tutorial yuj magnetism

7/26/2019 Tutorial YuJ Magnetism

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2005 KIAS-SNU Physics Winter Camp: Lectures on Magnetism Jaejun Yu,

Introduction to Magnetism and Magnetic Materials

Jaejun Yu

School of Physics, Seoul National University

[email protected]

c2005 Jaejun Yu

KIAS-SNU Physics Winter Camp

5-18 January 2005
mailto:[email protected]:[email protected]


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Abstract

In this lecture,1

I will give a brief introduction on a quantum mechanical view on mag-netism in real materials, especially, consisting of transition metal elements and their com-

pounds, and the physical principles for the applications of magnetic materials as magnetic

sensors and memory devices. Further, I will discuss the connection between magnetism

and superconductivity in highTc superconductors as an example.

1This lecture is support byKIASandSNU-CTP.
http://www.kias.re.kr/http://ctp.snu.ac.kr/http://ctp.snu.ac.kr/http://ctp.snu.ac.kr/http://www.kias.re.kr/


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Contents

1 What is all about condensed matter physics? 51.1 Macroscopic vs. microscopic objects . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Most condensed matter systems are quantum mechanical by nature! . . . . . . 111.3 How do we approach the condensed matter system in order to understand the

physics of ablack box? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Particle zoo in the condensed matter systems . . . . . . . . . . . . . . . . . . . 151.5 Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Magnetism in Real Materials 192.1 Magnetic Storages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Magnetic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 A quantum mechanical view of magnets:

Pauli exclusion principle and Coulomb interactions 34

3.1 Sources of Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Magnetic Moment of an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Magnetic Moments in Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 A Model for the Exchange Interactions in Solids . . . . . . . . . . . . . . . . . . 593.5 Transition Metal Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.6 Magnetic Impurity in a Metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


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4 Effective Hamiltonian and Phenomenological Theory 78

4.1 Heisenberg Model: Mean Field Solution . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Order Parameters: Description of Phase Transition . . . . . . . . . . . . . . . . 804.3 Bragg-Williams Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Ginzburg-Landau Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5 Second-Order Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.6 correlation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Giant magnetoresistance (GMR) and magnetic sensors 90

6 Magnetism and superconducitivity:

HighTc superconductivity 107

7 References 110


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1. What is all about condensed matter physics?

... How do we understand the physical properties of single particle systems in a classicalor quantum sense? In other words,what do we measure or observe?

O

x

p

?

... In classical dynamics, the state of a single particle is determined by the observables {x(t),p(t)}. It canbe extended to the system with many particles where the state of the system is described by the set of

observables {xi(t),pi(t)|i= 1, 2,...,N}. However, whenN 1025, it is practically impossible to tracethe orbits of all, even the part of, the particles. Here it is the point where the statistical physics comes into

playing a role. Instead of following each individual particles, we measure a quantity by an (ensemble or

time) average of the given quantities adopted in classical dynamics. In addition, now we have to deal withnew observables such as entropy, temperature, ....

The same thing applies for the case of quantum systems. The only difference is the dynamics state of the

quantum system is determined by a state vector |(x, t) for a single particle and |(x1,x2, ...,xN, t).


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1.1. Macroscopic vs. microscopic objects

Observablesfor the macro object consisting of more than 1020

particles:

specific heatcv entropy, temperature bulk modulus or compressibility pressureP polarization P, magnetization M electro-magnetic field E, B reflectivity, color, conductivity, ... etc.


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For an example, what physical property of the cube (or disk) makes this happen, i.e., acube float over a magnet against gravitation?

(Hint: This is the phenomenon called magnetic levitation, which is mainly at-tributed to the Meisner effect of superconductors.)


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Observablesfor the micro or nano object of order 102 particles:

conductanceG / electron tunneling current I force F magnetic Flux charge density distribution, electron cloud (bonding), ... etc.


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an AFM image an STM image a LEED image

What do we really see in these images?What physical quantities do they represent?

An example of the SEM System:


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MRI Image: What do we really see in this image?What physical quantity does it represent?

(This is an image of our brain probed by using the technique of the nuclear magnetic

resonance.)


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1.2. Most condensed matter systems are quantum mechanical by nature!

p d

Unfortunately, however, there is no quantum mechanical solution available except forfree (i.e., non-interacting) particle systems.

An example of exactly solvable models:

H = (a+a+1

2)

a+a|n =n|na|O = 0


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1.3. How do we approach the condensed matter system in order to understand

the physics of ablack box?

Experimental Methods:By disturbing the black box by phtons, phonons, electrons, and/or neutrons, try to get ahint on elementary excitations in the system.

Theoretical Methods:By guessing possible elementary excitations and working out the QM equation of mo-tions, see if we can predict the physics of the system. If wrong, correct the model forthe elementary excitations for the better description.

How to disturb the black box?


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Question: Measuring the dc resistance is a way of disturbing the system? What reallyhappens inside the black box when we apply a bias voltage?

Current-Voltage curve of a YBCO highTc superconductor


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Question: What happens when we shine a light on the matter?

Light scattering experiment: (Photoemission / IR spectroscopy)


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1.4. Particle zoo in the condensed matter systems

elementary excitations in solids

quasi-particles: electrons, holes, polarons, excitons, Cooper pairs, ... collective excitations: phonons, magnons, zero-sound, plasmons, ...


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1.5. Energy scale

The condensed matter system is merely a collection of atoms, where each atomconsists of electrons and a nucleus (me mN).From theuncertainty principlexp ,

p=

2meE

3mkBT

Thus, practically, the size of atom the size of electrons.

atomic unit: = e2 =me = 1

x aB = 2

me2 = 0.529177A (Bohr radius)

EB = 12

e2

aB= me

4

22 = 13.6058eV= 1Ry

In atomic unit,aB = 1,EB = 1/2,c = 1/ 137,kB 3.2 106K1B = 2.13 106T1 , ...

(= e2/c: fine structure constant)


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energy of an electron in a box of sizeL:

x L p L

Eo = p2

2m 1

2L2

speed of electrons in a box:ve

1

L

c

L electrons in solid Fermions:Pauli exclusion principle Degenerate Electrons:lowest possible excitationsnear the Fermi energy


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Energy Scale

Coulomb interaction: eV Magnetic ordering temperature Tc: 0.1 meV 50 meV Dipole-dipole interaction: 0.1 meV


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2. Magnetism in Real Materials

Magnets


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Solenoid vs. Bar Magnet


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Origin of Magnetic Moments

Magnetic permeability and origin of magnetic moments?


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Magnetic Diagram


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Magnetic Image of Our Brain

MRI (Magnetic Resonance Imaging) using the NMR technique

Mapping of Proton Spin Resonance Frequencies


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Levitating magnetic bar on a superconducting bed

Magnetic shielding by the superconducting current


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2.1. Magnetic Storages

Magnetic thin films


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2.2. Magnetic Sensors


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MFM (Magnetic Force Microscopy) Image of Nanomagnet Array(C. A. Ross, MIT)


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3. A quantum mechanical view of magnets:

Pauli exclusion principle and Coulomb interactions

3.1. Sources of Magnetic Moments

Magnetic moment:M =gBJproportional to the angular momentum J = L + S

Spin moment S

Orbital moment L


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Localized State

Since the rotational symmetry is preserved, it has the quantum eigenstates |jm:

J2|jm =j(j+ 1)|jm

whereJ = L + S

If the spherical symmetry is broken, e.g., inside a lattice, the orbital moment can bequenched due to the lack of rotational symmetry, i.e.,

L = 0


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Delocalized State

A free electron state such as|k may not be an eigenstate of the angular momentumoperator L.

Only |sms may contribute to the magnetic moment.

k

Exception:Diamagnetic shielding current in a superconducting state.


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Perfect Diamagnetism: Superconductivity

SC

Inside the superconductor:B = 0

Supercurrent induced by the current of Cooper pair electrons:

Js(r) = A(r)


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A Localized State Coupled to Delocalized States?

An impurity atom in a metal

k kd

Formation of a Local Moment Broken Symmetry State!Anderson Impurity Model


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3.2. Magnetic Moment of an Atom

s

p

d

m=0 m=1 m=2m=-1m=-2


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Hunds Rule

1. Maximum totalS= max Sz withSz =

i msiObeying the Pauli exclusion principle

2. Maximum totalL = max Lz withLz =

i mliMinimizing the Coulomb interaction energy

3. Spin-orbit interaction:

J=

|L+ S| if less than half-filled|L S| if more than half-filled


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Exchange Density

(r)a* (r)

b

(r)a

(r)b

Gaining more (negative) exchange energy by aligning spins:

Hunds 1st rule


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Overlap and Exchange Integrals of the atomicp-orbitals


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Minimization of Coulomb Energy TermUab

by letting(r)be separated: Hunds 2nd ruleYlm(, ) = (1)mYlm(, )


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Configuration of 3d Transition Metal Ions

L S

0

1

2

3

0 2 4 6 8 10

dn


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Spin-Orbit Coupling

determined by the sign of the coupling constantL S

where

dV(r)dr

L S

SL

high spin low spin


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3.3. Magnetic Moments in Solids


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Cubic Lattice Symmetry: Broken Spherical Symmetry

Crystal Field Splitting:

l=2

eg

t2g

|eg = {|x2 y2, |z2 r2/3}|t2g = {|xy, |yz, |zx}


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Kinetic Energy vs. Coulomb Energy

Uncertainty Principle:p

x

Kinetic Energy:EK=

p2

2m

2

2m

1

(x)2

Coulomb Exchange Energy:EC Ae

2

x

where A 0.1, an order of magnitude smaller than the direct Coulomb interactionenergy.

Whenx 1, the Coulomb energy dominates over the kinetic energy. Wigner solid:frozen localized electrons


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Wigner Crystal: Metal-Insulator Transition

x

x

x

x

Result of the competition between kinetic and Coulomb energy!

Here we dropped the interaction term between electrons and background positive charge.What is it?


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Delocalized State:Molecular / Band State

hA hB

o

A B

t=

o

A VAB B

Ho= hA+ hB+ VAB

Ho= o t

t o


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Single-particle eigenstates and eigenvalues:

| = 12 (|A |B) =o t

o

o

o+t

o- t

2t

Corresponding two-particle state:

2 = 12

[|+, |+, |+, |+, ]

Spin-singlet ground statewith single-particle molecular states |


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Energy Curve for a Hydrogen Molecule


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Consider two hydrogen atoms separated by a distance d. What are the ground states ofthis system in the limits ofd 0 and d ? What is the dominating factor in each limit?Try to make a qualitative argument in describing the physics.What about a lattice of hydrogen atoms with a lattice constant d? Discuss the possible phys-ical properties of the lattice with varying d from to 0.


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Three-Sites Molecular States

o+t

o- t

2to


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N-Sites States: Tight-Binding Energy Band

o+t

o- t

2to

These delocalized molecular state is stablewhen the Coulomb interaction energy U issmallerthan the kinetic energy gain W= 2t.

U= 2|VC(1, 2)|2

dr1dr2VC(r1 r2)A(r1)A(r2)

: on-site Coulomb interaction energy


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Pauli Paramagnetism

the change of the Density-of-states due to the Zeeman coupling:

D() =dN()

d


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Pauli Paramagnetic Susceptibility

N=

EFBB

1

2D(E+ BB)dE

EF0

1

2D(E)dE=

1

2BBD(EF)

N =

EFBB

1

2D(E BB)dE

EF0

1

2D(E)dE= 1

2BBD(EF)

M=B

(N

N) =2

BD(E

F)B =

3N 2B

2kBTFB =

s=

3N 2B

2kBTF

Substracting the Landau diamagnetic contribution L = N2

B

2kBTF, we have the Pauli Para-

magnetic Susceptibility:

P = N 2BkBTF


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3.4. A Model for the Exchange Interactions in Solids

WhenU

W, the localized states of

|o

becomes stable relative to the band (molecular)

state |2:|o = 1

2[|, s1|, s2 |, s2|, s1]

which is a combination of the localized orbitals |A,B instead of the molecular states |.Here the starting Hamiltonian Ho include the interaction term of U:

Ho=hA+ hB+ U

and the hopping termVAB should be considered as a perturbation.


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Many-Particle Excited States

Two-particle excited states:

|o(s1, s2) = |A, s1|B, s2|A(, ) = |A, |A, |B(, ) = |B, |B,

Ho|o = 2o|oHo|A = (2o+ U)|AHo|B = (2o+ U)|B


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hA hB

o

A B

o

o+U2


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Only remaining degrees of freedom of the ground state |o are

spins!|p(s1, s2) = |s1, s2Energy Correction via the Perturbation Theory with H1=VAB First-order correction:

s1, s2|H1|s1, s2 = 0

Pauli exclusion prohibits the double occupancy at the same site:

|A, |A, = 0

That is,|A, s1|A, s1 = 0


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Second-order correction:

, |H

1

|A

=

, |H

1

|B

=t

, |H1|A = , |H1|B = 0

E(2), = n

|, |H1|n|2En Eo = 0

E(2)

, = n

|,

|H1

|n

|2

En Eo = 2t2

U = JAF


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Anti-Ferromagnetic Superexchange Interactions

Heff = +JAFA B


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Ferromagnetic Exchange Interactions

What happens if additional degenerate (or almost degenerate) states exist at each atomwhileU W?

hA hB

o A Bo

1 B

A

oo

1 11


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Now additional excited states become available:

|A(

,

)

=|Ao,

|A1,

and

o(, )|H1|A(, ) = 0

Ho|A(, ) = (o+ 1+ U J)|A(, ) =EA(, )|A(, )If the exchange energy is larger than the difference |o 1|, i.e.,

J > |o 1|

we haveEA(, )< EA(, )

Therefore,

E(2), =

n |, |H1|n|2

En Eo

2t2

U J < E

(2), =

2t2

U


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Ferromagnetic Exchange Interaction

Heff = |JF M|A B


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3.5. Transition Metal Oxides

x

Cu

O

Cu

The largerx, the smallert, i.e.,W = more localized!


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FM Metal and AF Insulator

n=N

e

2N0 1/2 1

T

FM FM

AF

Band-filling controlled byband degeneracyor bydoping


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Question: Why should the half-filled state be an insulator? What can one imagine tohappen when an additional electron or hole is introduced in a half-filled system?


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Why Transition Metals (TM) and TM Compounds

r

rR(r)

0

3d4p

4s

both 3dand 4felectron wavefunctions areextremely localized:no node in radial wavefunctions


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Question:

Let us consider the transition-metal fluoride MnF2, which is observed to be antiferromag-

netic at low temperatures. In this system, the fluorine F becomes F

being fully ionized sothat the manganese Mn becomes Mn2+.

(a) Based on the Hunds rule, discuss the possible spin and orbital configuration of Mn2+

ions. (Note that Mn2+ has a 3d5 configuration.)

(b) Considering that the Mn ions of MnF2form a bcc lattice, guess the magnetic configurationof the ground state of MnF2.

(c) What difference would it make if we replace F ions by O2 ions, that is, the ground stateof MnO2? (Hint: Consider the crystal field effect in a cubic lattice.)

Now consider the long-range magnetic order in metals. We know that Cu is a good para-magnetic metal while Fe is a ferromagnetic metal. Explain the difference and similarity of thetwo systems. Why Fe is not anti-ferromagnetic instead of being ferromagnetic? (Note that Fehas a configuration of 3d64s2 and Cu has 3d104s1.)


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3.6. Magnetic Impurity in a Metal

low temperature: singlet ground state

before

after

high temperature,spin-flip scattering


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Kondo Effect

Spin-flip scattering between the free electrons of a metal and the local moment of amagnetic impurity = the Kondo effect.

Highly correlated ground state where the conduction electrons form a spin-polarizedcloud around the magnetic impurity.

Below Kondo temperature, = a narrow resonance at the Fermi energy: Kondoresonance.


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Friedel Sum Rule

Ntot= 1lm

lm,kF = 2l

(2l+ 1)l,kF

(r)k,l

(R)=0

Impurity

k,l


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Scattering Phase Shift and Change of Density-of-State

nk

nk n+1kn-1kn-2k

R

d

()

o ()

F

*d

0

0

(b)

(a)


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Kondo Resonance

d

()

F

d d0

TK

+U


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4. Effective Hamiltonian and Phenomenological Theory

4.1. Heisenberg Model: Mean Field SolutionH =

ij

JijSi Sj+ gBH i

Si

Introducing an effective field Heff,

H = ij

JijSi Sj+ gBH i

Si= gBi

Si Heff

where the effective mean field

Heff= H 1gB

j

JijSj

and the average magnetization

Si = VN

M

gB

Heff= H + M

= V

N

Jo(gB)2

M= NV

F

H =Mo(

HeffT

)


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For the case of H= 0,one can fine the magnetizationMby solving the equation:

M(T) = Mo(MT

)

o(T) =

MoH

H=0

=Mo(0)

T

where the Curies constant is determined to beCo = Mo(0).

For the case ofH

= 0,

=M

H =

MoHeff

HeffH

=o(1 + )

= o

1 o = CoT Tc

where the critical temperature Tc becomes

Tc=NV

(gB)2

3kBS(S+ 1)= S(S+ 1)

3kBJo


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4.2. Order Parameters: Description of Phase Transition

T > Tc T < Tc Symmetry

Supercondutor s(r) = 0 s(r) = 0 U(1)Magnets Ms(r) = 0 Ms(r) = 0 O(3)Ferroelectrics Ps(r) = 0 Ps(r) = 0 O(3)Liquid/Gas-Solid (G) = 0for G = 0 (G) = 0for G = 0 TROrder-disorder = 0 = 0 Z2

= nAA nBA


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4.3. Bragg-Williams Theory

Consider the Ising model with the spin =

| or

| . The order parameter m =

is the

average of the spin:m= (N N)/N

EntropyS:S= ln CNN = ln C

NN(1+m)/2

S

N =s(m) = ln 2 1

2(1 + m) ln(1 + m) 1

2(1 m)ln(1 m)

Average EnergyE:E= J

ij

m2 = 12

JNzm2

wherezis the number of nearest neighbor sites.

Bragg-Williams free energyf(T,m):

f(T,m) = (E TS)/N= 12Jzm2 +

1

2T[(1 + m) ln(1 +m) + (1 m)ln(1 m)] Tln 2


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The equation of state under an external field h:f

m= zJm +1

2Tln[(1 + m)/(1 m)] =h

zJm + Ttanh1

m= h

m= tanh[(h + Tcm)/T]

Note thatheff=h + Tcm.


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Mean field solutions for h = 0: nearT 0:

m= tanh(Tcm/T) 1 2e2zJ/T


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nearT Tc :

m (Tc/T)m 1

3(Tc/T)

3

m

3

(Tc/T)m 1

3m

3

m= [3(Tc T)/T]1/2


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4.4. Ginzburg-Landau Functional

Free energy nearTc

s(m) = ln 2 12m2 1

12m4 + ...

f(m) = 1

2(T Tc)m2 + 1

12m4 Tln 2 + ...

whereTc = zJ

Assuming(r)as a local order parameter,

F =

drf(T, (r)) +

dr

1

2c |(r)|2

wherefcan be expanded by

f(T, ) =1

2r2 w3 + u4 + ...

wherer = a(T Tc)Symmetry properties of the free energy functionalF!


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4.5. Second-Order Phase Transition

f(T, ) =1

2

r2 + u4

The equation of state:r + 4u3 =h

Forh = 0,=

0 if T > Tc(r/4u)1/2 if T < Tc


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(Tc T)

where= 1/2.

Susceptibility[r+ 12u2]

h= 1

=

h=

1/r ifT > Tc1/(2|r|) ifT < Tc

|T Tc|

with= 1.

Free energy densityff=

0 ifT > Tcr2/(16u) ifT < Tc

Specific heatcvcv = T

2

fT2 =

0 ifT > TcT a2/(8u) ifT < Tc


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4.6. correlation length

1

(r

,r

) =

2F

(r)(r)= (r+ 12u

2

c2

)(r

r

)

(q) = 1

r+ 12u2 + cq2

(q) =

1 + (q)2 =

1

c

2

1 + (q)2

where

= c1/2[r+ 12u2]1/2 =

(c/r)1/2

ifT > Tcc1/2/(2r)1/2 ifT < Tc

and the correlation length |T Tc| with= 1/2.

0=

c

r(T= 0)

1/2=

c

aTc

1/2


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5. Giant magnetoresistance (GMR) and magnetic sensors

What is Magnetoresistance?

In the free electron system, the conductivity is often described by the Drude model:

=ne2

m

UnlessH(magnetic field) is too large, the mean scattering timedoes not depend on H:

(H)

H = 0 + o(H2)


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H

l=vF


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Magnetic Multilayer Structure

Spin-dependent Density-of-State atF: Ds(F)

= s= e2

mns(H)

ns(H) Ds(F)


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E

EF

E


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Change ofDs()with an external magnetic field H

E

EF

E

s increases by the application ofH, i.e.,/H >0


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Giant magnetoresistance effect (GMR)

Giant magnetoresistance effect (GMR) in a junction between two magnetic electrodes Electrons undergo quantum tunneling through a thin insulating layer Conductances across the junction

s = 4e2

hN

DLs(F)DRs(F)


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Tunneling Magnetoresistance (TMR)

0 H

( )H

o

1


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TMR in Layered Manganites: La1+xSr2xMn2O7(x=0.4)


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Effective Model Hamiltonians for Doped Manganites

Double Exchange Model with Superexchange Interactions

H= L

ij,

tijc

+icj + h.c.

JH

Li,ab

Si abc+iacib+ JAFLi,j

Si Sj,

JH


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Formation of Ferromagnetic Domain:Kinetic Energy Gain by the Double Exchange Mechanism


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Application of a GMR Device


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Exchange Coupling in Magnetic Multilayers

The magnetic coupling of two magnetic layers depends on the thickness of the inter-vening non-magnetic spacer layer.

Oscillatory exchange coupling: observable quantum interference effects. Confinement of electrons in a quantum well formed in the nonmagnetic layer by the

spin-dependent potentials of the magnetic layers.


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(Mathon et al.)

MRAM (magnetic random access memory)


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AF FM FM

Insulator

A


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6. Magnetism and superconducitivity:

HighTc superconductivity

What happens when the half-filled AF insulator is being doped?


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Phase Diagram of HighTc Superconductors:Doped Cu-oxides Systems

T

x0

0.3

SCAF

pseudo-gap


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Anti-Ferromagnetic Spin Correlation vs. BCS State

Spin-order Ground State of an Anti-Ferromagnet:

|AF =i

c+(2i)c+(2i+1)|O

BCS State of a Superconductor:

|BCS = k

(uk+ vkc+kc

+k)|O

t-JModel for the HighTc Superconductivity:

Heff= ij

c+icj + Jij

si sj


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7. References

Basic Quantum Mechanics:

1. S. Gasiorowicz, Quantum Physics, (John Wiley & Sons, 1996).

2. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, (Hermann, 1977).

Magnetism:1. R. M. White, Quantum Theory of Magnetism, (Springer, 1983)

2. D. C. Mattis, The Theory of Magnetism, (Spinger, 1981) Correlated Electron Systems:

1. P. Fulde, Electron Correlations in Molecules and Solids, (Springer, 1995).

2. M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.70, 1039 (1998).

tutorial yuj magnetism

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