Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

3.3 4-State Nonradiative Multiphonon Model

The 2-state NMP theory is very efficient for describing BTI on large-area devices, where large ensembles of defects are simultaneously active. However, when looking at the single defect level (RTN, TDDS), there are certain phenomena which can not be explained with 2 states. A prime example for failing of the 2-state model is the observation of anomalous RTN (aRTN), in which the RTN signal is interrupted for longer periods of time [118]. This behavior can only be explained by introducing an additional third state [194]. As can be seen in Fig. 3.14, a defect captures and emits a charge with a high frequency transitioning between state 1 and 2. After a certain time, a transition between state 2 and \( 2^\prime \) happens and the defect stays for a comparably long period in state \( 2^\prime \).

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Figure 3.14: Anomalous RTN can be explained by introducing an additional third state.

This observation combined with the knowledge from DFT simulations (see Section 2) motivates the introduction of the 4-state NMP model. Additionally to the two stable states 1 and 2 which occur in 2-state NMP theory, two additional meta-stable states \( 1’ \) and \( 2’ \) are introduced. A direct transition from one stable state to the other stable state is only possible via one of the two meta stable states \( 1\rightleftharpoons 1’ \rightleftharpoons 2 \) or \( 1\rightleftharpoons 2’ \rightleftharpoons 2 \), see Fig. 3.15. Analogously to the stable states, one-meta stable state is neutral while the second meta-stable state is charged. The transition between a neutral and a charged state, which would be the transitions \( 1’\leftrightharpoons 2 \) and \( 1\leftrightharpoons 2’ \) in Fig. 3.15 are modeled using NMP transition rates, while transitions between a state and the meta-stable state with the same charge \( 1\leftrightharpoons 1’ \) and \( 2\leftrightharpoons 2’ \) are thermal transitions which are described within classical transition state theory leading to transition rates of the form \( k_{11’}=\nu _0\mathrm {e}^{-\beta \varepsilon _{11’}} \) where \( \varepsilon _{11’} \) is a constant, i.e. gate voltage independent barrier.

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Figure 3.15: The state diagram for the 4-state NMP model shows two stable states \( 1 \) and \( 2 \) and two meta-stable states \( 1’ \) and \( 2’ \). A charge transition between the stable states is only possible via the meta-stable states. A transition between a two states with the same charge (\( 1\leftrightharpoons 1’ \) and \( 2\leftrightharpoons 2’ \)) corresponds to a structural relaxation while a transition with a charge exchange (\( 1\leftrightharpoons 2’ \) and \( 2\leftrightharpoons 1’ \)) can be modeled as NMP transition. The corresponding configuration coordinate diagram (right) shows the potential energy surfaces for the 4-state model.

Using the more complex potential energy surfaces shown in Fig. 3.15 allows to express the four NMP and the four thermal transition rates as shown in [174]. This rates allow obtaining the overall capture and emission time from one stable state to the other. Using Markov theory, the probability \( p_i \) of a defect to be in state \( i \) can be derived enabling reliability simulations.