# Improved Simulation Of Breakdown

Simulating the breakdown region of semiconductor devices can present problems. This is not only because including impact ionization increases the nonlinearity and coupling of the carrier continuity equations. The discretization of impact ionization used in the original Stanford version of PISCES causes calculated breakdown voltages to be too high. The Stanford implementation also predicts non-physical impact ionization in certain situations. This leads to significant overestimation of output currents for some devices and bias conditions. Non-physical impact ionization can also impair, and in some cases prevent, numerical convergence.

Alternative discretizations of impact ionization
have been implemented by several groups. Most of these alternative
discretizations lead to more accurate predictions of breakdown voltages,
but do not solve the problem of spurious impact ionization. A new
implementation of impact ionization that is accurate, and that also
overcomes the problem of spurious impact ionization, has been implemented
in **ATLAS**. Some results are presented here to demonstrate
the capabilities of the **ATLAS** implementation.

A warning is appropriate at this point. The
literature contains many examples where breakdown voltages calculated
using PISCES are inaccurate. *In some papers calculated electrical
behavior that was caused by spurious non-physical impact ionization
has been misinterpreted as being a real physical effect.**
*

Accuracy

The accuracy of the **ATLAS** implementation is demonstrated
by calculations of the reverse characteristics of simple abrupt
junction np diodes. Figures 1 and
2 show the reverse characteristics calculated for diodes with p
region dopings of 10 and 2.0 10 cm respectively. The doping concentration in the n+ region is 10 cm in both cases.
The Crowell-Sze model of impact ionization[1] was used, and a lattice
temperature of 300 K was assumed. The breakdown voltages calculated
using the **ATLAS** implementation are around 13.1V and 8.5V
respectively. These values agree very well with the values for these
doping levels given by Sze2. The Stanford PISCES implementation
overestimates the breakdown voltage by a significant amount.

Figure 1. Calculated breakdown
voltage for a doping of 10 cm. |
Figure 2. Calculated breakdown
voltage for a doping of 2 10 cm. |

Elimination of Spurious Ionization

The importance of eliminating spurious impact ionization is illustrated by results obtained for an SOI transistor. The simulated device has a gate length of 0.7 µm, a channel doping of 10, a channel thickness of 0.16 µm, and a gate oxide thickness of 170 Å. The breakdown behavior of this structure was calculated using the ionization coefficients of Overstraeten and DeMan.[3]

Figure 3 shows the internal distributions of generation
rate predicted by the **ATLAS** and Stanford PISCES implementations
of impact ionization when Vg = Vd = 1 V. The **ATLAS** implementation
predicts significant generation only in the vicinity of the drain
junction. The Stanford PISCES implementation leads to spurious generation
in the channel and at the source junction. Figure 4 compares the
generation rate along a cross section under the gate at the Si-SiO2
surface.

Figure 3. Calculated generation rate distributions within the SOI transistor. |
Figure 4. Generation rate under gate at the Si-SiO2 surface. |

The spurious impact ionization can have a significant effect on the calculated terminal currents in the pre-breakdown region. The drain characteristics shown in Figure 5 show that the Stanford PISCES implementation of impact ionization causes significant overestimation of the drain current in the "prekink" region. The gate characteristics shown in Figure 6 show that spurious impact ionization has a significant effect on the calculated threshold voltage, causing a shift of around -0.2 V.

Figure 5. Calculated drain characteristics for the SOI transistor. | Figure 6. Calculated gate characteristics for the SOI transistor. |

Conclusions

A high quality implementation of impact ionization has been
incorporated into **ATLAS**. The implementation is accurate and
avoids the serious problems associated with spurious impact ionization.
**ATLAS** provides effective and reliable breakdown analysis
of modern semiconductor devices.

**References**

[1] C. R. Crowell and S. M. Sze,

"Temperature Dependence of Avalanche Multiplication in Semiconductors,"

Appl. Phys. Lett., vol. 9, pp. 242-244, 1966.

[2] S. M. Sze,

Physics of Semiconductor Devices, 2nd Edition,

Wiley-Interscience, 1981.

[3] R. Van Overstraeten and H. DeMan,

"Measurement of the Ionization Rates in Diffused Silicon p-n
Junctions,"

Sol. State Electron., vol. 13, pp. 583-608, 1970.