75th Anniversary

of the Transistor

History of Computational

Electronics and

Emerging Trends

Computational Electronics (CE) broadly spans topics related to the modeling and simulation of electronics and optoelectronics, although practically speaking, it mainly focuses on device modeling. It is closely related to Technology for Computer-Aided Design (TCAD), which hierarchically couples device and process simulation together with extraction of compact models for circuit simulation, as part of electronic design automation (EDA). Here we mainly focus on device modeling, which includes nanoscale transistor technology, beyond CMOS technologies (e.g. TFETs, spintronic devices, neuromorphic elements, etc.), wide bandgap power electronic devices, electromechanical devices (MEMS/NEMS), rf and high frequency devices, optoelectronic and energy conversion devices. The overarching goal of CE is to provide an understanding of the behavior of current device technologies and predictions of future new technologies.

Almost all approaches to device modeling require a self-consistent approach involving the simultaneous solution of coupled sets of equations representing charge/heat transport and the electromagnetic fields, solved over some spatial domain, as illustrated in Fig. 1. Here the transport kernel is informed by electronic structure and phonon dynamics. Transport includes both electronic transport and thermal transport, the latter being increasingly important in ultra-scale transistors and power electronics. For high frequency and optoelectronic devices, Maxwell’s equations must be fully solved, whereas for most devices, the quasi-static solutions of Poisson’s equation are sufficient. For device simulation, the transport and field equations are solved on a grid (structured or unstructured). Process simulation couples to define the structure, materials, doping, etc. defining the device. From simulation of the device characteristics (e.g. time dependent currents, potential and charge distribution), a nonlinear equivalent circuit model (compact model) may be extracted for use in circuit simulation at a higher level of abstraction beyond the CE approach.

The main differentiator between CE approaches is in the physical models used for the transport, illustrated in Fig. 2. While somewhat oversimplified, this figure reflects the tradeoff between increasing physical accuracy versus computational cost, going from classical continuum models to quantum mechanical descriptions [1]. The semi-classical description of transport is given by the Boltzmann transport equation (BTE), which describes the time evolution of the distribution function for the position and momentum of particles, including random scattering processes. Continuum models are derived from moments of the BTE [2], leading to partial differential equations for e.g. the charge density, which in its simplest form are the drift and diffusion-based semiconductor equations going back to Van Roosbroeck [3]. Monte Carlo particle-based simulation is the direct stochastic solution of the BTE at the semi-classical level. The most physically accurate models are based on quantum transport, which contains all the quantum mechanical correlations leading to non-classical behavior such as quantization, tunneling, and quantum interference, which we discuss in more detail below.

In the early days of semiconductor technology, the electrical device characteristics of MOSFET devices were described using simple analytical models, such as the gradual channel approximation, that relied on the drift-diffusion (DD) formalism. Numerical simulation of carrier transport in semiconductor devices was enabled by Scharfetter and Gummel [4], who proposed a robust discretization of the DD equations, which is still in use today. In the DD approach, the electron gas is assumed to be in thermal equilibrium with the lattice temperature, which is not true in nanoscale devices [5]. Due to the short dimensions, high fields, and short transit times, non-stationary (e.g. quasi-ballistic) transport may dominate, which is not described by the local DD equations. Bløtekjær [6] derived conservative hydrodynamic (HD) equations using higher moments of the BTE, which partially captures non-stationary effects like velocity overshoot. Since self-heating effects significantly influenced the transport characteristics of devices with channel lengths of 200 nm and below, heat flow equations have been added to the DD transport models, see for example Wachutka [5]. A fully thermal hydrodynamic model was introduced by Benvenuti and co-workers [7].

Historically, most CE software was developed at universities or corporate research laboratories, and later commercialized by TCAD companies, starting with DD/HD-based codes in the late 1970s and 1980s. The PISCES device simulation code [8] was developed at Stanford University (together with SUPREM for process simulation), and commercialized through Technology Modeling Associates (TMA) in 1979. At the same time, MINIMOS was developed at the Technical University of Vienna [9]. DESSIS [10], a fully hydrodynamic simulator developed between the University of Bologna, ETH Switzerland, and SGS-Thompson, was introduced in 1996. All these codes underwent continual improvements incorporating thermal modeling, 3D simulation capabilities, etc. Silvaco Inc. later licensed PISCES (then MEDICI) and SUPREM4 in 1989 to offer the fully integrated commercial platforms ATLAS and ATHENA, for device and process modeling. The successor to TMA was meanwhile acquired by Synopsys Inc., who later acquired the DESSIS code in the mid-2000s, resulting in the Sentaurus TCAD platform. Other TCAD companies include Comsol, Crosslight, Cogenda, Global TCAD, and others.

Ensemble Monte Carlo (EMC) methods were developed starting in the 1960s for the direct stochastic solution of the BTE to explain high-field phenomena such as the Gunn effect. The early developments of the method starting in the UK and Italy, have been extensively reviewed [11]. In contrast to the DD/HD approach, EMC is a microscopic simulation method, where the semi-classical trajectories of an ensemble of particles subject to driving fields are simulated using the computer random number generator, which is used to generate scattering events and the final state after scattering, based on the quantum mechanical probabilities due to different scattering mechanisms (phonons, impurities, etc.). Since transport is based on an underlying microscopic description, macroscopic observables such as carrier mobility and drift velocity are determined microscopically rather than from empirical models. Early work through the mid-1980s was based on simplified multi-valley band structures. Full-band EMC was first developed at the University of Illinois (see [12]), where the full electronic structure of materials was employed. DAMOCLES [13] was the most well-known full-band EMC code developed at IBM research labs in the early 1990s, which was also a device simulation tool. Device simulation with Monte Carlo is more complicated since one has to map a discrete charge representation into continuum solutions of Poisson’s equations [14]. In full-band EMC, there is a computational bottleneck in determining the final state after scattering within a complicated energy band structure. Computational methods have subsequently been developed based on optimized search algorithms [15] and using pre-tabulated scattering rates across the full Brillouin zone of electronic states in a Cellular Monte Carlo (CMC) approach [16], where greatly improved computational speed is obtained by using significantly more memory.

A recent trend in full-band device simulation has been to leverage the rapid advances in *ab initio* structure calculations over the past decade. There are now several commercial/open-source based packages (e.g. VASP, Quantum Espresso), which in addition to providing first principles electronic (phononic) structure information, also increasingly provide information scattering processes, such as first principles electron-phonon interactions [17], and impact ionization [18]. Full-band EMC is particularly important in recent years for wide bandgap semiconductors such as GaN and diamond for power electronic applications, where very high fields are experienced. Full-band simulation of non-bulk materials such as 2D materials and nanostructures is another trend in the field. Another recent trend has been to combine EMC and thermal transport. At nanoscale dimensions where dimensions are shorter than the phonon mean free path, Fourier’s law fails, and more exact models for phonon transport are needed. One approach is to combine moment expansion of the phonon BTE with electronic EMC for electrothermal transport [19]. More recently, full dispersion particle-based simulation of phonon transport coupled to EMC for electrons and holes has been reported [20].

As the characteristic length scales in semiconductor devices have shrunk into the decanano scale, quantum mechanical effects associated with the wave nature of matter became important for dimensions shorter than the electron *phase coherence length* [21], where quantum mechanical effects such as quantum confinement, tunneling, and quantum interference start to dominate. Real electron devices must possess at least two terminals, contacts, or leads, and hence every device is an *open system* with respect to carrier flow [22].

In nanostructures [24], electron transport is often ballistic, where electrons are transmitted through the device without undergoing energy relaxation, which occurs in the contacts. Early on, the Landauer—Büttiker [23], [24] formalism provided a quantum flux-based description of current in terms of transmission probabilities to describe this ballistic regime. Various transfer and scattering matrix approaches were developed in the 80s and 90s for modeling mesoscopic devices, where appropriate scattering boundary conditions for open systems need to be employed [25]. An efficient variant of the scattering matrix approach is the Usuki method [26].

Due to their connection to scattering matrices, Green’s function techniques were developed for quantum transport in mesoscopic systems, with the coupling to the leads being introduced via the self-energy. A very efficient and widely used algorithm is the recursive Green’s function method [27], which became the basis of the NEMO 1-D code that was developed at TI/Raytheon [28] based on the nonequilibrium Green’s function (NEGF) approach. It included full-band structure through the use of atomistic tight binding methods and elastic scattering processes, to successfully describe the behavior of resonant tunneling diodes. The incorporation of inelastic scattering processes into the NEGF formalism is critical for modeling real nanoscale technologies such as scaled CMOS, tunnel FETs, 2D materials, etc., and was implemented in NEMO-5 [29]. The NEMO-5 code was recently acquired by Silvaco TCAD for commercialization. Another important NEGF code was developed independently in Europe—the Atomistix ToolKit later QuantumWise combined a NEGF transport simulator with first principles DFT [30]. QuantumWise was recently acquired by Synopsys for integration with their Sentaurus TCAD products.

Finally, an alternate popular approach to quantum transport related to CE is based on the Wigner function approach [31], which is a kinetic equation approach similar to the BTE, except that the Wigner distribution is non-classical and can be negative. Scattering processes can be included through a collision integral similar to the BTE although non-local. The main advantage is that it is explicitly time-dependent and can describe entangled systems in general of relevance to recent advances in quantum information science.

To conclude, Computational Electronics has more than fifty years of history and continues to grow rapidly due to the enormous growth of computational power through hardware and efficient algorithms. We briefly touched on three main areas of active development in terms of continuum, particle-based, and quantum approaches to transport problems. A broad trend in recent years has been the emergence of TCAD companies and commercial tools, and the continued aggregation of tools across all levels of transport discussed in the present article, into commercial platforms. Certainly, new tools will emerge in the future to meet the challenges not only of a post-CMOS world but of dealing with the rapidly emerging quantum information technologies. It may even be the case that in the future, quantum computing-based CE tools will be developed to simulate what are presently intractable quantum problems.

Biography

**Stephen M. Goodnick** (M 1987; SM 1990; F 2004) is currently the David and Darleen Ferry Professor of Electrical Engineering at Arizona State University. He received his Ph.D. degree in electrical engineering from Colorado State University, Fort Collins, in 1983, respectively. He was an Alexander von Humboldt Fellow with the Technical University of Munich, Munich, Germany, and the University of Modena, Modena, Italy, in 1985 and 1986, respectively. He served as Chair and Professor of Electrical Engineering at Arizona State University, Tempe, from 1996 to 2005. He served as Associate Vice President for Research for Arizona State University from 2006-2008 and presently serves as Deputy Director of ASU Lightworks as well as the DOE ULTRA Energy Frontier Research Center. He is also a Hans Fischer Senior Fellow with the Institute for Advanced Studies at the Technical University of Munich. Professionally, he served as President (2012-2013) of the IEEE Nanotechnology Council, and served as President of IEEE Eta Kappa Nu Electrical and Computer Engineering Honor Society Board of Governors, 2011-2012. Some of his main research contributions include analysis of surface roughness at the Si/SiO_{2} interface, Monte Carlo simulation of ultrafast carrier relaxation in quantum confined systems, global modeling of high frequency and energy conversion devices, full-band simulation of semiconductor devices, transport in nanostructures, and fabrication and characterization of nanoscale semiconductor devices. He has published over 450 journal articles, books, book chapters, and conference proceedings, and is a Fellow of IEEE (2004) for contributions to carrier transport fundamentals and semiconductor devices.

References

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[31] D. K. Ferry and M. Nedjalkov, *The Wigner Function in Science and Technology*, IOP Publishing, 2018.

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daniel.tomaszewski@imif.lukasiewicz.gov.pl