# Analysis of Channel Potential Recovery According to the Back Pattern in 3D NAND Flash Memory
## 3 DCP Recovery Time
The flux was proportional to the carrier concentration gradient. The equation for the electron diffusion flux is given as:  
\[J_{n(diff.)}(x)=qD_{n}\frac{dn(x)}{dx}\]  
where \(q\) is the quantity of electric charge, \(D_{n}\) is the diffusion coefficient, and \(\frac{dn(x)}{dx}\) is the gradient of the electron concentration. According to the above equation, as the gradient of the concentration increases, the diffusion current increases. Therefore, if the electron diffuses a lot, the concentration of the electron is expected to decrease considerably. As shown in Figure 4a, when the adjacent WLs were programmed in a low state, less electron diffusion occurred. The reduced electron concentration of the floating channel is shown in Figure 4b. The significant point to note is that the electron concentration variation in WL8 was similar to the recovery of the channel potential shown in Figure 2b. Therefore, this means that the channel potential was recovered owing to the decrease of the electron concentration. In terms of electron diffusion, the electron concentration should decrease slightly when the adjacent WLs are programmed in a low state. However, the electron concentration decreased significantly, even though there was less electron diffusion when the adjacent WLs were programmed in a low state. Therefore, in the early stage, this is inconsistent with the trend of quick channel potential recovery and of many variations in the electron concentration of the floating channel when the adjacent WLs are programmed to a low state as shown in Figures 2b and 4b. This indicates that electron diffusion is not a major cause of channel potential recovery. The hole concentration of the floating channel at 0 s and after 5 \(\upmu\)s are shown in Figure 5a,b, respectively. Although the hole concentration decreased after 5 \(\upmu\)s in all patterns, the extent of hole concentration changed significantly in the P1 pattern. If the electron concentration and the hole concentration are high in the floating channel, EHP recombination can occur a lot. EHP recombination in the floating channel was examined, as shown in Figure 5c. When EHP recombination occurred, the hole concentration of the floating channel region decreased. Then, the electrons recombined with the holes, and the electron concentration decreased. Therefore, the channel potential recovery varied in each pattern owing to the extent of EHP recombination. EHP recombination occurred in the order of the patterns P1, P2, and P3, and the EHP recombination of the pattern P1 had the same appearance  
Figure 3: Electron concentration from WL6 to WL10 at (**a**) 0 s and (**b**) 5 \(\upmu\)s.  
as the hole concentration of the pattern P1. This indicates that the EHP recombination had a great impact on the extent of the channel potential recovery in the early stage. The adjacent WLs region had higher hole concentrations, and no significant change was observed in the hole concentration although EHP recombination occurred in the patterns P2 and P3. In addition, although much EHP recombination occurred in the pattern  
Figure 4: (**a**) Electron diffusion current from WL6 to WL10 at 0 s, which is proportional to the electron concentration at 0 s. (**b**) Variance of electron concentration in WL8 located in the floating channel. Figure 5: Hole concentration from WL6 to WL10 at (**a**) 0 s and (**b**) 5 \(\upmu\)s. (**c**) Recombination rate in the floating channel at 0 s. (**d**) Hole diffusion current from WL6 to WL10 at 0 s, which is proportional to the hole concentration at 0 s.  
P1, over time, the hole concentration was found to be similar to that of the others. This indicates that the hole diffused from the adjacent WLs to the floating channel. The equation for the hole diffusion flux is given as:  
Figure 1:0  
where \(q\) is the quantity of electric charge, \(D_{p}\) is the diffusion coefficient, and \(\frac{dp(x)}{dx}\) is the gradient of the hole concentration.