Explain the equations, approximations and techniques available for deriving a model with specified properties, for a general device characteristic with known qualitative theory

Apply suitable approximations and techniques to derive the model referred to above starting from drift-diffusion transport equations (assuming these equations hold)

Offer clues to qualitative understanding of the physics of a new device and conversion of this understanding into equations

Simulate characteristics of a simple device using MATLAB, SPICE and ATLAS / SYNOPSYS

Explain how the equations get lengthy and parameters increase in number while developing a compact model

List mathematical functions representing various non-linear shapes

Module no.

Module Learning Outcomes

No. of (Total) Hours

0

Motivation, Contents and Learning Outcomes

1

1

Introduction

At the end of this module you should be able to

State the constituents of a device model

Recognize the importance of approximations in a model

Recognize the various stages of IC design where device models are used

Distinguish among activities of analysis, modeling, simulation and design

Transform the equivalent circuit form of a device model into a mathematical form, and vice-versa

Recognize how the equations get lengthy and parameters increase in number while developing a model

1

2

Semi-classical Bulk Transport – Qualitative Model

At the end of this module you should be able to
explain qualitatively the following in semiconductors

The reason for terming certain mechanisms of carrier motion as semi-clasical

The concepts of scattering, effective mass and carrier temperature

The phenomena of ohmic transport, velocity saturation, velocity overshoot and ballistic transport of carriers

The series of approximations leading to the drift-diffusion carrier transport formulation starting from the concept of carriers as particles in random thermal motion

5

3

Semi-classical Bulk Transport – EM field and Transport Equations

At the end of this module you should be able to

Write the equations of electromagnetic field driving the device current, namely

Maxwell’s wave equations and their quasi-static approximation

Lorentz force equation

Recognize the four approaches of determining the device current, developed out of the individual carrier and ensemble viewpoints, in each of which the carrier can be treated either as a particle or as a wave

Recognize that the equations of carrier transport in semiconductor devices have a common form which manifests conservation of some physical quantity

Write the fundamental equations of determining the device current based on each of the following: Schrodinger equation, Newton’s second law and Boltzmann Transport Equation (BTE)

Write the equation for lattice temperature or heat flux, and recognize its necessity for determining the device current from the ensemble point of view

Derive the approximations of the BTE, namely:

carrier, momentum and energy balance equations

drift-diffusion and thermoelectric current equation

Apply the balance equations to derive expressions for the velocity-field and velocity overshoot characteristics

8

4

Drift-Diffusion Transport Model – Equations, Boundary Conditions, Mobility and Generation / Recombination

At the end of this module, you should be able to write, for the widely used drift-diffusion transport model,

its three coupled equations in electron concentration, n, hole concentration, p, and potential ψ

the conditions imposed on n, p and ψ at the contacted and non-contacted boundaries of the device, to solve he coupled equations

the equations for field dependent mobility in bulk and inversion layers

the equations for different generation-recombination mechanisms

5

5

Characteristic times and lengths

At the end of this module, you should be able to

State the characteristic times and lengths associated with

the bulk carrier population under equilibrium

the relaxation of disturbance in

* carrier momentum and energy
* excess EHP concentration
* space-charge

the transit of an average carrier across the device length

State the conditions (including those at the boundary) and the defining differential equation associated with each characteristic time and length

State the order of magnitude of, and factors governing, the characteristic times and lengths

State how the characteristic times and lengths are useful in

qualitative description of device phenomena

simulation and characterization of devices

validation of approximations

Derive the defining equations associated with the various characteristic times and lengths

Identify approximations which will simplify, decouple or eliminate any of the equations of carrier transport, e.g. DD equations, balance equations etc.

Express the qualitative analysis using graphs of n, p, J_{n}, J_{p}, E, ψ versus x, t

6

6

Energy band diagrams

At the end of this module you should be able to

Explain how the wave nature of electrons restricts the allowed energy, ε, of electrons subjected to a periodic potential, to certain energy bands

Outline the features, methods of determination and utilities of ε-k and ε-x diagrams of a semiconductor

Explain the concept of crystal momentum

Determine the effective mass, group velocity and crystal momentum of an electron, having an energy ε, from theε-k diagram

Sketch and explain the ε-k diagrams of Si and GaAs

Determine and sketch the ε-x diagram of a uniform semiconductor under the following conditions:

equilibrium, for any doping and temperature

uniform volume generation

applied bias

Determine and sketch the ε-x diagram of a spatially non-uniform semiconductor under equilibrium, in which the doping and composition change

abruptly at a point (hetero-junction)

gradually

Explain how the energy band diagram can be used to derive the exponential

increase in the diode current with forward bias

Sketch and explain the ε-x diagram of a p+ n junction under high forward bias

Sketch the 1-D band diagram (ε-x) and interpret a 2-D band diagram (ε -x, y) of any device

7

7

SQEBASTIP: The Nine Steps of Deriving a Device Model

At the end of this module, you should be able to

Describe the nine steps for deriving a device model

Apply the nine steps to derive the model of a spreading resistance

Name the requirements of an elegant model

Identify the variables, constants and parameters of a model

Organize the approximations associated with a device model into a specific tabular form

Express an equation in a normalized form

3

8

Types of Device Models

At the end of this module, you should be able to

describe the classification of device models based on the

time rate of change or frequency of voltage / current variation

amplitude of voltage / current variation

starting point of the derivation

attributes of the solution technique

attributes of the mathematical function

attributes of the parameters and constants

- application

interpret, from the jargon employed, the type of model being addressed in any literature on device modeling

use the appropriate jargon to convey a particular type of device model

2

9

MOSFET Model: Structure and Characteristics, Qualitative Model

---- Under development ---

10

MOSFET Model: Equations, Boundary Conditions and Approximations

---- Under development ---

11

MOSFET Model: Surface Potential based and Threshold based solutions

---- Under development ---

12

MOSFET Model: Testing, Improvement and Parameter Extraction

M. Lundstrom, “Fundamentals of Carrier Transport”, Cambridge University Press, 2000.

C. Snowden, “Introduction to Semiconductor Device Modeling”, World Scientific, 1986.

Y. Tsividis and C. McAndrew, “MOSFET modeling for Circuit Simulation”, Oxford University Press, 2011.

BSIM Manuals available on BSIM homepage on the internet.

T. A. Fjeldly, T. Ytterdal and M. Shur, “Introduction to Device Modeling and Circuit
Simulation”, John Wiley, 1998.

Y. Taur and T. H. Ning, “Fundamentals of Modern VLSI Devices”, Cambridge University Press, 1998.

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