5  Conclusion and Outlook

This chapter summarizes the main results of the previous chapters and gives an outlook on our future plans to refine those results.

5.1  The Microscopic Limit of the RD model for NBTI

The work presented in Chap. 3 shows that the reaction-diffusion model for the negative bias temperature instability, which has been used for nearly forty years to interpret experimental data, has a number of inherent assumptions on the underlying physics that lack any physical justification. Those are:

1. Continuous diffusion in the sub-nm regime. Diffusion of neutral hydrogen atoms and H2 proceeds via jumps between the interstitial sites of the host material. Positional changes that are smaller than about 4Å are atomic vibrations around an equilibrium position and thus not diffusive in nature. This is especially relevant as in the macroscopic modified H-H2 RD model, the onset of the power-law regime is quite discretization dependent.
2. Instantaneous well-stirredness along the interface. The one-dimensional macroscopic RD model, which gives the experimentally relevant t^1∕6 behavior inherently assumes that all hydrogen atoms that are liberated during stress instantaneously compete with other hydrogen atoms at the interface for available dangling bonds or dimerize with each other. However, at typically assumed dangling bond densities of 5 × 10¹²cm^-2, the distance between two dangling bonds will be about 4.5nm. At a depassivation level of 1% this means that the average initial distance between two hydrogen atoms is even in the range of 45nm. The reduction of this distance to the typical H2 bonding distance of 0.7Å [59] needs to be overcome by a diffusion step, which takes about 200s at a diffusion coefficient of 10^-13cm²∕s.
3. Rate-equation-based description. It is well established in chemical literature that bimolecular reactions are not sufficiently described by reaction rate equations if the particle numbers are small. In a reaction rate equation system it is for instance possible for 0.5 H atoms to form 0.25 H2, which is physically meaningless. An accurate description in the limit of small particle numbers is only obtained from an atomistic description.

We have implemented a stochastic three-dimensional modified reaction-diffusion model for NBTI to study the degree to which a more realistic description changes the predicted behavior. The model is theoretically well-founded on the theory of stochastic chemical kinetics and is understood as a consequent realization of the physical picture behind the reaction-diffusion theory.

The degradation predicted by the microscopic model features a unique new initial regime in which the motion of each hydrogen atom is completely independent from the others. This regime features a strongly increased power-law exponent that is not observed experimentally, yet it is a necessary consequence of the liberation of hydrogen during stress. Application of the atomic RD model to a real-world example shows that for a realistic jump width it is impossible to obtain the experimentally observed behavior due to the apparent diffusion limitation of the dimerization and passivation rates. The match of the microscopic model with the macroscopic version and experimental data can be improved by using an increased diffusion coefficient at the interface. However, the required diffusion coefficients are many orders of magnitude above 10^-9cm²∕s which leads to diffusion lengths way beyond the dimensions of individual microelectronic devices during stress. Consequently, interface diffusion coefficients of this magnitude would lead to cross-talk and a dramatically increased degradation due to the loss of hydrogen.

The recovery predicted by the microscopic model matches the macroscopic counterpart as soon as the previous degradation has entered the classical diffusion-limited regime. This behavior is due to the prerequisite that the system has to be equilibrated along the interface before the t^1∕6 regime can emerge. As the recovery happens on much larger time-scales than the stress duration, lateral equilibration effects are invisible in recovery traces. A distribution of arrival times as predicted by the simple estimate using different diffusion coefficients during recovery as in  [16] could not be found.

In summary, our study of the microscopic limit points out severe problems in the traditional mathematical formulation of the reaction diffusion model for NBTI, rendering all variants that are based on partial differential equations physically meaningless. In a physically meaningful microscopic version of the model, no experimental feature remains that can be accurately predicted. The apparent match of the RD models with experimental data must therefore be considered a mathematical artifact without any physical background. In the author’s opinion, the only way to add physical meaning to the reaction-diffusion model is to abandon the assumption that the power-law arises from the out-flux of the diffusing particles and move to a dispersive-hopping formulation. This path was taken by our group years ago and led us to the multi-state multi-phonon models we use today.

5.2  Atomistic Modeling and the BTI Defect

The work presented in Chap. 4 shows how the number of free parameters of the multi-state multi-phonon model for BTI can be reduced using a density functional theory based atomistic defect model. As examples, we investigate two well studied model defect structures, the hydrogen bridge and the oxygen vacancy in α-quartz. Both of these defects feature two stable structural configurations both in the neutral and the positive state, which are called dimer and puckered state for the oxygen bridge, and closed and broken state for the hydrogen bridge.

The activation barriers for the structural reconfiguration of the defect are calculated using the nudged elastic band method. The reconfiguration barrier of the hydrogen bridge is in agreement with the extracted parameters of the BTI defect. The predicted reconfiguration barrier of the oxygen vacancy for the 1′→ 1 transition is about 36meV, which is to too low for the BTI defect as it would be overcome instantaneously in the typical temperature range of BTI experiments.

The charging and discharging rates are calculated using the non-radiative multi-phonon theory. We have developed a method to calculate the NMP transition rates for the charging and discharging of a defect from an atomistic defect model and a macroscopic device simulation. For this purpose, we model the non-radiative transitions as quasi-optical transitions with negligible photonic energy. The derived expressions for carrier capture and emission rates consist of a tunneling expression, a line shape, and an empirical factor. For the rate calculation, the tunneling probability is taken from a macroscopic device simulation and the line shape is calculated from the atomistic model of the defect. Three approaches for the calculation of line shapes from the atomistic model are presented:

1. A quantum mechanical line shape based on Franck-Condon factors for approximate one dimensional parabolic potential energy surfaces in the different charge states. These line shapes are taken as the quantum mechanical reference calculations. They include nuclear quantization and tunneling. To get smooth lines it is necessary to introduce an empirical spread of the Dirac peaks. Our Franck-Condon-factor-based line shapes compare well with published analytic expressions. However, our method goes beyond previously published calculation methods as it allows to include a frequency change of the coupling mode. As our potential energy surface extractions show, this frequency change is a relevant feature of the atomistic defect models.
2. A line shape derived from classical statistical mechanics based on the same approximate potential energy surfaces. These line shapes are simple analytic expressions that can be easily implemented into a device simulator. For the temperature range in which typical BTI experiments are executed, these classical line shapes are a good approximation to the quantum mechanical line shapes.
3. A classical line shape based on molecular dynamics simulation. These line shapes consider the full potential energy surfaces obtained from the density functional calculations. Unfortunately, reasonably smooth line shapes require long molecular dynamics runs. For regions that are far (>1eV) away from the line shape maximum, this method quickly becomes unfeasible. For the region around the maximum, however, good agreement is found between the line shapes based on molecular dynamics and the approximate potentials.

For the evaluation of our model defects as candidates for the BTI, we concentrate on the initial hole capture transition. The calculated line shapes for this transition are energetically too low for the oxygen vacancy and too high for the hydrogen bridge. This leads to a much too high initial charging barrier for the oxygen vacancy and a too weak temperature dependence for the hydrogen bridge. However, these results heavily depend on the selected energy alignment scheme, which itself bears large uncertainties.

To illustrate how the line shapes obtained from the atomistic models can be employed to calculate rates, we calculate the hole capture rate using the density functional line shapes and an open boundary non-equilibrium Green’s functions device simulation. The presented results are meant as a proof-of-concept for extracting NMP parameters for device modeling from DFT and also serve as a benchmark for computationally less expensive approximations. The calculations have been compared qualitatively to experimental data obtained using the time dependent defect spectroscopy method on small area MOSFETs. The gate voltage dependence of the calculated capture time constants shows good qualitative agreement with experiment. Also, the reported strong temperature activation can be explained by the NMP model and good agreement is found for the temperature activation of hole capture rates based on classical and quantum mechanical line shapes down to very low temperatures.

The good agreement between the predicted and measured hole capture rates, however, is only made possible by a modification to the alignment scheme and is therefore of limited significance for the evaluation of the defects as candidates for the BTI. A method for the definitive alignment of the energy scales of the defect line shapes and the states in the device is not in sight at the time this document is written. However, for the search of the BTI defect it is expected that if both charging and discharging kinetics are taken into consideration, this will compensate the uncertainty in the energy alignment. Judging from the results presented here, the hydrogen bridge seems to be the more promising candidate for the BTI defect.

The work presented here is the fundament for future efforts to find the defect responsible for the bias temperature instability. The next steps on this path are the study of the reconfiguration barriers and line shapes of oxygen vacancies and hydrogen bridges in more realistic host structures, such as amorphous silica and Si-SiO2 interface structures. Different approaches to this target are tested at the moment as part of the EU project “Modelling of the reliability and degradation of next generation nanoelectronic devices” (MORDRED). The results of our α-quartz based calculations will also serve as a reference for these studies. For this purpose, however, our results have to be checked against larger supercells or embedded cluster calculations, to exclude artificial strain effects.

The inclusion of the calculation of emission rates into the device simulator VSP2 is being worked on heavily at the moment and first results are expected to be published soon. For a future inclusion of the non-radiative capture and emission model into standard TCAD simulation, it is necessary to find a compromise between physical accuracy and computational efficiency. Therefore, different levels of approximations will be compared for the NMP capture and emission rates, concerning both the tunneling expression and the line shape functions. In this context, the good agreement between the capture rates computed from the classical and the quantum mechanical line shapes provides a quite promising result.