Charge transport model to predict dielectric breakdown as a function of voltage, temperature, and thickness

Abstract

As the minimum pitch in interconnects continues to shrink, dielectric breakdown is becoming increasingly more difficult to qualify for each new technology node. Standard voltage-acceleration models provide quick, but general, assessments of the dielectric quality. Instead, a one-dimensional charge transport model has been developed as a tool to investigate the process of the dielectric breakdown and why it occurs. The model couples Poisson's equation with constitutive equations for mobile electrons, trapped electrons, and defects in the dielectric. Bonds in the dielectric matrix are weakened by the electric field, and broken by energetic electrons, creating defects. Failure occurs when a critical defect density is reached, causing trap-to-trap tunneling and an abrupt increase in the current.

The model successfully replicates electrical data for leakage current and dielectric failure as a function of voltage, temperature, and thickness. The activation energy for dielectric failure is shown to increase as the electric field decreases, resulting in much higher activation energies at operating conditions compared to testing conditions. The dielectric strength also increases for decreasing thickness based on a previous theory for planar dielectrics, and is shown to cause the failure vs. field slope to increase for thinner dielectrics.

Introduction

The never-ending pursuit to shrink integrated circuit nodes, in order to improve device performance and reduce die cost [1], is creating exciting possibilities for new types of advanced circuit applications. However, reliability has become a key impediment to continue to scale down device dimensions. Time-dependent dielectric breakdown (TDDB) represents one such example of the numerous types of circuit reliability that must be addressed and qualified for each new technology node.

TDDB has become an increasing reliability concern over the years due to several factors. The spacing between the interconnect wires at the most local levels is decreasing at a faster rate than the operating voltage, causing the insulating materials between the wiring to operate at increasingly higher electric field and temperature [2]. The introduction of copper wires and low-κ dielectric materials, in order to decrease power consumption and RC time-delay [[3], [4], [5]], yielded insulators with a lower dielectric strength than SiO₂ [3] and more susceptible to extrinsic types of failure, such as copper ions [6,7] and moisture absorption [8,9]. Finally, controlling the spacing of the dielectric has become critical as issues such as line overlay, via mis-alignment, and line-edge-roughness can each negatively affect the minimum spacing between wires and create enhanced electric fields which accelerate breakdown [[10], [11], [12]]. As a result, dielectric failure is now dominated by the insulator spacing for each chip [13,14].

Standard TDDB tests are conducted at high voltage and at elevated temperature, and empirical models are used to extrapolate the results to operating conditions, in order to assess if a product meets the reliability lifetime and fail rate requirements (e.g. 10 years and <10 ppm). There are several voltage-acceleration models that have been proposed and/or used over the years, including the E-model [[15], [16], [17], [18]], 1 / E model [19,20], power law (PL) model [21], E^1/2 model [22,23], and lucky electron (LE) model [[24], [25], [26]], where E represents the electric field and is interchangeable with voltage. These empirical models each justify the voltage dependence on the failure time with a different physical mechanism for dielectric breakdown. However, the fitting parameters associated with each model are at best, only loosely tied to the physical mechanism, materials, and process used to generate the data. Even worse, each model provides a drastically different lifetime prediction at operating voltage, and there is still dispute amongst reliability engineers over which model is most accurate. For temperature dependence, an Arrhenius relationship is used to predict failure when the operating temperature differs from the test temperature. Typically, a constant activation energy is used. However, studies over the past 30 years have consistently shown that the activation energy increases as the test voltage decreases [27]–[30]. As a result, fail rates from dielectric breakdown are over-estimated for chips that operate at lower temperatures than testing conditions.

The lack of a consensus voltage-acceleration model and the use of a conservative temperature-acceleration model impose a critical need for a more comprehensive TDDB model. Previously, a phenomenological, charge transport model was developed to predict intrinsic dielectric failure as a function of voltage for low-κ SiCOH [31] and high-κ SiN [32], two materials commonly used in integrated circuits. The charge transport model incorporates a set of fundamental mechanisms, including electronic conduction and defect generation, resulting in breakdown when a critical defect density is reached [31]. However, the model had several limitations. First, it over-predicted the temperature dependence for dielectric failure by approximately 1 eV (based on an Arrhenius equation fit to simulations at various temperatures). This was traced back to the electron temperature in the original equations, which was improperly defined to decrease as the defect density increased. Second, the model did not exhibit any thickness dependence for failure as related to the electric field. The dielectric strength of SiCOH films has been found to increase for decreasing thickness [33,34], and a theoretical explanation for this behavior was recently reported by McPherson to apply to all dielectric materials [35]. The previous model equations are refined in this article to improve the electron temperature definition and to better represent the increase in dielectric strength as a function of dielectric thickness. The article will present simulation results for low-κ SiCOH as a function of voltage, temperature and thickness, as compared to experimental data and will demonstrate improved agreement with the data.