Simulating the electronic properties of semiconductor nanostructures
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Marquardt, Oliver
Semiconductor heterostructures represent key ingredients for application in novel light emitters and detectors, energy harvesting, or quantum technology. Numerical simulations of the optoelectronic properties of heterostructures such as thin films, nanowires, and quantum dots facilitate both a detailed and systematic understanding of observations from experiment as well as theory-guided design of nanostructures such that they fulfill the requirements of a specific application. We will provide an overview of our modelling capabilities of the electronic properties of semiconductor heterostructures using continuum-based multiband k.p models, implemented within the plane-wave framework of the SPHInX library. We illustrate the applicability of our approach by showcasing some recent example studies. 1: sxrepo.mpie.de 2: S. Boeck et al., Computer Phys. Commun. 182, 543 (2011) 3: O. Marquardt, Comp. Mat. Sci. 194, 110318
 Okay, yeah, thank you very much for the introduction. As I had almost no idea who would be in the audience I tried to keep this as kind of a general overview talk. I may, if we jump into details at some point I may decide to dump some of the examples, but we'll see just let me know once I run into critical time issues. So, talking about semiconductor nanostructures, we have a whole field of different nanostructures that are of technological interest that can be thin films that are used for efficient light emission and detection, ranging from infrared to ultraviolet spectrum. People have been using quantum dots as artificial atoms due to their well defined energy spectra, potentially suited for single photon emission with application in light emitters quantum computing or cryptography. And people are also interested in nano wires as you can see here on this figure, due to their free site facets that facilitate elastic relaxation have a large surface to volume ratio and show some very unique material properties that you would not observe in a planar or a bulk system. And that said there is a number of very different systems of interest and correspondingly, we need simulation tools that are both reliable and efficient. They come with a high accuracy should be of course, once we are doing numerics computationally efficient robust, easy to use always something of importance. And if I can close that my own window down here I have no idea about that because it's part of the screen missing now. And they should come with a manageable number of parameters and output data to make sure we don't run into some stuff that is a mirror parameter fitting just to to experiment. So, basically we can distinguish these models into two fundamental approaches one is a stick modeling like empirical tight finding the empirical pseudo potential method or density functional theory. And the other is the same as the other approaches or atomic approaches come with single atom precision they are extremely highly flexible of course for any kind of mono structure that you may be interested in. But on the other hand, it can be very difficult to set up these simulations, and it is at hand that these these models become computationally more and more demanding once we go for larger structures. And so, these are the most continuous models, and the more generalized version of K dot P models that I will be talking about in a minute. These models neglect the underlying atomistic system they are also highly flexible. And the clear plus is that the computational effort is independent of the number of atoms in the system simply because there are no atoms it's basically the bulk materials that we have and you can already estimate that we are talking about a few tens of millions of atoms in such a system and that's clearly. It's not totally beyond the capabilities of atomistic approaches but it's something that you don't want to do on everyday basis. So, the basic idea of effective mass and K dot P models is what you want to know is the energy and the wave function of electrons and holds in some crystal or a structure. We basically start with the Schrodinger equation here and we take we as a periodic potential with the periodicity of the crystal letters. And then we can basically split the wave function using blocks, theorem into some highly oscillating plane wave and periodic function. And that basically allows us to rephrase the a perturbative Hamiltonian, which is this. And then we will just basically the perturbation theory to this method that we are using here. And then there's a second perturbation coming in namely the fact that we limit ourselves to a number of bands that are of interest, for instance here for example of a guy awesome I bought bands that one could take only the bottom conduction bands and the top three way lens bands because they're basically close to each other, which is the idea of the eight band K dot P models so basically we take these four bands with this been up and spin down component, which gives us eight bands in total. And then the second approach is that we say okay, we have a perturbation in case so a perturbation means this approach is valid only for the vicinity around the gamma point or technically speaking also around every other high symmetry point that is chosen for the development model but typically limit themselves to gamma point. And let me note that this stage that is well possible to extend the region where the model as well as to larger and larger regions of the region zone by taking more bands into account. This is what people do. But basically what you will see mostly is this so called eight band K dot the model, which is limited to the bottom conduction band and the top three way lens bands. So it's such a typical K dot P Hamiltonian looks like. Basically, we have the interaction here from the terms from.학 omega maten, the terms for the action that has like terms and then that is the new terms for Wayland's work relates treaties, or terms copy. So basically our Hamiltonian that that one can use for the description of such a band structure and if we want to do this now not for bulk materials but for heterostructures we want to know where for instance the electron in a heterostructure sitting at what energy it has a hole. So this can be combined with an envelope function approach so we make the effective masses potentials, the coupling parameters which we hear points. You can make one of the station. And what else can enter our contributions like strain that can be easily computed via continuum elasticity theory but also why as an optimistic model if you wish it can always be easily integrated. And also appears up here so electric potentials there may be external potentials or maybe charges that jump in and out and what is at some point, and also exotonic effects can be contained via self consistency. So this eight band K dot P model to date still represents the backbone of modern device, similar to the device modeling. It is computationally inexpensive as a small number of material parameters, most of which are well known. In most cases, not at all. This surprisingly reliable even for small structures despite being limited to small K where you use with correspond to larger structures. So it's straightforward combined with linear elasticity theory to incorporate strain and build an electrostatic potentials. And it's also quite straightforward to incorporate turn or ordinary or alternately alloys or alterations of crystal phases by parameter interpolation. And there are these disadvantages so it neglects the optimistic setup of the head of structure, certain symmetries can be simplified in an eight band K dot P model that would be resolved correctly in for instance type binding model. The limitation to one conduction and three rail and spends can be critical at some point as we will see. And it's inaccurate for K where you send the auto brilliant zone for instance, if we would go for a silicon germanium eight band K dot P as it is might not be the best choice because we basically have an indirect and get material where other well is become important. There are a number of software packages existing, for instance, a type of cat and next nano, which can be used. And this formalism has been applied to a really really wide field of very different materials and heterostructures quantum dots, nano wires nano plate lets you see these publications date back to the late 90s so it's something that has been in use for more than two decades now and still is news. What we are using is a plane wave based implementation. So we have linear electricity and multi band K dot P formalism implemented within the plane wave framework of the existing swing software package. And the neat thing about our approach is that we have basically a wire fast Fourier transformation we have all the real space properties of a heterostructure contained so we don't have to do any assumptions on symmetry and so on. And we can switch from reciprocate to real space and back that allows us to make use of very simple and efficient gradient representations in reciprocate space. And we could make use of highly optimized minimization routines that were already available in the original DFT package swings and the kinetic energy is a very very direct convergence criteria and so we can basically look at energies and our particles at the site whether they are converged or not. And one very big disadvantage we are using a plane wave code and that said we have periodic boundary conditions intrinsic. That's no problem for electronic states but it can be quite severe for this long range of directions like frame and built and potentials and correspondingly at some stage we might need larger super cells for our computations. And what what makes our code I think very unique is that we have generalized it to a wide extent. So we are not limited to any materials or compositions or shapes of the heterostructures or whatsoever, because all this is basically saved in a three dimensional map. We can apply also in the linear elasticity module arbitrary elastic tensors and polarizations which come can come in for instance for certain crystal orientations that are non standard. And that allow you to apply a symmetry suited to the system and not to set up a cell because you somehow have to describe the symmetry in a huge unit cell. And in particular we have a generalized Hamiltonian which allows us to control the level of sophistication so we can basically we can do a single band effective mass models we can do six and eight band K dot D we can go for more bands we have done for his own models for Silicon Germanium for instance, or this can be done. The Hamiltonian is basically something which the user can bring on his own. And we have an open source and we have no investor sometime to make it installable per package manager on most common Linux systems. It seems to work pretty well so far there are from from the users we have there's not too much complaints yet and if there are then we are happy to take care of this. And let me know that we have recently given a software tutorial on the usage of this code in March, April this year. So you can basically imagine the workflow so we start with setting up some three dimensional material map that contains basically the shape of the system and other compositions in the system. And also we need material parameters that contain the primary parameters. So if you're not concerned then we just compute bye a building on the inside in theory, the strain and to the strain. And search on the possess goal what cook through the like best potential. The set of systems the moreReally the set consistency moves of air space in some, some between continuing the business even the business is not contained in the core software. It's one of the most but it's not really on a permanent decision I have to say. And once we know it's mainly going to this moving back area, we're going to take people around one, two, around this next class, then offsets, probably around this. And we had also 50 sprouting, there's two bands and then you can see that the property can be used in set consistency or in the next one properties or some other direction you can also use the larger enemies if you then the optical spectrum of the system. So the results of our simulations are then strain distributions developed in potentials, single particle energies and wave functions, charge densities, possibly excitonic contributions and hopefully in the end some optical spectrum. There's a list of example studies that we have done with our code. I will limit myself to a few of them within the next minutes to just give you a flavor of what can be done with our software and what we are also very open for for anyone who's interested for some other information. So first example would be this quaternary alloy indium arsenide antimony phosphate, graded composition quantum dots grown by a colleague from from Yerevan using liquid trace ecotexy so that's a very, very ancient method but surprisingly they see some in fact 3D hetero structures applications would be in particular in the infrared for energy harvesting and gas sensing and so on. And what is very interesting in this LPE grown systems that they exhibit such a composition profile so basically you will see the amount of content decreasing and the amount of D increasing over such a what would be looking like a cone. And that's something very interesting which does typically not occur in the typical MBE or MOCVD systems. We have some information also thanks to collaboration with Institute for Crystal Girls and the Institute Berlin. We have some information on the height and diameter distribution of our systems. And our question was now can we provide a reliable model for such nano structures for theory guided design process so can we tell our experimentalists to make a system smaller, larger, do something to them, to get them into some certain range that is more of interest for you. And basically what we started up with was a single particle calculation for this relevant range of heights and diameters that they have seen. So we have basically set up such a conical shaped system. Used an 8-band K.p model for simple and semiconductor. It's pretty much the Hamiltonian that you have seen before with some simpler symmetries. In fact, so the inside here shows you basically how the system looks like. The composition range in the US, the whole state, it's a type 2 system so we only expect the whole state to be localized in this system, which would be something suited for some detectors for instance. So there are no exotonic properties here because the electron is kind of far away. We also applied some gap corrections to our transition energies that we have computed here for different heights of the system and for different diameters of these environments. This is basically what experiments have seen. And then we have simulated an ensemble spectrum by basically taking the diameter distribution from experiment and plugging this onto our energies for the correspondingly computed systems. And we have seen now this is really kind of disappointing that this picture is here because the emission peak that we have simulated was at 3.829 microns, whereas the one on experiment was at 3.83 so I would consider that quite good, surprisingly good agreement at that point. Really you have to believe it's really 3.829. So it's really close. So next example would be on the influence of random dopant fluctuations in three nitride nanowires. So the advantages of nitride nanowires is that in particular of indium-gallium nitrides is that they would potentially allow to access the whole visible spectrum by only adjusting the indium content of these insertions here but theoretically and another advantage is that they have a very large surface volume ratio which is perfect for light emission. The problem is, as I said, theoretically because what happens if you plug in more and more indium is that you increase the strain, you increase built-in fields and basically it's quite difficult to get light emission from indium-gallium nitride heterostructures and gallium nitride with high indium contents. The basic charge confining mechanism in such a system, of course, the bulk bend offsets, polarization potential that occur and there's a very, very large potential barrier at the site facets of such a wire. So order of a few electron volts, the work function and the activation energy plus band cap for the pulse rate. There is significant elastic relaxation that's very unique in these wires because they can relax their space for relaxation here. And what also happens is that you are always close to some site fesset to some surface and that means there are surface potentials emerging from ionized dopants that are of interest for us. What else happens is that you observe sometimes alterations of the crystal phase and other effects for very thin wires that come into account might be dielectric confinement because you have a very, very steep change of the dielectric of the material and the outside which would typically be a vacuum. So what we had a look at and we'll have a look at now is the effect of surface potentials that emerge from ionized dopants. So what our colleagues in the experiment have observed is that with higher indium content in the active layer of such a nano wire you increase the photo luminescence intensity. That's pretty much the opposite of what would happen in a planar system where more indium content means higher defect density, means stronger potentials that pull electrons and holds away from each other. But what they have observed is higher indium content and also larger thickness of this disc and they see more intensity but something quite unusual. We could basically answer this in a first approach looking at surface potentials that we assume okay, there is a number of dopants in the system, ionized dopants we have assumed first that they are homogeneously distributed and that just rise to a surface potential. The surface potential is attractive for the whole states, for the blue ones here at the side. So what happens is we basically get an interplay between the surface potential and the polarization potential which would increase the indium content and puts the whole closer and closer back to the center of the axis and therefore closer and closer into the electron keeping in mind that these things are to scale so the diameter is about 80 nano and this is about 1 to 10 nano meters that we have taken into account here. And this interplay between polarization and surface potential has in fact been useful to explain the reduction of the photo luminescence intensity that my colleagues have observed here for smaller indium contents or layer thicknesses. Now the problem is if you assume okay, we have nano wire of 80 nano meters, a dope intensity of maybe 10 to the 17 cubic centimeters and take a segment of 20 nano meters length. We have an average of 8 dopants in the system and that's quite far away from the assumption of a homogeneous doping charge density here. So what one would have to do instead is to consider individual randomly distributed donors in nano wire. Problem is doping is an atomistic feature as we are using a continuum approach here and I told you that there are no atomistic effects in a continuum approach. But on the other hand, so we decided we stick to our continuum model because we have a system that is not available. So we have a system of under consideration has about 7 million atoms. The typical donors that we would observe the unintentional doping that's always basically there are silicon and oxygen and both of them represent shallow donors. shallow donor means that they are well described via simply their coolant potential and what we have here is that the donor means that they basically provide additional potential to the wires. And then we have basically started rolling dice so this is what it looks like. The donor here is very much very much very much of these potential keys here and you see the electron and all are kind of another regime of these donors and then we have started to do some statistics and discuss the ensemble. So we have picked a handful of typical model configurations with some fixed indium contents or some fixed thicknesses of this insertion we have always had the diameter of the wire of 8 nanomers and around too many parameters at the same time. And what we have observed is that the variation of the emission wavelength is surprisingly more or less unaffected by the choice of indium content and thickness of this layer. What's also a side issue is that the energies that we observe are always smaller than what we assume are homogeneous. It's not too surprising that both these distributed donors give us deeper potentials and the transition energy buildup and of course they are also smaller than the dashed line here what we assume that will be in the world. And we have in fact been able to start the line of the randomly distributed donors of approximately 150 milli-electron volts. This is very very bad news for all those people thinking that you could use these systems for producing reproducible light emission because taking exactly the same indium content in yourself taking exactly the same wire diameter with the same layer thickness you still have 150 milli-electron volts of line width due to the fact that you have no idea where your dopant is going. So what else we had a look at the ensemble charge densities here. So the red is the electron again, the blue is the hole, the average hole looks in very good agreement with the assumption of a homogeneous charge but this is not as it looks like so the typical hole you will find will be on these corners of course sitting or six of them but the typical hole is sitting in one of these corners distributed like in the case of the homogeneous doping and in particular the electron localization is governed strongly by the dopant and you see there are very very strong variations we have done here 500 simulations and you see it's not even near converging so that's really the electron is all over the place determined by the electron. So third example that I will talk about is polytypism. Polytypism is something that occurs regularly in wires so you see here this is a gallium arsenide wire you see word side segments in a mostly zinc blender phase wire and you see also zinc and wire phases in mostly word side structures and that basically allows you to form crystal phase nanostructures crystal phase nanostructures so you have a nanostructure without changing the chemical composition of the material these systems are almost three of strength so they are very very well let this match because again it's the same chemical material it's just in a slightly different crystal phase there are of course no alloy fluctuations and what is particularly interesting is that they form atomically flat interfaces that's something highly interesting and the question is how can we model these systems and there are some problems coming with such crystal phase heterostructures so we have here changed to gallium arsenide which is the more interesting material you see on the left side the band structure of zinc blend gallium arsenide and on the right side you see the band structure of wood, like gallium arsenide and there is something nasty happening in the years of gamma-A and additional conduction band coming into play which is energy extremely close to the conduction band which is the equivalent which is typically contained in an 8 band K.p model and to make the situation even worse we have already seen similar plots in the last talk where such a process for a valence ends in small strains and change the character of our production and these strains even though they may not directly appear from this crystal phase lattice mismatch they can appear because typically the wires that you look at are distributed on some surface and therefore they will see interaction by the surface and they will see strain by the surface and therefore plus minus 1% strain can easily happen and so far the question which is the lowest conduction band is not answered and the question how this goes into play is also not answered and the main problem is that the common models our backbone of device model in the 8 band K.p model is limited to only one conduction band so this is simply not contained in the 8 band K.p model the answer to this is quite simple so we take an 8 band model and just add 10 more bands, that's something we can do in our code because basically the Hamiltonian is some input that the user can change and there is no problem between the bands they are technically speaking one that is not a code for them but it's very very small so this is between the 8 band and the top way in this band here it will play a role for optical recombination if there is no other part available which can happen and this band is not existing in the Zincland phrasal and therefore basically is a very very high potential for the inside the wood side phase and the one that has a Zincland has a Zincland wood side they have very different effective masses and the question is what happens if we go to superlattices first or if we move on to wires which I won't present in this talk today and our first step was to take a superlattice of about 14 nanometers length because it's interesting of a Zincland and a wood side segment and the one information that is always through the hole will be combined in wood sides and what is going on behind me it has a very heavy effective mass so it's typically well localized however for the electron what can happen is that it's either confined by this gamma-6 band in the Zincland phase or if I have a very very thin Zincland and a very thick wood side segment it will jump to the wood side phase where it's well confined due to the much heavier effective mass it has in the wood side phase and if this happens or not basically depends on the thickness of the segments and also on the strain state and due to the fact that there is a small but nonzero matrix element between the gamma-8 conduction band and the top wavelength band if there are no other paths available this will be the recombination that will happen at some time with some larger lifetimes and so on but something that can in fact be seen and we found that this can in fact potentially explain very very contradicting measurements that have been observed recently on the effective masses of electrons in the wood side phase of Gaia-Marsonite so it's something which is not really good not really simple to verify correctly but it's some potential explanation okay do I have time for this point or shall I okay okay so as already mentioned in the beginning for my fourth example now which is also part of our portfolio what we can do taking a K.p model the more bands you take the more accurate your band structure gets even to the fact that you can infect your full zone models for silicon germanium and so on so that basically shows you how this K.p model is even effective masses before the A.band is going to use your parabola and of course this case where I have time to spend on us the 6th band model is a in the action of 3 bands the 8th band model will get that description more bands and more infections 14 band model here for instance and you see how this structure gets more and more detailed and we can see that we can see that the band structure is not as common but sometimes not sufficient as we have seen for instance for polytypism for indirect band gap materials silicon germanium also for something which is called bike inversion and isotrotropies so basically taking into account that you have in fact a different symmetry that is not contained in this 8 band model sometimes additional high symmetry points become important for the 8 band model the question is quite simple answer there is something which I would consider the bible as a whole government paper has the emulating K.p models has been became across this work the state of today has 5477 citations brings over a thousand references so they have really driven an enormous effort in digging all these K.p parameters for 3.5 materials for the whole bunch of 3.5 materials for binary ternary and ordinary alloys selected and justified why they have taken a certain selection for some parameters so it's really been an enormous effort but it is limited to 8 bands and sometimes well you see that may not be enough and also there are other materials for instance these new crystal faces for some materials that are not contained here the 2.6 materials which are not there and sometimes more accurate parameters are simply available and it's 20 years old paper so we have seen this here for the wood side phase you can basically treat what we have done here take one additional conduction band into account you can take the 8 band model with some hybrid bands which simply gives you the wrong band coupling because it's not contained but if you want to have a pattern model of this additional conduction band you need more bands above that are not directly of interest but give you the more accurate structure of this new band in there and what our next idea was to take a 16 band model 3-ray lens bands one conduction band as we had before one more conduction band and 3 more like p-like conduction bands with their spin up and spin down components for geomarsenite in the wood side phase there's exactly nothing existing so far and that said we need some fitting procedure to fit all the two existing band structures that you can compute from DFT the possible fitting schemes available so you have a comparatively large parameter space in the wood side phase that would be 25 parameters that you cannot directly read from the band structure problem is with band structures and fitting band structures you definitely will end up in finding local minima you use some gradient approach you've seen all these valleys and come up with some first guess you will definitely have crossing somewhere that is not resolved anymore working with a grid wise search interval in 25 dimensions will take a while so certainly not what you would want to do and what our idea was then to use some crossing with the color approach so basically this is now the example for two dimensional parameter space you could either randomly distribute points in the parameter space then fit this whole range to the parameters that you are interested in and simply find the k.p model for the whole band structure and see what fits best and the problem is in random sampling you have some areas where you have clustering of parameters some of us are totally empty so you have a high chance of missing the best set of parameters and our alternative was some low discrepancy points sets here the soboil sequence which distributes in an arbitrary number of dimensions these parameter sets and that was basically what we started up with so basically we set up such a soboil sequence of for instance a number of 3000 let's say we take 3000 different points sets with 12 dimensions or with the 12 we mapped the soboil sequence 12 dimensional then to a predefined parameter range which means we need at least to have some idea where our parameters are sitting that's a shortcoming of the current method then we basically simply find out okay what Hamiltonian goes throughout the whole area is that we have throughout the point of density both and simply take our initial band structure and the eigenvalues from this parameter set and the corresponding k.p Hamiltonian and find the minimum value for it we can then start applying some priorities to certain k points or to certain bands that are of interest we have benchmarked this for a 14 band model for Bayon Arsonite in the Zinc band phase where we knew the answer and have just started with a more up to date HSE 6 band structure and the total timing for a good fit that came about with better results than they had was a bit more than two minutes single CPU performance we have added some more optimization on this so basically you start finding the best parameter set in some parameter space and if you see it's at the very site of the parameter space then you move the parameter space a bit then once you're converged go for some interval search to make the parameter space smaller there's still room for improvement so that's a very very new feature here's what it looks like for Woodside Bayon Arsonite so red is the density functional theory blue where the initial size is the size of the model and black is our fit black and gray where black was basically the interest for us it cannot be perfect because again it's a model so it would be surprising but it's looking already quite good. Okay with that let me summarize so I have presented you a widely generalized K.p model that we have been using it's generalized in terms of material composition, shapes, dimensions and in particular the Hamiltonian is in the sense that we can control the level of sophistication versus the computational costs that are of course coming with a more detailed model we are making use of a plane wave implementation within the existing Sphinx library and we also have continuum elasticity theory module to account for strain and build and potentials that are currently combined with our simulations the example studies I have presented you were on ternary indium arsenide and demonite phosphate graded composition quantum dots I have talked about random dopant fluctuations in 3.5 nanowire heterostructures polytypism in gallium arsenide and parameter fitting models for anything beyond the classical 8.2 scheme and short outlook so what we will be interested in now to apply our model to maybe silicon germanium heterostructures in collaboration with the Institute for crystal growth Berlin we are looking at exotonic effects currently in 3.5 nanowires with random alloy fluctuations which can also be surprisingly well described in a K.p model and also I have shown you this very novel parameter fitting scheme there is clearly room for improvement and we are working together with Fumvoit University Berlin group of Andrej Avada to see if we can maybe improve this fitting and with this thank you very much for your attention for the fitting algorithm what kind do you use it's only I mean if you have I mean if I understand it correctly you have your DFT bands or energy bands and then you have your solutions from your from a Hamiltonian for a different parameter set and then you can put some weight on the parameters and then you minimize this kind of that you have yes that's the cost functional just okay and the fitting is only we have this so both sequence we have like I don't know 3000 parameter sets and we pick the one with the best fit so there is no gradient search on that nowhere so if I don't have enough points in my so both sequence I won't find a good solution it's really there there is no gradients anywhere it's just I take a big box of parameters and take the best one okay so it's not that you change the parameters and then compare them again to the DFT no no what what we do have in mind and what is currently in development with Humbart University is to on top of that to plug in some gradient search so if we are already close to the minimum it would be good to move on with the gradient search and what may also help is we have a number of parameter sets in the end that give us comparatively good fit so it kind of sounds reasonable to start with some genetic algorithm on top of that to take a handful of parameters from this one and the handful of parameters from that then maybe you randomize a bit mix them and see if this gets better yeah but at the moment as I said there's room for improvement yeah can you kind of improve it in the sense that I mean when you start from a six band or the eight band you already know some of the let's say material parameters can you let's say tighten them a little bit to say that variations on them for instance the lateral parameters should be too large compared to the parameters this was in fact our hope so that's basically how we started with all that when we developed this 16 band model we said okay it's basically two eight band models and you can fit you can fit the parameters for one eight band model in Mathematica the solve the end solve function we'll do that in a second the same for the upper eight bands and then you switch on the coupling and that's it Mathematica won't do anything on that so starting with what we know is maybe helpful but don't expect that the looking at parameters are only modified to a small extent that's what I would expect but this is not the case not our priority so this can really this coupling can totally change your parameters so knowing the parameters from the lower bands does not necessarily mean that you have a good guess for the coupling I mean if you're talking about the typical three lateral parameters for the for the Zieg-Bennelette right it's a material property yes yes how can it be but then you have this classical loading up parameters and then you have the modified loading up parameters which come in with this modification with cane coupling and that's only for the conduction band so of course your loading up parameters may not be modified at all but the modified ones definitely will be and the question is if you want to keep this in having the unmodified parameters and having the modification that then affects the effect of masses somewhere else between all these bands or if you modify the parameters themselves and this is really a critical point so no matter if you have this in writing down new loading up parameters or new modified loading up parameters the problem remains that the change can be severe okay so in order for let's say transparency to better compare between 6, 8, 14 band models I would rather let's say put the changes into the modified parameters that's as much of the commonly known parameters more or less unchanged but this can come with also known cane parameters these optical matrix parameters that they deviate completely from what you would expect from the experiment so it really is that's for sure okay we are all straced so another question yeah basically maybe you try the effect of the violence band yes so is your software in probably targeting violence bands or not or just provide an answer what you mean so basically we have the wayland span landscape that goes in somewhere it's computed basically from white properties from strain distributions which can be three-dimensional from external potentials from polarization potentials and then you are holding back the table and the third-year time of consuming at your own level at every part of the space you can get really the coupling between how to continue when they drop there is another servered distribution here well basically that's this looping Hamiltonian basically this is the six-band model which comes with all these diagonal elements and what you can see is that the wave function of the hole will simply change its character and this is not happening at all so it will come in with more and more light hole and heavy hole contribution so we can see from the simulation how it comes from they do cross at some point but this is something you can see there is no crossing like this but you see that a light hole takes over and heavy hole takes over another position so you can follow the path of the single banded by using the same way yes not to be a jay and very bad 1 ok ok
 Okay, yeah, thank you very much for the introduction. As I had almost no idea who would be in the audience I tried to keep this as kind of a general overview talk.  I may, if we jump into details at some point I may decide to dump some of the examples, but we'll see just let me know once I run into critical time issues.  So, talking about semiconductor nanostructures, we have a whole field of different nanostructures that are of technological interest that can be thin films that are used for efficient light emission and detection, ranging from infrared to ultraviolet spectrum.  People have been using quantum dots as artificial atoms due to their well defined energy spectra, potentially suited for single photon emission with application in light emitters quantum computing or cryptography.  And people are also interested in nano wires as you can see here on this figure, due to their free site facets that facilitate elastic relaxation have a large surface to volume ratio and show some very unique material properties that you would not observe in a planar or  a bulk system. And that said there is a number of very different systems of interest and correspondingly, we need simulation tools that are both reliable and efficient.  They come with a high accuracy should be of course, once we are doing numerics computationally efficient robust, easy to use always something of importance.  And if I can close that my own window down here I have no idea about that because it's part of the screen missing now.  And they should come with a manageable number of parameters and output data to make sure we don't run into some stuff that is a mirror parameter fitting just to to experiment.  So, basically we can distinguish these models into two fundamental approaches one is a stick modeling like empirical tight finding the empirical pseudo potential method or density functional theory.  And the other is the same as the other approaches or atomic approaches come with single atom precision they are extremely highly flexible of course for any kind of mono structure that you may be interested in.  But on the other hand, it can be very difficult to set up these simulations, and it is at hand that these these models become computationally more and more demanding once we go for larger structures.  And so, these are the most continuous models, and the more generalized version of K dot P models that I will be talking about in a minute.  These models neglect the underlying atomistic system they are also highly flexible.  And the clear plus is that the computational effort is independent of the number of atoms in the system simply because there are no atoms it's basically  the bulk materials that we have and you can already estimate that we are talking about a few tens of millions of atoms in such a system and that's clearly.  It's not totally beyond the capabilities of atomistic approaches but it's something that you don't want to do on everyday basis.  So, the basic idea of effective mass and K dot P models is what you want to know is the energy and the wave function of electrons and holds in some crystal or a structure.  We basically start with the Schrodinger equation here and we take we as a periodic potential with the periodicity of the crystal letters.  And then we can basically split the wave function using blocks, theorem into some highly oscillating plane wave and periodic function.  And that basically allows us to rephrase the a perturbative Hamiltonian, which is this.  And then we will just basically the perturbation theory to this method that we are using here.  And then there's a second perturbation coming in namely the fact that we limit ourselves to a number of bands that are of interest, for instance here for example of a guy awesome I bought bands that one could take only the bottom conduction bands and the top three way lens bands because  they're basically close to each other, which is the idea of the eight band K dot P models so basically we take these four bands with this been up and spin down component, which gives us eight bands in total.  And then the second approach is that we say okay, we have a perturbation in case so a perturbation means this approach is valid only for the vicinity around the gamma point or technically speaking also around every other high symmetry point that is chosen for the development  model but typically limit themselves to gamma point.  And let me note that this stage that is well possible to extend the region where the model as well as to larger and larger regions of the region zone by taking more bands into account.  This is what people do. But basically what you will see mostly is this so called eight band K dot the model, which is limited to the bottom conduction band and the top three way lens bands.  So it's such a typical K dot P Hamiltonian looks like.  Basically, we have the interaction here from the terms from. 학 omega maten, the terms for the action that has like terms and then that is the new terms for Wayland's work relates treaties, or terms copy.  So basically our Hamiltonian that that one can use for the description of such a band structure and if we want to do this now not for bulk materials but for heterostructures we want to know where for instance the electron in a heterostructure sitting at what energy it has a hole.  So this can be combined with an envelope function approach so we make the effective masses potentials, the coupling parameters which we hear points.  You can make one of the station.  And what else can enter our contributions like strain that can be easily computed via continuum elasticity theory but also why as an optimistic model if you wish it can always be easily integrated.  And also appears up here so electric potentials there may be external potentials or maybe charges that jump in and out and what is at some point, and also exotonic effects can be contained via self consistency.  So this eight band K dot P model to date still represents the backbone of modern device, similar to the device modeling. It is computationally inexpensive as a small number of material parameters, most of which are well known.  In most cases, not at all. This surprisingly reliable even for small structures despite being limited to small K where you use with correspond to larger structures.  So it's straightforward combined with linear elasticity theory to incorporate strain and build an electrostatic potentials.  And it's also quite straightforward to incorporate turn or ordinary or alternately alloys or alterations of crystal phases by parameter interpolation.  And there are these disadvantages so it neglects the optimistic setup of the head of structure, certain symmetries can be simplified in an eight band K dot P model that would be resolved correctly in for instance type binding model.  The limitation to one conduction and three rail and spends can be critical at some point as we will see.  And it's inaccurate for K where you send the auto brilliant zone for instance, if we would go for a silicon germanium eight band K dot P as it is might not be the best choice because we basically have an indirect and get material where other well is become important.  There are a number of software packages existing, for instance, a type of cat and next nano, which can be used. And this formalism has been applied to a really really wide field of very different materials and heterostructures quantum dots, nano wires nano plate lets  you see these publications date back to the late 90s so it's something that has been in use for more than two decades now and still is news.  What we are using is a plane wave based implementation. So we have linear electricity and multi band K dot P formalism implemented within the plane wave framework of the existing swing software package.  And the neat thing about our approach is that we have basically a wire fast Fourier transformation we have all the real space properties of a heterostructure contained so we don't have to do any assumptions on symmetry and so on.  And we can switch from reciprocate to real space and back that allows us to make use of very simple and efficient gradient representations in reciprocate space.  And we could make use of highly optimized minimization routines that were already available in the original DFT package swings and the kinetic energy is a very very direct convergence criteria and so we can basically look at energies and our particles at the site whether  they are converged or not.  And one very big disadvantage we are using a plane wave code and that said we have periodic boundary conditions intrinsic. That's no problem for electronic states but it can be quite severe for this long range of directions like frame and built and potentials and correspondingly at some  stage we might need larger super cells for our computations.  And what what makes our code I think very unique is that we have generalized it to a wide extent. So we are not limited to any materials or compositions or shapes of the heterostructures or whatsoever, because all this is basically saved in a three dimensional  map. We can apply also in the linear elasticity module arbitrary elastic tensors and polarizations which come can come in for instance for certain crystal orientations that are non standard.  And that allow you to apply a symmetry suited to the system and not to set up a cell because you somehow have to describe the symmetry in a huge unit cell.  And in particular we have a generalized Hamiltonian which allows us to control the level of sophistication so we can basically we can do a single band effective mass models we can do six and eight band K dot D we can go for more bands we have done for his own models for  Silicon Germanium for instance, or this can be done.  The Hamiltonian is basically something which the user can bring on his own.  And we have an open source and we have no investor sometime to make it installable per package manager on most common Linux systems.  It seems to work pretty well so far there are from from the users we have there's not too much complaints yet and if there are then we are happy to take care of this.  And let me know that we have recently given a software tutorial on the usage of this code in March, April this year.  So you can basically imagine the workflow so we start with setting up some three dimensional material map that contains basically the shape of the system and other compositions in the system.  And also we need material parameters that contain the primary parameters.  So if you're not concerned then we just compute bye a building on the inside in theory, the strain and to the strain.  And search on the possess goal what cook through the like best potential.  The set of systems the moreReally  the set consistency moves of air space in some, some  between continuing the business even the  business is not contained in the core software.  It's one of the most but it's not really on a permanent  decision I have to say. And once we know it's mainly  going to this moving back area, we're going to  take people around one, two, around this  next class, then offsets, probably around this.  And we had also 50 sprouting, there's two  bands and then you can see that the  property can be used in set consistency  or in the next one properties or  some other direction you can also use  the larger enemies if you then  the optical spectrum of the system.  So the results of our simulations are then  strain distributions developed in  potentials, single particle energies and wave  functions, charge densities, possibly  excitonic contributions and hopefully in the  end some optical spectrum.  There's a list of example studies that we have  done with our code. I will limit myself to a  few of them within the next minutes to just  give you a flavor of what can be done with  our software and what we are also very open  for for anyone who's interested for some  other information. So first example would be  this quaternary alloy indium arsenide  antimony phosphate, graded composition  quantum dots grown by a colleague from  from Yerevan using liquid trace  ecotexy so that's a very, very ancient  method but surprisingly they see some  in fact 3D hetero structures applications  would be in particular in the infrared  for energy harvesting and gas sensing and  so on. And what is very interesting in this  LPE grown systems that they exhibit such  a composition profile so basically you will see  the amount of content decreasing and the  amount of D increasing over such a  what would be looking like a cone.  And that's something very interesting which  does typically not occur in the  typical MBE or MOCVD systems.  We have some information also thanks to  collaboration with Institute for Crystal  Girls and the Institute Berlin. We have some  information on the height and diameter  distribution of our systems.  And our question was now can we provide  a reliable model for such nano structures  for theory guided design process so can we  tell our experimentalists to make a system  smaller, larger, do something to them, to  get them into some certain range that is  more of interest for you. And basically  what we started up with was a single particle  calculation for this relevant range of  heights and diameters that they have seen.  So we have basically set up such a  conical shaped system.  Used an 8-band K.p model for  simple and semiconductor. It's pretty much  the Hamiltonian that you have seen before  with some simpler symmetries.  In fact, so the inside here shows you  basically how the system  looks like.  The composition range in the US,  the whole state, it's a type 2 system so we  only expect the whole state to be  localized in this system, which would be  something suited for some detectors for  instance.  So there are no exotonic properties here  because the electron is kind of far away.  We also applied some  gap corrections to our  transition energies that we have computed  here for different  heights of the system and for different  diameters of these  environments.  This is basically  what experiments have seen.  And then we have simulated an ensemble  spectrum by basically taking  the diameter distribution  from experiment  and plugging this  onto our energies  for the correspondingly computed  systems. And we have seen  now this is really kind of disappointing  that this picture is here because  the emission peak that we have simulated  was at 3.829  microns, whereas  the one on experiment was at 3.83  so I would consider that  quite good, surprisingly  good agreement at that point.  Really you have to believe  it's really  3.829.  So it's really close.  So  next example would be  on the influence of random dopant fluctuations  in three nitride nanowires.  So  the advantages of  nitride nanowires is that in particular  of indium-gallium nitrides is that  they would potentially allow  to access the whole visible spectrum  by only adjusting  the indium content  of these  insertions here  but theoretically  and another  advantage is that they have a very large surface  volume ratio which is perfect for light  emission.  The problem is, as I said, theoretically  because what happens if you plug in more  and more indium is that you increase the strain,  you increase built-in fields and  basically it's quite difficult to get  light emission from  indium-gallium nitride heterostructures  and gallium nitride with high indium contents.  The basic charge confining mechanism  in such a system, of course, the bulk bend offsets,  polarization potential  that occur and there's  a very, very large potential barrier  at the site facets of such a wire.  So order of a few electron volts, the work function  and the activation energy  plus band cap for the pulse rate.  There is significant elastic relaxation  that's very unique in these wires because they can  relax their space for relaxation here.  And  what also happens is that you are always  close to some site  fesset to some surface and that means  there are surface potentials  emerging from ionized  dopants that are of interest for us.  What else happens is that you  observe sometimes alterations of the crystal  phase and other effects  for very thin wires that come into account  might be dielectric confinement  because you have a very, very steep  change of the dielectric  of the material and the outside which would  typically be a vacuum.  So  what we had a look at  and we'll have a look at now  is the effect of surface potentials  that emerge from ionized dopants.  So what our colleagues in the experiment have  observed is that  with higher indium content in the active  layer of such a nano wire  you increase the photo luminescence  intensity.  That's pretty much the opposite  of what would happen in a planar system  where more indium content means  higher defect density, means stronger  potentials that pull electrons and holds  away from each other. But what they have  observed is higher indium content  and also larger thickness of this disc  and they see more intensity  but something quite unusual.  We could basically answer this  in a first  approach looking at  surface potentials  that we assume okay, there is a number  of  dopants in the system, ionized dopants  we have assumed first that they are  homogeneously distributed and that  just rise to a surface potential.  The surface potential is attractive  for the whole states, for the blue ones here  at the side.  So what happens is  we basically get an interplay between  the surface potential and the polarization potential  which would increase the indium content  and puts the whole closer and closer  back to the center of the axis and therefore  closer and closer into the electron  keeping in mind that these things are  to scale so the diameter is about 80 nano  and this is about  1 to 10 nano meters  that we have taken into account here.  And this interplay between polarization  and surface potential has in fact  been useful to explain the reduction  of the photo luminescence  intensity that my colleagues have  observed here for smaller indium contents  or layer thicknesses.  Now the problem is  if you assume okay, we have nano wire  of 80 nano meters, a dope intensity  of maybe 10 to the 17 cubic  centimeters and take a segment  of 20 nano meters length.  We have an average of 8  dopants in the system  and that's quite far away from the  assumption of a homogeneous  doping charge density here.  So what one would have  to do instead is to consider  individual randomly distributed  donors in nano wire.  Problem is doping  is an atomistic feature as we are using  a continuum approach here and I told you  that there are no atomistic effects  in a continuum approach.  But on the other hand, so we decided  we stick to our continuum model because  we have a system  that is not  available.  So we have a system of  under consideration has about 7 million  atoms.  The typical donors that we would observe  the unintentional doping  that's always basically there are  silicon and oxygen and both of them  represent shallow donors.  shallow donor means that they are  well described via simply  their coolant potential and what we  have here is that the donor  means that they basically provide  additional potential  to the wires.  And then we have basically started  rolling dice so this is what it looks  like.  The donor  here is  very much  very much  very much  of these potential  keys here and you see the electron  and all are kind of  another regime of these  donors and then we have  started to do some statistics  and discuss the ensemble.  So we have picked  a handful of typical model  configurations with some fixed  indium contents or some fixed  thicknesses of this insertion  we have always had the diameter of the wire  of 8 nanomers and around  too many parameters at the same time.  And what we have  observed is that the variation  of the emission wavelength is  surprisingly more or less  unaffected by the choice of  indium content and thickness of this layer.  What's also  a side issue is that  the energies that we observe  are always smaller  than what we assume are  homogeneous.  It's not too surprising  that both these distributed donors  give us deeper potentials  and the transition energy  buildup and of course they are also  smaller than the  dashed line here what we assume  that will be in the world.  And we have in fact been able to  start the line of the  randomly distributed donors of  approximately 150  milli-electron volts.  This is very very bad news for all those people  thinking that you could use these systems  for producing  reproducible light emission  because taking exactly  the same indium content in yourself  taking exactly the same wire diameter  with the same layer  thickness you still have  150 milli-electron  volts of line width  due to the fact that you have no idea where your  dopant is going.  So what else  we had a look at the ensemble charge densities  here.  So the red is the electron again, the blue is the hole,  the average  hole looks  in very good agreement  with the assumption of a homogeneous charge  but this is not as it looks like  so the typical hole you will find  will be on these corners  of course sitting  or six of them but the typical hole  is sitting in one of these corners  distributed like in the case of the homogeneous  doping and  in particular the electron localization is  governed strongly by the dopant  and you see there are very very strong  variations we have done here  500 simulations and you see  it's not even near converging  so that's really the electron is all over  the place determined by the electron.  So third example that I will talk about  is polytypism.  Polytypism is something that occurs  regularly in wires  so you see here this is a gallium arsenide wire  you see  word side segments in a mostly zinc blender  phase wire and you see also  zinc and wire phases in mostly  word side structures  and that basically allows you to form  crystal phase nanostructures  crystal phase nanostructures  so you have a nanostructure without changing  the chemical composition of the material  these systems are almost  three of strength so they are  very very well let this match because again  it's the same chemical  material it's just in a slightly  different crystal phase  there are of course no alloy fluctuations  and what is particularly  interesting is that they form  atomically flat interfaces  that's something highly interesting  and the question is how can we model these  systems and  there are some  problems coming with  such crystal phase heterostructures so  we have here changed to gallium arsenide  which is the more interesting material  you see on the left side  the band structure of zinc blend gallium arsenide  and on the right side you see  the band structure of wood, like gallium arsenide  and there is something nasty happening  in the years of gamma-A  and additional conduction band coming into play  which is energy  extremely close  to the  conduction band which is the equivalent  which is typically contained in an 8 band K.p model  and to make the situation even worse  we have already seen  similar plots in the last talk  where such a process  for a valence ends  in small strains  and change the character of our production  and these strains  even though they may not directly appear  from this crystal phase  lattice mismatch  they can appear because typically  the wires that you look at are distributed  on some surface and therefore they will see  interaction by the surface  and they will see strain by the surface  and therefore  plus minus 1% strain can easily happen  and so far the question  which is the lowest conduction band is not answered  and the question how this goes into play  is also not answered and the main problem  is that the common models  our backbone of device model in the 8 band K.p model  is limited to only one conduction band  so this is simply not contained  in the 8 band K.p model  the answer to this is quite simple  so  we take an 8 band model  and just add 10 more bands, that's something we can do  in our code because basically the Hamiltonian  is some input that the user can change  and there is  no problem between the bands  they are technically speaking one that is not  a code for them but it's very very small  so this is between the 8 band  and the top way in this band here  it will play a role for optical  recombination if there is no other part available  which can happen  and this band is not existing in the Zincland  phrasal and therefore  basically is a very very high potential  for the  inside the wood side phase  and the one that has a Zincland  has a Zincland wood side  they have very different effective masses  and the question is what happens  if we go to superlattices first  or if we move on to wires  which I won't present in this talk today  and our first step was  to take a superlattice of about 14 nanometers  length because it's interesting  of a Zincland and a wood side segment  and  the one information that is always through the hole  will be combined in wood sides  and what  is going on behind me it has a very  heavy effective mass so it's typically well  localized  however for the electron  what can happen is that it's either  confined by this gamma-6 band in the Zincland phase  or if I have a very very  thin Zincland and a very thick wood side segment  it will jump to the wood side phase  where it's  well confined due to the much  heavier effective mass it has in the wood side phase  and  if this happens or not basically depends  on the thickness of the  segments and also on the strain state  and  due to the fact that there is a small but  nonzero matrix element  between the gamma-8  conduction band and the top wavelength band  if there are no other paths available  this will be the recombination that will happen  at some time with some larger  lifetimes and so on but something  that can in fact be  seen and  we found that this can in fact  potentially  explain very very contradicting  measurements that have been  observed recently on the effective  masses of  electrons in the wood side phase  of Gaia-Marsonite so  it's something  which is not really good  not really simple to  verify correctly but it's some  potential explanation  okay  do I have time for this  point or shall I  okay  okay so  as already mentioned in the beginning  for my fourth example now which  is also part of our portfolio  what we can do  taking a K.p model the more  bands you take the more accurate your band structure gets  even to the fact that you can infect your full zone  models for silicon germanium and so on  so that basically shows you how  this K.p model is even  effective masses  before the A.band  is going to use your parabola  and of course this case where I have time to spend on us  the 6th band model  is a  in the action  of 3 bands  the 8th band model will get that description  more bands and more infections  14 band model here for instance and you see how this  structure gets more and more  detailed  and we can see that  we can see that the band structure  is not as common but sometimes not sufficient  as we have seen  for instance for polytypism  for indirect band gap  materials silicon germanium  also for something which is called  bike inversion and isotrotropies  so basically taking into account  that you have in fact a different symmetry  that is not contained in this 8 band model  sometimes additional high symmetry  points become important  for the 8 band model  the question is quite simple answer  there is something which I would consider  the bible  as a whole government paper  has the emulating K.p models  has been became across  this work  the state of today has  5477 citations  brings over a thousand references  so they have really driven an enormous  effort in digging all these K.p parameters  for  3.5 materials for the whole bunch  of 3.5 materials  for binary ternary and ordinary alloys  selected  and justified why they have  taken a certain selection for some  parameters so it's really  been an enormous effort  but  it is limited to 8 bands  and sometimes well you see  that may not be enough and also there are other  materials for instance these new crystal faces  for some materials that are not contained here  the 2.6 materials which are not  there and sometimes more accurate  parameters are simply available  and it's 20 years old paper  so  we have seen this here for  the wood side phase  you  can basically treat what we  have done here take one additional  conduction band into account you can take the 8 band model  with some hybrid bands which  simply gives you the wrong band coupling because it's not  contained but  if you want to have a pattern  model of this additional conduction band  you need more bands above that are not  directly of interest but  give you the more accurate structure of this new  band in there and what our next  idea was to take a 16 band model  3-ray lens bands one conduction  band as we had before one more  conduction band and 3 more like  p-like conduction bands  with their spin up and spin down components  for geomarsenite in the wood side phase  there's exactly nothing existing so far  and that said we need some  fitting procedure to fit all  the two existing  band structures that you can compute  from DFT  the possible fitting schemes  available so you have a comparatively large  parameter space in the wood side phase that would be  25 parameters that you cannot directly  read from the band  structure  problem is with band structures and fitting band  structures you definitely will end up in finding  local minima you use some gradient approach  you've seen all these valleys and  come up with some first guess you will definitely  have crossing somewhere that is not resolved  anymore  working with a grid wise search interval  in 25 dimensions  will take a while  so certainly not  what you would want to do and what our idea  was then to use some  crossing with the color approach so basically  this is now the example for  two dimensional parameter space  you could either randomly distribute  points in the parameter space  then fit this  whole range  to the parameters that you are interested in  and simply find the k.p model  for the whole band structure and see what fits best  and the problem is  in random sampling you have  some areas where you have  clustering of parameters  some of us are totally empty so  you have a high chance of missing  the best set of parameters and  our alternative was some  low discrepancy points sets here  the soboil sequence which distributes  in an arbitrary number  of dimensions  these parameter sets  and that was  basically what we started up with so basically  we set up such a soboil sequence  of for instance a number  of 3000 let's say we take  3000 different points sets with  12 dimensions or with the 12  we mapped the soboil sequence  12 dimensional then to a predefined parameter  range which means we need  at least to have some idea where our parameters  are sitting that's a shortcoming of the current method  then we basically  simply find out  okay what Hamiltonian  goes throughout the whole  area is that we have throughout the point of  density both and simply take  our initial band structure  and the eigenvalues from this parameter set  and the corresponding k.p Hamiltonian  and find the minimum value for it  we can then  start applying some  priorities to certain k  points or to certain bands that are of  interest we have benchmarked  this for a 14 band model for  Bayon Arsonite in the Zinc band phase where we knew  the answer and have just started  with a more up to date  HSE 6 band structure and the total  timing for a good fit that came  about with better results than they had was  a bit more than two minutes single CPU  performance we have added  some more optimization on this so  basically  you start finding the best parameter set in some  parameter space  and if you see it's at the very site of the  parameter space then you move the parameter space  a bit then once you're converged  go for some interval search to make the  parameter space smaller  there's still room for improvement so that's a  very very new feature  here's what it looks like for Woodside Bayon Arsonite  so red is the  density functional theory  blue where the initial  size is the size of the model  and black is our fit black and gray  where black was basically the  interest for us it cannot be perfect  because again it's a  model so it would be surprising  but it's looking already  quite good. Okay with that  let me summarize so I have presented you  a widely generalized K.p  model that we have been using  it's generalized in terms of material  composition, shapes, dimensions and  in particular the Hamiltonian is  in the sense that we can control the  level of sophistication versus the  computational costs that are of course  coming with a more detailed model  we are making use of a plane wave  implementation within the existing Sphinx library  and we also have  continuum elasticity theory module to  account for strain and build and potentials  that are currently combined  with our simulations  the example studies I have presented you  were on ternary  indium arsenide  and demonite phosphate graded composition quantum dots  I have talked about  random dopant fluctuations in  3.5 nanowire heterostructures  polytypism in gallium arsenide  and parameter fitting models  for anything beyond the classical  8.2 scheme and short  outlook so what we will  be interested in now  to apply our model to maybe  silicon germanium heterostructures  in collaboration with the Institute for crystal growth  Berlin we are looking at exotonic  effects currently in 3.5 nanowires  with random alloy fluctuations  which can also be surprisingly  well described in a K.p model  and also I have  shown you this very novel parameter  fitting scheme there is clearly room for improvement  and we are working together with  Fumvoit University Berlin group of  Andrej Avada to see if we can maybe  improve this fitting and with this  thank you very much for your attention  for the fitting  algorithm  what kind do you use  it's only  I mean if you have  I mean if I understand it  correctly you have your DFT bands  or energy bands  and then you have your solutions  from your from a Hamiltonian  for a different parameter set  and then you can put some weight on the parameters  and then you minimize this kind of  that you have yes that's the cost functional  just okay and the fitting is only  we have this so both sequence  we have like I don't know 3000  parameter sets  and we pick the one with the best fit  so there is no gradient search on that  nowhere so if I  don't have enough points in my so both sequence  I won't find a good solution  it's really there there is no gradients anywhere  it's just I take a big box of parameters  and take the best one  okay so it's not that you  change the parameters and then compare them again  to the DFT  no no what what we do have in mind  and what is currently in development  with Humbart University is to  on top of that to plug in some gradient search  so if we are already close  to the minimum it would be good to move on  with the gradient search and what may also help  is we have a number of parameter  sets in the end that give us  comparatively good fit  so it kind of sounds reasonable  to start with some genetic  algorithm on top of that to take a handful  of parameters from this one and the  handful of parameters from that then maybe you randomize  a bit mix them and see if this gets better  yeah but  at the moment as I said there's room  for improvement yeah can you kind of  improve it in the sense that I mean  when you start from a six band  or the eight band  you already know some of the  let's say material parameters  can you let's say  tighten them a little bit to say that  variations on them for instance the lateral  parameters should be too  large compared to the parameters  this was in fact our hope  so that's basically how we started with all that  when we developed this 16 band model  we said okay it's basically two eight band models  and you can fit  you can fit the parameters for one eight band model  in Mathematica  the solve the end solve function  we'll do that in a second  the same for the upper eight bands and then you switch  on the coupling  and that's it Mathematica won't do  anything on that so starting with what  we know is maybe helpful  but don't expect that the looking at parameters  are only modified to a small extent  that's what I would expect but this is  not the case not our priority  so this can really  this coupling can totally  change your parameters  so knowing the parameters from the lower  bands does not necessarily mean  that you have a good guess for the coupling  I mean if you're talking about the  typical three lateral parameters for the  for the Zieg-Bennelette right  it's a material property  yes  yes  how can it be  but then you have this classical  loading up parameters  and then you have the modified loading up parameters  which come in with this modification  with cane coupling and that's only for the  conduction band so of course  your loading up parameters may not be  modified at all but the  modified ones definitely will be  and the question is if you want  to keep this in having  the unmodified parameters and having  the modification that then  affects the effect of masses  somewhere else between all these bands  or if you modify the parameters themselves  and this is really a critical  point so no matter if you  have this in writing down new loading up  parameters or new modified loading up  parameters the problem remains  that the change can be severe  okay  so in order for let's say  transparency  to better compare  between 6, 8, 14 band  models  I would rather  let's say put the changes  into the modified parameters  that's as much  of the  commonly known parameters  more or less unchanged  but this can come with  also known cane parameters  these optical matrix parameters that they  deviate completely from what you would  expect from the experiment  so it really is  that's for sure  okay  we are all straced  so another question  yeah basically  maybe you try  the effect of the violence band  yes  so is your software  in  probably targeting violence bands  or not  or just provide an answer  what you mean  so basically we have the wayland span  landscape that goes in somewhere  it's computed basically from  white properties from strain distributions  which can be three-dimensional  from external potentials  from polarization potentials  and then you are holding back  the table  and  the third-year time of consuming  at your own level  at every part of  the space you can get really the coupling between  how to continue when they drop  there is another  servered distribution here  well basically that's this  looping Hamiltonian basically  this is the six-band model which comes with  all these diagonal elements  and what you can see is that  the wave function of the hole  will simply change its character  and this is not happening at all  so it will come in with  more and more light hole and heavy hole  contribution  so we can see from the simulation  how it comes from  they do cross at some point  but this is something  you can see there is no crossing like this  but you see that a light hole takes over  and heavy hole takes over  another position  so you can follow the path of the single banded  by using the same way  yes  not to be a jay and very bad  1  ok  ok
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