Tutorial: Computational Electronics –
From Semi-Classical to Quantum Transport Modeling
Dragica Vasileska,
Professor
Department of Electrical, Computer and Energy
Engineering
Continuing technological advances
make possible the fabrication of electronic devices with increasing structural
and conceptual complexity, and in an expanding variety of material systems. In
the field of Computational Electronics,
advanced modeling and simulation techniques are created, developed and employed
to assist in the invention, design and optimization of micro-, nano- and opto-electronic devices
and circuits. Research in Computational
Electronics draws upon knowledge from a variety of disciplines,
predominantly solid state physics, quantum mechanics, electromagnetics
and numerical algorithms, to achieve an accurate description all aspects of
device operation.
Birth of Computational
Electronics
Device structure, material
composition, and operating principles are all intimately related. For example,
the characteristic length scale of devices such as resonant tunneling diodes
and quantum dots which rely on coherent quantum effects, are constrained to
just a few nanometers. Most optoelectronic devices exploit heterojunctions
between two or more different materials for confinement of both charge carriers
and light; characteristic thicknesses of absorption or gain regions typically
vary from around one hundred nanometers to several microns. Power electronic
devices, on the other hand, may reach several millimeters in width due to their
current-handling requirements, and are increasingly fabricated using materials
other than silicon in a quest for superior thermal performance and breakdown voltage.
The wide variety of possible applications, material selections, and realizable
device structures make Computational Electronics a broad and exciting field.
Computational Electronics
There are several building blocks that comprise general
physically-based device simulator and these include (1) the Electronic
Structure Module which feeds the carrier dispersion to the (2) transport
module, which in turn is connected to, in general, (3) electromagnetic field
solver, which in the quasi-static approximation reduces to the Poisson
equation. The interchange of variables between the various kernels that
comprise a complete device simulator is schematically illustrated below.
A schematic description of the device
simulation sequence
The transport module, on the other hand, can range from
simple drift-diffusion model to the most complicated but at the same time in
principle most accurate Green’s functions approach. What transport kernel has
to be used depends upon the application at hand. For modeling, for example,
first and second generation solar cells, drift-diffusion models are quite
accurate. If the transport in the device is such that velocity overshoot
effects occur, one in principle has to use hydrodynamic approaches. These approaches
are currently extensively used in industry in designing next generation
transistors. The problem with the hydrodynamic/energy balance approaches is the
inability to properly determine the energy relaxation times as they are both
material and device geometry dependent parameters.
Fluid Transport Models
The use of particle-based device simulators which in the
long time limit solve the Boltzmann Transport equation eliminates these
problems and these approaches are accurate down to the ballistic limit of the
operation of the device. Quantum corrections like quantum-mechanical size
quantization effects and tunneling can be easily integrated into particle-based
approaches.
Monte Carlo Method for the Solution of the Boltzmann Transport Equation
When the problem is such that the device operates in such
way that quantum interference effects dominate (resonant tunneling diode) then
one has to reside on the validity of quantum transport approaches which include
density matrix, Wigner functions, Green’s functions and direct solution of the
many-body Schrodinger equation (which is still prohibitive even with today’s
computer power). Recursive Green’s function approach is applicable to devices
with two contacts. When the problem is such that one simultaneously has to
calculate, for example, the source-drain and gate leakage current, the Contact
Block Reduction method is a better choice. The Usuki
method in the ballistic limit is equivalent to the recursive Green’s function
technique.
In this Tutorial we will give an overview of both semiclassical transport approaches, quantum corrections to semiclassical approaches and quantum transport. The various
topics covered include:
1. Introduction to
Computational Electronics
2. Semi-Classical
Transport Theory
3. The
Drift-Diffusion Equations and Their Numerical Solution
4. Hydrodynamic
Modeling
5. Particle-Based
Device Simulation Methods
6. Modeling Thermal
Effects in Nano-Devices
7. Quantum
Corrections to Semi-Classical Approaches
8. Quantum Transport
in Semiconductor Systems
9.
Far-From-Equilibrium Quantum Transport
10.
Future Developments of Computational Electronics
