isella.txt

Okay, so in the past lectures we introduced the K.P method. The starting point is a generic formulation of the equation for crystals, where we simply use it as a solution function which is compliant with the block theorem. and then essentially if we decompose this p squared term which enters in the kinetic energy as two times the application of the momentum to a mean but in this easy mathematical procedure we get to this equation which contains only the periodic part of the block function And so, starting from this point, what is done in the K-dot-P method is assuming that K is a perturbation, so it's not actually a physical perturbation, it's more a mathematical perturbation. And so we assume that we know K equals zero, which means essentially we know the energies and the wave function of the different bands at K equals zero. and based on these quantities, so suppose we know u0 and e0, we build up our solution also for k different from 0. So from a visual point of view, essentially I suppose I know all these values and the different wave function for k equals 0. so whenever I want to find the score of k different from zero, I will represent it as a superposition of the different wave functions at k equals zero. So as a didactic tool, we imagine that we can describe a semiconductor simply using two bands. So this is k, this is the energy, as before there are two states, one is just half of the energy get below the zero, the other one is half of all this, just to make the computation be easier and we assume that this state here will have an s character and the state here will have a p character in the default. So our solution will be looked for like a combination of S character and of the P character state. So in the last lecture we beat up the matrix, the representation of the San-Victonian. So we just copy down here, so this is s8, 8, s8, 8. and what you obtain is the following the energy gap divided by 2 plus this is essentially the theta of p-therm that we express in terms of the PCV matrix element So essentially what we are saying is that we have this eigenvalue eigenvector problem where eigenvector components are the two coefficients we are looking for and k will be the various version of the two states. So we have two values for k, two values corresponding to the eigenvector. So first step is actually to find this value of E, one structure, essentially I need to set zero, that we know. This is minus E. I saw that this equation I will find the value of k. So we can write down product of the E times the diagonal this way. So essentially here when you do this multiplication you recognize that we have a product of a plus B times a minus B so it is relatively easy to solve and then this will be equal to the square of this okay so now we can essentially write down Terminal. But with plus minus. Square root of the formula. Ok. So we continue with some depth. so eventually we get energy gap of this here I can take out of the square root gap divided by 2 So I get 1 plus... Ok? So, the calculation will be even more complicated than the simple model we are using here. Instead of using PCD, the matrix element, we have some error. Instead of using these PCD terms, we use an energy. So we convert this into an energy, which is called k-energy-EV. and is defined essentially as two times times where divided by m0. So if you make this, you replace this expression, you actually get equation here. multiply by this. Okay? So now we can again make a look at what happens around the gamma point. As already pointed out in the past lectures so the higher is the number of state I put in my model the further away I can go in the three-dimensional state. You want a full description of the theoretical energy of the ends to have a full description. So here we have only two. So let's try to describe our system only in a whole region around gamma. So let's make the approximation that k is almost zero. So our square root here looks like something like this. One plus some quantity x and this quantity is approaching zero. So if I expand this series I get something like this. One plus one of x. Where x is always that way. So let's see this approximation. Now we get to this question here. And so eventually making the application. And as a final step, I can collect this homophobe here at 1+. plus minus 1 plus so essentially my structure will have two solutions so one is the one with the plus sign the other one This one is one with a minus sign. For the upper bend, I will have this kind of behavior. She is so positive curvature, so the bend will go upward. for the minus sign I will get downward parabola so even though the model is extremely simple I get the right version of two beds I'm sorry. Let me check. There is some... This is wrong and it's actually like this. sorry because I was actually putting the plus minus sign from this term which doesn't help essentially if you collect this common factor you should end up with something this kind so we can actually notice that E p in common semiconductor is typically a value which is which is between 25 and 20 eV so it's much larger than the energy gap so essentially we can forget this line here And we get to an event that is discussed. So the result before now is here that we have one bend, a positive curvature and one with an angle. So the results are summarized here, where you have the determinant and the plot of the two Bell dispersion taking the typical value of E p equal to 20 of the energy gap equal to one electron volt more or less. So the continuous line is essentially a dispersion without parabolic approximation where we still have a square root and the dashed line is is a parabolic approximation of the final expression we obtained where essentially this this dashed line here corresponds to this. Okay? So we see that, as expected, the parabolic approximation is a good approximation only around the gamma point. otherwise the dispersion is not necessarily parabolic. So from our calculation, we can also get some interesting conclusions and point out something that I've tried to be careful of, take care of during my derivation. From the first equation to the last one, I have always wrote here m0, indicating that this, all the mass we used so far, was the free electron mass, so it's not the effective mass of the electron mass. This is the one you would measure for an electron at rest. So now we can essentially see that this dispersion helps us in defining the effective mass of our system. We expect that 1 over m star, the effective mass of the electron inside the crystal, is actually 1 divided by m0, dp divided by the energy gap. So essentially we have that m star is expected to be proportional to the energy gap. So the smaller the energy gap, the smaller is the effective mass. So this is something that we pointed out in our, let's say, Python calculation of the band structure. If you remember, we looked at group 4 elements, so the effective mass of diamond, silicon, and germanium at the gamma point. Or was it, I don't remember which example we did, but it was some gamma point effective mass. and we notice this, say by eye, that the curvature was getting smaller and smaller, or the effective mass was getting smaller and smaller as the energy gap is reduced. Now here with the k2p we also recover this result. The next step we can do, now that we know the eigenvector, it's finding the eigenvalue so I can replace this expression here the structure I obtained with the plus sign for example and now we get two values of a s and a p so the two components of the eigenvector then I can replace the value of the minus sign and I can get two values of components of S and P in the conduction band. If you do this, if you do this, you get this kind of expression. so I did it again with a symbolic coding like Mathematica or Python is also something similar and this is the plot of the different value of A, S and AP I get. This is A S and A P squared for the conduction band So the bend is an output dispersion, the bend is a positive sign of our bend structure. What we notice in the session is that if I m k equals 0, the state is fully a s. And this was our original hypothesis. So we started with the idea that k equals 0, the character of the state was s-type, even state. So our starting point. And the opposite, of course, happened in the valence band. So we started with the idea that k equals 0, we had a p-type state. So at the beginning, we had 1. Probability of finding this electron in a p-state was 1. As I move away from k equal 0 outside the gamma point, you see that I have what is called band mixing. So, a generic state with a certain value of k possesses both characters. So, of course, it will remain mainly a s-type, but it will also have a contribution of p-type. Of course, if you make an addition of these two terms, you need to get one. So the probability of the electron to exist in some of the two states. And of course, the opposite happens when I consider the balance. As I move away from the gamma point, I will maintain predominantly B-type character, but I will also have some S component. So this phenomenon in Kp is called band mixing. So k gamma, k equals zero, the states have well-defined character, but as I move away from gamma, I mix up the different bands, and this is actually the approach used by Kp. So I just wanted to go back to this parabolic approximation. So again, this is what we obtain and we have that the effective mass is essentially proportional to this value, where that is the energy gap. and what you have here is a plot of experimental data of the effective mass of electron on one y-axis and of the bandgap on the right axis for different semiconductors. And we see that this linear behavior is perfectly followed by a one-gap semiconductor and not perfectly but this is followed in a satisfactory way by high band gap semiconductor. So we can ask ourselves two questions. The first one is why this is true for the electrons? Why I plotted the electron data and not the whole data, not the valence band? And the reason is that while this description of a single s-state of the conduction band is pretty close to reality, let's say, so because all the electrons in S state are made by a single S band. The approximation of the balance band with a single p-type state is very bad because we have learned that we have actually three different bands, heavy, light, tall, and split off, each one with a given dispersion. So this is why if you would know the same plot using the host, the active mass would just get values scattered around this plane and not a well-defined line. Second point is why is this working so nicely with more band gap semiconductor, and why it's working less nicely with wide band gap semiconductor. Because we need to remember that this is still a perturbation approach. So whenever... So this is still a perturbational approach. So if I want to describe the evolution of this state here, in this, the contribution will be given mainly of the states which are close in energy. You remember in equation theory, you always add that the weight of a different contribution of the state is over the difference between the energy of the state I'm considering and the perturbed one. So it is clear that if I want to describe this state here, if I have a small band with a semiconductor, this state will contribute a lot to describing this one, while if I consider a wide band with a semiconductor, this state will contribute less, so I will need to add more state, so there can be states in the conduction band which are closer. So essentially, the small bandgap semiconductor is closer to this condition, where the mass contribution is given by a balanced band state and not by other conduction band states. While if I have a large band gap, I will have a contribution in my perturbation on the states which are in the conduction band, which are closer in energy. an extreme case of this dependence of the energy gap on the mass is actually Rafini where essentially the gap goes to zero and the effective mass goes to infinity this is a bit of a not perfectly fitting example because in graphene the dispersion is not quadratic it's linear but in general we can think about graphene as an application of this general rule that we find in semiconductor that the smaller is the bending and the smaller is the effective mass ok ok so this is how the general principle of of k dot p in a toy model which is this one with S and P stable. More realistic one is the so-called 8-band K dot P, where the 8 bands are essentially indicated here. You see now we have a matrix with 8 columns and 8 rows. And if you look at this, how these columns are labeled, you will recognize S up is a conduction band state with spin up, S down the conduction band state with spin down and this labeling indicates the J and JZ angular momenta of the conduction band. So V and a half, V and a half is what we call the heavy hold, V and a half, one half is the light hold, and one half, one half is the split off. we have two versions, the one with J pointing up and the one with J pointing down. But the working principle is the same. So I solve this matrix and I can find the band dispersion and also the mixing of the different states in this case. again matrixes are relatively easy to solve using a simple computer so for example we can do the following I can write down the code where I have the relevant quantities. This is the case of the direct gap of germanium. The energy gap, this is with temperature dependence, but we don't need this now. Different matrix elements entering in our 8x8 matrix. Of course here we not just have PCT, we have a few more parameters. I have defined a few directions I want to calculate. and this is essentially our matrix that I want to solve. And the output of this function which is called k dot p 8 by 8, when I put in the direction, this is the strain that we see. So I want to see the dispersion of this direction, and I will use the p-value, using for the calculation, the eigenvalue and the eigenvector. So the different components of the different eight states I'm considering in my calculation. Then of course I can take different energy bands and plot them. There will be doubly degenerate, because I will have a band for spin up and spin down, or J up and J down, which are essentially identical. So I just plotted four bands out of the eight I get from the calculation. And if we do this from different directions, we get something like this, where we recognize the S-state conduction band, The heavy hole, the light hole and the split-off. So these are different directions along in the neocypical space and so you can see that the version is not spherical. We have different special in-hole bands, for example. The heavy hole is extremely heavy along the 0-0-1. It's a bit less heavy along the 1-1-1 because this energy goes down further. the light hole is actually light close to gamma but then it can kind of parallel the heavy hole and also you see it off the o1 direction as this dispersion after 10 percent of the the freeway zone and much stronger one along the 111. So you can recover the dispersion with a specific symmetry. You can also take then the vector, so the different components of the fix-up state and calculate them for example and plot them. Actually we can plot square of these values because we know they need to adapt to one, the probability for the state exists at all. If we do it we get a plot like this one. Okay, so these are eigenvectors with spin up, they don't mix with the one with spin or say JZ down, so if you do the same with JZ down you get just an identical plot. So and this is the conduction band, so we see that at gamma equals zero this state is a pure S state as before in our case because this is initial hypothesis of K of P that at the gamma point we have this value. And as I move further away in the Green-Wen zone along the OO1 direction in this case, it mixes up with a light hole, split off and doesn't mix with the heavy hole. The heavy hole stays to zero. And again, you add up these values you always get one. You have to get one. If we go and now look at the heavy hole, now these plots are obtained in the over one direction, which is the one we choose to project. It's the Z direction where we projected the angular momentum. So in this direction in the reciprocal space, essentially you don't have any mix. So the states are always purely O . If you go to the light-hole band, of course again in the beginning I need to have pure light-hole state, and then as I move farther away, the mix-up again with the conduction band and the split-off and the split-off mix up with the light hole and the conduction band and you see the area will stay completely separated from all the other states in this particular direction. So if I move and look at another direction in a simple curve space, for example the 111 I would get a different behavior course I mean here is it missing point which is the one exactly a zero where you get one the other direction of a different mix of my representation of the states of course depends on the direction of quantization I've chosen okay so these are some example of These are some examples of calculations plotted in a larger region of reciprocal space. Before we were essentially looking at the kernel, we are going further away. and for comparison in these two plots I have kept the over one direction on the right end and in one case we have the and in the other the . Looking at these plots we can also try to guess how the isoenergy surfaces would look like. So we see that in the conduction band, we have essentially the same dispersion in every direction. So this is not parabolic as I move away from gamma, but it's more or less the same whichever direction I look. So if I make an isoenergy cut in my band structure, I should get more or less spherical behavior. Instead, if I look at the heavy hole, essentially I have a very non-isotropic dispersion, so in three directions I have very different behavior, and what we can notice here is essentially that the expression along the OO1 direction corresponds more or less to this direction here. So we see that actually in this direction the modes are relatively light, the radius of curvature in this mode, while for example in the 110 direction the radius of curvature is relatively large and this corresponds essentially to this direction here. Same energy, minus 2 let's say, over a longer path in reciprocal space and so this means that they have a heavier mass. The 1 over 1 direction is something in between. cannot see it in this plot, it would be out of the spring, something in between. This is an example of a 40-bent k.p band structure where you can plot all the band structure essentially and but this is also I will maybe show you after break it is also something which if you have the code it's solved in a matter of seconds because it's again solving the eigenvector and the eigenvalue of a matrix okay so I think it's a good time to make a break because then we switch the arguments So let's make it like a bit earlier. So in the first two weeks of lectures, we have seen two main methods for calculating the structure. One is a tight binding. The other one is a k-to-p. And before leaving the topic of the lecture, quantity that you have seen several times in your studies, which is effective mass. But typically, most applications are just considered as a number, but you put in place of an electron mass to take into account, for example, the use of external forces or to calculate the density of state. What I would like to show you today, we have partially seen that actually this quantity is typically anisotropic because the dispersion is anisotropic so we will move from the concept of effective mass as a single as a scalar quantity to actually the effective mass tensor then we will also look at the how we can treat the effective mass tensor in multivalent semiconductors so when we have degeneracy so as you have seen that this is the case in silicon, in germanium and on and actually that depending of the physical problem you are addressing you might have a different combination of the components of this effective mass tensor for one case you describe conduction in a semiconductor or for example to describe the density of state. And then we will also, even though not, let's say, in a quantitative way, look a bit better at the whole effective mass dispersion in the balance plane. So let's start from a mathematical introduction of the concept of effective mass. So for a bit more general, let's consider a semiconductor which has a minimum which is not in the gamma point, but it's at some point K0 in the free-wire zone. For example, it can be the delta valley of silicon. So what you can do is that you can essentially express the energy dispersion around this point as making a failure fraction of the massages. we have value of the energy gap exactly k0 then in principle we add some linear term k0 is the x component of kx but since we are considering what's happening around the minimum of our band structure we know that these linear terms are zero because essentially this is the minimum even by the derivative equal to zero so this linear term can be neglected and so we end up with second order terms, where we have second derivative of the energy around kx, and so we have kx 0x and we will have the same for ky, ky, 0y, 0m, sorry, and also the same for kz. In principle, we also have mixed terms where we have derivative between kx and ky, but if we choose x, y, and z as the principal axis of our crystal, it can be shown that these terms are virtually zero. So we are only left with these terms. So in this expression, if we compare, for example, this expression here with the kinetic energy of three electrons, you can essentially define the effective mass simply as the 1 over h plus square double derivative can structure along the echelon direction. So this will be the effective mass along the x direction, but now we have one along the y direction and one along the z direction. we can essentially write down our energy dispersion around this filament at zero, this way. you Okay, so in a cubic crystal essentially I will need three effective masses to describe the dispersion of my system. So I don't need actually tensors, but only three diagonal terms are relevant. more complicated non-qubit structure I can have other terms or if I choose my axis in a odd way.