We report the voltage-dependence of voltage acceleration for ultra-thin oxides from 2.2V to 5V over a range of Tox values from 1.7nm to 5.0 nm. This unique behavior manifest itself as a power-law voltage-dependence for time-to-breakdown (TBD) over a variety of experimental observations. Using the concept of energy-to-breakdown, we explore the possible scenarios such as fractional energy or defect generation probability as a function of voltage to account for the increase in voltage acceleration with decreasing voltages.

It is proved that every density matrix is the limit, in the sense of weak operator convergence, of a sequence of operators each of which may be represented as an integral over projection operators onto coherent states (in the sense of Glauber) with a square-integrable weight function. This result is a special case of one that holds for all operators with trace and for overcomplete families of states other than just the coherent states. We prove our more general result, at no cost of complexity, within the more general framework of continuous-representation theory. The significance of our results for representing traces of operators is indicated.

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We consider direct-product representations of the canonical commutation relations. An irreducible representation is defined on each of the incomplete direct-product spaces (IDPS) of von Neumann. We prove that two such representations are unitarily equivalent if and only if the corresponding IDPS are weakly equivalent, for which simple analytic tests exist. The matrix elements of these representations, coupled with a Friedrichs-Shapiro type of integral, fulfill group orthogonality relations. This classification into unitary equivalence classes also applies to direct-product representations of the canonical anticommutation relations.