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

6.4 Multiple Defects

If several different defects are observed in measurements, additional steps are needed to construct the resulting Markov chain. The transition matrix of the system is then given by the Kronecker product of the individual defects (see [155]). For $N$ independent defects, the resulting transition matrix is:

$(6.26) \begin{equation} \underline {\bm {k}} = \underline {\bm {k}}_\mathrm {N} \otimes \underline {\bm {k}}_\mathrm {N-1} \otimes \dots \otimes \underline {\bm {k}}_\mathrm {1} \eqlabel {multi_k} \end{equation}$

Note, that the size of the system matrix equals the product of the matrix sizes of the individual defects. This can be a serious problem in terms of computation time, because the number of different states quickly grows to impractical sizes (for four three-state defects, the resulting system would already have $N=3^4=81$ different states). The number of different levels of each defect $j$ when changing its charge only depends on its electrostatic influence on the device (usually given in terms of (math image) or ) and the number of states with a charge transfer.

$(6.27) \begin{equation} \bunderline {\bm {V}}_\mathrm {th}^j = \lbrace q_i\cdot \dVth \rbrace ,\quad i = 1\dots (n-n_\mathrm {th})\eqlabel {multi_dVth_individual} \end{equation}$

Here, $\underline {\bm {V}}_\mathrm {th}^j$ is the vector of different charge levels for defect $j$ , (math image) is the electrostatic influence of one charge, $n$ is the number of states of the defect, $n_\mathrm {th}$ is the number of thermal states, and $q_i$ is the absolute charge state of the defect in state $i$ . Note that the levels for the thermal states correspond to the states they are tied to. That means that for each thermal state, a duplicate entry has to be created at the correct position in (6.27) so that the size of the set is again $n$ .

As an example, for a three-state defect without thermal state, the levels for $\dVth =\SI {3}{mV}$ would be $\bunderline {\bm {V}}_\mathrm {th}=\lbrace \SI {0}{mV},\ \SI {3}{mV}, \SI {6}{mV}\rbrace$ . On the other hand, the defect depicted in Figure 6.4 would possess the (math image) vector $\bunderline {\bm {V}}_\mathrm {th}=\lbrace \SI {0}{mV},\ \SI {3}{mV}, \SI {6}{mV},\ \SI {6}{mV}\rbrace$ .

To determine the corresponding levels for a system of $N$ defects according to equation (6.26), the Cartesian product of the sets defined in (6.27) can be used.

$(6.28) \begin{equation} \bunderline {\bm {V}} = \bunderline {\bm {V}}_\mathrm {th}^1 \times \bunderline {\bm {V}}_\mathrm {th}^2 \times \dots \times \bunderline {\bm {V}}_\mathrm {th}^N \eqlabel {multi_dVth_cartesian} \end{equation}$

Since the resulting subsets in $\bunderline {\bm {V}}$ contain the corresponding levels of all defects for every state in (6.26), the final step is to sum over all the entries in each subset to obtain the vector $\bunderline {\bm {V}}_\mathrm {th}$ with the actual voltage levels for all the states in $\underline {\bm {k}}$ (the indices here mark the $N^\mathrm {th}$ subset).

$(6.29) \begin{equation} \bunderline {\bm {V}}_\mathrm {th} = \bigg \lbrace \sum V_1,\sum V_2,\dots ,\sum V_\mathrm {N}\bigg \rbrace \eqlabel {multi_dVth_final} \end{equation}$

To clarify the construction of the (math image) levels in (6.27) – (6.29), an example of a system consisting of a two-state defect with $\dVth =\SI {5}{mV}$ and the three-state defect from Figure 6.4 with $\dVth =\SI {5}{mV}$ should be given.

Inserting $\bunderline {\bm {V}}_\mathrm {th}$ of the defect into (6.28) gives

$(6.30) \{begin}{align} \bunderline {\bm {V}} = & \big \lbrace \SI {0}{mV},\ \SI {5}{mV}\big \rbrace \times \big \lbrace \SI {0}{mV},\ \SI {3}{mV},\ \SI {6}{mV},\ \SI {6}{mV}\big \rbrace \notag \\ = & \big \lbrace [\SI {0}{mV},\ \SI {0}{mV}],\ [\SI {0}{mV},\ \SI {3}{mV}],\ [\SI {0}{mV},\ \SI {6}{mV}],\ [\SI {0}{mV},\ \SI {6}{mV}],\notag \\ &\qquad [\SI {5}{mV},\ \SI {0}{mV}],\ [\SI {5}{mV},\ \SI {3}{mV}],\ [\SI {5}{mV},\ \SI {6}{mV}],\ [\SI {5}{mV},\ \SI {6}{mV}]\big \rbrace . \eqlabel {multi_dVth_example} \{end}{align}$

Performing the summation for every sub-vector in (6.30) delivers the final defect levels of the combined system:

$(6.31) \{begin}{align} \bunderline {\bm {V}}_\mathrm {th} = & \big \lbrace \SI {0}{mV},\ \SI {3}{mV},\ \SI {6}{mV},\ \SI {6}{mV},\ \SI {5}{mV},\ \SI {8}{mV},\ \SI {11}{mV},\ \SI {11}{mV}\big \rbrace \eqlabel {multi_dVth_example2} \{end}{align}$

In principle, the combined transition matrix and the combined levels derived above are sufficient to train a HMM to find the most likely combined system (see Section 6.7). However, the entries in the combined transition matrix of the individual defects are not independent from each other. That means that two independent defects would converge towards one single defect with the combined number of states, which is of course unphysical (see Section 6.7.3). Another constraint is given by the (math image) levels of the defects depicted in (6.27) as they can only be multiples of the elementary charge for each state. As with the transition matrix for the combined system, the entries of $\underline {\bm {V}}_\mathrm {th}$ in (6.29) are also not independent of each other.

These two constraints require modifications for the parameter update in the HMM training which will be explained in Section 6.7.3.