# Reliability of NAND Flash Memories: Planar Cells and Emerging Issues in 3D Devices
## 3 NAND Flash Memory Reliability
### Random Telegraph Noise (RTN)
#### 3.2.3 Models
A constant PDF on a log-time scale was successfully assumed in [138], such as the one that will be used to describe detrapping after cycling in Section 3.4.2. It is worth noting that the constant trap density adopted in [122] implicitly provides a similar time constant distribution, due to the exponential nature of the tunneling probability, the only difference being the lack of correlation between \(\tau_{c}\) and \(\tau_{e}\) and between the time constant and the spatial location of traps. Within this framework, the expression for the single-trap \(V_{T}\) shift pdf, \(P(\Delta V_{T}^{1})\), becomes  
\[P(\Delta V_{T}^{1})=\int\int P(\Delta V_{T}^{1}|\tau_{c},\tau_{e})p(\tau_{c}, \tau_{c})d\tau_{c}d\tau_{e}, \tag{6}\]  
where \(p(\tau_{c},\tau_{e})\) is the probability for a trap to possess such capture/emission time constants. Once \(P(\Delta V_{T}^{1})\) is computed, multiple trap effects can be easily accounted for assuming statistical independence of their \(V_{T}\) shifts. This means that the overall \(V_{T}\) shift is the sum of the individual contributions, which translates into a convolution operation for pdfs. In particular, for \(N_{t}\) independent traps, each with the same \(P(\Delta V_{T}^{1})\) pdf, we have  
\[P(\Delta V_{T}^{N_{t}})=\bigotimes_{1}^{N_{t}}P(\Delta V_{T}^{1}) \tag{7}\]  
\[P_{10}(\tau_{c},\tau_{e}) =1-P_{11}=\frac{\tau_{c}}{\tau_{c}+\tau_{e}}\left(1-e^{-T/\tau_{ eq}}\right)\]0 PDF (Probability Density Function) of the single-trap threshold voltage shift \(P(\Delta V_{T}^{1})\) (**left**) and Markov process view of the single-trap RTN waveform between times \(t_{1}\) and \(t_{2}=t_{1}+T\) (**right**). and, since the number of traps per cell can be assumed to be statistically distributed according to a Poisson statistics \(p(N_{t})\), we finally get the RTN pdf:  
\[P_{10}(\tau_{c},\tau_{e}) =1-P_{11}=\frac{\tau_{c}}{\tau_{c}+\tau_{e}}\left(1-e^{-T/\tau_{ eq}}\right)\]1  
Typical comparisons between model results and experimental data are reported in Figure 13 for NAND [138] and NOR [122] arrays. The left-hand side shows that a nice agreement can be achieved for different technologies and cycling conditions. In particular, because of the exponential nature of \(P(\Delta V_{T}^{1})\), its slope \(\lambda\) matches what is obtained from array data and resulting from the convolution operation in (8). The right-hand size is instead related to data reported in Figure 9 (left), where RTN distributions for increasing read operations (i.e., time) are reported, and shows the probabilities at fixed \(|\Delta V_{T}|\) values as a function of the number of read operations. The increase in the RTN figure is well reproduced by the model and can now be explained: the increase in the elapsed time between the read operations (i.e., \(T\) in Equation (4)) allows for more and more traps to be activated, as their \(\tau_{eq}\) becomes comparable with \(T\). This increases the randomness of \(V_{T}\), enlarging the RTN distribution. This effect was nicely shown in [96; 122] as an extension with time of the oxide region whose traps are sampled by RTN. Finally, the temperature-independent RTN behavior observed in [138] was explained in terms of a lack in correlation between the time constants and their activation energy values; therefore, temperature will not affect the overall time constant distribution. Of course, different defects will contribute to the RTN fluctuations, as some are accelerated and will end up being filtered by the read time while slow ones will come into play as a result of thermal activation.