# Reliability of NAND Flash Memories: Planar Cells and Emerging Issues in 3D Devices
## 3 NAND Flash Memory Reliability
### Charge Detrapping
#### 3.4.2 Models  
We start this Section with the description of a compact model [221; 222; 226] that captures the main features of the data and can be used for some first-order extrapolations, while moving to more refined descriptions of the underlying physics later on. To this aim, we assume a log-time dependence of the \(V_{T}\) shift that, considering that the first read operation is performed at time \(t_{0}\) after the end of the cycling phase, leads to  
\[\Delta V_{T}(t_{B})=V_{T}(t_{0}+t_{B})-V_{T}(t_{0})=-\alpha\ln(t_{0}+t_{B})+ \alpha\ln(t_{0})=-\alpha\ln\left(1+\frac{t_{B}}{t_{0}}\right)\,, \tag{9}\]  
which is what we observe. The dependences of \(\Delta V_{T}\) on \(N_{C}\) and probability level (see Figure 19) can then be simply inserted into the expression of \(\alpha\). To account for the thermal activation of detrapping during bake, which results in a rigid shift of the detrapping curve in a semilog plot (see right-hand side of Figure 21), we can now modify the expression of the term \(t_{0}\), which is the zero-loss intercept of the log-time \(V_{T}\) curve (see Figure 21, left). This results in:  
\[t_{0}^{*}=t_{0}\,e^{E_{A}(1/k_{B}T_{B}-1/k_{B}T_{ref})};\quad\Delta V_{T}(t_{B })=-\alpha\ln\left(1+\frac{t_{B}}{t_{0}^{*}}\right)\,, \tag{10}\]  
where \(t_{0}\) is now the initial wait time at the reference temperature \(T_{ref}\) and \(k_{B}\) is the Boltzmann constant. Following this line of thought, even data in Figure 22 can be represented by a dependence of the intercept \(t_{0}^{*}\) on cycling conditions. To do this, it is assumed that damage creation during P/E cycles and recovery between them can be described as a fast-cycling experiment (i.e., with negligible delay between cycles) generating the whole oxide damage, followed by a single recovery phase of a certain duration. Combined with the thermal activation term, this results in the final expression  
\[t_{B}^{*}=t_{0}^{*}+At_{c}\,e^{E_{A}(1/k_{B}T_{B}-1/k_{B}T_{C})};\quad\Delta V_ {T}(t_{B})=-\alpha\ln\left(1+\frac{t_{B}}{t_{B}^{*}}\right), \tag{11}\]  
which allows to completely describe the \(\Delta V_{T}\) dynamics as a function of the various experimental parameters once the fitting parameter \(A\) is selected. This means that the term \(\ln(1+t_{B}/t_{B}^{*})\), called the _universal damage metric_ or UDM, should now account for all \(\Delta V_{T}\) transients measured on a specific technology, irrespective of cycling and bake conditions. This is demonstrated in Figure 23 (left). Figure 23 (right) shows instead the value of \(t_{B}^{*}\) for different cycling conditions. Figure 22: Dependence of \(\Delta V_{T}\) on cycling conditions, namely total cycling time \(t_{C}\) (**left**) and cycling temperature \(T_{C}\)[227], © 2016 IEEE (**right**). The experimental data presented so far and the physical understanding leading to Equation (11) led to the recognition that traditional testing procedures based on fast cycling and long retention times were magnifying the \(V_{T}\) loss with respect to real usage conditions, when cycling takes place over the functional life of the device (see the effect of increasing \(t_{C}\) in Figure 22, left). The concept of _distributed cycling_ conditions was then proposed in [221] as a way to better emulate the real array behavior by performing either a uniform cycling over a longer time or several groups of fast cycles preceded by bake times, usually at high temperature to accelerate the charge loss. However, the previous model shows its limitations when dealing with highly non-uniform cycling patterns, calling for a more comprehensive interpretation of the trapping/detrapping physics, that was pursued in [223; 224; 225; 227; 223]. Such a model was built upon a few phenomenological assumptions: a Poisson distribution for the number of trapped electrons \(n_{t}\) and a uniform distribution over a log-time axis of their detrapping time constant \(\tau_{d}\).