PX431 Structure and Dynamics of Solids

PART 2:

Defects and Disorder

Diane Holland P160 d.holland@warwick.ac.uk

2. Defects and disorder (10L)

crystal defects – point, line and planar defects; dislocations and mechanical behaviour

point defects and non-stoichiometry; radiation induced defects; thermodynamics and stability of defects; elimination of defects

influence of defects on: ionic diffusion and conductivity optical properties electronic properties

amorphous materials and glasses – formation and structure; structural theories; short and intermediate range order

techniques for structural analysis – diffraction and the pair distribution function; total scattering; local probes (NMR, EXAFS, Mössbauer, IR and Raman)

Conduction• Conductivity = nZe

n – number of charge carriersZe – charge - mobility of charge carrier

All of these are affected by the presence of defects

• electronic or ionic (S m-1)

• electronic metals 10-1 - 105

semiconductors 10-5 - 102

insulators < 10-12

• ionic ionic crystals < 10-18 – 10-4

solid electrolytes 10-3 – 101

• temperature dependence metals dec with Tall other inc with T

ElectronicSemiconductors - defects can:(a) Provide source of charge carriers – i.e.

inc. n and (b) provide traps for e/h – i.e. dec n, and

Impurity semiconductors - e.g. Si, Ge

Also many compoundsn-type U3O8, SnO2

p-type Ag2O, SnO, MnOamphoteric Si, SiC, UO2

in most cases, the mobilities of the e/h are too low to be useful

Insulators- Delocalised (band) model - extra states in

band gap can reduce activation energy for conduction – increases n

valence

conduction

E

Localised model (charges associated with specific ions)

TM compounds (a) mixed valency

(b) non-stoichiometry

Examples

(a) NiO oxidise by heating 1000oC/air Ni1-xO Ni2+1-3xNi3+

2xVxO

Thermally activated - electron hopping from Ni2+ to Ni3+

(b) Obtain same effect by doping

0.5xLi2O + NiO LixNi2+1-2xNi3+

xO

x = 0 ~ 10-10 S cm-1

x = 0.1 ~ 10-1 S cm-1 at 25 oC

hopping conduction v sensitive to T useful as thermistors

Effect of crystal structure e.g. Ni1-xO, spinels

Ni1-xO – NaCl structure Ni3+ and Ni2+ on adjacent octahedral sites

Fe3O4 inverse Fe3+T[Fe3+Fe2+]OO4 - easy hopping

Mn3O4 normal Mn2+T[Mn3+

2]O4 - greater separation

2+

2+

3+

3+

3+

e-

e-

e-

Ionic conduction (see earlier for solid electrolytes)

Depends on mobility of ions within material which is a function of:

-T = 0exp(-Em/RT)

Em – activation energy for ion motion

- structure

- size and charge on ion

-microstructure of polycrystalline materials (inc. mobility along grain boundaries)

e.g. NaCl- Na+ or Cl- ?

- vacancies or interstitials ?

at moderate T, conductivity by Na+ migration via cation vacancies(Callister: Materials Science and Engineering)

• To follow dotted arrow - Na+ would have to push two Cl- apart to pass through to vacancy

• Less energy needed to follow solid arrow

NB – remember that this is a close-packed lattice, so the Cl- ions are in contactNa+

Cl-

V- Na

migrating Na+

Em

length of jump

Intrinsic conductivity

– inc. exponentially with T as more vacancies created

NV exp(-ES/2RT)

ES – formation energy of Schottky defects

Include mobiity= A’exp(-Em/RT)exp(-ES/2RT)

Em – activation energy for migration of ions/vacancies along pathway through crystal structure to the next vacant site

At low T - few intrinsic defects formed-often exceeded by extrinsic defects due to impurities

-e.g. MnCl2 doped NaCl MnxNa1-2xVxCl

Only require energy to move these defects

Ln EXTRINSIC

Slope = Em

Inc. defects

INTRINSIC

Slope = Em+ ES/2

1/T

Get regions of different slopeIntrinsic slope = Em + ES/2Extrinsic slope = Em

In reality

= ATexp(-E/RT)

Pre-exponent factor AT = (1/T)

= (0/T)exp(-E/RT)

plot lnT rather than lnget slope of –Ea and intercept ln 0

0 contains n, Ze and information on jump frequency and distance

At low T, the formation of defect clusters may reduce the extrinsic mobility.

Ln (T)

1/T

EM and mobility depend on mechanismAgCl -dominant defects Frenkel

i.e. interstitial Ag+.Can look at how self-diffusion occurs

Ag Cl Ag Cl Ag

Cl Ag Cl Ag Cl

Ag Cl Ag Cl Ag

Cl Ag Cl Ag Cl

Ag Cl Ag Cl Ag

Ag

Ag

Mechanism 1. direct

Mechanism 2. indirect

Nernst-Einstein equation

D – self-diffusion coefficient - conductivityn – concentration of conductorsZe – charge

f – Haven ratio – dependent on mechanism different for 1 and 2

Mechanism 1. Direct self-diffusion distance = charge migration dist

f = 1Mechanism 2.

Indirect self-diffusion distance = ½ charge migration distf < 1

In practice observe Indirect but f also affected by defect concentration.

2

kTD

fn(Ze)

Effect of dopant

e.g. Cd2+ cation vacancies

AgCl +CdCl2 Ag(1-2x)CdxVAgCl

region 3. extrinsic defects VAg mobility dominates

region 2. more intrinsic defects (Agi) form but eliminate VAg

region 1 (high T). intrinsic Agi+ dominate

T at which these events take place depend on concentration of CdCl2 which creates defects

1

2

3

Log

1/T

[Cd2+]

log

12 3

T1

T2

T2 > T1

Colour centres• Crystals of alkali halides, when exposed to X-rays

become highly coloured

• Also happens with UV, neutrons, -rays

• Form F-centres (Farbenzentre)

• Colour characteristic of compoundNaCl – deep yellow-orangeKCl - violetKBr – blue-green

• Get same colours if heat crystal in vapour of alkali metal (doesn’t matter which). Intensity proportional to amount of excess metal

F-centre

• excess alkali atom diffuses into crystal

• halide vacancy associated with atom

• atom releases electron into vacancy

• the electron/vacancy pair are equivalent to an electron in a potential energy well

• transition between energy levels in well lies in visible

Cl Na Cl Na Cl

Na Cl Na Cl Na

Cl Na e Na Cl

Na Cl Na Cl Na

Cl Na Cl Na Cl

‘Blue John’ is mineral example of F-centres (CaF2)

H-centre

• Formed by heating alkali halide in halogen gas

• Cl2- ion formed

• If an F-centre meets an H-centre, they cancel

Many other colour centres exist

Cl Na Cl Na Cl

Na Cl Na Cl Na

Cl Na Na Cl

Na Cl Na Cl Na

Cl Na Cl Na Cl

Cl

Cl-

Density and Diffraction

1. Change in lattice parameters with compositionContinuous non-stoichiometricNo change 2-phase

2.

Exptl. density E = M/V

Diffraction density D = Z MM/VX

MM = molar mass of crystal

VX = volume of unit cell Z = number of formula units per unit cell

If E D then composition deviates from stoichiometry

Cel

l par

amet

er

Composition or Defect concentration

2-phaseNon-stoich

3.

Substitution - depends on relative atomic masses

Interstitials increase density

Vacancies decrease density

Frenkel – should not change density

Schottky – vacancies reduce density

Ignores lattice relaxation but changes are v. diff. to detect