I - uni-mainz.de · Basic Theory of Antiferromagnets I September 26, 2016 Antiferromagnetic Spintronics Waldhausen Schloss Helen Gomonay Johannes](https://reader030.vdocuments.net/reader030/viewer/2022040613/5f070ebf7e708231d41b16d2/html5/thumbnails/5.jpg)
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
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Hierarchy of atomic interactions
6
1µeV
1 meV
1 eV
energy, eV
k k
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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)
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Quantum state vs Neel state
,
ˆ ˆˆjk j k
j k
H J=∑ S S
Quantum state, T=0
8
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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
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Different experimental technique
ARPES
XMLD
RIXS
Raman spectroscopy
ND
10
Bryan Gallagher: afternoon session
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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
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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
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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
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Variety of AFM structures
α-Mn
NiONiMn
Mn3NiN
14
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Take-home messages
15
• AF = variety of exchange mechanisms • AF = metals,insulators and in between • AF = variety of structures • Macroscopic description = sublattice
magnetizations
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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
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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
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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
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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
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Spin-flop transition
20
w =H
kanis
8Ms
L2z
� H?anis
32Ms
�L4x
+ L4y
�
LLM1
M1M2 M2
Possibility for information coding!
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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
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Spin-flop transition
22
H
L MAF
H > Hsf =q2JinterH?
anis
Exchange enhancement
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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
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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
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Hierarchy of atomic interactions
25
1µeV
1 meV
1 eV
energy, eV
k k
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Magnetoelastic interactions
Covalent bonds ⇒ spin-orbit coupling ⇒ mag.-el.
Spontaneous striction:
26
uspon
= � �me
c0L⌦ L
wel =1
2cjklmujlukm
wme = �jklmujlLkLm
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Spontaneous striction
L
L
27
uspon
= �⇤
c0L2
uspon
=⇤
c0L2
uspon
= 0
uspon
= � �me
c0L⌦ L
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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
�
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Example: domain walls
spon1u
spon2u
L1 L2
X
ψ
ϕLY x
y
29
�Ad2�
dz2+
Kanis +
�2
c0f(z)
�sin� cos� = 0
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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
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Shape-induced effects
2shape 2 2destr x y
aW K L L
b⎛ ⎞= −⎜ ⎟⎝ ⎠
Inhomogeneous sample, destressing energypy
31
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Magnetostriction and stress
T>TN T<TN
σin
32
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Take-home message
33
• Magnetostriction = source of additional anisotropy
• Orientation domains = magnetoelastic • Shape anisotropy = magnetoelastic • Different effects in nano and macro-
samples
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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
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Spin Torques in antiferromagnet
35
dM1
dt= �M1 ⇥H1 + ⇤M1 ⇥ p1 ⇥M1
dM2
dt= �M2 ⇥H2 + ⇤M2 ⇥ p2 ⇥M2
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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
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Exchange enhancement
37
𝜞1N
𝜞2N
𝜞ex
𝜞ex
M1
M2
BN1
BN2
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Solid-like dynamics
38
inter, JHT <<
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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
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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
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Take-home messages
41
• AF: dynamics magnetisation • AF: inertia due to exchange • Dynamics = balance equation for
magnetizations
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Conclusions
42
• AF different from FM • Field effects => weak • Exchange interaction => important for dynamics • Strong magneto elastic effects
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Thank you!
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Surface vs bulk anisotropy
xy
44
Wsurf = Ksurf
Z
S(L · n)2dS
Wbulk = �1
4Kbulk
Z
V
�L4x
+ L4y
�dV
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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
⌘
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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
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Basic Theory of Antiferromagnets II
September 26, 2016 Antiferromagnetic Spintronics
Waldhausen Schloss
Helen Gomonay Johannes Gutenberg Universität Mainz
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Take-home message
2
• Exchange enhancement ⇒fast antiferromagnetic dynamics
• Antiferromagnetic states can be effectively manipulated by spin and charge current
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Motivation
3
19th century physics
• All-electrical control and manipulation of AF states
• Information and data storage with AF
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Outline
4
• Dynamics: spin-waves
• Dynamics: domain walls • Current-induced dynamics • Switching and STO
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Outline
5
• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • What is beyond?
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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
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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
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Circular polarised modes
8
M / i!l2
JinterM / � i!l2
Jinter
!± = ±�H + !AFMR
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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
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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
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Take-home messages
11
• Spin-wave spectra: many modes • Spin waves transfer magnetization • Exchange enhancement • Magneto elastic gap, size effects
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Outline
12
• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO
![Page 59: Basic Theory of Antiferromagnets I - uni-mainz.de · Basic Theory of Antiferromagnets I September 26, 2016 Antiferromagnetic Spintronics Waldhausen Schloss Helen Gomonay Johannes](https://reader030.vdocuments.net/reader030/viewer/2022040613/5f070ebf7e708231d41b16d2/html5/thumbnails/59.jpg)
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
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Anatomy of FM DW motion
14
𝜞an
𝜞ZM
BZ
⌧FM =1
�↵GHan
MFMDW =MsS
�2HanxDW
MFMDW
⌧FM=
↵GMsS
�xDW
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Above Walker breakdown in FM
15
Velo
city
Field
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No Walker breakdown in AFM
16
Velo
city
Field
I.V. Bar’yakhtar, B.A. Ivanov, Solid State Comm., 1980
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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
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Dynamics of DW in AF and FM
18
Velo
city
Field
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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 ✓
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Take-home messages
20
• No Walker breakdown • Small mass and exchange enhancement • Relativistic dynamics
Ulrich Rössler: today and tomorrow session
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Outline
21
• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO
![Page 68: Basic Theory of Antiferromagnets I - uni-mainz.de · Basic Theory of Antiferromagnets I September 26, 2016 Antiferromagnetic Spintronics Waldhausen Schloss Helen Gomonay Johannes](https://reader030.vdocuments.net/reader030/viewer/2022040613/5f070ebf7e708231d41b16d2/html5/thumbnails/68.jpg)
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
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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
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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
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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
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Magnetization⇒ rotation
const=+ AFMmδ
26
MAF
�m ! MAF / L⇥ L ! L(t)
MAF
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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
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Outline
28
• Dynamics: spin-waves • Dynamics: domain walls • Current-induced dynamics • Switching and STO
![Page 75: Basic Theory of Antiferromagnets I - uni-mainz.de · Basic Theory of Antiferromagnets I September 26, 2016 Antiferromagnetic Spintronics Waldhausen Schloss Helen Gomonay Johannes](https://reader030.vdocuments.net/reader030/viewer/2022040613/5f070ebf7e708231d41b16d2/html5/thumbnails/75.jpg)
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
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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
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Critical current
FM
AFM, uniaxial
anGFMSTT HH α=
AFM, cubic
31
HAFSTT = ↵G
pHanJinter
HAFSTT =
r↵2GHanJinter +
⇣H
kan �H?
an
⌘2
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FM vs AFM, possible dynamics near the critical current
32
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Take-home messages
33
• Spin transfer = magnetisation = dynamics • Exchange enhancement of spin • Different dynamics of FM and AF
![Page 80: Basic Theory of Antiferromagnets I - uni-mainz.de · Basic Theory of Antiferromagnets I September 26, 2016 Antiferromagnetic Spintronics Waldhausen Schloss Helen Gomonay Johannes](https://reader030.vdocuments.net/reader030/viewer/2022040613/5f070ebf7e708231d41b16d2/html5/thumbnails/80.jpg)
What’s beyond?
34
• Quantum fluctuations, quantum excitations • Large fields ~ intersublattice exchange • Ultrafast dynamics, magnetooptics • Small AF particles (mesoscales)
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Welcome to AF spintronics!