Pilling-Bedworth Ratio 

Physical metallurgy relies on a set of guidelines to identify protective oxide coatings for corrosion protection. Such coatings should be pinhole-free, exactly as the IC industry requires for high-k metal oxides. Accordingly, guidelines developed for protective oxide coatings in physical metallurgy can be applied to the protective metal oxides in integrated circuits.

In their 1923 paper "The oxidation of metals in high temperature" presented to the Institute of Metals, N. B. Pilling and R. E. Bedworth first correlated the porosity of a metal oxide with the specific density1. The Pilling-Bedworth ratio, (P-B ratio) R, of a metal oxide is defined as the ratio of the volume of the metal oxide, which is produced by the reaction of metal and oxygen, to the consumed metal volume:

M and D are the molecular weight and density of the metal oxide whose composition is (Metal)_a(oxygen)_b; m, and d are the atomic weight and density of the metal.

Pilling and Bedworth realized that, when R is less than 1, a metal oxide tends to be porous and non-protective because it cannot cover the whole metal surface. Later researchers found that, for excessively large R, large compressive stresses are likely to exist in metal oxide, leading to buckling and spalling. In addition to R, factors such as the relative coefficients of thermal expansion and the adherence between metal oxide and metal should also be favorable in order to produce a protective oxide.

Using the P-B ratio, Bruce Chalmers, Gordon McKay professor at Harvard University (Cambridge, MA), separated "protective" metal oxides from "non-protective" metal oxides. The table lists "protective" and "non-protective" metal oxides and their P-B ratios.

The list can be readily applied to the protective metal oxides used in integrated circuits. The intrinsic protective metal oxides, including the oxides of Be, Cu, Al, Cr, Mn, Fe, Co, Ni, Pd, Pb, and Ce, may be able to replace silicon oxide. On the other hand, a few popular metal oxides, e.g. Ti oxide and Ta oxide, are non-protective, suggesting a possible reason why these oxides have not been successfully used in commercial products after years of research. Besides oxides of elemental metal, the P-B ratio can be applied to oxides of metal alloy, metal nitrides and other metal ceramic systems.

Protective metal oxides can be produced by two classes of methods: growth and deposition. Growth methods include thermal oxidation, plasma oxidation, anodization, and implantation. Deposition methods include direct sputtering, reactive sputtering, and CVD. The IC industry does not have much experience with production of protective metal oxides. Still, manufacturing experience with sub-10 nm silicon oxide suggests that it is better to form the protective metal oxide using a growth method, or to form at least a portion of the protective metal oxide using a growth method and the remaining portion using a depositing method. Improved oxide thickness uniformity is one added advantage of such a composite layer.