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Advanced Electrical Characterization of Charge Trapping in MOS Transistors

3.3 The Defect Centric Model

The defect centric model [81, 79] describes the effects of defects on the threshold voltage shift in a statistical manner. It is based on phenomenological descriptions of the statistical properties of defect ensembles, such as the number of defects per device and their individual contributions. This allows both to obtain defect parameters from measured distributions of changes of the threshold voltage, as well as to estimate these distributions from defect parameters, obtained for example from single defect measurements.

In the model, the number of defects per device is assumed to follow a Poisson distribution with mean \( N \), while the effect of a single defect on the threshold voltage is assumed to be exponentially distributed with expectation value \( \eta \) [81], as shown in Figure 3.8.

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Figure 3.8: Distributions of the defects per device and the step height per defect in the defect centric model. Left: The average number of defects in the devices is assumed to be Poisson distributed around a mean value \( N \). Right: The impact of the single defects on the threshold voltage shift is given by an exponential distribution with the expectation value \( \eta \). While the majority of defects (\( 1-\frac {1}{e}\approx 63\% \)) show step heights below \( \eta \), some defects produce much larger steps. The variance in both the number of defects and step height per defect underlines the fact that for small devices with few defects on average, some devices can show degradation much worse than the average device.

From this, the distribution of threshold voltage shifts in a device can be calculated [79, 82]. One core assumption of the model follows from experimental observations and is the exponential distribution of the threshold voltage shift caused by single defects:

(3.31) \{begin}{`} f(\dvth , \eta ) = \frac {1}{\eta } \expb {-\frac {\dvth }{\eta }}. \{end}{`}

Assuming the defects to be independent, i.e. their total effect on (math image) is given by their superposition, the shift of the threshold voltage for \( n \) defects can be found as the \( (n) \) autoconvolution of the contributions of the defects:

(3.32) \{begin}{`} f_n(\dvth , \eta ) = f \ast f \ast \dots \ast f = \frac {\expb {-\frac {\dvth }{\eta }}}{(n-1)!} \frac {\dvth ^{n-1}}{\eta ^n}. \{end}{`}

Alternatively, with the gamma distribution for \( p \in \mathbb {N} \)

(3.33) \{begin}{`} \gamma _{p,b}(x) = \frac {b^p}{(p-1)!} x^{p-1} \expb {-bx}, \{end}{`}

with \( p = n \) and \( b = 1/\eta \) the function yields

(3.34) \{begin}{`} f_n(\dvth , \eta ) = \gamma _{n,1/\eta }(\dvth ). \{end}{`}

The probability of having \( n \) defects in a device is given by the Poisson distribution:

(3.35) \{begin}{`} P_N(n) = \frac {\expb {-N}N^n}{n!}. \{end}{`}

The total probability density function (PDF) of the threshold voltage shifts is thus the sum of Poisson weighted Gamma distributions for \( n \) defects:

(3.36) \{begin}{`} f_N(\dvth , \eta ) = \sum _{n=1}^{\infty } P_N(n) \gamma _{n,1/\eta }(\dvth ). \{end}{`}

Using this model, simple relations between the first moments of the distribution of threshold voltage shifts of a set of devices and the statistical properties of the defects can be found for the number of defects per device:

(3.37) \{begin}{`} N &= \frac {\left <\dvth \right >}{\eta }, \{end}{`}

and for the average impact of a defect on the threshold voltage:

(3.38) \{begin}{`} \eta &= \frac {\text {Var}(\dvth )}{2\left <\dvth \right >}. \{end}{`}

The goal is to apply the compact formulas to calibrate the model to measurement data. However, distributions measured using an MSM scheme on actual devices will have additional components due to RTN and measurement noise. This can be included in the model by convoluting the BTI contribution as outlined above with contributions due to RTN and Gaussian noise, as shown in Figure 3.9, which yields

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Figure 3.9: Left: Measured distribution of (math image) after stress in comparison with the defect centric model. Center: The one-sided BTI component gets smaller after stress due to decreasing \( N \). Right: The symmetric RTN component stays constant throughout recovery.

(3.39) \{begin}{`} p(\dvth ) =~ &f_N(\dvth ,\eta ) \\ *~ &g_{N\sub {RTN}}(\dvth ,\eta ) \nonumber \\ *~ &h\sub {Noise}(\dvth ,m,\sigma ). \nonumber \{end}{`}

The RTN component is similar to the BTI component, but symmetric in (math image):

(3.40) \{begin}{`} g_{N\sub {RTN}}(\dvth ,\eta ) = &\sum _{n=0}^{\infty } \mathcal {P}_{N\sub {RTN}/2}(n) \gamma _{n,1/+\eta }(\dvth ) \nonumber \\ * ~ &\sum _{n=0}^{\infty } \mathcal {P}_{N\sub {RTN}/2}(n) \gamma _{n,1/-\eta }(\dvth ). \{end}{`}

This is because an defect showing RTN can also be charged during the initial (math image)((math image)) sweep of the measurement sequence, and become neutral after stress. The component accounting for measurement noise is a normal distribution \( \mathcal {N} \), with mean \( m \) and variance \( \sigma \):

(3.41) \{begin}{`} h\sub {Noise}(\dvth ,m,\sigma ) = \mathcal {N}(m,\sigma ^2). \{end}{`}

The method, as shown above, describes a single defect distribution. If the oxide stack consists of more than one material, e.g. in high-k devices, the model can be extended accordingly, see for example [83]. In Section 6.2, this method is applied to find the statistical behavior of defects for various stress cases in a high-k/metal gate technology.