1

1

Diffusion

• Diffusion occurs from a concentration gradient• The difference between diffusion in metals and

in ceramics is that the diffusing species inceramics is often charged (vacancies,interstitials)

• The movement of a charged ion results in acurrent

• Thus diffusion and ionic conductivity arelinked

2

• Frenkel Defect --a cation is out of place.• Shottky Defect --a paired set of cation and anion vacancies.

• Equilibrium concentration of defects kT/QDe~!

Defects in ceramic structures

Shottky Defect:

Frenkel Defect

2

3

Schottky and Frenkel defectsSchottky defects in NaCl Both cation and anion are

missing from their regularlattice sites

At room temperature, 1 in1015 sites are vacant

200 kJ/mole (2.7 eV) creationenergy

Cation Frenkel defects inAgCl

Cation displaced fromregular lattice site ontointerstitial site

150 kJ/mole (1.6 eV)creation energy

4

• Impurities must also satisfy charge balance = Electroneutrality

• Ex: NaCl

• Substitutional cation impurity

Impurities

Na+ Cl-

initial geometry Ca2+ impurity resulting geometry

Ca2+

Na+

Na+Ca2+

cation vacancy

• Substitutional anion impurity

initial geometry O2- impurity

O2-

Cl-

anion vacancy

Cl-

resulting geometry

3

5

Crystalline point defects• If cationic impurities are introduced into a solid and the

dopant does not have the same valence as the cationit is replacing, extrinsic defects will be introduced– Ca2+, Y3+ in ZrO2 have anion vacancies– Ca2+ or Cd2+ in NaCl creates cation vacancies

• Real crystals contain both intrinsic and extrinsicdefects

• The dominate defect type depends on temperatureand doping level– Typically

• High temperatures– intrinsic

• Low temperatures– extrinsic defects

6

Kröger-Vink notation• Standard notation for defects in ionic crystals• Composed of 3 parts• Main body identifies the defect

– V = vacancy– M = metal– X = non-metal

• Subscript denotes site the defect occupies– i = interstitial– x = non-metal– M= cation site

• Superscript identifies the effective charge– • = positive charge– ' = negative charge

4

7

ExamplesConsider MgO

V

Mg

'' A vacancy on the Mg siteIt has a double negative charge since Mg is 2+

V

O

•• A vacancy on the O siteIt has a double positive charge since O is 2-

Al

i

••• An Al interstitialIt has a triple positive charge since it is 3+

Al

Mg

• An Al on a Mg siteIt has a positive charge since Al is 3+ and Mg is 2+

Li

Mg

' An L on a Mg siteIt has a negative charge since Li is 1+ and Mg is 2+

8

Defection associations andconcentrations

[V

Mg

'' ]

[Al

Mg

• ]

Concentrations given in brackets:

[e '] = n

[ !h] = p= concentration of electrons= concentration of holes

5

9

Defect reactionsReactions occur for defects, just like other chemicalspecies in the lattice

Consider Shottky defects in MX (M=cation, X=anion)

There is a random distribution of cation and anion vacancies

V

M

'+V

X

•! null

K

S= equilibrium constant = [V

M

' ][VX

• ]

From the definition of the equilibrium constant:

Ks= exp

!"gs

kT

#

$%&

'(= exp

!"hs+T"s

s

kT

#

$%&

'(= exp

"ss

k

#

$%&

'(exp

!"hs

kT

#

$%&

'() exp

!"hs

kT

#

$%&

'(

1

10

Defect reactions

K

S= equilibrium constant = [V

M

' ][VX

• ]

= exp!"g

s

kT

#

$%&

'(

When these are the only defects present, then

[VM

' ] = [VX

• ] = exp!"g

s

2kT

#

$%&

'(

Frenkel defects

M

M!V

M

'+M

i

•

K

F= [V

M

' ][Mi

• ]

Electronic defects

e '+ !h ! null K

e= [e '][ !h] = np

6

11

Rules for defect reactions

• These rules must be satisfied:– Mass conservation

• Not creating or destroying matter!– Electroneutrality

• The + and - charges must be balanced on each side ofreaction equation

– Site ratio conservation• Different crystal structures are not created

aiA

i! b

iB

i

k =

"i

"i

[Bi]b

i

[Ai]a

i

= exp#$G

kT

%&'

()*

12

Oxidization and reduction

M O M O M OO M O M O M

M O M O M O

O M O M O M

1/2O2 (g)2e

M (g)

2h Oxidation - generate holesReduction - generate electrons

V

M

' V

M

''

V

O

••

V

O

•

e

e

CB

VB

7

13

Oxidation and reduction

O

O!V

O

••+ 2e '+

1

2O

2(g)

KR= [V

O

•• ]n2pO2

1/2= K

R

o exp!"g

R

kT

#

$%&

'(

1

2O

2(g) +V

O

••!O

O+ 2 !h

KO=

p2

[VO

•• ]pO2

1/2= K

O

o exp!"g

O

kT

#

$%&

'(

14

Examples

1. Sodium tungstate bronzeNaxWO3 x: 0.32-0.93 perovskite with

V

Na

'

n-type for x < 0.25Metallic conductivity for x > 0.25

2. Ce3S4Ce2.67S4 ρ ~ 10-3 Ω-cmCe3S4 ρ ~ 109 Ω-cm

3. BaTiO3 heated and quenched in H2BaTiO3-x good semiconductorTi4+→Ti3+ + h•

4. ZnO sintering rate increases as pO2 decreasesformation of Zni

8

15

Impurity induced, ion compensated

There is no such thing as a 'pure' materialCan get 99.9999% pure (4 9's, Alfa Aesar, e.g.)

Concentration of impurities is 100 ppm or 10-4

Consider adding CdCl2 to NaClAssume Cd sits on Na site (not interstitial, too large)

Cd is 2+, for charge balance must form Na vacancies orCl interstitials (unlikely)

CdCl

2

2NaCl! "!! Cd

Na

•+V

Na

'+ 2Cl

Cl Na1-2xCdxCl

CaO

ZrO2! "!! Ca

Zr

''+V

O

••+O

OZr1-xCaxO2(1-x)

16

Frenkel defects

M X M X M

X M X M X

M X M X M

X M X M X

M

AgBr, CaF2

N = number of normal sitesN* = number of interstitial sites

nF= (NN

* )1/2 exp!Q

F

2kT

"

#$%

&'= number of Frenkel painr

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17

Assumptions

• Only have one type of predominate defect– Schottky or Frenkel

• Assume a dilute solution– Neglect interactions between defects

• Constant volume• Energy for defect formation independent

of T

18

Diffusion in lightly doped NaClConsider adding CdCl2 to NaCl

CdCl

2

2NaCl! "!! Cd

Na

•+V

Na

'+ 2Cl

Cl

The Na diffusion coefficient is

DNa

= [VNa

' ]!"2 exp#$G

VNa

*

kT

%

&''

(

)**

ΔGVNa* is the energy for migration of free vacancies

DNa

= [CdCl2]!"2 exp

#SNa

*

k

$

%&

'

() exp

*#HNa

*

kT

$

%&

'

()

At low temperatures, extrinsic behavior observed

10

19

At high temperatures, there are additional vacancies fromSchottky defects that swamp the effect of the impurity

DNa

= [VNa

' ]!"2 exp#$G

Na

*

kT

%

&'

(

)* = !"2 exp

#$sS

2k

%

&'(

)*exp

#$SNa

*

k

%

&'

(

)* exp

#$HS

2kT

%

&'(

)*exp

#$HNa

*

kT

%

&'

(

)*

-1/k(ΔHNa* + 1/2Δhs)

-1/k(ΔHNa*)

1/T

ln D

(cm

2 /se

c)

low Textrinsic

high Tintrinsic

20

Diffusion in cation-deficient oxides

The transition metal oxides are typically cation deficient

Ni1-xO, Co1-xO, Mn1-xO, Fe1-xOx↑ 3 x 10-4 10-2 at 1300˚C

Consider Co1-xO

1

2O

2(g) = O

O+V

Co K

1= [V

Co]a

O2

!1/2

VCo

=VCo

'+ !h K

2=

[VCo

' ]p

[VCo

]

VCo

'=V

Co

''+ !h K

3=

[VCo

'' ]p

[VCo

' ]

x = [V

Co]+ [V

Co

' ]+ [VCo

'' ]

Can get up to x = 0.15, thenform F2O3

electrical conductivity is p-type

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21

Diffusion in highly doped oxide - cubicstabilized ZrO2

CaO

ZrO2! "!! Ca

Zr

''+V

O

••+O

O

Usually 8-15% added

[Ca

Zr

'' ] = [VO

•• ]

Brouwer approximation:

Large concentrationof oxygen vacanciescompared with mostoxides

ΔG* = 1 eV (small)

Ca1-xZrxO2(1-x) fast ionconductor

single cubicphase

cubi

c +

tetra

gona

ldefectclustering

22

Electrical conductivity• Conductivity values range over 25 orders of magnitude

– Most insulating LiF (band gap > 12 eV)– Superconductors (no band gap)

• Electrical conductivity arises from– Movement of charged ions

• Ionic conductivity– Sensors, electrochemical pumps, solid electrolytes in fuel cells, high T battery systems

– Movement of electrons• Measured electrical conductivity

– From both ions and electrons– σtotal = σelec + σion– ti = transference number = σi/σtotal

• If telec > tion electronic conductor• If tion > telec ionic conductor

• Oxides that are easily reduced are n-type semiconductors– e.g. TiO2, SnO2, ZnO, BaTiO3

• Oxides that are easily oxidized are p-type semiconductors– e.g. transition metal monoxides (NiO, FeO, CoO)

12

2310-18

10-12

10-6

100

106 RuO2 (thick films)

SrTiO3 (photoelectrode)

TiO

V2O3•P2O3 (glass)

TiO2-x (oxygen sensor)

Al2O3 (substrate)

ZnO (varistor)

TiO2

SnO2•In2O3 (transparent elect.)LaNiO3 (fuel cell electrode)

fast ionconductor

solidelectrolyte

insulator

metallic

semiconducting

insulating

Na β-Al2O3 Na/S battery

ZrO2-Y2O3 (1000˚C) Oxygen sensor Li2O-LiCl-B2O3 (glass, 300˚C) KxPb1-xF1.75Primary battery

LaF3, EuF2fluorine ionspecific electrode

NaCl

SiO2

passivation onSi devices

IONIC CONDUCTORS ELECTRON CONDUCTORSYBa2Cu3O7-x

!

(Ω-cm)-1

24

Mobility

Mobility = velocity

driving force chemical, electric field, mechanical

In a chemical gradient, the absolute mobility given by

(ergs/mole) !µ

i

The chemical mobility = Bi' = Bi/NA

note: this is the chemical potential, notthe electrical mobility

Bi=

velocity (cm / sec)

force (ergs / cm)=

vi

Fi

=v

i

!1

NA

" !µi

"x

#

$%

&

'(

)

*++

,

-..

13

25

Mobility and diffusivity

Ji = civi = ciBiFi

Ji= !

1

NA

" !µi

"x

#

$%

&

'( B

ic

i

For an ideal solution,

!µi= µ

o+ RT lna

i= µ

o+ RT ln!

ic

i" µ

o+ RT lnc

i

d !µi

dx= RT

1

ci

#

$%&

'(dc

i

dx

Ji= )

1

NA

RT

ci

dci

dx

#

$%&

'(B

ic

i= )

RT

NA

Bi

dci

dx= )D

i

dci

dx

Di = kTBiNernst-Einstein relation

26

Fi(electrical) = z

ie

d!dx

= zieE

Ji= c

iB

iF

i= c

i

Di

kT

"

#$%

&'(z

ieE) =

ziec

iD

iE

kT

Ji= c

iv

i=

ziec

iD

iE

kT

vi=

zieD

i

kTE

µi=

vi

E=

zieD

i

kT= z

ieB

i=

ziFB

i

NA

= ziFB

i

' F = Faraday's constant =96,500C/mole = eNA

Instead of using a chemical potential (hard to measure), putthese expressions in terms of an electric field

relating electronic mobility with chemical mobility

14

27

Ionic conductivity

!i=

zi

2e

2D

ic

i

kT

µi=

zieD

i

kT !

i= z

ieµ

ic

i

Usually written in (Ω-cm)-1 or S-m-1

where S = Ω-1

e = 1.6 x 10-19 CD in cm2/secc in #/cm3

k in ergs (107ergs = J)

Conductivity depends oncarrier concentrationmobility of carriertemperature

At room temperaturenot many defectsmobility low

28

Diffusion and electrical conductivitymeasurements• Diffusion of a radioactive tracer element Na

was measured• The electrical conductivity was measured

D

1/T

tracer

conductivity

Difference is ~ 2 x 1011

cm2/sec

15

29

The electrochemical potential

• Gradients in chemical potential (concentration)and electric field mobilize defects

• Even in the absence of an external field,internal electric fields are present– Non uniform distribution of space charge

• Driving force for mass transport is theelectrochemical potential (η) instead of just thechemical potential

!

i= µ

i+ z

i"F

F = Faraday's constant = eNA = 96,500 C/mole

30

The force on the particle, Fi, is the negative gradient of ηi

Fi= !

1

NA

d"i

dx

#

$%&

'(

Ji=!c

iB

i

NA

d"i

dx

#

$%&

'(=!c

iB

i

NA

d !µi

dx+ z

iF

d)dx

#

$%

&

'(

Even a modest electrical field can offset the effectof the concentration gradient in the oppositedirection

16

31

Ambipolar diffusion

• Coupled transport of different charged species• Ionic crystals must maintain charge neutrality

– Long range charge separation must be avoided– Charge species are coupled

• Effect of slowing down faster diffusing species andspeeding up slower diffusing species

• Both diffuse with a common diffusivity– Chemical or ambipolar diffusion coefficient

!D

32

Consider MgO

VMg

''+V

O

••! null K

S= [V

Mg

'' ][VO

•• ]

e '+ h•! null K

i= np

OO!V

O

••+ 2e '+

1

2O

2(g) K

O=V

O

••n2pO

2

1/2

The flux of oxygen vacancies must be matched by anequivalent charge flux of electrons outward, holes inward

µe > µh

2JVO = Je

Using the ambipolar diffusion coefficient:

JV

O

••= ! !D

dcV

O

dx

"

#$$

%

&''

and Je '= ! !D

dn

dx

"#$

%&'

17

33

How does D depend on DVO and De?~

Rewrite Fick's first law in terms of ηi acting on the 2 defectsseparately, equating the fluxes to solve for the internal field.

2JV

O

•• = !2c

VO

DV

O

RT

" !µV

O

"x+ 2F

"#"x

$

%&&

'

())

flux is raised by internal field

Je '

= !nD

e

RT

" !µe

"x! F

"#"x

$

%&'

() flux is lowered by internal field

Then, rewriting the 2 expressions to get ∂φ/∂x

!"!x

=RT

F

De# D

VO

( )!c

VO

!x

De+ 2D

VO

JV

O

••= #

3DeD

VO

De+ 2D

VO

$

%&&

'

())

!cV

O

!x

!D =

3DeD

VO

De+ 2D

VO

the concentration gradients, !n

!x= 2

!cV

O

!x

n = 2cV

O

"

#

$$$

%

&

'''

!µ = !µ

o+ RT lnc( )

34

If De >> DVO, the D = 3DVO

Ambipolar diffusion rate is controlled by the slower speciesAmbipolar coupling causes rate to be enhanced by 3X

If DVO>>De, then D = 1.5 D

Slower species is rate controllingAmbipolar coupling increases effective diffusion coefficient

The ambipolar diffusion coefficient is greater than thatof the slower defect, due to charge-coupling to thefaster one

~

~