Tutorial: From Semi-Classical to Quantum

Transport Modeling

Dragica VasileskaProfessor

Arizona State UniversityTempe, 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?

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.

The need for semiconductor device modeling

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 Sharfetter-Gummel 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”, Solid-state 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 DENSITY2. velocity saturation

effect3. mobility modeling

crucial

106

107

1 10 100

Current simulationsYamada simulationsCanali exp. data

Dri

ft v

elo

city

[c

m/s

]

Electric field [kV/cm]

1. Particle density2. DRIFT VELOCITY, ENERGY DENSITY3. velocity overshoot effect

problems

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

Drain Voltage [V]

Dra

in C

urr

en

t [m

A/u

m]

1020 cm-3

1019 cm-3

0.1 ps

0.2 ps

0.3 ps

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?

1. Quantum MechanicalTUNNELING

2. SIZE-QUANTIZATIONEFFECT

3. QUANTUM INTERFERNCE EFFECT

The Van Dort model is activated by specifying N.DORT on the MODEL statement.

E0

E1

distance

Energy n(z)

z

z

CONVz

QMz

Classical density

Quantum-mechanicaldensity

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

Model Improvements

Compact models Appropriate for CircuitDesign

Drift-Diffusionequations

Good for devices down to0.5 m, include (E)

HydrodynamicEquations

Velocity overshoot effect canbe treated properly

Boltzmann TransportEquation

Monte Carlo/CA methods

Accurate up to the classicallimits

QuantumHydrodynamics

Keep all classicalhydrodynamic features +

quantum corrections

Approximate Easy, fast

Exact Difficult

Sem

i-cla

ssic

al a

pp

roa

che

sQ

ua

ntu

m a

pp

roa

ches

Quantum-Kinetic Equation(Liouville, Wigner-Boltzmann)

Accurate up to single particledescription

Green's Functions method Includes correlations in bothspace and time domain

QuantumMonte Carlo/CA methods

Keep all classicalfeatures + quantum corrections

Direct solution of the n-bodySchrödinger equation

Can be solved only for small number of particles

Model Improvements

Compact models Appropriate for CircuitDesign

Drift-Diffusionequations

Good for devices down to0.5 m, include (E)

HydrodynamicEquations

Velocity overshoot effect canbe treated properly

Boltzmann TransportEquation

Monte Carlo/CA methods

Accurate up to the classicallimits

QuantumHydrodynamics

Keep all classicalhydrodynamic features +

quantum corrections

Approximate Easy, fast

Exact Difficult

Sem

i-cla

ssic

al a

pp

roa

che

sQ

ua

ntu

m a

pp

roa

ches

Quantum-Kinetic Equation(Liouville, Wigner-Boltzmann)

Accurate up to single particledescription

Green's Functions method Includes correlations in bothspace and time domain

QuantumMonte Carlo/CA methods

Keep all classicalfeatures + quantum corrections

Direct solution of the n-bodySchrödinger equation

Can be solved only for small number of particles

D. Vasileska, PhD Thesis, Arizona State University, December 1995.

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