 You can follow along with this presentation using printed slides from the Nanohub. Visit www.nanohub.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. Let's talk about today. I'll get started on lecture 17 and today we'll talk about Hall Effect. It's a very famous experiment and as well as start on a new topic on diffusion and the last class we talked about drift, response of an electron under an electric field and drift and diffusion together will be the basis of the basis of the set of equations that we are interested in and once we know that set, how to solve it, we are set. In fact, you can analyze all devices be it transistors, lasers, microwave devices, all of them essentially just by this little set of five equations. That's all you need and so that will be very powerful. So let's get started on the problem of mobility measurement and first of all why we need it and the problem is the following. I mentioned it in the last class that if you have a semiconductor, chunk of semiconductor and shown here in green and let's say this is in 1, 0, 0 surface or 1, 1, 1 surface, one of those surfaces that we have talked about and I have put four contacts, the two red and the two yellow ones, the two red ones are current contacts. So you inject, you put a battery around it and you inject current through one, arrow coming in and take the current through the other one. It's not very important how much current you put, just any magnitude would be fine because it will cancel out later on and through those two yellow probes there's a voltmeter and you pick up how much voltage drop there is when the current is going through the red circles. The point is the reason you want to do this, not simply have the voltage measured between the red two points is because of the contact resistance issues. Many times when you put down two probes or two contacts, there is small resistance which is beyond what the green semiconductor provides, simply because of the contact with the metal because metal is a contact to the semiconductor there is extra resistance. In order to avoid it you have this four probe configuration because the yellow ones don't pull any current. So there is no problem with the drop voltage drop there. On the other hand the red one only pumps current, it doesn't measure any voltage so you do not have any contact resistance related issues. So that's why this particular more difficult configuration is important. I mentioned that the electric field which is given by the voltage that you measured through the yellow probes divided by the length or separation of them that electric field is of course proportional to current especially if the voltage is low. Remember what is the low voltage, critical voltage is 10 to the power 4, 10 kilo volts per centimeter below that everything is linear do you remember. So we are in the linear low voltage regime and we said that well the current J is combination of the drift of electron and hole current both responding to the electric field and if you just compare the first two equations you get a expression for resistivity and if it is primarily one type doped let's say n doped, donor doped or acceptor doped then you will simplify. Now here I made this substitution that n is equal to n d which region it is in the device is in extrinsic region remember this freeze out there is this extrinsic and the intrinsic region so when I made this substitution I assume that it is in extrinsic region but it need not be so I need to find that out properly how do I know I am in extrinsic region right that's the problem. So the problem is that I do not really know n d and n a to begin with and I cannot really say that n is equal to n d maybe I am in the freeze out region how do you know so that is where that is where the additional problem of or additional question of hall measurement comes in that will be next five six slides. Now again I mentioned that the resistivity as a function of the donor or acceptor density if you do log log on that expression then you can see it will have a slope of minus one and indeed the measurement going from all the way to the 10 to the power 13 to 10 to the power 20 does show a more or less linear one minus one dependence okay so the main point I want to discuss is how to measure carrier concentration independent of temperature without any assumption that's the goal okay so hall effect is a way of measuring carrier concentration and let me explain how this works it will be a few lines of math but follow the physics math you can do high school math nothing nothing more than that so the idea of a hall measurement is that you take a piece of semiconductor and with a certain will put down some axis xz and y for convenience and we'll say that the length of the semiconductor with and thickness these are all given by Lw and small d and you are putting it in a magnetic field now people seven semiconductors seems to like magnetic field a lot now the rear you remember why we applied magnetic field before also where we wanted to calculate the effective mass right one thing that is very good about this book is that you should notice after every book or this course in general after every theoretical concept there is an experimental measurement so when you read the book make sure that you understand that why is the theory and why is the experiment how do I measure it how do I understand it how do I measure it so this is a way we'll see how to get the carrier concentration and thereby the mobility so that's the purpose of this of this discussion now this magnetic field is a little weaker than the magnetic field we had to put in when we wanted to do the effective mass right cyclotron frequency that will be so it's a little bit weaker magnetic field and we'll pump a current through the semiconductor and see what happens well what's going to happen that as the current flows then the magnetic field will try to rotate it and rotate it to one side and if you put a voltmeter perpendicular to it then voltmeter you know has a big resistance right it doesn't allow much current to flow so you will see that there's no current flow but there is this voltage developing now in principle you see without the magnetic field you don't expect any voltage right because the current is flowing through there is no reason why in the transverse side there should be any potential difference potential difference reflects piling off of carrier or an electric field that you have applied you haven't applied any electric field without the magnetic field there is no reason to prefer it to the right or the left right this is all the same so therefore when the magnetic field is zero the hall voltage v sub h h fuzz for hall hall voltage is zero now when you apply a magnetic field what we'll see that gradually hall voltage is developing and the reason is and I'll explain mathematically in a second that the electrons will be pushed in one side there will a magnetic field because of this force that it has it will always try to push it let's say to the left if the magnetic field is on the on the top side and as a result there will be more charge on the left compared to the charges on the right and if you put a voltmeter you will see a voltage proportional to almost to the magnetic field higher it is more pushing going on and therefore more voltage and from that we'll be able to find out that whether it's electrons or holes moving and how many of them are moving now this actual the picture that is shown on the left is a cartoon of course the actual picture that you see is over here on the right the big two cylinders that you see what could that be that's the magnetic field these are electromagnet big electromagnet and the sample is sitting in the right in the middle right in the middle and from here you can essentially apply the magnetic field and then connect wire it up so that current can flow through the system now the system generally can work on a broad range it's not as restrictive as the effective mass measurement one that requires a higher field and it has to be at a lower temperature why because it has to complete the orbit right cyclotron frequency if it doesn't complete the orbit it cannot catch the microwave microwave signal so therefore here there's no microwave business going on so therefore it's a little less even I can do it these experiments I have done and their field is not terribly high one to about one Tesla or so most universities have this type of setup so no big deal so this is just the configuration one more time and one thing I wanted to point out that this is a magnitude electric effect that is the electric field is pumping the current from one contact to another and the magnetic field is doing something to that current and in fact last 1890 years this has led to at least probably four Nobel prizes directly just this configuration in some form or other and maybe there'll be another more from this class what do you say one is this fractional quantum hall effect this combination the quantum hall effect that one this combination the original one so this is a lot of very beautiful physics that comes out of this electric and magnetic field interaction so very powerful tool good to know how to how to work with this one okay so let's do some math I will start with something called a drew-day model it's a simple model of how to think about these experiments hall experiment is actually a old one this previous quantum mechanics I think it's about 1870 or so 1870-72 so the hall measurement came about because it was had some funny results that people couldn't understand and people couldn't understand it till they understood the concept of holes because holes has a positive charge and many times in p-dope material it was showing sign of the ball voltage that people at that time didn't understand because the concept of a hole until you know band structure then you have a known notion of a hole and so for many years hall measurement was a puzzling measurement not a characterization tool is only after the semiconductor physics allowed us to understand it properly then it became a very powerful tool right so the idea is this I'm to looking at a top view and a red electron is coming in and the black circle you see that is the magnetic field coming out of plane so it's a top view so you can see and when the electron is coming in then the magnetic field will try to turn it now you can easily see what is that how with the what strain does it turn it that's given in the first equation so the electric field in the absence of a magnetic field if it's an electron minus qe right the electric field the voltage is trying to push the electron around then the second term is due to the magnetic field v cross b again cyclotron frequency one this exactly what you saw right no new things here so that's the force first and second one as the electric and magnetic force working on the electron now what is the third term third term is because of the scattering this is from phonon scattering ionized impurity scattering that we discussed in the last class then the force will be balanced by that friction so that's why the force has the first two term and the last two term is a friction balancing each other so that electron moves in a continuous level now the last term was not present do you remember when we did the effective mass measurement there was no scattering there that's why we had to do it at a very very low temperature right and so that's it now we want to solve for this now this is a vector equation because you have v and b and not in the same direction in fact perpendicular to each other so in general you can cross multiply with tau get an expression but what I want to say is that if the magnetic field b you see see it's a implicit equation because you have v on both sides so I cannot immediately solve it I could make it in a vector and solve it but cannot solve it directly but if the magnetic field b is weak that's the assumption here then I can drop the second term temporarily and if I drop the second term and that's what I have written on the right hand side and from that right hand side so this is an approximate solution and from that right hand side just containing the balance of the electric field and the scattering and from that I can get an approximate expression for the velocity and I can put this expression of velocity back in there you see very simple and once I have done it then you can see that I can simply divide both sides by m star effective mass now right why is it effective mass do you remember magnetic field is weak and then electron is making a huge circle encapsulating a lots of atoms so it sees sort of all the potentials together and that's why it is a effective mass not the free electron mass so this is why the quantum mechanics is hiding in this classical experiment and you can see you can get an expression for velocity of the electron as it's moving through it is directly proportional to the electric field but that's the longitudinal force but the E cross give the B gives it a transverse force as well right so that's why the electron wants to move to the one side because of the second term and if you set B to 0 then electron goes straight it doesn't go left and right now generally this expression the weak B field I just have have written it here only if the second term is much smaller than the first term only then it will work right this is simple perturbation if the second term is comparable to the first one we shouldn't be using this expression okay so let's calculate a few things now we have the equation so let's calculate it so q is the current electron current is minus q m v v I just calculated the expression for the first term proportional to the electric field and second term proportional to the magnetic field right now once I have that I can write the q square the first thing sort of multiplying the electric field call it a name and that's the conductivity so I'll call that sigma not it's just a definition I haven't done anything I'll call that sigma not do you see that the second term has a mu sitting in there q tau divided by m star the last class we talked about the mobility right one over the effective mass and directly proportional to the scattering time so that's it nothing complicated here now I have written it as I mentioned is a it's a vector relationship because it's a two-dimensional electron is moving in two-dimension so electron will have a component in the x direction pushed by the electric field y direction well magnetic field is asking it to go to the y direction so J x and J y and you can quickly check it out e cross b you know you write it in that simple simple cross product form and that this would be this would be the result okay this is I haven't done anything I just wrote the writing definitions and making assignments in the next one I'll do something next slide okay so that's the expression I have now I'm going to do a few things I'm going to assume that the magnetic field is small this is an assumption so that I can replace the cross term you see yeah I can replace the cross term one of the cross terms so J x is sigma not e y right and then sorry sigma not ex J x directly proportional to the x component of the electric field and the J y has both the magnetic component and the electric component that's fine now this next step is very crucial this is where many times I set up exam problems here the J y I have set it to 0 why because look at this configuration I have put a voltmeter around I put a voltmeter around voltmeter draws no current right so therefore my J y is 0 I cannot set J x to 0 of course current is flowing along the x direction as a result I set J y to 0 and as soon as I do that I can get an expression by solving this and you should write out an expression for e y from here e sub y the y component of the magnetic field right and how do I get the y component I measure it you see this is a hall voltage I know the width of the device divide the hall voltage by the width of the device that gives me e y b sub z well I know that's the magnetic field remember this chunk of big electromagnet I know how much current I am putting so I know the magnetic field and J x J x my current that is I am putting it through this system so when I divide I get so I solve this problem I get a quantity which has a dimension of a resistance or as a factor which is called a hall factor r sub h I can measure that and amazingly if you just when you go home and do this two line of math you know just substitute take the first equation and the substitute the expression for e x in the second one then you will see that there is this amazing result do you see what what has happened that that r 8 which is something I can measure is equal to minus 1 over q n now q I know charge but this one directly telling me how many electrons I have in the semiconductor because it gave me n not giving me n d or na or anything directly telling me how many electrons I have it works for all regions freeze out extrinsic intrinsic everything it works I will ask you to do two things when you go home one is that redo this calculation for holes if you do it you will find this expression it will go exactly the same way because holes have positive sign right plus q this answer will eventually get out to be plus 1 over q p plus and that tells me immediately the hall voltage will be in the opposite direction for electrons and a holes and that immediately will allow me look just by looking at the sign whether it's n-doped or whether it's a p-doped material right just by looking at that expression otherwise somebody gives you a little piece of silicon how do you know it's n-type how many electrons you have whether it's p-type you actually know do not know anything this experiment tells you the doping of that semiconductor okay now another thing that is a little variant that you can play against with is that what if instead of putting a voltmeter instead of putting a voltmeter if I try to short it or put it through a resistance finite resistance I should still be able to handle it j y I should not set it to zero anymore right other than that this equation tells me everything for any configuration so if I give you a problem which you have to think about a little bit about how to handle slightly different configurations you should be able to do that right hopefully and the real in reality there is a slight difference between the actual theoretical and actual value between point five to two but you know when you're talking about 10 to the power 18 three times 10 to the power 18 whether you are up by a factor of two it's not a big deal but of course when we do 656 the other course I will explain precisely when it should be why it's point five or two and what range how it comes about there's a little complication associated with the band structure that we'll discuss in a separate place but for this introductory course here is what we're stopped now this is how this I cheated when I presented this one a little bit because you know I simply said n divided by Nd is the one and therefore there's an extrinsic region before that I have this freeze out region how do I know I didn't know Nd at that time I implanted device with a certain number of dopants I do not know how many actually are activated because there atoms has to go in a substitutional site it has to remove a silicon if it is sitting in a cage surrounded by silicon but not in a lattice site then it doesn't give any electron or anything so therefore I really did not know about Nd so the way people would do it they will do the hall measurement remember the experiment 4 degree to 300 degree Kelvin I said next to the experimental setup right and that range in fact you go around and from there you get the curve and whenever the part is flat well you say that's equal to Nd and that's how you backtrack Nd out and then the intrinsic and extrinsic also that that was the setup that's what I'm saying that this had the entire range of temperature and then you can map out map these things out okay so once you know how to calculate electron number then you can push it in and get the expression for mobility right remember mobility needed the carrier concentration number now you don't have to make that approximation N equals Nd you just put the right value of N in and then you will get the mobility from there so that was my whole purpose of how do people get mobility through hall measurements in a indirect way let's talk about the physics of diffusion a little bit so see we are making progress the last equation is Poisson equation we just calculated and understood the physics of mobility mu sub n or mu sub p now we will talk about the diffusion coefficient or diffusion the second term this term and that term we'll talk about that and see how it have how it works okay now diffusion is something that you know for a long time right this happens all the time now diffusion is something much before drift and people understood about this people understood about diffusion if you have pollens let's say on water you can see they gradually spreads out right from that you can actually calculate the rate at which spreads out the avogadro's number that's what Einstein did to calculate avogadro's number about how interpreting an experiment from a botanist about how the pollens spread out when you put it on a water right now this is same physics so if you have a drop of ink in water it spreads out and similarly if you have one point where you have photo excited electron and holes lots of but one point let's say then a little bit later if you come you will see that the electron and holes have essentially spread out from their original position so let's start with there now it's a one-dimensional thing highly simplified so if you pump it in electrons will be moving in both directions plus and minus k remember why band structure when you pump electrons out it can move up from the valence band to the conduction band either in the plus side or in the minus side no problem and so the electrons in the beginning will start moving like that and a little bit later few of them half of them will be moving to the right the ones that started moving to the left a little bit later they will fraction of them will change direction and half of them will keep going in the same direction and this will continue and gradually you will have a spread out profile of the electrons original set of electrons now of course I haven't drawn it to the scale if you just shine light and turn it off the height of the middle region will gradually come down and gradually get spread out because area under the car must be the same now there is a funny thing in here why does it is all in a sudden the electrons decide to reverse direction it must be because this scattering going on right so the electrons come in even when you in the background lattice what is shown here in white it is not white it is full of atoms which are vibrating in real temperature and this full of ionized impurity remember so the physics of diffusion and the physics of scattering are essentially very closely related because that is what makes half of them every once in a while half of them going one way and the half of going going the other way I'll explain why this is a fundamental consideration in a second okay so let's do a simple calculation for saying that what rate the electrons will move through that lattice no electric field here but simply because of this density gradient so I have taken a snapshot and the line here you see black line that's a function that's an instantaneous profile of let's say the electron concentration or whole concentration so let's look at the electrons in the red region you see in the red region half of the electron in that region this is a number but half of them moving to right half of them moving to the left similarly on the other side you have that many electrons again half and half going to the both sides so if you wanted to calculate current all you have to do is to sit down in a highway and at one point one point look how many cars going one way versus how many cars going the other way and if it is steady state that difference tells you the current if you know current at one point you know current at every point right okay so what we'll be doing we'll be sitting at x equals zero looking at how many blue electrons cross the net from left to right and how many red electrons go from right to left that calculation is what I am after so this will be a complicated expression but actually is very simple so let's go step by step let's just focus on one term forget about everything focus on one term what I have written within the bracket with the blue is the area under the car for the blue trapezoid do you agree see the one of the ordinate how do you calculate the area of a trapezoid the two ordinates right and then you divide by two multiplied by the distance right that's the area of a trapezoid p0 is one of the ordinate at x equals zero think about the blue now what about the other ordinate at minus l well if the slope is dp dx if I knew the slope and I know from here multiplied by l that tells me how much it has come down starting from the x equals zero point subtract from p equals zero that's my second ordinate divide by two and you can see a multiply by l right out of the right out of the square bracket so that's my area under the car so that and the minus q over two which is multiplying the whole thing minus q because electrons moving and two why is this two two because half of the blue electron is moving to the right so that is the first two and the second two is from area under the car right two different things well I shouldn't have to do the red one right the only thing I have done in the red one that since it is going to plus l so therefore instead of a blue dp dx has a minus sign and red dp dx has a plus sign a little bit more electrons so that's that's what you have now you see this first term has minus q over two going from left to right the second term has plus q over two going from right to left okay so this is the number of electrons but are they moving quickly or slowly that's the denominator how long does it take for them to cross because if they are moving crossing very slowly then you have a low current if they are moving very fast then you have high current so l divided by vth which is on the denominator tells me how long this electrons take to cross barrier why is it not two l over vth the reason is I will assume on the average although the blue electrons could be anywhere starting from zero to minus l on the average they are at minus l over two similarly the red electron on the average starting from plus l over two in the middle and so the difference is l not two l so you divide it out fortunately everything goes away and you have a very simple expression that tells you that the current is directly proportional to the gradient of course if you had no gradient 50% going this way and 50% going the other way gives you zero current so for diffusion you must have a gradient and there is a bunch of constant l multiplied by v divided by two well I will call it a diffusion coefficient just like for mobility we gave it a name we will give the name l multiplied by v divided by two the name diffusion coefficient okay why this quantum mechanics in all this we studied all those things and is it not in anywhere in this business do you see this this is actually do you see where it is this is isn't there this my daughter has this world why is Waldo type of thing so you should always ask yourself why did I lose my quantum mechanics quantum mechanics is coming with you in your pocket you may not see it quantum mechanics is hiding here thermal velocity is what is that half mv squared is equal to kt and what is that m effective mass that is what is through this vth is carrying through your diffusion coefficient the information about all your band structure all your velocity and everything so make sure that this is a semi classical thing of course people knew about diffusion 200 years ago but the interpretation of diffusion for semiconductor devices that is after quantum mechanics and other things that is a completely modern phenomenon remember that too scattering back and forth and so although this looks like that this doesn't know about scattering in fact because you know I don't see any tau sitting anywhere here but actually the diffusion coefficient knows about scattering because without scattering this half and half division at every stop that they are half going this way half going the other way that wouldn't have happened unless this scattering was going on at every step of the way now I'll talk about something is very important is called this Einstein relationship the Einstein relationship is says that you don't really need to know both diffusion coefficient and mobility separately you can just find one and we just did that hall measurement from that we calculated mobility so you don't have to do an diffusion coefficient anymore why because I just expressed lv divided by 2 that was the definition of diffusion coefficient and q tau over m star do you know that's the mobility expression right if you divide it out then l l is the distance it goes before scattering how do I know that because that is when it is half and half is scattering going the opposite way you see if 10 electrons started a little bit later 2 electron scatters back that is not what l is the l that I have to choose in that previous last slide calculation is a very specific l l up to which half the electrons have scattered back and moving in the opposite direction so very special l and that's called a mean free path and so therefore l is the velocity v multiplied by the scattering time average scattering time tau so you can see that's why these two things are related so if you put it in you will get half mv squared on the numerator and half mv squared is on the average it is kt and so therefore d over mu is kt over q it sort of rhymes right that's how I remember d over mu kt over q so that's how you remember that how how these two things relate now it's very important that understand why these are related why math says so because both are governed by scattering so what do you think if in modern transistors if you don't have any scattering in the device let's say electron going from one contact to the other without scattering will you have an instance relationship in between not really because there's no balance between the scattering is not really connecting them up together right so this is very important now one thing I want to mention that maybe I go I'll go to the next slide and derive it in a slightly different way and explain this that this is although it bears Einstein's name and therefore you might assume everything is obviously always correct this is an approximate relationship very powerful but approximate relationship where is the approximation the approximation is in equating half m not v square is equal to kt this happens only for non degenerate semiconductor if you had fermi dirac statistics then the electron number the way you calculated this average energy will not be equal to kt and so therefore in more degenerate semiconductor case in that case again this relationship will require modification but you will still have a relationship only that this is not the constant one which you can count on let me derive it in a slightly different way and that's we will see how it works so this is the same derivation of Einstein's relationship but in a slightly different way what I have drawn here is a diode semiconductor diode but forget about how I have drawn it and I'll explain that later what I have done especially is that I have the dotted line is a fermi level and I have drawn it flat now the flat anytime you have a system in equilibrium fermi level must be flat so that's why I have drawn it flat by definition you have to have flat fermi level now the ec1 is sort of the conduction band close to the fermi level and ec2 is the conduction band far away from the fermi level so do you realize that on the left hand side you have a lot of electrons right close to the fermi level you have a lot of electrons and on the right hand side you have very few electrons why far away from the conduction band so you remember n sub c the effective density of state e minus the e to the power ec minus ef over kt so it's far out few electrons so that's what the green one shows the green one shows that the carrier density is moving from n2 to n1 coming down now think about what would have happened to this to this density because of diffusion that we just learned on the left hand side you have humongous carrier concentration on the right hand side tiny amount so diffusion would immediately try to wash it out why can it not wash it out because you can see that there is a electric field going from ec1 to ec2 change in the potential there is a electric field and the electric field is pushing it back diffusion is trying to wash it away electric field is pushing it back and as a result in equilibrium no current so the drift pushing it back and diffusion try to go to them going to the right hand side together they make the current zero in equilibrium so therefore I could always write that expression that the derivative of n divided by n is given by that constant now I can integrate that equation so that I relate n2 to n1 but how it's changing I do that and let's say the mobility is a constant and the diffusion coefficient is a constant so all I have to do is to integrate the electric field across the region when it's moving from ec1 to ec2 what is that that's equal to the potential difference right between these two points the v so I now I now know how n2 is relates to n1 through this drift diffusion equation but I also know it in another way do you agree with this statement independent of the top three equations I could also have written it this way n2 is how far away the ec2 is with respect to the Fermi level right similarly n1 I can say how far away it is from the Fermi level for the ec1 if I divide it out the whole thing comes about e to the power qv over kt because the difference of ec1 and ec2 now you will see a magic that the Einstein relationship will come out of here because n2 and n1 this relationship I know in two different ways therefore I must be able to say that these exponents must be equal to each other and you see do you see that Einstein's relationship over d over mu is kt over q is coming in therefore this is another thing about Einstein's relationship that it's a property of the equilibrium in general right because you see the whole thing I derived through equilibrium and what it does is that we have seen this before we have seen this when we did the capture and emission in Shockley-Reed Hall right do you remember the detail balance electron coming down and the electron going up from the trap level so we have seen this relationship before so Einstein's relationship is a consequence of the detail balance Einstein relationship is a particular approximation of the detail balance detail balance is always true property of the equilibrium Einstein's relationship is a way to capture that and the main point is detail balance requires the Fermi level in equilibrium be flat everywhere because if it were not flat you can see n2 and n1 would have a different relationship Einstein's relationship would not have been satisfied so this is Einstein's relationship is simply a consequence of equilibrium property in detail balance that's something we need to know okay so we'll end here so the first two or three things we covered today first was the measurement of Hall effect hopefully you will understand it in detail so that you can answer complicated questions not too complicated hopefully but at least you will be able to answer non-trivial questions I told you about drift in the last class diffusion today recombination generation last two lectures or the two lectures before that and we now have everything we the only thing we'll have to do is the continuity equation and we are done with the set of equations that will allow us to solve all sorts of devices for electronic applications right so that's the in fact also the end of the first five weeks and the exam will be once we put it together that's where we stop and the first exam will be based on that material okay all right then thank you