# 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  
The experimental data just reported call for a consistent model able to explain the different features of the phenomenon. While data shown in Figure 7 can be reproduced by a simple exponential fit [98], the explicit time dependence captured by Figures 8 and 9 requires a time-dependent analysis, that was first put forward in [96; 122]. The starting point is the exponential distribution of \(V_{T}\) shift extracted from simulation results described in Section 3.2.1. Such a distribution describes a positive \(V_{T}\) shift between time \(t_{1}\) and \(t_{2}\) if an empty trap at \(t_{1}\) is filled at \(t_{2}\) and a negative shift if the opposite is true. If the trap state does not change, a zero \(V_{T}\) shift holds. The final time-dependent distribution is then the one shown in Figure 12 (left), where the positive and negative exponentials and the zero-shift delta function must be weighted by their occurrence probabilities, representing their areas. This is accomplished describing the RTN process as a two-state Markov process [141; 142], as depicted in Figure 12 (right). The two states correspond to the empty (low-\(V_{T}\), state 0) and filled (high-\(V_{T}\), state 1) trap. Considering constant capture and emission times \(\tau_{c}\) and \(\tau_{e}\), the transition probabilities between state \(i\) and \(j\), \(P_{ij}\), over a time interval \(T\) become:  
\[P_{11}(\tau_{c},\tau_{e}) =\frac{1}{\tau_{c}+\tau_{e}}\left(\tau_{c}\;e^{-T/\tau_{eq}}+\tau _{e}\right) \tag{4a}\] \[P_{10}(\tau_{c},\tau_{e}) =1-P_{11}=\frac{\tau_{c}}{\tau_{c}+\tau_{e}}\left(1-e^{-T/\tau_{ eq}}\right)\] (4b) \[P_{00}(\tau_{c},\tau_{e}) =\frac{1}{\tau_{c}+\tau_{e}}\left(\tau_{e}\;e^{-T/\tau_{eq}}+\tau _{c}\right)\] (4c) \[P_{01}(\tau_{c},\tau_{e}) =1-P_{00}=\frac{\tau_{e}}{\tau_{c}+\tau_{e}}\left(1-e^{-T/\tau_{ eq}}\right), \tag{4d}\]  
where \(\tau_{eq}^{-1}=\tau_{c}^{-1}+\tau_{e}^{-1}\). Note that these expressions describe the probabilities to find the trap in a certain state after a time interval \(T\), given the initial condition, irrespective of the number of transitions taking place in-between. Figure 11: RTN cdfs for two cycling schemes and room-temperature read (**left**) and for one cycling scheme and different read temperatures (**right**). No significant differences appear in all cases. From (4), the probability for a single trap to provide a \(V_{T}\) shift, \(\Delta V_{T}^{1}\), can be easily calculated:  
\[P(\Delta V_{T}^{1}>0|\tau_{c},\tau_{e}) = P_{0}^{t_{1}}P_{01}(\tau_{c},\tau_{e}) \tag{5a}\] \[P(\Delta V_{T}^{1}<0|\tau_{c},\tau_{e}) = P_{1}^{t_{1}}P_{10}(\tau_{c},\tau_{e})\] (5b) \[P(\Delta V_{T}^{1}=0|\tau_{c},\tau_{e}) = P_{0}^{t_{1}}P_{00}(\tau_{c},\tau_{e})+P_{1}^{t1}P_{11}(\tau_{c},\tau_{e}), \tag{5c}\]  
where \(P_{i}^{t_{1}}\) is the probability for the trap to be in state \(i=0,1\) at time \(t_{1}\). Once the analysis for a given set of \(\tau_{c}\), \(\tau_{e}\) is assessed, we must account for their statistical distribution among the traps. The original approach in [122] relied on the classical theory of direct tunneling into oxide defects [80; 81], explicitly linking \(\tau_{c}\) and \(\tau_{e}\) to the space and energy position of the trap inside the oxide. Later results have, however, shown that capture/emission times in MOSFETs are not compatible with such a description [143; 144], due to device variability effects on the time constants [145; 146; 147] and structural relaxation of defects [140; 148], generating a large spread in time constants even for traps located close to the silicon/oxide interface and disrupting the classical correlation between \(\tau_{c}\) and \(\tau_{e}\). The study of the microscopic properties of RTN traps is still an active research topic [149; 150; 151; 152; 153; 154; 155; 156], but for our purposes we will adopt a pragmatic approach, assuming given distributions of capture/emission time constants that may be consistent with observations, without linking them to any particular trap location.