defect chemistry – a general introduction

truls.norby@kjemi.uio.no http://folk.uio.no/trulsn

Department of ChemistryUniversity of Oslo

Centre for Materials Science and Nanotechnology (SMN)

FERMIOOslo Research Park (Forskningsparken)

Defect chemistry – a general introduction

Truls Norby

Brief history of structure, stoichiometry, and defects

• Early chemistry had no concept of stoichiometry or structure.

• The finding that compounds generally contained elements in ratios of small integer numbers was a great breakthrough!

• Understanding that external geometry often reflected atomic structure.

• Perfectness ruled. Non-stoichiometry was out.

• Intermetallic compounds forced re-acceptance of non-stoichiometry.

• But real understanding of defect chemistry of compounds is less than 100 years old.

Perfect structure

• Our course in defects takes the perfect structure as starting point.

• This can be seen as the ideally defect-free interior of a single crystal or large crystallite grain at 0 K.

Close-packing

• Metallic or ionic compounds can often be regarded as a close-packing of spheres

• In ionic compounds, this is most often a close-packing of anions (and sometimes large cations) with the smaller cations in interstices

Some simple classes of oxide structures with close-packed oxide ion sublattices

Formula Cation:anion coordination

Type and number of occupied interstices

fcc of anions hcp of anions

MO 6:6 1/1 of octahedral sites

NaCl, MgO, CaO, CoO, NiO, FeO a.o.

FeS, NiS

MO 4:4 1/2 of tetrahedral sites

Zinc blende: ZnS Wurtzite: ZnS, BeO, ZnO

M2O 8:4 1/1 of tetrahedral sites occupied

Anti-fluorite: Li2O, Na2O a.o.

M2O3, ABO3 6:4 2/3 of octahedral sites

Corundum:Al2O3, Fe2O3,Cr2O3 a.o.Ilmenite: FeTiO3

MO2 6:3 ½ of octahedral sites

Rutile: TiO2, SnO2

AB2O4 1/8 of tetrahedral and 1/2 of octahedral sites

Spinel: MgAl2O4Inverse spinel: Fe3O4

The perovskite structure ABX3

• Close-packing of large A and X

• Small B in octahedral interstices

• Alternative (and misleading?) representation

We shall use 2-dimensional structures for our schematic representations of defects

• Elemental solid

• Ionic compound

Defects in an elemental solid

From A. Almar-Næss: Metalliske materialer.

Defects in an ionic compound

Defect classes

• Electrons (conduction band) and electron holes (valence band)

• 0-dimensional defects– point defects– defect clusters– valence defects (localised electronic defects)

• 1-dimensional defects– Dislocations

• 2-dimensional defects– Defect planes– Grain boundaries (often row of dislocations)

• 3-dimensional defects– Secondary phase

Perfect vs defective structure

• Perfect structure (ideally exists only at 0 K)• No mass transport or ionic conductivity• No electronic conductivity in ionic materials

and semiconductors;

• Defects introduce mass transport and electronic transport; diffusion, conductivity…

• New electrical, optical, magnetic, mechanical properties

• Defect-dependent properties

Point defects – intrinsic disorder

• Point defects (instrinsic disorder) form spontaneously at T > 0 K

– Caused by Gibbs energy gain as a result of increased entropy

– Equilibrium is a result of the balance between entropy gain and enthalpy cost

• 1- and 2-dimensional defects do not form spontaneously

– Entropy not high enough.– Single crystal is the ultimate

equilibrium state of all crystalline materials

• Polycrystalline, deformed, impure/doped materials is a result of extrinsic action

Defect formation and equilibrium

Free energy vs number n of defects

Hn = nHSn = nSvib + Sconf

G = nH - TnSvib - TSconf

For n vacancies in an elemental solid:

EE = EE + vE K = [vE] = n/(N+n)

Sconf = k lnP = k ln[(N+n)!/(N!n!)]

For large x: Stirling: lnx! xlnx - x

Equilibrium at dG/dn = 0= H - TSvib - kT ln[(N+n)/n] = 0

n/(N+n) = K = exp(Svib/k - H/kT)

Kröger-Vink notation for 0-dimensional defects

• Point defects– Vacancies– Interstitials– Substitutional defects

• Electronic defects– Delocalised

• electrons• electron holes

– Valence defects• Trapped electrons • Trapped holes

• Cluster/associated defects

csA

• Kröger-Vink-notation

A = chemical species or v (vacancy)

s = site; lattice position or i (interstitial)

c = chargeEffective charge = Real charge on site

minus charge site would have in perfect lattice

Notation for effective charge:• positive/ negativex neutral (optional)

Perfect lattice of MX, e.g. ZnO

xZnZn

2ZnZn

-2OOxOO

ivxiv

Vacancies and interstitials

iZn

//Znv

Ov

//iO

Electronic defects

/ZnZn

/e

hZnZnOO

Foreign species

ZnGa

/ZnAg

/ONOF

iLi

Protons and other hydrogen defects

H+ H H-

OOH

iH

O(OH)

/iOH

OH

xiH

xMO2(2(OH))

How can we apply integer charges when the material is not fully ionic?

Ov

The extension of the effective charge may be larger than the defect itself

)v(4M OM

……much larger….

)v4O(4M OOM

…but when it moves, an integer number of electrons also move, thus making the use of the simple defect and integer charges reasonable

)v4O(4M OOM

O v

Defects are donors and acceptors

E

xOv

Ov Ov

Ec

Ev

ZnGa

/ZnAg

//Znv/

ZnvxZnv

iH

Defect chemical reactions

Example: Formation of cation Frenkel defect pair:

Defect chemical reactions must obey three rules:

• Mass balance: Conservation of mass

• Charge balance: Conservation of charge

• Site ratio balance: Conservation of host structure

i//Zn

xi

xZn ZnvvZn

Defect chemical reactions obey the mass action law

Example: Formation of cation Frenkel defect pair:

i//Zn

xi

xZn ZnvvZn

]][Zn[v]][v[Zn]][Zn[v

[i]][v

[Zn]][Zn

[i]][Zn

[Zn]][v

i//Znx

ixZn

i//Zn

xi

xZn

i//Zn

//

xi

xZn

iZn

vZn

ZnvF aa

aaK

RTH

RΔS

RTΔG

aa

aaK vib

vZn

ZnvF

xi

xZn

iZn

000

i//Zn

Δexpexpexp]][Zn[v//

Notes on mass action law

• The standard state is that the site fraction of the defect is 1

• Standard entropy and enthalpy changes refer to full site occupancies. This is an unrealisable situation.

• Ideally diluted solutions often assumed

• Note: The standard entropy change is a change in the vibrational entropy – not the configurational.

RTH

RΔS

RTΔG

aa

aaK vib

vZn

ZnvF

xi

xZn

iZn

000

i//Zn

Δexpexpexp]][Zn[v//

Electroneutrality

• The numbers or concentrations of positive and negative charges cancel, e.g.

• Often employ simplified, limiting electroneutrality condition:

Note: The electroneutrality is a mathematical expression, not a chemical reaction. The coefficients thus don’t say how many you get, but how much each “weighs” in terms of charge….

][Zn][vor ]2[Zn]2[v i//Zni

//Zn

][h][OH][Ga][v2]2[Zn][e][N][Ag][O2]2[v OZnOi//

O/Zn

//i

//Zn

Site balances

• Expresses that more than one species fight over the same site:

• Also this is a mathematical expression, not a chemical reaction.

) in ZnO 1 ( [O]][OH][v][O OOxO

Defect structure; Defect concentrations

• The defect concentrations can now be found by combining

– Electroneutrality

– Mass and site balances

– Equilibrium mass action coefficients

• Two defects (limiting case) and subsequently for minority defects

– Brouwer diagrams

• or three or more defects simultaneously

– More exact solutions

• …these are the themes for the subsequent lectures and exercises…