isella4.txt

 Okay. Okay. So in our first lecture, we discussed about the properties of alloy formed by putting together different semiconductors. And so in these lectures, I always try to make example of technological or interesting physical application of the topics you are covering during lectures. So today I would like to make with you an exercise which is dedicated to, let's say, a non-quantum, non-nanostructure application of different alloy semiconductors, which is the fabrication of multi-junction solar cells. So these are solar cells with very high efficiency, the typical market value is around 28%, but you can go to almost 40%. And they are made of different alloys of free-fire semiconductor, actually substrate is group 4, but then free-fire junction is used. and their main application is space powering up satellites so you want to have a very high efficiency because you spend a lot of money in building the satellite and sending it out of the atmosphere so you want the best solar cell possible and they are also becoming useful for terrestrial application in concentration systems so a concentration system is essentially a solar system working this way you have a solar cell which is small because it costs a lot of money so we are talking about 100 euros for a square centimeter of this 3,5 solar cell and instead of making a square to collect solar radiation from a square meter essentially you make a square meter mirror so you concentrate the light on your cell so basically you take out the best of the material and collect as much as possible light. Of course these are systems which are rather complicated because this mirror need to be oriented with the motion of the Sun but still it is an application that there are solar fields working on this principle. So let's go back to our topic and just to be all on the same page let's review quickly how solar cell works it's the semiconductor solar cell so silicon solar cell like the one you find in the roof of many buildings is essentially a substrate which is p-docter where we have an N diffusion and so we form a p-injection okay so we know that in equilibrium p-injection features this kind of band diagram where here we can recognize the N part which is degenerate we see that the Fermi level is in the conduction band and this is the P part okay so at equilibrium these devices doesn't produce any electrical power so what happens when I shine light on it so I have a photon being absorbed and what happens is that the holes positive charge will go one way collected by the built-in potential, built-in electric field and the electron will go the other way so let's imagine that I don't connect my solar cell to any external circuit I will just have an accumulation of negative charge on one side and on positive charge of the other so this means the band diagram is essentially the landscape of a potential energy felt by the electron so if I put a lot of positive charges the potential energy of the energy will go down and so essentially I see a difference now between the two Fermi level which is a photo voltage okay so I will have some output power so this condition where I have no current flowing corresponds to essentially what we call the VOC where OC stands for open circuit voltage of course in this way I will add voltage but I will not have any power electrical power is voltage times current and so I need to close my circuit and have current running in my external circuit. From the point of view of the characteristic of these devices so a p-n diode we know it's essentially described by the diode equation which is this one when I shine light on this device and I have some photo generated current this current adds up it's a photo current and adds up as a reverse current because you see that if you look at this diagram you see that the essentially all will flow and electron will flow as if you had a a reverse current so essentially what happens is that under illumination you have an IV curve like this which is essentially this one translated by this negative term and so if you want to operate your solar cell you have a maximal point where the product between the voltage and the current is a maximum so you typically want to operate in this working point okay and so this is how we generate current in a photovoltaic cell so the energy we are delivering as electrical power is coming from the Sun so if we look at the solar radiation spectrum we have different spectra depending on the point we are using for measurement so and this different spectra are typically indicated with this AM and then you have some number convention so AM stands for air mass because the amount of solar power reaching certain point on the earth will depend on how much of this light has traveled through the atmosphere so for example for space application we are outside of the atmosphere so we need to use this AM0 so no air mass which is essentially this spectrum here okay which is very nicely approximated by a black body with a temperature of 5800 Kelvin right so this is essentially the temperature of the external part of the Sun which is where the radiation is coming so the internal part the temperature is like 15,000 Kelvin but this radiation is been retrieved by the the Sun and what reaches the earth as essentially as you see the shape of the black body of this temperature around 6000 degree once the light travels in the atmosphere then I mean it depends on essentially the inclination of the Sun with respect to the point where we are so let's say if a Sun is at the zenith if it's here we will have this path if it's not at the zenith it's here we will have this other path and so this Hermas convention is to say to define a given position a given illumination let's say as a m and then 1 over the cosine of of theta of this value so a reference value which is used is AM 1.5 which corresponds to more or less 45 degrees so if you make a 1 over cos theta 45 is 1.4 now the inverse is 2 divided by root square of 2 and so in this case for the MAM 1.5 it's essentially this spectrum here and you see you have many absorption lines which comes from water vapor ozone and other molecules which are making up our atmosphere and which are absorbing light. So it is clear that we have a certain spectrum of the solar light and we have semiconductor with many different energy gaps. So if I take a semiconductor with a large gap I will be able only to get absorption say above certain values or absorb this energy and if I take a semiconductor with a smaller gap I will also be able to actually absorb the remaining part of the energy so but this is what I get as an absorber light so we will see that there is a counterpart which is the voltage my solar cell can deliver because if I just look at this plot the idea is okay let's make a solar cell with the metal I will absorb all the energy but actually the output bias as you will see will be zero so there is a trade-off between having a larger band gap which means a little of some power but a high output voltage as we will see in me or the opposite having a lot of current but the low output voltage okay well how does that voltage depends on the energy gap let's imagine that I have my bias my p-n junction this something like this so this is the condition under without an illumination so the Fermi level is the same in the n and in the p part when I illuminate what will happen is that I will have a photo voltage here appearing and so the band structure will become something like this okay so it will tend to flatten so ideally I would like to have mean the Fermi lab if we take this condition when the Fermi level is here at the conduction band in the end part and here in the valence band in the p-type part the maximum value of the photo voltage which I can get is essentially the energy gap divided by the electron charge so as a rough number, I cannot go above this value. Actually it is even worse, let's say, typically the output bias of the solar cell is 70% of the energy gap, but for our purposes, let's assume that we are in an ideal world and the maximum output voltage I can get is the energy gap. So you understand that if the energy gap is small, you will get a small bias, but a lot of current because you are absorbing a lot of photons. If the energy gap is large, you will have a high bias, but a low current because you are absorbing fewer photons. So our first goal would be to understand, is there an optimum energy gap? is there an optimal semiconductor which I can use to make solar cell which optimizes these two trade-offs and to answer this question we will speculate a bit on this spectrum so let's for the sake of simplicity assume that the solar spectrum is well represented by this black body curve so here we have energy on this scale and on this side we will have a quantity which is essentially the number of photons per unit area per unit second and also for unit energy so for let's say so I can represent this thing as plotted in the plot I was showing you as dn over dhv which is a photon energy so it's a number of photons per unit area per unit time and for a tiny energy interval okay and this is the radiation spectrum of our black body so it's something like this okay and this is a photon energy now we know that if I have a semiconductor with a given energy gap I will be able to absorb only the photons with an energy higher than this energy gap. So from this curve I can calculate the following function let's say. I will call it NF gap but this is not the number of photon with this energy gap. This is defined as the number of photons so per unit area and per unit time with an energy h nu which is larger than the energy gap so essentially this would be the total number of photons which I can absorb with a given semiconductor how do we get this function NFE gap well essentially let's say let's consider this is our energy gap the value of NF will be the area below this part okay so if I take another energy gap this other value of n photon energy gap will be the area below this curve so essentially this function is nothing but the integral of our spectral density that we defined before integrated over the energy for an energy which goes from energy gap to ideally infinite okay so this is essentially the integral of this curve so if I now plot this n photo energy gap which the unit now will be so it's a number of photon per unit area per unit time which have an energy larger than EG I will get something like this so at zero energy I will have essentially all the area below this curve I will have all this so it's a maximum value possible okay and that as I increase the energy I will get a monotonically decreasing function so we left something like this okay and this would be my qualitative description of this curve which is actually plotted here is this one in a more really taking the say integral of this uh am 1.5 curve so you see that we go also here because you are integrating this So now, so the question, our original question was, how does the power I get from solar cell depend on the energy gap? So let's suppose that I have an energy gap. So this is still an energy scale. So I take a semiconductor with an energy gap here. so this point here is giving me the number of photon I am absorbing so the maximum number of electron all pairs so it will be something proportional to the current and how much voltage is this absorption giving out well we have seen before that the maximum energy I can get is essentially of the order of the energy gap why this happens? so let's consider that we have this flat band condition where the photo voltage is essentially equal to the energy gap so we will never reach this thing you will always have something like this so let's assume that we are in this condition when this value is approaching the energy gap and now I can have two things I can absorb one photon which is giving me let's say an electron all pair so I will get this quantum of energy but also if I absorb let's say a photon like this what will happen is that of this larger energy I have absorbed it's called H nu I will still get out only the energy gap because before these two carriers reach the contacts they will thermalize so we'll have this will go down to here this will go down to here so even though the photon has a larger energy in a conventional solar cell you would get only an energy comparable to the energy gap because everything else is lost for thermalization so there are designs where you make the solar cell extremely thin and you use also these hot carriers so when they reach the contact they still have this larger energy and so in this case you get more energy but let's say in conventional solar cell typically you always get the energy gap energy and so this means that now essentially I have this value of sorry this value of n FG and each one of this carrier is giving me an energy which is the energy gap so the power I get out from my cell will essentially be this area so it will be the product between an photon energy gap times the energy gap if you express this energy gap in electron volt you get the power essentially in joule because you have a charge this will be essentially charge times the photon flux which is the current times the bias which is the photo voltage okay and so we can visualize the maximum power we get from our solar cells simply as in this area of this rectangular square so which would be the efficiency of this set so you can immediately understand that if you would say move around this point here and choose a material with a different gap you get something with a different area and so you can find the best solar cell possible the one giving you the maximum area below this point okay because if you say are at i a wide bank semiconductor you will have something like this if you use a small area semiconductor small bandgas semiconductor you will have something like this and so you visualize basically that there must be a point where you maximize your output power another interesting thing we can notice is that okay so this is the as we say the power that we get out of solar cell let's call it p s is given by this or is proportional to this quantity the efficiency of the solar cell is essentially the ratio between this power let's call it photovoltaic p photovoltaic is the power of the solar cell is the ratio between this p and p sun the total power conveyed transported by the solar spectrum and this is how can we still visualize it with this plot where we have our n-photon because essentially p-sun is the area below the old curve because you imagine that let's say I have an idea semiconductor which takes this photon this given band gap will absorb this one then I can add another semiconductor which absorbs the other bit and I will have this one and so on so the area behind behind this curve is essentially the total power available from the Sun radiation so essentially our efficiency will be the ratio between this rectangular plot which is a P photon and the area below this curve which is P Sun okay so if you make this so and this is just a plot of what we were saying so we are considering this ideal case where our solar cell as about an output bias of equal to the energy gap in a more realistic case where essentially you get less energy because you never reach this flat band condition you should consider let's say this curve and this curve but I mean for our purposes let's stick to the idea case so for example you can get let's say a curve like this one where you have depending on the energy gap of the semiconductor the maximum efficiency you can get with a cell made with a single semiconductor so we see that the ideal material would be a material with an energy gap around 1.5 okay this will give the best efficiency here you see two curves one is c equal one which means essentially we have no concentration so we have no mirror we are just collecting if the cell is one square centimeter we are getting one square centimeter of light of sunlight and this is a concentration of 1000 so where i get essentially a big mirror multiplied by a factor 1000 the solar radiation and i mean having more carrier for some internal mechanism of recombination not only gives more energy, let's say, from the same cell, but also more efficiency. So you are more efficient because you are, let's say, illuminating more intensively your device. So 1.5 is, for example, gallium arsenide or cadmium telluride, this kind of material. They are intensive, of course. Meaning, all the cells we see on our roofs are made with silicon, which is less efficient. so silicon is around 1 point something but of course it's I think it's fourth more abundant element in the earth crust so it's easy to find it's not it's not dangerous if you had some roof getting fire and you have gallium arsenide on top it's a problem because you have arsenic around while with silicon this is less relevant, so this is why one or three use silicon for making solar cell ok, now we know that, for example the idea set is made with gallium arsenide but actually we can also use this plot in a different way so because now I know that I can put together different semiconductor materials so one possibility is doing the following it's making the structure where I have materials with different energy gap so let's say from the large one to a smaller one and I illuminate from this side okay so in our diagram what will happen is the following let's say this first material will absorb only photons with an energy higher than a given value let's say it's this one be more extreme let's say this one and so we know we will get this kind of power out now the second cell will absorb also photon with a lower energy so photons with this energy and so the one with the very large so with the energy larger than the red one will not be absorbed but here I will also get all this energy here so essentially we will get this additional part so some part of the photon will be absorbed by say the red semiconductor but I will add more absorption from the green one and then eventually I have a smaller one absorbing photon like this and this will give me this additional part so eventually my solar cell will cover better the solar spectrum I mean from the physical point of view what happens is that essentially if I had I minimize the losses due to thermalization because a high photon energy which would give me a great loss in the small bandgap material will actually be absorbed by the one with the larger bandgap so we'll have fewer thermalization so you can make this game and calculate efficiency for different number of junctions and you would get a block like this let's say for one band so here we are considering the concentration of the thousand sands you would get 37% which is essentially this value here and the best band gap is around 1.5 now you can put two bands and you will have something like this okay so these are two bands and you get 50% then you can go with the three bands which would be this one and you would get 56% and that means you can go 46 band which is basically technological impossible and you would get to 72% okay with a super high or very high number of gaps you can reach a maximum efficiency which is one minus say the temperature of the earth divided by the temperature of the Sun because whatever we do we need to mean comply with Carnet theorem so our hot source is the Sun is a 6000 Kelvin eventually the cell will dissipate on earth so temperature so this is a maximum efficiency you can get which is of the order of in something like 96% 97 so in reality what happens is that the solar cell I was mentioning before which are using the space application commercial ones are typically triple junctions over cells so you have three materials and these three materials are indicated there so we have a germanium subset which is used as a piece as a wafer then we have indium gallium arsenide and then we have indium gallium phosphide so we can understand why people have chosen this material looking at this table here which is again another version of our chart with band gap and energy gap so germanium is here so it will be the bottom cell the one with the smaller gap so the order of 0.6 electron volt and then when I go up you have essentially gallium arsenide actually to have a better lattice match you add a little bit of indium okay and then you go up again and you have indium gallium phosphide so something like here okay aluminum arsenide we have seen it's in indirect gap semiconductor so it's a better light absorber so it is not used so and in this way you essentially cover back to the solar spectrum in this way so the bottom cell will take the near-infrared path the one in orange small photon energy central cell will take the visible part let's say around it so I think it's more shortwave infrared and near infrared and then the top cell will take the visible and part of the ultraviolet part and so in this way you get to efficiency which as I told you are of the order of typically 28 for space application Okay, so this was to give you an idea of an interesting application of alloy semiconductor, how they are used since the 50s basically for this purposes. Only a few companies in the world produce this kind of cell, also because the demand is not that high it's just used by the aerospace industry and one is actually a few kilometers from here so if you go on the tangentiale est you see CHESI a big building with CHESI written on top CHESI stands for Centro Elettrotecnico Sperimentale Italiano so we do many things in let's say electrotechnical technology and one of the things they do there is a small group of people that are producing this kind of triple junction solar cells so they are grown with a chemical deposition technique and they essentially are used mainly for powering up satellites okay but if we go back to our master plan our idea was to use alloys mainly for forming nanostructures also putting together materials with different band gaps to confine electron holes in a nanometer size region so now let's address this problem what happens at the interface between two semiconductor how do the band align and so this will be our next topic so first of all let's start from what we already know and we discussed a few minutes ago we know that if we put together two semiconductor with the different doping let's say p type semiconductor so where this is a Fermi level and this is a conduction band balanced band and if I put it close to an n-type material we have a built-in potential forming okay so an alignment an electrostatic alignment of the bands done in this way now we want to address and this is what we call an homo junction so homo junction means same material same semiconductor let's say and different doping Now we want to see what happens in the case of an aether junction. So we have now possibly different doping and also different materials, different energy gaps. Okay? So in this case, I will have, let's say, the superposition of two effects. one, let's call it an electrostatic effect which is what I have here also in the case of the homojunction but then I will also have a more complicated problem which is the interface between two different materials actually what the first message I want to give you is that these two problems can be to a great extent completely decoupled so I can consider them in a separate way and the reason is the following so when I form a homojunction so when I look at the electrostatic effect the band variation can cover a distance which goes from, I don't know 10 nanometer if I have a super heavily doped p-n junction to several micrometers ok, so this means that our change in the band diagram takes place over a distance which is several times the unit cell instead if we think about what's happening and we try to prove to you experimentally when I put together together two semiconductors. Essentially, in a very naive approach, but a perfectly correct one, I can say that I have a material A on one side, or one and two, as it's called here. And material one on one side, where the atom has a certain atomic potential, and material two, where the atom has a different atomic potential, which or in any case be column-like. And if you make the superposition, like what we did in the tight binding approach of this potential, you will see that the variation from one potential landscape to the other takes place in a few unit cells, in a very small number of unit cells. And so what we will do in the following is essentially try to calculate the band alignment as if we didn't have any electrostatic effect and then we will add the electrostatic effect on top. Let's say electronic engineers they essentially plot diagrams like this where you have this part which is the curvature of the band due to the electrostatic effect and then I have essentially vertical lines indicating the bundle linings this in reality if we would zoom in this line would not be vertical but it will be covering only a few angstrom a few lattice parameter okay so what we will do now for the moment is focus on this so what happens at the interface so the way we will proceed then is essentially the following let's assume that that they have a material like this. And with the method that I will tell you, let's say that we have calculated this kind of band alignment. So we have a strong band offset in the conduction band and the small one in the valence band. And then I decide that this material has this doping and this material as this other doping, then we apply the electrostatic effect and that will add something like this. I will have strong bending, then this, and another part of the bending. Here I will add something like this, my bend offset, and this thing. So the two fermilayers will be aligned. But the two effects can essentially be considered, in most cases, as two separate problems. And for the moment, we will focus on the first one. How do we calculate the band alignment? First of all, I want to show you that what I'm telling you is actually true. So I was showing essentially this cartoon of the atomic orbital and say, if you try to make this calculation and superpose a different atomic potential, you will see that after a few small region, you would get essentially the alignment for semiconductor one and the one for semiconductor two. I would like to show you that this is true experimentally using this paper from a few years ago where actually you can visualize the transition from one material to the other. So this was done in a special class of material which allowed the experimental in this. So what we have here is a diagram shown on the left part. We have N gallium arsenide, so gallium arsenide which is doped n-type. And then we have a layer of 5 nanometers of gadolinium oxide, which is essentially an oxide so that you can take it as a very wide bandgap semiconductor. and on top you have amorphous gallium arsenide, so A stands for amorphous, it's essentially a layer made to protect the surface of gadolinium oxide. The beauty of this, of 3-5s, is that if you cleave your material, so if you scratch your sample and then break it, the cleavage plane will be very sharp and you will expose atomically flat plane because you have certain direction which are typically the 1 0 0 in in gallium arsenide where you expose perfectly flat plane this is also a feature that allowed the fabrication of the first 3 5 laser so the mirrors in the first 3 5 laser were obtained simply by cleaning your sample because you obtain a layer which is atomically sharper so a very flat mirror in group four it's not so true when you get always some more complicated facetings so what these people did so they grew the sample growing it in this direction so multi-day structure and then they cleave it bend it down and did the scanning tunneling microscopy on the cleaved face so this is a cross-section and you have this cleaved face and to prove that what I was telling is true that you have super sharp cleavage plane there is this zoom here which is a made with atomic resolution scanning tunneling microscopy and you see the single atoms just to show you that this face it's I'm exposing is really close to idea okay so now that we do this this microscopy step we also can do scanning tunneling spectroscopy so in the microscopy version I do the following let's say that I have my potential on the surface due to the atoms which is like this I go with the tip I apply some bias between the tip and the sample and at some point I will have some tunneling current okay and the tunneling is so dependent on the distance that essentially the tunneling takes place between the last atom on the surface and the last atom on the tip this is what gives me atomic resolution okay and depending on the atom I'm very sensitive to the distance so if I move a bit farther away the current will drop to zero if I'm too close the current will increase so in the microscopy mode I keep this current constant so I have the same distance always between my tip my last atom of the tip and the last atom of the sample and move around maintaining the current constant so we'll have something like this okay and I have a feedback loop controlling my piezo positional in such so the position is moved to maintain the current constant and in this way I recover the topographic information so I recover for example this image here okay the other way of doing it is actually sitting on say one position on the surface okay at a fixed distance and changing the bias okay between the tip and the sample and so in a semiconductor I can have actually two different configuration which are shown here so in this graph what do you see so S is a sample T is the tip and this is essentially the applied bias okay and so the tip is a metal where I have essentially electron up to here and in this graph let's say the sample will be a semiconductor so we'll have some state here and then other states further up so if I apply positive bias between the tip and the sample I will inject electron from the tip to the available state in the sample if I do the opposite if the bias is negative I will inject sample from the essentially sample to the tip so this essentially means that if I have a positive bias in a semiconductor I am probing the conduction band okay because the valence band will be filled and so there is no place for this electron to to enter in the semiconductor so I'm looking at the states of the conduction brain if I do the opposite if I have a negative bias I am essentially probing the valence band where I have electrons and put in this here okay so what you see here this different color coded plot on the left hand on your right hand you You have the sample bias positive or negative and to actually see this current better people instead of plotting the current itself plus the derivative plots the derivative of the current and so what we see here that we have mean lines with different colors which corresponds to this different position in the sample okay so the gray color here corresponds to the gallium arsenide so where I am basically I'm looking in the bulk of the material and far from the interface and so you see that you have a region where the current is high this one which means that you are probing the valence band then this derivative gets flat and then it gets high again so essentially this flat part corresponds to the energy gap of gallium arsenide so when you have essentially this derivative flat it means that the current is not changing you are not adding any any electron holes anymore this curve has been translated for the sake of to have it more clear. On the other side I have this black line and this is a gadolinium oxide so you see the same effect but this time the energy gap is much larger because it's an insulator. So what we are interested in is what's happening in the middle and this is this color plot so the A red line is taken here then we have the orange and so on so we are getting closer and closer to this interface to see this better essentially you take this plot and you reshape it in this way, so this is the distance away from the interface this is the sample bias and this is, let's say the current here is written as the density of state, so essentially say the red line here corresponds to something drawn in this way where you have a high intensity then zero and then low intensity essentially in this way you are essentially seen with your own eyes the electron state distribution between these two semiconductors so you will see a left part where you have of gallium arsenide with a certain band gap of around 1.5 and the right white part which is a gadolinium oxide part and you notice that the variation between one state and the other one band gap and the other, it's really taking place in a few angstroms because it's essentially this region here ok so as we kind of guessed adding together different current potential so it's true that we can actually treat the problem in two separate steps so considering only the band alignment which takes place at the atomic scale and then adding the electrostatic effect so unfortunately this is the only, there are few systems where you can do this, where you have such a nice cleavage plane so if you want to understand the band alignment between silicon and germanium you cannot do this kind of experiment so people use a different approach for this and this approach is typically based on making essentially photo emission typically in two different regions with two different techniques so using x-ray photoelectron spectroscopy so XPS and ultraviolet photoelectron spectroscopy so UPS and we go back to this scheme later and so essentially what you can do is the following so let's suppose that I want to measure the the band offset between material X and Y I need essentially three different samples to perform these experiment so in one case I measure sample X so let's call it X bulk and essentially I get by combining the XPS and UPS spectra I get some peak coming from the core level from XPS and some peaks and with UPS essentially I see the valence band so I can basically determine where the Fermi energy is of my system okay and this is done by UPS and so I can combine these two techniques I can measure the difference in energy of these two levels so let's say for material X I can get the energy of what we will call core level which is measured by XPS and the position of my valence band of Fermi level which is measured by ups so the reason why of course mean even with XPS you get information you can get electron from the valence band but typically the resolution is not good enough for our purposes so in the say most of the work you use two different they've seen recently that let's say instrumentation for making XPS has kind of reach higher resolution so you can use one single tool but the basic idea for what we are saying now is that in this way I can measure the distance between these to energy level okay then i can do the same for the bulk y so also here i will have a core level and the valence band level that i can measure and the third sample is essentially a sample where I do I form my aether junction so I deposit say a layer of Y on top of X or the other way around it's the same and this layer so let's call it epilayer must be thin enough that when I do my XPS measurements I can get signal both from this material here and also this material here so now what will happen when I put together the two semiconductor is essentially I will have a charge redistribution in this region so you can imagine that I have a dipole for example let's say if I have more charge in one way than the other and this charge will shift the energy level of the two systems in a rigid way and so essentially what I will do when I form my heterojunction and put together the x and y system sorry my x and y system so this is the valence band and this is the core level I can measure very precisely the difference between these two core levels so let's call it delta core level this is measured in the yx sample ok and then I retrieve from the measurements of the y sample the difference between the valence band and the core level so essentially this delta VB core level is measured in sample Y and also here delta VB core level is measured in sample X and then I measure the difference between the two core levels making the x, y and x sample what is the benefit of proceeding like this is that core level have very sharp transition energy so they have sharp peaks in XPS spectra and so the accuracy that you can get is very high if you would measure the valence band of the mixed sample you would have a superposition of the two valence bands so we'd have a superposition of something like this and something like this and it would be extremely impossible essentially to measure this difference so in this way you get all the information so everything is summarized in this plot here which is actually more complicated than necessary, there's many indications but essentially you see that you have a core level of one material and the valence band the collateral of the other material and the balance band. Once you have calculated the balance band alignment, if you know the band gate, you can get the conduction band alignment. So let's make one example. Actually, you will see that if you look at the exams of past year, I mean, every year you find one or two times this kind of exercise. So it's very, you can try to already now do this kind of work. Okay. So this is one example I also gave as an exam a few years ago. so it's a paper from not long ago where people deposited lead telluride on indium antimony okay and they report this experimental work where they say which is a band alignment between these two semiconductors so this is one of the spectra the spectra of the bulk in human timonite actually you see that there are two spectra because essentially they these authors pointed out say a cleaning problem so in the black curve there was some say carbon contamination that was shifting the core level also and so they did some annealing treatment, cleaning treatment and they get the after spectrum which is a red one which is the reliable one where you don't have this carbon contamination okay so in this peak you can see some relevant core level so this inset Is essentially a zoom Or this part here around 500 and something Okay, so we are basically this peak here. And so you can see this this D states Of the antimony atom in our in human team right and then they do the UPS spectrum and so they locate the position of the valence band and you can accurately measure the difference between this level okay again in book in embarking the on it. Now you do the same in bulk lead telluride and again you have some XPS part where you use this D peak of lead and the valence band so you get essentially this number and eventually you make your lit telluride indium antimonide sample and you just look at the core level here there was some mistake in the in the manuscript so they were measuring the four level and then they indicated the frame in 3d 5 you don't have a this kind of orbital you need at least four and you maybe not in any case it's wrong here because your way indicates the deep and so you get this distance between the core level and eventually you build up your diagram so to summarize once more this is obtained from the lead telluride indium antimonide sample this is obtained from the indium antimonide sample and this is obtained from the lead telluride sample so you make that with this very simple geometric relation and you get the valence band offset between the two then if you want the conduction band offset you will need to know the energy gap you add them up and you also get the conduction band offset. So this way we get a band alignment of this kind. So for this material we had this kind of band alignment which is called type III where essentially the valence band of one material is in contact with the conduction band of the other material and these are the other two cases that can actually happen and they are essentially with very very little imagination they are typically called type 1 type 2 and type and the difference is essentially they are divided in depending on where the carrier stay in this heterojunction so in the type 1 band alignment both electrons and holes will stay in the same material in the type 2 electron will stay on one side and hole will stay on the other side in the type 3 I will have this electron and the electron of one material can tunnel in the valence band of the other or vice versa okay so this is as far as experiments are concerned is it possible to calculate the the band alignment in a simple way so one possible approach is this one which is called Anderson rule and essentially the idea is I have a material with an energy gap the material with a different energy gap and they will have some electron affinity so electron affinity key is the difference between the conduction band the bottom of the conduction band and the vacuum level and if they share the same vacuum level I can easily calculate the conduction band as the difference between these two electron affinity if you make a check with experimental data and we plot on one axis the electron affinity and on the other axis the conduction band offset we see that we have one good data point which is aluminium gallium arsenide and this was the system where people worked with for a decade at least because it was the first three five to be synthesized and as I told you we have this perfect almost perfect lattice matches easy to produce but essentially it's the only point on this line maybe this one gallium arsenide and germanium everything else is scattered all around okay and actually there is no real reason why this should work because we know that i mean the vacuum level is not a common reference level for all material because whenever you have an interface you have some charge accumulation and this will change your vacuum in fact in also in common elements for example if I have a tungsten depending on the surface of time thing I'm making I'm considering I have different vacuum level let's say so the alignment of the vacuum level is something which is not really ever really physical ground and so we will see in the next lecture how we can address this problem in a more complicated way so before actually finishing today lecture I will just would like to go back to something we discussed a few lectures ago. Let me share this. Thank you. Okay, so in the lectures on the spin-orbit interaction, I wrote down this Klebsch-Gordan transformation. And every year I give this lecture, I'm very unsatisfied because I'm not able, I mean, I wasn't able to actually derive them. So I annoyed a few people in the department, like Professor Cicacci, Andrea Piccone, and so on. And we tried to solve this problem in a, let's say, mixed way, with some theory, some good theory input, and also numerically. so now I have updated the nodes and we decided to take let's say a bit brute force approach with matrix calculation of this problem so you remember we had this L dot S spin orbit so this you can write it down as the tensor product of the L matrices for the orbital angular momentum and of the Pauli matrices of spin angular momentum. And so you can calculate this Hamiltonian using this, adding up this tensorial product, and you get this kind of Hamiltonian. Here you find the eigenvector, and the eigenvector are expressed essentially in this base, which is also the tensorial product between the different orbitals and the spin up and down and eventually these are the eigenvalue that you get in this space where you have orbitals, I mean spherical harmonics and spin up and down and then you simply replace This spherical harmonics with the PXP wire presentation which is better to describe selection rules because we have well oriented orbitals So you can essentially look at this text and see if everything is clear enough and of course We did I mean I didn't do all this calculation. Let's say By hand okay, so you can find another, let's say, Python notebook, where you can use SymPy, which is a symbolic toolbox, and you can essentially define matrices and do symbolic calculation. So all these inverted matrices which I would not be able to do by hand by any means, it's done in a matter of minutes and so you basically relieve the whole problem of making this paperwork. And here you can basically run this code and find the Hamiltonian and so on. And then you can also see if our eigenvectors are orthogonal or not as one projects on the other. So next student here will profit about this. Now I'm a bit late. I didn't do it on time, but I think you can still, you can download the new version of the booklet and also you will find a new code you can play with to understand better this spin-orbit interaction. So for today, that's all. We meet next Monday.