PX431 Structure and Dynamics of Solids

1. PX431 Structure and Dynamics of Solids PART 2: Defects and Disorder Diane Holland P160 d.holland@warwick.ac.uk

2. 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)

3. Conduction • Conductivity  = nZe n – number of charge carriers Ze – 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 T all other  inc with T

4. E conduction valence Electronic Semiconductors - defects can: • Provide source of charge carriers – i.e. inc. n and  • provide traps for e/h – i.e. dec n,  and  Impurity semiconductors - e.g. Si, Ge Also many compounds n-type U3O8, SnO2 p-type Ag2O, SnO, MnO amphoteric 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

5. 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

6. e- 2+ 3+ 3+ 2+ e- e- 3+ 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

7. 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)

8. Na+ Em Cl- V-Na migrating Na+ length of jump NB – remember that this is a close-packed lattice, so the Cl- ions are in contact • To follow dotted arrow - Na+ would have to push two Cl- apart to pass through to vacancy • Less energy needed to follow solid arrow

9. INTRINSIC Slope = Em+ ES/2 EXTRINSIC Slope = Em Ln  Inc. defects 1/T 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 Get regions of different slope Intrinsic slope = Em + ES/2 Extrinsic slope = Em

10. Ln (T) 1/T In reality  = ATexp(-E/RT) Pre-exponent factor AT = (1/T)  = (0/T)exp(-E/RT) • plot lnT rather than ln get 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.

11. Ag Ag EM and mobility depend on mechanism AgCl -dominant defects Frenkel i.e. interstitial Ag+. Can look at how self-diffusion occurs Mechanism 1. direct Mechanism 2. indirect

12. Nernst-Einstein equation D – self-diffusion coefficient  - conductivity n – concentration of conductors Ze – charge f – Haven ratio – dependent on mechanism  different for 1 and 2 Mechanism 1. Direct self-diffusion distance = charge migration dist f = 1 Mechanism 2. Indirect self-diffusion distance = ½ charge migration dist f < 1 In practice observe Indirect but f also affected by defect concentration.

13. 1 2 3 T2 log • T2 > T1 T1 [Cd2+] 1 Log  2 3 1/T 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

14. 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 compound NaCl – deep yellow-orange KCl - violet KBr – blue-green • Get same colours if heat crystal in vapour of alkali metal (doesn’t matter which). Intensity proportional to amount of excess metal

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

16. Cl - Cl 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

17. Cell parameter Non-stoich 2-phase Composition or Defect concentration Density and Diffraction 1. Change in lattice parameters with composition Continuous  non-stoichiometric No 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

18. 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