# Data-Driven Modeling of Low Frequency Noise Using Capture-Emission Energy Maps
## 2 Theoretical Model
The contribution of multiple-trap levels leads to a noise power spectrum expressed by the superposition of a single Lorentzian noise as follows [27]  
\[S_{N}(\omega)=\sum_{i=1}^{N_{t}}S_{N,i}(\omega)=4\sum_{i=1}^{N_{t}}\Delta N_{i} ^{2}\frac{\tau_{eff,i}}{1+\omega^{2}\tau_{eff,i}^{2}} \tag{5}\]  
where \(\Delta N_{i}^{2}=A_{i}^{2}\tau_{c,i}\tau_{e,i}/\left(\tau_{c,i}+\tau_{e,i} \right)^{2}\) is the variance of the fluctuation \(\Delta N\) in the number of carriers \(N\), \(A_{i}\) is the amplitude, \(N_{t}\) is the number of traps following a Poisson distribution, and \(\omega\) is the angular frequency. With the assistance of capture-emission activation energies, a change in charge state can occur even if the energy levels of the traps are higher or lower than the Fermi level compared to a few \(k_{b}T\), resulting in \(1/f\) noise [31, 32, 33, 34, 35]. Assuming that \(A_{i}\) is random variables identically distributed and the effective times constants are statistically independent, the noise power spectrum of the occupancy \(P_{occ}(E_{c},E_{e},t_{s},t_{r})\) of the traps in the energy element \(\Delta E_{c}\) and \(\Delta E_{e}\) is given by [34, 35]  
\[S_{N}(\omega)=4N_{t}^{*}A^{2}\frac{\tau_{c}\tau_{e}}{(\tau_{c}+\tau_{e})^{2}} \frac{\tau_{eff,i}}{1+\omega^{2}\tau_{eff,i}^{2}}P_{occ}(E_{c},E_{e},t_{s},t_{ r})\Delta E_{c}\Delta E_{e} \tag{6}\]  
where \(N_{t}^{*}\) is the trap density per unit area. Given that the trap distribution is described by the normalized density \(D_{r}^{*}(E_{c},E_{e})\) of the recoverable component, the average noise power spectrum for a device with a width \(W\) and length \(L\) is converted to an integral expression  
\[\overline{S_{N}(\omega)}=4N_{t}^{*}WLA^{2}\int_{0}^{\infty}\int_{0}^{\infty} \frac{\tau_{c}\tau_{e}}{(\tau_{c}+\tau_{e})^{2}}\frac{\tau_{eff}}{1+\omega^{2} \tau_{eff}^{2}}D_{r}^{*}(E_{c},E_{e})\ P_{occ}(E_{c},E_{e},t_{s},t_{r})\ dE_{c}dE_{e} \tag{7}\]  
where \(\tau_{eff}=(1/\tau_{c}+1/\tau_{e})^{-1}\) is the effective time. Note that the term \(P_{occ}(E_{c},E_{e})\) is similar to the occupancy of traps \(f_{T}(1-f_{T})\) described by a Fermi function \(f_{T}\), reflecting the fact that traps with higher occupancy probability contribute to noise power. The equivalent gate voltage noise power spectrum of MOSFETs in the linear region is given by \(S_{V_{g}}(\omega)=q^{2}\overline{S_{N}(\omega)}/(W^{2}L^{2}C_{ox}^{2})\) where \(C_{ox}\) is the gate oxide capacitance per unit area. Under cyclo-stationary square wave excitation, the noise behavior is governed by the time during the on-state, \(\tau_{c,on}\) and \(\tau_{e,on}\), and the times during the off-state, \(\tau_{c,off}\) and \(\tau_{e,off}\). The effective and characteristic times are written as [28]  
\[\tau_{eff}^{*}=\left(\frac{\alpha}{\tau_{on}}+\frac{1-\alpha}{\tau_{off}} \right)^{-1},\ \ \tau_{c/e}^{*}=\left(\frac{\alpha}{\tau_{c/e,on}}+\frac{1-\alpha}{\tau_{c/e, off}}\right)^{-1} \tag{8}\]  
with \(\tau_{on}=(1/\tau_{c,on}+1/\tau_{e,on})^{-1}\), \(\tau_{off}=\left(1/\tau_{c,off}+1/\tau_{e,off}\right)^{-1}\). For an accurate estimation of CEE distribution, a GMM of the distributions appears to be a natural choice for two components of Gaussian distributions. Given a two-dimensional random variable \(E_{a}=(E_{c},E_{e})\), with mean \(\mu_{i}\) and covariance \(\Sigma_{i}\), the GM density function of two Gaussian components is given by [36, 37, 38, 39]  
\[p(E_{a})=\sum_{i=1}^{2}\pi_{i}\mathcal{N}(E_{a}|\mu_{i},\Sigma_{i}) \tag{9}\]  
where \(\pi_{i}\) is the weight of the \(i\)-th mixture component, subject to \(0\leq\pi_{i}\leq 1\) and \(\pi_{1}+\pi_{2}=1\). The \(i\)-th independent normal distribution is  
\[\Delta V_{th}(t_{s},t_{r}) =\int_{0}^{\infty}\int_{0}^{\infty}D_{r}(E_{c},E_{e})P_{occ}(E_{c},E_{e},t_{s},t_{r})dE_{c}dE_{e} \tag{2}\]0  
The expectation maximization (EM) algorithm is used to compute the parameters of each Gaussian component [36, 37, 38].