Advanced Physical Models for Silicon Device Simulation

The description of transport in semiconductor devices requires models for both the interaction processes and the embedding system. These models have different form depending on the transport equations used, but on every level one needs expressions for the scattering of charge carriers with elementary excitations of the crystal as well as with each other, with impurities, device boundaries or interior interfaces, and models of all generation-recombination processes. The environmental system is given by material parameters, e.g. the band gap and the intrinsic carrier density, by external quantities like the doping concentration, defect profiles, boundaries, and others.

The hydrodynamic (HD) transport scheme [2.8, 2.56] has become a standard device simulation tool with the capability for describing nonlocal and nonstationary phenomena. Computation times compare favorably with those of Monte Carlo (MC) simulations [2.23] and methods based on a spherical-harmonics (SH) expansion of the Boltzmann transport equation (BTE) [2.59]. Thus, it is suitable also for more sophisticated applications like power-device, multi-device or 3D-device simulations.

In 1934 Zener [3.118] proposed band-to-band tunneling as explanation for the electrical breakdown. A modified Zener theory was used by McAffee et al. [3.75] in 1951 to describe the breakdown of reversed biased pn-junctions, called Zener diodes since then. However, experimental work [3.76–3.78] in the following years showed that in such diodes with wide junctions the breakdown is not caused by tunneling, but by impact ionization. Only in narrow junctions, where the width of the transition region is less than 50 nm, the necessary field strength for tunneling is reached before the avalanche effect sets in. This was first clearly demonstrated by Chynoweth and McKay [3.22] in 1957 by the absence of microplasma noise and by the temperature coefficients of reverse and forward characteristics of junctions with different breakdown voltages. In the same year Esaki [3.32] discovered that narrow pn-junctions between degenerate regions can have forward characteristics with a portion of negative differential conductivity, and that the tunnel “hump” is only weakly temperature dependent. Esaki’s work initiated intensive experimental and theoretical investigations. Holonyak et al. [3.53] and Hall [3.45] observed structures in the I (V)-characteristics of heavily doped Si-junctions at 4.2 K, which they attributed to the momentum-conserving phonons in indirect band-to-band tunneling. Various phonon energies could be resolved in these characteristics. Chynoweth et al. [3.19, 3.20] then found evidence that the excess current in silicon Esaki junctions, i.e. the current between the tunnel “hump” and the normal forward injection current, is essentially caused by the process of field ionization of impurity levels.

The metal-semiconductor interface is among the most challenging problems in the field of solid-state theory and device physics. A variety of physical phenomena, e.g. the influence of interface states on barrier height [4.20, 4.34], the effect of interfacial layers (dipole, oxide, or contamination) [4.2, 4.6, 4.15, 4.18], inelastic scattering events [4.16, 4.22], recombination, trapping [4.5, 4.10] and trap-assisted tunneling [4.9], vertical and lateral potential fluctuations [4.8], barrier height fluctuations [4.33], interface roughness [4.30], band-state mixing [4.12], realistic image forces, hot carrier effects, and some other issues make the theoretical modeling a complicated task. Simplified contact models, e.g. suitable for device simulation, have to neglect most of all these effects.

In modern microelectronics the transport of carriers across thin and ultra-thin dielectric barriers is of considerable interest. Well-known problems are the highenergy injection of carriers into gate oxides of MOSFETs [5.13, 5.43] leading to a long-term shift of their threshold voltage (so-called degradation), the strong tunnel currents during the erase mode of electrically erasable programmable read only memories (EPROMs) [5.68], the current-voltage characteristics of metal-insulator-semiconductor (MIS) solar cells [5.15, 5.25, 5.69, 5.78], or the tunneling leakage occurring in memory cells [5.5, 5.33]. Apart from a realistic distribution function, the simulation of the current requires a good knowledge of the quantum-mechanical transmission probability.

The presented review of physical models for device simulators which rely on moments of the Boltzmann equation has shown a remarkable gap between the demands for highly accurate and efficient TCAD tools from the side of semiconductor industry and the availability of models that meet these requirements, partly caused by a lack of the fundamental physical understanding. This is somewhat surprising, because silicon has been the basic material of semiconductor research over more than three decades and is now the basic material of the second largest industrial branch. Among the deficiencies the following items are striking: A fundamental quantity like the intrinsic density of silicon is not precisely known. Heavily doped silicon is scarcely understood. This holds true for the bandgap energy, the mobility, and all recombination channels. Device models of bandgap narrowing disagree significantly in their quantitative predictions. There is no unique theory-based bulk mobility model which covers ultra-high doping concentrations and strong compensation. The actual recombination channel at high doping concentrations is not really known. Furthermore, local models of impact ionization or band-to-band tunneling have only a very limited value in modeling the strong nonlocal effects typical for sub-quarter-micron devices. Temperature-dependent models in EB or HD equations provide an approximate nonlocal description, but they also fail as soon as the high-energy tail of the distribution function becomes responsible for the physical effects.