Chapter Twelve Ferromagnetism and Antiferromagnetism

In some materials, there are important interactions between magnetic moments resulting in ordered magnetic phases in the absence of H.

0M >>Parallel alignmentFerromagnetism

0M =Anti-parallel alignmentAnti-Ferromagnetism

0M >Anti-parallel

Ferrimagnetism

What are the interactions ?

(1) Dipole-dipole interaction

produces local field → align adjacent very weak(2) “Exchange” interaction

QM interaction due to overlap of electron wavefunctions (Spin dependent)

µ µNot Important

∑ •−=ij

ji SSJ2U

Ferromagnetism

H

M

Ms

Hc

The typical magnetization curve of the ferromagnet

A hysteresis loop of magnetization is present.

Ms : saturation magnetization

Mr = M(0) residual magnetization

Hc coercive field M(± Hc)=0

T=300K

High T (T>Tc) : paramagnetic

Low T (T<Tc) : ferromagnetic

Ms

TTc

χ

Ferromagnet Paramagnet

θχ

−=

T1~

HM

Curie-Weiss law

Curie temperature

T(K)

Ms

(em

u/g)

1473K

1273K

1073K

273K

627K

1388K

1043K

FCC

Mo

Approaching Tc (Curie temperature)

High T (T>Tc) : paramagnetic

Low T (T<Tc) : ferromagnet

( ) γχ −−∝ cTT

( )βTTM cs −∝

In mean field approximation : γ=1 and β=1/2.

Fe

Co

Ni

Gd

CrO2

CuBr3

EuS

1.33

1.21

1.35

1.3

1.63

1.21

0.34

0.42

0.37

0.33

1043

1388

627

292

387

33

17

Experimental data γ β Tc(K)

Ni

Tc=354oC

T/Tc

Ms/Mo

How will Ms vary with temperature ?

What is Tc ?

( )a1acoth)(

MM

o

−== aL

Recall Langevin function

TkHa where

B

µ= in the theory of

paramagnetism

Weiss suggested that a “molecular field HM” in addition to His acting on material. : H + HM= H + γM

When H=0,o

o

BB

M

MM

TkM

TkHa µγµ

== aMTk

MM

o

B

o

=

µγ

M/Mo is a linear function of a with a slope proportional to T.

a=µH/kBT

M/M

o

T1

T2

T3

Three straight lines:

w/. T3>T2>T1

aMTk

MM

o

B

o

=

µγ

P The intersection P gives spontaneous fractional magnetization Ms/Moachieved at that temperature.

Ms decreases with increasing T and reaches 0 at T2.

When T is higher than T2, the spontaneous magnetization vanishes implying that T2 is Tc.

The Curie temperature Tc :

31

MTk

o

cB =µγ B

oc 3k

MT µγ=

Hence, a3TTa

MTk

MM

co

B

o

=

=

µγ)(

MM

o

aL=intersects w/. at P

determining Ms/Mo.

The law of corresponding states :

All ferromagnetic materials, which naturally have different values of Mo and Tc have the same value of Ms/Mo for any particular value of T/Tc.

Very nearly, but not exactly, correct

M in the calculation is magnetic moment per unit volume, emu/cm3 .n: # of atoms per unit volume, changes with temperature.Exact statement : take M[emu/g] = M[emu/cm3] / ρ[g/cm3] instead of M.All materials have the same values of Ms/Mo for the same value of T/Tc.

aT3TaTk

co

B

o

=

=

MMΜ

µγρ

T/Tc

Ms/M

o

The Weiss’s prediction of a law of corresponding states is verified.

But the shape of the curve is wrong.Weiss-Langevin : J=∞

As discussed in the previous chapter, Ch.11.

QM need to be taken into accounted for the source of M.

( )

−

++

==2Ja'coth

2J1

2J'a12Jcoth

2J12J)(a'BJ

oMM

whereTkH

TkHJ'a

B

H

B

B µµ==

g

a'MTk

oH

B

o

=

γρµMM

The straight line :

The slope of BJ(a’) at the origin is :

+

J1J

31

( )B

oB

B

oHc 3k

1JJ

1J3k

T MgM γρµγρµ +=

+

=Curie temperature

a'T3T

J1J

co

+

=MΜ

=

c

os

o

s

T/T/tanh MM

MΜ

J=1/2

in fairly good agreement with experiment

J=1/2 implies that the magnetic moment is due entirely to spin, g=2 and there is no orbital contribution.

Fe

Co

Ni

Cu2MnAl

Ni78Fe22

Ni79Mo5Fe16

2.12

2.22

2.2

2.01

2.07

2.12

2.08

2.18

2.09

2.00

2.11

2.10

Experimental data g* g+

Ferromagnetism is due essentially to electron spin

with little or no contribution from orbital motion of the electrons.

At 0K, the spins on all the atoms are parallel, in one direction. At a higher temperature, a certain fraction of the total, determined by the Brillouin function, flip over into the other direction; the value of that fraction determines the value of Ms.

At 0K

At T≠0

Reduction of Ms (T≠0) is due to excitation of spin waves.

In the calculation shown above, we only consider the effect of the molecular field and put the applied field H equals to zero.

Now we apply a magnetic field H, ( ) ( )Tk

HTk

HHa'B

H

B

MH Mγρµµ +=

+=

ooH

B

B

H

oH

B

oH

B

o

Ha'TkTkHTka'Tk

MMMMMΜ

γργρµµ

γρµγρµ−

=

−

=

M/M

o

a’o

HMγρ

B(a’)

J=1/2Curve is the Brillouin function for J=1/2.

Lines 2 & 4 represent the molecular field alone.

The dashed lines 2’ & 4’ represent the molecular and a applied field.

Above the Curie temperature T>Tc

For instance, T=1.2Tc, the effect of the applied field move the intersection point of the field line and the magnetization curve from the origin to point B.

Near the origin a'3J

1J

o

+

=MΜ

( )( ) Jk3/1JT

Jk3/1JH BoH

BoH

+−+

==Μ

ΜΜγρµ

µχSusceptibility θ−=

T'C

( )Jk3

1J'B

oH +=

ΜC µ ( )Jk3

1J

B

oH +=

Μγρµθand

θ is the temperature at which susceptibility becomes infinite.

Tc is the temperature at which the spontaneous magnetization becomes 0.

+

=J

1J3k

TB

oHc

Mγρµ identical

Below the Curie temperature T<Tc

For instance, T=0.5Tc, the effect of the applied field shift the intersection point of the field line and the magnetization curve from point P to point P’.

Since the magnetization curve is nearly flat in this region, the increase in relative magnetization from P to P’ is very slight.

M/M

o

T(K)

paramagneticferromagnetic

spin“molecular field HM” : γρMs

so

cBsM

TkH MM

MHµ

γρ ==

( )( )( )( ) ( )218emu/g

221.9emu/gerg/Oe102.061043Kerg/K101.38

20-

-16

××

=

Oe106.9 6×=

For Fe, µo=221.9emu/g

Oe109.11H Co 6M ×=

Oe107.41H Ni 6M ×=

J=1/2

These fields are very much larger than any continuous field yet produced in the laboratory.

However, the molecular field is in no sense a real field, but rather a force tending to make adjacent atomic moments parallel to one another.

Fe

Co

Ni

Mo[emu/g]

221.9

162.5

57.5

µH[µB]

2.22

1.72

0.60

Ferromagnetic

ANμHo =M

N : Avogadro’s number

A : Atomic mass

The huge difference between a ferromagnetic and paramagnetic is due to the degree of alignment achieved and not to any large difference in the size of moment per atom.

erg/Oe10927.0 20B

−×=µ

Exchange energy

Play an important part of the total energy of many molecules and of the covalent bond in many solids.

Heisenberg showed that it also played a decisive role in ferromagnetism.

Ferromagnetism is due essentially to electron spin and reduction of Ms (T≠0) is due to excitation of spin waves.

Quantized spin waves : Magnons

∑=

•−=N

1ji,ji SSJ2U where J is called exchange integral and

spin angular momentum of atom i : Si

In general, consider N spins

If J is positive, U is a minimum when the spins are parallel (φij=0) and a maximum when they are anti-parallel (φij=π).

If J is negative, U is a maximum when the spins are parallel (φij=0) and a minimum when they are anti-parallel (φij=π)

Treat Si as classical vector

As we have already seen, ferromagnetism is due to the alignment of spin moments on adjacent atoms.

The ground state of a simple ferromagnet has all spins parallel

J>0 for ferromagnetism

∑=

+•−=N

1i1ii SSJ2U

2

o JNS2U −=

What is the energy of the first excited state?

One particular spin is reversed.

2o JS8UU +=OR

we can form an excitation of much lower energy if all spins share the reversal

The elementary excitations of a spin system have a wave like form and called magnons.

Analog to lattice vibrations or phonons

A spin wave on a line of spins

Side view

Top view

A classical derivation of the magnon dispersion relation

For the pth spin ( )1p1pp SSSJ2 +− +•−

The magnetic moment at site p : µp = - gµBSp

( ) ( )

+−•−=+•

−− +−+− 1p1p

Bp1p1p

B

p SSJ2SSJ2

µµ

µµ

ggEffective magnetic field or exchange field on the pth spin Bp.

ppp B

dtSd

×= µ

Classical mechanics

( ) ( )1pp1-pp1p1-pB

pBp SSSSJ2SSJ2S

dtSd

++ ×+×=+

−×−=

µ

µg

g

In Cartesian coordinate Assuming that SS and S,S S,S zp

yp

xp =<<<<

( )y1p

zp

y1-p

zp

z1p

yp

z1-p

yp

xp SSSSSSSSJ2

dtdS

++ −−+=

( )z1p

xp

z1-p

xp

x1p

zp

x1-p

zp

yp SSSSSSSSJ2

dtdS

++ −−+=

( )x1p

yp

x1-p

yp

y1p

xp

y1-p

xp

zp SSSSSSSSJ2

dtdS

++ −−+=

( )y1p

y1-p

yp SS2SJS2

+−−=

( )x1p

x1-p

xp SS2SJS2

+−−−=

0=

A trial solution set

( )[ ]( )[ ]

SS

tkpaexpvS

tkpaexpuS

zp

yp

xp

=

−=

−=

ω

ω

i

i

( ) ( )( )

( ) ( )( )ukacos1JS4u2JS2v

vkacos1JS4v2JS2u

ikaka

ikaka

−−=−−−=−

−=−−=−

−

−

eei

eei

i

i

ω

ω

( )( )

( )( )0

kacos1JS4

kacos1JS4

=−−

−

ω

ω

i

i

( )( ) 0kacos1JS4 22 =

−+−

ω

( )( ) vu and kacos1JS4 i−=−=

ω

For a set of solution

( )( )

SS

tkpasinu S

tkpacosu S

zp

yp

xp

=

−=

−=

ω

ω

At long wavelengthka<<1

( ) ( ) 222

k JSa22

kaJS4 =≅ω

0.00 0.25 0.50 0.75 1.000.0

0.5

1.0

1.5

2.0

ω/4

JS

ka/π

Dispersion relation for magnons in a ferromagnet in 1D w/. nearest-neighbor interactions

( )kacos1JS4

−=ω

k2 may be determined accurately by neutron scattering or by spin wave resonance in thin films.

( ) 222 Dkk JSa2 ==ω

Fe Co Ni

281 500 364 meVÅ2

At long wave length limit,

T= 295Kby Neutron scattering

D

Magnon spectrum obtained by neutron scattering experiment

k

'nk

nk

kn

'2n

2

n

2n

2

ω2M

k 2M

k

+=

Quantization of spin waves kkk 21n ωε

+=

In thermal equilibrium the average value of the number of magnons

( ) 1Tk/exp1n

Bkk −=

ω Planck distribution

The number of magnons from ω to ω+dω ( ) dkk421dD 2

3

ππ

ωω

=

( )

ωπ

ωπ

πω

ππ

ω

2/3

22

2222

222

3

JSa241

JSa4JSa221

k2JSa2

k2

1

ddkk4

21D

=

==

=

( )k2JSa2

dkd

k JSa22

22

=

=

ω

ω

( )∫∫∞∞

−

=

0 B

2/3

220

k 1Tk/expJSa241dn)(Dd

ωω

πωωω

Total number of magnons

Tkx

B

ω=Let ( )2

0x 40587.0

1xdx π=∫−

∞

eand

( ) 1Tk/expTk/

TkdTk

JSa241n)(Dd

B

B

0 B

2/3B

2/3

220

k −

= ∫∫

∞∞

ωωω

πωω

3

2/3B

a1

JS2Tk0587.0

=

2/3Bk

JS2Tk

SQ0587.0

NSn

)0(MM

=∑=

∆

( )240587.0 π=

The number N of atoms per unit volume:

Q/a3

SC BCC FCC

1/a3 2/a3 4/a3

Fractional change of magnetization

Bloch T3/2 law In agreement with experimental data

NiTc=627KMo=510 gauss

At T=60K ~ 0.1Tc

3

o

102 M

M −×≈∆

Bloch T3/2 law 3/2

o

ATM

M=

∆ A=(7.5±0.2)x10-6 K-3/2 Ni

A=(3.4±0.2)x10-6 K-3/2 Fe

This application of band theory to magnetic problems was made by Stoner, Mott, and Slater in 1933. (the collective-electron theory)

Why µH(0K) are 2.22, 1.72, and 0.6 µB for Fe, Co, and Ni, respectively?

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

3d 0 0 1 2 3 5 5 6 7 8 10 10

4s 1 2 2 2 2 1 2 2 2 2 1 2

3d+4s 1 2 3 4 5 6 7 8 9 10 11 12

Electron distributions in free atoms

The occupation of energy levels is in accordance with the Pauli exclusion principle.

When atoms are brought close together to form a solid, the position of energy levels are profoundly modified.

overlapping of electron clouds

Splitting of levels

Interatomic distance d

When d=do, the 3d levels are spread into a band extending from B to C, and the 4s levels are spread into a much wider band extending from A to D.

Why does the band spread wider in 4s than 3d?

A: 4s electrons are farther from the nucleus.

An important and difficult problem of the band theory is to calculate the “shape” of energy bands, i.e., DOS : N(E) for the band.

Density of states N(E)=dN/dE

The states are full of electrons, up to the Fermi energy at 0K.

The total number of spins in the subband with spins up is different from that in the subband with spins down.

Net magnetization Molecular field or exchange interaction

The nonintegral values of the magnetic moments are easily explained by the complex shapes of the bands.

dEdN

dEdN

dEdN

dEdN

Co Fe

Ni

3dFilled-10electrons4s

3d↑ 3d↓5 electrons 5 electrons

4s and 3d bands in Cu

4s and 3d bands in Ni

3d↑ 3d↓

above Tc below Tc

3d↑ 3d↓

Fermi surface

4s

0.54 electron

4.73 electron

0.27 hole

4.46 electron

0.54 hole

0.54electron

5 electron

7.1eV2.2eV

3.46eV

3d

For Ni, µ=0.6µB,

x=0.6

Hence,

atomper electrons 3d ofnumber x natomper electrons 4s ofnumber x

atomper electrons 4s 3d ofnumber n

=−=

+=

The magnetic moment per atom for this series of transition metal is predicted to increase as number of n-x of 3d electrons increase up to n-x=5 and then decrease to zero as number reach 10.

At saturation (T=0K), the net moment is the difference between spin-up states (n-x=5) and the spin-down states, n-x-5:

( )[ ] [ ] BB xn105xn5 µµµ +−=−−−=

[ ] Bn6.10 µµ −=

The agreement is generally good for Fe, Co, and Ni.

Magnetic alloys : Alloys of the transition metals or rare earth metals

↓↑ += NNZChemical valence Valence electrons

( ) BNN µµ ↓↑ −=Magnetization of the atom ( ) BZN2 µµ −= ↑

For the transition metals, N↑ is determined from the fact that the spin-up d bands lie entirely above or below the Fermi level.

ZN2Z dm −= ↑Magnetic valence

Other contribution from the 4s band (actually 4sp hybrid band) Bs 3.0N µ=↑

( ) ( ) BmBsm 6.0ZN2Z µµµ +=+= ↑The magnetic moment per atom

Ex. Fe0.8Co0.2 alloy :

( ) BB

m

4.26.08.18.12.018.02Z

µµµ =+==×+×=

The average magnetic valence concept can then be used to predict the resulting magnetic moment when transition elements are alloyed.

The Slater-Pauling curve :

The curve of magnetic moment per atom versus electron-to-atom ratio.

Assuming 0.3µB for the sp band spin-up electrons.

Magnetic valence Zm=2Nd+-Z

Band model contribution to magnetic moment per atom

Ericksson et al. considered the contributions of the 3d and 4sp hybrid bands and obtained the magnetic moment resulting from the excess holes (Nh) in the spin-up and spin-down 3d bands..

Parameter Fe Co Ni

Excess holes (Nh) 3.4 2.5 1.5

µspin3d 2.26 1.64 0.64

µspin4sp -0.07 -0.07 -0.02

µspintotal 2.19 1.57 0.62

µorbit3d 0.09 0.14 0.07

µ total 2.28 1.71 0.69

Experimental value 2.20 1.72 0.6

Criteria for the existence of ferromagnetism in a metal :

The electrons responsible must lie in partially filled bands in order that there may be vacant energy levels available for electrons with unpaired spins to move into.

The density of states in the band must be high, so that the increase in energy caused by spin alignment will be small.

The atoms must be the right distance apart so that exchange force can cause the d-electron spins in one atom to align the spins in the neighboring atom.

Ruling out inner core electrons

Ruling out valence electrons

Only Fe, Co, and Ni of transition metals are ferromagnetic.

Many of rare earths are ferromagnetic below room temperature due to spin imbalance in their 4f bands.

1839

1945

1843

1879

18851803

1907

1885

18961880187818861879

18281878

Exchange forces are responsible for magnetism, as well for antiferromagnetism, and ferrimagnetism.

Bethe-Slater curve:

postulated variation of J with the ratio ra/r3d.

ra :the radius of an atom

r3d : the radius of its 3d shell of electrons

Spins are ordered in an antiparallel arrangement with zero net moment.

A small positive susceptibility present at all temperatures but varies in a peculiar way with temperature.

Antiferromagnetism

T > TN, θχ

+∝

TC

like Curie temperature w/. Tc = -TN

ParamagneticT < TN, anti-ferromagnetic order

0M =

Below TN :Néel temperature

a weak cusp at T=TN

Anomalous paramagnetic

1904-2000

1970

Some antiferromagnetic materials

material TN(K) θ(K) χ(0)/χ(TN)

MnO 122 610 0.69

FeO 198 570 0.78CoO 293 280NiO 523 3000 0.67

Fe2O3 950 2000

Cr2O3 307 1070 0.76

FeS 613 857

FeCl2 24 -48 <0.2

FeF2 79 117 0.72

MnO2 84 116 0.93

α-Cr 310

Most are ionic compounds :oxides, sulphides, chlorides, and the like.

Manganese Oxide MnO: Ordered arrangements of spins of the Mn2+ ions.

A

A

B

B

Antiferromagnetic arrangement of A and B sublattices in 2D

Considering only the nearest neighbor interaction AB

AmB

BmA

MHMHγγ

−=−= Molecular field acts on the ion A due to magnetization of B

Molecular field acts on the ion B due to magnetization of A

Above TN,TH

Μ C==

ρχ

( )( )AB

BA

MH'TMMH'TMγργρ

−=−=

CC

Curie law ( ) ( )( )

( ) H'2M'M'H'2MT

MM2H'TMM BABA

ρργργρ

γρ

CCTCCC

=+−=

+−=+

Therefore,ργρ

χ'T'2

HΜ

CC

+==

When a field is applied above TN, each sublattice becomes magnetized in the same direction as the field, but each sublattice set up a molecular field in the opposite direction to the applied field, tending to reduce both MA and MB.

Hence, the susceptibility χ is smaller than that of an ideal paramagnetic in which the molecular field is zero.

Below TN, Each sublattice spontaneously magnetized, in zero applied field, by the molecular field created by the other sublattice.

0MMM BA =+=

BNA M'TM ργC−=At T=TN, where H=0

NNB

A TTMM' =−=ργC

The Néel temperature at which χ(T) is the maximum equals to θ.

Below TN, each sublattice is spontaneously magnetized to saturation just as a ferromagnetic is.

=

TkH J,B

B

µ

oA

A

MM Brillouin function

For the spontaneous magnetization in the absence of applied field,

AABmA MMH Mγργγ ==−=

=

Tk J,B

B

AH

oA

A MMM γρµ

oA

A

MM

oB

B

MM

−

NTT

The net spontaneous magnetization is zero below TN.However, an applied field can produce a small magnetization. χχ depends on the angle bet. the applied field and its spin axis.

A

A

B

B

Spin axisH

H

χ┴

χ║

MnF2

Field at right angle to spin axis

Assuming that the applied field turn each sublattice magnetization away from spin axis by a small angle α.

This rotation immediately creates a magnetization M in the direction of H.

The spins will rotate until thatcancel the applied field H.

MB MA

HmAHmB

M

Hm

Hα

αγρα sin2Hsin2H AmA M==

mBmA HH

+

αsin2 AMM =and Mγρ=H

Therefore,θγρ

χ2

1H

CM===⊥ a constant

Independent of temperature

MB MA

H∆MB

∆MA

Field parallel with spin axis

Assuming that the applied field H increase the zero-field value of the A-sublattice magnetization by ∆MA and decreases the corresponding value of the B-sublattice by ∆MB.

A net magnetization in the direction of H

BABA MMMMM ∆+∆=−=

M/M

o

a’=µHH/kBT

B(J,a’) )'a(J,B''a ooAA MM ∆=∆nearly linear

oA

A

MM∆

oB

B

MM∆

( )

( )AB

H

BB

H

Tk

Tk'a

MH

MH

a

a

∆−=

∆−=∆

γρµ

γρµ

( ) )'a(J,B'T2k oA

B

2H

A MHn

M ag ∆−=∆ γρµ

)'a(J,B'Tk2)'a(J,B'2

H2

H o2HB

o2H

a

A

a11 γρµ

µχ

g

g

nnMM+

=∆

==Therefore,

temperature dependence

It becomes at T=TN.

θχ

+∝

TC

It reduces to at high temperatures.θ

χ2C

=

It approaches to zero as T approaches 0.

For a powder specimen, there is no preferred orientation of the crystals.

⊥

⊥

+=

+=

χχ

θχθχχ

32

31

sincos

11

2211p

Theoretical calculation for J=1

Ferrimagnetism

Ferrimagnetic materials exhibit the phenomena of magnetic saturationand hysteresis, like ferromagnetics.

The most important ferrimagnetic materials are certain double oxides of iron and another metal, called ferrites.

The ferrites were developed into commercially useful materials. 1933Si0.03Fe0.97 (Silicon ferrite) ….

Until 1948, ferrimagnetics were separated from ferromagnetics.Néel provided the theoretical key to understand the ferrites.

They fall mainly into two groups with different crystal structures :

Cubic : MO•Fe2O3 where M is a divalent metal ion, Mn, Ni, Co, Fe,…Ironferrite FeO •Fe2O3 is the oldest magnetic material to man.

Hexagonal : BaO •6Fe2O3 Barium ferrite

Octahedral B siteTetrahedral A site

Cubic ferrite

metal ion metal ion Oxygen ion

Hexagonal ferrite

Barium ferrite

Key: Lattice with bases of two magnetic atoms

Moments anti-align, but do not cancel

Eg. Magnetite Fe3O4=FeO • Fe2O3

Fe2+ Fe3+ antiparallelparallel

0M ≠

Why is it different from ferromagnetic ? How to know this fact ?

Fe3+ ions have a spin of 5/2 and should contribute 5µB

Fe2+ ions have a spin of 2 and should contribute 4µB

expectation Effective moment (2×5+4) µB = 14 µB at T=0K

results It show only 4.1 µB only from Fe2+ ions

confirmed by Neutron diffraction

Magnetization : complex behaviors

FeO • Fe2O3

T > Tc, paramagnetic

HMM BA

ρχ += Let CA and CB be Curie constants for ions A and B.

( )( )ABB

BAA

MHTMMHTMγργρ

−=−=

CC

For non-zero solutions MA and MB

At H=0BAc CCT ργ=

( )2c

2BABA

TTCC2TCC

−−+

=ργχ

Below Tc

T(oC)

M(e

mu/

g)M

/Mo

MO • Fe2O3

Cubic ferrite

Below Tc

M3+ is a trivalent metal ion.

Y3+ is a diamagnetic ionAt T=0K, resulting in 5µBM decreases w/. increasing T and reaches to zero at T=Tc.

Rare earth M3+ :are paramagneticand magnetized (c) opposite to the resultant of the Fe3+ ions (a+d).M drops rapidly w/. increasing T Due to weak c-a and c-d coupling.Then M passes through zero andincrease again due to Fe3+.

M3 Fe5O12

The compensation temperature

at which the magnetization crosses zero.

3 Fe3+ ion on tetrahedral site d2 Fe3+ ion on octahedral site a3 M3+ ion on site c

Iron Garnets

T/Tc T/Tc

A neutron can be inelastically scattered by the magnetic structure,With creation or annihilation of a magnon.

magnon spectra

Neutron magnetic scattering

An X-ray photon see the spatial distribution of electronic charge, whether the charge density is magnetized or unmagnetized.

A neutron sees two aspects of a crystal:

the distribution of nuclei and the distribution of electronic magnetization.

Neutron

knkn

’

Magnonk

n

'2n

2

n

2n

2

M2k

M2k ω

+=

Neutrons are uncharged, easily penetrate electron cloud, and are scattered only by nucleus.

If the scattering atom or ion has a net magnetic moment, that moment will interact with neutron beam, because the neutron has a small magnetic moment, ~0.3µB.

Neutrons M=1.67x10-27kg

E(eV)0.28)Aλ( = When E=80meV , λ=1Å

µ=5.4×10-4µB

MnF2

MnF2 TN=67K

T=23K

T=300K

BCT structure

Body-centered tetragonal

MnO

T=80K

T=293K

TN=122K

The first substance to be clearly recognized as antiferromagnetic in 1938.

However, the first direct evidence came from Neutron diffraction experiment by Shull and Smart in 1949.

MnO

Magnetic Domains

Ferromagnetic materials are not uniformly magnetized, but break up into regions called domains. The magnetization for each domain will, in general, have a different orientation.

Origin of domains – magnetic, exchange, and anisotropy energies

Magnetic field energy

∫= dVH2

2oE µ

Introducing

more domains

decreasing the magnetic field energy

However,

it increases the wall energy.

For a 180o magnetization reversal in one step, an exchange energy per wall area has to be overcome: 2

2ex JSa2W =

For a 180o magnetization reversal in N steps, an exchange energy per wall area is reduced:

22

2ex NJS

aNW

=π

a: lattice constant

Domain wall orientations

Bloch Wall Néel Wall

Anisotropy Energy

The magnetic moments in a magnetic material tend to line up preferentially along certain crystallographic direction.

The exchange energy depends on the orbital overlap of electronic wavefunctions between electronic orbits on different sites.

With crystal anisotropy the rotation away from the easy axis costs extra energy

where k1 and k2 are the magnetic anisotropy constants

( ) 23

22

212

21

23

23

22

22

211K KKU ααααααααα +++=

N spins in a Bloch wall with lattice constant a

KNaUK =2

22ex N

JSaNW

=π

The energy per unit area of the wall

KNaN

JSaNUW

22

2KexW +

≈+=πσ

0KaNa

JSN

22W =+

−=

∂∂ πσMinimizing σW,

3

22

KaJSN π

= aKJS2

2

W πσ =and

For Iron, σW ≈ 1 erg/cm2 (180o wall, N≈300)

Saturation magnetization

Saturation magnetization

Coercive field

Demagnetized state

Remanent magnetization

Magnetic hysteresis

T

1/χ

T

1/χ

Diamagnetism Paramagnetism

Ms

TTc

χ

Ferromagnet Paramagnet

θχ

−=

T1~

HM

Curie-Weiss law

Curie temperature

Ferromagnetism

A

A

B

B

Anti-ferromagnetism

Ferrimagnetism

Ms

TTc

1/χ

Ferrimagnet Paramagnet

Curie temperature

Néel temperature

TN

A

A

B

B

Magnetic Devices

Permanent Magnets

Transformers

Magnetic Amplifiers

Data Storage

Spin Electronics+Spin

a multidisciplinary field including magnetism, semiconductor physics, optics, mesoscopic physics, superconductivity and new connections to other fields

The central theme is how to manipulate the spin degree of freedomwhich interact with the solid-state environment.

Spin Photonics

Magneto-Electronics

Quantum Electronics

Spin-dependent transport

Conventional transistors make use of current or voltage to control the transmitted current.

Nature 404,918 (2000)

Use the spin configurations to control current

Spintronics

Electronics

The most commonly built structures for spin-dependent transport make use of

(1) Giant magnetoresistance effect (GMR)

(2) Tunneling magnetoresistance effect (TMR)

M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988).

Tunneling magnetoresistance effect (TMR)

Tunneling conductance across the barrier is proportional to the product of density of states on both sides.

↓↓↑↑ +∝ 2,1,2,1, NNNN G

parallelR parallelantiR −<

GaMnAs/AlAs/GaMnAs tunneling junctionM. Tanaka and Y. Higo,

Phys. Rev. Lett. 87, 026602 (2001).

Datta-Das Spin field-effect transistor

How to maintain spin coherence ?

Datta and Das,Appl. Phys. Lett. 56, 665 (1990).

gate

1989 - IBM scientists made a string of key discoveries about the "giant magnetoresistive" effect in thin-film structures.

2000 - IBM and Infineon established a joint MRAM development program. 2002 - NVE Announces Technology Exchange with Cypress Semiconductor. 2003 - A 128K bit MRAM chip was introduced, manufactured

win 0.18 technology.

2004June - Infineon unveiled a 16M bit prototype based on 0.18 September - MRAM becomes a standard product in Freescale,

which has began sampling MR

MRAMMagnetic RAM chips use magnetic rather than electrical structures to store information, so they do not need to be constantly powered to retain data, like current RAM technologies.much faster and less expensive

IBM MRAM images

NEC, Toshiba claim

Magnetorestitive RAM could replace flash memory and DRAM by as early as 2010

One issue involves the size of MRAM cells, which tend to be bigger than those of other memory types.

New Technique could allow them to develop 256M bit MRAMs by early 2006