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8.4 Step Height Distribution Function

In order to study the average contribution of single traps to the (math image) shift, the complementary cumulative distribution function (CCDF) the of step heights can be calculated from a large number of devices. The CCDF plot shown in Figure 8.19 demonstrates that the step heights are exponential distributed and can be described by the probability distribution function (PDF)

(8.7) \begin{equation} f(\dVth )=\frac {1}{\eta }\myexp ^{-\frac {\dVth }{\eta }} \end{equation}

with \( \eta \) the mean threshold voltage shift induced by a single charge.

(image)

Figure 8.19:  The step height distribution function for (math image) pMOSFETs can be described with a unimodal CCDF. From the CCDF normalized to the number of devices the active trap density can be directly obtained as the intersection between the extension of the CCDF and the ordinate.

Furthermore, the cumulative distribution function (CDF) of the step heights is given by

(8.8) \begin{equation} F(\dVth )=1-\myexp ^{-\frac {\dVth }{\eta }}. \end{equation}

To study the step height distribution, the complementary cumulative distribution function can be used and normalized to the number of devices

(8.9) \begin{equation} \mathrm {CCDF}(\dVth ) = \frac {1-F(\dVth )}{N_{\mathrm {devices}}}=\NT \myexp ^{-\frac {\dVth }{\eta }}. \label {equ:unimodalCCDF} \end{equation}

The advantage of this formulation is that the number of traps per device (math image) is directly accessible from the plots. In recent investigations exponentially distributed amplitudes have been found for RTN signals [159, 160, 161]. These findings strengthen the link between RTN and BTI [162, 122]. Furthermore, the average contribution of a single trap to the threshold voltage shift \( \eta \) plays an important role in the context of device variability in deeply scaled devices [161, 163, 164].

In pMOSFETs employing a high-k gate stack the step height distribution functions are found to be bimodal, see Chapter 12 [165, MWJ1]. Such a behavior can be described by the sum of two unimodal CCDFs given by

(8.10) \begin{equation} 1-\mathrm {CDF}=A_1\myexp ^{-\frac {\dVth }{\eta _1}} + A_2\myexp ^{-\frac {\dVth }{\eta _1}}. \label {equ:bimodalCCDF} \end{equation}

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