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7 Modeling of Bias Temperature Instabilities

As discussed in previous chapters, BTI has a detrimental impact on the transistor characteristics by (i) shifting the threshold voltage \( \Vth   \), and reducing (ii) the device transconductance, (iii) the device saturation current and (iv) the channel mobility. As a consequence, increasingly large delays are introduced in high performance CMOS circuits which affect the timing of the transistors. Furthermore, defects can act as TAT centers and thus endanger the reliable operation of SRAM devices. In order to correctly understand the experimental data and to be able to provide meaningful lifetime predictions, a thorough and systematic theoretical analysis of the experimental data is required. Therefore, a variety of theoretical models for BTI have been suggested, starting from empirical fit expressions over more or less accurate compact models up to detailed TCAD models.

Because the impact of BTI on a transistor is typically expressed in terms of an equivalent threshold voltage shift, models developed around these instabilities rely on reproduction of the (math image) measured for different stress and recovery biases and at different temperatures. Furthermore, the threshold voltage shift caused by BTI is observed to be the superposition of a recoverable and permanent component. Thus a suitable model necessarily has to capture both contributions to the measured threshold voltage shift.

In the following, empirical models which are due to their simplicity regularly used to describe BTI are presented. As empirical descriptions can not reflect the physics behind BTI, namely the trapping kinetics of single defects, modeling of single defects is introduced next. Finally, recent advances on how to properly explain the permanent threshold voltage shift via the hydrogen release (HR) model are discussed.

7.1 Empirical Models

Empirical models provide the simplest description for experimental data and are often used if the detailed mechanism behind an observation is not known. In the context of device physics experimental data can often be modeled using a power law or exponential-like functions [90, 91]. The advantage of such empirical formulations is that they provide a fast and simple method to quantify the results with a reduced parameter set. Although such models are used to compare different technologies they have to be treated with care as they do not provide a physics-founded description of the underlying mechanism and thus extrapolations may be inaccurate.

Next, the BTI modeling using a power law is briefly discussed, followed by the reaction-diffusion (RD) model and its extensions.

7.1.1 Power Law based modeling of Bias Temperature Instabilities

At its simplest, the impact of BTI on the device threshold voltage of large-area devices can be expressed by comparing the (math image) characteristics before and after the transistors have been subjected to BTI stress, see Figure 7.1.

(-tikz- diagram)

Figure 7.1:  (left) For pMOSFETs subjected to NBTI stress the (math image) characteristics are shifted towards larger absolute threshold voltages with increasing stress times. To determine the threshold voltage shift, the gate voltage leading to a defined drain-source current is calculated for all stress times. (right) The stress bias dependence of the threshold voltage shift can be modeled using a power law.

As can be seen, the larger the stress time gets, the larger the observed threshold voltage shift will be. The observed threshold voltage shift can be described by a power law in time

(7.1) \{begin}{align}   \dVth = A\tS {n} \{end}{align}

with \( A \) being a stress voltage dependent prefactor and \( n \) the power law exponent in the range of 0.1 and 0.25 [92]. The prefactor itself is stress bias dependent and given by

(7.2) \{begin}{align}   A \propto \Eox ^\gamma \approx \left (\frac {V_{\mathrm {ov}}}{\tox }\right )^\gamma \{end}{align}

with the overdrive voltage \( V_{\mathrm {ov}}=|\VGStress -\Vth | \) and the exponent \( \gamma   \) typically in the range of 2.5 to 3 for NBTI in Si devices [92]. Furthermore, the temperature dependence follows an Arrhenius’ Law

(7.3) \{begin}{align}   \dVth \propto \myexp ^{-\beta \EA } \{end}{align}

with the activation energy \( \EA   \) reported to be in the range \( \SI {60}{}-\SI {80}{\milli \electronvolt } \) [93]. The combination of all three equations leads to the simple analytic expression

(7.4) \{begin}{align}    \dVth = A \left (\frac {\VGStress -\Vth }{\tox }\right )^\gamma \myexp ^{-\beta \EA } \tS {n} \{end}{align}

for the threshold voltage shift. Based on the above equation for instance the device lifetime can now be estimated. However, an extrapolation based on such an empirical description is usually very inaccurate. To provide a more accurate explanation for BTI a physically correct BTI model is needed. The most promising approach appears to be based on the study of single defects, which much more clearly reveals the underlying physics. For this, nanoscale devices have to be used as they allow to study single defects individually.

7.1.2 Reaction-Diffusion Model

For a long time the reaction-diffusion (RD) model was the most successful model to explain NBTI observed in large-area devices [94]. The RD model assumes that interface states, which are dangling bonds present at the Si/(math image) interface, are the culprit for threshold voltage shifts introduced during normal device operation. During the fabrication process interface states can be passivated by hydrogen (H) thereby forming a Si-H bond. However, in the RD model it is assumed that this formation can capture a hole which weakens the Si-H bond. As a consequence the H can be released from the Si atom leaving an unsaturated, electrically active bond behind. The hydrogen itself is then assumed to diffuse into the gate dielectric. Conversely, hydrogen atoms can turn around to passivate the dangling Si bond and neutralize the interface state.

By analyzing (math image) characteristics under different stress conditions [94] observed that the creation of interface states follows a power law of the form \( \Nit \propto t^{1/4} \). This behavior was then described by considering passivation and depassivation of the Si dangling bonds according to [94]

(7.5) \begin{equation} \frac {\partial \Nit }{\partial t} = \kf (\NO -\Nit )-\kr \Nit \Hit \label {equ:RDoldA} \end{equation}

with (math image) and (math image) the forward and backward rate, respectively, (math image) the interface state density and (math image) the interfacial hydrogen concentration. Note that in this one-dimensional mathematical framework the hydrogen concentration is a function of \( H(x, t) \) and \( \Hit =H(0,t) \). The diffusion of released hydrogen is described by

(7.6) \begin{equation} \frac {\partial H}{\partial t} = -D\frac {\partial ^2 H}{\partial x^2} \label {equ:RDoldB} \end{equation}

with \( D \) the diffusion coefficient.

Nearly forty years little attention was paid to BTI until the continued scaling of the transistor geometries reached the sub-micrometer regime. During that time the RD model was used to describe BTI.

As the impact of BTI on (math image) was revisited using more detailed stress/measure sequences, the threshold voltage shift was found to relax immediately after stress release and the relaxation was observed to continue for very long recovery times. To describe both observations, a modified RD model was proposed [95, 96, 97, 98, 99]. In this modified model the one-dimensional motion of hydrogen was replaced by three-dimensional hydrogen distribution and diffusion. Furthermore, the diffusion H and H\( _2 \) molecules and their inter-conversion is considered. Based on (7.5) and (7.6) the modified RD model is described by

(7.7–7.10) \{begin}{align}   \frac {\partial \Nit }{\partial t} & = \kf (\NO -\Nit )-\kr \Nit \Hit \\ \frac {\partial H} {\partial t} & = -D\frac {\partial ^2 H}{\partial x^2} -\kH H^2 + \kHtwo H_2\\ \frac {\partial
H_2}{\partial t} & = -D_2\frac {\partial ^2 H_2}{\partial x^2} + \frac {\kH }{2} H^2 - \frac {\kHtwo }{2} H_2\\ \{end}{align}

with (math image) and (math image) the reaction rates for dimmarization and atomization of hydrogen, respectively. However, closer inspection revealed that not even this extended RD model can describe the dynamics of BTI [100, 101, 102, MWJ8, 103].

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