 You can follow along with this presentation using printed slides from the nano hub. 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. So this is 606, Solid State Devices. If this is the course for which you are in, then you are in the right room. Now, I'm Muhammad Ashraf Lalum and I'm a professor of electrical engineering. And I have been teaching this course for several years now. I have come after a long industrial experience at Bell Laboratories for about 10 years. And so I have a broad industrial background. And I have been teaching it for a few years now, three, four years now. And so this is one of the courses that defines sort of the language by which people in our area, micro-electronics nanotechnology area, they communicate with each other. Because many of you come from different backgrounds, some from electrical engineering, some from physics, some from different countries. And you will meet here. This course will define the language. And then, of course, in your PhD or master's thesis, you will be doing different things. And this sort of will define the broad framework of how you will discuss these topics among yourselves later on. And most important, there are two additional reasons why you take this course. One is this qualifying exam. People say that this is something, this qualifying exam is based on or you'll have to take this course. And the other is when you have job interviews, almost invariably, they will not ask you about any advanced courses. The only thing they will ask you is what you learn in this course alone. So therefore, this is a very important course both from intellectual perspective and also from practical career perspective. Okay. So let me begin by giving you some broad course information. And then we will go through the other topics of the day. The books, the two books we'll be using, these are advanced semiconductor fundamentals. This will be for first five weeks. And the second book is semiconductor device fundamentals. This will be for the next 10 weeks. So the first five weeks, we'll talk about the basics, the fundamentals, quantum mechanics, statistical mechanics, transport theory. These are general things. Now, in the next five weeks or next 10 weeks, we'll apply this whatever we have learned, the basic equations we have learned. And we'll be applying them to understand how semiconductor devices work, like diode, bipolar transistor, MOSFET, how they work. Now, these two things, what we'll do in the first five weeks, you'll see, we'll go through a lot of things. But at the end, we'll encapsulate it only a very few concepts, three or four concepts. And that encapsulation would allow us to then explore such complicated devices. These are very complicated things, diode, bipolar, and MOSFET, but in a very simple way. And that is the essence of these two books. Now, the course website will be at this cobweb.ecn.purdue at EE606. Notice the tilde. And the office hours, as I said, I'll be available after the class, which is supposed to end at 8.20 till 9.15. Either here, we'll stay here as long as possible, and then I'll be available in my office. This office is at EE320, upstairs. Now, in terms of the bad things, which is the homeworks and exams, there'll be nine homeworks posted, four will be graded, and this four will be randomly chosen. And there'll be three exams as we, you know, this first part of the basic transport theory. We'll, after we finish that, we'll have one exam, and the second will be after we finish the bipolar transistors, and the third is the final, which we'll do when we have discussed MOSFET. Now, let's get a bar-side view of what this course is about. And this course essentially has, let me first explain to you that if you wanted to understand a modern electronic device like that computer that you have on the left-hand side, then you understand it's a very complex thing. It has lots of ICs. It has a lot of the display itself. It has millions of transistors, 10-film transistors in the display. The microprocessor sits underneath the keyboard. The hard disk drive, there's ICs for those, analog, digital, all sorts of complicated things. But regardless of what the final product is, it always starts in the foundation of physics. And that foundation of physics is in quantum mechanics, as in statistical mechanics. And when you combine these two, and I'll explain what we mean by this as we go on. Now, these two, when you combine them, we'll get something called a transport theory that when you apply an electric field, then how the electrons move in a semiconductor. If you put excess electron hole pair in some place, how they diffuse away from that point. So this is the basic that creates the foundation. And that's the first five weeks, right? The first book, first five weeks. Now, once you know how electrons respond as a function of the density gradient, more density one place, less density another place, how they diffuse away, or how they respond in presence of electric field, then you are ready to talk about how different devices work. For example, how registers work, how semiconductor diode work. These terms you may have heard from your undergraduate years, bipolar, and how MOSFET works. So that's, will be, this would be the foundation on the right hand side, and the middle column, this would be the devices that we'll think about. Once you know how this individual components work, that's not the end of course. Then you will have one transistor, one diode, that's good for nothing. These days in a microprocessor you have at least a billion transistors. So there are of course additional issues. And the additional issues are of course how to connect them together, how to make them work as an aggregate. And this would be 10-film transistor for the display. As I mentioned, the display on your computer on the back plane, millions, millions of transistors, almost as large as a microprocessor. Then of course you have the microprocessors, you have disk drives that reads off your data, and then there are memes for the read heads. So all sorts of other things that you can get by combining the basic devices, right? And finally, of course, you have the, when you put them together, you have a system. So what is 606? 606 is this rightmost two columns. This is 606 and as you go through your graduate education, you will do the third and the fourth column by other courses, right? So that's what we are after. So what are the courses? Well, this is how in Purdue things are arranged. In Purdue, we have the 606 and 604. These are in our area, which is microelectronics and technology area, two basic courses. No matter where you come from, whatever your background is, you start with sort of these two courses provides you a foundation. But of course, this is a foundation which is not deep enough for graduate school. This is deep enough for probably undergraduate students. Now, for the first five weeks, remember, we said quantum mechanics, statistical mechanics, the transport theory, right? We said that's the first five weeks. If you wanted to know more, especially people doing work on the theory and modeling area, then in that case, you will have to do, take a whole set of other set of courses, like 656. This is a transport theory, semi-classical transport theory. 659, quantum transport theory. These are transport theory, expanded version of the first five weeks. Now, you can take all these courses for the foundation, but those would be doing, let's say, more experiment or systems. Then they would go to the other side, which is the next 10 weeks, right? And then they will go to the system side. And then at Purdue, you have excellent set of other courses. For example, the 654. This is, this should be 654 is solid state 2, or this may have a slightly different name here. Solid state 2 is a more advanced version where, you know, whatever you have learned, the basics of bipolar, you will learn about lasers. You will learn about microwave transistors, more advanced devices. But if you know 606 well, this course well, you see, that shouldn't be a problem. You should very easily be able to learn the other materials. 612, remember the last three or four weeks, we'll be talking about MOSFET. 612 is essentially all about MOSFET. And also bipolar transistor, but that you can take in a more advanced and expanded form in 612. And then there are courses on system design and other issues that you will be able to take. And you should take depending on your interest. But then that gives you the context that if you understand 606 well, then you can see that will create a foundation for all the other courses that you will be taking subsequently. So this is very important, right? Now, that's all about courses. But let me explain to you first that why this course, apart from the requirement, 606 is a required course, but why electronics now is a very exciting place to be, or a topic to learn. You know, last century, people say that this was a century of electronics and started from this radio transmission across Atlantic, great achievement. Even though it takes weeks to go across the Atlantic in ships, but you could cross the Atlantic in seconds by even less by radio waves. And you know how where electronics have come today. But you might not have realized that this 100 years is actually, there was a series of revolutions that happened. And these revolutions are based on electronic devices and these were very fundamental. It may or may not have happened. And if it didn't happen, then we wouldn't be where we are today in terms of electronics industry. Now, this one on the left side from 1906 to 1950s is vacuum tube diode. Even these days, you have vacuum tube diodes in very high power electronics. Then 1947 onward, you can see that transistor and by the end of the course, you will know exactly why the transistor has this wave shaped form and how it functions. And that was the next 30 years. And then there is this MOSFET from 1960s onward and that is what we still have in our microprocessors today. Now, each one of them, each of the transitions of course were noted or distinguished by several Nobel prizes. So these were big things. This is a lot of interesting things. And what I'd like to convince you today that there are several Nobel prizes waiting to happen. So maybe one of you. So as this one also comes to an end, the MOSFET, the time of MOSFET is also coming to some sort of saturation and we have a great opportunity for the future. So why did this transition occur? That you need to understand. The transition occurred because initially you have one vacuum tube. Well, you have some power dissipation. You have to hook up with the battery. But very soon, by the end of Second World War, when you put the vacuum tubes together in the system, it looked like that a huge amount of them were failing because it used to be so hot that they used to say that one third of the American Navy during Second World War was in for repair because of some form of failure and electronic failure was often a big deal because when you have a racks full of diodes, so hot that any of them, it's like light bulbs, any of them can fail at any time. And when one fails, you have to take it out, replace it, a big problem. So the power dissipation when it became so high, the temperature became so high, you couldn't have vacuum tube electronics go any further. So then came the bipolar transistor and it immediately bought down the temperature because it works at lower voltage. You can put them more together, close it together, take away the power, power came down. But then, of course, what happened is, you know, you crammed more and more and more. And by 1980s, the supercomputers had to put in water in order to take away the heat. You could see, literally see, and this I have seen in Cray 2. These were the supercomputer that bubbles are essentially coming out. It's so hot. And of course, therefore, the days of bipolar transistor was gone and then you had to do it MOSFET. And I don't have to tell you that MOSFET is also getting hotter. Laptops, they sell in Walmart special holders so that it can take your heat out. So the laptop is also getting too hot and think about all the data centers that Google and other people have. Huge power dissipation. And this cannot continue. And so the question is that what is next? So what this course does, actually, is try to understand this history of 100 years, the essence of this 100 years in essentially 14 weeks, and then try to set you up to see for the next generation. And that's what you are here for, right? You want to do research. You want to work on different topics. And there are many new things. People are talking about spintronics, biosensing displays, wall-mounted displays. That's like a paper, thin sheet of paper, all sorts of things. But if you understand how the previous generation worked, what their problems were, then you are ready to take the next step. So that's what 606 is, a physics of the last three generations. Okay? Now let's start by thinking about the current flow in semiconductors. I have a green region here, which is a semiconducting region, or some conductor. It could be metallic also. Two contacts, and I have applied a voltage. And you know what will happen. The current will flow. Now current flows, let's say the electrons shown here in red, flows in one direction, and corresponding current goes in the other direction. Why? Because electron has negative charge. So whichever direction it goes, the current goes the other way. Now, of course, this depends on how much current will go, depends on the resistivity of the material. Now different materials have different resistivity. So that, of course, depends on the chemical composition, chemical structure, temperature, and doping, all sorts of things. And we'll discuss that in a little bit. Now one thing is we could make a huge humongous table, measure the resistivity of every material. And that is, in fact, up till 1900s. That's what people used to do. Ohm's law, they used to make a resistance table of a huge amount of material. Just go in the lab, make measurements. And then you are done. But the thing is that that only works for known material. If you wanted to try out something slightly different, then, of course, it's better to have a theory for it, that why resistance changes as a function of composition or materials in a particular way. So that would be better rather than making a big table. Then we won't have to carry a big table with us. Now as you know, the current through Ohm's law, current is proportional to the voltage. But the other way you could express current, you could say the current involves a certain amount of charge q. N is the carrier density. However number of carriers you have multiplied by the velocity rate at which they're moving from one point to another. And A is the cross-sectional area. Now what we'll do in this course in the first five weeks is that first for essentially two or three weeks, we'll talk about the physics of carrier density. How do I compute as a function of materials and material structure? How do I calculate the number of electrons I have available for conduction? And that will involve quantum mechanics and equilibrium statistical mechanics. And that's chapter one through four. That's the one through four of the first book we are talking about. The first book has probably six chapters. And the next two chapters, the remaining two chapters, we'll talk about how when you apply an electric field or if you have density gradient, high density here, low density on the other side, what rate or how fast do the electrons move? That will be the next two chapters, which is chapter five and six. And that's the transport theory. We'll think about scattering as the electrons go along, how they scatter with the lattice, how they scatter with other impurities around. That we will talk about in. And our ultimate goal will be a drift diffusion theory, which I'll explain. And that will, the whole thing we'll use to understand diodes by polar transistor MOSFET. So that's it, right? First four chapters, calculating carrier density N and last two chapters, calculating velocity. You're done. Now the main thing is that, you know, quantum mechanics and statistical mechanics seems like a very complicated thing. But what will be amazing that you will see just two concepts. One is this effective mass and another is something called a Fermi factor. These two will be sufficient. After reading all this, that is the two things we'll carry over. And similarly for transport, all we'll carry over is a notion of a mobility and a diffusion coefficient. So we'll do all this, but at the end, carry over only a very few things for our next stage. And we'll explain why such a few things describe such a complicated problem so well. So let's talk about the three things that determine the carrier density. One is this atomic composition, arrangement of the atoms, and for periodic arrays like crystals, the concepts of unit cells, and how they are classified. So different types of material systems. So of course we understand that carrier density could be different, depending on number of electrons you have in a material. Like silicon have a certain number of electrons. And then let's say copper has a different number of electrons. So the electrons could be different depending on the number of the material, original material you have. And so let's start there. So this is a truncated periodic table for electrical engineers. For us, our periodic table used to be historically just silicon, nothing else. But it's a little bit expanded version of the periodic table, and you can see the materials of interest. What are the materials of interest? Like from group four, you see in the middle, from the group four materials like silicon is of course very important, and carbon. Carbon in the form of graphene, you may have hard silicon, carbon nanotube, these type of materials are of great interest for electronics industry. So the group four, and the first transistor was made of germanium. And at the end of the lecture, at the end of the semester, make sure you know why it was germanium not silicon. And we should be able to answer that question. Then there are also combinations of group four materials that are also huge in significant use. For example, all your microprocessors, the IC in your microprocessors in Pentium, these days have silicon germanium and you can find them 14 and 32 in the middle column, both of them. And also for high temperature electronics, you know many times for space shuttles and other things, silicon carbide is a material of great interest. Now there are also materials, this you might know, from three and five, from column three and column five, like indium phosphide, gallium arsenide and so on and so forth. Every time you talk in your phone, then the optical fiber carry the signal and the laser that sends the light through that optical fiber are made of indium phosphide. It used to be red laser is gallium arsenide base, but the indium phosphide, all telecommunication lasers are indium phosphide base. You can also have a more exotic combination like indium gallium arsenic phosphide and again from three and five, from group three and group five. And this of course have, each one of them have different properties. We'll get to that a little later. Now from two six, you have cadmium terrarium. From two and six you have cadmium terrarium. Do you see cadmium? That's 48 on the left side and 52 on the right most column you can see. And the late sulphide, well this was the first when the transmission, transatlantic transmission of radio signals, the first diodes they used to have, these are all based on late sulphide. This is a yellow paste that used to have this rectifying properties. So the first transistors are all based on late sulphide. So very soft material. Now of course all combinations are not possible. If you like to put anything in any combination it is not possible because of lattice mismatch that all materials do not have the same spacing among the lattice. So you cannot just put them anything with anything else. And you have to think about how to spend their lifetime in terms of developing this material. That's what material science department is for. Now that's material. But more important or equally important to material is the arrangement of atoms. And the arrangement of the atoms are in solid crystals. The atoms are in specific positions and with specific orientation. Now it is possible to have specific positions but random orientation that's like in plastic crystal doesn't conduct as well. You will be seeing any significantly lower compared to solid crystals. Then there is this liquid crystals as I mentioned the displays where you have the orientational order but no positional order. You see this these are in random places and for liquid you have neither positional order nor orientational order. And this fourth one will doesn't carry too much current. So available number of electrons you will see is very few. On the other hand for solid crystals this significantly more. So order is very important both orientational and positional order. Now in addition of this type of division there are this type of divisions here. This is a cross section of a MOSFET and we'll talk about MOSFET much later but I just wanted to illustrate a point. Look at the bottom side on the silicon substrate. These are individual atoms of silicon. Look at them just everybody in their place. This is how perfect modern processing is. You have billions of these transistors and almost this perfect throughout your entire IC. Now that's crystalline. We use a lot and I'll explain why the polysilicon gate and the polysilicon is polycrystalline silicon where you have crystalline domains of different order. That's one type of material that we should be talking about. The other material which is like a liquid in terms of random orientation is this amorphous oxide and without this the modern electronics would be impossible and third one is of course the crystalline semiconductor ordered nice arrays. Now how do we think about let's start with crystalline semiconductors. Well ordered and try to define them but what I want to point out we do it because it is easy but at the end it is not the whole story of course but many of the conclusion will be reaching based on understanding the crystalline semiconductor they will apply to amorphous silicon they will apply to polycystalline silicon and I'll explain why but the point is we are sort of solving this because it is easy to solve not because this is the main story you see. So let's talk about crystalline semiconductor and classification of crystals. So this is a periodic array of points it could be ions it could be atomic ions for example or for this matter here for this case just assume that these are geometrical points a series of array of geometrical points shown here in 2D it could also be in three dimension in periodic array. Now how will you build it up? It's an infinite array so one way to build it up is to have a unit motif of a rectangle shown here in yellow and you can see one fourth of the corners is included of each point is included in the yellow and this you can see that you can repeat and build up the whole crystal possible right you can just repeat them but you can also see that this is not unique you could start something like this which has the same volume or same area do you see that because say trapezoid height multiplied by the base and it's the same as before so this is an equivalent description and you could also use this one to build up the whole crystal that's possible you could even do this you could take one in the center put a rectangle around repeat this then you can build up the whole system right so this motif if you study the property of any one of them we will see that is sufficient to understand the property of the whole crystal that's why we'll be talking so much about the crystalline order or unit cells that's the amazing thing think about a semiconductor that may have let's say 10 to the power 23 atoms impossible to solve a quantum mechanical problem with 10 to the power 23 atoms we shouldn't even try no computer in the whole world can solve that many atoms problems with that many atoms but what is amazing is that if we can just solve one and patch it up in the right way we'll see we can have the solution for the whole thing and that's the power of the unit cell and that's why we'll be spending a few moments talking about the unit cell now of course unit cell need not be primitive meaning the smallest volume it could be something big and you could also repeat that the blue square and if you have used that same motif you could figure out the rest of the things so I think I have just discussed that the unit cells are not unique you can have various variation of it it's primitive and non-primitive primitive means the smallest one but even for the primitive one you can see the yellow one the trapezoidal one or one that has the atom in the center all three are primitive cells but they are of different types so even the primitive cells need not be unique and the property of one as I mentioned defines the property of the whole and that's why we want to understand them with some in some detail now one very good way so because it can be done the primitive cells can be done in various ways one prescription that you can follow over and over again is something called a Wigner site cell so this is a way of defining the primitive cell one of them but you are fine if you define one that's fine and this is how it works the chooser reference atoms so these are set of atoms shown here in blue circles choose the reference atom that's the red one in the center then the second step is connect to all these neighbors so by straight lines so you see here the blue the red has been connected to the blue and do that for all of them everybody however want you want in the neighbor first neighbor, second neighbor, whatever you want connect them all with straight lines but of course you have to take care of all the nearest neighbors then in the next step you see you should draw lines which is a perpendicular bisector of the connected lines so here the red and the blue on the left side you see I have drawn a vertical line which is a perpendicular bisector for that line right and then you should do it for all the connected lines all of them now you can see the atoms that are further away the intersected points the bisector happens at a point which is further down and the primitive cell well primitive cell is the smallest volume enclosed in this area so you see the one that happens in the further out which is the sort of the right top blue blue point the perpendicular bisector is further out so when you take second or third or fourth nearest neighbor connecting them doesn't really affect your primitive cell so you can take them it doesn't really matter but this would be the primitive cell so you can realize that in the previous slide when you had the atom in the center and the square around it that was the Wigner site primitive cell now there are now we know about unit cells the basic definition let's say how we can construct different lattices so one particular structure I have shown here it has a lattice constant of A and B a lattice separation in the vertical direction A and the other is B and we'll see how many different ways we can arrange this A and B in order to make crystals and the angle in between so that is the definition of a Breville lattice so for example the first definition let me step back the first definition is the Breville lattice is that if you stand in on any of the atoms in the lattice your environment around it should look exactly the same so for example if you take if you sit on the one that has the crosses in it all the neighbors around it should look exactly the same and I'll give you examples in a little bit later so that's one definition of Breville lattice the other definition is that you should be able to go from any atom right any point to any other point by this multiplying this unit vectors which is A and B you know the sides the two sides multiplying it with factors H and K which are integers so for example the one vector that I show what is the value of H I have 1, 2, 3, 4 so H is 4 and K is 2 because this goes vertically up you cannot have 4.32 or you cannot have K equals like 2.5 you cannot have that it has to be integer values and by providing the integer values you should be able to go to any points in the lattice now what about this is it a Breville lattice according to this definition well it is not for this very simple reason is because you see if you sit on the yellow yellow atoms then if you take to the right you have the green atoms very close by but on the left but for the green atoms you see there's nothing not a yellow atom on the right that is looks exactly the same there's a yellow atom on the left of course that's exactly the same so this is not a Breville lattice because in Breville lattice remember what is the definition every position that you want be that the environment should look exactly the same it doesn't look so but you can convert it to Breville lattice by combining the yellow and green into one point and then you can see if you sit on the blue points then the entire environment is the same but within each blue point there is this blue and there is this green and yellow subatomic structure that is the basis then you multiply the lattice with the basis you get actual actual lattice structure so in 1D one dimension you can see from the top you have atoms periodic series of atoms that's the Wigner side cell right and how many you have in 1D you have just one type because you have a line you just cut it in the middle you will just have one 1D these are actually one dimensional and these are of course in recent days problems of great interest many chemical molecules, polymers are actually one dimensional chain and in that case like the DNA DNA would be in one dimensional chain in some effective way and you can assign a basis and you can then expand it out so my main point I want to make that in 1D there is only one type Bravais lattice right what about 2D will be more right because 2D has two sides A and B and the angle in between so there will be more and it turns out there are exactly 5 in 2D there are exactly 5 and these are the 5 if the two sides are the same and the angle between is 90 degrees then that's one type two sides the same angle not equal to 90 degree you have a second type two sides unequal angle 90 degrees two sides unequal angle not equal to 90 degree and the general form and you can see the square in the blue what is that that's the Wigner side cell for 2D and you should convince yourself that this indeed is a Wigner side cell you say we take the neighbors bisect the connecting lines and that would be my Wigner side cell but main point is 5 types no more than 5 types and these would be basis of many many work now this is an example this is again not a Breville lattice very important material this is silicon graphene a material of great research and a lot of interesting work these days maybe some of you already work on this topic this is again not a Breville lattice because if you sit on the yellow then the blue is on the right if you sit on the blue there's no way corresponding yellow on the right there's a yellow on the left so this is not a Breville lattice because it's not equivalent all points are not equivalent but you could make it equivalent by doing this by combining a and b into basis replacing them with one point and if you combine all the yellow and the blue in one point you can see that this will become a rhombus like structure you connect them and that will again become one of the five Breville lattices right so that's something you should convince yourself that this indeed is the case now many of these things that you see in classical tiling these are all can be easily converted into Breville lattices you can see for the Kepler tiling if you put the green one and then bisect all the neighbors squares then you can have a square region which would be a unit cell of a Breville lattice and that you could repeat so obviously this would be a Breville lattice with a basis based on that whatever is inside the green region but there are many modern materials I mean many people make the mistake in assuming that all modern materials must be described by these five types it's not really it's only when you have one type of motif periodically repeated then of course there are five types but if somebody allows you two types or three then there are many more materials that can happen so for example these are called penrose sizes and you can see that the green one is a little rhombus and the blue ones are a slightly different rhombus almost like a square by putting this together you can cover the whole surface it's not periodic but essentially it's almost periodic and because now you have two types of unit cells if you allow two on the right hand side if you allow three or four then you can have many more many more types and so the point I wanted to make yes not if you see a lattice which is not a Breville lattice you shouldn't automatically think I just combine some atoms in some random way I'll always get back to Breville lattice Breville lattice is very important it just drives a huge number of materials but that's not the whole universe there are lots of other materials also that do not follow in the Breville lattice and cannot be reduced to one you see and there are important materials in fact people didn't know its existence until 1986 and then all of a sudden a research group first publishes the result of material of that type of structure nobody believes them in the beginning and then others start finding it so it's almost less than 20 or 25 years old the existence of many of this material what about 3D well life is getting more complicated 1D, 1, 2D, 5 well in 3D I have 14 so 14 let's start with the top row 14 you can understand because there are 3 lattice vectors right height, width and the length and the angle, pair of angles in between of course there will be more so you can see starting from the top the triclinic has this alpha beta and gamma these are the 3 pairs of 3 angles among a pair of lattice vectors none of them are equal to 90 degrees so that's the most general form then you can have cubic all 3 sides the same angles 90 degrees no problem tetragonal and you can see the 2 sides are the same the base is the same height isn't the height is a little longer and then then onward you have now why this 7 this 7 is associated with the rotation angle or various symmetry operation you can see you can take a cube rotate it 90 degrees it will exactly remain a cube rotate another 90 degrees so you can exactly have a cube but you can see and you can do this vertically also if you rotate on the vertically then again you have a symmetry operation but you can see in a tetragonal one yes rotating is sidewise is fine along a vertical axis is fine but if you try to rotate it on the horizontal axis around horizontal axis when you rotate it's no longer the same right and so it will look different so therefore these whole classification the 7 people spend their life on in fact of how to define these things then there are 7 but in addition if you go vertically along the line let's go on the cubic the second column you will see that there are possibilities where electrons are or the atoms are only in the corner that's a normal cubic then in the second one marked as I there's one atom in the middle of the body that's body centered cubic makes sense right one atom in the body and the third one is face centered cubic because there are 6 faces and you have one atom in all the 6 faces so that's the face centered cubic now if you think about it this is 4 in the vertical column and 7 in the horizontal column how many there should have been 28 right but you can see that there is 14 and there are also lots of blank white spaces why why did that happen why are these blank spaces so for example let's focus on this tetragonal one the third element there is no face centered cubic on the tetragonal one but there is a cubic face centered that's that element is there why not because it turns out that the second and the third element that body centered and face centered are exactly the same so you are double counting and let me explain why in the next slide but in order to avoid that double counting we don't put it there anymore do you see that this is a tetragonal and this is a face centered tetragonal because on the six faces I have the six atoms and you can see that I have marked the two atoms A and C and see what these two atoms do as I move this around so what I have done here that was one unit I just copied another one you can see just copied another one to the extended to the left and also on the top and bottom that's exactly what I've done you can see where the C is C was in the face so the face is being shared by two neighbors and so this is being common on both the neighbors now and look at the A now you see do you see so the C now you see is still it was on the face but if I just look it in the corner the same crystal instead of looking face on if I look it diagonal on then that becomes a body centered cubic and as a result then you can see that therefore this is not a new lattice because old lattice viewed in a different way so you should put it in one of the amazing things and that's where I'll end is that this counting was done before people knew about atoms simply based on the fact that all these crystals the jewelry people and others they had been studying this for a long time you want to give a king a nice diamond so that he gives you a lot of money back so people knew about crystal planes a lot and by 1840s people have already figured out all these answers 14 type of Breva lattices but actually he counted 15 the first person is not Breva the first person is frankenheim in 1842 he counted 15 just one more and few years later Breva is of course a very famous scientist he did lots of other things but and Breva pointed out the two of the lattices that frankenheim said and therefore Breva's name remains and frankenheim is forgotten you see just one mistake and 150 years later here we are talking about Breva frankenheim gives our sympathy but doesn't get any prizes okay and let me conclude with this the Wigner site cell for BCC and cubic lattices one thing I want to point out this 14 cells that you saw these were not necessarily primitive cells these were the cells that were easiest to deal with but if you wanted to do primitive cell then you should do what this how this has been done for example you have a center atom look at the right one first the cubic one in the green center atom and you can see the neighboring atoms have been connected and bisected by a plane now instead of a line it has been bisected by a plane that's the blue transparent the cube that has been generated that's the that's the Breva lattice that's the primitive cell for this structure in many cases for the cubic is exactly the same but for many other systems you will see especially for face centered cubic it's not the same for example in the face centered cubic I'll show you that Breva lattice have a volume of factor of factor of 4 compared to a primitive cell you'll do it in your homework you'll figure it out okay so let me conclude so I try to explain to you that first five weeks we'll be talking about two basic concepts number of electrons and then we'll be talking about the velocity of the electrons in order to calculate current we're talking about number of electrons and we've spent some time in talking about different types of materials available we just find you know we expect gallium arsenide to be different from silicon we understand that but we are now also talking about periodicity and structures and the periodicity and structures are very important because if there were no periodicity there's no hope no matter what computer of solving the quantum mechanical problem that will give us the number of electrons so we do it because of convenience but later on we'll see even when since I'm not periodic many of the conclusion that we reach here will remain essentially the same that's why we'll spend so much time on the periodic structure okay that's it