5.3 The Monte Carlo Method

Numerical techniques allow a solution of the Boltzmann transport equation with a rigorous treatment of the collision terms. Among them, the Monte Carlo (MC) method is well established [Kosina00b]. In the MC method, trajectories of charge carriers undergoing scattering events are calculated numerically both in real- and in momentum space under the influence of an external electric field. The method uses random numbers to select a scattering event and to determine the time of free flight. Because of its statistical nature, the calculation of a large number of trajectories is required to determine the average values of interest with a certain precision. In fact, this is the main disadvantage of the MC method as compared to deterministic solution methods of the Boltzmann transport equation. However, especially for the simulation of hot-carrier transport phenomena in two- or three dimensions MC is the method of choice, since complex physical models can be taken into account.

After the pioneering work of Kurosawa in 1966 [Kurosawa66], who was the first to apply the MC method to simulate carrier transport in semiconductors, a significantly improved MC method was successfully applied to transport calculations in a variety of semiconductors [Jacoboni83]. For electrons in Si, the most thoroughly investigated case, it is believed that a satisfactory understanding of the band structure and of the basic scattering mechanisms has been achieved giving rise to a "standard model" [Fischetti96b].

For predicting performance of modern CMOS bulk and SOI devices an accurate MC evaluation of carrier transport properties in inversion layers is of primary importance. Due to the strong confinement of carriers in the inversion layer of MOSFETs or due to the geometric confinement in multi-gate FETs the carrier motion is quantized in one or two confinement directions giving rise to the formation of subbands. The MC approach may incorporate the subband structure to describe the quantized carrier motion in the direction orthogonal to the current. The subbands are calculated by the self-consistent solution of the corresponding Schrödinger and Poisson equation. The free carrier motion within each subband may still be considered semiclassical and therefore can be well described by the corresponding Boltzmann equation for the subband distribution function $f_n({{\ensuremath{\mathitbf{r}}}},{{\ensuremath{\mathitbf{k}}}},t)$, where $\mathitbf{k}$ is a 2D wave vector. Because of possible carrier scattering between different subbands, the collision integrals on the right-hand-side of the Boltzmann equation have to include the terms responsible for the intersubband scattering processes. The transport in the inversion layer of a MOSFET is finally described by a set of Boltzmann equations for every subband, coupled by the intersubband scattering integrals. The set of the subband Boltzmann equations for $f_n({\bf r},{\bf k},t)$ is conveniently solved by a MC method. Therefore, this approach combines the advantages of a quantum description in confinement direction with a semiclassical description in transport direction.

In the following, from the set of scattering models included in the simulator VMC only the electron phonon scattering mechanisms and their adaptation for the 2DEG are discussed in detail, whereas we refer to the documentation of VMC [VMC2.006] for a complete description of implemented scattering models. Additionally, the models for surface roughness scattering and dielectric screening of the 2DEG will be described.

Subsections

• 5.3.1 Bulk Scattering Mechanisms
  • Acoustic Intravalley Scattering
  • Optical Intravalley Scattering
  • Intervalley Scattering
  
  
• 5.3.2 Scattering Mechanisms in the 2DEG
  • Acoustic Intravalley Scattering
  • Intervalley Scattering
  • Surface Roughness Scattering
  
  
• 5.3.3 Coupling to the Schrödinger Poisson Solver

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology