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Charge Trapping and Single-Defect Extraction in Gallium-Nitride Based MIS-HEMTs

6.3 Multi State Defects

The procedure presented in the previous section can easily be generalized to more than two states. The Master equation for state \( i \) is given by the probabilities to go from state \( i \) to state \( j \) together with the probability to stay in state \( i \) [153]:

(6.236.24) \{begin}{align} P\left \lbrace X( t+\dif t) = q_j | X( t ) = q_i \right \rbrace & = k_{ij}\dif t \\ P\left \lbrace X( t+\dif t) = q_i | X( t ) = q_i \right \rbrace & = 1 - \sum _{i\neq j}{k_{ij}\dif t} \eqlabel {multistate_prob} \{end}{align}

From those probabilities, the Master equation is again obtained analogous to equation (6.10) using \( \dif t \rightarrow 0 \).

(6.25) \begin{equation} \od {p_i(t)}{t} = -p_i(t)\sum _{i\neq j}{k_{ij}} + \sum _{i\neq j}{k_{ji}p_j(t)} \eqlabel {multistate_master} \end{equation}

Since the charge has to be in one of the states, from \( N \) equations, only \( N-1 \) are independent from each other. The PDF of the first passage times for multi state defects can again be derived from the solution of the Master equation. For a three-state defect this is done in [110], while [153] gives more general approaches to derive analytic expressions for that problem in Chapter 6. In principle, the same procedure as presented in Section 6.2 can be used to to derive the equilibrium first passage times for neighbouring states, if the correct occupancies are inserted. For non-neighbouring states, the PDFs are the normalized differences of exponential distributions, where the faster defect truncates the distribution of the slower defect [110]. Since both of the extraction methods presented in Sections 6.6 and 6.7 do not rely on the knowledge of the PDFs, the analytic expressions are omitted at this point.

Typical candidates for multistate defects are three-state defects producing two or three level anomalous RTN [108, 112, 113], see also Section 4.2.1. In the case of two-level anomalous RTN as seen in Figure 6.3, the third state possesses the same charge state as the second one. Such states are attributed to a structural relaxation of the defect in the NMP four-state model and will be referred to as thermal states throughout the remainder of this work. There is no straight-forward way to determine a thermal state from the measurements with the histogram methods described in Section 6.6. With respect to Markov models they are also called tied states because they share the same emissions (and thus the same PDF for the emissions) with the state they are tied to.

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Figure 6.3: Left: The Markov chain of a three-state defect with a thermal state, producing anomalous RTN. The charge state is neutral if the defect is in state \( 1 \) and negative if in state \( 2' \) or \( 2 \). Right: Simulated emissions of the three-state defect for \( k_{12'}=k_{2'1}=\SI {10}{\per \second } \) and \( k_{2'2}=k_{22'}=\SI {0.5}{\per \second } \). Note that there is no straight-forward way to determine the thermal state directly from measurements. The simulations were done with the Hidden Markov library presented in Section 6.7, with a small amount of Gaussian noise added to the emissions.

An example for an even more complex RTN signal commonly called three-level anomalous RTN, is given in Figure 6.4. Here the Markov state \( 2 \) is charged negatively, while the states \( 3' \) and \( 3 \) are charged double-negatively. Again, the states \( 3' \) and \( 3 \) cannot be separated easily, because they share the same charge state.

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Figure 6.4: Left: The Markov chain of a three-level defect with a thermal state. The charge state is neutral if the defect is in state \( 1 \), negative if in state \( 2 \) and double negative in the states \( 3 \) and \( 3' \). Right: Simulated emissions of the three-level defect for \( k_{12}=k_{21}=\SI {2}{\per \second } \), \( k_{23'}=k_{3'2}=\SI {10}{\per \second } \) and \( k_{3'3}=k_{33'}=\SI {0.5}{\per \second } \). The simulations were done with the Hidden Markov library presented in Section 6.7, with a small amount of Gaussian noise added to the emissions.