Defects in Solids

John Dalton; "Atoms of different elecment can be combined by chemical reaction to form chemical copounds in fixed ratios"

Initially assumed all solids were perfectly stoichiometric, but all ionic compounds are strictly not stoichiometric with no simple ratios. Even crystals contain defects.

All solids contain defects; 

• Intrinsic; stoichiometric (Schottky and Frenkel defects)
• Extrinsic; non-stoichiometric

In the formation of compounds, at 0K, dG = dH and therefore a perfect crystal is formed. However, compounds arn't made at 0K, so we have to consider entropy. dH relates to the lattice energy, and the energy required to move an ion into the less favourable site. 

3rd Law of Thermodynamics; crystals can only have complete atomic order at 0K, at any other temperature, we must have disorder.

Schottky Defects; Vacancies on both cation and anion sites (ions removed from both)

Frenkel Defects; Displace either a cation or anion from it's ideal site into a new interstitial site (more commonly a cation, as these are usually smaller) Exception is the Fluorite structure, which favours anion Frenkel defects.

These increase entropy, as the defects can occur at any site within the crystal (resulting in an inherent randomness). 

• Schottky and Frenkel are specific to stoichiometric crystals - don't use for non-stoichiometric solids. 
• Overall compound is charge balanced

Consider dG = dH - TdS to see if defects will actually form (favourable dS and unfavourable dH).

Linear increase of dH -> ndH, but as we introduce more vaccancies, the entropy stabilisation decreases. The type of defect that exists depends on the relative size of dH_v and dH_f, where dH_f is the energy required to create a Frenkel defect.

For a rock salt structure (NaCl); Frenkel defect involves movement of an octahedral cation into a tetrahedral site (highly endothermic / very unfavourable). Therefore Schottky defects are more favourable, as dH_f > dH_v except AgCl, AgBr, and AgI (ZnS structure).

Favour Frenkel defects as Ag is more covalent than Na, which has a preference for tetrahedral. It is less unfavourable to Ag⁺ to occupy tetrahedral sites than Na⁺. (It's not due to size)

Actual number of defects cannot be easily predicted at low temperatures. The number of defects is small, but essential for solid state synthesis. 

Normally dH_v is large, so there is only a small number of defects.  However, some metallic oxides (e.g. TiO and VO) have a very small dHv, resulting in highly defective structures. 

TiO and VO are both metallic, due to the overlap of t2g orbitals. This means most of the energy is not electrostatic, and dH_v doesn't really relate to the lattice enthalpy, as much of it is lost by the creation of defects. 

We can sometimes introduce vacancies by doping. If you dope with an ion of the different charge, the stoichiometry adapts to maintain a charge balance.

Aliovalent dopant; doping with an ion of a different charge

Isovalent dopant; doping with an ion of the same charge.

For example; doping Y³⁺ into NaCl, 3Na⁺ = Y²⁺ + 2V (where V = vacancys). This gives us the formula Na_1-3xY_xCl, with V_2x. For 10% defects, we want x = 0.05, giving us Na_0.85Y_0.05Cl.

When either NaCl or KCl is heated in K or Na vapour, it gives a yellow solid. Additional cations enter the structure, which contains electrons trapped in anion vacancies in order to balance the charge. This is caled an F centre;

KCl + xK -> K_1+xCl e^-_x

Despite having the formula K_1+xCl, it is not because we have extra K. it is because we have Cl missing (so a better form of notation it would be KCl_1-x).

The solid produced is non-stoichiometric, and the colour can be related back to the quantised energy levels. 

Composition of solids depends on the partial pressure of the component elements in equilibrium with the solids. As before, the driving force is the gain in entropy from the formation of defects. Enthalpy is unfavourable.

If enthalpy is small, we would expect very non-stoiciometric solids (large imbalance between cations and anions). The electron in the anion site is very enthalpically unfavourable, but is a requirement to remain charge-balanced. 

Transition metals, with their variable oxidation states, allows the compound to remain charged balaned without having extra electrons. For example;

FeO -> Fe_1-xO + x Fe   (some Fe is reduced to Fe⁰); disproportionation occurs. 

dH is low for FeO, as the loss in lattice energy from vacancies is offset by higher cation charge (see Born Lande equation). This results in a large deviation from stoichiometry.

All materials are non-stoichiometric, but some compounds are more non-stoichiometric than others.

• Chemically significant composition range; occurs when dH is small. To be chemically significant it must be analytically measureable. 
• Vegard's Law; deviations often occur at high defect concentrations
  • a plot of compositon vs a₀
  • Linear decrease in unit cell size as concentration of Fe³⁺ increases - effect of ionic radii
  • Deviation from linearity at low Fe content due to larger concentrations of defects. Interactions between the defects result in non-linear behaviour.

Enthalpy and Entropy dictates the chemistry of non-stoichiometric phases.

• Entropy favoures high concentration and random distribution of defects
• Enthalpy favours small concentration and ordered distribution of defects

At low temperature, enthalpy is most important, there is a narrow range of stoichiometry and defects tend to order to give new structures. As temperature increases, entropy becomes more important, there is a wider range of non-stoichiometry, and a random distribution of defects.

From the unit cell, the structure is primative (P). 

In most materials, oxygen vacancies are essentially random/show no order, however in SrFeO_2.5, the vacancies order to form Brownmillerite (a new structure). (Transition occurs at approx 850^oC).

Vacancies in alternate layers; octaheadral then tetrahedral. 

Structure also adopted by BaInO_2.5, but this is a stoichiometric system and In can't vary it's oxidation state like Fe (transition occurs at approx 950^oC).