MSE 7025 Magnetic Materials (and Spintronics)cfpai/MSE7025/MSE7025_Lecture5_Interactio… · MSE 7025 Magnetic Materials (and Spintronics) Chi-Feng Pai [email protected] Lecture 5:

MSE 7025 Magnetic Materials

(and Spintronics)

Chi-Feng Pai [email protected]

Lecture 5: Interactions, Episode I

mailto:[email protected]

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


– Connection to “exchange interaction” and total Hamiltonian

EH M(Remember Hw2?)

2000

mf

,

îj 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


applied c

M C C

H T C T T

Ferromagnetism


• 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


– 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



– 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



/ sM M / sM M

Ferromagnetism

• Curie-Weiss law – Curie temperature of various ferromagnetic materials

Ferromagnetism

• Curie-Weiss law – Curie temperature of various ferromagnetic materials

Ferromagnetism

• Bloch’s law – Magnons / spinwaves

3/2( ) (0) 1 /s s cM T M T T

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


• Anti-bonding orbitals – Favor ferromagnetism

– Antisymmetric wave fn

• Bonding orbitals – Favor superconductivity

– Symmetric wave fn

Spatial Spin

Spatial Spin



• 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



• Hamiltonian

1 2H S S

*

S S S 1 2Ê H d d r r

*

A A A 1 2Ê 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


• 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)…

,

îj 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


• 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


• 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

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982)

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


53d53d

Superexchange

• Example: MnO (Manganese oxide, antiferromagnetic) – Rocksalt structure

– Antiferromagnetic


(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

Spinel ferrite

• Spinel structure

• Possible distributions of Fe ions

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



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)



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

Magnetization and susceptibility revisit

• Susceptibility revisit

http://nptel.ac.in/courses/113104005/81

Documents

MSE 7025 Magnetic Materials (and Spintronics)cfpai/MSE7025/MSE7025_Lecture5_Interactio… · MSE 7025 Magnetic Materials (and Spintronics) Chi-Feng Pai [email protected] Lecture 5: