Transcript Document

Tutorial:
From Semi-Classical to Quantum
Transport Modeling
Dragica Vasileska
Professor
Arizona State University
Tempe, AZ USA
Outline:
 What is Computational Electronics?
 Semi-Classical Transport Theory
 Drift-Diffusion Simulations
 Hydrodynamic Simulations
 Particle-Based Device Simulations
 Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators
 Tunneling Effect: WKB Approximation and Transfer Matrix Approach
 Quantum-Mechanical Size Quantization Effect
 Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum
Moment Methods
 Particle-Based Device Simulations: Effective Potential Approach
 Quantum Transport
 Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical
Basis of the Green’s Functions Approach (NEGF)
 NEGF: Recursive Green’s Function Technique and CBR Approach
 Atomistic Simulations – The Future
 Prologue
What is Computational
Electronics?
The need for semiconductor device modeling
1. Increased costs for R&D and production facilities, which are becoming too
large for any one company or country to accept.
2. Shorter process technology life cycles.
3. Emphasis on faster characterization of manufacturing processes, assisted by
modeling and simulation.
With permission from Intel Corp.
Computer simulations, often called technology for
computer assisted design (TCAD) offer many advantages
such as:
1. Evaluating "what-if" scenarios rapidly
2. Providing problem diagnostics
3. Providing full-field, in-depth understanding
4. Providing insight into extremely complex
problems/phenomena/product sets
5. Decreasing design cycle time (savings on hardware
build lead-time, gain insight for next product/process)
6. Shortening time to market
Some TCAD Prerequisites Are:
 Modeling and simulation require enormous technical
depth and expertise not only in simulation techniques
and tools but also in the fields of physics and chemistry.
 Laboratory infrastructure and experimental expertise are
essential for both model verification and input parameter
evaluations in order to have truly effective and predictive
simulations.
 Software and tool vendors need to be closely tied to
development activities in the research and development
laboratories.
R. Dutton, Stanford University, the father of TCAD.
Historical Development of Device
Simulation:
 1964: Gummel introduced the decupled scheme for the
solution of the Poisson and the continuity equations for a
BJT
 1968: de Mari introduced the scaling of variables that is
used even today and prevents effectively overflows and
underflows
 1969: Sharfetter and Gummel, in their seminal paper
that describes the simulation of a 1D Silicon Read
(IMPATT) diode, introduced the so-called SharfetterGummel discretization of the continuity equation
H. K. Gummel, “A self-consistent iterative scheme for one-dimensional steady state transistor
calculation”, IEEE Transactions on Electron Devices, Vol. 11, pp.455-465 (1964).
A. DeMari, “An accurate numerical steady state one-dimensional solution of the p-n junction”, Solidstate Electronics, Vol. 11, pp. 33-59 (1968).
D. L. Scharfetter and D. L. Gummel, “Large signal analysis of a Silicon Read diode oscillator”,
IEEE Transaction on Electron Devices, Vol. ED-16, pp.64-77 (1969).
Coupling of Transport Equations to
Poisson and Band-Structure Solvers
D. Vasileska and S.M. Goodnick, Computational Electronics, published by Morgan
& Claypool , 2006.
What Transport Models exist?
Semiclassical FLUID models
(ATLAS, Sentaurus, Padre)
Drift – Diffusion
Hydrodynamics
1. PARTICLE DENSITY
2. velocity saturation
effect
3. mobility modeling
crucial
Drift velocity [cm/s]
1. Particle density
2. DRIFT VELOCITY, ENERGY DENSITY
3. velocity overshoot effect
problems
Drain Current [mA/um]
8
7
0.3 ps
6
0.2 ps
5
0.1 ps
4
1020 cm-3
3
2
6
10
Current simulations
Yamada simulations
Canali exp. data
1019 cm-3
1
0
7
10
1
0
0.2
0.4
0.6
0.8
Drain Voltage [V]
1
1.2
10
Electric field [kV/cm]
100
What Transport Models Exist?
Semiclassical PARTICLE-BASED
Models:
Direct solution of the BTE Using
Monte Carlo method
Eliminates the problem of Energy
Relaxation Time Choice
Accurate up to semi-classical limits
One can describe scattering very well
Can treat ballistic transport in devices
Why Quantum Transport?
2. SIZE-QUANTIZATION
EFFECT
The Van Dort model is activated by specifying N.DORT on the
MODEL statement.
1. Quantum Mechanical
Energy
TUNNELING
n(z)
Classical density
z
E1
E0

Quantum-mechanical
density
z CONV
z QM
z
distance
3. QUANTUM
INTERFERNCE EFFECT
What Transport Models Exist?
Quantum-mechanical WIGNER
Function and DENSITY Matrix
Methods:
Can deal with correlations in space
BUT NOT WITH CORRELATIONS IN
TIME
Advantages: Can treat SCATTERING
rather accurately
Disadvantages: LONG SIMULATION
TIMES
What Transport Models Exist?
Non-Equilibrium Green’s
Functions approach is MOST
accurate but also MOST difficult
quantum approach
FORMULATION OF
SCATTERING rather
straightforward,
IMPLEMENTATION OF
SCATTERING rather difficult
Computationally INTENSIVE
Quantum approaches
Semi-classical approaches
Approximate
Easy, fast
Model
Improvements
Compact models
Appropriate for Circuit
Design
Drift-Diffusion
equations
Hydrodynamic
Equations
Boltzmann Transport
Equation
Monte Carlo/CA methods
Good for devices down to
0.5 m, include (E)
Velocity overshoot effect can
be treated properly
Accurate up to the classical
limits
Quantum
Hydrodynamics
Keep all classical
hydrodynamic features +
quantum corrections
Quantum
Monte Carlo/CA methods
Keep all classical
features + quantum corrections
Quantum-Kinetic Equation
(Liouville, Wigner-Boltzmann)
Accurate up to single particle
description
Green's Functions method
Includes correlations in both
space and time domain
Direct solution of the n-body
Schrödinger equation
Can be solved only for small
number of particles
Exact
D. Vasileska, PhD Thesis, Arizona State University, December 1995.
Difficult
Range of Validity of Different Methods
Summary
 Different transport models exist with different
accuracy and different computational needs for
modeling the wide variety of devices that are
used in practice every day
 The goal of Computational Electronics is to
teach you what models are appropriate for
modeling specific device structure and what are
the limitations and the advantages of the model
used