202 CHAPTER 10 Nanosystems and their Design 10.3 DEFECTS IN NANOGRAINS The simplest kinds of grains (nanoparticles) are made from a single material, e.g. ger- manium or silicon. Generally the only issues are whether the surface has a different chemical composition, either due to the adsorption of impurities from the environ- ment or due to the generation of lattice defects at the surface. In the bulk, the two kinds of defects are vacancies and interstitial ions. In compound materials, formed from a cation C and an anion A, the occurrence of defects is governed by the constraint of the need to maintain electrical neutrality; i.e., vacancies must be compensated by interstitial ions of the same sign (Frenkel type) or by vacancies of the opposite sign (Schottky type). More specifically, they may be cations on interstitial sites and vacancies in the cation half-lattice (Frenkel) or anions on interstitial sites and vacancies in the anion half-lattice (anti-Frenkel); cations and anions on interstitial sites (anti-Schottky) or vacancies in the cation and anion half- lattices (Schottky). Compounds can be created deliberately non-stoichiometric; e.g., by growth of a granular metal oxide film via physical vapor deposition (PVD) in an oxygen-poor atmosphere, thereby assuring an abundance of defects. Notation (Schottky). The superscript represents the electrical charge of the defect, relative to the undisturbed, overall electrically neutral lattice: 0, , represent nega- tive, positive and neutral (zero) excess electrical charge. Subscript is a vacancy, subscript is an interstitial. Hence C is an interstitial cation, and C 0 is a cation vacancy, `null' signifies the undisturbed lattice, and CA represents the addition (at the surface of the nanoparticle) of a new lattice molecule. A filled circle is used to represent the substitution of a native ion by a foreign one, e.g., X .C/ would be a divalent cation X on a cation lattice site, Cu .Ag/ would be (mono)valent copper on a silver lattice site in (say) AgCl, etc. Relations between Defects. The following relations are possible [71] C C C 0 A C A C C A 0 null C A 0 0 null null CA (10.1a) (10.1b) (10.1c) The above three equations can be used to derive some more: C 0 C A C CA CA C A CA C C 0 (10.2a) (10.2b) (10.2c) To each of these there corresponds a mass action law, i.e., for the last three, writing x for the mole fraction, identified by its subscript: x C 0 x A D K x A =x C D K 1 x C 0 =x A 0 D K 2 (10.3a) (10.3b) (10.3c)