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

6.5 Coupled Defects

This section investigates the modeling of correlated RTN from two defects which are coupled to each other. While the physical origin of the coupling mechanism is of minor importance at that point, previous studies identified possible mechanisms as the Coulomb interaction between the defects or percolation path effects due to random dopants [AGC5][112, 113, 156, 157]. The RTN traces of such systems can help to determine empirical coupling factors, which then can be validated using TCAD simulations.

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Figure 6.5: The Markov chains of a system of coupled two-state defects. The capture and emission rates of defect \( \mathrm {A} \) are multiplied by \( \gamma _\mathrm {c} \) and \( \gamma _\mathrm {e} \) if defect \( \mathrm {B} \) captured a charge.

To determine the behaviour of coupled defects, the most basic case of two two-state defects \( \mathrm {A} \) and \( \mathrm {B} \) as seen in Figure 6.5 is chosen, but can be used for any type of Markov chain. The dashed line denotes the coupling with the factors \( \gamma _\mathrm {c} \) for electron capture and \( \gamma _\mathrm {e} \) for electron emission. The individual transition matrices of the two defects are:

(6.32) \begin{equation} \underline {\bm {k}}_\mathrm {A} = \begin{bmatrix}[1.5] 1-k^\mathrm {A}_\mathrm {12} & k^\mathrm {A}_\mathrm {12} \\ k^\mathrm {A}_\mathrm {21} & 1-k^\mathrm {A}_\mathrm {21} \end {bmatrix}, \quad \underline {\bm {k}}_\mathrm {B} = \begin{bmatrix}[1.5] 1-k^\mathrm {B}_\mathrm {12} & k^\mathrm {B}_\mathrm {12} \\ k^\mathrm {B}_\mathrm {21} & 1-k^\mathrm {B}_\mathrm {21} \end {bmatrix} \eqlabel {multi_coupled_init} \end{equation}

The transition matrix of the system is then given by:

(6.33) \begin{equation} \underline {\bm {k}} = \underline {\bm {k}}_\mathrm {B} \otimes \underline {\bm {k}}_\mathrm {A} = \begin{bmatrix}[1.5] (1-k^\mathrm {B}_{12}) \cdot \underline {\bm {k}}_\mathrm {A} & k^\mathrm {B}_{12} \cdot \underline {\bm {k}}_\mathrm {A} \\ k^\mathrm {B}_{21} \cdot \underline {\bm {k}}_\mathrm {A} & (1-k^\mathrm {B}_{21}) \cdot \underline {\bm {k}}_\mathrm {A} \end {bmatrix} \eqlabel {multi_coupled} \end{equation}

Let’s assume that if for example defect \( \mathrm {B} \) captures an electron, the capture and emission rates of defect \( \mathrm {A} \) change. The modified transition matrix of defect \( \mathrm {A} \) with the coupling factors \( \gamma _\mathrm {c} \) and \( \gamma _\mathrm {e} \) then is [AGJ6]:

(6.34) \begin{equation} \underline {\bm {k}}_\mathrm {A}' = \begin{bmatrix} 1-\gamma _\mathrm {c}k_\mathrm {12}^\mathrm {A} & \gamma _\mathrm {c}k_\mathrm {12}^\mathrm {A} \\ \gamma _\mathrm {e}k_\mathrm {21}^\mathrm {A} & 1-\gamma _\mathrm {e}k_\mathrm {21}^\mathrm {A} \end {bmatrix} \eqlabel {multi_coupled_Aprime} \end{equation}

Finally, the modified transition matrix \( \underline {\bm {k}}_\mathrm {A}' \) has to be inserted into (6.33) for the states where \( \mathrm {B} \) has captured an electron. The transition matrix of the coupled system then becomes:

(6.35) \begin{equation} \underline {\bm {k}} = \begin{bmatrix}[1.5] (1-k^\mathrm {B}_{12}) \cdot \underline {\bm {k}}_\mathrm {A} & k^\mathrm {B}_{12} \cdot \underline {\bm {k}}_\mathrm {A} \\ k^\mathrm {B}_{21} \cdot \underline {\bm {k'}}_\mathrm {A} & (1-k^\mathrm {B}_{21}) \cdot \underline {\bm {k'}}_\mathrm {A} \end {bmatrix} \eqlabel {multi_coupled_final} \end{equation}

Figure 6.6 illustrates the differences between an uncoupled system and systems with various different coupling factors for electron capture and emission rates.

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Figure 6.6: The RTN signals of a system with two coupled two-state defects according to Figure 6.5. The two defects \( \mathrm {A} \) and \( \mathrm {B} \) have step heights of 2 mV and 5 mV respectively. When the slow defect \( \mathrm {B} \) captures an electron, the emissions rates of defect \( \mathrm {A} \) are changed by a factor of \( \gamma _\mathrm {c,e} \). If the coupling factors \( \gamma _\mathrm {c} \) for capture and \( \gamma _\mathrm {e} \) are different from each other, the average occupancy between the coupled and uncoupled emissions changes. On the other hand, if \( \gamma _\mathrm {c}=\gamma _\mathrm {e} \), the average occupancy is unchanged but the capture and emission times will change accordingly.

This method will be used in Section 7.2.4 to estimate the electrostatic coupling factors for correlated RTN signals recorded on GaN/AlGaN fin-MIS-HEMTs.