凝聚態(tài)物理導(dǎo)論

有關(guān)固體磁性的基本概念和規(guī)律在上個(gè)世紀(jì)電磁學(xué)的發(fā)展史中就開(kāi)始建立了。19世紀(jì)中期：分子電流為基礎(chǔ)——最初關(guān)于磁性介質(zhì)的理論；19世紀(jì)后半期：發(fā)展了鐵磁磁化現(xiàn)象的試驗(yàn)方法——確立磁化規(guī)律的基本要素——分子場(chǎng)初步假說(shuō)和順磁磁化的Curie定律；20世紀(jì)初期：發(fā)展了順磁性的Langevin理論和鐵磁性的Weiss理論；20世紀(jì)前半期和中期：量子力學(xué)的提出和整個(gè)物理學(xué)的發(fā)展——鐵磁性，反鐵磁性及 鐵磁性理論的發(fā)展，并發(fā)展了許多新的物理實(shí)驗(yàn)技術(shù)，如電子磁性 ，核磁 及鐵磁

等?！?.1

Magnetism

of

atoms(1)

Electronic

states

in

atoms

or

ionsSingle

electron

Hamiltonian(2li+1)-fold

degeneracySpin-orbit

couplingInclusion

of

Coulombinteraction

(CI)(2L+1)(2S+1)-fold

degeneracy(2J+1)-fold

degeneracyL-S

coupling;

(CI>LS)J-J

coupling.

(CI<LS)(2)

Hund’s

rulesThe

rules

that

determine

the

ground

state

of

an

atom.Under

the

condition

that

satisfies

Pauli

exclusion

principle,

Stakes

its

um;Under

the

condition

that

satisfies

Pauli

exclusion

principle,

Stakes

its um

where

L

is

the

largest;If

the

number

ofelectrons

inthe

outer

s is

less

than

the

half-filling,

thenJ=|L-S|;

if

the

number

of

electrons

in

the

outer

sis

larger

than

the

half-filling,

then

J=|L+S|.Cr+3

(3d3):

the

ground

state(S=3/2,

L=3,

J=3/2)2S+1LJ(3)

Atom

in

a

magnetic

fieldp→p+eAIntroduce

a

magnetic

field

B0

along

z

direction,Take

gauge:satisfyingLz(without

inclusion

of

spin)Presume

B0

is

weak,

the

perturbation

energy

up

to

theorder

isE=E0B0=0E=E0+ΔEB0≠0Zeeman

splitting321ML=0-1-2-3Magnetic

moment:Intrinsic

orbitmagnetic

moment(indep.

B0)Induced

magnetic

moment:(dep.

B0)Origin

of

diamagnetismThe

case

including

spin:(Spin

moment-magnetic

fieldinteraction)Total

magnetic

moment:The

perturbation

energy

up

to

theorder

is(Landé

g-factor)Intrinsic

magnetic

moment:§6.2

Magnetism

in

solidsFive

basic

types

of

magnetism

have

been

observed

and

classified

onthe

basis

of

the

magnetic

behavior

of

materials

in

response

to

magneticfields

at

different

temperatures.

These

types

of

magnetism

are:ferromagnetism,

ferrimagnetism,

antiferromagnetism,paramagnetism,

and

diamagnetism.(1)

Diamagnetism

of

saturated

electronic

structuresThe

ionic

and

covalent

solids,

similar

to

noble

gases, have

filledelectron

structures.Paramagnetic

moment:(2)

Paramagnetism

ofbandcarriersElectrons

in

the

conduction

band

of

semiconductors

are

less,

and

onemay

presume

they

satisfy

Boltzmann

statistics.The

average

moment

of

band

electrons:At

room

temperature

T,(3)

Paramagnetism

of

impurities

and

defects(ESR,

EPR)Measuring

gfactorFrom

web(4)

Pauli

Paramagnetism

and

Landau

diamagnetismEFEFH=0H≠0,nonequilibriumH≠0In

Chapter

3,

we

gotPauli

paramagnetismPauliEnergy

levels

of

an

electron

in

a

magnetic

field

are

called

Landau

levels:Landau(5)

Knight

shiftThe

Knight

shift

is

a

shift

in

the

nuclear

magnetic

resonance

frequencyof

a

paramagnetic

substance

published

in

1949

by

the

Americanphysicist.The

Knight

shift

is

due

to

the

conductionelectrons

in

metals.

Theyintroduce

an

"extra"

effective

field

at

the

nuclear

site,

due

to

the

spinorientations

of

the

conduction

electrons

in

the

presence

of

an

externalfield.

This

is

responsible

for

the

shift

observed

in

the

nuclear

magneticone

is

the

Pauliponent

waveresonance.

The

shift

comes

from

two

sources,paramagnetic

spin

susceptibility,

the

other

is

thefunctions

at

the

nucleus.Depending

on

the

electronic

structure,

Knight

shift

may

be

temperaturedependent.

However,

in

metals

which

normally

have

a

broad

featurelesselectronic

density

of

states,

Knight

shifts

are

temperature

independent.Knight

shift:§6.3

Theory

of

paramagnetismCurie

lawCurie-Weiss

lawAtomicmomentLangevin

theory:PierreCurieAverage

moment:(Brillouin

function)Paul

LangevinMarcel

Louis

Brillouin1854-1948Curie

lawCurie

constantVan

Vleck

paramagnetismQuench

of

orbital

angular

momentumBy

Patrik

Fazekas§6.4

Theory

of

ferromagnetismMagneticsMagneticwallsTfθ

θp1/χCurie

lawFMmagnetic

hysteresis

loopMsMrHsHc(1)

Weiss

molecular

field

theory

on

spontaneous

magnetizationTwo

assumptions

for

ferromagnets:The

existence

of

an

internal

field---molecular

field;The

existence

of

magnetic

s.Pierre-Ernest

Weiss(1865-1940)x0T1BJ(x)T2MT3(T3>

T2>

T1)(2)

Paramagnetism

at

high

temperatures(3)

Localized

electron

model

for

spontaneous

magnetizationGeneralization:Werner

Heisenberg#

of

nearestneighborsmagnetizationWeiss分子場(chǎng)的實(shí)質(zhì)來(lái)源于原子間的交換作用，而交換作用來(lái)源于Pauli不相容原理。(4)

Spin

waves1930年，Bloch基于Heisenberg

model提出了自旋波的概念，用于

在低溫下自發(fā)磁化強(qiáng)度與溫度的關(guān)系，得到了M(T)隨T3/2變化的規(guī)律，這就是著名的Bloch

T3/2定律。Felix

Bloch根據(jù)Heisenberg

model，鐵磁體的基態(tài)是所有自旋沿同一方向排列。在低溫下，有一部分自旋將處于激發(fā)態(tài)，最低的激發(fā)態(tài)對(duì)應(yīng)于一個(gè)自旋反轉(zhuǎn)。由于同近鄰的自旋間有耦合，一個(gè)自旋的反轉(zhuǎn)必定引起整個(gè)系統(tǒng)自旋的不同程度的反轉(zhuǎn)，產(chǎn)生集體激發(fā)，這種自旋的集體激發(fā)被稱為自旋波(spin

wave)，其對(duì)應(yīng)的準(zhǔn)粒子為磁振子(magnon).A

boson經(jīng)典的自旋波理論是利用自旋角動(dòng)量S在磁場(chǎng)中的進(jìn)動(dòng)關(guān)系，可以求得一維單原子鏈的自旋波的色散關(guān)系:自旋波的量子理論是利用Holstein-Primakoff變換關(guān)系，將自旋算符變換成磁振子算符，即可求出上述色散關(guān)系。和聲子類似，自旋波得能量是量子化的：在低溫時(shí)，波數(shù)為k的自旋波的平均粒子數(shù):注意到每激發(fā)一個(gè)磁振子相當(dāng)于一個(gè)自旋反向，則有(D:

spin-wave

stiffness)At

low

TRandy

S.

Fishman

et

al,

Phys.

Rev.

Lett.

99,

157201

(2007).Spin

Wave

Excitations

in

a

Frustrated

Magnet

CuFeO2Due

toantiferromagneticinteractions

betweennearest-neighbor

Fe3+spins

in

each

hexagonalplane,

CuFeO2

is

ageometrically-frustratedantiferromagnet.(5)

Itineran ectron

magnetism

(band

magnetism)金屬磁性材料中原子磁矩并不是整數(shù)，例如鐵是2.21μB

，鈷是1.70

μB

，鎳是0.6μB

，它們與 原子磁矩的大小相差甚遠(yuǎn)。局域電子模型不能說(shuō)明金屬磁性材料的磁性，而能帶模型卻比較成功地說(shuō)明了金屬磁性材料的磁性及原子詞句的非整數(shù)性。能帶理論認(rèn)為，過(guò)渡金屬中3d與4s帶是交疊在一起的，3d電子雖然存在能帶結(jié)構(gòu)，但它們又相域，電子間的交換作用使自旋簡(jiǎn)并的電子能帶發(fā)生?？紤]電子間交換作用后，能帶 成不對(duì)稱形式，可以看出自旋向上的電子比自旋向下的電子數(shù)目多，在3d能帶中形成未被抵消的自發(fā)磁矩，因而可以發(fā)生自發(fā)磁化。54.420.583d10According

to

Stoner

model,

the

conditionshould

be

satisfied,

where

U

is

the

on-site

Coulomb

interaction

betweenelectrons.金屬磁矩的非整數(shù)性可以這樣解釋：一般認(rèn)為S帶的電子對(duì)鐵磁性沒(méi)有貢獻(xiàn)，d帶貢獻(xiàn)的大小依賴于能帶的性質(zhì)。如鎳，有10個(gè)價(jià)電子，飽和磁化說(shuō)明每個(gè)原子只有0.58個(gè)電子磁矩，能帶模型認(rèn)為，它是9.42個(gè)價(jià)電子處在d帶，0.58個(gè)電子處在s帶，9.42個(gè)電子中，5個(gè)電子自旋向上，4.42個(gè)電子自旋向下。這即解釋了為什么沒(méi)個(gè)原子只有0.58個(gè)電子磁矩?！?.5

Antiferromagnetism

and

ferrimagnetism(1)

Antiferromagnetism相鄰磁矩反平行排列，大小相等，方向相反，互相抵消，對(duì)外呈現(xiàn)出總磁矩為零。在溫度TN時(shí)，自發(fā)的反平行排列

了，成為Neel溫度。在Neel溫度以上，

順磁性。T>TN:Louis

Néel(1904-2000)NT

反鐵磁性是靠什么機(jī)制產(chǎn)生的呢？Cramer和Anderson先后用超交換模型即使了MnO晶體的鐵磁性，超交換作用有時(shí)也稱為間接交換作用。TχparamagnetismχTferromagnetismTccomplexTχantiferromagnetismTN-θCurieCurie-WeissNeelPhilip

Warren

Anderson(2)

Ferrimagnetism鐵磁性實(shí)際上是一種特殊的反鐵磁性，在研究其自發(fā)磁化時(shí)，需要將晶格分為兩個(gè)子晶格，然后按照鐵磁性的理論在每個(gè)子格子上進(jìn)行，鐵磁體具有

溫度。四氧化三鐵是典型的

鐵磁體，以及其它的鐵氧體：Fe(A-Fe)O4型，A=Mn2+、Fe2+、Ni2+、Zn2+。鐵磁體具有兩個(gè)主要特點(diǎn)：(1)有相當(dāng)大的磁化強(qiáng)度，但比鐵磁體里的磁化強(qiáng)度小；(2)這類材料的電阻率都相當(dāng)大，具有半導(dǎo)體的性質(zhì)，可用鐵氧體材料來(lái)制作微波元件等?！?.6

Low-dimensional

magnetismLow

spatial

dimensionality:

D

<

3One-Dimensional

(1D)Systemschainswireszigzagladdersalternating

chainsrandom

chainsTwo-Dimensional

(2D)

Systemssquaretrianglebrick-wallKagomébherringboneTwo-Dimensional

(2D)

Systems?

depleted

square

latticeRectangular

latticeInterpenetratedb

latticeTurtle

back

lattice?+=

10.8-11.2

AstronMolecular

magnetsMn12-acFe8(1980)V15

:

low

spin

molecule

with

spin

1/2S=1/2J~-800KJ'~J1~-150

KJ''~

J2

~

-300KAFM

couplingNi12-WheelS=12[Ni12(chp)12(O2CMe)12(THF)6(H2O)6]Cr8:

S=0(8

AFMcoupledS=3/2

Cr

centers)[Cr8F8(O2CCMe3]1612

FM

coupled

S

=

1nickel

centresNi24-wheel:AFM,

but

notdiamagneticMn6Cr4Cr8Laboratoire

Louis

Néel,

Wolfgang

Wernsdorfer1D

&

2D

Ising

model

can

be

exactly

solved.(Si

:

up

or

down)1D

exact

solution

(Ising1925):Magnetization:When

H?0,

M?0.

No

spontaneous

magnetization

for

T>0.Specific

heat:Susceptibility:Ising

ModelErnst

IsingZero-field

susceptibility

for

Ising

chain

with

spin

?

(Fisher

1963)Specific

heatMeasure

parallel

to

determinethe

sign

of

J.2D

exact

solution

(Onsager

1944)Specific

heat

on

a

square

lattice:T/Tc1It

is

found

that

CH=0(T)

is

logarithmic

divergent

at

T=Tc:M/NgμBMagnetization

(1952):for

T<TcCritical

pointLarsOnsager2D

Ising

model

exhibits

a

phase

transition

at

T>0:OnsagerBlote

et

al

(SC,1969)Critical

exponentsExact

solutions

of

2D

Ising

model

establish

the

foundationof

modern

theory

of

the

critical

phenomenon.Critical

temperatureHeisenberg

ModelExact

resultsHeisenberg

S=1/2

AFM

chain

by

Bethe

ansatz(Bethe

1931,

Hulthen

1938)Energy

of

the

ground

state:For

anisotropic

exchange

integrals

(Orbach

1958,

Walker

1959)Werner

HeisenbergHans

Albrecht

BetheExcitation

energy

(des

Cloizeaux

&

Pearson

1962)xyFor

S=1/2

AFM

chain,the

excitationfrom

the

ground

state

(singlet)

to

theexcited

state

(triplet)

is

gapless:-0Susceptibility

at

T=0

(Griffiths

1964,&1966)Spin-spin

correlation

function

(Shanker

et

al

1990)Power-law

decayLieb,

Schultz

and

Mattis

TheoremConsider

a

1D

AFM

chain

with

the

Hamiltonian,satisfying

the

periodic

boundary

condition

SN+1=S1,

N=even.

ForS=odd

integer/2,

e.g.,

S=1/2,

3/2,

…,

the

excitation

from

theground

state

to

the

excited

state

is

gapless.Ground

state

for

Heisenberg

antiferromagnet(Anderson

1951)Ground

state

for

Heisenberg

ferromagnetFor

the

Heisenberg

ferromagnet

(J<0),

the

fully

ferromagnetic

stateФFM

is

one

of

the

ground

state

multiplet,

whereMagnetization

plateaus

(Oshikawa

et

al

1997)Consider

zero-temperature

quantum

spin

chains

in

a

uniformmagnetic

field,

with

axial

symmetry,For

integer

or

half-integer

spin,

S,the

magnetization

curve

can

haveplateaus,

and

the

magnetizationper

site

m

is

topologicallyzed

as

n(S-m)=

integer

atthe

plateaus,

where

n

is

the

period

1/6of

the

ground

state

determined

bythe

explicit

spatial

structure

ofHamiltonian.mH/JConfirmed

both

theoretically

and

experimentallyHida

1994plateaun=3,

S=1/2(3-site

translation

invariant)Mermin-Wagner

TheoremFor

the

quantum

Heisenberg

model,

with

theshort-range

interactions

satisfying,there

cannotexist

any

magnetic

(including

FM

and

AFM)long-range

order

at

any

nonzero

temperature

in

one

andtwodimensions.CorollaryIf

there

exists

a

gap

from

the

ground

state

to

the

excitedstate,

there

will

be

no

magnetic

LRO

in

1D

and

2D

atzero

temperature.Goldstone

TheoremIf

there

exists

a

magnetic

LRO,

then

the

excitation

fromthe

ground

state

to

the

excited

state

will

be

gapless.Jeffrey

GoldstoneN.

David

MerminMagnetic

Long-Range

OrderHeisenberg

AFModelMagneticLong-RangeOrder

in

theGround

StateMagneticLong-RangeOrder

at

T>0D=1No(owing

to

quantumfluctuations)No(owing

tothermalfluctuations)D=2S=1/2,

Yes(numerical)S≥1,

Yes(rigorous)No(owing

tothermalfluctuations)D=3S=1/2,

YesS

≥1,

Yes(rigorous)S=1/2,

Yes(?)S

≥1,

Yes(rigorous)Heisenberg

Model

on

Square

LatticeModified

Spin-wave

theory

(Takahashi

1989)Uniform

susceptibilityFor

quantum

Heisenberg

chains

with

spin

integer,there

will

be:the

ground

state

is

unique;there

exists

a

gap

between

the

singletground

state

and

the

triplet

excited

state;the

ground-state

spin

correlation

functiondecays

exponentially.Within

the

continum

limit,

Haldane

observedSROCorrelation

lengthΔ

c-1Haldane

gap:c:

spin-wave

velocity~JSHaldane

Scenario

(1983)F.DuncanM.HaldaneAKLT

Model

(1987)S=1

Heisenberg

AFM

spin

chain

with

biquadratic

interactions(Affleck,

Kennedy,

Lieb,

Tasaki)The

ground

state

is

a

valence

bond

state

(VBS)(<i,j>:

nearest

neighbors)It

can

be

proven

thatAs

the

eigenvalues

of

H

0,

the

VBSis

the

unique

singlet

ground

state.They

proved

it

exists

an

excited

gap

from

theground

state:

=0.75JConfirmation

of

Haldane

conjecture!Bilinear-Biquadratic

ModelS=1

Heisenberg

AFM

chain

with

bilinear

and

biquadratic

interactionsCritical

points

separatingHaldane

phase

fromother

phasesΔ=0Haldane

phase:-

/4<

θ

<

/4Δ

=0.411J

at

θ=

0for

S=1,

=6Δ

=0.085J

at

θ=

0for

S=2,

=49Δ

=0Can

half-integer

spin

chains

have

a

gap?(Oshikawa

et

al

1997)Translationally

invariant

spin

chains

in

anapplied

field

can

be

gapful

without

breakingtranslation

symmetry,

onlywhen

themagnetization

per

site,

m,

obeys

S-m=integer.Such

gapped

phases

correspond

to

plateaus

atthese zed

values

of

m.Half-integer

S

spinchainscanhave

“Haldane

gap

phase”

undersome

conditions.S=3/2,

m=1/2Quantum

Phase

TransitionsA

quantum

phasetransition

(QPT)

is

aphase

transition

betweendifferent

quantum

phases(phases

of

matter

at

zerotemperature).QPT