Defects in solids

All solids, even the most ‘perfect’ crystals contain defects. Defects are of great importance as they can affect properties such as mechanical strength, electrical conductivity, chemical reactivity and corrosion. There are several terms used to describe defects which we must consider:

Intrinsic defects – present for thermodynamic reasons.

Extrinsic defects – not required by thermodynamics and can be controlled by purification or synthetic conditions.

Point defects – Occur at single sites. Random errors in a periodic lattice eg absence of atom from usual place (vacancy) or atom in a site not normally occupied (interstital).

Extended defects – ordered in one, two and three dimensions. Eg errors in the stacking of planes.

Every solid has a thermodynamic tendency to acquire point defects, as they introduce disorder and therefore increase entropy.

The Gibbs free energy, G = H – TS, of a solid, is contributed to by the entropy and enthalpy of the sample (fig. 14). Entropy is a measure of disorder within a system, hence, a solid with defects has a higher entropy than a perfect crystal.

Intrinsic point defects:

Point defects are not easy to directly detect. Several techniques have been used to study them. Two physicists, Frenkel and Schottky used conductivity and density data to identify specific types of point defects.

Schottky defect – Vacancy in an otherwise perfect lattice. Point defect where atom / ion is missing from its usual point in the lattice. Overall stoichiometry usually unaffected as there is normally equal numbers of vacancies at both M and X sites preserving charge balance.

These defects are encountered more commonly when metal ions are able to easily assume multiple oxidation states.

Frenkel defect – Point defect where an atom / ion has been displaced into an interstital site. eg In AgCl some Ag+ ions occupy tetrahedral sites (fig. 16) which are normally unoccupied. Stoichiometry is unchanged.

Encountered in open structures (wurtzite, sphalerite, etc) where coordination numbers are low and open structure provides room for interstital sites to be occupied.

Extrinsic point defects:

These are inevitable because perfect purity is unattainable in crystals of any significant size.


Oxford Dept of Chemistry
http://www.chem.ox.ac.uk/vrchemistry/solid/Page17.htm



Oxygen vacancy


While the surface of the anatase phase is well known for efficient photocatalytic effects, the high dielectric constants of the rutile phase (ɛ =30–80) have made it a candidate material as a nanoscale insulator such as an ultrathin gate oxide in field-effect transistors or a dielectric layer in capacitors for dynamic random access memory.

Positive charges at the vacancy site are not strong enough to hold electrons locally or create an F center within the band gap. This is in part due to the ionic displacements, especially those of nearby Ti atoms. In fact, when atoms are fixed at the bulk position, we find that the oxygen vacancy creates a defect state within the energy gap. The outward relaxation of Ti atoms effectively screens the positive charge of the oxygen vacancy. The shift of localized defect levels with increasing supercell size is similar to the case of an oxygen vacancy in SrTiO3 as recently reported. The dispersion in the defect state for the 3x3x5 supercell indicates
that one still needs a larger supercell to accurately characterize the localized level. However, the qualitative nature of the defect state, such as the relative position from the conduction minimum, is well addressed in the 3x3x5 supercell It is well known that ionized oxygen vacancies effectively dope TiO2 with electrons, resulting in a n-type transport behavior. In our calculations, the oxygen vacancy is created simply by taking out one oxygen atom from the supercell. After relaxation, the total energy is lowered by 1.69 eV from the initial energy and surrounding Ti atoms are displaced by 0.27–0.30 Å outward from the vacancy site. This is due to
the effectively positive charges of the vacancy site which interact repulsively with nearby cations. We calculate the electronic population at each atomic site by integrating the total electronic charges inside a sphere centered on each atom with effective ionic radii, the so-called Shannon-Prewitt radii. They are 0.61 and 1.36 Å for Ti and O atoms, respectively. We find that the electron population of three Ti
atoms surrounding the oxygen vacancy substantially increases from the bulk value of 0.968e to 1.021e while those at other Ti atoms change less than 0.02e, consistent with the positive character of the vacancy charge.

The analysis of the occupied states in the conduction band indicates that they are uniformly distributed with Ti d character. In other words, the additional electrons
at nearby Ti atoms are contributed by valence states. To confirm this, we carry out a calculation with two fewer electrons. This shifts down the Fermi level to be within the original band gap. However, the charge accumulation at Ti atoms around the vacancy is almost unchanged, indicating that extra electrons mainly originate from the small polarization of the valence states.

To determine the relative stability between point defects, we compare defect formation energies (Efor) of two point defects using the following formula:
Efor = Etot(defect) − nTi μTi − nO μO,

where Etot(defect)the total energy of the supercell containing a defect. ni and μ
i are the number and chemical potential of the constituent atoms, respectively, satisfying the relation μTi+2 μO=Etot (bulk) /2. Assuming μO is half the total energy
of an oxygen molecule, the formation energies are 7.09 and 4.44 eV for the Ti interstitial and oxygen vacancy, respectively, indicating that formation of the oxygen vacancy is energetically favoured. However, the relative stability is reversed
at μO=Etot(O2)2−2.64 eV, corresponding to a Ti richer environment, where the formation energies of the two defects are equally 1.8 eV. It is noted that the defect level for the Ti interstitial is of a localized d character and unphysical
self-interactions of occupied electrons may influence the formation energy of the Ti interstitial unfavorably. We also add that our calculations are for neutral defects in the sense that charge neutrality is maintained without uniform background charges. However, the delocalized characters of doped electrons imply that charge states of point defects are more consistent with +2 for both the oxygen vacancy and Ti interstitial. The Ti2+ interstitial found in our calculation is rather at variance with the traditional picture of this defect, i.e., Ti4+ or Ti3+. There are several possible causes for the discrepancy, such as the defect clustering or the accuracy of the local density approximation LDA.

In summary, we perform first-principles density functional calculations on the oxygen vacancy and Ti interstitial in rutile TiO2. The defect level associated with the oxygen vacancy is not identified within the energy gap while the Ti interstitial gives rise to a defect level that can be related to the infrared experiment.


First principal study, PHYSICAL REVIEW B 73, 193202, 2006
http://drm.kist.re.kr/CSC/publication/pdf/p-63.pdf