MSE 7025 Magnetic Materials
(and Spintronics)
Chi-Feng Pai [email protected]
Lecture 5: Interactions, Episode I
Course Outline • Time Table
Week Date Lecture
1 Feb 24 Introduction
2 March 2 Magnetic units and basic E&M
3 March 9 Magnetization: From classical to quantum
4 March 16 No class (APS March Meeting, Baltimore)
5 March 23 Category of magnetism
6 March 30 From atom to atoms: Interactions I (oxides)
7 April 6 From atom to atoms: Interactions II (metals)
8 April 13 Magnetic anisotropy
9 April 20 Mid-term exam
10 April 27 Domain and domain walls
Course Outline • Time Table
Week Date Lecture
11 May 4 Magnetization process (SW or Kondorsky)
12 May 11 Characterization: VSM, MOKE
13 May 18 Characterization: FMR
14 May 25 Transport measurements in materials I: Hall effect
15 June 1 Transport measurements in materials II: MR
16 June 8 MRAM: TMR and spin transfer torque
17 June 15 Guest lecture (TBA)
18 June 22 Final exam
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
EH M
applied appliedEM H H H H M
(Remember Hw2?)
2000
applied
C M
T H M
applied c
M C C
H T C T T
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
– Connection to “exchange interaction” and total Hamiltonian
EH M(Remember Hw2?)
2000
mf
,
ˆij i j B i B i
i j i i
H J g g S S S B S B B
mf 0 0E B B H H H M
exchange Zeeman
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
applied c
M C C
H T C T T
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
• Curie Temperature Tc
c
C C
T C T T
2
0 eff
3
vc
B
nT C
k
(From Tc and mu_eff we can estimate lambda, Hw.3)
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
– Again, borrow from theory of quantum paramagnetism
( ) ( ) tanhv B j s j sM n g j B x M B x M x
0 applied/ /B B B Bx g jB k T g j H M k T
s v BM n g j
applied applied
0 0
1 1B B
B B
k T k TM x H x H
g j
1/ 2j
1/ 2j
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
– Again, borrow from theory of quantum paramagnetism
– Find M(x) by plotting these two functions!
/ tanhsM M x
applied
0
1/ B
s
v B B
k TM M x H
n
applied
v B
H
n
tanh x
/ sM M
Ferromagnetism
• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field
– Again, borrow from theory of quantum paramagnetism
/ sM M / sM M
Interactions: From atom to atoms
• Dipolar interaction – The magnetic field induced by a magnetic dipole moment μ
– If there are two moments, then the dipolar interaction energy is
0
3 2
3
4 r r
r μ rB r μ
1 201 23 2
3
4E
r r
μ r μ rr μ μ
2310 J
(Hw.3)
1 2 μ μ (~SOI energy, Hw.2)
Interactions: From atom to atoms
• Electronic structure of atoms
Polar bond Covalent, metallic bonds
(delocalized) (localized)
Another quantum mechanics in one slide
• Linear combination of atomic orbitals (LCAO)
(higher E)
(lower E)
Keywords: Identical particles, exchange symmetry, Pauli exclusion principle, fermions
Another quantum mechanics in one slide
• Anti-bonding orbitals – Favor ferromagnetism
– Antisymmetric wave fn
• Bonding orbitals – Favor superconductivity
– Symmetric wave fn
Spatial Spin
Spatial Spin
Keywords: Identical particles, exchange symmetry, Pauli exclusion principle, fermions
Another quantum mechanics in one slide
• Anti-bonding orbitals – Favor ferromagnetism
– Antisymmetric wave fn
• Bonding orbitals – Favor superconductivity
– Symmetric wave fn
S 1 2 2 1
1 1
2 2a b a b r r r r
A 1 2 2 1
1 1
2 2a b a b
r r r r
1s
0s
Keywords: Identical particles, exchange symmetry, Pauli exclusion principle, fermions
Another quantum mechanics in one slide
• Hamiltonian
1 2H S S
*
S S S 1 2ˆE H d d r r
*
A A A 1 2ˆE H d d r r
* *
S A 1 2 2 1 1 2ˆ2 a b a bE E H d d r r r r r r
1/ 4
3 / 4
1s
0s
(Why? Hw.3)
(eff) S A S A 1 2
1ˆ 34
H E E E E S S
Another quantum mechanics in one slide
• Exchange integral J
(eff) S A S A 1 2
1ˆ 34
H E E E E S S
* *S A1 2 2 1 1 2
ˆ2
a b a b
E EJ H d d
r r r r r r
spin S A 1 2 1 2ˆ 2H E E J S S S S
0J S AE E 1s
0J S AE E 0s
Exchange Interaction Hamiltonian
• Heisenberg model
– If more than two spins (and let us just consider spin-1/2)…
,
ˆij i j
i j
H J S S
ˆ 2 ij i j
i j
H J
S S
Note that the S here is unitless (in previous classes we used them with unit of h_bar)
2110 JJ
(Hw.3)
Molecular field theory re-visit
• Assumed that J is the same for each nearest-neighbor pair
– For a particular atom i
mf 0 mfˆ 2i i
ex i j i B i
j
E H JS S B g S H
mf
0 0
2 2j j
jB B
J zJH S S
g g (sum over z nearest
neighbors)
v B jM n g m
mf 2
0
2
v B
zJH M M
n g
Molecular field theory re-visit
• Curie temperature
220 eff0 eff
2
0
2
3 3
vvc
B v B B
zJnnT C
k n g k
2
eff
2
2 2 ( 1)
33c
BB B
zJ zJ s sT
kg k
2
0
2
v B
zJ
n g
Molecular field theory re-visit
• Curie temperature (general form)
2
2 1 ( 1)
3
j
c
B
zJ g j jT
k
(general form)
2
2
0
2 1j
v B
zJ g
n g
2
( 1) 1jG j j g De Gennes factor
Exchange stiffness constant
• From a semi-classical approach
22zJsA
a
2 2 2ˆ 2 cos const.ex ij ij ij
i j i j
E H J S J S
2 222ij
ij ijex
ij ij
E J S Ma A A
V V x x M
11~10 J/m
effA K
(domain wall width/thickness)
Exchange stiffness constant
Category of exchange interactions
• Direct exchange – In fact this only plays a minor role in determination of magnetic
properties
– Extremely small effect
• Indirect exchange – Superexchange (oxides)
– Ruderman-Kittel-Kasuya-Yoshida interaction, RKKY (metals)
• Anisotropic exchange – Dzyaloshinskii-Moriya interaction (DMI)
– DMI and its correlated spintexture (chiral domain walls, skyrmions) are still hot topics in the spintronics community
Superexchange
• Superexchage – Indirect exchange
– a.k.a Kramers–Anderson superexchange
– Goodenough-Kanamori rules
• Example: MnO (Manganese oxide, antiferromagnetic)
http://chemwiki.ucdavis.edu/Core/Materials_Science/Magnetic_Properties/Antiferromagnetism
53d 53d
Superexchange
• Example: MnO (Manganese oxide, antiferromagnetic) – Rocksalt structure
– Antiferromagnetic
– Goodenough-Kanamori rules
53d53d
Superexchange
• Example: MnO (Manganese oxide, antiferromagnetic) – Rocksalt structure
– Antiferromagnetic
– Goodenough-Kanamori rules
(strong)
(weak)
Iron oxides (ferrimagnets)
• Commonly seen Fe-oxide structures
– Rocksalt (FeO)
– Spinel (AFe2O4)
– Garnet (A3Fe5O12)
– Perovskite (AFeO3)
• Structure and magnetic properties of iron oxides
3 5Fe 3 5 Bd
2 6Fe 3 4 Bd
Spinel ferrite
• Spinel structure
Note that antiferromagnetic coupling between A and B sites
A O B 125
B O B 90
A O A nonexist
A
B
T-dependence of M in ferrimagnets
• Competition between moments of two sublattices
Example: Iron garnets
compT CT
T-dependence of M in ferrimagnets
• Competition between moments of two sublattices
Example: NiFe2-xVxO4
Néel model for ferrimagnets
• Curie-Weiss model
• Néel model – Consider two sublattices: A sites and B sites
EH M
A
E AA A AB BH M M
B
E BB B AB AH M M
applied appliedE
CM H H H H M
T
applied c
M C C
H T C T T
appliedA
A AA A AB B
CM H M M
T
appliedB
B BB B AB A
CM H M M
T
Néel model for ferrimagnets
• Néel model – Consider two sublattices: A sites and B sites
AB AA appliedA
A B
CM H M
T
appliedB
B A
CM H M
T AB BB
c A BT C C
2 2
applied
2A B A BA B
c
C C T C CM M
H T T
(Hw.3)
Néel model for ferrimagnets
• Néel model – Consider two sublattices: A sites and B sites
2 21
2
c
A B A B
T T
C C T C C
Néel model for antiferromagnets
• Néel model (antiferromagnetism)
– Néel temperature
– Susceptibility
NT CA BC C C
2
N
C
T T
p
p
C
T