welcome to the course on semiconductor device

modeling i am shreepad karmalkar i have about 25 years of experience in the area of semiconductor

devices i have been doing teaching and research in this area it will be my pleasure to share

with you the enthusiasm and expertise i have gained in this area over 2 decades of research

work i believe that as a teacher my job is to remove any fear you may have about the

subject dispel confusions bring about intellectual clarity and arose curiosity so please feel free to raise doubts ask questions

or make comments whenever they arise in your mind the first 2 lectures will be introduction

in this first lecture we shall answer the question what and why what is a device model

and why i study device modeling then this brief discussion will clarify what is the

target audience of this course and what is the background preparation required to go

through this lectures after this we will discuss the contents in

detail we will point out what will be the take away from this course what skills will

you develop at the end of this course in other words these are called learning outcomes we

shall look at some of the unique features of the approach adopted in this course and

then we will close the lecture with a set of reference books now let us look at the question what is a

device model we shall consider several examples example 1 you would have come across what

is called the ideal diode model in the first level course on solid state devices what is

this model let us look at the static model you apply a voltage v to the diode and as

a result you have a current i this is the plot of the dc current i as a function of

voltage you can express these current voltage characteristics

in terms of an equation which is called the dc equation i=is into exponential of v/vt-1

where is = q times ni square into square root of dn/tau n/na+ square root of dp/tau p/nd

into the area of the diode dn and dp are the diffusion coefficients tau n and tau p are

the minority carrier lifetimes and na and nd are the doping levels on the 2 sides of

the junction ni is the intrinsic concentration and q is the electronic charge let us look at the ac model so here you have

applied a dc voltage in series with the small signal ac voltage consequently you get a dc

current i and a small signal current i the ac equivalent circuit expresses the relation

between i and v this relation is expressed in terms of an equivalent circuit so if you

apply a voltage across this combination of resistance and capacitances then whatever

current result is the small signal ac current the resistance component of this equivalent

circuit is the so called diffusion resistance it is = the thermal voltage / the dc current

i+is which is referred to as the reverse saturation current similarly one component of the capacitance

is diffusion capacitance which is approximately = dc current i + reverse saturation current

is multiplied by tau p which is the lifetime of holes in the n type region so called like

a dope region which governs the dc characteristics and also the ac characteristics divided by

the thermal voltage the depletion capacitance on the other hand is given by the permittivity

of silicon epsilon s into the area that is the area of cross section of this device divided

by the depletion width xd so width of this region let us look at another example the drift diffusion

model of a general device the model that we discussed just now for a diode was such that

it can only be used for that particular device namely the diode now we are going to look

at a set of equations using which we can derive the current voltage characteristics of any

device you would have come across these 6 equations out of which these 2 are referred

to as a current density equation for example this is current density of electors

jn = q times the diffusion coefficient of electrons into gradient of n this quantity

is the diffusion current plus this quantity is the drift current which is q into electron

concentration into mobility of electrons into the electric field similarly this is the equation

for the whole current density since this model is based on drift and diffusion currents it

is called drift diffusion model look at these equations here these 2 equations

are called continuity equations for example for electrons it is dou n/dou t = 1/q into

diversions of jn+g which is the volume generation rate this is excess generation because of

light because of avalanche multiplication and so on minus the excess electron concentration

by the lifetime of the region in which we are applying this continuity equation is the

lifetime of minority carriers the excess electron concentration is equal

to the electron concentration minus the equilibrium value of the electron concentration at that

point in the device similarly this is the whole continuity equation these 2 are the electrostatic equations e=

– gradient of psi and diversions of e=rho/epsilon s rho is the space charge given by q times

the hole concentration + the concentration of ionized donors – the electron concentration

– the concentration of ionized acceptors psi here is the potential now using these equations

you can derive the current voltage characteristics of any device in the first level course you would have used

these equations made some approximations and then derived the characteristics of diodes

bipolar transistors mosfets and so on now while solving these equations you need

boundary conditions on 6 variables because there are 6 equations there are 6 variables

electron and hole concentrations electron and hole current densities the electric field

and the electrostatic potential so these boundary conditions are decided by device structure

and applied bias now how do you get the current the current

is obtained by integrating jn+jp over the contact area which is ds there is a dot product

here because you must consider the current flow perpendicular to the contact area to

find out the total current from the current density similarly the potential applied across

the 2 terminals can be calculated using this formula negative of the integral of e dot

dl so this is how you can get the current i for any voltage psi so in this approach you will take different

values of psi and for each value of psi you will solve these 6 equations as a function

of space and time and then at any instant you will find out the current for that psi

using this formula let us look at one more example the energy

band model of a semiconductor here is the semiconductor across which a voltage has been

applied now the conditions within the semiconductor can be represented using this diagram which

is called the energy band diagram where this e0 is the vacuum level ec is the conduction

bandage and ev is the valence bandage so this diagram plots the electronic energy

in the y direction as a function of distance this is the 1-dimensional energy band diagram

now you find that the lines are somewhat curved because the conditions in the semiconductor

are not uniform so even a diagram is a kind of a model graph also is a kind of model similarly

a table could also represent the model of a device therefore we can summarize by saying that

a device model is a representation of the characteristics of or conditions in the device

in the form of an equation an equivalent circuit or a diagram or graph one could also include

table in this category however please note that while the equation or the equivalent

circuit or diagram or graph is the face of the model the model contains more than this

face just as a body is not nearly the face so what is that something more that is the

reasoning and assumptions or approximations leading to the representation so you must

remember this very important point that the representation together with the reasoning

and approximations that lead to the representation constitute the model this point about the

approximations and the reasoning we shall elaborate in the second lecture let us address the question why study device

modeling different people have different motivations for doing things let us list some of the motivations

one motivation i think for studying device modeling is that it is intellectually and

emotionally satisfying the study of device modeling satisfies our curiosity about how

things work and stimulates thinking up to the highest level now you might say well i am a more practical

person can you tell me what are the practical reasons for studying device modeling so device modeling is also practically useful

example device models are used in analysis simulation and design of devices and circuits

in the next lecture we shall clearly understand the difference between analysis simulation

design and modeling simulation cuts down fabrication costs significantly so you are using models

in simulation and the reason why one must study simulation is because simulation cuts

down the fabrication cost for example for fabricating a test structure

which consists of a 6 mask process on a 150 mm diameter silicon wafer let us look at the

2 approaches and compare them if you do a physical fabrication involving processing

packaging testing and analysis actually go through the steps then the time required is

2 months and the cost is about rupees 60 lakhs which is equivalent to in us dollars about

120k on the other hand you can get the same information

that you get out of fabricating this test structure and then doing a testing and analysis

using simulation you replicate the working of the test structure in a computer that is

what a simulation for doing this and getting the same information as you get out of this

physical exercise you take less than a month and you just end of paying about rupees 2

lakhs or us dollars 4k because these 60 lakhs is the cost of material

processing energy and so on this 2 lakhs is actually the cost of employing 2 device engineers

right for about a month let us look at importance of simulation apart

from cost efficiency simulation helps in assessment of new device concepts now how does it do

that simulation yields accurate distributions of charge current density field and potential

within a device these distributions provide insight into the device operation allowing

development of approximations for analytical device modeling let us look at an example a recent example

a practical example this regarding a high electron mobility transistor the particular

type of high electron mobility transistor that we are considering in this example is

called field plate hemt will explain what a field plate hemt means now you might wonder

how will i understand this example if i do not understand the operation of a hemt well to understand what i am going to talk

about you do not need to know the operation of a high electron mobility transistor in

grade depth you have been exposed to a field effect transistor in your first level course

hemt is another field effect transistor okay let us look at its structure it consists of

a heavily doped n-type aluminium gallium nitride a thin layer deposited over a gallium nitride

substrate which is relative with it but since the thickness of this gallium nitride is still

not several tens of micron which is required from mechanical handling reasons you deposit

this gallium nitride on a thick sapphire substrate which can help you handle the whole device

mechanically now electrons are transferred from this heavily

doped region aluminium gallium nitride into the gallium nitride region as it happens in

any junction and this high concentrations of electron at this surface forms a channel

from source to drain if want an analogy of this device to the mosfet this would be the

oxide of the mosfet right this would be the gate and these 2 would be the source and drain

you can look at it in that form now let us look at the half state operation

of this device you know that any field effect transistor nowadays is used for either switching

or amplification so we are looking at the high electron mobility transistor as a switch

now what should a switch do it should be able to withstand as high a voltage as possible

in the off state in the on state it should be able to conduct as much current as possible let us look at the off state and let us see

how we can increase the breakdown voltage of this structure so this is where the simulation

helps so what you do is you get the field distribution within the device by simulation

it will look something like this you might have come across a field distribution like

this in your first level course near the drain end of the gate in a mosfet so this electric field distribution this is

the end of the gate and this is the drain it is directed from drain to source so you

know that the electric field can have a vertical direction and also it can be horizontal because

you are applying a voltage in this direction because of the gate and you are also applying

a voltage from this contact to this contact so there is a field in this direction as well

as vertical direction horizontal and vertical and so the relevant component of the field

here is the field in the horizontal direction area under this horizontal component of the

electric field gives you the drain voltage now if you want to improve the breakdown voltage

of the device so that this device can withstand a high off state voltage then the way to go

about doing it is to reduce the peak electric field because if the peak is high the device

will breakdown for a lower voltage so the issue is how do you reduce the peak

the way to reduce the peak is to spread the field over a wider distance so that since

the area under the field distribution is the applied voltage for the same voltage or for

a same area if the base of this triangular like region is wide the height which is the

peak will reduce now one way to do that is to use a field plate

that is use a protrusion of the gate over an insulator such as silicon nitride in this

fashion this protrusion is called the field plate because a field from this particular

plate modulates the electric field in the channel now you see what happens is that the

electric field gets distributed and it develops 2 peaks it is easy to understand that if you have

a protrusion of the gate and you expand it in this direction stretch it here then the

field will get distributed and at the 2 edges the field will be maximum there is this edge

and this edge therefore you get 2 peaks now the area under this expanded field distribution

is the same as the area under the dotted field distribution which was the distribution in

the absence of this protrusion or absence of the field plate so clearly the peak has reduced now this feature

can be exposed very nicely in a simulation so you take the device structure and you simulate

and then you see how much is the reduction in the peak now in fact this idea of field

plate was conceived intuitively but at the time it was applied to the high electron mobility

transistor it was not clear just how much enhancement in breakdown voltage it will give so since first people did not do a simulation

of this device structure and they started fabricating the new structure to see the enhancement

in breakdown voltage it turned out that they got only about 20% to 30% enhancement in breakdown

voltage they were quite happy about it but they could not figure out just how much more

they can improve now the technology which was available at the time this field plate

idea was applied to the hemt allowed you to deposit only thin silicon nitride insulators and with that thin silicon nitride insulator

and some intuitive idea of this length people got about 20% to 30% enhancement it is only

a simulation which showed that by proper choice of this thickness of the insulator and the

length one could enhance the breakdown voltage by a factor of 2 or 3 so rather than 20% or

30% it could be 200% or 300% now this could not have been possible by fabrication because

the technology at that time did not permit you to deposit a thick silicon nitride insulator

on this particular device structure however when the simulation showed that you

can get 200% to 300% enhancement then there was a motivation to develop the technology

of depositing this silicon nitride film of sufficiently large thickness to get the required

or to get the maximum potential out of this particular structural modification okay so

these are examples where the simulation helps you to work on the technology and improve

the technology right so as to achieve an improved device performance let us look at a few more practical reasons

for studying device modeling ic designers using circuit simulators example spice without

an understanding of device models spend 30% longer time in circuit design a similar comment

applies to device designers using device simulators example atlas without an understanding of

the transport and other physical models which together constitute a device model finally a very important practical reason

is that we understand and react to the world by building mental models of its various aspects

so you can regard the human being also as a device right and you want to model human

behavior so here also you are getting your encountering the activity of modeling so when

we react to the world we react by building mental models of its various aspects so our relation with others depends on our

models of how others are likely to behave with us transfer of an expertise in device

modeling by analogy to modeling of other aspects can improve our understanding of the world

and help us react to it in a better manner so this is a very very important offshoot

of studying about modeling in general and device modeling in particular now this description should clarify the target

audience for this course who should do this course so the first set of people who should

do this course are students in electrical engineering or science streams such as senior

bachelors masters and phd students pursuing a degree in area such as microelectronics

integrated circuit design solid state technology and semiconductor devices then engineers and scientists in semiconductor

and ic industries and research laboratories such as developers of models for device simulation

developers of models for circuit simulation device physicists and designers using device

simulation and ic designers using circuit simulation so all these kinds of people working

in industry can benefit by studying about device modeling now let us look at the background preparation

required who can do this course so all those who have studied solid state devices covering

these topics concentration and transport of carriers in semiconductors analysis of diodes

bjts and mosfets leading to simple current-voltage equations for these devices now we come to the course contents this course

is actually a modular series of lectures so the lectures are organized into various modules

the module 0 which consist of just one lecture that is the present lecture it talks about

motivation contents and learning outcomes of the course and module 1 will introduce

the course and its remaining contents qualitative model of semi-classical bulk transport

is what we are going to discuss in module 2 in module 3 we shall discuss the electromagnetic

field and transport equations of this semi-classical bulk transport now any modeling exercise starts

with a qualitative aspect and then goes on to quantitative aspects that is why here we

are first discussing the qualitative model of transport and then quantitative model further

we are going to restrict our self to semi-classical transport we shall elaborately explain what is the meaning

of the word semi-classical basically it means that we do not consider current such as tunneling

okay we restrict ourselves to drift diffusion maybe little bit of thermoelectric current

and so on then we shall look at the drift-diffusion transport model equations boundary conditions

mobility and generation/recombination now if you would like to know what are the

various things that will be covered in these modules the simple approach for you would

be to go to the very first lecture of the module and look at the first couple of minutes

of the lecture these couple of minutes will give you in detail what are the learning outcomes

of the particular module similarly each module has a summary at the end which is fairly detailed

for example in some modules almost the entire lecture is devoted to the module summary in module 5 we shall look at the characteristic

times and lengths which are used in device modeling in module 6 we look at the energy

band diagrams which is a very very important tool both for representing conditions in a

device and for analyzing conditions in a device in 7th module we shall look at the 9 steps

of deriving a device model and these are abbreviated in terms of this 9 letters which can be spelled

as sqebastip so here s stands for structure and characteristics

of the device that is the first step you should know the structure and device characteristics

the next step is qualitative physics or qualitative understanding then comes equations and boundary

conditions e and b then you summarize all the approximations that you have made during

your qualitative model development and any other approximation that you may make to equations

and boundary conditions so after that you have summarized all the

approximations that are possible in the given situation then you go on to solve the equations

under the given boundary conditions so s stands for solution then t stands for testing you

check the solution whether it is right or not there are several ways of testing the

solution after a testing if you see a scope for improvement then the next step is i that

is improving the model and finally after your model is complete then

you extract the parameter so p stands for parameter extraction so extract the parameters

of the model for the given device so that you can use this model to make calculations

about circuit performance or any other thing in module 8 we shall discuss types of device

models next we shall discuss modeling of the mosfet

in detail in this module 9 will discuss structure and characteristics and qualitative understanding

of the operation of a mosfet in the tenth module we shall look at the equations and

boundary conditions together with the approximations that are used in modeling of mosfets then

we shall look at 2 broad approaches of modeling of mosfets namely surface potential based

and threshold based solutions and then finally we shall look at testing

improvement and parameter extraction steps of the mosfet model so here we are going to

take you through all the 9 steps for the mosfet now this is the modular series of lectures

right now what will be uploaded will be up to models of the mosfet later on we may record

more lectures on models of other devices like bjts then passive devices and so on and then

we will go on expanding this modular series on semiconductor device modeling general learning outcomes now apart from specific

outcomes related to each of the modules we expect that a student undergoing this entire

series of lectures will develop some general abilities now what are those general abilities

this is what is listed here so at the end of this course you should be able to explain

the equations approximations and techniques available for deriving a model with specified

properties for a general device characteristic with known qualitative theory so suppose you are given a device and then

you know its qualitative operation how do you derive a model set of equations and so

on right for any device you must know a road map to derive the equations starting from

a knowledge of the qualitative theory another ability you should develop is apply suitable

approximations and techniques to derive the model referred to above starting from the

drift-diffusion transport equations assuming these equations to hold we have already remarked that our course will

be based on the drift-diffusion model next offer clues to qualitative understanding of

the physics of a new device and conversion of this understanding into equations so this

is a fairly advanced ability right the first 2 abilities that we discussed were related

to converting a known qualitative theory into equations but now what we are saying is suppose

there is a new device right and you want to develop its theory right from

beginning then if you have done this device modeling course perhaps you would be able

to offer some clues as to how to start developing the model

another ability you would develop is to simulate characteristics of a simple device using matlab

spice and atlas or synopsis so spice is a circuit simulator and atlas/synopsis is a

device simulator and matlab is a program you know it can be used to make calculations using

any set of equations now you will be able to explain at the end

of the course how equations get lengthy and parameters increase in number while developing

a compact model a compact model is a model of a device used in circuit simulation right

what are the details of this kind of a model what are the features we shall discuss them

in this course okay but you will know why the equations are lengthy and large number

of parameters are used okay in a compact model for example mosfet model right used in circuit

simulation so called bsim model has a few 100 parameters why should there be so many

parameters right how do so many parameters arise so this is something that you will be

able to appreciate by the end of this course and finally you will be able to list mathematical

functions representing various nonlinear shapes let me highlight some unique features of the

approach adopted in this lectures problem solving or theory development what

are we talking about here we are talking about both now let me give you an example now you

are aware of the kind of the preparation you do for entrance exam such as the joint entrance

exam to the bachelors program in iits or the gate examination which is an examination which

helps you to enter into the post graduate programs in iit now the kind of preparation you do there is

we can call as problem solving preparation so preparation to solve problems so this kind

of preparation involves understanding the key principles in a sufficient depth so that

you can apply these principles to solve problems however how those principles are developed

from scratch is not something that you bother about much how those principles are developed

from scratch so that is called theory development the goal of this particular series of lectures

is to cover both these aspects problem solving as well as theory development so therefore

our approach will start from scratch and develop some of the equations that are used in device

model and it will also discuss how the equations can be applied to develop the models for different

devices as a result we are going to talk about both foundations and developments what does this mean let me give you an analogy

analogy of a building supposing you think the first level course on solid state devices

which you have undergone is the ground floor of the building then in this course we shall

talk about both the higher floors of the building as well as the basement and foundations on

which the first floor stands right so we are going to go a little bit deeper down as well

as move up okay so both these aspects will be covered so starting

from drift-diffusion model developing the equations for iv characteristics of mosfet

would constitute developments right it is like the higher floors of a building whereas

looking at how the drift-diffusion model is obtained starting from electron as a particle

or a wave is analogous to or it amounts to exploring the foundations of the drift-diffusion

model physics or modeling now you might say why

this should be an issue this is the course on device modeling in fact if you see all

of physics is also some sort of modeling it is arriving at some sort of the representation

for various phenomena in the world okay so therefore this course has a lot of physics

right a lot of material that people would call as physics though it all comes in modeling so in other words the so called foundations

maybe likened by people to the physics and these developments could be liking to modeling

however please note that both foundations and developments are some sort of modeling

just as the physics is also some sort of modeling now is it a course of some 3 or 4 credits

or is it a modular series this is an important issue the way the course is fashioned as i have

remarked earlier it is an expanding series of several modules we have given importance

to explanations of some key ideas and therefore we have not tried to put restriction on the

time for example if i take a topic such as energy band diagram i do not put restriction

that i must cover this topic let us say in 3 hours so that the entire course is restricted

to about 42 hours or say 52 hours okay so this set of video lectures is like an aide

for understanding this area teachers who would like to use these lectures to teach a time

bound course should use these lectures to get explanations of many key ideas and then

they can decide what portion of the course they would like to actually teach in the classes

so the focus is on explaining a concept in sufficient detail and the time constraint

has been relaxed a little bit therefore the number of lectures is going

to be large another aspect of the course is related to explanation and organization of

materials so here we are providing some explanations which are not readily available elsewhere

similarly we have organized the material in a fashion that is not available in the same

form in many of the texts the material as such is available but the explanations and

organization maybe somewhat different for example characteristic lengths and time one set of lectures discussing all the characteristic

lengths and time which are used in modeling now this kind of a module or a set of lectures

you will not find in books but in modeling use of characteristics lengths and time is

very important that is why we have chosen to devote a module for this topic so this

is how some of the organization of the material is different and the topics the way they are

covered are also different about 6 lectures are devoted to energy band

diagram okay now you may not find in books a separate chapter

such as energy band diagrams they might cover different devices and while developing models

of different devices they might include a discussion of the energy band diagram of that

device however here we have discussed several energy band diagrams in one place and how

the energy band diagram arises first of all from first principles right so both foundations and developments because

we feel that ability to draw energy band diagram of a general device is an important skill

that a person who wants to become a good device model developer should have video lectures

versus book many books are available on device modeling so why would one look at the video

lectures one reason why people like to look at video lectures is because nowadays they

are readily available on internet where much of the other information is available so since the computer is the window to the

world nowadays many people would like to look at whatever is available on the internet and

work with it as far as possible so this is one of the reasons why video lectures are

preferred over books another reason people might view video lectures is to get the feel

of a classroom when you read a book it is you who has to pay attention to what is being

read there in a video lecture however there is a teacher

there is an audio visual effect and so on to help you understand the subject and therefore

learning becomes somewhat more easy and more comfortable another reason for video lectures

is that if you compare it to a live class in a live class if you do not understand a

subject or a topic or a small part of the class you might hesitate to ask the teacher

to repeat okay whereas a recorded video lecture you can pause

at any point and then you can repeat that portion which you would to like to go through

again and again so this freedom is available it is one more reason why people like video

lectures finally i would like to mention that we have

extensively used slides in the video lectures it is true that students like blackboard because

things can be developed in a very nice way starting from some simple things on the board however when we want to impart a large amount

of material efficiently slides can be very useful there are certain diagrams and so on

which take a long time to draw on the board but that much of time need not be spent in

imparting the knowledge about that diagram to the student so that amount of time that

is spent on drawing the diagram on the board and giving as an example can be saved if the

same thing is developed quickly in a slide so in this manner the information transfer

becomes much more efficient you can transfer more information in the same time that is

why slides have been used apart from blackboard let me end the lecture with the set of reference

books one nice book for this course or rather the transport phenomena part of the course

is the book by lundstrom on fundamentals of carrier transport many people might think

that this topic falls in the area device physics rather than modeling but as i mentioned physics

is also about modeling another point is today the device sizes have shrunk a lot okay and many types of devices are coming where

lot of different types of phenomena occur and therefore it is very important to get

a good thorough understanding of electron transport and therefore apart from drift-diffusion

model there are other models which are required to understand the modern devices such as balance

equations and in fact some of the cases you have to go to as basic equations as schrodingers

equation or boltzmann transport equations and so on and therefore discussion of these topics is

very important so that a person who is exposed to this topic can then branch off to modeling

in any direction nano or high frequency or high power or anything of the sort therefore

a good idea of carrier transport is important and this is a good book then coming to applications

device simulation christopher snowden this book introduction to semiconductor device

modeling discusses how devices can be simulated how models are used for device simulation coming to circuit simulation tsividis and

mcandrew mosfet modeling for circuit simulation this book is very good for understanding the

mosfet models employed in circuit simulation compact models finally really coming to the

application and bsim manuals available on bsim homepage so here we are starting from

foundation and going to applications right so this is a fundamental book and this is

simulation this is further moving towards application this device simulation and this

circuit simulation and finally this really about the model used in circuit simulation

the practical aspects of the model right if this is theoretical aspects of the model these

are really practical aspects of the model i would also like you to look at 2 other books

one of which is a book by fjeldly ytterdal and shur on introduction device modeling and

circuit simulation this book gives models of various devices for circuit simulation

whereas the book by tsividis concentrates on mosfet this considers various devices and

another book taur and ning talks about fundamentals of modern vlsi devices so with that we come to the end of the first

lecture of this course to quickly summarize what we have discussed we first answered the

what and why question what is a device model and why study device modeling then we looked

at the contents of the course we looked at what abilities you will develop as a consequence

of going through these set of video lectures and then we pointed out some unique features

of the course and finally a set of reference books in the next lecture we shall give a more formal

introduction to this course or this modular series of lectures on semiconductor device

modeling

# Mod-0 Lec-01 Motivation, Contents and Learning outcomes

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