# Simulate Advanced Submicron Devices Use the S-Pisces 2B Energy Balance and Hydrodynamic Models

*Dr. Y. Apanovich, Dr. E. Lyumkis,
Dr. B. Polsky, A. Shur and Dr. P. Blakey*

Advanced submicron technologies require accurate
simulation. The conventional drift-diffusion model of charge transport
neglects non-local transport effects such as velocity overshoot
and energy dependent impact ionization. The consequences may be
minor for large devices used in older technologies, but they are
often significant for submicron devices. Drift-diffusion calculations
of substrate currents and breakdown voltages can differ qualitatively,
as well as quantitatively, from measurements. SILVACO has responded
to the industry's needs by adding two non-local transport models
to **S-Pisces 2B**. Great care was taken to ensure that the new
models are accurate, robust, general, and genuinely applicable to
the needs of industry.

The New Models

Poisson's equation, particle continuity equations, and either of two forms of particle current and energy flux equations are solved, for electrons and holes, for steady-state and time-domain transient conditions. Transport parameters such as mobility and impact ionization coefficient are treated as functions of carrier temperature rather than local electric field. The models are consistent with the drift-diffusion model in the limit of large devices. The two models are obtained from energy balance[1] or hydrodynamic[2] perspectives. The relative merits of the two formulations are still being researched by academic groups. SILVACO decided to provide customers with both models.

Implementation

An improved spatial discretization of the energy flux equations was developed. Expressions for the energy flux densities were reformulated and associated Scharfetter-Gummel type approximations were derived. The spatial oscillations encountered with previous approaches were thereby virtually eliminated. Reliable time-stepping was obtained using an extension of the absolutely stable scheme[3]. Convergence assessment of steady-state solutions based on changes of temperature alone, which is used in some implementations of hydrodynamic and energy balance models, leads to errors of up to 20% in calculated output currents. More reliable convergence criteria were therefore developed and implemented.

Results

Some examples illustrate the importance of the new transport models. The first is the classic test case provided by the n+nn+ "ballistic" diode. Figure 1 shows the electron velocity distribution in the diode for an applied bias of 1.5V. As was previously reported by numerous authors the hydrodynamic model (HDM) gives a very large non-physical velocity spike at the collecting nn+ junction. (The spike is non-physical in the sense that it is not present in more detailed calculations performed using Monte Carlo simulation.) The Energy Balance Model (EBM) gives a more realistic velocity distribution that is in much better agreement with Monte Carlo results.

Figure 1.

Figure 2 shows substrate currents calculated using the EBM and DDM for a 0.7 µm channel length LDD MOSFET at a drain voltage of 3V. The EBM results are similar to measured data. The DDM results predict too much substrate current and have the wrong trend (i.e. increase instead of decrease for the higher gate voltages considered). Figure 3 shows on a linear scale the substrate currents predicted by the EBM and the HDM. The HDM predicts significantly lower currents than the EBM. Figure 4 shows the drain currents predicted by the EBM and the DDM. The EBM predicts higher currents due to the occurrence of velocity overshoot.

Figure 2.

Figure 3.

Figure 4.

Another example is provided by the realistic bipolar structure shown in Figure 5. This structure has an effective base width of 0.15 um. Figure 6 shows the base current as a function of the collector-emitter voltage with the base-emitter voltage held at 0.75V. Such curves are used to estimate BVCEO. The DDM overestimates the amount of impact ionization and so predicts an earlier drop off in the base current. The EBM predicts somewhat higher breakdown voltages, and the HDM predicts breakdown voltages that are higher still.

Figure 5.

Figure 6.

We believe that the discrepancies between the EBM and the HDM are associated with overestimation of thermal diffusion currents by the HDM. The n+nn+ ballistic diode solutions and the behavior of calculated drain currents certainly suggest that the EBM results are more realistic than the HDM results. Comparisons with experimental measurements on substrate current and breakdown voltages are needed to confirm that the EBM is uniformly better than the HDM. If this proves to be the case, then the HDM, which predicts lower substrate currents and higher breakdown voltages than the other models, would lead people to designs that are insufficiently conservative with respect to breakdown criteria. SILVACO is the only vendor that offers the energy balance model.

Summary

Non-local transport effects that are neglected
by the conventional drift-diffusion model can have a significant
impact on the electrical behavior of advanced submicron devices.
Models that take these effects into account have been added to the
general purpose device simulator** S-Pisces 2B**. These models
are available in the Energy Balance (EB) module, the latest in a
series of Industrial Application Modules developed at SILVACO. Reliable
simulation of advanced submicron devices is now accessible to everyone
in the industry!

Acknowledgment

We thank Dr. D. Chen, Dr. Z. Yu, and Professor R. Dutton of Stanford University for interesting discussions concerning differences between the hydrodynamic and energy balance models.

References:

[1] R.Stratton, Phys.Rev., 126 (6) p. 2002, 1962.

[2] K.Blotekjaer, IEEE Trans., ED, 17, p.38, 1970.

[3] B.Polsky, J.Rimshans, Solid-State Electron., 26 (4) p. 275, 1983.