Making Simulation Fast, Accurate

Computer simulations of complex electronic devices and new materials, when executed prior to the start of development, can save money and unnecessary process times. “Unlike silicon, many of the new materials are extremely expensive and very difficult to obtain,” said Marco Saraniti, director of the computational electronics group and associate professor of electrical engineering and physics at the Illinois Institute of Technology (Chicago). The impact of advanced materials has made its way into SEMI's current estimates for the materials market with an expected growth of 9% in 2007, reaching $24.0B in sales. The ability to determine material quality and purity before a crystal is grown and, for particular applications, to assess suitability and effectiveness of the design is becoming a necessary part of the development process.

Computer modeling is not new, and performing more simulations is not enough. The continued decrease in device geometries, currently in the realm of tens of nanometers, makes the modeling of device behavior increasingly more complicated. Phenomena such as quantum mechanical interference and single electron effects must be considered. The issue becomes one of obtaining accurate representations with reasonable computer times — on the order of hours instead of days. Researchers at the Illinois Institute of Technology, in collaboration with Arizona State University (Tempe, Ariz.), have created a simulation algorithm, the cellular Monte Carlo (CMC), to meet these stringent requirements.

Methods of simulation

The two types of simulation commonly used are flux-based, also known as drift and diffusion, and particle-based. Most conventional methods are flux-based and considered to be fast, but lack the accuracy needed for newer applications. The flux method solves a flow of charge, rather than points or particles, and can be represented as a mathematical function of the density of charge at a particular velocity. Similar to hydrodynamics, the flux is a continuous function defined on a domain, which is the device. In this sense, flux-based simulations are less-microscopic representations and normally require solving differential equations.

Particle-based simulations provide a more realistic representation of individual electrons localized and in motion. The first particle-based software, Monte Carlo (MC, or stochastic) method, was developed by Enrico Fermi and John von Neumann at Los Alamos and first applied to semiconductor devices by Kurosawa in 1966. It was later extended to realistic band structures by the University of Illinois (Urbana-Champaign) and developed at IBM (Yorktown Heights, N.Y.). The MC technique is a numerical method that simulates non-equilibrium transport in semiconductor materials and devices. The interaction of electrons among themselves, the crystal and impurities present are expressed in quantum mechanical terms, with the decision whether an electron moves left or right following an interaction based on probability.

The MC algorithm generates a random number for each particle in motion, a process similar to that of a roulette wheel in gambling casinos — hence the name Monte Carlo. The trajectories of electrons are computed as sequences of “free flights” occurring under the effect of fields generated by an external bias and the spatial distribution of charges in the system. At the end of each “free flight,” the simulation identifies, through the generation of a random number, the type of scattering (Fig. 1 ) occurring and accordingly changes the final energy and momentum of the electron. The numbers generated are compared with a calculated probability on which a decision is based.

1. Scattering mechanisms in a typical semiconductor.

CMC

The MC algorithm was initially designed with a non-parabolic approximation of the band structure using analytical calculations to relate electron momentum and energy. However, because of the inherent inaccuracies, it is used primarily for low-energy transport applications.

Today, full-band simulations are commonly available, providing a truer representation of material and device behavior. For example, high field effects, such as impact ionization, can be reliably included in the model. To do this, full-band simulations replace analytical approximations with computed band structure information and scattering probabilities stored in tables. An example of the equi-energetic surfaces of the first conduction band of an hexagonal gallium nitride crystal is shown in Figure 2 . The greater accuracy and detail, however, come with a price — long simulations requiring days of computer time. Typically, for every picosecond of simulated device time, hours of computer time are needed.

2. Energy surfaces of the first conduction band of hexagonal GaN are simulated using the cellular Monte Carlo algorithm.

The joint effort of Saraniti's group and the researchers at Arizona State went a step further with their CMC method, taking advantage of the lowering cost of computer memory. Inspired by work done in cellular automaton, CMC applies an old rule in computer programming: memory translates into speed.

Prior to running a simulation, all possible dynamic electronic configurations are stored in a table, eliminating time-consuming computations. Using up to 16 Gb of RAM, the probability of scattering is pre-computed and stored from each initial state to all possible final states. Scattering mechanisms are simply modeled by randomly choosing a final state compatible with the carrier momentum. With CMC, this is done sequentially for each possible outcome. For example, if an electron can be scattered in four possible states (a, b, c and d), the probability of scattering in direction a is calculated and stored, next the probability of direction b is calculated and stored, and so on through all possibilities. In contrast, traditional MC stores the sum of all four probabilities in a considerably smaller table, and subsequently needs to determine the final state after each scattering, a computationally costly process. Finally, with all the probabilities of all the final configurations stored, the random number is compared and the highest probability — the final state and new configuration — is efficiently selected.

Other time-reducing schemes were developed, such as on-the-fly compression techniques. In this way, the 16 Gb table can remain stored and accessed while in a compressed state, similar to a ZIP file. This elegant full-band technique has reduced simulation times by as much as 50× compared with the traditional MC method, providing an equivalently accurate representation of device dynamics.

CMC device simulation

CMC simulations have been performed on various electronic devices and new materials; in particular, high electron mobility transistors (HEMTs), based on hexagonal gallium nitride (GaN). The behavior of a representative portion of the population of electrons, typically 100,000, is modeled with each electron simulated with the full-band CMC scattering algorithm.

In HEMTs, extremely low-noise and high-frequency operation is obtained by confining the transport of charge in a narrow undoped region under the gate. The confinement is obtained by joining two different semiconductors in a heterojunction, with the difference in the respective energy gaps generating a potential well that confines the electrons. The position of electrons within a high-power, high-frequency GaN HEMT is shown in Figure 3 .

3. Simulation of the distribution of electrons in the conduction and valence band profile of a GaN HEMT uses colors to indicate electrostatic potential. Most electrons are confined under the drain (inset).

Of particular interest to the telecommunications sector are GaN HEMTs. Most commonly used semiconductor materials, whether silicon, germanium, GaAs or InP crystallizes, are in a cubic structure, also referred to as Zincblende or diamond structures. Nitride semiconductors, such as InN and GaN, however, can crystallize in cubic or hexagonal crystal structures (Wurtzite structure). A few groups in the world have particle-based simulation tools that can simulate devices built with hexagonal semiconductors, while most others are designed to handle only cubic crystals.

In either crystal structure, the chemistry of the nitrides remains the same. However, different structures will produce extremely different electrical behavior. The Wurtzite GaN, for instance, has a much larger energy gap than the Zincblende one, and is likely to perform better. This is verified with CMC simulations of MESFETs fabricated on each of the two structures. The electrical properties proved dramatically different with better, more interesting results coming from the Wurtzite crystal, which demonstrated higher gain, cut off frequencies and maximum operating frequency.

Future directions

The limit of the MC method is the quantum limit. Electrons can be modeled as quantum-mechanical waves, but when the characteristic size of the devices becomes comparable with the wavelength of electrons, a fully quantum mechanical model will be needed for modeling the charge transport mechanisms.

Despite the effectiveness of algorithms such as CMC, its full range of capabilities has yet to be realized. For example, one area creating a bottleneck in the development of modern VLSI technology is the generation and transport of heat within highly integrated devices. The MC approach may prove crucial in understanding the microscopic mechanisms of heat generation by directly modeling the population of phonons interacting with electrons, and which mediate the exchange of heat between electrons and the crystal lattice, according to Saraniti. With faster simulations times, similar studies will likely be undertaken.