1. Natural Local Self Boosting(NLSB)

NLSB is a unique phenomenon of 3D NAND compared with 2D NAND. In 2D NAND, the soft programming problem in the inhibited string(unselected string) is a severe problem. Since the boosting potential is mainly determined by the pass voltage, the soft programming caused by Fowler-Nordheim(FN) tunneling of selected WL is problematic because the voltage difference between the program voltage and the boosting channel potential is quite large ⁽³⁾.

Fig. 1(a) and (b) show electric potential contour of the inhibited 3D NAND string in TCAD and the channel potential of the selected string to be programmed and unselected string not to be programmed. The programming cell(selected WL) has the higher channel potential than that of pass cells(unselected WLs) naturally in the 3D NAND due to the electron flow and the change of the state over time without using the local self-boosting method ⁽⁶⁾.

This phenomenon reduces the soft programming caused by FN tunneling of the programming cell in the inhibited string. However, due to NLSB, hot carrier injection can occur in the 3D NAND near the programming cell as well as at both ends where the DSL and SSL are located.

2. Program Disturbance by Hot Carrier Injection

Fig. 2(a) and (b) show the inhibit bias scheme and assumed program-verify(PV) states for modeling and the energy band diagram. In Fig. 2(a), because the programming operates in page units, the only difference between selected string and unselected string is bit line voltage(V$_{\mathrm{BL}}$). And we define that the programmed and unselected WL above the selected WL is a neighboring cell of the programming cell.

Also, V$_{\mathrm{th}}$ of the programmed state is 3 V, V$_{\mathrm{th}}$ of the erased state is {--}2 V. And single programming pulse is used.

Since the PV state of the neighboring cell of the programming cell is dominant in generating maximum electric field at the direction of the channel, the following PV states are assumed.

As can be seen from the Fig. 2(b), the energy band near the programming cell due to NLSB is considerably curved and the tunneling distance between the valence band and the conduction band is considerably enough close to cause BTBT.

Due to BTBT, electron-hole pairs are generated, and some of the electrons, which have a large energy, tunnel into the charge trap layer near the programming cell and cause an unintentional V$_{\mathrm{th}}$ shift.

Fig. 3(a) shows the device that performed the simulation. Trapped electron charge (eTrapped charge) in nitride by HCI are located between the programmed state of the unselected WL and the erased state of the selected WL. Also, these electrons are seen to be located close to the tunneling oxide in the y-direction.

As can be seen in Fig. 3(b), the peak point of the electric field appears in the spacer region of the programming cell due to the voltage difference between the boosting potential of the programming cell and the boosting potential of pass cells. And its peak point is coincided with the peak point of BTBT.

On the other hand, in Fig. 3(c), the peak point of trapped electron charge in nitride by HCI is located in the direction of the programming cell rather than the peak of BTBT. It is because electrons generated by BTBT must obtain sufficient energy with the lateral electric field(x-direction) for being injected as hot carriers. When NLSB occurs, the trapped charges injected as hot carriers show a peak in the spacer region slightly distant from the gate of the programming cell. This differs from the injection by FN tunneling where trapped charges are located at the center of the gate of the programming cell.

The value of the maximum lateral electric field increases as the pass voltage is lower and the program voltage is higher. And V$_{\mathrm{th}}$ shift of the programming cell also has the same tendency as can be seen in Fig. 4. V$_{\mathrm{th}}$ shift due to hot carrier injection has been confirmed that there is depenency of the electric field. This suggests that the modeling of HCI should be based on accurate trends of the electric field.

3. Channel Potential Lowering

The electrons generated by the BTBT are accumulated in the channel under the programming cell by the high program voltage. Over time, the electrons continue to gather and consequently lower the channel potential of the programming cell.

As shown in Fig. 5, the channel potential of the programming cell gradually decreases over time.

Fig. 6 shows that the maximum channel potential of the model with the BTBT is gradually lowered as electrons accumulate continuously during the duration time of programming pulse after rising time, whereas that of the model without BTBT maintains constant.

This programming cell channel potential lowering should be considered to describe HCI exactly.

4. Modeling

Because the hot carrier injection is strongly dependent on the electric field, the channel potential is used to describe the exact electric field. Finally, a new HCI current modeling based on channel potential in the programming operation of 3D NAND has been developed as follows.

V$_{\mathrm{pgm(ch)}}$, V$_{\mathrm{pass(ch)}}$ mean the channel potential of the programming cell and pass cell respectively. And V$_{\mathrm{pv(ch)}}$ is the term to express that the PV states of the programming cell and the neighboring cell affect the channel potential.

The first and second term of Eq. (1) are the terms to express NLSB, and the third term is the expression of lowering by accumulated electrons generated by BTBT. Also, Eq. (2) have been confirmed that when NLSB occurs, the channel potential of the pass cell coincided with V$_{\mathrm{pass}}$ by TCAD. C$_{\mathrm{ch}}$ means the capacitance of channel.

Eqs. (3, 4) are the maximum lateral electric field and the maximum oxide electric field respectively expressed by channel potential. The change of channel potential due to the programmed state of the neighboring cell of the selected WL and the erased state of the selected WL is expressed as V$_{\mathrm{pv(ch),P-E}}$ in the numerator term of the lateral electric field. In the maximum oxide electric field, we added V$_{\mathrm{pv(ch),E}}$ which means channel potential change by erased state of selected WL. C$_{\mathrm{r}}$ means the capacitance coupling ratio of O/N/O. Also, t$_{\mathrm{spacer}}$, t$_{\mathrm{ox}}$ are the length of the spacer and the thickness of the tunneling oxide respectively.

Eq. (5), BTBT current for 3D NAND flash memory, is represented through modifying the well-known BTBT current modeling of the PN junction ⁽⁸⁾.

Eq. (6), HCI current is based on the lucky electron model, and the SPICE-friendly NAND HCI model ^(7,9). Through TCAD, it has been confirmed that HCI occurs more frequently when more BTBT occur, and it has been considered. Also, as the mean free path becomes longer, the electrons become less scattering and the probability of obtaining energy increases. And when the lateral electric field becomes stronger, the probability of electrons getting energy increases. These have been considered by putting into the denominator term of exponential term. Besides, when the electric field of the tunneling oxide becomes stronger, the electrons in the channel feel lower energy barrier than 3.1 eV that is the original energy barrier between silicon and oxide. This has been also considered by putting into the numerator term of exponential term. Finally, through iteration of Eqs. (1-5), HCI current is calculated numerically.

$\gamma, \alpha, \beta, \lambda$ are model parameters. ${\gamma}$ is 0.7 as mentioned in ⁽⁶⁾. ${\alpha}$, ${\beta}$, ${\lambda}$ are parameters based on the lucky electron model. ${\alpha}$ is 2.6$\times $10$^{-4}$ [(V${\cdot}$cm)$^{\mathrm{1/2}}$], ${\beta}$ is 3.0$\times $10$^{-5}$ [(V${\cdot}$cm$^{2}$)$^{\mathrm{2/3}}$], ${\lambda}$ is 8.9$\times $10$^{-7}$ [cm]. A, B, C, D are fitting parameters.

Fig. 7(a)-(d) show fitting results of the extracted data by TCAD when V$_{\mathrm{pass}}$ = X+2, V$_{\mathrm{pgm}}$ = Y+2 and modeling. Modeling successfully describes HCI current according to the programming time.