To describe the impact of defects present within a device on its function and behavior, the charge trapping kinetics of the defects and their interaction with the device have to be modeled. For this, empirical and physics based models have been proposed. While empirical models typically rely on simple analytic formulas which are used to describe the measurement data, physics-based approaches aim at the explanation of the behavior of single defects and describe the observed behavior by the superposition of the impact of many such defects. While the former is a more straight-forward approach and much faster to calculate, it lacks in accuracy and secondary effects like, e.g. the saturation of the drift of the threshold voltage with increasing stress time, are usually not considered in such models. Thus, physics-based approaches are preferred for the scientific investigation of the defects, and will be used this work.
In the first section of this chapter, models describing the trapping behavior of individual defects are discussed. Section two then outlines how defects can be studied and used in computer simulations. Finally, in section three, the defect centric model, which phenomenologically describes the statistical distribution of the threshold voltage shift observed in an ensemble of devices, is discussed.
Charge trapping models aim to describe the charging and discharging dynamics of individual defects. Common among all defect models which are discussed within the scope of this work is the usage of a continuous-time Markov chain [46] to mathematically describe the defect. The fundamental difference between the models is in the interpretation of the transition rates between the individual states, as well as in the number of possible states in the Markov chain. Modeling the defects using Markov chains implies that the Markov property is assumed for the defects. This is that the defects retain no memory of their past and their behavior depends solely on the state they are currently in—they are memoryless. Makrov chains consist of a finite number of states, in any of which the defect has to be at any time. Mathematically, the Markov chain can be described in terms of probabilities. The probabilities of being in any of the states is given by the state vector . It follows that the expectation values of being in any state have to sum up to unity:
The transition rates from state i to j are and can be written as the transition matrix . An example of a Markov chain in equilibrium is shown in Figure 3.1.
The temporal evolution of can be described by the so-called master-equation [47]:
This equation contains the individual transition rates between the states, which are described by the physical defect model. By assigning the diagonal elements in , related to the dwelling times, i.e. the time that no transition occurs,
the master equation can be written in vector form:
with being the transpose of . With the general mathematical formulation at hand, a link between its parameters and the physical behavior of the defects has to be established. Thus, in the following the physical models describing the states in the Markov chain and the transition rates between them are discussed.
First proposed in 1952, the Shockley–Read–Hall (SRH) model [48] provides a statistical description for the recombination of electrons and holes in a semiconductor through a recombination center in the band gap. In this model, defects are assumed to have a charged and a neutral state. Defects which are electrically negative in their charged state are termed acceptor-like traps and defects which can be positively charged are termed donor-like traps. However, the defects are also frequently called electron- or hole-traps, depending either on their charge in the charged state, or with which band they primarily interact with^{1}. The model itself distinguishes between four separate processes, as shown in Figure 3.2:
• Hole capture: A hole moves from the valence band to the trap. This is equivalent to an electron moving from the trap to the valence band and recombines with a hole there.
• Hole emission: A hole moves from the trap to the valence band. This is equivalent to an electron moving from the valence band to the trap, thereby generating a hole there.
• Electron capture: An electron moves from the conduction band to the trap.
• Electron emission: An electron moves from the trap to the conduction band.
Using the hole capture process as an example, the capture probability for a defect located within the band gap is modeled simply as the product of the thermal velocity of the carriers in the reservoir and a capture cross section :
For the thermal velocity, is the Boltzmann constant, the lattice temperature, and the effective mass of the carrier. From this, the capture rate can be obtained by integrating over the interacting band:
Here, is the hole occupancy probability, given by the Fermi-Dirac distribution in thermal equilibrium, and the density of states. Similarly, the emission rate can be expressed as
Using and the property of Fermi-Dirac statistics
with leads to
Assuming thermal equilibrium, the principle of detailed balance must hold in equilibrium, which means that the probability of the defect being neutral and capturing a carrier at a specific energy must be balanced with the probability of it being charged and emitting a carrier at that energy:
With this, the emission probability can be related to the capture probability and the energy of the captured carrier :
If Boltzmann statistics hold, can be expressed as
with being the valence band weight. Combining Equation (3.9) and Equation (3.11c), yields
Finally, by approximating all carriers to be located at the valence band edge, which is known as band edge approximation [49], and substituting Equation (3.12) in yields simple analytical expressions for the charge capture and emission rates:
The model is frequently used to model interface and bulk defects, but has also been extended for oxide defects. To model defects located in the oxide, a Wentzel–Kramers–Brillouin (WKB) tunneling coefficient is commonly included in to account for elastic tunneling between the defect and the carrier reservoir(s), i.e.
The transmission coefficient for a trapezoidal potential barrier can be written as [50]:
with the start and end potentials and of the barrier, the electric field and the effective carrier mass in the oxide . The issue with using this model to describe oxide defects is that its neither able to correctly describe the bias-dependence nor the temperature-dependence of the defects observed in measurements. As compared to experiments, calculated transition times are often magnitudes too small. To model defects located energetically above or below the band gap, modified equations have to be used [50].
^{1} Note that this leads to some ambiguity in terminology, as an electron trap emitting to the valence band and a hole trap capturing from the valence band describe the same process, for example.
It was found that the SRH model does not adequately describe the temperature dependence observed in measurements of oxide defects. An extended approach to explain charge trapping at oxide defects has been developed by Kirton and Uren. They realized that the model has to account for the structural relaxation of the defects and they supposed that the thermal barriers required for the description stem from multiphonon processes. To account for this, they incorporated a phenomenological Boltzmann factor in the effective cross section [52, 51]:
Here, is the capture barrier of the defect responsible for the thermal activation, as illustrated in Figure 3.3. This additional term allows to model the temperature dependence of 1/f noise and the average charge transition times. However, it introduces a correlation between capture and emission time which can not be observed in measurements. Further, the barrier is independent of gate bias and thus still does not correctly describe bias dependence of charge trapping.
To describe the bias dependence as observed in measurements the phenomenological energy barrier as introduced by Kirton and Uren is inadequate, and has to be replaced by one calculated in dependence on the effective trap level of the defect which is shifted with the gate bias. Thus, the model has been extended accordingly to the non-radiative multi-phonon (NMP) model. The NMP model [53, 54, 55] is based on the physical description of the system consisting of all electrons and nuclei involved in the charge transition.
Due to the difference in timescales at which the electrons and nuclei act, the transition can be split into an electronic and a vibrational part using the Born-Oppenheimer approximation [56]. In this approximation, the nuclei move in a potential given by their positions, the adiabatic potential energy. Local minima of this surface correspond to the previously mentioned states in the abstract Markov-chain description. The transition from one state to another is associated with a thermal barrier which is determined by a transition state on the potential energy surface (PES), i.e. the highest point along the minimum energy transition path. While the potential energy surfaces can be calculated using DFT simulations, they are not directly accessible experimentally. To obtain a usable model, the situation is thus simplified. The multi-dimensional PES is reduced to one dimension along a reaction or configuration coordinate , described by the lowest energy path between the states. This PES is then replaced by a Taylor expansion around the minimum, truncated after the quadratic term, which enables a description of the defect states as harmonic oscillators. This results in a configuration coordinate diagram with parabolic potential energy curves as shown in Figure 3.4.
Following from the Born-Oppenheimer approximation, the Frank-Condon principle [57, 58] and Fermi’s golden rule [59], the transition rate from one state to the other can be written as [60]
with the electronic matrix element between the initial and final electronic states and the line shape function . The electronic matrix element cannot be calculated^{2} for the systems concerned, and is thus commonly approximated using a WKB tunneling factor together with a prefactor. The line shape function can be calculated, but is commonly approximated by neglecting nuclear tunneling [63] below the highest point in the minimum energy path. In this classical limit [64], transitions happen at the intersection point of the PES and the line shape function is written as a Dirac function at the energy of intersection. Thus, the energy barrier between the two states can be found by the intersection of the barriers as shown in Figure 3.4. In the semi-classical regime—i.e. if tunneling below the barrier can be neglected—the transition can be modeled using a Boltzmann factor using this barrier energy.
With all these simplifications the model effectively reduces to two parabolas, which needs to be parametrized. A commonly used parametrization of the parabolas consists of
• the ratio of curvatures ,
• the relaxation energy with the Huang-Rhys factor ,
• the ground state energies and .
Note that this parametrization eliminates the reaction coordinate between the states, which would be necessary for calculating nuclear tunneling but is not needed in the semi-classical approximation. From these parameters, and using , the energy barrier from state 1 to 2 can be calculated as [50]
Note that for , Equation (3.20) has a singularity, and in this case the energy can be calculated as
And finally the backward barrier can be calculated from to
Using again the valence band as an example, and including the band edge approximation, leads to the final capture and emission rates for the two state NMP defect model:
^{2} Recent works [61, 62] propose a methodology to do so using DFT, for this, however, the detailed atomic structure of the defect has to be known.
Single defect measurements, such as RTN and TDDS measurements have shown that individual defects can exhibit more than one characteristic dwelling time in a single charge state. When RTN signals are studied, this behavior can be observed as anomalous RTN (aRTN), where periods of inactivity of the defect follow periods of RTN signals [22]. In TDDS measurements it has further been observed that individual defects seem to disappear for a number of measurements and later reappear [65]. These observations, together with knowledge gained from DFT calculations [45, 55] for suitable defect candidates have led to the introduction of the four-state defect model. In this model, the Markov chain describing the defect has a meta-stable and a stable state for both the charged and neutral charge state, as shown in Figure 3.5a.
The transitions between the neutral and charged states, i.e. and , are modeled as NMP transitions, while transitions between states of the same charge are modeled by purely thermal barriers of constant height. The resulting energy profiles along the multiple reaction paths of such a defect are schematically illustrated in Figure 3.5b.
In this picture, a defect can transit between its charged and neutral states using two separate pathways. This enables the description of complex capture and emission time behavior as observed in RTN and TDDS measurements. In particular, this enables to distinguish between fixed and switching traps [66]. Fixed traps show bias-dependent capture times, but at low gate bias the emission time can become bias-independent. This behavior can be explained by them using the 1–2–2 path for both capture and emission. In this case, the emission time is determined mainly by the bias-independent barrier between the states 2 and 2. Switching traps on the other hand use the same 1–2–2 path for charge capture, but the 2–1–1 path for emission, which yields bias-dependent rates for both processes. Using this model the charge trapping kinetics of a number of defects extracted from planar Si MOS devices [50, 67, 68], from devices employing high-k gate stacks [69], but also defects extracted from 2D devices [BSJ6] has been explained.
While the four state model seems to be able to describe the recoverable component of BTI, measurements also reveal a smaller “permanent” degradation of the threshold voltage of the devices [70, BSC3]. The defects which are considered responsible for this permanent component can not be explained using the four state model. However, to explain the permanent component of BTI a hydrogen release mechanism has been recently proposed [70]. In silicon devices, hydrogen is purposely introduced during manufacturing to passivate dangling bonds at the Si-SiOinterface. It is thus readily available in varying concentrations throughout the device. Various studies have suggested possible interaction between hydrogen and interface or near interface states even after passivation [71, 72, 73, 74]. With two of the suspected defect candidates—the Hydroxyl E center and the Hydrogen-bridge—incorporating hydrogen, it has recently been suggested [74] that such defects may be activated or disabled by available atomic hydrogen during operation. The basic idea of the hydrogen release model [70, BSC4] relies on this statement and assumes that hydrogen-related defects may be able to surrender their hydrogen atom in the neutral state. This hydrogen can then activate preexisting defects at other locations in the oxide or create additional defects at the interface by depassivating Si dangling bonds, i.e. . In addition, more hydrogen might be released from the gate into the oxide at harsh stress conditions, thereby increasing the number of active defects in the oxide or at the interface. An illustration of the model is shown in Figure 3.6. Due to the high hopping rate of atomic hydrogen in SiO [75] the hydrogen in the oxide is available to all precursors. Together with low barriers for hydrogen capture [41], this leads to a system governed by the release of hydrogen from neutral defects or the gate.
For a mathematical description of the model, the very low diffusion barrier of interstitial hydrogen can be approximated as zero. This allows to treat each material as a single reservoir for free hydrogen. The total amount of hydrogen, which is assumed to stay constant, is thus the sum of the hydrogen in the gate reservoir , the interstitial hydrogen in the oxide and hydrogen trapped at each defect site :
The exchange between the gate reservoir and the oxide can be formulated as a rate equation:
with temperature dependent diffusion rates and to and from the oxide, respectively. To simplify the implementation, the defects may be decoupled from the hydrogen system. For this, the defects are implemented by using a regular four-state defect model and a probability of possessing hydrogen is assigned to each defect (or defect site). The expectation value of hydrogen atoms at the defect site can then be calculated using , the probability of the defect being in the stable neutral state, as
with hydrogen capture and emission rates and , and the maximum number of hydrogen at the defect site ( for single defects).
The effective charge impacting the device is determined by the probability of the defect to be active (to possess hydrogen) and the probability of the active defect to be in a charged state. Thus, to calculate the charge of the defects or defect sites, their probabilities of being charged need to be weighted by their expectation value of attached hydrogen:
To give an example, in Figure 3.7 results of a long term experiment characterizing the permanent component of the degradation of a pMOS transistor are shown, together with the results simulated with the hydrogen release model. During the experiment the gate voltage has been cycled between −1.5 V and 0 V, and also the temperature has been varied. As can be seen, the degradation saturates at 300 °C, but increases again at the next stress phase of 350 °C due to release of hydrogen from the gate. This temporal behavior can be explained by the proposed hydrogen release model.