In chapters 4–9, we studied the statistics of strength and lifetime of quasibrittle structures. The underlying approach was anchored by the weakest-link theory, in which the structure is statistically modeled as a chain of representative volume elements (RVEs). Because the size of material inhomogenieties (or, equivalently, RVE size) is not negligible compared to the structure size, this chain is not infinitely long. This causes the probability distributions of structural strength and lifetime to deviate from the classical Weibull distribution. As a result of the finite-chain model, the probability distribution functions of structural strength and lifetime vary with the structure size and geometry.
The finite weakest-link model is a general mathematical model, which is not limited to the modeling of quasibrittle fracture. There exist other failure processes that follow the weakest-link model but require considering a finite number of links. One is the breakdown of high-k gate dielectrics, an important problem for the semiconductor industry. In this chapter, we use the same theoretical framework as developed in Chapters 4 to 7 to study the lifetime statistics of high-k gate dielectrics, based on a mathematical analogy between the breakdown of dielectrics and the fracture of quasibrittle structures (Le, Bažant, & Bazant 2009; Le 2012).
Deviation of Lifetime Histograms of High-k Dielectrics from the Weibull Distribution
High-k gate dielectrics, such as Al2O3, HfO2, Si3O4, ZrO2, and so on, have recently been adopted in the design of metal-oxide-semiconductor field effect transistor (MOSFET) as an attractive alternative to the conventional SiO2 native oxide gate dielectrics, in order to reduce current leakage and increase the gate capacitance. These high-k dielectrics are known as “trap-rich” materials. The trapping of electrons in the gate oxide layer induces a trap-assisted tunneling process, which leads to the gate leakage current at a low voltage (Wilk, Wallace, & Anthony 2001; Kim & Lee 2004; Chatterjee et al. 2006). When the trap (or defect) density reaches a certain critical value, a weak localized breakdown path between the gate electrode and the substrate is formed, which is called the soft breakdown (SBD). The Joule heating in the local breakdown path then causes lateral propagation of the leakage spots and eventually leads to a significantly increased tunneling current passing through the layer. This is called the hard breakdown (HBD) (Chatterjee et al. 2006).