Ok welcome back. So, if you recall the last lecture we had ended with the introduction of mobility. What is electron hole mobility and how does it depend on effective mass?
We wrote down the expression of mobility and also we tried to discuss what physically it means it is the slope of the velocity and field characteristics. So, if you accelerate by applying more and more field an electron will move faster and faster that slope of how fast it is moving the velocity. Field is called mobility. So I told you that mobility is a very important concept because mobility dictates how much current can flow and how much current flows will also decide how the device performs and mobility depends strongly on temperature.
So if you cool down a device or if you heat up a device, mobility will change and that change will also affect the current that is flowing and that will affect the device that is working. Hence, you always have a range of temperature over which you can operate devices, okay. There are other important reasons for the temperature limitation regarding from the circuits point of view but fundamentally from a device point of view okay the mobility is very sensitive to temperature okay.
An electron or whole mobility is a very important parameter in understanding devices that will always come okay. Many things are tied together I told you you know drift the term drift and diffusion are something we are using although we have not extensively discussed that is something we will also discuss. So we will resume this lecture with where we left and that is you know the definition of mobility and How to interpret mobility in terms of temperature dependence ok.
So, we will come to the whiteboard again if you recall from the last lecture I told you that mobility depends very strongly on phonons which is the vibrating atoms. There are different kinds of vibration I told you acoustic and optical there are branches actually of phonon vibration sorry acoustic and optical ok. And acoustic and optical phonons are essentially two sort of a branches two ways in which atoms can vibrate they affect a lot how the electrons mobility the value of electrons mobility and also ionized impurity can scatter electrons and that is why they affect the mobility ok. So, these are the two things now we will see how their effect you know translates into temperature dependence sort of a thing for mobility ok.
I told you that unit of mobility we denote it with mu and the unit is centimeter square per volt per second ok. What is what are the typical values of mobility? For example, if you take an n type dope silicon a moderately n type dope silicon then your electron mobility mu n would be approximately you know around say 300 centimeter square per volt second hole will be much lower.
If you also take other semiconductors the mobility value might be much larger much slower smaller you know. For example, you have probably heard of the word graphene graphene the The exfoliation and the synthesis of graphene had gotten the Nobel Prize in Physics in 2010 to Novoselov and Gaim of England. And graphene is like a two-dimensional sheet of one sheet of carbon atoms. Electron mobility in graphene can be very high.
It can be, if I recall correctly, something like 50,000 or even like 100,000 or even more centimeters square per volt second. There are other ways you can also even go up to probably a million, you know, if you suspend the graphene. But those are other things.
So, you know depends on the material very strongly. So, in silicon typically which is what we are going to discuss this devices in this course, you have a mobility of say around 300, it depends on also temperature. So, this is a good number we should keep in mind and whole mobility will be smaller.
If you recall mobility depends on the charge times the scattering time and the effective mass. So, if your effective mass becomes smaller your mobility becomes larger. So, you want a smaller effective mass material, if you have a smaller effective mass then your mobility will become better.
That is a good indication if your effective mass is high your mobility will become low and this scattering time this is your mean scattering time or you can say the yeah the mean collision time this mean scattering time in a way in a very simplistic way the time between two adjacent collisions this mean scattering time depends as I told you on phonons and on ionized impurity. So I told you phonons are nothing but vibrating atoms we have the different acoustic and optical modes. You know actually how we define acoustic and optical is I told you we can think of atoms as like their spring essentially you know their vibrating atoms.
You can also talk about two dimensional that is another thing though but if any if you take about one dimensional at you know string of atoms they will vibrate either they can vibrate this way or one can vibrate this way one can vibrate this way. So, there are different ways in which they can vibrate right they can vibrate out of phase in phase. So, that is why these things come let us not go into the details of that but let us see how.
You know temperature affects mobility. How temperature affects mobility? So, if your temperature is you know raised, if your temperature is raised then atoms sorry then atoms will vibrate more right atoms will vibrate more.
If your temperature raises the atoms will vibrate more, if your atoms vibrate more then they will they will scatter the electrons more or electrons will collide more ok. So, the scattering of the electrons will be more and more. will be really more and so the mobility will come down so if I plot mobility purely because of phonons only because of phonon for example and with respect to temperature okay with increasing temperature your mobility will come down okay suppose this was 300 this is 200 this is 100 this may be say room temperature which is 300 kelvin this may be 400 kelvin this may be 500 kelvin and so on I am just giving a qualitative sketch.
The values need not be accurate here. What I am saying is that as you increase the temperature your mobility because of phonons will keep coming down because as you increase the temperature as you increase the temperature in this direction you have more scattering because your atoms are vibrating more they are going to collide you know the electrons are going to collide more they will scatter more. Let us keep this picture in mind.
Next ionized impurity scattering I told you ionized impurity scattering works by the fact If you have a say N D a positively charged ionized impurity an electron was moving in this direction this because of the coulombic force because of the coulombic force your ah ionized impurity will set up a potential that will scatter that will perturb the electrons path like this ok that is how it scatters and it does not physically collide per say so if the temperature is low if the temperature is low. What will happen? That means electrons have lower energy, okay.
If the temperature is low electrons have lower energy and if electrons have lower energy then they will spend more time in the vicinity of this ionized impurity. In the vicinity of this ionized impurity they will spend more time or they will take more effort to overcome this columbic force sort of. If you increase the temperature the electrons will have more energy, they will have to spend less time here, they can overcome this scattering more.
which means if I decrease or lower the temperature then electron will spend more time around it and it will scatter more which means the mobility will come down when temperature is lowered. So if I plot mobility purely limited by ionized impurity then this is temperature on the x-axis okay 300 kelvin maybe 400 kelvin this is 200 kelvin maybe. and you know keep decreasing 10 Kelvin and so on. Then with this impurity limited scattering this mobility will reduce when your temperature is reduced why because at lower temperature electrons have lower energy so they will not be able to overcome this columbic potential so well they will have to spend more time so mobility actually falls down its lowering temperature so it goes like it goes something like this okay. So you see impurity ionized impurity scattering goes like this.
and your phonon scattering goes like this. So, the total scattering will therefore go how the total mobility not scattering the same thing actually the total mobility because of both ionized scattering and phonons you know scattering with respect to temperature will therefore, have a shape like this it will go up and then come down one of them is limited by phonon. at high temperature, at low temperature it is limited by ionized impurity. This total mobility is actually the mobility that you can experimentally measure, this is the net mobility that you can experimentally measure. So, this is the total mobility of the electrons or say holes, okay.
And typically in silicon this should ideally fall off in the range of t to the power minus 3 by 2 and this should rise up in the range in the way of 3 to the power 3 by 2, there are some theory and also experimental data to sort of support this claim. It may not be exactly 3 by 2 because of many impurities and many other imperfections in the crystal and so on but loosely speaking this should be the trend okay this should be the trend and if you should remember that actually the total mobility suppose the mobility because of phonon only is say you know 400 centimeter per centimeter square per volt second sorry and mobility because of ionized impurity purely is suppose 500 centimeter square per volt second these are mobility. purely because of phonon purely because of ionized impurity.
Then the total mobility that you will actually measure in the sample, you cannot measure this directly directly you know generally so easily. What you measure actually is the total electron mobility, it depends on both this and this, it is not a summation of both of them, it is actually 1 by total mobility is equal to 1 by mobility due to phonon plus 1 by mobility due to ionized impurity. So the total mobility will be 1 by 400 in this case plus 1 by 500. and then you take the inverse.
So, the total mobility will be 400 into 500 by 400 plus 500. So, that will be your 2000 by 9 which is roughly 220 centimeter square per volt second. So, this will be 220 centimeter square per volt second will be the net mobility if you are phone on and impurity limited mobility at this at a particular temperature, okay. So, that is with respect to you know temperature.
So, for example, You know if I take doping for example, you know doping also affects mobility by the way all this while we have been talking about sort of very moderate doping, so very moderate doping. If your doping is very high, if your doping is very high then ionized impurity concentration increases, your ionized impurity concentration will increase. If your ionized impurity concentration increases with very high doping.
then your ionized impurity scattering limited mobility will reduce and this will be the dominating factor. If your doping is very high, then more than phonon ok more than phonon your ionized impurity scattering will be so high, you have so high doping. So, your electrons will scatter more rapidly and more more more repeatedly and frequently from the high density of ionized impurity.
And if that is the case then the mobility due to ionized impurity will be so low. That phonon mobility will not be an important contributor. Remember 1 by mu total is given by 1 by mu phonon plus 1 by mu ionized impurity. If this because of very high doping this ionized impurity you know limited mobility might become very small and you know that if you have 1 by X plus 1 by Y then the smaller of the two dominates in this thing right. So this will dominate then this will not dominate so much.
And if your doping is very low. If your doping is very low then you know if your doping is very low then your extremely low doping then your ionized impurity contribution will decrease and your phonon limited mobility will be the dominating mobility. So what I mean is that if I dope it very low then this quantity the ionized impurity limited mobility will be very large number because it is not limiting it so then this will dominate right so that you should keep in mind.
So for example if I do a suppose I plot the total mobility I plot the total mobility versus say doping nd for n type semiconductor plotting doping suppose this is 10 to the power 13 which is almost no doping anyway silicon I am talking about silicon 10 to the power 14 10 to the power 15 10 to the power 16 probably 10 to the power 17 10 to the power 18 10 to 10 to the power 19, 10 to the power 20 and so on, this is a high doping by the way 10 to the power 19 and doping. So, the total mobility initially the total mobility because of both phonon and impurity scattering will remain constant initially why, because at lower doping your ionized impurity concentration is not so high. So, the scattering because of ionized impurity scattering is not going to be dominant factor it is very small I mean the mobility is very large.
So, the scattering is very low and entire value here is dominated by phonon scattering and phonon scattering does not depend on doping more or less. But as your doping starts increasing your ionized impurity collision will now increase so much that it will start playing an important role. So, it will start decreasing eventually at 10 to the power 18, 10 to the power 19 the mobility will decrease very much mobility will decrease very much. So, that is what is going to happen. So, if I am giving you a plot like this and suppose this is value is 400, this is 300, this is 200, this is 500, then you know that this value of 400 corresponds to purely optical phonon limited mobility.
This is purely optical phonon limited because in this range at low doping range your ionized impurity scattering is going to be very low. is very infrequent or very the scattering is very weak they are not so much scattering this scattering becomes important only at high doping or even moderate doping it will have some effect but at very low doping it will probably not have so much of effect ok so that we should keep in mind so now we know mobility and if I mention current density this is current density. So, it is normalized to the area for example, current actually is nothing but charge that you are carrying times the flux of the carriers that is flowing.
So, charge of course, is q the charge of electron right and your flux is nothing but total number of electrons that you have times the velocity which is there moving. I told you velocity is in this case drift velocity and drift velocity this is drift velocity and drift velocity will come only when there is a field in semiconductor like an n type doped semiconductor just a uniform semiconductor this field that you need may come from external application, external bias. There may be cases like in p n junction where the field might exist internally, but here I am talking about just the uniformly doped semiconductor. So, the drift velocity to come you need a field, the field generally comes from external bias and now I can write q times n, I can write velocity as mobility times the external field. So, let me write again j is equal to q n mu field, this is field, this quantity is actually called sigma which is called conductivity.
Conductivity of the sample, so J is conductivity times field, you probably know this, this comes also from Ohm's law, field is actually your voltage by the length of the sample in a way, so you actually have a direct relation between the current and the voltage that you are applying, it is a linear relation, comes from Ohm's law, it also testifies Ohm's law, this is your current density that is proportional to field, proportionality factor is called conductivity, conductivity is actually q mu n. Ok depends on mobility, depends on the charge density if you remember and this is purely a drift current because it depends on the field right, this is the field, it depends on field and it is a purely drift current and drift current depends on mobility and charge as well ok. It only exists when there is a field, the field may be externally applied or it might also internally exist in the semiconductor ok. So now mobility defines your field.
Before I go to the concept of diffusion, we will wrap up one last thing and that is I told you the velocity if I apply with plot with respect to field it increases linearly so the slope is called mobility I told you right. So, it cannot increase indefinitely after sometime it will saturate, it will saturate which means with increasing field the velocity is no longer increasing, but it has saturated at a value and that value is called V that is called V sat or it is called saturation velocity and saturation velocity is actually a very important parameter of a material saturation velocity is you know it is a very important parameter for a material silicon for example I think has a saturation velocity of close to 2 into 10 to the power 7 centimeter per second close to that may be slightly lower than that ok gallium arsenide might actually have more than this. some other materials might have lower than this so depends actually ok.
So, this saturation velocity is very important and this will set a limit to how fast electrons will move cannot move faster than this ok. This I told you this region where the velocity and field are having a linear relation is called a low field regime and the moment the velocity has saturated right the velocity has saturated this region is called high field regime. Now, what is the physical region for the velocity saturation?
The velocity has saturated here, it cannot increase more than that. The physical reason is that as you are increasing the field more and more, you know this is the increasing field. As you are increasing field more and more, the electrons or holes for example are getting more and more accelerated, they are gaining more and more energy. But eventually, you know because they are gaining more and more energy, they move faster and faster and they collide more also.
As if you are getting accelerated more, if you are getting accelerated more. By getting more energy from the field, you also get collided more. You can imagine thinking of this like a vehicle, like a car moving in traffic.
If the car moves much faster, it will also collide and have accidents more and more with other vehicles on the road, right. So if you accelerate a particle more and more, it will move faster, it will also collide more. Eventually what will happen is that the rate at which the electron is gaining energy from the field will be basically dissipated away to phonons.
to the vibrating atoms. The electrons that are gaining energy from the field will be dissipating away the energy to phonons particularly optical phonon you know optical phonon is a type of phonon vibration okay so electrons that are gaining energy and getting accelerated from the field will now basically start dissipating a large amount of the energy to the phonons and so the phonon scattering. Or you can say you know the fact that electrons are now giving away energy to the phonons will be limiting the velocity at which you can the maximum velocity you can achieve, okay.
So in the velocity saturation regime electrons are colliding so much by getting so much energy that it has put a limit, okay it has put a limit to the velocity it can go because it is starting to dissipate away that gained energy to the crystal, to the crystal that is. to the vibrating atoms of the crystal lattice is going to dissipate out that energy, so it cannot accelerate more that is why there is a limit of velocity saturation here, okay. There is a limit to the velocity saturation here, semiconductors do display velocity saturation, metals do not display this velocity saturation because the energy you need to achieve this velocity saturation in metals is so high that it will actually melt the metal, it will be so high temperature eventually, okay because it will heat up with so much of current. That is secondary.
So semiconductors do have a saturation velocity, different materials have different velocity saturation and when the velocity is saturated we call it a high field transport because the field is very high and the energy that the electrons are gaining, remember the electrons that are gaining energy from the field are now losing the energy to phonon, it is basically dissipating the energy away to phonon, we call it phonon emission actually if you want to know the exact term, it is called phonon emission which means electrons are giving away the energy to phonons. They are giving energy to phonon, they are no longer accelerating, okay. So you should remember this term and it is velocity, you know high field transport comes in velocity sort of a saturation regime, okay. So now we know drift transport that comes in when there is existence of a field and we will now come next come to a fact called diffusion current. Okay so once we have studied diffusion so we have already started drift and now we will study diffusion okay.
So drift and diffusion will basically complete our picture of electron transport and current flow. So drift comes from field I told you already drift comes from field and diffusion comes from concentration gradient. It comes from Concentration gradient.
So, if there is a gradient of electron concentration then there will be diffusion. What do you mean by gradient? There is a slope. That means there is a higher concentration, there is a lower concentration.
For example, if I spray a deodorant in one corner of the room, after few minutes the whole room will smell the deodorant because the the deos the spray molecules of the deodorant will now diffuse to areas where there are lower concentration of the deos you know molecules. So, if you have too many electrons here, too many electrons are here. very few electrons here for example then under you know this normal conditions the electrons will try to diffuse from a region of higher concentration to a region of lower concentration. This is called diffusion and it is a natural tendency of even gas molecules and particles right so electrons will diffuse.
So concentration whenever there is a difference in concentration whenever there is a concentration gradient there is going to be diffusion current and whenever there is a field. will be drift current. So in ideal situation drift and diffusion might balance each other for example like in a PN junction. So in equilibrium you will not get any current because although there is a field in some sometimes you know the diffusion might basically exactly balance the drift component and so on. So that is one thing that will come when we discuss PN junction but we should introduce the concept of diffusion now it is high time and you will find out that diffusion current actually depends on the derivative of the concentration gradient.
For example, I take electron concentration n, this is n, okay, this is the total electron concentration I am plotting as a function of x, x is distance, okay. So electron concentration, suppose electron concentration is constant like this, then there will be no diffusion because electron concentration is constant, there will not be any flow of electron because of concentration gradient, there is no concentration gradient. Now suppose I have a concentration gradient like this, which means electron concentration is high here, so this is maybe high electron concentration.
And this is electron concentration is low here. So electrons will try to move from this region to this region by diffusion, okay. Electrons will try to move from the higher concentration region to a lower concentration region and we have to capture that mathematically.
So how do you do that, okay? How do you capture the mathematical picture of diffusion, you know? For example, I take n and I have x.
I take and this is the electron concentration for example, I take any point say x naught and you know electrons collide and I am talking about a one dimensional picture here. So, it will only have it can electrons can either go in plus x or it can go in minus x there is no y and z direction. So, electrons collide of course, they collide in all kinds of you know when they move they will collide in any direction there is no field here and we will say that the mean collision time is tau. This is different from the tau we defined in mobility, this is you know there is no field here per say, so electrons will collide and mean collision time is a tau and the distance that electrons move between two successive collision is L, sort of the mean free path okay like you know the between two collisions electron will at least move L okay, it is the mean free path you can say it is the mean path they will go and say if L is say 10 nanometer it means electrons will collide on an average every 10 nanometer.
So I will take a line here. And I will draw a box very infinity similarly thin box here you know which is at X naught plus L, one mean free part okay and on the right side I will take a box slightly larger than the other side in height because of course you know on the left side your concentration is high, on the right side your concentration is low, so it is gradually increasing, so this is X naught minus L the distance okay. I am taking two boxes. Why am I taking two boxes?
I am taking two very thin boxes of thickness only L, the mean free part and I am going to take the electron concentration here. So this is an electron concentration is varying, right, electron concentration is varying there. So suppose this value, this value is, this is on the right of X, X naught, so I will say nR, this is the electron concentration.
So suppose this is 10 to the power 18, this is 10 to the power 17 and so on and so on, right. So this is nR, some value here, okay, this is on the right side of X naught. On the left side of X naught this is suppose nL, left L. Of course nL is greater than nR because it is decreasing function here, right.
Now at any given point in time the electrons that are there in this box half of them will move this side, half of them will move that side. It is random. Similarly electrons that are there in this box half of them will move this side, half of them will move this side because it is random.
We are interested in considering or finding the flux of electrons that are actually crossing the plane of X naught. This plane of X naught here, what is the fraction, what is the flux of carriers crossing X naught from left to right. So I told you half of N L is crossing from left to right, you agree and half of, so N by 2 and half of N R is going from right to left, okay. The net that is actually flowing, the net that is flowing to the right, what is the net concentration or the net you know the flux of electrons that are flowing from left to the right, from left to the right is nL by 2 minus nR by 2 because half of this block are going to right, half of this block are going to left, so the difference of that, okay.
So the net Net flux of carriers moving across the plane at x naught will be given by the charge here, I mean the total number of carriers here times the velocity. So the flux of carriers that are crossing the plane x naught is given by the total carrier which is nL minus nR by 2 into the velocity and velocity is nothing but distance by time. The mean distance from centroid to centroid is actually L.
So it is L. I told you mean free path by the time it taking which is tau tau is the means free time you know the mean collision time so in time tau is going that way right so now I can say the flux phi is equal to n l minus n r half into l by tau this is your flux now if you recall your high school physics you know if you recall your high school physics I have a function f of x and this is suppose x This is function x you know if I take one point here x2 and another takes a point x1 very small very close to each other they are infinitesimally close almost then x then at this point you have f of x2 at this point you have f of x1 so let me write it down in a different slide probably it is a high school physics high school math actually this is f of x this is x this is say x2 this is say very close to it x1. and this is going like that.
So, x 2 is this, x 1 is this, this is f of x 2. This is f of x 1. You notice the similarity between this figure and the figure before. If you notice the similarity between that figure and the figure before, you will see that this is actually n L which is like f of x 2 and this is like n R the right box which is like f of x 1. So, remember this function n L minus n R by 2 this is like excuse me this is like f of x 1 minus f of x 2 by 2. What it means is that if I take this then f of x2 minus f of x1 can be written as minus df x by dx into this is delta x x1 minus x2 is delta x delta x can you write that the slope times this we can write that right. So, similarly in the previous function nl If you look into this, if you look into the previous slide here, nL minus nR that can be written as how you know that can be written nL minus nR can be written as minus d sorry it can be written as minus dn because n is a function of x by x into L.
because l was the mean the delta you know the difference. So, let us me write down the flux again the flux the flux of carriers moving across the plane if you remember the flux of carriers moving across the plane here was given by this quantity right this quantity. So, I will write that quantity as minus now I will use this d n by d x into l okay what was that by 2 into l by tau by 2 into l by tau so this is minus l square by 2 tau d n by d x flux is given by minus l square by 2 tau d n by d x i can call this quantity as d n which is diffusion coefficient it is a coefficient it is a constant depends on temperature depends on temperature depends on also doping to some extent ok.
But more or less I can define it to be d n and so I can write the flux as minus d n d n by d x down. So, this is the derivative actually this the derivative of the concentration gradient. So, if your n the electron concentration gradient is varying like this, there is a definite concentration gradient so there is a diffusion. If on the other hand if you are there is no gradient like constant then this quantity will become 0 and so there will be no diffusion current. So diffusion current diffusion current is given by there is a charge of electron is negative right so minus Q times the flux which is minus dn dn by dx that is equal to Q dn This is a diffusion current okay of course this is current density so if you multiply by area it will become amp otherwise it will become amp per centimeter square.
Similarly for holes you can define a diffusion current and because the charge of hole is positive right the charge of hole is positive your hole diffusion current I will call it Jp hole diffusion current is actually minus Qdp. hole concentration by dx the slope will decide of course slope will decide and your electron concentration of diffusion current will be actually Q dn dn by dx. So the total diffusion current if both electrons and holes are present if both electrons and holes are present and both of them are diffusing then the total diffusion current okay will be given by Q dn. by dou x minus Q dp dou n by the dou p by dou x this is the total diffusion current if both electrons and holes are going to diffuse.
Remember this sign and this sign is important because you have to also consider the slope this slope of this and this for example if you have n going like this then electrons you know certainly will diffuse from here to here. direction and so the current will flow in this direction because electron direction is opposite to current direction right and that you can capture here you see. Your this slope, dou n by dou x will be negative because it is coming like this. So, negative slope if you put here, it is negative, this quantity will become negative and that is indicated by this, the current, the total current. On the other hand, if you have holes here p and the holes will then diffuse also from here to here because higher concentration of hole to lower concentration of hole, but hole direction is the same as the electricity direction, so current will flow in this direction.
So, in this case, the hole slope is also negative here. If you put a negative term here, then this negative that negative will cancel out, so it will become positive which means this is the positive x direction in which the current will flow. So we shall wind up the lecture here today, okay.
We have introduced diffusion current, diffusion of electrons and diffusion of holes and we derived the expression for the diffusion current in terms of the derivative of the concentration gradient, introduced both the hole and electron concept and also before that we have also discussed on the velocity saturation. temperature dependence of electron mobility, whole mobility is similar the trend right. So now we know drift and diffusion, so now we can actually go ahead and discuss about current flow. These are actually current equations by the way, now we will discuss certain relation between drift diffusion and we are ready to now understand devices, but before that the slight few more things we will have to discuss okay.
So we will take up many of these things in the next class including the relation between drift and diffusion for example and also how A high drift current means a high diffusion current also without those things there is something called Einstein relation, we will come to that Einstein relation and from there we will take on a few other things before we can touch on the devices, okay. So we will end up the class here, thank you for your time and we will again meet on the next class.