# Reliability of NAND Flash Memories: Planar Cells and Emerging Issues in 3D Devices
## 3 NAND Flash Memory Reliability
### Random Telegraph Noise (RTN)  
#### 3.2.1 RTN Amplitude  
Random telegraph noise (RTN) in MOS transistors is an abrupt two-level fluctuation in drain current \(I_{D}\) following trapping and release of a single electron by an oxide defect. This well-known phenomenon has been observed and studied since the 1980s [78; 79; 80; 81; 82; 83; 84; 85; 86], resulting in a prediction for the RTN amplitude [79] equal to  
\[\frac{\Delta I_{D}}{I_{D}}=\frac{q}{C_{ox}WL(V_{G}-V_{T})}+\frac{\Delta\mu_{n}} {\mu_{n}}\frac{\Delta A}{WL}, \tag{2}\]  
where \(q\) is the electron charge, \(C_{ox}\) the oxide capacitance per unit area, \(W\) and \(L\) the channel width and length, respectively, and \(\mu_{n}\) the electron mobility in the inversion channel, which experiences a fluctuation \(\Delta\mu_{n}\) over an area \(\Delta A\) as a consequence of the additional scattering with the trapped electron. Typical values for the first term are smaller than 10%, while the scattering part is typically negligible in the subthreshold regime, where NAND cells are usually read; therefore, RTN has long been regarded as a curious scientific phenomenon not constituting a noticeable reliability threat. This held true even if unusually large values for the RTN amplitude had been consistently reported on some devices, together with complex time behaviors [79; 82; 87; 88; 89], that seemed to defy Equation (2). When Flash memories made a large quantity of data readily available, making it possible to pinpoint and study anomalous behaviors occurring with extremely low probability, larger and larger fluctuation amplitudes emerged [90; 91], and RTN became a serious reliability constraint in static random-access [92; 93; 94] and Flash memories [90; 91; 95; 96; 97; 98; 99], prompting an intensive research effort to understand its root cause and impact on memory array operation. An example of the relevance of RTN is shown in Figure 4, where results obtained on a selected decananometer Flash cell are reported [91]: current fluctuations up to 60% were detected, which cannot be explained with the above theory. Though early explanations pointed to multiple carrier emissions [88; 89], change in defect properties [87] or interacting quantum wells [82], a different approach based on the percolative nature of the current conduction around the charged trap site had been mentioned in [100] and thoroughly revisited in [101], exploiting a concept applied to amorphous semiconductors [102]. The authors pointed out that no uniform conduction in a MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) channel could account for the anomalous RTN amplitudes. Rather, a non-uniform conduction with filamentary current flow was needed as a starting point; therefore, an RTN trap could effectively modulate a current-crowding path, resulting in a large current fluctuation. Fixed oxide and interface charges were proposed as sources of percolation, following earlier studies [103; 104]. This is now regarded as the correct interpretation of the RTN amplitudes, with the caution to add the actual main source of percolation: random dopant fluctuation (RDF). In fact, statistical variation in number and position of dopants was known to affect MOSFET parameters such as \(V_{T}\) and subthreshold slope [105; 106; 107; 108], and was shown to have a significant impact on RTN in [109] and in [110; 111], where large amplitude values compatible with experimental data were reported. For the evaluation of the RTN effects in Flash memories, the statistical distribution of its amplitude is needed, accounting for realistic device geometry and doping. This task was undertaken in [112; 113], where 3D Monte Carlo simulations of electron transport were carried out, accounting for randomized doping and trap positions. Simulations were conducted within the drift-diffusion framework, neglecting the field dependence of mobility to avoid unphysical effects close to the ionized dopants and RTN trap.