Magnetism - ecourse2.ccu.edu.tw

1

Magnetism

Magnetism

Quantummechanics

Electron-Electron

Interactions

2

Outline

• Magnetism is a purely quantum phenomenon!

• Diamagnetism

• Paramagnetism

• Effects of electron-electron interactions

Hund’s rules for atoms

Atoms in a magnetic field – Curie law

Magnetism in transition metals, rare earths

• Magnetic order and cooperative effects in solids

Transition temperature TC

Curie-Weiss law

• Magnetism: example of an “order parameter”

3

Definitions

Magnetization M = the magnetic moment per unit volume.

• Diamagnetic material: < 0

• Paramagnetic material: > 0

• Ferromagnetic material: M 0 even without external field

Magnetic susceptibility per unit volume = 0M/B

B: macroscopic magnetic field intensity,

0 : permeability of free space

4

Introduction to Solid State Physics by Kittel (2005)

5

Hund’s rules and electron interactions

• Hund’s rules:

1st rule: maximum spin for electrons in a given shell

=> Electron-electron interactions

Reason – parallel-spin electrons are kept apart because they

must obey the exclusion principle –thus the repulsive

interaction between electrons is reduced for parallel spins!

2nd rule: maximum angular momentum possible for the given

spin orientation

Reason – maximum angular momentum means

electrons are going the same direction around the nucleus –

stay apart – lower energy!

6

Hund’s rules and electron interactions

3rd rule: the total angular J=| L±S | with the sign being

determined by whether the shell of orbitals is more than

half filled (+) or less than half filled (-).

Reason

i i

i

H l L S = =

always favoring L counter-aligned with S

When the shell is more than half filled, the additional spin

still want to counter-align with its own angular momentum,

but then L would be aligned with the net spin.

7

External field2

0 ( )2

pH V r

m= +

2( )

( )2

B

eAp

cH g B V rm

+

= + +

adding an external magnetic field:

1

2A B r=

22

2

2

1( ) ( )

2 2 4 2

B

p e eH V r B r p B r

m mc mc

g B

= + + +

+

8

External field2

22

2

1( ) ( )

2 2 4 2B

p e eH V r B r p B r g B

m mc mc = + + + +

( ) ( )2 2 2

B

e e ep B r B r p B l B l

mc mc mc = = =

22

0 2

1( )

2 4B

eH H B l g B r

mc = + + +

paramagnetic term

diamagnetic term

9

Diamagnetism

• Consider a single “closed shell” atom in a magnetic field (In a

closed shell atom, spins are paired and the electrons are

distributed spherically around the atom, i.e., no orbital angular

momentum → there is no total angular momentum.)

• Diamagnetism results from an electric current being set up in

atoms due to an external magnetic field

• Lenz’s law – when the magnetic flux is changed, an induced

current is set up in such a direction as to oppose the flux change.

10

Larmor Diamagnetism2

2

2

1

2 4

eH B r

mc=

• Susceptibility 2 2

0

26

e r

mc

−=

2 2 2 2 22

2 2 2

2 2 28 8 12

e e B e BE B r x y r

mc mc mc = = + =

22

26

E emoment r B

B mc

= − = −

For large conductive molecules, this term would be very large.


11

Larmor Diamagnetism

https://www.youtube.com/watch?v=KlJsVqc0ywM

BH g B =


12

Free spin ½ (Curie or Langevin) paramagnetism

exp( ) exp( )B B

B B

B BZ

k T k T

= + −

lnBF k T Z= −

tanh( )BB

B

BFM

B k T

= − =

13

Free spin ½ (Curie or Langevin) paramagnetism

2

0

0lim B

HB

nM

H k T

→

= =

Curie law

exp( ) exp( )

tanh( )

exp( ) exp( )

B B

B B BB B

B B B

B B

B B

k T k T BM

B B k T

k T k T

− −

= =

+ −


1 2( ) BM N N = −

14

Free spin J (Curie or Langevin)

paramagnetism

( )BH B l g = +

2 2( ) [ ]

L J S JB L gS B J g

J J+ = +

2 22 2 2 2

2 2[ ]

2 2

J L J L J S J SB J g

J J

+ − − + − −= +

2 2 2 2 2 2

2 2[ ]

2 2

J L S J S LB J g

J J

+ − + −= +

' Bg B J=

1 1 ( 1) ( 1)' ( 1) ( 1)[ ]

2 2 ( 1)

S S L Lg g g

J J

+ − += + + −

+Landau g-value

15


paramagnetism

'exp( )

Z

JB Z

J J B

g BJZ

k T

=−

= −

exp( )

exp( )

J

J

J

J J

m J

J J

J

m J

m m x

m

m x

=−

=−

−

=

−

' B

B

g Bx

k T

=

1J

Zm

Z x

= −

' B JM ng m=

16


paramagnetism

exp( )J

J

J

m J

Z m x=−

= −

exp( )(1 exp[ (2 1) ]) sinh[(2 1) / 2]

1 exp( ) sinh[ / 2]

Jx J x J xZ

x x

− − + += =

− −

2 00

(1 )(1 ...)

1

na ya y y

y

−+ + + =

−

17


paramagnetism

y xJ=

' ( )B JM ng JB y=

2 1 2 1 1 1( ) coth( ) coth( )

2 2 2 2J

J JB y y y

J J J J

+ += −

Brillouin function

1/21/ 2, tanh( )J B y= =

sinh[(2 1) / 2]

sinh[ / 2]

J xZ

x

+=

1J

Zm

Z x

= −

18


paramagnetism

2

0 0 ( ' ) ( 1)

3

B

B

M n gM J J C

H B k T T

+= = =

2 1 2 1 1 1( ) coth( ) coth( )

2 2 2 2J

J JB y y y

J J J J

+ += −

y is around 2*10-3 at RT.

31( )

3

Jy y

J

+ +

' ( 1) '

3

B B

B

ng J g JBM

k T

+=

C: Curie constant

19


20


21


Quenched

orbital

moment due

to crystal field

J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)

23

zL x yi i y x

= = −

should be real.

zn L n must be purely imaginary…

Since the wave functions under crystal field are real functions,

zn L n

L is purely imaginary, but is Hermitian.

0zn L n =All components of the orbital angularmomentum of a non-degenerate state arequenched.

Quenched orbital moment due to crystal field

24

Van Vleck paramagnetism

For J=0 in the ground state (the filled shell), there is no

paramagnetic effect for the first-order perturbation.

2

0

0

0 ( )B

n n

B l g nE

E E

+ =

−

0 ( ) 0 0B B l g + =

However, there would be finite contribution from the second-order

perturbation.

It is positive and temperature-independent.

22

0

0 ( )2 z zB

n n

L gS nN

V E E

+=

−

Van Vleck

paramagnetism

25

Pauli paramagnetism for free electron gas (1925)

• Spin up electrons (parallel to field) are shifted opposite to spin

down electrons (antiparallel to field).

26


Density of states for both spins

One spin orientation

BE B =

• Energies shift by

2

( )

(1 / 2) ( )2 ( )

B

B F B B F

M N N

D E B D E B

= −

= =

• Magnetization

27


2

( )

(1 / 2) ( )2 ( )

B

B F B B F

M N N

D E B D E B

= −

= =

2(3 / 2) / ( )B B FM N B k T= Independent of T!

• Free electron gas, D(EF)=3N/2kBTF

• This is a way to measure the density of states!

(Note: There are corrections from the electron-electroninteractions. )

• Unlike the paramagnetism of magnetic ions, here the magnitudeof ～ diamagnetism’s (suppressed by a factor of kBT/EF (Pauliexclusion principle))

• The first method for attaining temperatures much below 1K.

28

Adiabatic demagnetization (proposed by Debye, 1926)


Thermal contact at T1 is provided by He gas

(a→b), and the thermal contact is broken by

pumping the gas (b→c).

• Freezing is effective only if entropy contribution from spin is

dominant (usually need T << TD, i.e., entropy contribution from

lattice vibrations is negligible.)

29


2 1( / )T T B B=

Can reach < 10-3 K

Adiabatic demagnetization

30

Ferromagnetism and

Antiferromagnetism

31


32

Magnetic materials

• What causes some materials (e.g. Fe) to be ferromagnetic?

• Others like Cr are antiferromagnetic (what is this?)

• Magnetic materials tend to be in particular places in the

periodic table: transition metal, rare earths.

• Starting point for understanding: the fact that open-shell atoms

have moments.

33

Modern Physics for Scientists and Engineers by Thornton and Rex (2013).

34

Understanding magnetic materials

• In most magnetic materials (Fe, Ni, ….) the first step in

understanding magnetism is to consider the material as a collection

of atoms where each atom has a magnetic moment

• Of course the atoms change in the solid, but this gives a good

starting point – qualitatively correct

Small!

• Ferromagnetism is not from magnetic dipole-dipole interaction,

nor the spin-orbit interaction. It is a result of electrostatic

interaction!

Dipole-dipole interaction:

35

When are atoms magnetic?

• An atom MUST have a magnetic moment if there is an odd

number of electrons – spin ½ (at least)

• “Open shell” atoms have moments – Hund’s rules

1st rule: maximum spin for electrons in a given shell

2nd rule: maximum angular momentum for the given spin

orientation

Mn2+:3d5

Fe2+:3d6

36

Exchange interactions

1 1 2 2 1 2 1 2( , , , ) ( , ) ( , )orbit spinr s r s r r s s =

1 2

1( , ) : , , ( ),

2spin s s +

1( )

2 −

1 2

1 1( , ) : ( 12 21 ), ( 12 21 )

2 2orbit r r + −

e-

e-

37


sin 0

12 [( 12 21 ) ( 12 21 )]

2gletE V= + + +

0

12 [( 12 21 ) ( 12 21 )]

2tripletE V= + − −

sin 02gletE K J= + +

02tripletE K J= + −

12 21J V=

12 12K V=

exchange integral

direct integral

38

Exchange interactions2 2 2

1 2 1 21 2

( )

2

S S S SS S

+ − − =

1 1 2 21 2

( 1) ( 1) ( 1)

2

S S S S S SS S

+ − + − + =triplet

1 3 1 31 2

12 2 2 2

2 4

− −

= =

1 2

1 3 1 30 1

32 2 2 2

2 4S S

− −

= = −singlet

sin 02gletE K J= + +

02tripletE K J= + −0 1 22 2

2

JH K J S S= + − −

39

Spontaneous Magnetic Order

,

2 ij i j B i

i j i

H J S S g B S= − +

• In an insulator (electrons could not hop from atom to atom)

• Exchange energy J

0ijJ Spins are anti-aligned.

Spins are aligned.0ijJ

The couplings between spins would drop rapidly as the distance

between spins increases.

Only consider nearest-neighbors.

40

Spontaneous Magnetic Order

,

2 i j

i j

H J S S

= −

For an uniform system without any applied field

,

2 ij i j B i

i j i

H J S S g B S

= − +

Heisenberg Hamiltonian

In the case of a ferromagnet, there could ordering of magnetic

moments even in the absence of any applied magnetic field:

spontaneous magnetic order.

41

Mean field theory (Weiss 1906)

For the ith spin of all the other spin in the solid, taken as the mean

effect field of the FM system

,

2 ij i j B i

i j i

H J S S g B S

= − +

i B i effH g S B= −

2 ij j

j

eff mf

B

J S

B B B Bg

= − = +

mean filed or

molecular field

mf mfB M=

42


Ferromagnetism

' ( ') ( ')B J S JM ng JB y M B y= =

' ( )'

B mf

B

g B My J

k T

+=

' ( )B JM ng JB y=

2 1 2 1 1 1( ) coth( ) coth( )

2 2 2 2J

J JB y y y

J J J J

+ += −

Paramagnetism

' B

B

g By J

k T

=

43


Solid State Physics by Schmool (2017)

44

Curie temperature

' 1y

31( ') ' ( ' )

3J

JB y y y

J

+ +

1' '

3 '

B CS

B mf

k TJM M y y

J g J

+= =

Without external magnetic field:

''

B mf

B

g My J

k T

= '

'

B

B mf

k TM y

g J =

' ( 1)

3

B mf S

C

B

g J MT

k

+=

45


Solid State Physics by Schmool (2017)

46


' ( )1

3

B mf mfC

S B mf S

g B M B MTM JJ

M J k T T M

+ ++=

CT T

( )mf C CM T T BT − =

/C mf

C

TM

B T T

= =

− Curie-Weiss law

47

Example of a phase transition to a state of new order

• At high temperature, the material is paramagnetic. Magnetic

moments on each atom are disordered.

• At a critical temperature Tc the moments order. Total

magnetization M is an “order parameter”

• Transition temperatures: 1043 K in Fe, 627 K in Ni, 292 K in Gd


48

Symmetry Breaking

,

2 i j

i j

H J S S

= −

It is rotational symmetric: the magnetization could point in any

direction and the energy would be the same.

Anisotropy energy

If the anisotropy energy is extremely large, it would force spin to

be either SZ=S or –S.

2

,

2 ( )z

i j i

i j i

H J S S S

= − −

Ising model

In some materials, the spin are lying in the xy plane.

XY model

49

Easy axis


50

Spin wave



51

Ferrimagnets

52

Antiferromagnetism (predicted by Neel, 1936)

• Magnetic moments can also order to give no net moment –antiferromagnetism

• Transition temperature Ttransition = TNeel

MnO (O2+ not shown)


53

Antiferromagnetism (predicted by Neel, 1936)

Phys. Rev. B 58, 11583 (1998)

54

Antiferromagnetism (AFM)

' ( )1

3

B mf mfN

S B mf S

g B M B MTM JJ

M J k T T M

− −+=

( )mf N NM T T BT + =

/N mf

N

TM

B T T

= =

+Curie-Weiss law for AFM

ij j

j

eff mf

B

J S

B B B Bg

= − = +

mf mfB M= −0ijJ

Magnetization as a function of field in AFM

materials


Magnetization of CoV2O6 as a function of field

Zhangzhen He et. al., J. Am. Chem. Soc. 131, 7554 (2009).

57


58

Reciprocal susceptibility of magnetite, Fe3O4


Direct exchange

59

Direct exchange

If the electrons on neighboring magnetic atoms interact via an

exchange interaction, this is known as direct exchange.

Usually direct exchange cannot be an important mechanism in

controlling the magnetic properties because there is no sufficient

overlap between neighboring orbitals.

60

Indirect exchange

indirect exchange: super exchange, double exchange, RKKY

The exchange interaction is normally very short-ranged, and thus the

longer-ranged interactions is called super-exchange interaction.

Superexchange can be defined as an indirect exchange interaction

between non-neighboring magnetic ions which is mediated by a non-

magnetic in-between ion.

61

Superexchange

Because superexchange involves the oxygen orbital as well as

metal ion, it is a second-order process and is derived from second-

order perturbation theory.

The matrix element is controlled by a parameter called the

hopping integral t, which is proportional to the band width of the

conduction band or the bandwidth in a simple tight-binding model.

Goodenough-Kanamori rule

The size of the superexchange depends on the magnitude of the

magnetic moments on the metal atoms and the metal–oxygen (M–

O) orbital overlap and the M–O–M bond angle.

AFM

FM

J. Stöhr and H. Siegmann, Magnetism (2006).

, ,0 ,

, 0 0

,0 0

, 0 0 0

t

t U

−

−

2

1

2

2

3

0

tE

U

E

tE U

U

= −

=

= +

1

2

3

, ,0

,

,0 ,

t

U

t

U

= +

=

= −

The antiferromagnetic coupling lowers the energy of the system

by allowing these electrons to become delocalized over the whole

structure, thus lowering the kinetic energy.

64J. Stöhr and H. Siegmann, Magnetism (2006).

65

Double exchange (Zener, 1951)Clarence Zener introduced double exchange to explain the

magneto-conductive properties of mixed-valence solids, notably

doped Mn perovskites.

A. Urushibara et al., Phys. Rev. B (1995).

66

Phase diagram of La1-xSrxMnO3

A. Urushibara et al., Phys. Rev. B (1995).

67

Double exchange (Zener, 1951)

Because in O2− the p-orbitals are fully occupied, the process has to

proceed in two steps by “double exchange”.

The movement of an electron from O to one ion followed by a

transfer of a second electron from the other ion into the vacated O

orbital.

Zener proposed a mechanism for hopping of an electron from one

Mn to another through an intervening O2−.

68


J. Stöhr and H. C. Siegmann, Magnetism (2006)

, , ,

, 0

, 0

, 0 0

U t

t U

U

−

−

1

2

3

0

E U t

E

E U t

= −

=

= +

1

2

3

, ,

,

, ,

= −

=

= +

The double exchange interaction favors a ferromagnetic alignment.

69


J. Stöhr and H. C. Siegmann, Magnetism (2006)

The ability to hop reduces the

overall energy and thus LSMO

ferromagnetically aligns to save

energy.

The FM alignment then allows

the eg electrons to hop through

the crystal and thus the material

becomes metallic.

Fe3O4

Fe3+

Fe3+

Fe2+

double-exchange

Super-exchange

71


In Fe3O4, a double exchange interaction ferromagnetically aligns

the octahedral Fe2+ and Fe3+ ions, while the superexchange

between tetrahedral and octahedral Fe3+ is antiferromagnetic.

Therefore, the two sets of Fe3+ ions cancel out, leaving a net

moment due to the Fe2+ ions alone. So, Fe3O4 is a ferrimagnetic

system.

In general, two metal atoms which are bonded through O may have

a valency that differs by one, such as for the cases Mn3+ (3d4) and

Mn4+ (3d3) in La1−xSrxMnO3 (LSMO) and for Fe2+ (3d6) and Fe3+

(3d5) in Fe3O4.

The electron is thus delocalized over the entire M–O–M group and

the metal atoms are said to be of mixed valency.

72

Solid-State Physics by James Patterson and

Bernard Bailey (2010)

Ruderman-Kittel-Kasuya-Yosida (RKKY)

interactionThe spin polarization of the conduction electrons oscillates in sign

as a function of distance from the localized moment and that the

spin information was carried over relatively large distances.

73

Screening can exist for either spin or charge scattering and results

in oscillations of the charge or spin density around the scattering

center

The oscillations in the charge density around a point-charge

impurity were first derived in 1958 by Friedel and hence go by the

name Friedel oscillations.

When conductive electrons are scattered by an atom, they will

rearrange themselves in order to minimize the disturbance. This

process is called screening.

Ruderman-Kittel-Kasuya-Yosida (RKKY)

interaction

74

Friedel oscillations

dI/dV as a function of distance

from a monatomic step.

M. F. Crommie et. al., Nature 363, 524–527(1993)

RKKY exchange

In the 4f metals the indirect mechanism involves the outer 5d

electrons which partly overlap with the 4f shell.

In contrast to the case of super-exchange, the indirect coupling

between two atoms thus proceeds through the outer electronic

states of the atoms themselves rather than through the electronic

states of a third atom.


RKKY exchange

The RKKY exchange coefficient J(R) is found to be oscillatory with

distance R according to

2 4

3 2 3 4

16 cos(2 ) sin(2 )( )

(2 ) (2 ) (2 )

e F F F

F F

A m k k R k RJ R

k R k R

= −

It makes a damped oscillation with distance from positive to negative

values.

2

3 2 3

2 cos(2 )1, ( )

(2 )

e F FA m k k R

R J RR

=

Therefore, depending upon the separation between a pair of ions their

magnetic coupling can be ferromagnetic or antiferromagnetic.

RKKY exchange


Ni80Co20 layers across a Ru spacer layer


Basic Aspects of the Quantum Theory of Solids by Khomskii (2010)


Basic Aspects of the Quantum Theory of Solids by Khomskii (2010)

80

Magnetic Domains and Hysteresis

81

Magnetic Domains and Hysteresis

In an actual ferromagnet, the material does not really break apart,

but different regions would have magnetization in different

directions to minimize energy.

The regions where the moments are aligned in one given direction is

called as a domain. The boundary of a domain is known as a domain

wall.

82

Hysteresis in Ferromagnet

Solid-State Physics by Ibachand Lueth (2009)

83

Hysteresis in Ferromagnet


X-ray microscopy


X-rays offer capabilities, such as elemental and chemical state

specificity, variable sampling depth, and the capability to follow

ultrafast processes on the picosecond scale.

One tunes the photon energy to a resonance and fix the photon

polarization and the magnetic contrast depends on the orientation of

the photon polarization relative to the magnetic orientation.

If now the sample contains microscopic regions with different

magnetic orientations, the signal from these regions will vary because

of the dichroic absorption effect.

Scanning Transmission X-ray Microscopy

(STXM)


In this approach the energy resolution is

given by the monochromator in the

beam line and the spatial resolution is

determined by the size of the X-ray spot.

The resolution is typically about 30 nm

with resolutions down to 10 nm or less.

They are “bulk” sensitive, in the sense

that the transmitted intensity is

determined by the entire thickness of the

sample.

Transmission Imaging X-ray Microscopy

(TIXM)


The spatial resolution of 15 nm has

been obtained.

Modern microscopes use a

monochromatic incident beam with

ΔE/E ~ 1/5,000 which also allows

spectroscopic studies of the detailed

near-edge fine structure.

X-ray Photoemission Electron Microscopy (X-

PEEM)


The third imaging method is based on X-

rays-in/electrons-out.

The sample is illuminated by a

monochromatic X-ray beam that is only

moderately focused, typically to tens of

micrometers, so that it matches the

maximum field of view of a photoelectron

microscope.

Most PEEM microscopes do not incorporate

an energy analyzer or filter and thus the

secondary electrons provides a suitably large

signal.

TIXM images of Fe/Gd multilayer


Exchange BiasExchange bias, arises if a thin film of a ferromagnet (FM), such as Co,

has a common interface with an antiferromagnet (AFM) such as CoO.

The size of the effect could only

be explained by assuming an

AFM–FM exchange interaction.

Many AFMs are best described by

two identical sublattices. In each

sublattice, the spins are parallel

generating a magnetization just

like in a FM.

MnO (O2+ not shown)Introduction to Solid State Physics by Kittel (2005)

The occurrence of exchange bias due to a “bias field” arising from the

antiferromagnet.

Exchange Bias


If a FM is deposited on an AFM in the absence of an external field, the

magnetization loop will still be symmetric and exhibit uniaxial

anisotropy.

Cooling FM-AFM systems across the TN, the magnetization loop may

be shifted horizontally in either the negative or positive field direction.

Exchange BiasThis case corresponds to a unidirectional magnetic anisotropy, since

the positive and negative external field directions are no longer

equivalent.

The field HB is called the transferred exchange field or the bias field.


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