# Data-Driven Modeling of Low Frequency Noise Using Capture-Emission Energy Maps
## 3 Results and Discussion  
The devices used for measurements were fabricated using a manufacturable remote plasma nitride oxide (RPNO) process. The gate oxide of devices used for measurements was grown with both furnace and rapid-thermal based processes. Following N\({}_{2}\)-plasma nitridation and post-nitridation anneal of the gate oxide, an undoped poly-Si gate layer was deposited. For p+ gates, B was implanted, and rapid-thermal annealing was then performed in N\({}_{2}\) for different durations. The threshold voltage shifts are obtained on pMOSFETs (\(W=10\) um, \(L=0.14\) um, \(t_{ox}=2.2\) nm), performing the ultrafast measure-stress-measurements [9; 10]. With a measuring delay time of 1 us, the temperature-accelerated measurement scheme was used to extend to the experimental stress-recovery window of 107 us. Noise measurements are conducted using an HP 3582A spectrum analyzer (1 Hz-25 kHz) and a Brookdeal 5004 low noise voltage amplifier connected at the output terminal. A bias resistance at the output terminal is chosen to be 10 times larger than the small signal channel resistance. The noise power spectrum is adjusted to subtract the noise due to the measurement system, which consists of the noise due to the amplifier and the bias resistance, and the thermal noise of device under test. Figure 1a shows experimental recovery data of \(\Delta V_{th}\) following stress time with numerical data for \(V_{gs}=-2.0\) V and \(T=125\) \({}^{\circ}\)C. Numerical data of \(\Delta V_{th}\) are obtained by integrating the CEE distribution in (2), which is calculated by taking the derivatives of measured data, as shown in Figure 1b. Figure 2a shows a scatter plot of GMM data using the soft clustering method. The cluster membership posterior probabilities in the interval (0.45,0.55) are members of both components with six shared data points. The estimated parameters of permanent component are \(\mu_{c1}=1.593\) eV, \(\mu_{c1}=1.604\) eV, \(\sigma_{c1}=0.203\) eV, \(\sigma_{c1}=0.098\) eV, \(\rho_{1}=0.035\), and those of recoverable component are \(\mu_{c2}=0.908\) eV, \(\mu_{c2}=0.997\) eV, \(\sigma_{c2}=0.376\) eV, \(\sigma_{c2}=0.317\) eV, \(\rho_{2}=0.8537\). Using the estimated parameters of GMM model, the CEE distributions are produced in the extended energy space exceeding experimental window, as observed in Figure 2b. The inset of Figure 2b shows a plot of the relative error values which are less than 0.5%. To investigate how the trap distribution influences RTS noise, the integrand of \(\overline{S_{N}(\omega)}\) for the recoverable component in (7) is numerically calculated and visualized as a function of \(E_{e}\) and \(E_{c}\) under stationary DC condition at a particular frequency. It is shown in Figure 3a that a peak is distinct at \(E_{c}=1.10\) eV and \(E_{e}=0.65\) eV for \(f=1\) Hz, and moves to relatively low activation energies \(E_{e}=0.75\) eV and \(E_{e}=0.45\) eV as the frequency increases to \(f=1\) kHz. As illustrated in the inset of Figure 3a, these results arise from different RTSs, indicating that faster components of shallow trap states for \(f=1\) kHz are situated close to lower activation energies [40]. To examine the behavior of cyclo-stationary RTS, the times are modeled as \(\tau_{coff}=m\cdot\tau_{con}\) and \(\tau_{coff}=\tau_{con}/m\), where the factor \(m\) is assumed to be greater than one [28; 40]. In the off state, \(\tau_{c}\) and \(\tau_{e}\) are multiplied and divided by \(m\), as they increase and decrease with a decrease in the gate voltage, respectively [28; 40]. For \(m=50\), a peak moves to higher emission energy \(E_{e}\), as shown in Figure 3b. Figure 1: Threshold voltage shifts \(\Delta V_{th}\) and capture-emission energy (CEE) map. (**a**) Numerical data with measured data of \(\Delta V_{th}\). (**b**) Comparison of numerical data with measured data of CEE map. \(1/f\) noise power is calculated by using (7) with the recoverable component obtained from GMM with soft clustering.