3.1 A Hierarchy of Simulation Strategies

Gordon Moore, one of the founders of Intel Corporation, predicted an exponential increase in computing power available in the mid 1960s [55]. He stated the transistors on a chip to roughly double every eighteen months. This rule, which has become known as Moore's law, has turned out to be true for the last four decades.

Technology Computer-Aided Design (TCAD) has become crucial to maintain the continuous gain in transistor performance, which itself enables more and more powerful simulation tools for engineering applications. While device design solely based on an experimental basis results in long development cycles and enormous costs, simulation offers a fast and inexpensive way to check and optimize device structures as well as their fabrication processes. With the tremendous progress in mainstream CMOS design as a strong background, TCAD has also reached the development process of novel and -- from a CMOS point of view -- more exotic devices.

The tools for numerical simulation in semiconductor device modeling can be separated in three categories of hierarchical order as sketched in Fig. 3.2. Accurate device simulation is based on an appropriate description of the underlying materials. Careful measurements of basic physical and chemical material properties are a valuable foundation of several simulation strategies for material description. Pseudopotential methods [56,57] deliver the band structure as well as its temperature dependence later used in bulk full band Monte Carlo simulations, which themselves calculate parameters like carrier mobility and energy relaxation time for high order transport models. Numerical simulation in material science not only performs parameter extraction for several device simulation strategies, but also enables directed research for a possible synthesis of certain materials with desired parameters. Practical application of first principle methods based on density functional theory (DFT) has been made possible in recent years by the ongoing increase of available computing power [58].

Depending on the needs of the device structure to be described, several models of different complexity can be derived. While all approaches have Poisson's equation in common, the transport description can be based on differently sophisticated physical principles. The simulation of devices in the nanometer regime demands quantum-mechanical formulations, which are either based on the Schrödinger [59,60,61] or the Wigner equation [62,63]. Quantum effects can be safely neglected in larger devices, and thus a semi-classical approach based on Boltzmann's equation may be chosen [64]. The Boltzmann transport equation can be either solved by Monte Carlo methods [65], by the expansion of the distribution function into its spherical harmonics [66,67], or by moment-based methods [68,69]. In contrast to the other approaches, the Monte Carlo method is based on a computationally rather expensive statistical approach. The Full band Monte Carlo method is by now the physically most rigorous approach to solve the Boltzmann transport equation, because it relies on the exactly calculated band structure, and is thus frequently used as a reference [70,71].

Moment-based methods can be carried out to different orders. Depending on the number of equations derived, the drift-diffusion model, energy transport models, or models of even higher order can be achieved. The drift-diffusion model has been TCAD's workhorse for many years due to its outstanding numerical robustness and performance.

Figure 3.2: Hierarchy of simulation approaches. Physically rigorous simulation approaches as well as measurement data are used to parametrize device simulators, which can be based on either quantum mechanical or semi-classical approaches. Device simulation results itself can be used to develop compact models for circuit simulation.

From physically-based device simulation, empirically derived compact models can be derived for lumped simulation of entire circuits. Their parameters are obtained by parameter extraction from device simulation. Mixed-mode simulation tools combine the approaches of device simulation and compact modeling. Thereby, device level simulation of one or more devices coupled with a set of devices described by compact models is possible.

M. Wagner: Simulation of Thermoelectric Devices