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

6.2 Two State Defects

The most simple case of a RTN producing defect follows a two-state Markov chain with the defect being neutral in state 1 and negatively charged in state 2, see Figure 6.2. 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. The Master equation of such a defect can be constructed from equation (6.3). For that, the chain is assumed to be in state $1$ at time $t$ . The conditional probability to go from state $1$ to $2$ is

$(6.7) \begin{equation} P\left \lbrace X( t+\dif t ) = q_2 | X( t ) = q_1 \right \rbrace = k_{12}\dif t. \eqlabel {jointCDF12} \end{equation}$

On the other hand, the probability of staying in state $2$ is

$(6.8) \begin{equation} P\left \lbrace X( t+\dif t ) = q_2 | X( t ) =q_2 \right \rbrace = 1 - k_{21}\dif t. \eqlabel {jointCDF21} \end{equation}$

The probabilities $p_1(t)$ and $p_2(t)$ to be in state $1$ and $2$ at time $t$ together with equations (6.7) and (6.8) give the probability to be in state $2$ at time $t+\dif t$ :

$(6.9) \begin{equation} p_2(t+\dif t) = (k_{12}\dif t)p_1(t) + (1 - k_{21}\dif t)p_2(t) \eqlabel {masterp2_1} \end{equation}$

After rearranging one obtains

$(6.10) \begin{equation} \frac {p_2(t+\dif t) - p_2(t)}{\dif t} = \od {p_2(t)}{t} = k_{12}p_1(t) - k_{21}p_2(t) \eqlabel {masterp2_2} \end{equation}$

At any time, the charge needs to be in one of the states, giving $p_1(t)+p_2(t)=1$ . With the same approach for $p_1(t)$ , the Master equation of the process can be written as [110]:

$(6.116.12) \{begin}{align} \displaystyle {\frac {\diff p_1(t)}{\diff t}} & = k_{21}( 1-p_1(t)) - k_{12}p_1(t) \\ \displaystyle {\frac {\diff p_2(t)}{\diff t}} & = k_{12}( 1-p_2(t)) - k_{21}p_2(t) \{end}{align}$

The solution of the Master equation describes the probability over time for the defect to be in a certain state.

With the initial probabilities $p_1(0)$ and $p_2(0)$ and the characteristic time constant $\tau = (k_{12}+k_{21})^{-1}$ of the defect it reads:

$(6.136.14) \{begin}{align} p_1(t) & = \frac {k_{21}}{k_{12}+k_{21}} + ( p_1(0) - \frac {k_{21}}{k_{12}+k_{21}}) \exp {\left ( -\frac {t}{\tau } \right )}\\ p_2(t) & = \frac {k_{12}}{k_{12}+k_{21}} + ( p_2(0) - \frac {k_{12}}{k_{12}+k_{21}}) \exp {\left ( -\frac {t}{\tau } \right )} \{end}{align}$

In the limit $t\rightarrow \infty$ , the well-known results for the equilibrium probabilities (i.e. occupancies) of the two states are retrieved:

$(6.156.16) \{begin}{align} p_1 & = \frac {k_{21}}{k_{12}+k_{21}} \\ p_2 & = \frac {k_{12}}{k_{12}+k_{21}} \{end}{align}$

The Master equation can easily be expanded to describe more complex defects, as will be shown in Section 6.3. However, in most cases one is not interested in the temporal evolution of the probabilities for being in a certain state, but rather in the PDF of the time it takes to change from one state to another. These are known as the first passage times and can also be obtained from the Master equation by setting the respective reverse rate to zero and the initial probability to be in the starting state to one. The first passage time to capture an electron is directly calculated from the Master equation:

$(6.17) \begin{equation} \displaystyle {\frac {\diff p_1(t)}{\diff t}} = -k_{21}p_1(t) \quad \Rightarrow \quad p_1(t) = \exp (-k_{21}t) \eqlabel {mastertauc_1} \end{equation}$

Equation (6.17) can be used to calculate the random variable $\tau _{12}$ , which is the point in time where the transition actually takes place. The probability of the defect already being in state $2$ at a certain time is given by $p_2(t)=1-p_1(t)$ . Therefore $\tau _{12}$ must also be smaller than $t$ .

The probability (and thus the CDF) of being in state $2$ is then given by [110]

$(6.18) \begin{equation} F(\tau _{12}) = P\lbrace \tau _c \leq t\rbrace = p_2(\tau _{12}) = 1 - \exp (-k_{21}\tau _{12}). \eqlabel {mastertauc_2} \end{equation}$

Exactly the same method can be applied to obtain the CDF of the emission time. Once the CDFs of the randomly distributed variables for capture and emission is known, the PDF can be obtained by taking the derive of the CDF. For the two-state defect, the PDF of the random variables for capture and emission $\tau _{12}$ and $\tau _{21}$ are given by [110]:

$(6.196.20) \{begin}{align} g(\tau _{12}) & = k_{12} \exp {\left ( k_{12}\tau _{12}\right )}\\ g(\tau _{21}) & = k_{21} \exp {\left ( k_{21}\tau _{21}\right )} \eqlabel {twostate_exp} \{end}{align}$

The random variables for the charge capture ( $\tau _{12}$ ) and emission ( $\tau _{21}$ ) times are thus exponentially distributed. The mean values of the observed times are given by the expectation values of the exponential distributions, which are the inverse rate constants of the process:

$(6.216.22) \{begin}{align} \bar {\tau }_{12} & = \int _0^{\infty } \tau _{12} g(\tau _{12}) \dif {\tau _{12}} = \frac {1}{k_{12}}\\ \bar {\tau }_{21} & = \int _0^{\infty } \tau _{21} g(\tau _{21}) \dif {\tau _{12}} = \frac {1}{k_{21}} \eqlabel {twostate_means} \{end}{align}$

An estimator for the mean values can be obtained directly from measurements and are commonly used to specify the characteristic time constants of defects. This is done simply by averaging over the observed capture and emission times for a single defect. Naturally, the variance and the confidence interval of this estimator become smaller the more observations are available for averaging.