 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. Okay, so let's get started. Today it will be lecture 10 of 6.06 and today I am going to talk about potential electric field and charges, EK diagram, a set of information that we will need often over the next few classes or throughout the rest of the semester. So the first thing is as I mentioned potential field and charge, how they are related to each other, how to calculate them and so on and so forth. EK diagram is the solution of the Schrodinger equation, right? That is that for every value of k between pi over a to minus pi over a, the Brilloian zone, we have come to learn or understand that there are a set of states in which electrons can sit. That's the EK diagram for a bulk material. If I take this table, there is a EK diagram for the table, right? There is everything that you have. In principle, all electronic materials, there will be an EK diagram for it, but band diagram and EK diagram is not the same thing. Band diagram is a position resolved information within the bulk. So one end of the table will might have a different band diagram compared to the other end of the table. Then we'll talk about the basic concepts of donors and acceptors, why we need them and finally conclude. Consider that I have a semiconductor and what you see on the top side, on the top fixture, the potential moving up and down in this rectangular type potential going up and down. And this reflects, if you remember, these are Coulomb potentials associated with the nucleus of the atoms. And so an electron shown here in red that moves through this crystal field of potential repeatedly being pulled in and out. Now, when we discussed all this, we did not discuss about its potential energy. We just assumed that it is sitting grounded with zero potential and we wanted to look at its energy with respect to that potential. But in principle, of course, you can take that same semiconductor and put an external electric field or external potential and pull the entire energy up and down. This is just like having a person being in the first floor. And then you take an elevator and put the person in the 10th floor. Now everything about the person has remained the same except by this external potential that you have moved its overall energy up. So, there is a distinction between external potential shown here in a battery so that you are pulling the whole thing up versus the internal potential which is coming from the Coulomb interaction from them. Now, from this you remember that what we have done that we have gotten rid of the internal crystal potential by this ek diagram because the ek diagram essentially contains the information about how the spacing of the atoms, the potential depth of the atoms, the ek diagram knows about them all. So, now if you want to look at this in the ek diagram consider an electron sitting at a certain k away from the center. So, naturally it will have with respect to the bottom of that band it has a certain amount of kinetic energy h square k h bar square k square divided by 2 m naught. That is the amount of kinetic energy it has. But in addition of course it has the potential energy this point EC. Now, this potential energy could be zero. This I could have a reference or if I put an external battery put the whole thing in a cage which has a higher potential then I can move the EC the bottom of the band arbitrarily up and down. And my total energy is therefore will be EC minus e reference and notice this sign because we are always talking about electrons where the electrons the electrons energy and so on so forth. So, we have a minus q multiplied by the potential V. Now, the e reference what is the value of e reference it does not matter because at the end of the day whatever value you choose whatever value you choose we will always be talking about the difference potential difference and therefore whatever value you choose for e reference eventually this will all disappear as a difference. So, we should not worry about that too much. Okay. So, these two things and potential energy and kinetic energy together it makes of course the total energy of the electron. Now, one thing I want to mention here that assume that a semiconductor is 1 centimeter. I have a E k diagram if it were 2 centimeter cube or 5 centimeter cube you see when I look at the E k diagram I essentially have the same E k diagram because eventually I will do it per unit volume per unit length. So, I will actually take care of that. However, one question you could ask that at what point how big does a solid have to be I mean no solid is infinite. How big does a solid have to be before it develops the E k diagram. That means that you have the periodic condition all the energy levels are essentially equally the space in bands how long does it have to be. It turns out on the order of maybe 5 to 10 atoms is enough. 5 to 10 atom as far as you are thinking about E k diagram is actually more than enough. Otherwise and I will explain that later in the course if 5 to 10 were not enough then the computer you are using would be unusable because the computer you use has a very thin layer called gate oxide which prevents carrier tunneling remember carrier tunneling through the gate. That is these days is 1 nanometer half the size of a DNA. That is how things are single strain equal to the size of a single stranded DNA and even that can prevent the leakage current essentially through it. So that means that within that 10 or 1 nanometer which is about 6 atoms or so it has developed the full band band gap that's why it's preventing currents from going. You will understand the statement a little bit later but I what I'm saying that essentially on the 10 to 15 would be more than enough to develop the E k diagram. So let me then break the solid break the solid up into the units of maybe 20 atoms let's say I break it up and that if each of the sections each of these 10 atom units are all at the same potential then I can say that I have the E k diagram for each section for each 20 atoms I have E k diagram and they are all in the same potential so I have drawn them all linearly on the same horizontal line right potential is away from the how far is it up over the reference energy so therefore they are all at the same energy because they are all in the same potential. Now if you take that same semiconductor one dimensional semiconductor but instead now you do the following that ground the left hand side left hand corner which is V equals 0 and then apply a voltage on the right hand corner which is V equals V 1 then of course you realize that if it is one side is 0 volt another side is 5 volts then in between the potential will gradually change you know 0.2 volts 0.4 volts it will gradually change to 5 volts and here I have just divided into 5 sections for simplicity of drawing so if it is 5 volts you can say that the green one that region is approximately at 1 volt the slight blue one light blue one 2 volts and so on so forth. Now what would be the corresponding E k diagram for such a system do you see that this should be the corresponding E k diagram why because each section is sitting at a slightly different potential if I assume that on the left hand side that green E k diagram is sitting at potential 0 then everywhere every other E k diagram will gradually shift down now it's not exact diagram of course but this is the idea that gradually the it will gradually go down why does it go down because I have applied a plus V and remember the energy is minus Q V so when something is that plus 5 volts that means electron energy has gone down by minus Q V so that's why I have drawn the right hand side lower than the left hand side okay so that is what the energy of potential energy is now you see the potential energy is changing as a function of position. Now the E k diagram will love of course if you have a big solid there will be lots of points like this so that's not a problem. Now let's talk about potential fields and charges let's say somebody has given you this stop diagram where the potential then this is the energy means minus Q V so therefore if you just wanted to calculate the potential V then this is how it must have gone down why is it think about it for a second the left hand corner on the very top E k diagram I'm setting that to be 0 and the energy on the right hand side of the E k diagram you know right hand the last figure blue and red the last one is at minus Q V that is how much the energy have gone down if you just wanted to know V not minus Q V then you will divide the top diagram with minus Q and if you do that you will see the whole thing will flip upward so that's the potential you can see the for there is a change between let's say 1 to 2 from 2 to 3 is approximately the same it's just a diagram and from 3 to 4 there is a jump up in the potential because I'm just dividing it up on the other side if this is my potential what is the electric field do you agree that this has to be my electric field why because electric field is a derivative of the potential with a minus sign think about the green curve for a second from 0 to the first point where it changes its slope if I take a derivative the derivative is 0 right so the electric field from that point to the first box in the magenta one that's 0 now you can see that from first point to the second point there is a change in the slope take a derivative change in the slope is the magenta line but this time do you see that it has gone below 0 why below 0 because the electric field has a minus sign minus dv dx and then you can easily understand how the rest of the thing proceeds now what is charge charges one step more that take the derivative of the electric field with respect to position and that gives you the charge so you could do the same for example in the magenta car take this derivative of the line it was 0 to begin with independent of position it stays there for derivative is 0 right now you can see the magenta car makes a rapid change at that point at the first point first transition point and so there'll be a delta function in the derivative then from let's say find from this point to this point then there is constant field derivative is 0 you get the idea that's how you calculate it now in reality what will happen people will give you the blue diagram on the bottom and we'll see that and what you will have to do is go up in the directions that they will give you the blue charge distribution then you will calculate the electric field then you will like calculate the potential that's a very good idea to learn to do it graphically you can always do a complicated integral with you know boundary condition here and there and all sorts of things but you see all you have to do from going from one diagram to another is to integrate the area under the curve for the previous one up to a given point x right if you do that you will get constructed easily learn to do it because that's for how will make your life significantly simpler okay so this is the ek this is the charge potential and electric field how you calculate each other calculate them now let me talk about this ek diagram versus band diagram how they are different it's a conceptual difference it's not really convenient and let me show you you remember from last class about carrier distribution that we have a density of state for each band of course and then we have multiply the by the probability that his state is occupied and then we get the certain number of electrons and similarly we get if you multiply with 1-f then you get a certain number of holes right this we have seen and this we have we can easily calculate so the other day for intrinsic semiconductor do you remember the value we calculated you remember this was ni with this geometric mean of nc and nv the effective density of state and e to the power the gap divided by 2 divided by kt right that's what we have half the value so that value that is how much we calculate now it's very important that you understand how small ni is and I'm going to talk about that in a few minutes ni if you put it put some numerical values in 1 ev band gap put this the number will come out for silicon on the order of 10 to the power 10 per centimeter cube right 10 to the power 10 per centimeter cube how many atoms do we have per centimeter cube approximately 10 power 22 or so so we have 10 power 22 atoms 10 power 10 number of electrons so that means I have one electron every trillion 10 to the power 12 every trillion atoms so you can see why when we did for me to rock we didn't have to think about Coulomb interaction because one electron there's a trillion of neighbors before it sees another electron from the other one right far far apart about 10,000 atoms in each side like 5,000 angstrom before it sees another electron so it's a very very dilute electron number and that's why intrinsic semiconductors are not very good conductor to begin with that few electrons what you're going to do now that information will also be useful a little bit later but let me now point out to this point that I also made this argument that most of the electrons sit within a few kt of the bottom of the band right so instead of really thinking about how they are spread out and everything why not I take those states the red one over there which is density of state multiplied by few kt that gives me the effective density of state and squish them all down to the bottom of the band and that is what the NC and NV is these are delta functions which I have integrated over a few kt multiplied by the density of state try this out the density of state take that expression multiply with kt don't do anything else you will see you are almost getting the expression for the effective density of state because that's where the area under that rate curve is or the blue curve is all right so I have this effective density of state NC and NV with one line I represent all the energy resolved information right now I have that picture you remember but I don't want to keep drawing all sorts of ek diagram and that will take forever but should I then do this I can in fact for the all the ek diagram remember only the bottom little bit of kt is being occupied for the valence band all in little bit is being occupied so I'm going to replace all those things with NC and NV at energy EC and AV throughout this region so I'm going to in fact not even do that I'm just going to draw a continuous line but remember each point on that line hides in it you're supposed to see in your mind that there's a ek diagram hanging on each point of this continuous curve because that point actually involves the information about all the quantum mechanics effective density of state all those things are actually hiding in this line why two because I'm just most of the time it's a conduction and valence band that are the two bands that I am interested in don't think that it is just two of course there are lots of bands and if I needed in sometimes there may be four bands carrying current then instead of two lines I'll have four lines following through this because the ek diagram just encapsulated in a slightly different form so from here anytime you have the top picture without the ek going like that you will exactly do the same thing and you can see in a therefore why potential is just the flip of the conduction band energy or the valence band energy and that's how the band diagram that top picture will be called a band diagram and the band diagram is related to potential with just minus q that's it and then you can do all the calculations as before now up to this point I have told you all this these are just preparatory information we'll use it over and over again you will see but for the time being let me start by talking about another additional piece of information now I just made this argument a few slides back that the intrinsic concentration of electrons in a semiconductor is very small 10 to the power 10 for silicon for 10 to the power 22 atoms one in a trillion for gallium arsenide it is 10 to the power 6 part 10 to the power 22 one in every 10 to the power 16 atoms you have one electron sitting and in a silicon dioxide like the window glass not even one electron if you made the whole earth made of silicon dioxide 10 to the power 27 is per centimeter cube is a volume of the earth if you made the whole thing about with you will not find one electron in the conduction band because the band gap is 9 ev so you can see there is no hope of finding anything you find it for a different reason but so the point is normal semiconductors most of the time will carry zero current no electrons so what is going to carry so therefore you have to dope it with additional things these are called donors and acceptors now in thinking about donors and acceptors people will be drawing a picture that is a little bit different from what you have learned from the crystal so for example this is a gallium arsenide face centered cubic do you remember face centered cubic lattice one fourth along the diagonal you have the other atom the sitting there you know all those now if you take any one of the corners and you realize that this is a tetragonal side it has it's connected to four neighbors right it's connected to four neighbors and actually you should convince yourself every atom here is connected to four neighbors now drawing three dimensional diagram is always a challenge so therefore what people do instead of drawing this in a four-dimensional diagram they press it in on a two-dimensional picture like this so for example you take the gallium gallium one which is a center magenta one you will see that it is being connected to the four neighboring blue arsenic ones right and every one is actually connected to four different neighbors so actually what they are doing because we want to simplify the pictures that we draw the top picture is a simplified equivalent of the bottom picture and here these are covalent bonds because electrons are being shared one from arsenic one one from gallium or for silicon for example and therefore you will see a pair of lines connecting the atoms the blue and the magenta atoms with a pair of lines there's sharing of the electrons for example for silicon this is a perfect sharing it's a pure covalent bond so that is a picture we'll be drawing but remember actually when we are drawing that picture we are actually drawing the three-dimensional crystal that's what we are actually doing now think about what happens when i have silicon so i have my silicon picture again you know it's not gallium arsenide so i don't draw two types of magenta and blue these are all silicon so i just draw blue here i realize that a number of intrinsic concentration minuscule so i have to do something about it what can i do i can put a throw in a bunch of phosphorus atoms in here now first thing to notice that if i put in a phosphorus atom and this is a very important statement every atom every one of them carries a certain number of protons certain number of electrons and generally they are the same proton has also the same number of electrons and same number of protons it has one extra electron but it also has one extra proton so when i sprinkle a bunch of phosphorus atoms in the whole thing is steel charged neutral do you see because i have bought in extra electron but i also bought in an extra proton so the whole thing is steel charged neutral charge is steel conserved however the one electron extra electron in proton the phosphorus that may be available for conduction for example i have shown in the bottom that extra electron in phosphorus phosphorus is other electrons that i haven't shown but the extra electron i have shown here in the in the blue red point and of course there is a corresponding extra proton sitting there also in the background of the silicon network do you see this it's a very important statement because people often get tripped on this this idea they think that phosphorus is an extra electron therefore i always have an extra charge not true because it also bought in a an extra proton along with it now this electron how do i know where would the electron sit because it's not silicon anymore there's a few extra phosphorus in where will the electron sit i can think about it this way i can think about it a phosphorus doped silicon as being pure phosphorus pure phosphorus plus i'm sorry pure silicon plus one extra proton sitting in the center and one extra electron going around it is that right because phosphorus has just one extra so i anything extra i put it in the right hand side like an hydrogen like diagram and everything that is almost like silicon i put it on the left hand side with a bunch of blue things now let me assume at this point and i'll prove that assumption in a second that this extra electron can be viewed that it is swimming in the background of silicon atoms that there are lots of silicon atoms so i take the left hand side and make it a gel because it's so big so many atoms there that essentially they can all be viewed as a background material and i'm thinking about this extra electron going around the extra proton over there for phosphorus right okay so i can calculate this energy level can i how do i do that because i already know from hydrogen levels where the level has to be this formula you have seen before in fact you have seen maybe in lecture two i pointed out when we said hydrogen levels and remember 13.6 divided by one over n squared i'm thinking about the first level so my n is one so you don't see that one divided by n square here but apart from that if you open your book it's exactly the same formula q to the power four m star and everything why is it m star because in the hydrogen case the hydrogen atom was moving in a vacuum so it was a free electron there and so therefore i put a m not there but this time however this extra electron moving in a sea of silicon atoms right and therefore it is filling the potential of the silicon atoms it's being bounced by those potentials therefore the effective mass has to be m host whatever the host material is on the other hand also on the bottom side you see i have the dielectric constant of silicon why because when the proton is attracting the electron this potential is being mediated by this the dielectric constant of the material host material right so therefore i have that that number okay there's no rocket science here very simple now if you want to express it in terms of hydrogen things hydrogen expressions that we already know so i have taken everything i have multiplied with m not and divide by m not the red m not and the black m not so it multiplied and divide and similarly you can see that i have taken the kappa for the host to the right hand side the k for the host on the right hand side so the left hand side the red red thing in red i already know do you remember that it was 13.6 in the hydrogen atom case that was 13.6 and that's why i pointed it out let's quickly calculate this number what is the typical effective mass for a for electron let's say one tenth of the free mass so that will be point one and what is the typical dielectric constant of a material silicon and others on the order of 10 also on the order of yet so so you have one over 10 squared and m host multiplied by m not is point one let's say so i have to divide 13.6 by a 1000 how much is that so it's with 13 mille electron volts you see very very small and therefore when you look at the level where the this atom sits it's a bound level so it's a little bit lower and this is about let's say 13 mille electron volts very close what is the band gap band gap is about a e v and that this electron sits about 13 mille electron volts from the band and that's why this electron is so easy to detach had it been a pure hydrogen he would need 13 volts to make the electrons go here 13 mille electron volts room temperature kt is 25 mille electron volts right so even in room temperature most of them will escape from the bound level you see now i made another approximation and let me the another approximation is that the electron moves so far out that it encamps encapsulates or encompasses a lot of silicon atoms right that's what i said that's how i made the blue almost like a continuum right is that true well again i can calculate this is again from that hydrogen atom picture you can see the effective mass and the dielectric constant has been replaced calculate the whole thing and you can convince yourself that for this material approximately the radius of the electron moving around is about 13 angstrom on the order of 13 to 15 angstrom how far away does the hydrogen atom electron a hydrogen electron go you know about a 0.5 to 1 angstrom so that's how close it is because the charge is very strongly being pulled now this is moving around about 13 angstrom around and the effective at the a for the lattice is on the order of a divided by 2 this spacing which is a few angstrom that means it can encapsulate encapsulates hundreds of atoms in its circuit you see inside it is moving around in a big big orbit and lots of silicon atoms are sitting there now you should see table 1.6 and 4.1 and put some begin to put some numbers in to convince yourself that the argument is true this is not always true and i can always give you an exam problem where this is not true and just to see whether you know the difference between real effective mass when you should use it and when you shouldn't but just like hydrogen atom you can have one level there are some other donor levels where you can have two levels below the gap there are some other levels shown here on the right on the yellow where you can have three levels out a level deep in the in the gap now there is a convention that anytime you have a donor which is in the upper half of the gap red and the blue ones the word donor will not be written and in this case you will always assume that this has an extra electron it is it gives the electron out to the conduction band it only talks to the conduction band for anything that is below people always assume that it should talk to the valence band but if you want to suggest otherwise that it only donates electron in that case there's you should write a small d to indicate that it is actually a donor although it is sitting way down in the energy gap it's just a convention it's no physics here you can do the same thing for boron and which is an acceptor atom again remember this is charge neutral it has one less electron one less proton charge neutral and again you can think about the same calculation one extra electron and you can think about that there's a hole which is moving around like this so the extra electron will be like the core of this and the hole will go around it same calculation i will not go through it and again the same ideas that do you see that anytime i'm talking about a donor or sorry acceptor generally it talks to the valence band and anytime i want to suggest that a level which is up but still is a actually an acceptor i will put a symbol on the next to it just to indicate that that's a donor level i'm sorry an acceptor level and finally this sometimes there are donors and acceptors which are sometimes there are atoms which in a material behaves like a donor sometimes behaves like a donor and sometimes behaves like an acceptor so it has a split personality now depends on the environment of course so for example i'm thinking about gallium arsenide thinking about gallium arsenide and let's say i take germanium or silicon and i put sprinkle down silicon in gallium arsenide now from a gallium's perspective from gallium's perspective silicon has an extra electron and an extra proton so it's a donor as far as gallium is concerned now as far as arsenic is concerned well it has one less electron and one less proton so as far as arsenic is concerned this is like a acceptor level right and so depending on which atom the silicon you know silicon you introduce it's either going to kick out a gallium and take its place or kick out an arsenic and take its place right it's going to do either one of the two if it kicks out a gallium the red one if it takes a kicks out a gallium then it has for the overall semiconductor it has one extra electron to give one extra electron uh one extra proton to give right so that's that will be a donor level on the other hand if it kicks out a arsenic and sits in there that place right then overall it has one less electron one less proton so it will be an acceptor time now this moving one to another many times you know you have the LEDs there are these white LEDs that is being being replaced in many places in the street signs and others one of the problem and many professors spend their lifetime in doing this is making sure that it preferentially goes to one place versus the other there are many materials which you can it only wants to go to let's say gallium one side doesn't want to go to the other side and sometimes it will go equally on both sides and eventually you don't have anything constructive because some gave some extra electron others took it out and at the end you're back to your intrinsic semiconductor what is what is that so these type of things at dopants are called amphoteric doping and controlling this i'm sure if so many of you will probably have research projects where controlling this will be a part of your thesis right so it's a very important topic the thermodynamics of it and how this goes but that will be covered in a different course and different time but remember this definition now i want you to read this table because be able to read this table because this is something you will be using in the next homework do you see let's just focus on germanium 0.66 ev on the band gap on the top side you can see the gap center dashed land line going throughout and you can see various dopants atoms lithium and then what is that tin sb is thin and then phosphorus arsenic and so on so forth now you can see a number written like 0093 that means this is 0093 ev below that level and at room temperature 25 millilectron volts that level will certainly be all ionized it will give away its electron and therefore it's a good donor and the phosphorus you see 0.012 we even five seconds ago i calculated right what was it i just calculated about 0.013 right 13.6 divided by a thousand that is how that number 0.013 came about and going down you can see 0.14 and 0.28 and in that case it has two levels from which it can donate the 0.14 will be easy to donate 0.28 will be more difficult right and do you see in some you have indicated by d do you see any one of them not probably on that one but you can see in silicon on the right hand side for tungsten there's a d 0.31 and d that means this is not an accepted this level actually talks to the conduction band this is a donor very bad donor of course because it is sitting so deep you'll really need a lot of energy to let it go but with light and other things you can let it go now you can see correspondingly all the accepted levels the boron we just discussed 0.01 ev just above the valence band and so on so forth you can also see that there are things marked as an a so for example copper in germanium on the top side on the right hand side you can see it is a rate 0.26 but underneath an a that means that that level that one level talks to the valence band and it is actually an accepted not a donor so these informations we will need over and over again when you do actual calculation okay so what I had to discuss today was the bulk electron density can be calculated by density of state and fermi drag statistics right that we that we discussed and we also talked about this notion of a band diagram and how to construct it from a ek diagram don't forget it you know many people actually learned these things but never see the connections explicitly so anytime you have a new material people often get tripped this is something people have done for 50 years for silicon so all the diagram and everything has been all been said you can almost memorize them but many times as in a new electronics your material will be very different your ek diagrams will be very different and so unless you know the algorithm by which they were constructed it is unlikely that you will calculate them correctly or draw them correctly okay now the intrinsic carrier concentration is very small this is something therefore most of the semiconductor must be doped in order to make them useful in order to have a certain number of carriers because think about it if you dope silicon with 10 to the power 16 number of phosphorus 10 to the power 16 they will all give a their electron so in the background you had 10 to the power 10 right it's own intrinsic concentration for silicon was 10 to the power 10 just by putting 10 to the power 16 atoms in you now get 10 to the power 16 number of free carriers available for conduction 10 to the power 16 is not a large number you see because the total number centimeter cube is 10 to the power 22 per centimeter cube right so you are putting one phosphorus atom in a million but steel that the or its own asset is so few silicons or germanium so few that even this minuscule contribution from phosphorus steel controls the conduction process in this material so doping is central for semiconductor for good conduction and i also try to explain why the dopant atom behaves like a hydrogen atom and with the modified dielectric constant and effective mass right and this is something you should check out put some numbers in because when i say of course i know these things i have walked this out but unless you start putting some numbers in the calculator yourself this will not sink in in you that how dramatic how dramatic these energy levels are compared to an hydrogen atom how large the radius of the orbit is with respect to the hydrogen atom okay all right so i will end here and next class we will continue