isella5.txt

 In the past lectures, we actually experimentally determined the band alignment between two different semiconductor. The idea is that essentially we can couple the problem in two parts. One is what happens far away from the interface and this is the static effect that is treated later in a similar way as you do the junction. taking into account bend-off. So the other thing is what happens really a few atomic planes on the two sides of the of the ateron junction and we have seen that this can be for example I mean the way of determining this bend alignment is using photoelectron spectroscopy so you typically use XPS to address the core level state and ultraviolet spectroscopy to determine the position of the valence band of the last state. So let's make an exercise in this to to cap and this is one example of the many exercises you will find in the past example so in this case we have an indium gallium arsenide indium phosphide interface and these are the data which are needed for drawing the band alignment the Venn diagram in this case. So as we discussed last week, we have essentially three sets of measurements. Okay? So the first set is related to the bulk in the Indium Gallium Arsenide. And here we have some core level and balance band position. The second set is related to the indium phosphide sample, and again we have some core levels and the panel span, and then we have a last sample which is actually our heterojunction. So here we have something like a indiophosphate substrate with a thin helium-dallium-arsenide epilayer and some thickness is typically a few nanometers. measurement i want to extract electron from the indium phosphide so i will have this some core level in this case is for the cadolinium is essentially the ingus sample the rate of state and then I will add the P state of the side layer. With this number I can essentially draw my band alignment. So some of the relevant numbers are already shown here. So here we have the position of the binding energy for the balance band in the two layers, I had the difference between the core level, here it's a bit difficult to read and also to get an accuracy high enough to obtain the correct result, so in the text it was essentially written here, which are the energy differences between this peak and the corresponding balance band. So this is around 120 something, so it's actually 12860 ee. The difference between this peak, 2 peak, 3 and a half, and the balance band. So we can maybe add it here. So this is 128.6 electron volt, while the other value here between the t state and the Farad's value, if you can imagine, is 17.02 electron volt. So how do we draw this diagram? First we divide our paste in one part which will be the indium phosphide substrate level, the other one will be this galium arsenide with this alloy composition then essentially we start from core levels so here we see that the core level of indium plus 5 is way below the one of indium galium arsenide because this is 128 db below the band gap or better, we get this information from the server. So you see that here we have essentially negative value, so this number of ROV auto 120 something and this is for the offset. So essentially what this plot is telling us is that if this is the p3.5 level of the indium of the indium phosphide, the indium dilumar senide will be somewhere here, and this will be the cal. 3T, 5.5, and this energy separation, 110537 eV. Now, on top of this, I need to have the energy separation between the bulk state and the balance band. So I need to position essentially here the balance band of Gallimards into Gallimards and I and of B. To prove this, I essentially need to take the difference or let's call it the the Indian-Italian Arsenal and this will essentially be the difference between 17.02 and 1.11. and, if I'm not mistaken, we get 8.91 eV. While the position of the value span of each value, as compared to the corresponding core level, is 128.6 minus 1.59 So 1.9 and this should be 127.01. So now we add these two values here and we get something like this. This will be our endpoint 91 and on the other side I get something like this. Our VZ is 127.1. So our conduction, our balance band offset will be this value here. So, in this drawing we get that this must be 127.1 minus 11573 plus 15.5. So, in this way we get 0.44 electron volts. This will be our balance membrane. Now, this is wrong, I think it's wrong, very wrong. I was correct the first time or the second time? This one is correct. So here, essentially, we need to block the sign. Now, if we want to calculate the conduction band offset, we just need to add a band gap. one is Van Gogh's data which we are not asking the particular example but for his composition of indium gallium arsenide the energy gap is 0.7 for electron volt and for 2.5 for we have 1.34 electron volt. Okay? So now add this to value. Yes. Your conduction bed oxide should be approximately 4.6 mm. So you will find, if you browse through the different stamps, you will find several different examples and typically the question is even the XPS data drop depending on the alignment. so this is how we can proceed experimentally what happens from the theoretical point of view so how does its calculation can be performed so we say that actually one of the attempts to get an easy answer to this question was this electron affinity group called after Anderson and we see that this is actually just more or less okay for the gallium arsenide, aluminum gallium arsenide class of material, everything else is outside this line. And the reason is that essentially this model doesn't take into account that in fact I will have some charge distribution here in this interface because the atomic potential is changing so the vacuum level will not be the same for all materials. Just like in a p-n junction I can have something like this where the vacuum level is bent by the built-in potential. Here it's the potential that plays, the new atomic plays. So actually we need a more refined calculation to do this. And one problem here is that, for example, even if you do more complicated band structure calculation, like a binding one, in this calculation you never get any level from the outside world. I mean, if you start from the idea that you have an infinite crystal, there is no reference level outside your crystal. but you need to put in your model some kind of interface itself, otherwise you will never get a meaningful result, because we can scale, we can take the band structure of silicon and set the zero at the valence band, at the bottom of the conduction band, wherever we want, and we will get the same measurements from any absorption or transfer measurement, because we basically are sensitive to the difference between the levels. So I need to put my interface inside the model. So in this case, what happened recently is that, say, theory took inspiration from experiment. So this is a schematic drawing of, say, how we can proceed with the calculation of the band alignment using ab initio calculation. So density functional theory. So let's start from this central block and then we will see what happens, why we have these other blocks around. So what I'm doing here is, you know that in EFT you essentially build your cell, which can be, for example, a cell like this, let me draw it horizontally, not vertically. So I can make a block of material A and I basically in a simple calculation when I have only material I take the unit cell and apply periodic boundary condition and I do my DFT calculation of the band. And I can do the same of course with material B. This will give me essentially the band structure isolated of the two materials. So the information content is as much as the one I can get with tight binding. I still don't have any interface in my system. What the step 4 we can do is actually take these two things put them together, form an ether interface and now my cell will become this one. So I will apply boundary periodically condition on the two border. So my unit cell instead of being made of, let's say, one fundamental cell of the lattice, will be made of many cells. Okay? So because I need to have a hyperphase, then a few states, a few planes on the two sides in such a way that they recover the normal behavior, the back behavior of my system. So now this becomes my periodic cell. So you perfectly understand that instead of having a few atoms in your unit cell, that you have separate elements, you need to have a lot of atoms. So the computational cost and effort is much larger. So this is one of the first work to report in this kind of approach in 2009, from the 50s like over 70s like Dave Bundy so then once I have this block you see that this is the calculation which is done and it's very similar to what we have seen in the experiment so I calculate the position of some core level and of valence band then I calculate the position of the other four level of the balance band and then again I when I form the heterojunction for the same reason that it would be maybe I am extremely sensitive to the position of the balance band I use the same kind of approach that I use yes so I take this number from bulk material And then when I calculate the larger cell, I look at the energy difference between the correlators because with the ab initio calculation, you get all in the atomic level, the energy level of your system. Neither one does not take part in the overlap and to the correlation there. So whether we are sitting on one atom, you can calculate them. So this is essentially mimicking what we did with XPS. So what's the meaning of these other blocks here? Essentially, here we are, this way we are addressing the ideal case where two materials have the same lattice parameter. Because if I'm building this larger cell, I am essentially assuming that they have an identical lattice parameter. And this is just a few lucky cases in the periodic table. So in most cases, I have some mismatch. So what is done is that, let's say, I start from material A, which has a smaller lattice parameter, and then material B, which has a larger lattice parameter, and I change the size of the unit cell in what we will call an hydrostatic wave. The strain is applied equally in all three principal directions. So that I change the volume of the cell, but I'm not changing the symmetry of my system. So if I have a sixth-general state, like in a silicon conduction band, and I shrink or inflate the unit cell equally by the same amount in all directions, I will still have a sixth-general state. And so you inflate the material A to make it larger, and you shrink the material B to make it smaller. And as we will see towards the end of this lecture, this transformation is relatively easy to treat from the theoretical point of view. This hydrostatic strain change is essentially a coefficient you apply to the different energy levels and you get the final result. So once they are, let's say, of the same size, then I can perform the calculation. and I get this what I need the conduction the valence band offset. So once I get this for this fake system where I have two material equal as you will see it's relatively easy to go back and understand what would happen if one is straight and the other not or if both are straight and so on. But to start with, we need this value here. So in this work, the author presented the result in this way, which is rather common in this kind of articles and topics. So as you see, we have three different cases. One is 4-4, so group 4 element, the other one is 3-5 compounds, and the other one is 2-6 compounds. And typically in every family of semiconductors you take one reference. So for example, you set to 0. Thank you. Thank you. . . . So you take as a zero reference the top of the valence band for diamond, carbon in a diamond like structure. And so this valence band of silicon would be 2.4 eV, the germanium would be 4.24 in 5.26. So if you want to actually get the balance band offset for example between silicon and germanium, you essentially take this difference here. And the same is true for all the other semiconductor classes. So essentially the energy, the balance band offset between these two aviation semiconductor is the height of this step. Another way of plotting this is for example a table like this one where essentially you take the difference between germanium and silicon you will have 0.38 meaning that you may use the bulb for silicon and aluminum phosphide will be minus 1.24 which means that the value of silicon is below that of sorry of having aluminum phosphide is below that of silicon so you can get data in this different way in this kind of way. Actually, here is a note taken from the paper, where they say, to enforce the transitivity relation, here from the calculated offset between A and B, the offset between A and C can be compared. The final offset presented in P2, which is V1, are obtained from the least square feet over the number of direct and transient pairs with the standard deviation of plus minus zero one. What does this mean? This means essentially that here we are assuming that if I take the carbon diamond, Thank you. Thank you. Carbon, silicon, and you get the time of the silicon, you get the certain time of the silicon. to get a certain balance band offset. Then, you know, you want to calculate the balance band offset between germanium and silicon. Okay. Let's take a break and then we'll try to restart. Thank you. Thank you. Thank you. Amen. okay so let's see if now I'll finish working so we were analyzing this sentence here so what it is with transitivity rule so it's the idea that if I have for example the interface between diamond and silicon I can calculate some band offset then I have the one between silicon and germanium and I will calculate another band offset and if now I want to calculate the one between carbon and germanium, well carbon-diamond and germanium, I would simply have to consider the adding up in this case the two band alignments So I don't need to repeat the calculation for all the different couples. So this is what we call the transitivity rule. I can get essentially what's behind essentially these plots here. So this is kind of obvious as natural, but actually you see here that the authors say that they need to enforce it. So this is actually not a rule that I can derive from the law of physics. because for example even if I take different interfaces so if I make my heterojunction along the 110 direction or 111 direction or 110 direction I would get slightly different results so since the variation is not that much typically we trust in this rule and so what we did here is essentially we brought this terkes for example was already the calculation between aluminum nitride and then gallium nitride, but then they just didn't do only the one between gallium nitride and helium nitride. So they investigated all the different possibilities of the different couples. That's it. So, they investigate all the different couples and they make a square fitting to get this staircase which more or less satisfies this transitivity, which is, as I told you, there is no physical reason why it should be like this because what we are saying is that the key point is what happens at this particular interface, now we are saying that actually it doesn't really matter because I can match different materials. So, since I always like to show experiments, so this is one case where this transitivity rule holds, it's actually experimentally verified within the, let's say, experimental error of determination of the balance band offset, which in some cases is relatively large, so is of the order of 0.20 eV. So I cannot really go very often below this. So this is essentially experimental band offset of the interfaces of C, silicon gemelium gallium arsenide, and helium gallium arsenide with two different oxides, charge zirconium oxide and aphneum oxide. So this was investigated a lot over the past 10 years or so. bit more because they have a very high dielectric constant so they are what we call typically high k dielectrics and so they are a good replacement of silicon oxide in microelectronics and the other region The reason is that silicon oxide is very good if you have silicon so you can form a thermal oxide. But since the idea of future CMOS technology is to use silicon for all the integrated part but then put in the channel of our N-MOSFET or P-MOSFET a material with better performances. So the idea is replacing silicon with germanium as far as holes are concerned. if you look at the table where we show the effective masses you will see that the whole mass of mass of holes in germanium is very low it's probably the lowest in the periodic table and the gallium arsenide has a much more electron mass so the idea is replacing this material so then you need an oxide which is no longer silicon oxide something else so this is why here where the g-oxide and the D-oxide are invested. So what do you see in this plot? You see, so you have a silicon where you put all this g-oxide and you calculate the conduction, band offset, and the balance band offset, and you do the same for apnea. And then you do the same for a germenium. And then you do the same for another, let's say, example where you have germanium and on top of this you have a thin layer of silicon and then you have the oxide but in this case again the balance band offset is calculated as the one between germanium and the oxide the silicon is so thin that it's not measured in this experiment So what happens here is that if this transitivity falls, if I put in an interlayer between my main semiconductor and the other side, I shouldn't see any difference in the conduction and balance band offset. And this is what, for example, what happens in this case, because you see that these numbers are identical. So even though I put in an interlayer, an effect of this transitivity rule is that I don't get actually any effect if I measure the management offset between these. Okay? And the same is essentially true, more or less, if I consider, for example, the case of indium-garium arsenide with and without germanium. in this case for measurements of for tonium oxide and also for gallium arsenide even without gemini. This is one example where this transitivity rule essentially holds. And this is a counter example, sorry but this got a bit shrinked. So the title of this paper is essentially lack of band offset transitivity for semiconductor interjunction with polar orientation. Zinc selenide on germanium, germanium on gallium arsenide and zinc selenide on gallium arsenide. So essentially what they did here, they determined these different band offsets. The first one is between germanium and zinc selenide. The second one is between germanium and gallium arsenide. And then they were to look at the interface between these two. And essentially, so this is A, B, and C. And if you make this calculation, so the first number here is the one between... here, so it's between A and B, second number is between B and C, and you get a value which is different from the one to get the direct interjunction. And here the claim is essentially, well, in the first cases we are considering the interface between non-polar semiconductor meaning that it's a purely covalent bond system so we have a group 4 element and two elements where instead I have also an ionic component. One is gallium arsenide and the other one is zinc selenide. So in this case it is experimentally proven that this transitivity rule doesn't work. And this is also the reason why in the previous paper essentially they were classifying elements in the different they were separated in group 4, group 5, group 6. In this case we can actually trust and rely this data when we want to calculate this study. So just one example of some relevant bend alignment that we can have. So you typically find it blocking this way so here you see you have Indium Gallium Arsenide, Indium Phosphate, and this is a matching composition. That is matching. And we have this here, here we have floccid balance band and conduction band. Here it's reported again, this is Indium Aluminum Arsenide, and again this is the same gallium arsenide. This thing is reported twice just for the sake of clarity. Here we can recognize the different types of band alignment. For example, we have type 1 band alignment here. So you remember type 1 means that we have electron and boron in the same material. So this is a typical combination of quantum well laser and invariant mercenary pipe. and here we have type 2 band alignment which is for example good if I want to make a transistor which works only for holes, the right carrier, the carrier I want in one material separated from the other And these are different sets of semiconductor. And we have this type III, which is less used, but actually in recent years, a new class of devices emerged which are called interband cascade lasers. So you probably know about the quantum cascade laser, but also this interband cascade laser. and we will see later on that they exploit this kind of type 3 band alignment. So essentially, so far what I told you is that if you want to calculate band alignment between two different semiconductors you need to look at the paper and look up at the table. This is not so satisfying. So, in the remaining part of the lecture, I would like to introduce you to a model, which is called, about this, I mean, the scientists who proposed it, the Yarros model. And you will see that this is actually a way that with paper and pen and a pocket calculator allow us to estimate, typically with a good approximation, the band alignment between two different semiconductors. So, let's see which are the hypotheses and approximations behind this model. So the first step is essentially the following statement. So we have seen that semiconductors have a valence band which is more or less always the same. Principal bands are all at the same main characteristics, but the conduction band is actually varying from one system to the other. So this is one first complication that we have been making, calculating the band alignment. So the Yarrows model has a first hypothesis saying, What did he say? Let's take the simplest pen structure as possible, calculation as possible. This is actually the so-called nearly free electron model. So how does this work? So a free electron model, we all know how it works. We have a dispersion of an electron with an energy. The electron is free. The only energy there is the kinetic energy. So we would have a band dispersion like this. Now I put a perturbation in the path of my electron which is a periodic array of atoms. So I will have a periodic potential like the one shown here. So if this potential is weak, what I'm assuming is that essentially the electron with a k vector which is far away from the giving diffraction effect with the lattice will be not strongly perturbed so we continue to have here an electron which is essentially free as the one i had before but when the wavelength of my electron becomes comparable with the periodicity of my system, I can have actually two situations. Okay, so I can have a condition where I will form a standing wave with the electron sitting close to an atom, or the same standing wave defaced in such a way that the electrons are are mainly between two different atoms. And these two states will open up the gap because they will have a clearly different energy. So this one where the electron is closer to the central atom we have a lower energy and we give the minus band. and in the other case I will have here a conduction band stable so energy here will be different, so this open up so even this small interaction the periodic array will form the peltier which will always form at the border of the free-loan zone, this position here because it will always happen when I have a diffraction condition between electrons traveling freely in my system and the atoms which are equally spaced ok, so whatever semiconductor I take, if I describe it as a nearly free electron system how we always have the same structure of the valence of the conduction. As you see, it's very different from what we get from for example, but this is, let's say, the starting point of our Yarros model. And essentially, which would be the difference between one semiconductor and the other? Well, essentially only the extension, the width of energy gap so I'm treating this periodic potential as a weak perturbation so depending on how weak it is I will have an anterior with this bandgap let's say another one here with a larger bandgap okay so now when I form my a hetero junction, this is typically I can hold it back to the first finger and so on I will essentially have something like this. And since essentially the dispersion is too bad, it's the same So they all come flat as I approach the end of the billowing zone. So the energy of the system is automatically aligned. It will always be in the middle between these two energy lines. So in a system like this one, I have energy gap 1 and energy gap 2. balance band offset will essentially be energy gap 2 minus energy gap 1 divided by 2. So I will have this value, also the conduction band will have the same value. So now we run into a first, let's say, kind of unsatisfying result, because we would have all the material, all the alignment of type 1, and I have an equal distribution of the difference between the energy gap between the conduction band and the balance band. So let's skip this point here for the moment. because then essentially in Yaro's model the idea is that okay this approximation here is actually good for the valence band because I know that all states from the valence band are similar but it's bad for the conduction band So we see later that essentially I use this here with the electron model and simple formula to calculate the balance band offset, but then to calculate the conduction band offset I will put back the real band gap of my material. So we use this only to calculate the balance band. Now the second set is the following. How do I relate this nearly free electron model to reality? So if I take a band structure, so this is essentially what I was saying. Now if I take the band structure of a different material, which would be this nearly free electron band is this one this one it's a direct one is really the fundamental one so for every material I have various options and which is it the good one so on this point the Yaro's model we take the answer from experiment because we can notice that this is something that we didn't look at when we discuss how the main properties of semiconductor varies moving around the periodic table in the first lecture. We can see that there is a very nice systematic behavior between the refractive index and, for example the band gap wavelength so this is the energy wavelength you get if you divide by 1.239 the energy gap in electron volt actually yes and you get the wavelength in micrometer and you see that we have a very nice behavior where essentially the refractive index is smaller for material with a larger band gap, which is a shorter wavelength. It's less nice to see, but you can also get the same kind of behavior if we plot directly the dielectric constant as a function of band gap. so here you get this trend which is approximately 1 over the energy gap and the floor is proportional to what we plot before the lambda pattern. okay and so the answer to our previous question was which band gap do we need to use to calculate the finance band offset in the model the answer is we need to correlate and in this band to the electric constant of a material so So we will see how we can extract this average energy gap, which is essentially an energy gap of an equivalent nearly free electron system. what will be obtained from the static electric constant So the static dielectric constant essentially is the quantity that creates the polarization vector of dielectric with an external electric field. We all know from electromagnetism that the polarization vector is equal to the the electric field or for our purposes relative constant minus one which is the electric field times Now the next step would be relating this calculated polarization vector for our semiconductor and the value of this one. Okay? So again the idea is keeping things as little as possible. So which is the simplest model we can use to simulate the absorption of the system like this? So we have a system where essentially I can have light absorption at this energy, which is the energy gap. And the simplest semi-classical model reproducing this is the Lorentz model. So we schematize this absorption and energy gap, which will replace some specific value of photon energy, with a resonator, so a mass spring. Imagine that we have a finger, in touch we have our mass, so there is a potential to take a coordinate like this, so we have a restoring force, minus kx and our model what we do essentially we say this mass here has a charge which is the electron charge and so this k this spring here is essentially the force of restoring the position of the electron on the shell so we know that if we have an equilibrium condition, which is when the system is not perturbed, I have a minimum of the energy. So if I have a minimum of the energy, I can always describe it in a smaller than enough environment, say, interval, as a linear force restoring my system. So if I try to perturb my electron direction by a small amount, it goes back here and the other way around. This minus K will contain essentially the interaction of the electron. Done for a small perturbation. And then I would have an external electric field. So, if I apply an electric field at 13 cos omega t, the variable 1, I will have a force acting on the electron causing the opposite of the catapult acting in minus E for the vector E and so the Lorentz model I saw the Newton equation for the system so I have the mass times the acceleration put the force acting on my system and in this frame of reference minus kx minus because of the table so the equation can be written down this way So, Km, we all know that it's essentially a square. of the frequency of oscillation of our system. So in our case, this omega 0, as we will see later, is essentially the energy gap divided by h1. So for the moment, let's just go to omega 0. So write down our system. So, this is now considered that this is a frequency of free oscillation of our system related to the energy gap and this omega is instead frequency at which I'm driving my oscillator. So the solution of this equation can be down in terms of this quantity here. So we take it x as an amplitude times the sine of omega t. In principle there should be also an initial phase we don't care about it so we have a second order regression equation I would need two constants one is a and the other one would be some initial phase we set it to zero for the sake of simplicity so if you replace this value equation you get something like this minus a omega squared cross omega t plus omega zero squared a cross omega t equal to minus zero divided by m cross omega t. omega t. We see that this is the original order system. We can see that it is and finally amplitude which is the only unknown in our case. This will be minus t, amplitude of interp divided by omega zero squared minus omega squared divided by that. Okay? So our x will be equal to this point. Okay? So In this case we see that this thing has a divergence, it goes to infinity, omega which is resonant with omega zero. We could have considered a damping parameter that is not necessary for our treatment. But for the sake of the narrative, We would have had something like this, where tau is essentially the scattering time dumping our . For the moment, what you see is that essentially we have recovered this value of X. And now, so let's not forget that our final goal is calculating the polarization. Okay? So now we know that the electron is moving by a quantity x. So the dipole is more associated to this oscillation in the sense that the electron charge, which times our x which is the value that we have calculated. So this will be plus square divided by m omega 0 minus omega square here again cos omega t let's put let's say the person of our axis here of okay and so we know that essentially p equal to the polarizability alpha times electric field so this means that since quantity here is electric field our polarizability alpha is essentially this energy okay now this is polarizability it's related to with the dipole moment of one electron. The polarization vector, P capital, is essentially the dipole moment of every vector multiplied by the number of dipole for unit volume. Okay? And so this is essentially n 4, which is the electron density in our case. Okay? So now we have p is equal to p small times n alpha times n times e and alpha is square by m omega 0 squared omega squared and as I was recalling you in the beginning of this part this is essentially epsilon r minus 1 times epsilon 0 times Epsilon zero. So eventually, we get this result, where Epsilon r one plus square n divided by m Epsilon zero omega zero squared minus omega squared. Right? So this is the outcome of the Tudor model. Now we say that this experimental collection of experimental data I'm showing you, this is essentially the steady state of the electric constant. So the one at omega equals zero. So now we take our formula, we set omega equal to zero, and I get epsilon r, one plus three. So we say that the whole objective of this treatment was to get some average band gap that we can use to calculate the balance band offset out from some experimental data. So now what we simply need to do is multiply this by h-y square and we get 1 plus h-bar square m square divided by m epsilon zero which multiplies and this will be essentially our energy gap. ok now since this is an energy actually the inverse of an energy epsilon r is unit less than 2 it is also this quantity needs to be an energy the ratio between the two energies to get the pure number between the two energies to get the pure number. So which energy is this? Actually the square of an energy, sorry. So it means that n is the root square of this. This needs to be some kind of frequency, omega, and this is actually the plasma frequency. Okay? So, the plasma frequency is what you would get if you would calculate the same problem, say oscillation force oscillation on a system like this let's say you have perfect an ideal metal without any loss you have the fixed ion in your metal and then you have a cloud of electrons if you drive the system with an electric field this electrode cloud will oscillate following the field and you would get this kind of oscillation which is resonant to this plasma frequency by the expression here. Okay? So I guess you will get a little bit already another part of your studies. So eventually what we get is that epsilon r is equal to 1 plus h bar omega p squared divided by energy gap squared. And what we actually want to do is knowing the electric constant getting out the value of this energy gap. So retrieving from the experimental data of a electric constant with average value. So to it we get the energy gap h bar omega p divided by the root square epsilon r minus 1. So eventually, when we have caused a problem like this, we have a material A with an energy gap epsilon r A and the material B, the average of the dielectric constant, sorry, epsilon r B, this Yaro's model tells me, okay, if you want to calculate the balance band offset, you need to get out from these two materials the average band gap of A and B starting from the derivative constant and the value of the plus frequency and once you know this the balance band offset d1 minus the other divided by 2. So this is the point where we are. So for the moment, it seems that here we have an important parameter to know what differentiates one and the other. one semiconductor, the other is this value of the electric constant and let's have a look in more detail at the plasma frequency so if in the plasma frequency there is something which is related to a semiconductor, the difference between the two semiconductors making up our heterojunction so we say that omega p the sentient value here. So we have E squared, which is the electric charge of the electron, so it's a constant. Epsilon zero is a constant. M, so we never actually specified whether this is an effective mass or the, say, the free electron mass. If we would put now here the effective mass, we will run in a series of complications, because which one should I take? The original transfer of the connectivity. So it would be actually putting more variables in our system. So let's say, let's try the simplest first. And we take this also as a free electron mass. So all these three things are constant. And n is the number of electrons per unit volume. Okay? So the number of electrons, how it is calculated in the Yarrows model. We see that we have always this tetragonal coordination in a semiconductor. So let's say that this number of electrons will be 4. The S2P2 electrons that we have seen in the first lecture, 4 electrons in the balance band, multiplied by the number of atoms I have in each unit cell. divided by the volume of the unit cell. So how many atoms do we have in a unit cell? If you plot the diamond structure and you count how many atoms you have in so of course you need to count also the fact that some are only partially in. Each atom on a corner will count only by essentially one it will be only a fraction we need one eighth we count only by one eighth and the atoms which are the faces count only by one half and the one which are inside will count by one. So if you make this computation you will get that you have eight atoms per unit cell total. And then you need to divide this essentially for the volume of the unit cell which is the lattice parameter. So you would get that this n, this density of electrons is 32 divided by the dimension of the unit center. So this is the second physical quantity discriminating one semiconductor to the other. So as an input to the Yaris model, I will need not only the electric constant, but also the lattice parameter of semiconductor A, the lattice parameter of semiconductor B. in the hypothesis that I have an FCC cell. If I have a zinc-blende structure, this number is, let's say, we stick to the cubic case of a diamond or zinc-blende structure. So let's try now to calculate which kind of numbers comes out from this energy gap. Yes, all of them. You're right. So let's try to make an example to have an idea of the other The lattice parameter is 5.65 and the electric constant we will need later is 10.9. So the first step is calculating the density of electrons, n, so I put this magic number 2, and then my recommendation is that you use for all this calculation units from the international system of units, so cubic meter and joule for energy, and only at the end you convert everything into electron volt. Otherwise, you typically, at some point or the other, you do some mistake. Okay? Okay. So this means that here I need to put 5.65 times 10 to the minus 10 for angstrom, and And in this way I will get the electron per cubic meter as the circle equals correctly 0.9. If you do this, you get the number which is 1.77 times 10 to the 29. So this is more or less always the same order of magnitude. The variation in the denominator is not never very large. So if you get a number which is different from low 10 to the 29 high 10 to the 28 there is some mistake in your calculation. So let's see which is the omega p coming out of this. I put the value then I put the electro charge square then I have Epsilon zero, 8.85 times 10 to the minus 12, again in unit of the system. And I have the electron mass in kilogram. So this number will be 2.7 times 10 to the 16. and the unit of omega is radiant for its frequency is actually equal don't measure in hertz rather than fuller second and again this is more or less the kind of value we get so now let's finish our calculation and we have the energy gap for H bar omega p divided by square root of epsilon f minus 1 and if you now at this point calculate h bar omega t should be 2.7 times 10 to the 6th times h bar which is 1.5 10 to the minus 4 and you get 2.49 49 10 to the minus 18 down only now electron volts will be divided by the electron charge to get 15.6 electron volts again this is more or less the order of magnitude of the energy of the plasma, H bar times omega. So now the energy gap will be essentially automatically obtained in electron water. So, as I said before, 10.9 minus 1 4.95 eV. So you see that it's a value which is substantially larger than the fundamental gap of Dalai-Marsenal, which is of the order of 1.5. Of course the idea became that this value, kind of the attempt of summarizing in a single number, an average of the value here. All this value which will give me, allow me to say project my complicated band structure into one single energy gap of a nearly free element. Okay? So you can repeat the same kind of This is the summary of our information. And this is the comparison between average band and the semiconductor. So what we did now was a wonderful guide. Here we got this around 500 point something value. And this is compared with the fundamental energy gap. So we see that they are trained, of course, on the Van Gap material, on an average Van Gap, but we also have the fact that this value is always larger. Okay? And so this is a table blocking the value you need to calculate the value. So, now let's go back to the final step. So let's suppose that I want to calculate, let me see, the average displacement offset between germanium and cadmium arsenide it means I take this value here for germanium lattice parameter and dielectric constant to the LROS model and I calculate energy gap I do the same for cadmium arsenide and I calculate this energy gap and now I need to make this average. So the first thing to keep in mind is that in the Yaro's model, in this nearly clear electron model, we always assume that we have a type I bandline. We have a material with a smaller gap, the one with the larger gap will have the balance band below the first one. In this case, this is the Geminium Gallium Arsenide example. Gallium Arsenide has 4.97 and Geminium has 4.05. So this means that we have, if this is germanium, this is gallium arsenide, I will have a pen alignment like this, where the material with the larger pen gap has a balance band below the one with the smallest pen gap. So this VBO, now, given by average value of senile minus average of premium, divided by 2. And if you do this, you get, in our case, the number reported in the table above, you get this value, which is 0.7, which is to be, for example, compared with the experimental value, which is 0.56. As I told you, we are within this 0.1 EV accuracy. Here we started from a kind of a complicated problem because we have a polar material with a non-polar one. Let's have a look at aluminum arsenide, gallium arsenide. Also, let's say in this case, the agreement is not that bad. Let's have a look, for example, at gallium arsenide, In Geo-Marcinite, in this case, the limit force, but if you browse through this table, you always see that we have considering the computation of the fork, we have two main results which are kind of hidden. One is hidden here. The first one is that we always get the band alignment right. We always get the fulfilled statement that I was pointing out here. The material with the larger average gap has always a balanced band below the other one. So in this table here you see variation between these numbers, but you don't see sign differences. We always get the right Vandana line, which is the first fundamental step. And secondly, in most cases, we get reasonably good agreement of our calculation with the experimental data. But some cases were actually pretty good. So, since the one idea is also that we, I think in this lecture, I would like you to be able to understand the fundamental physical properties of a nanostructure without going through several papers to find exactly the right number. This is something, I mean, this Yaro's model is something that with a pocket calculator or with an Excel spreadsheet, you can get in a matter of minutes and you already 90% of the time you have an answer which is good enough to understand physical properties of this. Okay? So, let's make an example of how we can use this. So, we did a lot of pages with formula. Let's have a look at the broader picture of what's happening in our world. So this is a photograph of a data center. So we typically, as consumers of electronics, we can put things on the cloud, really makes them kind of disappearing. We don't use paper, we don't use archives. But actually, the infrastructure where our data are stored is actually enormous. So we are talking about very large factories, which are filled up with servers, which essentially store and treat our data. And if you can imagine that basically a lot of it is videos and photographs of stuff which is actually not so relevant for our lives, but there is also other data which are actually taking less of an effort to be stored. Now our data come in bank accounts and things like that, but they are very relevant, so they are duplicated many times. If you look at the energy consumption of a system like this, you see that there is an energy consumption from the storage devices, so this is memory, essentially, where we store our data. This is the energy which is actually powered by the power supply of the system. And one thing which is actually very large is this yellow line. And this is energy consumption for refrigerating power systems. Every, just past lectures, every, let's say, fan cap of semiconductor determines to a larger extent temperature which it can work, because if I go to a... If you have a small band gap, as the temperature increases, you will have a lot of intrinsic carriers, so the doping will put in the spices won't be effective anymore. So typically, even though silicon can operate around 120 degrees C or even more, if you want to have humans working in this environment, you need to cool down your system. And so for example, one solution is the one that is employed in many data centers to be close to rivers. So for example, one of the biggest data center of Facebook is in Finland, close to two rivers that can be used to cool down power consumption and also have a cooler environment, let's because the average temperature is lower. An interesting experiment you can find here, done by Intel. So instead of having a larger computer infrastructure in the same building, they had this kind of vessel which was put 20 meters below the water. Again, I think it was in Thailand. Everything is filled with nitrogen. It's interesting because they took a certain number of servers, put them down for a few years, and had identical servers in a farm, doing the same work, the same job, and this one performed better because the environment was the same. But then every few years you need to take it out, and since this thing becomes warm, have wild mussels and other vegetables growing on your system and you need to do it. Okay so which is a possible solution to reduce this power consumption? So essentially power consumption is given by the Gauss law, that they have certain power which is dissipated because I have a resistance and a current flowing in my system. Now, if I, instead of using electron, I use photons to at least transfer information, I don't have joule heating anymore. Because if a photon runs through an optical fiber, there is no joule effect in this case. I would have some dissipation in the laser, but it's typically much less than what you get here. And the second advantage that I have is that when I have a lot of cables close together, packed together, I will also have some capacitors coupling between these cables and so we'll have an RC limitation to the speed of the bandwidth of my communications. I will not be able to go above some gigahertz, let's say. Again, if I use photons, I can use much larger bandwidth because I can use frequency modulation, different photon energies. So one of the idea behind this thing which is called silicon photonics is actually to replace the communication in a tunnel to charge flowing through a copper cable using photons. So please we don't want to change things radically, so the computer will still be using electrons, will be similar to the one I'm using here. So I will have a CPU or a server which has an output as some current, J. But now what happens is that I want something to transform this current into a photon flux which can run on a cable without all this dissipation effect and with a larger bandwidth. And on the other side, I want something that takes in this photon and again gives me out an electrical signal. So the first kind of device making this was this cable from Luxera, which has now been acquired by Inixis. It looks like a normal cable, but actually inside here you don't have a copper wire, but a glass fiber. So you have photon traveling. And this plug is not just an electrical plug, but it's quite an interesting piece of technology. It's something like this. So this is a plasma die, this is what's inside. On every side, essentially, you have two elements that I was plotting here. You need something to convert current in a photon flux, flux so you need a laser yeah okay this chip here is a distributed feedback laser So 155, 1500 nanometer. So these are DC lasers, they emit the constant power. And then you need to encode your signal to make one and zero. And so you have a mode MZI instead of Max-Zehnder mode. This is the T-Base. And so this might send a modulator modulator light signal and send it out light the other end of the cable. So when this same device needs to receive, it gets light coupled here and then I will I need a photodetector which is capable of absorbing light at 1500. So I need a photodetector which is able to absorb, I mean, detect light this way. And then this is the chip addressing and coding the activity of this material. And this is called silicon photonics because the idea is that we want to do this on a silicon platform because then we can take profit of foundries making the chips that we have in our computers and mobile phones and so on. And so as much as possible we would like to use silicon compatible material. So this is not possible for the laser, we need a free-fired material, so this is made in a different foundry, one which makes only this distributed laser, and then they are chip bonded, placed in the right position on the silicon chip, but actually, as you see now, The detector can be a material which is absorbing the infrared and is compatible with silicon and this is germanium. So actually, the HoopSfera was essentially founded on the possibility of the development of this new technology of growing germanium or silicon for this purpose. okay and so the basic let's say ingredient that we will treat today and where we discuss about the heterojunction is a device like this one so it is a photodetector where we have an absorption layer which is made of germanium the reason why in the germanium is explained here where we have the absorption coefficient of germanium and for comparison the absorption coefficient of silicon so you see that at 15 50 we get a strong absorption engine. Since we digged a lot into the physics of this semiconductor, leaving aside this technological factor, it is interesting to see the behavior of silicon engineering. You see that in both cases, we have an initial part of absorption which is extremely weak of tens of inter-centimeter and this is what? this is the absorption coming from So this transition here is silicon, the indirect one, and this transition here is germanium. It's an indirect transition, so it's weak because we know it's an indirect transition. But what happens in silicon is that this indirect transition continues for several energies. Okay, while in germanium, if I increase the energy a little bit, I get immediately a direct direct gambit, direct bandgap. So this is why these two plots of the absorption engine, the absorption coefficient. We essentially see that in silicon we have this smooth increase of several nanometers and we have this sharp edge which is the one that we use for the cause of battery direct transition. Okay, so this is a device which can be illuminated from the top. And so, for example, it's good for imaging, but in our silicon chips we actually prefer waveguided devices. So where we have a waveguide, because one thing we have here is this couplers, so light is coming through the fiber and then guided to the silicon until it reaches this waveguide photodetector. So we want these devices to get light this way, and this has several advantages, because essentially in a device like this one, the amount of light which I can absorb is given by the thickness, so if I want to absorb a substantial amount of photons, I need to put micrometer or more of material instead of the waveguided devices this is where the silicon is coming from the silicon waveguide what happens is that my germanium has a larger absorption coefficient so the mode is transferred to the germanium layer through this tapered structure and now the absorption say length is not the thickness of my layer but it's the length of my waveguide so it depends on how long so even if i deposit here only a few hundred nanometer a 200 nanometer of material i can get essentially absorb all the photon possible So this is a typical structure of a layer like this. We have a silicon oxide layer below the silicon. The waveguide is made of silicon, so if you see here, we have an increasing dielectric constant as we go from the bottom to the top. to the top so initially when there is no gemini the mode is confined to the silicon layers so travesty here the optical mode because as you can see silicon is perfectly transparent the wavelength we are using and then when we get through this taper and we get in contact the optical mode will be transferred here and then it's absorbed and we get our signal. And in this case two contacts are made on the two sides of the device and one is made on top. So there are these two large contacts here. So you see that these two are connected together in both form of the bottom contact this one is the one at the top and this is the SEM image of the contact area this one okay so this was the introductory part But let's see now how this did bring any criticalities due to the fact that I used two different semiconductors in designing my device. So I will basically take as an example the vertical illuminating structure which is easier to understand. so let's assume that I have our beacon upset at the top of it I have a germanium layer to the last part which is the p-type. Okay. we want to know is what happens at the interface between Germanium and silicon. To do this, we can calculate the depend alignment. For example, we can start with the Larros model. We need the dielectric constant for the two materials. So we have 16.2 and 7. Then we need the lattice parameter, which is 5.65 and 5.43. So from this I will send you the computation but you can calculate plasma frequency for for the two materials, I get 0.57 and 2.52, in both cases is multiplied by 10 to the 17th And from this value I get this average energy. So this will be 4.01 by 0.07. If we now construct this band alignment, we will take into account both effects. We will have the alignment due to the heterojunction and an alignment due to the electrostatic field, the doping. So first thing is that we tear your border, the other one, and the alignment between the of germanium and silicon is followed. Let's say that we divide into two pieces of silicon, here we will get the final aspect of germanium and we will have the value of the system and this BBO will be equal to 5.7 minus 4.01 times 12. And this number is equal to 0.53 Okay? So now we can construct the alignment of the conduction band. And here we use this kind of our trick from the Janos model. We say, okay, now we use the real band gap to build the band alignment. But again, we have an additional thing to consider. which band gap I am considering. The one which is formed between the valence band and the delta valley, the one with the L valley, the one with the gamma point. Let's have a look at all of them. Let's first consider what happens at the delta valley. So in silicon, we know that the energy gap separating the valence band and the delta valley is 1.20 p. and in your menu this is 0.85 essentially and so 0.53 this is the delta valley of gemini 0.65, ok The other valley we can draw is the L-valley for example. So let's take it in the column. In the medium column, this is 0.6 electron volt. In silicon, this is 2 electron volt. So now, which kind of bend alignment do I have to consider? Do I have to consider the one between similar states? So this one, or this one, or for example the one between the lowest state, which one I Well actually the answer is it depends on the problem we are addressing. So now we are interested in what? In understanding how a photodetector works. So in our photodetector what will happen is that we have a state here, which is a state gamma point of germanium which is at our 0.8 b, our 1500 nanometer. So when I want to absorb these photons I will have electron holes, electron years, electron year forming at this So now this electron, what will do? They will actually be able to scatter down to the level at lowest energy. So this happens in typically hundreds of phantosemites. What will happen is that I will have absorption. direct gap but the transport will essentially take place in the L minimum so and this electron when it reaches the interface it will also go down to the level of lowest energy so it will find some phonon like this one needs a funnel to go from gamma to the L point, and then you will find another funnel going from L points to delta. So this electron can actually continue to travel. So from this particular problem, where I have electrical transport to take into account, my relevant conduction belt offset is not the green one, it's the red one, but it's the one between the L and the delta state. with the minimum in one material and the minimum in the other material. So our conduction band offset is relevant. In this case, it is essentially the one between the delta and L state. So essentially what we need to do is following the L. We subtract 0.53 plus 0.66, which is this value here, and subtract the energy gap of silicon, and we should get the value which is very small, 0.27 electron volt. So let's say an electronic engineer probably not aware of what's gamma, l and delta, but even through, bend alignment like this one. Okay? With a substantial conduction balance band offset of 0.53 and this is a tiny offset of 70 mV. and this is the gap of silicon, and this is the one of zinc. Okay? Okay? So this value is so small that it's actually, depending on the value of the balance band offset you can choose, you can go from this type 2 band alignment to a type 1. If you get this number a bit larger, this thing will change sign. This is for example what is reported in this work here. Okay, where the balance band offset was taken a bit larger, the conduction band offset came out the other way around. The important thing is that also in this calculation is CBO let's say much smaller than carbon energy okay or comparable with thermal energy so it's not a big problem let's say for transport okay but so our Yaro's model told us that essentially it's most of the the difference between these two materials, which means the silicon falls on the balance band. So now the next question is, can this be a problem for transport? Because depending on the doping I do in my layer, so if I do a system like this, where I have n-silicon, n-P-germanium, or a system like this where I have n-germanium, n-P-silicon, I can have an asymmetry because this is no longer so. In a silicon diode, the p-n-diode or the n-P-diode are exactly the same. you don't have any, you cannot tell the difference. In this case, there can be a difference, because now we have an asymmetry in the band alignment. So let's try to draw by hand what would happen if I had a system where I have N-germinium and P-silicon. So how do I have to proceed? So let's draw first a Fermi level far away from my interface. This is the germanium layer. NJ means that the Fermi level is closer to the conduction band. And then I continue this. And this will be my silicon, where instead the Fermi level is close to the balance band. Okay? So now, in an ideal without any heterojunction, say for here I will get silicon, and here I will get this one, okay? But what happens in this case is that at some point I need to plug in my conduction and balance band offset. So here I will need to stop where I have my interjunction and consider my balance band offset. So then, it will continue more or less the same, it's not exactly the same because of the dielectric constant. And here, I will get something like this. And so, in our case, we should have something like this. In the paper, I was showing you that it was slightly different. we would have something like this. Okay? So this is how I draw the case of N-germanium with silicon. Now let's see, is this good or bad? So when I shine a photon, I will have a hole in the detector, and in the photon detector, these two carriers, they need to go where they are a majority carrier. So the electron needs to go to the left, to my left, your right, and the hole needs to go to the silicon part. And we see that in this motion, for the electron I see no troubles, it moves path, while here I have this large offset to overcome. so let's see what will happen in most days of exercise the opposite case so I throw a level now I am so case where the city and silicon this is the Fermi level this will be my menu part for silicon now with z-type I would have something like this or in a simple case I would have a like this one now again I need to plug in my conduction numbers at some point I will have this continuity at some point I will also have this okay so now I will add that electron which is generated need to go this way but you have see this barrier is tiny actually unclear if it is like this or like this we say and all can go the other way without having to overcome this barrier here. So this is shown in more detailed calculations. This plot where we have the case of the PIN and we have the case of the NIP. And these are essentially the two photo currents. So in the case of PIN, you see that the photocurrent actually gets to the value of the industry. So at 1550 we have a photocurrent. In the case of NIP, which is this one, you see the holes crashing into this barrier. Basically, this is a silicon photodetector. So this is a response that we get from the underlying silicon. You see that it stops at more or less the energy gap of silicon. A response coming from the silicon. So the chem generated energy cannot go from this barrier and so it can't have any problem. So in this case, what you have to do is go for the any time you have a heterojunction, you need to consider that you are putting symmetry in your system and therefore you need to take into account this symmetry. One trick, in case your process cannot be done with the right doping, the upper one. If you use every dope silicon substance, so if you use B++, 10 to the 20 carrier, what will happen here is that you will have barrier like this, which is essentially small enough to have tunneling, so even if you make a B++ doping on the bottom contact, you would get a working photodetector and they exist, they are done in this way. Okay, so here I have drawn everything by hand to show you how to put on top of the continuities calculated due to the heterojunction formation and the electrostatic. If you are interested you have online also basically everything worked out. I will calculate the band alignment considering the different band offices. So first is the HOMO junction, then it's the hetero junction. something that you have seen in electronics. It's a bit tedious. If you're interested, you can see how you can do it. As the last five minutes, again, something just interesting from the cultural point of view. so we have seen that you can do waveguide integrated photodetectors to use in telecom another interesting application is imaging so you can do imaging in the infrared and you can do insert in a single We are a photo detector which is grown on top of a single where below you already put integrated electronics. So for each pixel, for example, you have a transceiver, an amplifier which is a charge integrator, but collecting different charges. So this is a work that we did a few years ago with the CSCM in Switzerland. You see many pixels are not working. This was one of the best devices, but it was done in a tonnage, almost foundry, using a small size wafer and all the processes. But the basic idea is following the table. I have a scene like this. If I look with a visible camera, I am blinded by the light of the car. the car, but if I could illuminate the same scene with the infrared and look just at the infrared, I would be able to see, for example, the behind the car. So this is something which has been under development for automotive. So actually these images are not taken with the G-Media PhotoDetect, so they still do not exist on the market. with the Indian gallium arsenide. If you look at the chart with the band gap and the lattice parameter, you see the Indian gallium arsenide is also very good absorber in the infrared. The problem is that being a material which is not group 4 is extremely expensive. So the Indian gallium arsenide camera costs more than a panda car. so if you want to put it on a car it means that the car can only be a Ferrari more or less because it pays 20,000 euro just for the camera if you want to make it so the idea is making it cheaper in the material which is compatible with the construction as a last thing an interesting thing we did just around the COVID period so we essentially said Okay, why we use only the germanium layer? Why don't you use also the silicon one to absorb? And if you illuminate from the bottom here, you essentially get something similar to the double junction, multi-junction solar cells. So you will absorb the photon at a larger energy, the visible one in the silicon, and the one at smaller energy in germanium. So with the same pixel, you can detect the two events of the image. So we will show here. So if you apply one bias, you get absorption in the visible range, and in the other, you get absorption in the infrared. So we were doing this in the lab, then we had the COVID shut down the lab, and so my student Andrea brought home amplifiers, optical table, and things like that. And in the garage, we went on with this experiment. For example, we managed to do this image. So this is a bottle, a blue bottle made with a camera of your mobile phone, so a silicon camera. And this is the same image taken with our device, so the resolution is worse because we have a larger pixel. But here we see that basically when we switch on the silicon, so we have one bias, You see the blue bottle. When we use the germanium, the plastic is transparent, the water is an absorber. We can see the water level inside the bottle itself. And so this is again the same thing that we repeated a few years later with germanium tin and germanium. And so for example, these are bottles with two solvents. I think it was acetone and toluene, something like that. And so they look perfectly transparent, it could be water. If you take the image in one band or the other, they look, to our eye, pretty similar, but actually they are not. If you make a difference between two images, you get this false color image telling you that in this model there is a material difference from the other one. So for example, you can train a neural network system to recognize the different shadows of corners, tell you what's in the bottom without opening it, which is important for example for an airport. So now Andrea, who is in the work in this garage in the COVID, is the CEO of a company trying to promote this technology to an industrial level. but everything is based again on this band alignment of GPMC also we have two Andrea Andrea Vallabio is the CEO and Andrea De Jacopo is one of the partners so we did this work in collaboration with Università di Roma DRE so you know Andrea and Lorenzo ok yes so we are exactly okay so this is enough we meet next Wednesday