 you can follow along with this presentation using printed slides from the nano hub visit www.nano hub.org and download the PDF file containing the slides for this presentation print them out and turn each page when you hear the following sound enjoy the show okay we'll get started now this is lecture 16 on carrier transport by carrier I mean electron and hold transport finally they will be moving in response to an electric field or a density gradient I will start with an overview I want to bring you back to the bigger picture of where things are and we'll talk about drift current as one component of current drift current is something when you apply an electric field electrons and holes respond to the electric field and so they begin moving and that's drift current and in relation to drift current we'll discuss the physics of mobility mobility is scattering dominated transport when the electric field says go as fast as possible but everything around it says like friction that slow down and that physics of slowing down is is encapsulated in mobility and we'll see that mobility is generally associated with low field transport meaning when you apply a small electric field but these days you know the MOSFET or the transistors we talk about they have for 20 nanometer channel they apply a volt and many times on a gate oxide which is maybe one nanometer thick there's one volt on it I mean tremendous amount of field it is like a thousand times more than the power line the high voltage power line that carries electricity to our homes it is 1000 to 10,000 times more of that field that anything survives is a remarkable thing and we'll see that but I want to show you at high fields there's many more interesting things happening and some physics of that we'll discuss and then conclude so where are we we started this semester by discussing that if you put a semiconductor or a metal in between two contacts and apply a voltage then the current flow depends on the material itself silicon germanium gallium arsenide count the atoms and then you'll try to see how many electrons you have and in fact that is what you have tried to do by using quantum mechanics it told us where the electrons can sit density of state remember effective density of state and equilibrium statistical mechanics that is I'm not applying any external electric field no light shining on the material in that case we were able to calculate the electron number n by first finding the Fermi level in the in response to various donors and acceptors and from that Fermi level by using the charge in neutrality condition remember n and p that all this must sum up to 0 that we use to calculate the electron density so we were happy with that because that allowed us to calculate electron density at multiple temperatures right do you remember but in that discussion there was no mention of electric field so that's why you see I keep using the word equilibrium statistical mechanics and now we are talking about velocity the other ingredient of calculating current and the velocity calculation of velocity assumes non equilibrium statistical mechanics you shine light excess electron whole pair as a function of time these are disappearing through various channels so that is not any longer in equilibrium so that is something we discussed in the last class today we will expand on this a little bit more so chapter 5 was when we had a semiconductor sitting there shine light and it is like one contact because light is coming in exciting electron whole pairs and then letting it go light is coming out or the heat is coming out as a result today we'll talk about flow of electrons when they are in between two contacts and the two contacts have created an electric field in the semiconductor one thing I should mention although I didn't draw it it's very important that anytime one draws a picture that one of the terminal is grounded because without a ground or without a reference point later on you'll see sort of difficult to keep track of the all the electric field and potential and charges so ground anyone doesn't really matter which one it is and we'll set that one to zero if you ground for example to the right hand side set it to zero then the left hand contact will be at negative potential if you ground the left hand contact then the left hand is zero but the right hand is at a positive potential potential difference is what it matters absolute value of potential doesn't mean anything so you can ground any one of them but ground one to make sure that we are talking about the right reference okay so what are you going to do at this end of maybe three more lectures what we're going to do is to derive a set of equations and if you know this set of equations you know every semiconductor device that is made either for your laptop or for the interplanetary mission there when they send things out in space physics essentially essentially there are some stickations but essentially will be described by the equations I talked about just little five equations that's it nothing more and everything will come out of it and the rest of the course is actually the application of this equation solution of this equation to various cases analogy is like Schrodinger equation once you know Schrodinger equation you solve it for wide variety of cases similarly this transport equation once you know it then you apply it and solve it for various approximation in various cases so the first equation is called an Poisson equation which relates the electric field e or the displacement d with the charges now when you have homogeneous then of course the right hand side of that equation is zero so no variation in the electric field homogeneous okay now when things are not homogeneous of course that's when things become interesting then the electric field as well as displacement d that keeps changing from point to point so we'll have to solve for that Poisson equation is number one second is the continuity equation for electrons for example there are lots of electrons in the conduction when moving around and I will explain each one of the term now the term that is j sub n is the electron current and you can see g g sub n is a generation rate shine light last chapter then that will move all the physics that we talked about is hiding in g sub n r sub n is a recombination oj recombination direct band gap recombination you know all those recombinations we talked about in last two classes those are hiding in there and we'll derive that in the next class perhaps now this is these two are for electrons and these two are for holes same equations you will notice a little bit change in the sign in one case you see this you have minus 1 over q in another case the second equation you have plus 1 over q y well electron has minus q charge other one has plus q charge and you can also see that the gradient of n and gradient of p in the third and the fifth equation they also differ in sign why because this is diffusion we'll show and diffusion doesn't care about whether you are charged or not charged right because you put anything whether it drop of ink in water uncharged or electron in one point or hole in another point they diffuse essentially using the same physics so therefore they have different signs so that once you put it in the second and the fourth equation they have the same sign we'll come back to that we'll see how it how it works so today my goal is to explain the physics of that term Poisson equation presumably we have talked about it before remember from potential we drew the electric field and from electric field we did the charges you remember this so that was my discussion about Poisson equation we'll not do that anymore we'll apply it and you'll see how it works but today I will talk about the drift of electrons in response to an electric field e and mu sub n is the mobility that we'll talk about also so let's talk about drift current now before I move on to discussing drift current let me remind you that what approximations we are making because this is for the first time we're really going away from equilibrium to non-equilibrium case applying an electric field currents are moving so we'll have to think back a little bit about the approximations we made and why we can continue to make those approximations because it's not really obvious anymore you remember this question about free electron mass and effective mass when a free electron with the red circle they're there moving with a free electron mass of m0 in a periodic set of potential I compared that situation a few days ago in terms of as if there is a runner sprinter going over a set of hurdles and instead of keeping track of the hurdles and solving the whole problem correctly then and the whole the blue potential where the electron coulomb potential is attracting repeatedly the electron as it's passing through is shown is given in u crystal u sub crystal that's the one and u external where is what is that that came from the battery so we now have a battery so we have to put that extra piece of potential in the Schrodinger equation if we solve this equation properly then actually we should have started going back to ground zero started the Schrodinger equation all over again but that will be too much torture so we'll not do that but rather what we'll say and and then I'll explain the approximation in a second that we'll forget about the blue potential going up and down bobbing up and down and hide everything hide everything all the physics of it the interference of it all in the effective mass so therefore we'll not talk about the u crystal anymore and therefore you will replace you see previously in the first Schrodinger equation I had m0 the free electron mass in the second Schrodinger equation this is called an effective mass equation I have m0 star because that hides the crystal potential within it right okay now the u external is of course there now one thing I want to immediately point out that we are sort of heading into a little bit of trouble here because remember when can we define when can we define an effective mass it required the solution of the Schrodinger equation solution of the Schrodinger equation required that the potential of each little piece is periodic right that is how we folded back the equation periodic boundary condition do you remember that is how we got the effective mass now as soon as you put an electric field on it then of course the periodicity of the potential is gone right and so this at extra space extra transformation of effective mass in the presence of an external potential is not really 100 correct but what we'll ask you to do when you go home calculate the magnitude of these two quantities calculate when you have an electron one nanometer that's how far away the electrons move around one nanometer away from a proton the field will be on the order of 10 to the power 8 volts per centimeter on that order you know Coulomb Coulomb electric field so calculate that compare that with the maximum electric field your pentium is handling which is maybe on the order of 10 to the power 6 5 times 10 to the power 6 volts per centimeter so any potential that you can conceivably put on your device is actually far smaller than the potential the electron sees within the device and as a result even if we make approximation this little potential u external is not going to change the periodicity in a fundamental way you'll have to go about 100 or 200 atoms before you see a significant change in the potential that's why we are still able to keep this view even though the whole system is no longer in equilibrium okay so let me now very quickly discuss this issue let's say i have i have drawn the conduction band diagram i haven't drawn the valence band diagram here and i see that the potential or the energy c the conduction band energy is changing with position that means i have an electric field here and you always remember right that at each point really there's a ek diagram sitting there when we draw a continuous line actually we are folding back all those information in the effective density of state but in your mind anytime you see a line there should be this thing sort of in the back of your mind will not draw it all the time and then there are of course some electrons going in the positive direction you can see on the positive branch of the of the line and some going in the slightly negative direction and those electrons as it moves along they will scatter and they will go down the hill it's like somebody a skier is going down a downhole slope now instead of looking at this complicated picture we will transform it something simple we'll say well we don't have that many electrons that many electrons everybody going in their different way we'll assume sort of an effective representative electron which on moves with an average velocity so that it sort of reflects the chaotic motion of all the electrons on the top because we know that after you apply an electric field everybody's as a whole as a crowd is going to move downhill but instead of thinking about this swarm of electrons moving downhill we'll take a representative one and then think about how it is moving downhill and from that back calculate what the group is doing so here i show instead one electron moving through the effective density of state downhill representing the previous case now put a bunch of crosses there the cross simply means that the electrons cannot go on its own without scattering scattering with what i will tell a little bit later it is like friction when you put a put a pedal on your pedal put extra gas in the car then initially it accelerates but very soon the friction and your acceleration they balances and you move with a constant velocity so that is this extra crosses is representing that friction the nature of friction i'll discuss in a minute how the electron in response to an electric field what is the electric field by the way because this potential is changing from a top value to a smaller value gradient of that potential is the electric field you can see the gradient is constant and so i'm assuming that there is a constant electric field pumping the electrons along now the equation i have written is an equation for an equation for this is newton's law right i've said that change in the momentum as a function of time is equal to the force minus qe why minus electron right minus qe and if you have friction then the momentum will relax if you didn't put any electric field in it eventually it's going to stop completely the third term on the second equation or the second term after the equal sign that's the momentum loss because of the friction and the physics of tauss of m we'll talk about in a little bit but do you see why we have put m and star because we have hidden all the crystal crystal information in that m star and so the only extra electric field we are thinking about is qe that e is coming from the battery do you see that and the crystal potential are all gone so we have that now this is not a equation that we are afraid about of we can solve this and if you solve this equation you can very quickly see if you plug it in in the other one that you will get something in equation like this how does this equation look like you can immediately see that when time is long how long we'll say in a second when time is long then the second exponential term that will drop out right that will drop out and then we'll have a term which is proportional to the electric field which makes sense right after a long time friction has equal to the force and we are moving at a constant velocity now what about at a very early phase in the time you know that exponential will be expanded in 1 minus t divided by tau right e to the power x when x is small is essentially 1 plus x and in that case you will see that in very early phase when you're just accelerating 0 to 60 in 6 seconds don't they say you know when they sell a car that is what they're talking about that when you pump the gas then initially it will go linearly then the friction will catch up with it and then if you'll go with a constant velocity so that is the essence of this there's no it's a high school physics of a ball rolling down a fixed rolling down a inclined with friction on it and so that is what I have written here or shown here on the bottom right picture in which I plot velocity versus time note very carefully that the x axis now is time I'm going to show you another figure a little bit later where the x axis will be electric field but now it is time I'm thinking about starting a car pumping the gas that's the initial linear part then friction catching up and the velocity saturating now that's the and the saturation velocity is something I'm going to think about again now what is that time tau I will explain what that time is but right now or where the time constant comes from but that time is on the order of one to two picoseconds because within a picosecond everybody catches up that this guy is moving very fast we have to take some energy from it so scattering begins within a few picosecond and this is a number I have told you before do you remember when do you remember I said when the electron and holes see each other every time they see each other and bump into each other they don't recombine they scatter on the order of a picosecond the same number they scatter against each other on the picosecond but only when their wavelengths are just right only then they can recombine and that is 10 to the power minus nine seconds or a millisecond for indirect band gap indirect recombination so that is the same time here they see each other very frequently and therefore this time is so within a picosecond you essentially can approximate the velocity within something being proportional to the electric field and you don't have to worry about the early part at all you see now picosecond how fast a computer can you make with a picosecond it will be on the order of a thousand gigahertz let's say and you don't have a computer for thousand gigahertz so every computer you make you can assume that the early part as if doesn't exist right we have in these days maybe 100 200 gigahertz that's individual transistors and pentiums maybe 10 20 gigahertz something like that at most so as a result we will always worry about just the steady state but i'll give you an example where you have to think about the very early phase so instead of carrying around this q tau n divided by m star term around we'll call that constant a mobility so in for the mobility q we know charge effective mass well we spend many classes on that the only thing we don't know physics of tau of n and if i somehow know the physics of tau of n then i am set i can calculate mobility i can say when you put a battery how fast the electron goes okay so next few slides hopefully we'll talk about the physics of mobility or equivalently the physics of scattering how fast they scatter against each other now one thing you can easily see that before we move on that if your electric field is e1 then the velocity at which you will saturate that will be a little higher if you put a little bit less gas or and then it will move it is a little bit saturated it is a little bit slower velocity right that we know and so if we are just interested in the final velocity you know for our typical transistors then we realize that the velocity is almost linearly proportional to the electric field so the saturation velocity you will see if we just interested on those points we can pick up the red line the saturated part the blue line and plot it also as a function of e1 and e2 and that will also give me a characteristics that i'll be interested in but here they're saying the current is proportional to the electric field now if i keep putting a higher and higher electric field do you think i can get infinite amount of current well that's not happening and we'll explain why okay physics of mobility let's talk about that so physics of mobility means physics of the scattering the why something scatters and how fast it scatters again that same picture of electronic potential moving up and down and then electron trying to move it this was the original picture and we replaced it with the effective mass now the first type of scattering that we'll be talking about is ionized impurity scattering or impurity scattering and i'll explain the physics but let's first look at it from this point of view assume that i had a series of identical atoms i take out one atom put back a new atom in when i put back a new atom in its potential is not the same as before right obviously not so in that case let's say i move back that electron in now if i wanted to keep my description of effective mass then what do i need to do then i need to say okay i really i have the red potential but i will approximate it with my original blue one and put that potential in the effective mass and anything remaining the difference between the red and the blue i will put it as an extra potential you see at this extra potential when this electron with this effective mass is moving as if it will scatter against this extra potential and when it does so then it will cause the friction and this will slow down the electron okay now how does it look physically one way to think about it is think about a piece of semiconductor right donors sitting at random points decorating it as random points the donors have given away its electron now it is positively charged right so it's like a pudding with raisins sitting on top all positively charged now an electron is coming in as soon as it comes in close to a donor it says that don't go directly it will try to bend it towards itself right and the electron motion will continuously be bent by this positive charges and this is so it's perturbing its momentum and this is a scattering and this is called a ionized impurity scattering because only when it's ionized then it's going to scatter same for the acceptors negative charge when electron comes in it says the negatives are sitting in the space and it scatters okay and we will discuss a little bit more on that now this is something you are not supposed to learn in this course you will you learn in detail in another course in 656 for example but the point is this extra piece of potential with a little bump in the speedway that potentially if i call it u it is possible to easily calculate how frequently electrons scatter by a specialized formula and that formula is called Fermi golden rule now all i'm saying that there is a way to calculate it i'm not telling you exactly how to calculate it it will not be in the exam so don't worry about it but the point is this is how people would calculate this this type of scattering so there is two types of scattering that are of great interest that slows down the electron right one is called a phonon scattering and the second is one called ionized impurity scattering now ionized impurity i just explained right when you have these charges and they try to deflect the electrons slow them down in the process ionized impurity scattering and the ionized impurity scattering is inversely proportional to the number of ionized donors does it make sense if you have more right it will scatter more frequently and therefore the time will be shorter right for the scattering this inversely proportional to n sub d if you have n sub a acceptors then it should be a proper inversely proportional to n sub a now why is it just n sub d why is it not n sub d squared or something else the reason is remember the donors atoms are very few relatively speaking 10 to the power 22 number of atoms silicon atoms right how many donors 10 to the power 18 maybe so they are far apart and so once the electron scatter scatters with one then it forgets about that electron that scattering all together it goes a long way before scattering one more time therefore it's linearly proportional if they were very close together heavy dense a heavy doping effect then i couldn't simply say it's one over n d you see now why t to the power three halves well that has to do with screening of the electrons because anytime a positive atom sits it doesn't sit alone it sort of brings around it a bunch of other electrons to screen it and so when a new electron comes in it not only scatters with the positive one but it's sort of surrounding it's like a king and the core sort of the core and you know if you want to see the king or get scattered by the king then you have to sort of get through that electron cloud or the people around them and so that is where this t to the power three halves comes in but that will you will understand a little bit later has to do with screening with other electrons around a positive core that does this modifies the scattering okay so that's one part and this is the extra piece i just told you about now there's something else phonon scattering what is a phonon phonon is this lattice vibration of the vibration of the lattice and what happens when an electron comes in the lattice is vibrating sometimes it can steal a little bit of energy from the vibrating lattice and go up in energy a little bit or if the electron has a lot of energy then the lattice once it scatters with the lattice the lattice starts vibrating on its own and even it doesn't give back the electron its original energy but rather it dissipates it in the environment right so it's stealing electro energy from the electrons setting itself into motion and letting the motion dissipate in air as a result as far as the electron is concerned this is a friction which is taking away its energy now why is it inversely proportional to t because at higher temperature there is no more phonons so it can scatter more often and that's why it's inversely proportional to t what about these three halves because after scattering the electrons essentially has to stay almost at the constant energy surface and that's proportional to kt the density of state so this point isn't very clear i will i'll probably insert another slide later on but for the time being let's say the phonon scales with t and therefore the scattering time goes down as inversely proportional to t and the phonon essentially is a vibration of the of the lattice atoms and that you will so as far as the effective mass electron is concerned it will see as if a potential moving up and down once you subtract the blue the equilibrium position out then you will see as if the extra piece of potential is bobbing up and down and that's what this scattering is all about so when you have instead of one scattering if you have a bunch of them then what people do is they say that the scattering sum up linearly as one over does it make sense if something scatters every once in 10 seconds and something else squares scatters every other second together do this scatter faster or slower than 10 seconds it will be less than two seconds right because most of the time it will scatter with two seconds once in a while the 10 second one will come in and so the it has to be inverse of the scattering time do you see not just summing it up summing it up will give us a long physics now beyond this there's no justification of this rule this is the old rule before almost quantum mechanics and so this is really a empirical rule no physics here and because the electron can scatter with phonons with ionized impurity with all sorts of thing we sum them all up and the mobility the net friction will be a combination of everybody because everybody is trying to slow slow it down now if you see that one over mobility is also proportional to one over the scattering time so if the scattering times comes as the inverse the mobility of various pieces will also come at inverse right if there are just phonons one over mu phonon just ionized impurity one over ionized impurity but of course everybody is trying to slow it down so the net electron mobility is a inverse of all this let's flip it if I flip it then that's my mobility now this one I could have ended it here but most people write it this form the subtract of a mu minus a mu minimum plus added and subtracted now this could be an arbitrary value and they write this piece the second piece as something that is proportional to the number of ionized impurity n sub i do you see in the bottom that and then an n now these constants depend on energy where is the phonon scattering hiding it is hiding in mu not so it will hide in mu not and it will also hide in n not and where is the ionized impurity number of ionized impurity remember n sub d that is that n sub i because i could be acceptor or donor so therefore I have just written n sub i alpha is a constant between one and two okay so this is it and this is called a matheson's role and this is experimental values in the x-axis we have n a or n d which in that expression is equivalent to n sub i right so you start with mu minimum on the right hand side why do I start here because when n sub i is large infinity right very large then the second term will drop out only first term will remain right so therefore on n d equals infinity 10 to the power 19 then we start with mu minimum now as you go make n i smaller and smaller in the limit of n i equals zero then your final answer will be mu minimum plus mu not right that will be little bit more so you can see the curve going up and in between there is a transition so the initial early part is completely phonon dominated no ionized impurity the later part is a combination of phonon and for phonon and ionized impurity scattering now let's look at some numbers so what would you say the maximum velocity electron can go or mobility in electron can have in silicon on the order of a thousand right in silicon for the holes it is a little bit less by a factor of two that's why when you design transistors many times your n mos we'll talk about what n mos and p mos is later on but one type of transistor that depends on electron is actually half the thickness half the width of the other other one because holes move slowly holes effective mass is more right that's why it moves slowly so that's the result we'll have to take care of it in the circuit to that they all provide the same current okay so that's it for the time being we'll just stay here but the physics i cannot even begin to touch of it touch it that is so beautiful hopefully in other courses you'll learn about it similarly a temperature dependence is very important again why this t to the power minus three halves is coming from at high temperature lots of phonons phonon scattering dominates and that's what i said in a few seconds ago right few minutes ago that the phonon scattering goes as t to the power minus three halves right do you remember and that's why this mobility is also proportional to goes as t to the power minus three half so high temperature electron doesn't move as fast it's scattering too often with phonon that makes sense right that there's no problem and you can see in the inset that the exact experimental value is not minus three over two minus three over two comes from theory but if you look at the inset in the figure and in a log block plot it shows t to the power minus two two point three so experimentally it's a little bit more lots of people work on this that why it's different but that's a separate story now we'll start in the next 10 minutes talk about high field effects whatever i said now is sort of true if you put a kilo volts per centimeter on that order maybe one kilo volt per centimeter and maybe in in apollo days or maybe you know when the apollo went to moon maybe in those days we had that type of field for last 30 years those type of fields are not in present our devices so whatever i said is good but not good enough for the electrical engineers what happens when you increase the electric field too much then what happens at some point no matter how much you put the put on the gas you cannot increase the velocity anymore so we'll see why not what i have plotted this time is velocity as a function of electric field and as i said in the early part it is directly proportional to the electric field now unfortunately i have a typo there it should have been mu mu is proportional mu is a constant but the velocity is linearly proportional to the electric field oh i had it that's fine but if you exceed a critical electric field and for every material the critical electric field is slightly different if you exceed a critical electric field no matter how much you pump in the electric field electrons are not going anymore and that number called saturation velocities on the order of 10 to the power 7 centimeter per second almost for all materials that's why changing silicon to gallium arsenide in modern technology may not always help very much the thing is that this velocity saturation velocity is a number which is almost independent of material we'll explain why it is coming from that is the electrical critical electric field so anytime you do a calculation you should always check whether you are above the critical electric field or below it once you are above the critical electric field then the expression i show on the equation on the top right v is proportional to electric field of course but look at this expression in the denominator it has e minus ec so when e is equal to ec then that's one right so then it will be mu not divided by two but if it is much higher much higher than ec then do you see that the electric field will cancel do you see that electric field will cancel and what would be my velocity mu not multiplied by ec the critical electric field constant independent of electric field so that is how it will saturate now what is the physics of all this right i mean why why does it do so let's focus on the silicon conduction silicon electrons which is starting from 10 to the power five or so centimeter per second and at a low electric field and then you can see that curve is saturating on the order of 10 to the power seven more centimeter per second why does it do so so this is the picture when the electric field is zero you have same number of states on the positive side same on the negative side right no reason to go one way or other remember the Fermi level both sides equally feel we are done current is zero because plus and minus cancels if you apply an electric field but a low one then of course you can see that the some of the electrons will move on one side depending on the electric field and the ones that are going opposing to the electric field that number might change the blue and the red one the number is no longer the same so therefore the current has increased right now you have current if you apply a little bit more then of course you have more red and less less blue so current is even more that's that's very good but the thing is and you can see how i'm going along with the arrows on that silicon curve but if you scatter try to put anymore then it's not taking it anymore because what happens that when it has that much energy it is constantly scattered back because anytime it's going at a very high energy it's like a high energy collision you know it's when it scatters many times it's the car completely flips around and goes in the opposite direction so what happens beyond a certain point after scattering the reverse going one catches up with the one that is going in the forward direction so beyond that point the net flux doesn't change things given get frozen in the proportion of red and the blue one it cannot do anymore you can see the black arrow i have shown for the red electrons that is to indicate that at high field constant scattering back essentially creates two streams which are one optical phone on a way and these two streams the proportion of it cannot change and as a result current cannot change and the whole thing saturates now there's another very interesting feature of mobility or velocity for gallium arsenide and that again happens at high field not low fields low field you see everybody is the same moving proportional to the electric field is only in the high field regime and what is the high field regime here about 10 kilo volts per centimeter right beyond that you are sort of talking about high field regime this interesting phenomenon occurring now i'm talking about this gallium arsenide there is a bump so at certain electric field lot of velocity you put more not only it saturates but it actually goes down why is that i mean why putting too much is bad the reason it happens is shown on the right hand side which is the ek diagram for gallium arsenide do you recognize it this is a direct band gap material you can see both at the k points are coincident on k equals zero the gap is coincident on k equals zero in the beginning all the electrons sit in the little gamma valley so a spherical gamma valley in gallium arsenide you start putting an electric field just like in the previous picture the red electrons and the blue electrons starts moving up in the ek diagram right with force the k changes or momentum changes it goes starts going up now when it reaches a level which is shown here in the red arrow where it can not only scatter among itself but this energy is high enough that it can go to the neighboring valleys so there are valleys in the x direction for example also valleys in the l direction what it will do that now it has many places to go right after scattering because they are all in the same energy so after a phonon scattering phonon has a lot of momentum do you remember indirect band gap recombination i said that the phonon has a lot of momentum it can allow you to go from gamma point to the x point or gamma point to the l point very easily so here the phonon essentially can allow you to go all the way to the edges of the bernabryloin zone now look at the effective mass the effective mass once the electron has gone there it was happy to go there because it looked like a good place to go lots of density of state but as soon as it goes there what's this mass this mass is very heavy right so it came in very happy but as soon as it lands there it realizes that my old effective mass which is the curvature right second derivative is much lower here so now as if the mass of the car or mass of the electron effectively has increased and as a result what will happen since velocity is inversely proportional to the mass as a result the velocity will go down and that's why this is called a velocity overshoot it's a very important phenomena in many oscillators people actually use this effect in order to microwave oscillators use this effect because there's a negative differential region where you increase the electric field velocity goes down and you can make oscillators out of it and that's a very important phenomena that one has to understand okay and this is the last slide in which we just very quickly talk about how to measure it in the last all this is I talked about this theory how do you measure mobility well you put four contacts two red ones that you can see it's a top view of a sample of a silicon 1 1 0 let's say the green one and you have two contacts electron you pump in electron through one contact and taking out taking it out through the other one and the yellow points are the voltmeter where you measure the voltage between them and you can see the electric field when it's purely drift current is proportional to j and proportional to resistivity row and the resistivity because if you have both electron and holes is given by that formula and so the resistivity if you equate it you will get a relationship but let's say it's primarily n-doped or primarily p-doped material the green one then one will be n or p will be much larger than the other so for n-type material nd is much larger let's say so if I measure the resistivity as a function of nd then I can calculate the mobility right so that is the plot on the right hand side that's what you see that doping on one side so they prepare a set of samples green samples of various doping levels and they make the measurement of row by measuring the current and the electric field electric field they obtain from the voltmeter by taking the voltage and divide by the length of the yellow probes the difference between the yellow probes and from here they get it now do you see that this is reflected in the experiment take log log on both sides this is a log row y-axis log nd on the x-axis what should be the slope minus one right it should be minus one and you can see indeed this slope is minus one so experimentally the theory we just developed is sort of reasonable but in the next class we'll say that putting it like nd what I said and this saying it on the x-axis this is nd it's not so easy how do you know nd actually you have a implanter you implant some dopants in how do you know some of them didn't go here and there all went to the place you wanted to how do you count nd now that's not a easy thing it's not like counting atoms in a silicon so what I said this axis this x-axis is not so easy to make and that will be our discussion in the next class that how do we make understand the x-axis so that's it this is the summary we started by thinking about various types of equations that we'll need in future classes poisson and the drift diffusion equation we talked about drift today and the essential element is drift is mobility we talked about mobility but we thought of thought at the end that we know how to calculate mobility but the trouble is we don't know n sub d or n sub a and unless we know that we really don't know mobility so that will be the discussion in the next class okay thank you