# Data-Driven Modeling of Low Frequency Noise Using Capture-Emission Energy Maps
## 2 Theoretical Model  
The capture-emission process of charge carriers is consistent with non-radiative multi-phonon theory based on linear electron-phonon coupling [1]. In this theory, the capture-emission times are expressed as \(\tau_{c,e}(E_{c})=\tau_{0}\exp(E_{c,e}/k_{b}T)\) where \(\tau_{0}\) is the temperature-independent pre-factor, \(k_{b}\) is the Boltzmann constant, \(T\) is the temperature, and \(E_{c}\) andare the capture and emission activation energy, respectively. The NBTI-induced threshold voltage shifts \(\Delta V_{th}\) is related to the capture-emission processes of individual defects with a broad distribution of characteristic times. The step height observed in \(\Delta V_{th}\) for small-area devices is due to individual defects, while for large-area devices similar defects are grouped together using a suitable distribution of defects [1, 19]. This distribution is visualized in the CEE map, which can be directly obtained from experimental recovery curves of \(\Delta V_{th}\) for various stress conditions by using \(D(\tau_{c},\tau_{e})=-\partial^{2}\Delta V_{th}(\tau_{c},\tau_{e})/\partial \tau_{c}\partial\tau_{e}\)[1]. The defect density map is represented in agreement with bivariate Gaussian distribution, describing the permanent and recoverable components. In the activation energy space, the density \(D(E_{c},E_{e})\) is constructed by the joint probability density function [9, 10, 14, 15]  
\[D(E_{c},E_{e})=\frac{D_{0}}{2\pi\sigma_{c}\sigma_{e}\sqrt{1-\rho^{2}}}e^{[- \frac{1}{2(1-\rho^{2})}(\frac{(E_{c}-\mu_{c})^{2}}{\sigma_{e}^{2}}-\frac{2\rho (E_{c}-\mu_{c})(E_{c}-\mu_{c})}{\sigma_{c}\sigma_{e}}+\frac{(E_{c}-\mu_{c})^{2 }}{\sigma_{e}^{2}})]} \tag{1}\]  
where \(D_{0}=(V_{s}/V_{0})^{\gamma}\) is a voltage-dependent amplitude with the stress voltage \(V_{s}\) and the constants \(V_{0}\) and \(\gamma\). \(\mu_{c,e}\) and \(\sigma_{c,e}\) are the mean values and standard deviations of \(E_{c,e}\), respectively. A correlation coefficient is defined as \(\rho=\text{covariance}(E_{c},E_{e})/\sigma_{c}\sigma_{e}\). For defects being charged up to the stress time \(t_{s}\) and not yet being discharged at the recovery time \(t_{r}\), the threshold voltage shift is calculated by integrating over the activation energy map as follows [9, 10, 14, 15]  
\[\Delta V_{th}(t_{s},t_{r}) =\int_{0}^{\infty}\int_{0}^{\infty}D_{r}(E_{c},E_{e})P_{occ}(E_{c},E_{e},t_{s},t_{r})dE_{c}dE_{e} \tag{2}\] \[+\int_{0}^{\infty}\int_{0}^{\infty}D_{p}(E_{c},E_{e})P_{occ}(E_{c},E_{e},t_{s},t_{r})dE_{c}dE_{e}\]  
where \(D_{r}(E_{c},E_{e})\) and \(D_{p}(E_{c},E_{e})\) are the recoverable and permanent component, respectively, and \(P_{occ}(E_{c},E_{e},t_{s},t_{r})\) is the occupancy probability map for the applied stress and recovery waveform. The occupancy probability \(P_{occ}\) after \(t_{s}\) and \(t_{r}\) is described by [1]  
\[P_{occ}(E_{c},E_{e},t_{s},t_{r})=\left(1-e^{-\frac{\tau_{c}}{\
u_{c}(E_{c})}} \right)e^{-\frac{\tau_{c}}{\
u_{c}(E_{c})}} \tag{3}\]  
For AC stress and recovery patterns with the signal duty cycle \(\alpha\), and the signal frequency \(f\), the probability to be occupied after the high gate-source voltage \(V_{gs}^{th}\) will be written as [30]  
\[P_{occ}(E_{c},E_{e},t_{s},t_{r})=\frac{1-e^{-\frac{\tau}{\
u_{c}}}}{1-e^{- \frac{\tau}{\
u}(\frac{\tau}{\
u_{c}}+\frac{1-\alpha}{\
u_{c}})}}\left(1-e^{-t_ {s}(\frac{\tau}{\
u_{c}}+\frac{1-\alpha}{\
u_{c}})}\right) \tag{4}\]  
Trap-related processes are the most likely cause of the fluctuation in the number of charge carriers.