Even though higher-order transport models like the energy-transport model were proposed more than forty years ago, the classic drift-diffusion model remains the most commonly used macroscopic transport model today. This can be attributed to the fact that available energy-transport models contain many more unknown parameters and do not give significantly better predictions compared to the simpler drift-diffusion model. Interestingly, even with the best parameter choice available, the energy-transport model gives an improvement over the drift-diffusion model only for channel-lengths in the range from 300 nm down to 100 nm. Below 100 nm, however, the energy-transport model delivers too high terminal currents which are as inaccurate as the underestimated currents delivered by the drift-diffusion model. Unfortunately, though, the distributed quantities inside the devices, like the average velocity, are not well reproduced by the drift-diffusion model because non-local effects are neglected.

To overcome these limitations we considered macroscopic transport models based on the first six moments of Boltzmann's equation, which are a natural extension to the drift-diffusion model (two moments) and the energy-transport models (three or four moments). In addition to the solution variables of the energy-transport model, which are the carrier concentration and the average energy, the six moments model provides the kurtosis of the distribution function, which indicates the deviation from a heated Maxwellian distribution. The knowledge of the kurtosis allows us to model non-equilibrium processes like hot carrier tunneling and impact ionization with significantly improved accuracy compared to lower-order models.

All model parameters are obtained from bulk Monte Carlo simulations, which give a fit-parameter free transport model and leaves us with "no knobs to turn." Having too many adjustable parameters is a particular inconvenience inherent in many energy-transport models based on analytical models for the mobilities and relaxation times. This was found to be essential for higher-order models since the interplay between the various parameters is highly complex and the numerical stability of the whole transport model depends significantly on the choice of these parameters. In particular, the Monte Carlo based model outperformed its counterparts based on analytical mobility models significantly, both in terms of its numerical properties and in the quality of the simulation results.

A comparison of simulated terminal currents obtained from the drift-diffusion, energy-transport, and six moments models with full-band Monte Carlo simulations for a 50 nm double gate MOSFET is shown in the figure. The typical terminal current overestimation of the energy-transport model and the underestimation of the drift-diffusion model are clearly visible. The six moments model, on the other hand, stays close to the Monte Carlo results, even at such short channel lengths. The internal quantities like the velocity show a much better agreement with Monte Carlo data, which makes the six moments model a good choice for TCAD applications.