A reversible first-order dispersive model of parametric instability

https://doi.org/10.1016/j.microrel.2013.10.020Get rights and content

Highlights

  • A new parametric instability compact model for microelectronic devices.

  • Periodic stimuli of arbitrary waveform are addressed.

  • Model implementable in commercial simulators without resorting to external software.

  • Provided with a NBTI case study with gate bias and temperature dependence included.

Abstract

A general purpose instability model is derived for the variation of device parameters which is related to the activation–deactivation of statistically independent microscopic defects, with reversible first-order reaction kinetics and distributed rate constants. The model is aimed at predicting the parametric instability of electronic devices under periodic AC stimulus of arbitrary waveform over a wide time-scale range covering the whole device lifetime. As a practical application, we extracted a model for the negative-bias temperature instability of a p-channel type silicon MOSFET, including both the recovery effects and the voltage–temperature dependence. The model can be implemented in commercially available tools for the compact simulation of integrated circuits.

Introduction

Operation of electronic devices typically implies the ageing of elementary components that can be described as the variation of their characteristic parameters with time. In order to evaluate those parametric variations early in the product design stage, modern CAD circuit simulators of microelectronic devices are conveniently featured with device degradation tools [1], [2] aimed at predicting the parametric evolution of elementary devices under the foreseen device operation profile. The task of device wear out simulation gets more complicated if recovery effects are present [3], i.e. if a device parameter appears to recover towards its initial value upon ‘relieving the stressing condition’. Such behaviour is called parametric instability: its modelling is the main subject of this work.

One of the most important concerns in modern electronic devices is the threshold voltage instability of the p-channel type silicon MOSFET under negative gate bias and high temperature, known as Negative-Bias Temperature Instability (NBTI). The debate about the microscopic processes behind NBTI and about the modelling of its kinetics is still presently open. According to the recent literature, up to two [4], [5] or even three [6] different microscopic processes are suggested to cause NBTI, depending on the gate dielectric material. Recent advances [7] have revealed that a relevant fraction of the threshold voltage instability is due to the reversible capture-emission of charge carriers, at pre-existing defects of the gate dielectrics, associated with the microscopic structural relaxation of the dielectric lattice. The capture-emission process was described in terms of the field-assisted non-radiative multi-phonon theory. Accordingly, both the capture and the emission would be thermally activated, at least in the common operating temperature range, with distributed, bias-dependent, activation energy. It was shown [8] that a reversible first-order dispersive kinetic model derived from a simplified application of those findings was adequate for describing the threshold voltage instability of a p-channel MOSFET with 2.2 nm thick plasma-nitride (gate) oxide under a wide duration range of step-wise stress–relax, and low-frequency AC NBT stimuli. By resorting to a different physical model, but still based on the same kinetics, other authors [9] were able to account for the time and temperature dependence of NBTI stress and recovery, including the effect of the thickness and nitridation recipe of the gate oxide, in p-channel MOSFETs with gate dielectric thickness between 2.2 nm and 15 nm.

The laboratory characterization conditions typically comprise relatively high oxide electric field and rather limited cumulative stress duration. Should reversible first-order dispersive kinetics be suitable for describing the NBTI of the actual device operating conditions and lifetime scale too, one could predict, in principle, the parametric instability of devices under time-dependent NBT stimuli of arbitrary waveform. In order to be effective in supporting the design of microelectronic circuits, such model should be (i) suitable for implementation in common CAD software environment and (ii) able to produce its predictions with reasonable computational efforts.

With the continuous decrease of electronic device size the number of microscopic defects per individual component may well decrease to a few units. The related stochastic effects may lead to non-negligible device-to-device and time-to-time variations in the instability behaviour. Recently, a noticeable effort has been devoted to account for the stochastic effects in the circuit simulation of bias-temperature instability of very small devices [10], [11], [12]. Those models are based on an ‘atomistic’ or ‘defect-based’ approach and address the case of ideal ‘digital’ waveform consisting of a two-level periodic stimulus. The focus of this work is somehow complementary: while we will consider only large size devices such that all the stochastic variations can be assumed to be averaged out, the aspect of continuous variation of the stimulus conditions is included. The latter feature may be of interest for the purposes of digital modelling since the switching rise and fall times may well be a significant fraction of the clock period.

We will show that if the generic device parametric instability actually results from the additive effect of a large number of statistically independent activation–deactivation microscopic events following reversible first-order kinetics with distributed reaction rate constants then an exact kinetic solution can be actually found, yielding the variation of the device parameter under time-depending stimuli of arbitrary waveform. We will then propose suitable manipulation and approximation of the exact solution to make it compatible with its implementation in commercial CAD simulators. The proposed formulation allows for the calculation of a parametric drift induced by any periodic stimulus with arbitrary waveform over the whole device lifetime.

The instability model explained in Section 2 can be considered an extension of the first-order dispersive ageing model developed by some of these authors [13], [14] where a parametric degradation was treated as the superposition of different first order kinetics. It is worth to mention that different physical models have been proposed for explaining dispersive instability kinetics, which do not follow the present approach. That would be the case, for instance, of the well-known reaction–diffusion NBTI model [4], or even the case whether the charge capture-emission process contributing to the NBTI would take place on defects possessing more than two states, like in the ‘switching trap’ model proposed for explaining the high-frequency properties of NBTI [15]. We will present in Section 5 the extraction of the model parameters for the threshold voltage instability of a p-channel MOSFET under homogeneous NBT stress, with the gate voltage and temperature dependence included. The purpose of the extraction example is to highlight practical aspects and to prove the model effectiveness, not to bring any experimental evidence or to advance any conclusion about the microscopic mechanisms behind the NBTI.

Section snippets

Model

The instability model developed in this work relies on the following assumptions: (i) the generic device parameter drift Δ of the device under stress is proportional to the fraction of a large number of microscopic defects which gets activated independently, (ii) the distribution D of the defect parameters relevant to the reaction rates does not change upon stressing (and no extra defects are generated), and (iii) the activation–deactivation reaction obeys reversible first-order kinetics.

Implementation in a circuital simulator

A peculiar feature of device reliability practice is that the test stress conditions should just accelerate the parametric variation rate without triggering additional degradation phenomena, unimportant at the milder operating conditions, which would solely complicate the relevant acceleration laws or lead to wrong device lifetime predictions. This point might be the rationale behind the mathematical structure of the classical BERT reliability tool [1], now adopted by the most used commercial

Application to NBTI

The modelling approach described above is well suited for the description of NBTI in p-channel MOSFETs. Such a physical picture is similar to the one proposed in [5]. In this case, forward and backward reactions correspond to the transition of a pre-existing defect from a neutral state to a charged state and vice versa. The fraction of defects that is activated in the charged state leads to a variation in the MOSFET threshold voltage.

Consistently with the common observation of a rapidly

Discussion

Although the fitting accuracy of this model to the experiments is quite fine, no conclusion is drawn about the NBTI details, due to both the limited stress-recovery gate voltage combinations and the rather limited experimental data in the short stress-recovery time domain. Nevertheless, it appears that the observed threshold voltage instability of a silicon p-channel MOSFET with thermally grown, thermally nitrided SiO2 gate dielectrics can be described in terms of the cumulative effect of two

Conclusions

Starting from well-defined assumptions, a general purpose reversible first-order instability model has been developed allowing the prediction of parametric instabilities under any time-dependent stimulus. The model is applicable to the cases where both device-to-device and time-to-time stochastic variations can be neglected (large devices).

The relevant computation of parametric predictions up to the lifetime of real devices under bias waveforms of most practical interest, including the rapidly

Acknowledgements

The authors thank F. Pozzobon, G. Pizzo and G. Sommaruga for providing the package-level stress outcomes. The final manuscript of this work got substantially improved thanks to the reviewer’s valuable suggestions.

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