1.1 History of Three-Dimensional Device Simulation

In 1964 Gummel [18,19] suggested for the first time a fully numerical model of a one-dimensional bipolar transistor based on partial differential equations proposed by van Roosbroeck [20]. From this time numerical solutions have been extensively investigated and continuously improved. After a short break of a few years, in 1968, De Mari adopted this approach to $p\,n$-junctions for the steady state case [21] and for the transient case [22]. The first two-dimensional numerical simulation was done by Loeb [23] and, in parallel, by Schroeder and Muller [24] solving Poisson's equation for a MOS transistor. In the following years several thousand papers focusing on device simulation have been published for various devices. The first papers using three-dimensional simulation were published in 1980 studying narrow channel effects of MOS transistors [25].

One of the first three-dimensional device simulators was the FIELDAY program [25,26,27] of IBM announced in 1981 which solved Poisson's equation and both carrier continuity equations. The physical models included are, e.g., Shockley-Read-Hall and Auger recombination, avalanche and photo-generation, band-gap-narrowing, and Schottky contacts [8]. Moreover, various mobility models for FET and bipolar applications are available. The FIELDAY program was enhanced later on to include, e.g., the hydrodynamic energy-balance equations, Fermi-Dirac carrier statistics, the lattice energy equation, and incomplete ionization. Thus, both drift diffusion-simulation and hydrodynamic simulations were possible. Additional models have been announced like Fowler-Nordheim gate oxide tunneling or position dependent minority carrier lifetimes [28].

The input data have to be generated by external tools in a pre-processing step. The FIELDAY program uses a finite element mesh composed of prismatic and tetrahedral elements. The mesh is generated by tools like MESH3D [28]. The doping is applied to the mesh by another tool called DOPING [28] which can be used to generate analytical doping profiles as well as profiles from measured data. Simulation results from FIELDAY can be visualized by separate post-processing programs like FEMPLOT, VIDS [28], or IBM Data Explorer [29].

The FIELDAY program has been used for typical applications such as simulations of narrow channel effects in IGFET structures [25], VMOS structures [30], and trench-bounded MOSFET[31], which can be found in DRAM cells. The devices may consist of up to $100,000$ nodes and are typically simulated on IBM workstations like RS/6000.

In the same year, the device simulator WATMOS [32,33] was announced. A finite difference scheme was used for a numerical solution of Poisson's equation. WATMOS was used for investigations on narrow channel effects of MOSFETs.

In 1984 SILVACO was founded and presented their parameter extracting tool UTMOST. Today SILVACO is one of the larger companies for development of TCAD software. Their device simulator ATLAS is discussed in Section 1.2.4.

Another simulator was developed by Toshiba in 1985, called TOPMOST [34,35]. TOPMOST was designed to analyze MOS structures and solved the semiconductor equations for the drift-diffusion case. Previous investigations have been reported in [12]. One- and two-dimensional simulations could be performed by pseudo-one- and pseudo-two-dimensional models of the device by reducing the number of points in the omitted directions to $2$, respectively.

TOPMOST uses a nonuniform mesh with automatic refinement for critical regions such as below the gate oxide and in $p\,n$-junctions. The mesh may consist of boxes, prisms or tetrahedras.

The simulator TOPMOST was used to study the influence of the gate structure on the output characteristics and subthreshold swing in three-dimensional MOS devices [34,35].

At the same time CADDETH (Computer Aided Device DEsign in THree dimensions) [36] which had been developed at Hitachi for several years was announced as a three-dimensional simulator designed to run on a supercomputer. Both Poisson's equation and two current continuity equations are solved using Conjugate Gradient-based methods for non-symmetric linear systems. CADDETH distinguishes between three different materials, namely semiconductors, insulators, and metals. The physical models included are Shockley-Read-Hall and Auger recombination, avalanche and photo-generation. Other models used are band-gap-narrowing and impurity or field dependent mobility.

CADDETH consists of four subsystems: INPUT, PRE, MAIN and POST. The module INPUT transforms a free input format into a fixed format and performs syntax checking. The mesh structure, doping profiles, and materials are generated by the tool PRE. The actual simulation is performed by the MAIN module which writes the simulation results to output files. The visualization tool POST reads the files written by MAIN and displays the device structure and the output quantities graphically. CADDETH was applied to several structures like MOS and bipolar transistors and on memory cells.

In 1987 Thurner and Selberherr announced a new three-dimensional device simulator MINIMOS Version 5 [37] which is the successor of the two-dimensional simulator MINIMOS Version 4. At that time, MINIMOS contained most sophisticated models for carrier transport in MOS transistors and as one of the first three-dimensional device simulators [14,16,17,37], it is still one of the fastest device simulators for MOSFET structures, SOI transistors, and Gallium Arsenide MESFETs. MINIMOS Version 6 supports, e.g., transient analysis and a Monte Carlo model to replace the drift diffusion approximation in critical device areas. The fundamental semiconductor equations, consisting of Poisson's equation and two carrier continuity equations, are solved numerically in two- or three-dimensional space, respectively. Finite differences are employed for space discretization, and for time discretization the Backward Euler method is applied. MINIMOS is able to simulate planar and non-planar structures as well. Various carefully chosen models are implemented like carrier mobility, carrier generation/recombination, and band-to-band tunneling.

MINIMOS consists of a grid generator which generates the initial three-dimensional grid by solving Poisson's equation and supports automatic grid refinement. Moreover, AC-small signal analysis and transient simulations with an adaptive time step control can be performed. MINIMOS has been mainly applied to investigate narrow channel effects and subthreshold behavior of MOS transistors.

The tool SMART [38,39] was developed in 1987 at Matsushita Electric Industrial Co. and was a combined three-dimensional process and device simulator. SMART solved Poisson's equation and the current continuity equations. Physical models included Shockley-Read-Hall and Auger recombination, several mobility models, impact ionization, and band-gap-narrowing. The simulator uses tensor product grids and has been applied to , e.g., avalanche breakdown simulations of LOCOS isolated MOSFETs.

In 1988 the device simulator SITAR, which was developed by Siemens to investigate trench-type device structures, was introduced. It solves Poisson's equation and both carrier continuity equations, including steady state and transient analysis [8]. SITAR was mainly applied on leakage investigations of DRAM Cell structures.

In 1989 another three-dimensional device simulator was developed at Microelectronics and Computer Technology Corporation, called MAGENTA [40]. In its first version it simply solved Poisson's equation; versions including the carrier continuity equations and models for generation and recombination processes have been announced for version 3. MAGENTA has been used to study three-dimensional effects in floating-gate EPROM devices.

In the same year Texas Instruments presented their three-dimensional device simulator SIERRA [41] which has been initiated in 1987. It solves Poisson's equation and the carrier continuity equations for static, AC-small signal, and transient cases. The simulator was developed to investigate integrated circuit devices and reliability issues. Physical models such as impurity and field dependent mobility and impact ionization have been included. The SIERRA program consists of a grid module which uses triangular prism elements, only.

In 1990 Technology Modeling Associates (TMA) announced DAVINCI, their first three-dimensional device simulator, which was based on SIERRA. DAVINCI has been used to investigate influences of radiation on DRAM cells or narrow channel effects in MOS structures. In January 1998 Avant! merged with TMA and took over the simulator which has been the starting basis of TAURUS-DEVICE (see Section 1.2.3).

At the University of Bologna, the device simulator HFIELDS has been developed within an ESPRIT project in 1989. Its successor HFIELDS-3D [42,43] is capable to solve Poisson's equation and both carrier continuity equations. For the solution of the linearized equations iterative linear solvers, namely, ICCG and ILU-CGS are used. HFIELDS-3D allowed the definition of up to 10 semiconductor and insulator segments to describe the device structure [8].

The equations are discretized following the Box Integration Method which has been applied on prismatic grid elements. For the grid generation HFIELDS-3D uses a quad-tree-based grid generator with prismatic elements. The grid is first created in a two-dimensional plane using the program ATMOS [42]. In a second step the grid is dispersed by a one-dimensional grid in in the third dimension using a one-dimensional grid generator. Therefore, the shape of the devices can only be described in a two-dimensional plane. This shape proceeds in the third dimension. Grids with up to about $21,000$ grid points have been reported [44]. HFIELDS-3D has been used to study narrow channel effects in MOS transistors and floating gate EPROM and E2PROM cells [43,44].

Another three-dimensional device simulator is SECOND [45,46,47] released in 1990 which is the successor of MEDES, both developed at the ETH Zürich. SECOND solves Poisson's equation and both carrier continuity equations in steady state and for the transient case. The physical models implemented include Shockley-Read-Hall and Auger recombination, band-gap-narrowing, impurity and field dependent mobility models, and a global device temperature.

The grid is generated by the external tool OMEGA [48,49] which is the successor of GEN. OMEGA is an octree-based mixed element grid generator and uses several different types of elements such as tetrahedras, prisms, pyramids, or boxes. The simulation result can be visualized using PICASSO which is the successor of DISP. SECOND has been used to simulate narrow channel effects and the corner effect of MOS transistors. Simulations with about $50,000$ grid points are reported [50].

In the same year STRIDE [51], the three-dimensional device simulator of Stanford University, has been introduced. STRIDE solves Poisson's equation and both carrier continuity equations. Physical models like impurity dependent mobility or band-gap-narrowing are included. STRIDE has been implemented for message passing multiprocessors and is capable to solve structures with about $100,000$ grid points as published in [51]. Typical applications have been latch-up analyzes of MOS transistors.

In 1991 Adamsone, Polsky, and Shur announced another three-dimensional device simulator ALPHA-3 [52] which solves Poisson's equation and both carrier equations and enables steady-state and transient analysis. Models like Shockley-Read-Hall and Auger recombination, band-gap-narrowing, impurity or field dependent mobility are implemented in the simulator. A tensor-product grid is used. ALPHA-3 has been used to study narrow channel effects of MOS transistors with a grid of approximately $14,000$ points.

One year later the development of DESSIS (see Section 1.2.2) was started within an ESPRIT project. In 1993 ISE (Integrated Systems Engineering AG) was founded as a spin-off by the IIS ETH Zürch members and took over the simulator.

The three-dimensional device simulator FLOODS [53] has been developed at the Department of Electrical Enginieering of the University of Florida by Law and his students in 1994. FLOODS and FLOOPS (Florida Object-Oriented Device/Process Simulator) [54] rely on SUPREM, which is a process simulator originally developed at Stanford University. FLOODS uses an object-oriented approach and uses TCL/TK extended with flexible and modular libraries for an object-oriented view of the problem under consideration.

Some commercial device simulators have been announced which have been continuously improved and are part of in-house TCAD packages. These simulators are discussed in the following section.

All simulations described so far are based on analytical solutions of Boltzmann's equation in the form of the drift-diffusion or the hydrodynamic equation. A different approach is shown in [55]. Instead of solving the semiconductor equations Monte Carlo device simulations have been performed for three-dimensional MOSFET-simulation. Due to the high computational effort the simulation was carried out on a $65,536$ processor Connection Machine.

The latest reported three-dimensional simulator is ATAR [56], developed in 2001 at the Department of Electronics at Carleton University. ATAR is a thermal simulation tool which automatically generates three-dimensional models of the device to be simulated from layout information. The mesh is automatically generated and is based on a rectangular octree grid.

With the ongoing reduction of the feature sizes, atomistic simulations become reasonable. They allow to study the effects of random dopand distribution in the channel region, as depicted in [57,58,59].

In general, three-dimensional problems give large system matrices. The solving process is very memory and time consuming compared to two-dimensional simulations. Only ten years ago, when personal computers have been to weak to carry out those simulations, super-computers have been chiefly used [28,60,61]. Moreover, many investigations have been done to parallize the solving process [9,51,60,62,63].