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Czochralski Growth of Ge Computed by Using the Finite Element Tool Elmer
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Sprachtranskript
00:00
thank you for the introduction so before I come to this but a clear topic I would be a little bit more general this is the I'm from the
00:09
lattice Institute for a crystal growth symbol and so I will general overview what we are doing
00:15
and I just noticed that I can give you the link between the 1st talked very of the 2nd talk today on
00:23
so I added something here because from send all probably from the Deloitte was quantum dots so what is the sense centers and it contains silicon silicon as a major ingredient for a 4 hour electron and devices today because you put silicon crystals you mix of structures and so on OK so this is some basics to go in the end for quantum dots still many steps in between of course and yes I will give Europe and but this is the outline of the complete talks so you would will be more and more specific by time of
00:58
this what we do in our institute so we grow about crystals maybe so can but the Cisco ramazan so all the oxide crystals because they are always doped that they look this nice was the colors those that you see the applications for everything is as optics it's so all the Reddick Tronics that's every sink what you can expect from aporetic drawing insisting is so what you do not only you Cox's into 1st and then on the way 1st you do epitaxial growth though we are out of about every text about sin grows yesterday in the condensed matter a session upstairs Let's although what are doing and part is also the nanostructure so was it going to quantum dots in a certain way so this is a very general while outline what we are doing and I set and we do all as numeric that's and and what I will talk about
01:59
related but 1st of all OK this is the real silicon crystal has grown another Siltronic cows and 1 of the major players for silicon crystal grows this just to have a look on it because of the how we grow the crystals is very simple you have the melt here of the and and the cruiser both on and then you pull out of the crystal as on this is he also you can see them the melted a little bit there you see the crystal on there yeah that's it that's that's a technique so it's for a simple looks very simple button and practices of course a little bit more complicated speakers if you want to go to quantum dots and the and you don't want have many defects or at least no defects in your crystal services that is a challenge for crystal growth the old for the numerical modeling of
02:55
this process yeah you normally start with the entry of furnace like this the grower is interested in the thermal field that's that's the main point for and so typically you use axis summary calculations for this purpose and then you get a color picture shown use the temperatures on that's what you learned on then if you go online to detail would cause you have a melt your interesting them melt flow because it says all the name impact and the sum of field but of course also on the transport of dopants and so that's of major interest I personally that mouth row is typically street the and so you have to do with you want to model it correctly you have to go to Sri the calculations even so used to geometry as axis symmetric I will not talk about this to this is another topic and also here we have a a collaboration was spheres poly on the on this part on the compilation of the surmount for certain systems the now
04:05
I start with the space of missing topic i want to speak about once again very general do this crystal grows it's not so we to smart for such an industrial application this furor it is for a scientific application was for the projects called gather and Legent and the aim is to find a neutrino lost but at the say this 1 so why why is it so interesting 1 no that like trees have a modest but knows only the relative masters absolute 1 so so they want to find the absolute mass and the 1st will find it a rule that is expected said that gives another nobel prize so everyone is running to that the what they do is they make an expiry meant calmly at Grunseth's while it if you see the size that is the evidence human being here on their germanium detector so germanium single crystals I am Leonard a useful detectors currently this city kilogram and you want to go to 2 1 at Utica as you use uses detectors in more detail so there's a lot of maturity you want to use to find and the and assisted Matrine was better say and the germanium disburse you this is the target and it is always a the detector or so that
05:34
the challenges to grow the single crystal ok this is not really the challenge you can grok single crystals of all sorry the main problem
05:42
as you need a wary Harris's it were resistivity as which means you
05:52
need extreme purity on that means and the and you're restricted was your furnace on you cannot use all this equipment you normally use or let's see next story for lot under the main parties all year that you need inductive heating on not resistant teachers or curse that would be to God so at CERN the story and
06:18
now you can see there we'll while this is over the foreigners where we are growing germanium on so it looks inside won the game the main plot is here you have a mountain right on the crystal and blue and it's a travesti method as mentioned previously you pull lots this crystal in that direction and then hopefully you get a crystal of good quality OK that already be achieved but the high purity is really the challenge but I will not go Bartell bottom highpurity I will tell about how to model this process so
06:57
now what we go further down to the process once again it's axis systematic compilation are you see this all the all the parts which are going into the simulation on what we need to Juris of K we want to calculate the temperature and and addition also later the stress field that means the sum of stress field in the crystal the wall but just going to temperature of the what we haven't fared physics we have a heat conduction It's clear that simple wary of all the radiation waterwall radiation between all the parts here which is a little bit more complicated away the because is goes to the temperature to the 4th power so it makes all numerical kind of problems may be but we have in the woods situation because germanium it is not a high melting material hopes that's always
07:55
1 again we also need to compute surmount convection a simpler way here you might also be interested the gas convection all cases of this if this sport and all come see specific poured on crystal growth what to do OK and we you'll you'll computes inductive heating here but if you some heat and put in the graphite and but in the experiment you know OK if the occur has the shape of the brew wine what you do and the America see of a threephase comma decimal 3 face because it's crystal melted and gas unhcr at exactly at this point you know the chip is a melting point temperature Pierre on the experiment what you do you will make a controlled by an worse on my by by a PID controller to to June Z all the power that you gathered melting point temperature you the next step is of course you all the want to have a solid melt interface shape computed and easily we say here it is just a modern point temperature so you go looking for the temperature highs line well that's the way we compute now you see this is 1 situation at 1 time of the of the growth of the gross is rather slow as that some media meter per hour so it's a slow process but it's only 1 time shot so if you
09:24
want to make a transition Capellas' node 2 possibilities either you make a really trends in compilation on then all of the is squeezing on and on Gates' great so that's something you Leonard becomes worried dance year after a while and well it wider affairs that's 1 possibility I will not go to this kind of calculation I will speak about a series of steady state top elections somebody's serious so but in the end you then have this same kind of
09:55
video bold of not terms solving strategy exactly for that problem I'm I'm facing here but soulful 1st stab is always that we try to validate the experimental data later but what I mean by x from beta therefore we use exactly the shape as observed in the experiments so after the next moment you got this crystal and you see it's not optimal the radios was not a constant here received from some sparkling but OK be put in this shape of the crystal in the into the compilations course you can give by by hand and optimal 1 and and see what to do with with this 1 but I will only of tell you results for this 1 so now comes a little bit more complicated starry because of that seller we we we say this is a course length 0 OK and then we start 1st the computer how much melt I need so the mel tied to you a few crucible and there's a certain melt time went because you know the crystal weights I think in that case it was 9 1 rich a grammar so you know how much much should be in the beginning on Banyan's stodgier computation of K right is melt doesn't a phrase this is a small crystal so we had length L and so on a whole and the next time step over at the time step is not really time stupid delaying step it's always delta L of the crystal is longer by delta l but as they say you have appalling right it's boiling knowledge in that direction but also be the cost was growing the melt height is going down the the mel type is now at the next step here on from that you can get an our equation well delta l as I said this is the this is our time step so to say but the real time is this 1 here on from this 1 which I cannot compute because this is a given that is a given a parameter this I can compute and the like I know the growth velocity of my crystal gross loss at the hour of this interface but I can get here why is it important it is important the cost when when the causal is growing than it really is the latent heat that's the 1 which is going into a compilation because we have 2 of body in the latent heat which is coming between that step and that step so in a steady compilation we we'll let releases C. O. he here according to this of aqui set of equations the important point is typically what people do is they you always Our for computing the LAN and he only that well you off they putting gloss T but of course if the amount is going down speaker at in it will be not the same and we get a different results so yeah
13:22
so this is done for this computation for the velocity is done was small steps and then we can think of K 1 when we want to make a real compilation of the temperature field so we do it maybe every 10 sorry 2nd step so at what we want to have a bomb and so we perform as I said of studies that calculation we computer geometry so from that geometry way it we gather matched by G match so for every calculation we get a new match which is adopted to the new situation from crystal and melt and then we compute by the open source software package alma I had they had wanted said the developers are well also interested in crystal growth processes so there's almost everything in what we need for describing crystal growth processes since
14:18
yeah so it's pretty bad so it's the more the physics simulation software based on find elements are you see where it was developed in 10 years later say they make it open to the public this is also still we had the discussion that this still written in Fortran and yes you can put in some mathematical expressions for of years this kind of at C x though with the contains already solvers written by some users so you don't recompile everything so you have a compact program and just call the soloists in all the soul was you need our way lable this looks just like this so this is for the mount you can have the so was this 1 this 1 that 1 in other domains like crystal I can have different solids because I want to solve that stress all the but I need a different 1 so just by put out here the heat equation so OK say it's the distrust the name but this is a procedure that it really was of course the rights over and then during the density open the
15:33
temperature is you've arable here and you can give you as you see you can give your own names for a room for the velocity feud for the pressure so this is a story you can write down this as the input file on
15:46
then start to pick where everything so I think I have to start the video in arts it's a bit slower as though this is not just a sequence steady state solutions you see right hand rule which means different in the different domains yet just means temperature and this in the melting pot and the crystal party at has different meanings I you have the find is a stressed on this side so this is something which you compute directly in in your solver red means you have a high so almost stress which the back because then you will you get dislocations or multiplication of dislocations and but OK this is the not the full story follow them would indication of dislocations be cursed dislocations modification depends also on the temperature by Arrhenius factor and this is this is some kind of pre factor for the multiplication and this looks slightly better a little bit different but of course it was standards parts are near to the interface but also that the at that point I have also to stress the point how how important is the shape of the interface in principle the because you see the stress here is the highest and also the arenyautes others factor is the highest here love that's very important what you our houses a shape as yeah so 1 more to the
17:24
risk to the mount convection but for all those computations I I choose this small higher risk all the teeth and the physical 1 this is a physical 1 but in the end I choose this 1 for the computations to be clear that this is still a steady state and because of that you can start and this is once again how you can use this month expressions here and the beginning if there's no flow of then you start was a higher risk prosody and you know it because you already have a solution as the input from next iteration and you can do it by these expressions on this is the time variable here on so if your lesson 20 iterations you increase at a linear rate to be to get a stable solution to the problem now comes a point link to the
18:19
experiment's but what you can do and the experiments you cannot occur really into we saw the growing crystals but you cannot measure anything you don't see the interface but what you do you cut let afterward you cut cuts the personal and then using the socalled Lps method to to visualize our differences in the resistivity and this corresponds sent to all of the interface shape during growth ball that's this lines here and then all at the well laid yellow lines is the a calculated interface licenses agree quite well except that certain regions where rare you over the radius was changing dramatically on so it this is a quite good results so the other thing you can compare is there the poly the input power not directly because in your experiment you have some some outer loop for Europe the actor electricity so you have to to ideas subtract here something to to come to this curve for users this 1 is X primed born but the right 1 is the computed that have this peaks as speakers when the radius was changing to sharply than I have some some peaks but it and in the end this follows the x from a blind and now you see what happened if you would choose the of the gross at the identical to the boiling velocity you would get a completely different story even on them in the picture of power all it makes it important for this particular problem to take into account the real velocity
20:10
I think this was just the video to visualize some flow here and that it is changing in time on so the main contribution to the lower is the when the convection but you also have Marangoni convection on the surface and you also see well you have this cell line here but between the gas and melt is a fixed line which is computed beforehand to heaven I say I don't know if it's I have it here know so I have it somewhere OK
20:48
sorry because this angle between the causal on this line is certain degrees this is known from experiment so the line is just computed to match says and this equation by held to do this so this is done before I start the steadystate calculation for every step in this is computed the other thing what you might
21:13
be interested then because of the wall we need this high purity and those a gas connections a different story can all the computed really in this framework here those at 2 different you see through different configurations and 1 you you get a gas convection yeah true the cost 1 the other March of that I don't want to go into detail here yeah here is
21:37
sorry he was it's exactly the picture I was looking for on this curve is yeah this is given as I said why the important this angle here is sodium degrees for on and OK man should now well how the problem was Elmer because of a case this is the this is the boundary condition for all Marangoni convection as our IT mention that you have some flow here on the from the earth on the heart is go to her and her all the and T A K 4 we would principle I like to seperate get them out from the gas flow but means I have a little son to solve the numbers tilts equation the red part here with this boundary condition and separately I want to solve the NavierStokes equation and the rest with the gas as I have just received another boundary conditions only the problem is was of course I can write it a new function and recompile everything but we see recompiled system that is I can use to numbers don't so versus no rename it but even the Develo I or wharf that that that to this really that 1 can do with it the only problem is there the sum field I can as of the universe the velocity field cannot enter the cell of your because the sum of it I want to to compute globally as just 1 heat equation over the entire domain so in brand principle you want to do what you want to do is to give the advection term in the in the melt domain is the metal city and and here it is a gas SolarCity I and I didn't find any way to do it and since the developer peter rabbit was even astonished that I can use to to NavierStokes solar in parallel OK he didn't always in the know an answer for that question
23:39
yeah but this is only 1 part OK the other 1 is to be more precise with the computation of this the gross velocity that don't take into account the actual shape of C. the interface what we do is what you saw as a flat interface and we should we look for the difference in the form of the 2 flat interfaces and then computes the the overall gross velocities so of course it would be more accurate if you compute that locally because of the change or sometimes is a drastic changes the shape use between 1 step and the next step and of course it says although Leary's of Latin heat or a con the other way around the consumption of light and heat and of course the main point is all the computation of dislocation density which is an extra step so you through to you put all this the results and this as of this almost breast field and the temperature are you with us function the pre function of the vector function as shown before into another compilation where you computed involution of the dislocation density but you can also from a mathematical point sink about other things be cursed every sink is tunable were allowed to open source card 1 can think about it some kind of optimization of optimization loop outside around whatever we are looking for but you know I mean can you find something a deal looking for all I want to look for a or the maybe also over the entire process so a minimum of the dislocation stress over the entire run or something else and then you can parameterize that in the way that you change something in the system of to to be better for that are to achieves this off to minimize that so too and without
25:45
though the main part of course it in this work where well excremental 1 you see all the people involved in it and yes this is of this the scanner projected our is although supported by BMBF and thank you for staying here to the almost and the end of the talk so there will be test Kushan afterward but right to the end of the
26:09
talks thank you
00:00
Unendlichkeit
Numerisches Modell
Verschlingung
Verbandstheorie
Element <Mathematik>
00:22
Sinusfunktion
Numerisches Modell
Mereologie
Kartesische Koordinaten
Technische Optik
Kantenfärbung
Technische Optik
Gerichteter Graph
Quantisierung <Physik>
Strategisches Spiel
01:58
Stellenring
Prozess <Physik>
Gewichtete Summe
Kartesische Koordinaten
Temperaturstrahlung
Physikalisches System
Rechnen
Massestrom
Fluss <Mathematik>
Temperaturstrahlung
Numerisches Modell
Kugel
Reelle Zahl
Wärmeübergang
Mereologie
Körper <Physik>
Kantenfärbung
04:04
Transfinite Induktion
Numerisches Modell
Dichte <Physik>
Betafunktion
Ruhmasse
Ruhmasse
Betrag <Mathematik>
Schlussregel
Kartesische Koordinaten
Projektive Ebene
Methode der kleinsten Quadrate
Extrempunkt
RaumZeit
Topologie
05:39
Transfinite Induktion
Inklusion <Mathematik>
Fläche
Extrempunkt
Dichte <Physik>
Wärmeleitung
Betrag <Mathematik>
Methode der kleinsten Quadrate
Temperaturstrahlung
Extrempunkt
Rechenbuch
Symmetrische Matrix
Numerisches Modell
Theoretische Physik
Betafunktion
Ruhmasse
Leistung <Physik>
Normalspannung
06:17
Fläche
Prozess <Physik>
Gewichtete Summe
Physikalismus
Wärmeleitung
Schmelze <Betrieb>
Kardinalzahl
Rechenbuch
Richtung
Temperaturstrahlung
Numerisches Modell
Spieltheorie
Theoretische Physik
Minimum
Leistung <Physik>
Inklusion <Mathematik>
Addition
Güte der Anpassung
Temperaturstrahlung
Symmetrische Matrix
Rechter Winkel
Mereologie
Körper <Physik>
Leistung <Physik>
Normalspannung
Wärmeleitfähigkeit
Normalspannung
07:51
Fläche
Prozess <Physik>
Punkt
Prozess <Physik>
Gruppenoperation
Wärmeleitung
Reihe
Temperaturstrahlung
Rechnen
Rechenbuch
Reihe
Symmetrische Matrix
Numerisches Modell
Hauptidealring
Theoretische Physik
Meter
Fließgleichgewicht
Gerade
Normalspannung
Leistung <Physik>
09:54
Geschwindigkeit
Resultante
Einfügungsdämpfung
Länge
Punkt
Gewicht <Mathematik>
Momentenproblem
Schmelze <Betrieb>
Gleichungssystem
Term
Rechenbuch
Strategisches Spiel
Richtung
Numerisches Modell
Geometrie
Umwandlungsenthalpie
Beobachtungsstudie
Parametersystem
Matching <Graphentheorie>
Physikalischer Effekt
Fläche
Rechnen
Menge
Rechter Winkel
Strategisches Spiel
Körper <Physik>
Geometrie
14:17
Wärmeleitungsgleichung
Geschwindigkeit
Stereometrie
Multiplikation
Gerichteter Graph
Schmelze
Mathematik
FiniteElementeMethode
Zeitbereich
Physikalismus
Mathematik
Element <Mathematik>
Gleitendes Mittel
Variable
Dichte <Physik>
Trigonometrische Funktion
Arithmetischer Ausdruck
Druckverlauf
Numerisches Modell
Körper <Physik>
Massestrom
Optimierung
Normalspannung
Gleichungssystem
15:46
Subtraktion
Folge <Mathematik>
Unterring
Punkt
Schmelze
Sterbeziffer
Iteration
Massestrom
Multiplikation
Arithmetischer Ausdruck
Numerisches Modell
Iteration
Fließgleichgewicht
Indexberechnung
Verschlingung
Zeitbereich
Fläche
Schlussregel
Teilbarkeit
Variable
Arithmetisches Mittel
Rechter Winkel
Mereologie
Normalspannung
Standardabweichung
18:16
Geschwindigkeit
Resultante
Radius
Subtraktion
Kurve
Fläche
Massestrom
Polygon
Loop
Numerisches Modell
Flächentheorie
Rechter Winkel
Schnitt <Graphentheorie>
Einflussgröße
Innerer Punkt
Gerade
Leistung <Physik>
20:45
Einfach zusammenhängender Raum
Erweiterung
Wärmeausdehnung
Numerisches Modell
Physikalischer Effekt
Winkel
Gleichungssystem
Reelle Zahl
Rechnen
Konfigurationsraum
Variable
Gerade
21:34
Fläche
Geschwindigkeit
Wärmeleitungsgleichung
Resultante
Subtraktion
Prozess <Physik>
Gewichtete Summe
Punkt
Dichte <Physik>
Extrempunkt
Minimierung
Zahlenbereich
Gleichungssystem
StokesIntegralsatz
Bilinearform
Massestrom
Term
Gerichteter Graph
Strategisches Spiel
Loop
Numerisches Modell
Strom <Mathematik>
Ganze Funktion
Grundraum
Lineares Funktional
Erweiterung
Prozess <Physik>
Kurve
Mathematik
Winkel
Zeitbereich
Fläche
Globale Optimierung
Physikalisches System
Vektorraum
Dichte <Physik>
Randwert
Mereologie
Körper <Physik>
Lateinisches Quadrat
Normalspannung
Geschwindigkeit
25:43
Numerisches Modell
Exakter Test
Mereologie
Messprozess
Metadaten
Formale Metadaten
Titel  Czochralski Growth of Ge Computed by Using the Finite Element Tool Elmer 
Serientitel  The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018 
Autor 
Miller, Wolfram

Mitwirkende 
LeibnizInstitut für Oberflächenmodifizierung e.V. (IOP)

Lizenz 
CCNamensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35358 
Herausgeber  WeierstraßInstitut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB) 
Erscheinungsjahr  2018 
Sprache  Englisch 
Produktionsort  Leipzig 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik 
Abstract  The Czochralski method is widely used for growing single crystals as e.g. silicon crystals. Typically, the thermal field and related ones such as thermal stress is computed for certain stages of the growth. Here we present the successive computations starting from inital state up to nearly the end by combining a perl script and the finite element solver Elmer. The shape of the crystal is given by input and can be taken either from experimental result or by own definition. The numerical computed interface is compared with the one observed by LPS at the crystal grown. 