 So, now let us begin first of all with semiconductor device. So, the obvious first question is what are, so what are semiconductor? In fact, for a long time semiconductor materials were considered totally useless. They did not, they did not conduct as well as the metals, therefore could not be used for wires and they were not, they were quite leaky, they did conduct a little bit. So, they were no good as insulators. So, you could not use them as wires nor could you use them as the sheath around the wire. So, what good are these materials? Well, these materials lie somewhere in between and once methods of controlling their conductivity both chemically and through electrical means became known. Then their full potential was released and these are the semi conductor materials and devices. That means they do not conduct quite as well like metals do nor are they blockers of current like insulators. They are somewhere in between and their conductance, their conducting properties can be controlled through chemical means through doping and so on, but also through electrical means by applying voltages or biases and so on. So, much of semiconductor materials and devices is the science of understanding how that happens. So, let us first discuss conduction. Now, in order to have conduction you need to have charges which can move obviously. So, therefore you need to have mobile charges and let us understand where these mobile charges come from. Now, in a semiconductor if you have essentially considered an absolutely pure semiconductor, semiconductors have an average valency of 4 generally. Elemental semiconductors are like silicon, germanium and indeed now even silicon carbide and diamond are being used as high temperature semiconductor. But also you could have compound semiconductors like gallium arsenide like gallium aluminum arsenide or gallium phosphide, indium phosphide these are called 3, 5 semiconductors where you have complexes of elements from the third group and elements from the fifth group so that the average valency is still 4. Similarly, you have 2, 6 semiconductors like zinc oxide and cadmium telluride and cadium selenide and so on. These are 2, 6 semiconductor materials and their average valency is also 4. In this course at least in this set of lectures we will mostly be concentrating on elemental semiconductors and in particular whenever I take examples I will be talking of silicon. So consider absolutely pure silicon, silicon has a valency of 4 each atom has 4 electrons and it has 4 neighbors. It shares an electron with each one of its neighbors and therefore that forms essentially a covalent bond and a lot of energy is required to break this bond. So, this bond makes this element stable and a lot of energy would be required to break this bond. However, if this bond could never be broken then silicon would be an insulator because all the electrons are tied and cannot move about. Obviously, that is not the case silicon as we know is a semiconductor and therefore the amount of energy required to break these bonds is indeed available at least at room temperatures and thereabouts. So now we have essentially an equilibrium between energy the electrons which have become free because a few bonds have been broken and electrons which are bound because if bonds are broken then free electrons can then be captured back by these positions to form a bond again. So it is like a chemical process you have some free electrons they carry a negative charge you also have those places from where electrons have gone away leaving a net positive charge at that place. Now remember we must make a distinction between an iron which is the nuclear material which is absolutely fixed in this lattice and the absence of this electron which is a hole. So, suppose an electron has gone away leaving this area positively charged this atom can pull an electron from its neighbor and now the neighbor is deficient and this guy is neutral. In short if an electron has moved from my left to me then I am no more positive and my neighbor is now positive. So, while the electron has moved from my neighbor to me the charge appears to have moved from me to my neighbor earlier I was positively charged now my neighbor is positively charged. This kind of movement of charge is called hole conduction these are called holes and if you apply a field because it makes it possible for me to steal electrons from my neighbors by the way these neighbors need not be the nearest neighbor. So, because it allows me to steal electrons from my neighbors this kind of movement is called hole conduction and holes are positively charged. So, in a material I have electrons these are free electrons because the bond got broken I have holes which are positively charged and these are essentially locations from where electrons have been removed and this location can move about because the electrons can be captured from nearby neighbors. It is almost like you know there are places where you have to take your chappals off and it is considered in semiconductor that if somebody has walked off with your chappals you steal somebody else. So, this is the movement of chappals which happens in these atoms and the chapel deficient person moves from one to the other and that is how the conduction takes place. So, you have these conducting materials electrons and holes the presence of this these electrons and holes is in fact caused by the presence of energy. If there was no energy around then the bonds would have never been broken and we would not have these electrons and holes. So, therefore, the third element is energy. Now, the way an equilibrium is maintained between them is that energy causes the generation of an electron and a hole pair the bond breaks the electron walks off leaving an electron deficient place. So, now in this material you have one electron and one hole. So, one unit of energy which should be sufficient in value to break a bond results in the generation of an electron and a hole. Now, suppose there is an electron deficient phase and a free electron. Now, I am not talking of stealing an electron from a neighbor. A free electron happens to wander by then energetically it is quite attractive for this electron deficient guy to catch this this person. As a result now this electron has vanished it is not free anymore it is now bonded and that hole has vanished because this place is no more electron deficient. So, you have the electron and the hole has vanished this system has become more stable and therefore, it emits energy essentially it is this process in reverse. So, in short there is a net equilibrium of electrons and holes electrons and holes recombine to give off energy this energy shows up as vibration or what have you and then this energy in turn at some other place perhaps cause the generation of electrons and holes. It is this equilibrium which determine how many electrons and how many holes would be there at a given time. This is almost like a chemical reaction and therefore, the law of mass action applies. Let us say that the concentration of electrons in a material is represented by n this is the first order reaction and let us say that the concentration of holes is represented by p because on the right hand side there are no physical reactants. Therefore, if we write down the chemistry chemical equilibrium for this then we get that the product of electrons and holes should be constant. This implies that the temperature is constant because the amount of energy is constant therefore, the product of n and p should be constant. This implies that as electrons increase holes should decrease and the other way round this makes sense. Suppose, there were double the number of electrons then the probability of recombination would increase and the number of holes will therefore, decrease. So, therefore, this reaction essentially maintains a an equilibrium between the number of electrons the number of holes and the energy in the system and at a given temperature the product of n and p will not change in equilibrium provided equilibrium can be maintained. This now let us apply to an absolutely pure semiconductor. What happens therefore, is that as we had seen in this reaction whenever an electron is generated a hole is also generated and therefore, it stands to reason that the concentration of electrons would be the same as the concentration of holes because they are simultaneously any reaction which generates an electron also generates a hole any reaction which destroys an electron also destroys a hole. So, as a result at any given time the concentration of electrons and holes should be equal. Let us represent that equal concentration by n i and this tells us that this constant that we are talking of that is equal to n i square. n i is constant at a given temperature it will it is a function of temperature because the amount of energy available is a function of temperature. So, now we have found out this constant and therefore, we can write down the equilibrium relation which governs statistics of carriers in a semiconductor saying the product of n and p will be equal to n i square. However, a pure semiconductor is not that interesting if it was only a pure semiconductor then we would have that old picture that nothing much can be done with this material. However, we can control the concentration of electrons and holes by doping this semiconductor by adding impurity. If we add for example, phosphorus to silicon in very very small quantity these quantities are very small part per million or so. In that case the overall material properties do not change. However, every phosphorus atom has 4 neighbors, but 5 electrons. So, one of the electrons cannot be stabilized by sharing with the neighbors this electron is very easy to plug. Now, you do not need the energy to break a bond there was no bond for this fifth electron. So, therefore, it is with very little energy this electron can be detached and at room temperature practically all phosphorus atoms have lost this electron. These phosphorus atoms now are perennially positively charged because 4 electrons are shared with neighbors and the one electron which was not shared has walked off has detached from phosphorus and gone away leaving this phosphorus positively charged. Such impurities are called donors and because it has lost an electron it becomes positively charged and contributes an electron. Notice that now we have a process in which you do not have simultaneous generation of an electron and a hole. When this electron has gone away it has left an atom all 4 bonds are satisfied it has no motivation to pull an electron from its neighbor. As a result we do not have the generation of a hole simultaneously and therefore, in this material there will be many more electrons than there are holes. Similarly, if you had boron boron has only 3 it is trivalent it has only 3 electrons, but it has 4 neighbors it can share an electron with each of its neighbors if it had 4. Therefore, it is always interested in capturing an electron from somewhere so that it can form a pseudo bond with its neighbor to stabilize the lattice to stabilize its energy. Therefore, it is like a hole this hole has been created without the simultaneous creation of an electron this hole is just there because boron is there. So, now adding this by the way is called an acceptor because it has taken an electron and the electronic charge is more than its nuclear charge it is negatively charged. So, acceptors are fixed iron and which are negatively charged and phosphorus are fixed iron which phosphorus is positively charged boron is negatively charged phosphorus has donated an electron it is a donor boron has accepted an electron to become negative and therefore, it is an acceptor. So, this is roughly the kind of charged materials that you find inside the semiconductor. Now, there are donors there are acceptors there are electrons and there are holes. All of these are charged of these donors and acceptors are charged, but they are fixed in space electrons and holes are also charged, but these can move around in the semiconductor and can contribute to current. Therefore, whenever we discuss electrostatics then we will be talking of all four. However, whenever we talk of currents we will talk only of electrons and holes. So, this now tells us of this equilibrium, but also notice that when you add let us say I have got silicon and I add phosphorus to that. Now, I have many more electrons that is that makes sense. However, because of that chemical equilibrium if you have added this impurity not only would you have more electrons, but you will have much fewer holes in this material. So, why should holes be affected if you add phosphorus to a material? Well the answer is simple the probability of a holes existing is now lower, because there are so many more electrons available. So, these holes will often capture these electrons the electrons are available much more widely now. Therefore, the probability of an electron being available for being captured is much higher and therefore, the equilibrium number of holes is much smaller. So, we are still governed by that same chemical reaction n times p is the same constant n i squared the same reaction applies, but n and p are no more equal normally we add only donors or only acceptors to a semiconductor. If we add donors like phosphorus then it becomes n type, because there are electrons in it if we add only acceptors like boron then it becomes p type, because it has only acceptors and not donors. The kind of carriers which are predominantly in heavy numbers in this semiconductor those are called majority carriers and the other kind of carrier is called minority carrier. Let us take an example by the way we are not discussing only content we are also talking of teaching students. So, be warned there is a source of confusion it just so happens that phosphorus has the chemical symbol p, but it makes the semiconductor n type. So, when we talk of a p type semiconductor we are not talking of a semiconductor which is phosphorus in fact, we are talking of a semiconductor which might have boron in it. So, this confusion may be may sound trivial, but often can bother students who are just starting out make sure that you get rid of this confusion right in the beginning of the class as I have done now. So, if you put in phosphorus which is a donor then it makes the semiconductor n type and it has many more electrons then there are holes. If you add for example boron by the way phosphorus is not the only available donor there is also arsenic another material. If you add boron that is acceptor and it makes the semiconductor p type then in this material if you have added boron then this would be a p type and holes will be the majority carriers. If you have added phosphorus which is a donor then the electrons will be predominant and then that will be called n type and electrons will be called the major majority carriers. I have a bit of a bit of an objection to this terminology unfortunately it is standard, but to call these majority carriers and minority carriers you know somehow in our mind in this democracy of ours majority and minority are still quite comparable to each other. What I would like to establish now are orders of magnitudes of these quantities. Now the typical value for n i at room temperature in silicon if is of the order of 1.5 into 10 to the power 10 per centimeter. By the way in semiconductor devices it is still quite common to use the c g s system that is to say centimeter and we will do it do so here. Let us be lazy let us drop that 1.5 for orders of magnitude calculations and say that n i is of the order of 10 to the power 10. Therefore n i squared which is actually 2.25 into 10 to the power 20 and that will be per centimeter 6 that determines the product of electrons and holes. Now the number of atoms per centimeter cube is of the order of 10 to the power 20 to 23 they are about in a solid and if you dope 1 part per million then we will get a doping 10 to the power 20 to 1 part per million therefore we will get a doping which is of the order of 10 to the power 16. So let us consider this material which is p type. Remember as I had said p type does not mean phosphorus it means boron like material. It has a doping of boron and the concentration of this dopant this impurity is of the order of 10 to the power 16 per centimeter cube. This is about part per million every millionth atom is not silicon but boron. We now find out how many holes are there in this material. Remember p type predominantly boron. So there would have been of the order of 10 to the power 10 electrons to begin with but now we have added 10 to the power 16 holes to this material and therefore this number is so much larger than 10 to the power 10 that we can approximate p to the same number. So the number of majority carriers is then 10 to the power 16 or so. It will be 10 to the power 16 plus 10 which will be 1.000001. So what is 1 millionth between friends? We will agree that it is just of the order of 10 to the power 16. How many electrons are there in this material in that case? So we have p times n equals 2.25 times into 10 to the power 20 and we know that p is of the order of 10 to the power 16 and that gives us that the number of electrons in this is of the order of 2.25 into 10 to the power 4 per centimeter. There are only about 22000 electrons in this material. There are 10 to the power 16 holes and only about 22000 electrons. We have about 1300 people participating in this exercise and that means 20 times that number just that is the concentration of electrons. That is nothing. You could probably give an individual name to every electron and get to know them personally. This is a very small number of electrons. Indeed what is the ratio of holes to electrons? The ratio of holes to electrons is 10 to the power 16 divided by let us say 2.25 not tropics into 10 to the power 4 which is roughly 4 into 10 to the power 11. This ratio is of course dimensionless. This is you know these majority carriers are majority like nobody is business. There are 4 into 10 to the power 11 holes for every electron in this material. So this number you know 4 into 10 to the power 11 we get somewhat blasé about these numbers. What does it mean? So just imagine what is the population of the world? Well our population is a few 10 to the power 9 and we know that every 6 or 7th person in the world is an Indian. So the population of the world is about 10 to the power 10. 1 in 4 into 10 to the power 11 means that if I was to go around the world meeting every man woman and child and if I did it 40 times then I will meet one person of the other kind. This is the ratio between electrons and holes and this is not a particularly heavily doped material. Had I taken even higher doping this number would be even more scintillating. Therefore what I would like to say is that the majority carrier even in ordinary cases over help the major minority carriers and it is quite a good approximation to consider only the charge of the majority carriers because the minority carriers are really really in very small numbers. Therefore if you are talking of conduction in a semiconductor very often unless you have broken the equilibrium by introducing a larger number of minority carriers than usual. The number of minority carriers can be considered absolutely negligible. It is not like majority and minority in a parliament. It is like 40 times the population of the world to one. This is this is a huge differential. So now what does our semiconductor contain? It contains a number of donors. These are impurities which had an electron to spare. This electron has been detached from them and has walked off and therefore the atom itself has now become an ion which has a positive charge. You have I use capital letters here to show that these are fixed in space. You also could have acceptors which accept an electron and therefore become negatively charged. These are also static. They cannot move about in this semiconductor. You have holes which are positively charged and these can move when you apply a field. You have electrons which are negatively charged and these will also move when you apply a field. Therefore for conduction these two will contribute whereas for electrostatics the charge of all is important. We also know that the number of electrons and holes is not independent. They are constrained by this relation. The concentration of electrons is represented by n. So that is constrained by this relation that n times p is n i square and at a particular temperature this n i square is indeed constant. Now this is this is a this is a bit you know multiplication is tough. We do not like multiplication. Whenever there are multiplication what do we do? We took up take log because that reduces multiplication to addition. So essentially if I was to take log of these quantities then what it says is that log of n plus log of p that is constant. So now I have a logarithmic representation for n. I am representing electrons and holes on a log scale log scale of distance and what it says is that the distance representing electrons and the distance representing holes in the same material the sum of these two distances these distances are proportional to the logarithmic of electron logarithm of electrons and holes and what we are saying is that the sum of these two distances is always constant. Now if I change them the number of electrons and holes what it says is that I could have much smaller number of electrons maybe I could have it here a very small number of electrons but then the number of holes would be correspondingly much larger and as a result this distance will always remain the same at a given temperature. Therefore we can represent both electrons and holes by just one free variable. Once we have decided the number of electrons the number of holes can be calculated as we did in the in that case that when we calculated that n to p ratio was 4 into 10 to the power 11. So it is possible to calculate one from the other therefore there is only one independent variable. So we can call the position of this divider as that independent variable. Now if we map these two fixed ends so these two fixed ends if we map this to the band gap of this material this is the solid it has a band gap if we map this to band gap then this divider is called the Fermi level. So now you have the conduction band the balance band electrons which are bound to their atoms have energies lower they are more stable they are bound electrons which become free require a certain amount of energy to go from the balance band to the conduction band and free electrons will have energies close to the edge of this conduction band. And now we have introduced an artificial imagined level in between which simultaneously determines the concentration of holes and concentration of electrons. In this particular case this represents this represents this represents the number of holes and this represents the number of electrons and in short if holes are more then the Fermi level is much closer to the balance band because it is far away from the conduction band. Now the presence of this Fermi level alone is enough to simultaneously evaluate the number of electrons and number of holes in this material. So now we have understood the statistics of the availability of carriers just for simplicity let us say that only one kind of mobile carriers is available let us take electrons as the current carrying elements that means we have an n type semiconductor. And the concentration of the majority carrier is easy to determine because it is given essentially by the concentration of the dopants that you put inside. Now suppose I apply an electric field as the result these mobile carriers will start moving with a certain velocity. So if I have a semiconductor which is n type and which has a concentration n of electrons and let us say I apply a field in this direction making this end negative and the hidden end positive then I have applied a field here. The electrons will be attracted towards the positive side and repelled from the negative side. As a result electrons will start moving in that direction. Now if I put a unit area across and want to find out how many electrons will cross that plane so let us put a unit area somewhere in between. We put a unit area somewhere in between and we want to find out how much charge crosses from right to the left in this material in a given amount of time per unit time. That after all is the current charge per unit time is the current. So if I allowed them to move for one second then the amount of charge will be the charge which was contained in this block whose size is equal to the velocity right. Because velocity into whatever is the time that I have taken that is the distance travelled by an electron. So every electron which is in this block which was originally in this block of silicon would have crossed over this plane in time t. Just at time t this farthest electrons at this particular point would just drift over and just cross this. All the other electrons will cross this earlier therefore the total amount of charge which would cross over from this plane from the right to the left for me that is equal to the total charge contained in this volume. Now because this area is unit and the time is also unit therefore the total charge is q where charge is the carried by one single electron times n. Now n is per unit volume. What is the total volume of this? The total volume of this is area times v t and if area is unit and time is unit that means how much charge crosses over per unit area per unit time which is the current density. Therefore what we get is that j is n times q times v velocity. Now if the velocity and this will perhaps take much longer to establish we will have a brief discussion about it. But if the velocity is dependent on to the field it is dependent on the field which is causing these electrons to move and if it is linearly dependent on that field then the velocity is proportional to and therefore there is a proportionality constant here mu times the electric field. Notice that this curly E is the field and not energy. Indeed in case of negative charge the velocity will be in the opposite direction that determines the direction of the current. So if the electrons go to the left the currents go to the right which is the same direction as the field. Therefore finally what we have is that j is n times q times the mobility times the field. In other words the conductivity of this material is given by this fact. So now we have had a brief look at the electrostatics and the current conduction in this. If you have both kinds of carriers then the total current will in fact be n times q times the mobility of the electrons plus p times q times the mobility of the holes times the field and this will then be the conductance. This total thing would be the conductance of this. But as we have seen under ordinary circumstances either n is much larger than p or p is much larger than n and therefore often we are able to simplify this equation to contain only one term and use only the majority carrier term. However, notice that all of this applies only in equilibrium and you can disturb this equilibrium by putting in extra charges extra electrons and holes over and above the equilibrium condition and that is called injection. You can inject electrons into a semiconductor introduce electrons into a semiconductor and that will disturb this imbalance. This concentration will now be changed and now we have. So now we have a non-equilibrium concentration which is not given by n times p equal to n i square. This applies in equilibrium only. We will take a break at this point. We will continue this discussion. We have studied only the semiconductor materials here but now we have a gut feel of what happens in this material and how conduction takes place. We will see how we make use of this in order to make actual devices.