1  Introduction

Point defects in solids have been in the focus of scientific interest since the very beginnings of solid state theory due to their large influence on the macroscopic mechanical, electrical, and optical properties of the host material. In semiconductor technology, which is itself largely based on the intentional introduction of impurities into a host material, defects affect the motion of the electrons and the holes by scattering, carrier trapping, or as recombination centers. Under certain operational or environmental conditions, or just over time, point defects in a given semiconductor device might be created, destroyed, repositioned, or modified in their ability to interact with the carriers. As these changes usually alter the characteristics of the device in an undesired manner, detailed knowledge of the defect physics is of utmost importance in the field of semiconductor device reliability, within which the present work has been carried out.

1.1  The Bias Temperature Instability

The main focus of interest for this thesis lies in the bias temperature instability (BTI) of metal-oxide-semiconductor (MOS) transistors. This instability is usually encountered during burn-in tests or high-performance operation of MOS devices [1, 2] at temperatures in the 100oC–250oC range, when a large voltage is supplied to the gate contact (≈ 7–10MV∕cm oxide field) while all other terminals are grounded, see Fig. 1.1. BTI degradation shows as a shift of the threshold voltage and a degradation of the channel mobility. Bias temperature instabilities have been known for forty years  [3, 4] and, due to their minor influence on early semiconductor technology, were mainly of academic interest. With the aggressive shrinking of feature sizes down to the nanometer regime, oxide fields need to be increased with every new technology generation, to keep the sub-threshold leakage of the transistors at reasonable values [5]. Additionally, the increasing concentration of currents during device operation into smaller and smaller volumes leads to an increase in the thermal power density due to Joule heating and thus raises the operating temperature of the MOS transistor. These effects brought bias temperature instabilities to the industrial agenda in the early 2000s. As the problem is getting worse with every new technology node, partially also because of the introduction of new materials to the semiconductor process, BTI has evolved to a major reliability issue in modern, highly scaled MOS transistors [2]. Although present in both n- and p-channel devices with both polarities of the bias voltage, the most pronounced BTI is observed when negative bias stress is applied to p-channel MOS (pMOS) transistors. The corresponding instability is termed negative bias temperature instability (NBTI).

1.2  BTI Modeling

Lifetime predictions for BT instabilities are usually made based on accelerated (so-called burn-in) tests, where the devices are operated at temperatures and voltages that are beyond their specifications, aiming at the amplification of one specific degradation mechanism as much as possible without damaging the device otherwise. The degradation observed in these tests is then extrapolated to the normal operation conditions by using empirical acceleration factors [6] to predict the lifetime of the devices under these conditions. As the acceleration factors are purely empirical expressions, they bear large uncertainties, requiring equivalently large reliability margins, and tend to be overly pessimistic. An accurate and physics-based model could help to reduce reliability margins thus making more efficient circuit designs possible.

In order to obtain accurate long-term (ten years or more) predictions for the amount of degradation induced by a certain mechanism, a detailed physical understanding of this degradation process is required. The identification of the relevant physics, however, proves to be difficult in practice due to the limited experimental observability. These limitations arise from the small structural dimensions of the microelectronic devices together with the low density of defects present in the materials. The resulting number of defects generated during stress in a commercial device is below the detection limit of most measurement methods that give hints to the atomic structure involved, like electron spin resonance (ESR, also known as electron paramagnetic resonance EPR) or optical absorption. Additionally, the defects generated during BT stress recover fast, so any delay between the removal of the stress and the measurement of the generated defects influences the outcome of this measurement. As a consequence, BTI degradation models are quantitatively evaluated solely against electrical measurements. Other experimental data is considered only in a qualitative fashion.

1.3  The Reaction-Diffusion Model for NBTI

Because of its impact, most modeling has been done on the negative bias temperature instability. The first model for NBTI was put forward by Jeppson and Svensson in 1977  [7]. Their model was based on the following ideas, which are illustrated in Fig. 1.2. Due to the lattice mismatch between silicon and silicon dioxide, some of the silicon atoms do not have an oxygen neighbor. A silicon atom in this situation has one unpaired valence electron, which is called a dangling bond. This dangling bond is visible in electronic measurements as it gives rise to states within the band-gap [2]. During the manufacturing process the wafer is exposed to a hydrogen-rich atmosphere so that hydrogen atoms can penetrate through the oxide and passivate the silicon dangling bonds, leading to a removal of the band-gap states.

During stress, the presence of holes at the interface and the increased temperature leads to a liberation of the hydrogen atoms. The remaining silicon dangling bonds become electrically active carrier traps. According to the model, the depassivation and repassivation of dangling bonds at the interface reaches an equilibrium in a very short time [8, 9], and it is the constant flux of hydrogen atoms (or some hydrogenic species) away from the interface that determines the temporal evolution of the degradation. Because of the two proposed stages — the electrochemical reaction at the interface and the subsequent diffusion of the hydrogenic species — this model bears the name reaction-diffusion (RD) model.

The mathematical framework the of model is based on a macroscopic description using a rate equation for the interface reaction and a Fickian diffusion equation for the motion of the hydrogen in the oxide. Central actors are the density of depassivated silicon dangling bonds at the interface Nit = [Si^*], and the concentration of hydrogen in the oxide H = [H](x,t) and at the interface Hit = [H](0,t). During degradation, a fraction Nit of the initially passivated silicon dangling bonds N0 = [SiH]₀ is depassivated according to

(1.1)

with the depassivation (forward) rate kf and the repassivation (reverse) rate kr. The hydrogen liberated at the interface then diffuses into the oxide as

= -D

(1.2)

(1.3)

This power-law degradation corresponded well with experimental results of the seventies.

Figure 1.3:

Basic features of the degradation predicted by the RD model for NBTI. In the initial phase, the depassivation reaction with rate kf dominates, giving rise to a degradation that increases linearly with time. After the depassivation and repassivation reactions have reached an equilibrium, the degradation is determined by the flux of hydrogen away from the interface, which gives rise to a power-law with an exponent of 1∕4.

In later experiments, power-law exponents were found that differed from the 1∕4 prediction of the model. These findings led to a modification of the original RD model to account for different diffusing species such as H2 [10]. For almost four decades, the reaction-diffusion idea was the unquestioned standard interpretation for NBTI until around 2005 NBT recovery moved into the focus of the scientific attention. The experiments showed that NBTI recovery starts immediately (even before a microsecond) after the removal of stress and extended over several decades, continuing even after more than 10⁵s  [11, 12]. This behavior stands in strong contrast to the predictions of the RD model, which predicts a recovery that proceeds within four decades, centering around the duration of the preceding stress phase [13, 14]. A comparison of a typical experimental NBT recovery trace and the corresponding prediction of the RD model is shown in Fig. 1.4.

Figure 1.4:

Typical recovery trace as predicted by the RD model for NBTI using equations (1.4)–(1.6), which is similar for all variants of the RD mechanism. The comparison with experimental data  [12] shows that the RD predicted recovery occurs much too late and proceeds much too fast.

Several extensions to the RD model have been put forward, such as dispersive transport of the hydrogenic species  [11, 9], but none could give the observed experimental behavior. The current state-of-the-art RD-based modeling supplements the RD theory with empirical hole-trapping expressions. It is assumed that short-time (1s) degradation and recovery is dominated by hole trapping into oxide and interface defects, while the long-term degradation and recovery are determined by the RD mechanism [15, 16, 17, 18]. The RD theory employed in these modeling efforts is the modified RD model  [19, 20, 21] that has been developed as an extension of the classical RD models and explicitly considers diffusion of H and H2 and their interconversion reactions. Classical models assume an instantaneous transition between the liberated interfacial hydrogen and the diffusing species, usually H2 [10]. The reactions present in the modified RD model are the interface reaction SiH ⇌ Si^* + H, the dimerization reaction 2H ⇌ H2, and the diffusion of both species. The mathematical framework is an extension of (1.1) and (1.2) [21, 20],

	= kf(N0 - Nit) - krNitHit,	(1.4)
	= -D - kHH2 + k H2H2,	(1.5)
	= -D2 + H2 -H2,	(1.6)

The combination of this modified reaction-diffusion model with empirical hole-trapping somewhat improves the match with experimental DC and AC stress data. The failure of the RD model to properly describe NBT recovery is shifted out of the time window of some experiments, but essentially remains.

Most recently, it has been argued that the shortcoming of the reaction-diffusion based model concerning the prediction of NBTI recovery comes from the one-dimensional description of the H and H2 diffusion and that a proper description of the three-dimensional atomic motion would lead to the experimentally observed long recovery tails  [16, 17]. Additionally, some groups are working towards a microscopic formulation of the reaction-diffusion model which would then be applicable to small-scale effects like random telegraph noise (RTN) and nano-scale devices in general [23].

We have also derived and implemented a microscopic formulation of the RD model  [24, 25], in order to study the behavior of the RD mechanism on the atomic scale. Our calculations, which are more deeply explained in Chap. 3, have disproved the claims in  [16, 17] and raise strong doubts of the validity of the proposed RD mechanism and its modeling using rate equations.

1.4  Defect-Centered Models

The obvious inability of the reaction-diffusion model and its variants to accurately predict NBTI recovery as well as other properties, such as the universal scaling of BTI degradation [27], led to increased interest in alternative descriptions. A promising approach was found in 2008 in BTI models based on dispersive reactions between two or three states  [27, 28], termed double-well or triple-well models. Although the central actors were still believed to be hydrogen atoms, these models brought a fundamental reinterpretation of the physical process behind the BTI. The collective diffusion process that determines the degradation and recovery in the RD models was replaced with a dispersive hopping of isolated particles. Even better accordance with experimental data was found later that year with a model that coupled a two-state description of the ‘hydrogen atom’ with an also statistically distributed thermally activated hole capture process  [29]. The introduction of more complex experimental techniques such as rapid gate voltage and device temperature switches created new testing-grounds for BTI modeling. Early 2009 a refined model could be devised that was also able to explain the more complex experimental data with striking accuracy [26]. This model brought along a complete reinterpretation of the degradation process that no longer assumes the hole capture to happen at the Si-SiO2 interface, but instead at defects within the gate-oxide. Upon hole capture, these defects would undergo complicated reconfiguration, as explained in Fig. 1.5, and eventually offer a bonding state to a hydrogen atom. Interfacial hydrogen atoms would be thermodynamically more stable at this defect site than at the interfacial silicon dangling bond, thus creating an interface defect. In contrary to its predecessors this model is based on a concrete microphysical picture, where the structural reorganization is described as a transition over a barrier and the hole capture is understood as a field-accelerated multi-phonon process. This behavior was inspired by models for irradiation damage, which assume the oxygen vacancy defect in SiO2 as the central actor  [30]. In its neutral state, the oxygen vacancy is assumed to exist in a dimer position, where the silicon atoms adjacent to the vacancy form a bond. The positively charged variant of this defect structure is usually identified with the E′_δ paramagnetic center, which is visible in electron spin resonance (ESR) measurements of amorphous silica. Upon hole capture the bond of the dimer is weakened and one of the silicon atoms eventually relaxes through the plane of its oxygen neighbors and forms a weak bond with a nearby oxygen atom, which is then threefold coordinated. The resulting position is called the ‘puckered’ state, which is also paramagnetic and usually identified with the E′_γ center.

While the microscopic picture behind the two-stage model was based on a broad and rigorous literature study, the mathematical description was formulated in a way that only captures the basic behavior arising from the microscopic theories but that did not fully implement the physical details of those theories.

1.5  A Multi-State Multi-Phonon Model for BTI and RTN

Figure 1.6:

(left) In small-area devices, NBT recovery proceeds in discrete steps which is accounted to the discharge of single defects. The careful analysis of the step-curves has led to the development of the time-dependent defect spectroscopy (TDDS) method [31, 32]. In TDDS experiments a number of stress-recovery cycles are measured and the steps are detected. The times and magnitudes of those steps are binned into a two-dimensional histogram which is called the TDDS spectral map (right). In these spectra, defects show up as isolated clusters. The variation of the stress and recovery conditions makes the experimental study of the trapping behavior of single defects possible.

A further advance of the defect model was stimulated by experiments taken on small-area devices, where the recovery in response to NBT stress proceeds in discrete steps instead of a continuous curve  [31]. The careful analysis of these steps led to the development of the time-dependent defect spectroscopy (TDDS) method  [31, 32, 33], which makes a detailed investigation of the charging and discharging of single defects possible  [32]. A typical TDDS spectral map is given in Fig. 1.6. The results of the TDDS experiments served as a testing ground of unprecedented detail for BTI models and allowed for the refinement of the physics in the two-stage model. The resulting multi-state defect model not only fits experimental data from BTI and TDDS experiments very well, it also links these phenomena to other phenomena such as random telegraph noise (RTN) and flicker noise [34, 35, 36, 32, 37].

As an extension to the original two-stage model, the multi-state NMP model considers four states of the oxygen vacancy, as illustrated in Figures 1.5 and 1.7, and rests upon a more detailed physical theory for the hole capture and emission rates, which will be more deeply discussed later.

The mathematical formulation of the defect based model assigns a probability p_α(t) to every state α of a defect with ∑ _αp_α(t) = 1. The dynamics of the defect state is then described using rate equations of the form

(1.7)

The transient behavior of a defect arises from the potential energy surfaces in its various charge states. The potential energy surface Ei() corresponding to a certain charge state α of the defect assigns a total energy to each atomic configuration . In our model, the defect is assumed to have two relevant potential energy surfaces, corresponding to the neutral and the positive charge state of the defect. Fig. 1.7 shows the gradual change of the total energies as the system moves in configuration space and the resulting energetic minima which give rise to the states of the defect (denoted 1, 1′, 2, and 2′). The model knows two types of transitions: the transitions between the structural configurations (1 ↔ 1′ and 2 ↔ 2′) and the transitions between the charge states of the defect (1 ↔ 2′, 2 ↔ 1′). The former are treated as adiabatic barrier-hopping transitions of the form

(1.8)

where ν_α is the attempt frequency and E_αα′ is the activation energy associated with the transition. The charge state transitions are understood as non-radiative multi-phonon transitions and are modeled as

(1.9)

where σ is the capture cross section of the defect, v_th is the thermal velocity of the carriers, p is the hole density at the defect site, and E_αβ′ is the energy barrier defined by the crossing of the potential energy surfaces.

In our model, the defect is initially in state 1, which is electrically neutral. During normal device operation, the defect will remain there due to the large energetic barriers separating state 1 from all other states. When large negative bias is applied, the relative energetic shift of the neutral potential energy surface to the positive ones changes as indicated in Fig. 1.7, resulting in a decrease of the transition barrier for hole capture from the silicon valence band. After the transition to the positive state (1 → 2′), the defect undergoes structural relaxation and moves to state 2 by overcoming a small energetic barrier. In its new (secondary) structural configuration, even after the bias is removed, the defect can change its charge state much easier, due to the smaller barriers for interaction with the silicon. From the neutral secondary state 1′, the defect can return to the initial configuration 1 by again overcoming a thermal barrier.

In electrical measurements of large area transistors one observes an ensemble of the described defects and, due to the amorphous nature of the gate oxide, every defect moves on a different energetic landscape. This means that all the state energies and barrier energies are statistically distributed and thus each defect shows a different transient behavior. The macroscopically observed degradation is the superposition of a large number of defects changing their state from 1 to 2, e.g. due to electrical stress, as the charged state of these defects influence the charge carriers in the silicon substrate. Similarly, if an increased number of defects undergo the transition 1′→ 1 or 2 → 2′→ 1, a recovery of the device characteristics will be observed macroscopically.

1.6  The Search for the BTI Defect

Although the central defect for BTI has been illustrated as an oxygen vacancy up to now, to date there is no direct experimental evidence of the actual microscopic nature of this defect. Defects in SiO2 have been predominantly studied with electron spin resonance (ESR) measurements [38]. From these studies, two types of defects have been identified in the MOS system: Pb- and E′-centers [39, 38, 40, 41, 42].

Judging from the properties of their ESR-spectra the Pb-centers are located at the Si-SiO2-interface. They have been identified with fast amphoteric interface states  [43, 44, 39, 45, 46, 2]. Their connection to NBTI is still actively debated, but the experimental evidence suggests that the Pb-centers are only generated after long-term bias-temperature stress [26], although the generation process is still unclear. A complete picture of BTI degradation will certainly involve the Pb-center dynamics in some form, most likely as a slowly recovering, or permanent damage component. The present work, however, concentrates on the defects that are dominant for BTI, which are described using the multi-state NMP model and show strong recovery. These defects have often been suggested to be E′-centers, which are paramagnetic centers inside the SiO2. Several centers of this class have been detected in MOS structures, especially in experiments involving irradiation damage [43, 47, 48, 49, 50, 46, 51, 52, 40, 53]. Atomistic models for many of the E′-centers have been put forward [54, 55, 56, 57, 50, 58, 51, 59, 60, 61, 62]. The best agreement between theory and experiment has been found between oxygen vacancy (VO) models and the E′γ-center, which is the most abundant dangling bond center in amorphous SiO2 [40]. For this reason, a huge amount of literature exists that deals with oxygen vacancies in SiO2  [54, 55, 56, 50, 58, 59, 63, 64, 65, 66, 62, 67, 68, 69, 70, 71, 72] spanning four decades of research. Apart from radiation damage, oxygen vacancies have been linked to 1/f-noise  [73], high-field stress response  [74], leakage currents  [75], and time-dependent dielectric breakdown  [76] in the context of MOS reliability. The most interesting property for our BTI modeling efforts, however, is the bistability of the oxygen vacancy  [57, 58, 59, 77, 63, 65], which can be identified with the two structural states of our BTI defect. In the so-called dimer configuration, the two silicon atoms adjacent to the oxygen vacancy relax towards each other and form a strong bond. This configuration has been identified with the E′δ-center by some authors  [59, 40, 65]. The second stable configuration is the “puckered” position, where one of the oxygen atoms relaxes through the plane of its oxygen neighbors and bonds to a close oxygen atom. The puckered position has been associated with the important E′γ-center in amorphous SiO2 and to its crystalline counterpart, the E′1-center in α-quartz [57, 59, 78, 68]. Due to its involvement in different types of degradation effects and its well-studied bistability, the oxygen vacancy has been viewed as a likely candidate for the BTI defect  [26, 32].

In addition to oxygen vacancies, also defects involving silicon dangling bonds and hydrogen have received some attention. Studies showed that atomic hydrogen is produced in quartz under heavy irradiation conditions  [79, 80]. Although hydrogen is routinely used in the production process of MOS structures to passivate dangling bonds, it has been shown that exposure to hydrogen can also induce degradation  [81, 80, 82]. Particularly interesting for the present work is the hydrogen-complexed oxygen vacancy, which has been identified with the E′2 and the E′4 center in α-quartz  [56, 59]. This defect, which is also sometimes called hydrogen bridge, has been linked to hot-carrier degradation  [83] as well as stress-induced leakage currents  [84, 59]. The hydrogen bridge is also the only hydrogen-containing defect described so far that has two stable structural configurations, which correspond to the two configurations of the oxygen vacancy  [56, 59]. In the following, the states corresponding to the dimer and the puckered oxygen vacancy will be called closed and broken hydrogen bridge, respectively [85, 86]. Both the oxygen vacancy and the hydrogen bridge are illustrated in Fig. 1.8.

Unfortunately, electron spin resonance measurements can only give hints on the atomic structure of the defect and electrical measurements can only show the capture and emission behavior of the defects under different operation conditions. Thus from the experimental findings up to this point no conclusion can be drawn regarding the microscopic origin of the BTI. On the modeling side, the great success of the multi-state NMP model for BTI in the explanation of experimental data is somewhat spoiled by the large set of parameters and the unknown defect structure. The model parameters are usually determined by calibration to measurement data. Especially for the modeling of BTI and RTN, however, this method is somewhat unsatisfactory as the measurements show a broad spread in transition rates  [34, 87, 31], which requires assumptions on the statistical distribution of the defect parameters. Some insight in the behavior of single defects is gained from the TDDS [31]. However, the measurement of TDDS spectral maps is a quite demanding task and few devices have been analyzed using this technique up to now. Thus, there is not yet enough data available to give an estimate about the statistics of the observed defects.

The goal of the present work is to extend our modeling efforts to the atomic scale, in order to make the prediction of the BTI-related parameters from an atomistic model of a point defect possible. The methods developed and described in this work can be used to evaluate atomistic models of defect candidates against parameters obtained from calibrations to experimental data. In order to find an atomistic model of the BTI defect, different defect candidates can then be evaluated against the available parameter sets. A suitably designed atomistic model of the amorphous MOS oxide could then be used to study the statistics of the BTI defect and make predictions on its stability and its dependence on processing conditions.

At first, we study the foundation of our BTI model in the framework of physical chemistry to get a detailed understanding of the microscopic processes behind the states and transitions described above. This includes an attempt to shed some light on the huge amount of literature available for multi-phonon transitions. In Chap. 4, the microscopic description of the point-defect is put into the context of a semiconductor device and a multi-scale modeling method is developed, which combines the description of the defect at the microscopic level with a macroscopic model of the MOS structure. At the time this document is written, the search for a defect candidate that explains the behavior seen in BTI experiments is still ongoing. For the reasons stated above, we study the behavior of atomistic models of the oxygen vacancy and the hydrogen bridge in crystalline SiO2 as an example. The discrepancies arising from these models with respect to BTI experiments are pointed out and possible directions for a future search of the BTI defect are given.