Magnetism and Magnetism and Magnetic Materials Magnetic Materials

Chemistry 754Chemistry 754Solid State Chemistry Solid State Chemistry

Lecture #23Lecture #23May 23, 2003May 23, 2003

Magnetic Moment of an Magnetic Moment of an ElectronElectron

Magnetism in solids originates in the magnetic properties Magnetism in solids originates in the magnetic properties of an electron.of an electron.

SS = g[S(S+1)] = g[S(S+1)]1/21/2 [(eh/(4 [(eh/(4mmee)])]

BB = (eh/(4 = (eh/(4mmee))

SS = g[S(S+1)] = g[S(S+1)]1/21/2 BB

– S = ½, the spin quantum numberS = ½, the spin quantum number– g ~ 2, the gyromagnetic ratiog ~ 2, the gyromagnetic ratio

– BB = 9.2742 = 9.2742 10 10-24-24 J/T, the Bohr magneton J/T, the Bohr magneton

So that for a free electronSo that for a free electron

SS = 1.73 = 1.73 BB

Magnetic Moments of Atoms & Magnetic Moments of Atoms & Ions Ions

Almost all atoms have multiple electrons, but most of the electrons are Almost all atoms have multiple electrons, but most of the electrons are paired up in orbitals with another electron of the opposite spin. When paired up in orbitals with another electron of the opposite spin. When all of the electrons on an atom are paired the atom is said to be all of the electrons on an atom are paired the atom is said to be diamagneticdiamagnetic. Atoms/ions with unpaired electrons are . Atoms/ions with unpaired electrons are paramagneticparamagnetic..

Diamagnetic IonsDiamagnetic Ions = There is a very small magnetic moment = There is a very small magnetic moment associated with an electron traveling in a closed path around the associated with an electron traveling in a closed path around the nucleus.nucleus.

Paramagnetic IonsParamagnetic Ions = The moment of an atom with unpaired = The moment of an atom with unpaired electrons is given by the spin, S, and orbital angular, L and total electrons is given by the spin, S, and orbital angular, L and total momentum, J, quantum numbers.momentum, J, quantum numbers.

effeff = g = gJJ[J(J+1)][J(J+1)]1/21/2 BB Full treatment:Full treatment:Accurate for LanthanidesAccurate for Lanthanides

effeff = [4S(S+1)+L(L+1)] = [4S(S+1)+L(L+1)]1/21/2 BB Neglecting spin-orbit couplingNeglecting spin-orbit coupling

effeff = 2[S(S+1)] = 2[S(S+1)]1/21/2 BB Spin only valueSpin only value

Atomic Moments in Atomic Moments in CompoundsCompounds

Unpaired electrons and paramagnetism are usually associated with the Unpaired electrons and paramagnetism are usually associated with the presence of either transition metal or lanthanide (actinide) ions. In presence of either transition metal or lanthanide (actinide) ions. In many transition metal compounds the surrounding anions/ligands many transition metal compounds the surrounding anions/ligands

quench the orbital angular momentum and one needs only to take into quench the orbital angular momentum and one needs only to take into account the spin only moment. Consider the following examples:account the spin only moment. Consider the following examples:

IonIon ee-- Config. Config. S S SS((BB) ) S+LS+L((BB)) obs obs ((BB))TiTi4+4+ d d11 1/21/2 1.73 3.01 1.7-1.8 1.73 3.01 1.7-1.8VV2+2+ d d22 1 1 2.83 4.49 2.8-3.1 2.83 4.49 2.8-3.1CrCr3+3+ d d33 3/23/2 3.87 5.21 3.7-3.9 3.87 5.21 3.7-3.9FeFe3+3+ d d55 (HS) (HS) 5/25/2 5.92 5.92 5.7-6.0 5.92 5.92 5.7-6.0NiNi2+2+ d d88 (HS) (HS) 1 1 2.83 4.49 2.9-3.9 2.83 4.49 2.9-3.9CuCu2+2+ d d99 1/21/2 1.73 3.01 1.9-2.1 1.73 3.01 1.9-2.1

Deviations from the spin-only value can occur for the following reasons:Deviations from the spin-only value can occur for the following reasons:Orbital (L) ContributionOrbital (L) Contribution

– Can arise for partially filled (not ½ full) tCan arise for partially filled (not ½ full) t2g2g orbitals orbitalsSpin-orbit CouplingSpin-orbit Coupling

– Increases the moment for dIncreases the moment for d66, d, d77, d, d88, d, d99

– Decreases the moment for dDecreases the moment for d11, d, d22, d, d33, d, d44

Magnetic OrderingMagnetic Ordering

ParamagnetiParamagneticc

AntiferromagnAntiferromagneticetic

FerromagnetFerromagneticic

FerrimagnetiFerrimagneticc

Interaction between an Applied Interaction between an Applied Magnetic Field and a Magnetic Magnetic Field and a Magnetic

MaterialMaterialThe interaction between an external magnetic field (H) and a The interaction between an external magnetic field (H) and a

material depends upon it’s magnetic properties.material depends upon it’s magnetic properties.

Diamagnetic Diamagnetic Repulsive Repulsive Paramagnetic Paramagnetic Attractive Attractive

Magnetic Flux LinesMagnetic Flux Lines

B = H + 4B = H + 4II

BB = Magnetic Induction (field strength within the sample)= Magnetic Induction (field strength within the sample)

HH = Applied magnetic field (field coming from external source) = Applied magnetic field (field coming from external source)

II = Magnetization Intensity (field originating from the sample) = Magnetization Intensity (field originating from the sample)

Magnetic SusceptibilityMagnetic SusceptibilityB = H + 4B = H + 4II

The permeability, P, is obtained by dividing the magnetic induction, B The permeability, P, is obtained by dividing the magnetic induction, B (total field in the sample), by the applied field, H.(total field in the sample), by the applied field, H.

P = B/H = 1 + 4P = B/H = 1 + 4I/H = 1 + 4I/H = 1 + 4

Where Where is the is the volume susceptibilityvolume susceptibility (extrinsic property). To obtain the (extrinsic property). To obtain the intrinsic material property, intrinsic material property, mm, molar susceptibility we multiply by the , molar susceptibility we multiply by the

Formula weight, FW, and divide by the density, Formula weight, FW, and divide by the density, ..

mm = = (FW)/(FW)/

Typical molar susceptibilities areTypical molar susceptibilities are

– Paramagnetic Comp. ~ +0.01 Paramagnetic Comp. ~ +0.01 BB

– Diamagnetic Comp. ~ -1Diamagnetic Comp. ~ -11010-6 -6 BB

– Ferromagnetic Comp. ~ +0.01-10 Ferromagnetic Comp. ~ +0.01-10 BB

– Superconducting Comp. ~ Strongly negative, repels Superconducting Comp. ~ Strongly negative, repels fields completely in some instances (Meisner effect)fields completely in some instances (Meisner effect)

SuperexchangeSuperexchangeIn order for a material to be magnetically ordered, the spins In order for a material to be magnetically ordered, the spins on one atom must couple with the spins on neighboring on one atom must couple with the spins on neighboring atoms. The most common mechanism for this coupling atoms. The most common mechanism for this coupling (particularly in insulators) is through the semicovalent (particularly in insulators) is through the semicovalent superexchange interaction. The spin information is superexchange interaction. The spin information is transferred through covalent interactions with the transferred through covalent interactions with the intervening ligand (say oxygen).intervening ligand (say oxygen).

FeFe3+3+ d dx2-y2x2-y2

½ Filled½ Filled

FeFe3+3+ d dx2-y2x2-y2

½ Filled½ Filled

O 2p O 2p

The covalent interaction through the O 2p orbital stabilizes

antiferromagnetic coupling.

FeFe3+3+ d dx2-y2x2-y2

½ Filled½ Filled

CrCr3+3+ d dx2-y2x2-y2

EmptyEmpty

O 2p O 2p

Here the oxygen based electron will spend some time on Cr3+ and due to Hund’s rule polarize the t2g

e- leading to ferromagnetic coupling.

Magnetic Susceptibility vs. Magnetic Susceptibility vs. TemperatureTemperature

Paramagnetic substances with localized, weakly interacting Paramagnetic substances with localized, weakly interacting electrons obey the Curie-Weiss law.electrons obey the Curie-Weiss law.

mm = C/(T+ = C/(T+))

• mm = Molar magnetic susceptibility= Molar magnetic susceptibility

• C C = Curie constant= Curie constant• = Weiss constant= Weiss constant

A Curie-Weiss plot is a plot of 1/A Curie-Weiss plot is a plot of 1/mm vs. temperature. Ideally it vs. temperature. Ideally it should give a straight line if the C-W law is obeyed. From such should give a straight line if the C-W law is obeyed. From such a plot we can then extract the Curie constant from the inverse a plot we can then extract the Curie constant from the inverse

of the slope and the Weiss constant from the x-intercept.of the slope and the Weiss constant from the x-intercept.

mm = (T+ = (T+)/C = (1/C)T + )/C = (1/C)T + /C /C

Curie-Weiss PlotCurie-Weiss Plot

The Curie constant is equal to the The Curie constant is equal to the inverse of the slope. It gives us the inverse of the slope. It gives us the

size of the moment per formula unit.size of the moment per formula unit.

C = (NC = (NAA/3k)/3k)22

= (3kC/N= (3kC/NAA))1/21/2 = 2.84 C = 2.84 C1/21/2

NNAA = Avogadro’s Number = Avogadro’s Number

k = Boltzman’s constantk = Boltzman’s constant

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

0 50 100 150 200 250 300 350

Temperature (K)

Inve

rse

Mo

lar

Su

scep

tib

ility

, (1/

m

) Fe3+ (d5) = 5.9 mB

= 0

= 50K

= - 50K

The Weiss constant is equal to the The Weiss constant is equal to the x-intercept. It’s sign tells us about x-intercept. It’s sign tells us about

the short range magnetic the short range magnetic interactions.interactions.

= 0 = 0 Paramagnetic ParamagneticSpins independent of each otherSpins independent of each other

> 0 > 0 Ferromagnetic FerromagneticSpins tending to align parallelSpins tending to align parallel

< 0 < 0 Antiferromagnetic AntiferromagneticSpins tending to align antiparallelSpins tending to align antiparallel

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 50 100 150 200 250 300 350

Temperature (K)

Mo

lar

Su

scep

tib

ility

, m

Fe3+ (d5) = 5.9 mB

= 0

= 50K

= - 50K

Ferromagnets & AntiferromagnetsFerromagnets & Antiferromagnets

FerromagnetFerromagnetThe susceptibility increases The susceptibility increases

dramatically at the Curie temp. dramatically at the Curie temp. As the T decreases further the As the T decreases further the

magnetic ordering and the magnetic ordering and the susceptibility increase. susceptibility increase.

AntiferromagnetAntiferromagnetThe susceptibility begins The susceptibility begins

decreasing at the Neel temp. decreasing at the Neel temp. As the T decreases further the As the T decreases further the magnetic ordering increases magnetic ordering increases

and the susceptibility and the susceptibility decreases. decreases.

The Curie-Weiss Law characteristic of a pure paramagnet is typically The Curie-Weiss Law characteristic of a pure paramagnet is typically only obeyed when T only obeyed when T 3T 3TCC in a ferromagnet. Deviations also occur in in a ferromagnet. Deviations also occur in

AFM systems above TAFM systems above TNN..

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 100 200 300 400

Temperature (K)

Mo

lar

Su

scep

tib

ility

, m

Ferromagnet

TC =100 K(T=0) = 3 B

TC

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 100 200 300 400

Temperature (K)

Mo

lar

Su

scep

tib

ility

, m

Antiferromagnet

TN =100 K

TN

Other Classes of MagnetismOther Classes of MagnetismSpin GlassSpin Glass – A random orientation of frozen spin orientations – A random orientation of frozen spin orientations

(in a paramagnet the spin orientations are fluctuating.) Can (in a paramagnet the spin orientations are fluctuating.) Can occur when the concentrations of magnetic ions are dilute or occur when the concentrations of magnetic ions are dilute or the magnetic exchange interactions are frustrated.the magnetic exchange interactions are frustrated.

Cluster GlassCluster Glass – The spin orientations lock in with magnetic – The spin orientations lock in with magnetic order in small clusters, but no order between the clusters order in small clusters, but no order between the clusters (similar to a spin glass).(similar to a spin glass).

MetamagnetMetamagnet – There is a field-induced magnetic transition – There is a field-induced magnetic transition from a state of low magnetization to one of relatively high from a state of low magnetization to one of relatively high magnetization. Typically the external field causes a magnetization. Typically the external field causes a transition from an antiferromagnetic state to a different type transition from an antiferromagnetic state to a different type (such as a ferromagnet).(such as a ferromagnet).

Superparamagnet Superparamagnet – A ferromagnet whose particle size is too – A ferromagnet whose particle size is too small to sustain the multidomain structure. Thus the particle small to sustain the multidomain structure. Thus the particle behaves as one large paramagnetic ion. behaves as one large paramagnetic ion.

Magnetic Ordering in Rock Salt OxidesMagnetic Ordering in Rock Salt OxidesIn the rock salt structure the primary mechanism for magnetic exchange In the rock salt structure the primary mechanism for magnetic exchange is the linear M-O-M superexchange interaction. In all of the compounds is the linear M-O-M superexchange interaction. In all of the compounds

below the ebelow the egg orbitals are ½ filled, so the exchange interaction is AFM and orbitals are ½ filled, so the exchange interaction is AFM and overall magnetic structure is AFM as shown below. overall magnetic structure is AFM as shown below.

M-O M-O DistanDistan

cece

TTNN (K)(K)

MomeMoment (nt (BB))

MnOMnO dd55 2.22 Å2.22 Å 121200

~5~5

FeOFeO dd66 2.15 Å2.15 Å 191988

3.33.3

CoOCoOdd77

2.13 Å2.13 Å 292911

3.53.5

NiONiO dd88 2.09 Å2.09 Å 535300

1.81.8(M-O distance (M-O distance ) ) (Covalency (Covalency ) ) (Superexchange (Superexchange ) ) (T (TNN

))

The moments given in this table (and the ones that follow) were The moments given in this table (and the ones that follow) were measured at low temperature using neutron diffraction. Under such measured at low temperature using neutron diffraction. Under such circumstances the moment should roughly be equal to the number of circumstances the moment should roughly be equal to the number of

unpaired electrons.unpaired electrons.

Spin up Spin up NiNiSpin Spin

dowdown n NiNi

OxygenOxygen AFMAFMSESE

AFM Magnetic Ordering in PerovskitesAFM Magnetic Ordering in PerovskitesThe perovskite structure has even simpler magnetic interactions (in rock salt there is The perovskite structure has even simpler magnetic interactions (in rock salt there is

a competing interaction across the shared octahedral edge). Some magnetic data and a competing interaction across the shared octahedral edge). Some magnetic data and the most common AFM structure for perovskite is shown below. The coloring of the the most common AFM structure for perovskite is shown below. The coloring of the

magnetic structure is analogous to the crystal structure of NaCl. It is called the G-type magnetic structure is analogous to the crystal structure of NaCl. It is called the G-type magnetic structure. magnetic structure.

M-X M-X DistancDistanc

ee

TTNN (K)(K)

MomenMoment (t (BB))

LaCrOLaCrO33 dd33 1.97 Å1.97 Å 282282 2.82.8

CaMnOCaMnO

33dd33 1.90 Å1.90 Å 110110 2.62.6

LaFeOLaFeO33 dd55 1.99 Å1.99 Å 750750 4.64.6

KNiFKNiF33 dd88 2.00 Å2.00 Å 275275 2.22.2

Note that the superexchange that involves ½ filled eNote that the superexchange that involves ½ filled egg orbitals (LaFeO orbitals (LaFeO33) is ) is much stronger than the corresponding interaction of ½ filled tmuch stronger than the corresponding interaction of ½ filled t2g2g orbitals. orbitals.

Also upon going from oxide (LaFeOAlso upon going from oxide (LaFeO33) to fluoride (KNiF) to fluoride (KNiF33) the covalency ) the covalency decreases, which weakens the superexchange and lowers Tdecreases, which weakens the superexchange and lowers TNN. .

Spin up Spin up FeFe

Spin Spin down down

FeFe

OxygenOxygenAFMAFMSESE

Magnetic Ordering in LaMnOMagnetic Ordering in LaMnO33

An interesting example that shows where the An interesting example that shows where the superexchange rules do not always lead to an superexchange rules do not always lead to an

AFM structure, with ferromagnetic nearest AFM structure, with ferromagnetic nearest neighbor interactions is the magnetic neighbor interactions is the magnetic

structure of LaMnOstructure of LaMnO33. Which contains HS Mn. Which contains HS Mn3+3+, , a da d44 ion with one electron in the e ion with one electron in the egg orbitals. orbitals.

Mn

La

O

½ filled dz2 type orbitals

This structure is This structure is called the A-type called the A-type AFM structure.AFM structure. 2.18 A

1.90 AOverlap of ½ filled and Overlap of ½ filled and empty eempty egg orbitals gives orbitals gives

FM coupling and FM coupling and stabilizes FM layers.stabilizes FM layers.

Double Exchange in Double Exchange in FeFe33OO44

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

eeg g

tt2g 2g

Oct. Sites FerromagneticOct. Sites FerromagneticMetallic Double ExchangeMetallic Double Exchange

Oct. Sites Oct. Sites AntiferromagneticAntiferromagnetic

Localized Localized electrons/insulatingelectrons/insulating

FeFe2+2+ tt2g2g

4 4 eegg22

FeFe3+3+ tt2g2g

3 3 eegg22

FeFe33OO44 is an inverse spinel, with Fe is an inverse spinel, with Fe3+3+ on on the tetrahedral sites and a 1:1 mixture the tetrahedral sites and a 1:1 mixture of Feof Fe2+2+/Fe/Fe3+3+ on the octahedral sites. It on the octahedral sites. It is ferrimagnetic, with the octahedral is ferrimagnetic, with the octahedral

sites and the tetrahedral sites aligned sites and the tetrahedral sites aligned in different directions. The in different directions. The

ferromagnetic alignment of the ferromagnetic alignment of the octahedral sites is necessary for octahedral sites is necessary for

delocalized carrier transport of the delocalized carrier transport of the minority spin tminority spin t2g2g electron. This electron. This

mechanism is called double exchange.mechanism is called double exchange.

SummarySummaryMagnetic Ordering in SolidsMagnetic Ordering in Solids

– DiamagnetismDiamagnetism: No unpaired e: No unpaired e--

– ParamagnetismParamagnetism: Unpaired e: Unpaired e--, disordered and fluctuating, disordered and fluctuating– FerromagnetismFerromagnetism: All unpaired e: All unpaired e-- spins aligned parallel spins aligned parallel– AntiferromagnetismAntiferromagnetism: Unpaired e: Unpaired e-- aligned antiparallel aligned antiparallel– FerrimagnetismFerrimagnetism: Unpaired e: Unpaired e-- aligned antiparallel but don’t fully aligned antiparallel but don’t fully

cancel outcancel outMagnetic SuperexchangeMagnetic Superexchange

– Unpaired electron spins couple through covalent interactions Unpaired electron spins couple through covalent interactions with intervening ligandwith intervening ligand

– ½ filled metal orbital – ½ filled metal orbital, AFM SE½ filled metal orbital – ½ filled metal orbital, AFM SE– ½ filled metal orbital – empty metal orbital, FM SE½ filled metal orbital – empty metal orbital, FM SE– Strength of superexchange interaction increases as covalency Strength of superexchange interaction increases as covalency

increasesincreasesDouble ExchangeDouble Exchange

– Spins on neighboring atoms must be aligned in a certain manner Spins on neighboring atoms must be aligned in a certain manner (usually ferromagnetically) in order to allow carrier delocalization(usually ferromagnetically) in order to allow carrier delocalization

– Magnetism and electrical transport are intimately linkedMagnetism and electrical transport are intimately linked