An investigation of electromigration induced void nucleation time statistics in short copper interconnects

The stress evolution model (SEM) of Korhonenet al., is used to calculate the void nucleation time in a large number of short interconnects lengths up to 50 um. Finite element calculations show that the effect of the nonlinearity in the SEM model is small, and that a mesh size of the order of the grain size is quite adequate to give accurate simulation results. Via failure is the only mode considered in the current calculations, however the gain in simulation time over other solution methods means that more complex situations, possibly including void dynamics, may be modeled in future in this way. Using normal mass-lumping methods the analysis is isomorphic to the voltage development on a random RC chain, so standard methods from very large scale integrated static timing analysis may be used to obtain dominant time constants at each mesh point. This allows the distribution of nucleation times to be obtained as a function of the distributions of line parameters. Under the assumption of a lognormal grain size distribution and a normal distribution of diffusion activation energies, the nucleation time distribution is shown to be close to lognormal

Diffusivity variation in electromigration failure

Electromigration driven void dynamics plays an important role in the reliability of copper interconnects; a proper understanding of which is made more difficult due to local variations in line microstructure. In simulations, the parameter incorporating these variations best is the effective atomic diffusivity Deff which is sensitive to grain size and orientation, interface layer thickness, etc. We examine a number of experimental results and conclude that, to explain observations using current theoretical models, Deff values must vary significantly along the interconnect, and that such variations are enough to yield encouraging simulations of resistance variations under bidirectional stress

Analysis of multistate models for electromigration failure

The application of a multistate Markov chain is considered as a model of electromigration interconnect degradation and eventual failure. Such a model has already been used [ Tan et al., J. Appl. Phys. 102, 103703 (2007) ], maintaining that, in general, it leads to a failure distribution described by a gamma mixture, and that as a result, this type of distribution (rather than a lognormal) should be used as a prior in any Bayesian mode fitting and subsequent reliability budgeting. Although it appears that the model is able to produce reasonably realistic resistance curves R(t), we are unable to find any evidence that the failure distribution is a simple gamma mixture except under contrived conditions. The distributions generated are largely sums of exponentials (phase-type distributions), convolutions of gamma distributions with different scales, or roughly normal. We note also some inconsistencies in the derivation of the gamma mixture in the work cited above and conclude that, as it stands, the Markov chain model is probably unsuitable for electromigration modeling and a change from lognormal to gamma mixture distribution generally cannot be justified in this way. A hidden Markov model, which describes the interconnect behavior at time t rather than its resistance, in terms of generally observed physical processes such as void nucleating, slitlike growth (where the growth is slow and steady), transverse growth, current shunting (where the resistance jumps in value), etc., seems a more likely prospect, but treating failure in such a manner would still require significant justification

Analysis of critical-length data from electromigration failure studies

An accurate estimation of the Blech length, the critical line length below which interconnect lines are immortal, is vital as it allows EDA tools to reduce their workload. In lines longer than the Blech length, either a void will inevitably nucleate and grow until the line fails, or the line will rupture. The majority of failure analyses reveal voiding as the failure mechanism however recent analysis suggest Blech length failures are characterised by simultaneous [6] voiding and rupture, and a non-zero steady-state drift velocity. This paper provides an alternative interpretation of results

The influence of microstructure on the probability of early failure in aluminum-based interconnects

For electromigration in short aluminum interconnects terminated by tungsten vias, the well known “short-line” effect applies. In a similar manner, for longer lines, early failure is determined by a critical value Lcrit for the length of polygranular clusters. Any cluster shorter than Lcrit is “immortal” on the time scale of early failure where the figure of merit is not the standard t50 value (the time to 50% failures), but rather the total probability of early failure, Pcf. Pcf is a complex function of current density, linewidth, line length, and material properties (the median grain size d50 and grain size shape factor σd). It is calculated here using a model based around the theory of runs, which has proved itself to be a useful tool for assessing the probability of extreme events. Our analysis shows that Pcf is strongly dependent on σd, and a change in σd from 0.27 to 0.5 can cause an order of magnitude increase in Pcf under typical test conditions. This has implications for the web-based two-dimensional grain-growth simulator MIT/EmSim, which generates grain patterns with σd =0.27, while typical as-patterned structures are better represented by a σd in the range 0.4 – 0.6. The simulator will consequently overestimate interconnect reliability due to this particular electromigration failure mode

Modeling the electromigration failure time distribution in short copper interconnects

The electromigration EM lifetime in short copper interconnects is modeled using a previously developed means of generating realistic interconnect microstructures combined with the one-dimensional stress evolution equation of Korhonen et al. J. Appl. Phys. 73, 3790 1993 . This initial analysis describes the void nucleation and subsequent growth in lines blocked at one end and terminated with a pad at the other. For short copper interconnects, the failure time is largely spent on void growth, and, for sufficiently short lines (≤ 50 mm), the growth is largely steady state. This allows for the development of a simple expression for the variation of the failure time with microstructure. Assuming that the diffusion activation energies are normally distributed, the permanence property of summed lognormals leads to a roughly lognormal distribution for EM failure times. Importantly for EM design rules, linear extrapolation on lognormal plot is found to slightly underestimate interconnect reliability

Electromigration voiding in nanoindented, single crystal Al lines

We consider the interpretation of some theoretical and experimental work regarding electromigration voiding in nanoindented, single crystal aluminum lines. A recently suggested voiding criterion of a critical accumulated flux divergence is found, in fact, to be identical to the widely accepted critical stress criterion. The inclusion of the stress dependence of the atomic diffusion coefficient is shown to be vital when the steady state is characterized by J ≠ 0, such as in the case of a void growing at a constant rate. It is found, for example, that the stress required for steady void growth, within single crystal Al lines, is probably significantly smaller than previously suggested

Operational reliability calculations for critical systems

Reliability theory deals with the effect of mean time to repair upon overall system failure rates, but for critical systems such calculations are not what is required because an important performance criterion relates to operational failures, which are fundamentally different to unsafe failures: essentially they are the result of the system-level response to avoid unsafe failures. This paper introduces the particular problem for critical systems in general, presents an analysis of some of the relevant conditions and provides some simulation results in the context of a railway active suspension application that illustrate the overall effects and trends

VThreads: A novel VLIW chip multiprocessor with hardware-assisted PThreads

We discuss VThreads, a novel VLIW CMP with hardware-assisted shared-memory Thread support. VThreads supports Instruction Level Parallelism via static multiple-issue and Thread Level Parallelism via hardware-assisted POSIX Threads along with extensive customization. It allows the instantiation of tightlycoupled streaming accelerators and supports up to 7-address Multiple-Input, Multiple-Output instruction extensions. VThreads is designed in technology-independent Register-Transfer-Level VHDL and prototyped on 40 nm and 28 nm Field-Programmable gate arrays. It was evaluated against a PThreads-based multiprocessor based on the Sparc-V8 ISA. On a 65 nm ASIC implementation VThreads achieves up to x7.2 performance increase on synthetic benchmarks, x5 on a parallel Mandelbrot implementation, 66% better on a threaded JPEG implementation, 79% better on an edge-detection benchmark and ~13% improvement on DES compared to the Leon3MP CMP. In the range of 2 to 8 cores VThreads demonstrates a post-route (statistical) power reduction between 65% to 57% at an area increase of 1.2%-10% for 1-8 cores, compared to a similarly-configured Leon3MP CMP. This combination of micro-architectural features, scalability, extensibility, hardware support for low-latency PThreads, power efficiency and area make the processor an attractive proposition for low-power, deeply-embedded applications requiring minimum OS support

Quantum systems engineering: a structured approach to accelerating the development of a quantum technology industry

The exciting possibilities in the field of new quantum technologies extend far beyond the well-reported application of quantum computing. Precision timing, gravity sensors and imagers, cryptography, navigation, metrology, energy harvesting and recovery, biomedical sensors and imagers, and real-time optimisers all indicate the potential for quantum technologies to provide the basis of a technological revolution. From the field of Systems Engineering emerges a focused strategy for the development cycle, enabling the existence of hugely complex products. It is through the adoption of systems thinking that the semiconductor industry has achieved massive industrial and economic impact. Quantum technologies rely on delicate, non-local and/or entangled degrees of freedom — leading to great potential, but also posing new challenges to the development of products and industries. We discuss some of the challenges and opportunities regarding the implementation of Systems Engineering and systems thinking into the quantum technologies space