 Good morning. In the last lecture we were talking primarily about the chemical reactions that can happen in a CVD reactor and the fact that under the operating conditions of a normal CVD reactor the chemical equilibrium assumption is usually a valid one particularly when you are operating at high temperatures and high pressures but even under the simplifying assumption of chemical equilibrium prevailing the calculation of the species compositions present can still be very complicated because there are hundreds of chemical reactions going on. So what we said was that in order to obtain the equilibrium composition of the system a simpler process would be to use a free energy minimization approach in any system as it approaches equilibrium conditions the free energy of the system is minimized just as entropy is maximized. So if you can write an expression for the free energy and then use a conventional minimization algorithm you should be able to obtain the chemical equilibrium compositions. So if you look at the expression for Gibbs free energy, Gibbs free energy calculation itself can be quite complicated in a multi-component multi-phase system because remember that in a CVD reactor you have hundreds of species not only in the gas phase but also in solid phase and there could be some liquid formation as well. If there is liquid formation in a CVD reactor it is usually unintended but sometimes especially in a hot wall system you can have localized condensation of nucleated droplets for example so it becomes a three phase system. So what that means is that when you consider the expression for the Gibbs free energy you have to essentially summarize the free energy over all the chemical species that are present in the system and also over all the phases that are present in the system and then you write this as N i k times mu i k where again N is the total number of species, pi is the total number of phases, N i k is the mole fraction of ith species in the kth phase and mu i k is the chemical potential of the ith species in the kth phase, okay. So you write the expression for G and you minimize G to obtain N i k. Now how do you, you can further expand this and write this as this is equal to summation N equal to i to N k equals 1 to pi N i k and you can write mu i k in terms of the free energy of formation of species i plus RT ln the ratio of, so the delta G i k is the free energy of formation of species i in phase k and the f values are the fugacities. So f i k is the partial fugacity of species i in phase k and f i 0 is the fugacity of pure species i. So once you have written it this way delta G f is typically available in various handbooks and so you can essentially, I mean in order to solve the system of equations A you need to have a good numerical algorithm and actually the technique that is most widely used is what is known as direct search optimization which enables you to home in on the global minimum very quickly. One thing you have to be watchful about is that because the expression for free energy is quite complex there are many local minima. So you have to be able to differentiate between the local minima and the global minimum and home in on the global minimum. So you need to choose numerical techniques that essentially allow you to do that. There are commercial packages by the way which have been written for this purpose. The one that I have used a lot is was actually written by NASA that was back in the 80s. It is called the NASA CEA program. I think it stands for chemical equilibrium analysis program. It is really downloadable from the web but even the more conventional packages like Kempkin also have modules that deal with calculation of equilibrium compositions using the free energy minimization approach. The other part of this is you need to have a good database because of the thermo chemical properties of all possible species in the system because that is really what this right hand side is. You for every i you need to know these values. What is delta G F i and F i and F i 0 for species present in various phases gas, liquid, solid and so on. So you need a very extensive thermo chemical database and there is something called the JANF thermo chemical database which is coupled with programs like CEA and Kempkin. There are few more packages. Actually if you go on the web and just search for free energy minimization you will be surprised at how many hits you get. People have really spent a lot of time coming up with very efficient numerical algorithms for doing this but where they all succeed or fail is in the completeness of this database because unless you have a good way to accurately estimate the thermo chemical properties of the species and be know all the species that are present in your system this method of solving will not yield you correct answers at the end. For example if you severely underestimate the number of product species that can form then obviously your analysis is going to be incomplete and sometimes you can overestimate it as well. So the way that these programs work they start with an overestimation approach. They basically take the elements that are present look at all possible combinations of the chemical elements and propose that all these chemical species can be present in the system and then they go through this free energy minimization algorithm and pretty soon you can start weeding out species that are very unlikely to form and then you start kind of homing in on the right set of components. That is a better approach than going the other way where you start with a limited number of species because that can lead to these local minima type of problems. You may think that you have free energy minimized solution but it may be only because you have taken an incomplete set of chemical species to consider. So just as important as the selection of your numerical algorithm is the selection of your database with which you are going to do your analysis and also your identification of possible species that can be present in the system. So you need to know something about the process. This approach is what I would call more of an engineering approach where you do not just treat the CVD reactor as a black box and do this entirely mathematically. If you know that certain species are likely to form, you can actually force the program to treat those as more likely to form than other species. So that way essentially the convergence will be faster. So you can use a few tricks like that depending on your insight into the system to get your results in a more efficient manner. Now what are the, in any minimization algorithm as you know, there have to be constraints, right. So what would be the constraints in this case? Can you think of some? Yeah, obviously the number or the mole fractions of the species have to be non-negative, right. That is one constraint. The other is mass conservation. So when you talk about mass conservation, here is where the differentiation between PVD and CVD becomes very evident. In physical vapor deposition, there is no transformation of one species into another, right. If you put in SiH4 in a physical vapor deposition reactor, it will stay as SiH4. There will be no conversion to various other compounds in the system. So in that case, the imposition of the mass constraint would simply say that summation of k equals 1 to pi of Ni k must be equal to some Ni total which is the amount of species i that has been fed to the reactor through the feed stream, right. So the mass constraint is just placed on the species. In a CVD reactor, how would this constraint change? It is not enough. In fact, this equality would not apply in a CVD reactor because the species will not stay intact, right. They will be transformed into other species. So how do you handle that? How would you impose a mass conservation condition in a CVD reactor? Any ideas? What is it? Even in a CVD reactor, what is it that does not change? What is it that cannot be created or destroyed? Total mass but that does not help us much. Anything else? Elements. Elements. Unless it is a nuclear reactor, right, the total amount of elements that you feed into the system must remain. You cannot transform one element into another. So in the case of a CVD reactor, the mass constraint is placed on elements and essentially the way you would write that is to say that the summation of element, you essentially do the summation over all, actually you do this for every element that is present. So you would still sum over n equals 1 to total number of species capital N of A E i times n i k equals some B E where A E i is the mole fraction of chemical element E in species i. So when you sum that over all and by the way this summation should also be over all phases k equals 1 to pi summation n equals 1 to n and this is the total amount of that element that was fed into the system, right. So essentially a CVD reactor must obey element conservation principles and total mass conservation principles. Whereas a PVD reactor must obey total mass conservation, element conservation and species conservation because the species do not get transformed. So that is a key difference that you should try to remember. So essentially given these constraints and the non negativity constraint you can proceed to do a constrained minimization of the free energy and so the program will essentially give you a set of product compositions for a given essentially what you feed into the program will be the feed composition and the operating conditions of the reactor, the temperature distribution and the pressure distribution. Once you feed that in at each location in the inside the CVD reactor the free energy minimization algorithms will yield the equilibrium compositions prevailing under the local conditions. Now if the program is running correctly and your reactor has been designed properly through the entire reactor you should only see gas phase species and the second phase should only start appearing at the substrate and therefore you do not really have to worry about multi phase calculations through 99% of the CVD reactor space. It is only the vicinity of the substrate that you are going to see a multi phase situation happen and start condensing solid films on the surface. These subroutines are also very good not only for pure condensates but also for mixed condensates. So you can have situations where instead of depositing a simple film of silicon you may actually be depositing a composite film of silicon germanium for example. Here these algorithms can predict appropriate compositions but the database again becomes important. You have to know about the phase equilibrium characteristics between the two condensed phases. You have to know when silicon and germanium form a compound what is the phase diagram look like and that data has to be fed into the database in order for the predictions to be accurate. So the thing to remember is free energy minimization algorithm is a very elegant and very efficient method to achieve convergence towards equilibrium compositions but the predictions are only as good as the database. The stronger your database, the more comprehensive your database, the more confidence you have in the predictions. But still even with an inadequate database these predictions are likely to be more accurate than one where you try to assume certain chemical reactions and then try to calculate equilibrium compositions based on that because you can never really guess all the chemical reactions that are going to happen and it is purely guess work, right? You do not really know what reactions are taking place. All you know is these were my feed materials and this is my product composition. You do not really know how they got there, right? You can make certain assumptions but they are just as likely to be wrong as they are likely to be right. So bottom line is if you are going to do chemical equilibrium compositions, the free energy minimization algorithm not only for a CVD reactor in any chemically reactive situation where the conditions are such that you will have a large number of chemical species forming, you are much better off using a free energy minimization algorithm approach rather than something that requires you to specify reaction pathways. Of course the chemical equilibrium assumption may not be valid in many reactions that you deal with. For example in the bio area, most reactions happen close to room temperatures because high temperatures will kill many organisms and so on. So in those cases, your reactions are very much rate limited and this kind of a chemical equilibrium approach would not work but in situations that involve high temperatures such as CVD reactors, combustion systems, even this incandescent bulb that I talked about earlier, these are all cases where there is a very high temperature, high pressure and the likelihood of reaching equilibrium is very, very high. In such cases there is no question that the free energy minimization algorithm is the way to go. So any questions on that before we move on to a discussion of some basic transport phenomena? I will also send some reading materials on the free energy minimization approach today and take a look at it, you know review it and then if you have any questions you can you know maybe bring them up in the next class. So now that we have a reasonably good handle on how to deal with the chemistry in CVD reactors, the second important aspect is the flow dynamics. Now the reason that flow dynamics are important is not only because they establish local pressures, local velocities and local temperatures but also because they have a very direct influence on the transport of the reacting species into the reactor and then the transport of the daughter species to the vicinity of the substrate on which you are trying to deposit the CVD film. So it is very important to characterize flow dynamics and the simplest way to do it is to assume that the CVD reactor is essentially a tubular reactor in which flow is happening under laminar flow conditions. As I mentioned earlier most CVD systems are designed to have laminar flow and therefore making the assumption of laminar conditions is perfectly valid. So when we look at a CVD reactor the way we have sketched it in the past is something that looks like this. So there is a substrate which is mounted into the reactor such that deposition happens on the substrate and it is provided some source of energy whether it is thermal or laser or photochemical or whatever and the flow typically happens in this direction and the exit gases leave in this direction. So if you are trying to model the CVD reactor from a transport view point the simplest way to model it is just assume that this is a tubular reactor and that flow is happening through this reactor and the other assumption which is very important to modeling of transport phenomena inside the reactor is to assume that the flow enters the reactor essentially as a plug flow most likely because hopefully your flow input is designed such that you have plug flow going into the reactor but by the time it leaves the reactor what kind of profile do you think the velocity will have? Remember we said it is going to be laminar. So in a tubular flow under laminar conditions what kind of velocity profile would you get? Parabolic right. So the basic assumption in all of our transport modeling is that flow will enter as a plug flow and leave as a parabolic flow. Under what conditions is this assumption likely to be valid? What are some major assumptions you make about CVD reactors and by the way the assumptions I am going to talk about will not apply under special conditions such as extremely low pressure CVD reactors plasma enhanced or plasma assisted CVD reactors because the first assumption we will make is continuum. So what does continuum flow mean? What are we saying? When we say that there is continuum flow inside a system or a reactor what are we talking about? The system is not equal to the length scale of the reactor well it is not just it is not equal to it is much much less. So the mean free path divided by a characteristic dimension of the reactor let us say that we take that as the length of your substrate. So this must be much much less than 1. So that will yield continuum conditions and by the way all the equations that you are familiar with that you have studied in your courses have all been derived under continuum conditions you know whether you talk about Navier-Stokes equations or you know Bernoulli equations or any mass momentum energy conservation equations that you have dealt with in your coursework so far have all been under continuum conditions right. Have you have you dealt with free molecular flow or transition flow okay. The second assumption that we will make is that it is incompressible what does that mean? What do we mean when we say incompressible? Density does not change with pressure I mean in the system essentially density stays constant not with time but actually along stream lines right it can actually change with time by the way in an incompressible system density is allowed to change with time it is only it is not allowed to change with the stream lines of flow but so what decides whether a fluid is compressible or not is that something you can claim based on the velocity of flow the requirement for incompressible flow is that the velocity of fluid flow divided by the speed of sound must be much smaller than 1 okay. The third assumption we will make is that it is viscous flow what does that mean? It just means that the fluid has a finite viscosity and in fact that is what you need in order for this structure to be obtained right. If you have a completely inviscid flow you would not have a parabolic structure. So you are assuming that the fluid has a finite viscosity which leads to the parabolic flow. Another assumption that we have mentioned even earlier is laminar so again the laminar flow is what gives you that parabolic configuration but in terms of the surfaces that the fluid comes in contact with what does that imply what do we mean when you say laminar flow right. So when you say laminar flow the flow under the entering flow is at a uniform velocity of U infinity and that velocity drops to 0 everywhere along the solid surfaces that the fluid comes in contact with. Now in a conventional CVT reactor obviously the transition from laminar to turbulence is decided by what number Reynolds number right. So let us say that you have a Reynolds number which is defined as some U infinity L by nu where U infinity is your entering velocity and nu is your kinematic viscosity which is mu by rho dynamic viscosity divided by density. This number has to be less than 2300 in a CVT reactor with hydrogen as the primary diluent. As long as you keep the Reynolds number below 2300 in a CVT reactor with hydrogen as the flowing gas you can be sure that laminar flow will be obtained. The conditions for laminar flow will change depending on the fluid so this is only for the case where hydrogen constitutes 90% of the fluid that is flowing. So under such conditions how do we write an expression for the parabolic flow do you remember? So you write it as V as a function of R being equal to 1 by 4 mu times dp by dz times a squared minus r square where mu is the dynamic viscosity. dp by dz is the pressure variation in the normal direction and A is your tube radius and r is the radial distance. Now once you have the expression for V of r you can also calculate the flux as integral of 2 pi r V of r dr. So it is essentially the area times the velocity will give you the mass flux and the other critical parameter here is obviously the boundary layer thickness. As we mentioned earlier it is the diffusion through the boundary layer that ultimately transports the reacting species as well as the species that should be adsorbed on the surface to the substrate. So the final resistance to that is the diffusion through the boundary layer. So the boundary layer thickness becomes an important parameter. The variation of the boundary layer thickness as a function of x distance is what? I mean what will be the dependence of boundary layer thickness on x? Does it go as delta is proportional to x or x to x squared or x to the power half? See we are talking about a flat plate flow past a flat plate and so we are trying to understand how this boundary layer thickness will change as a function of downstream distance x. What would the shape look like? Will it look like this or will it look like this? The second one right. So essentially the dependence is it is proportional to square root of x where x is this distance and this is your boundary layer thickness or numerically this can be written as 5 times square root of U infinity x by nu. Now this is an important equation because it tells you that A there is going to be a non-uniformity in the boundary layer thickness which we talked about earlier right. As flow occurs over a flat plate where the normal to the plate is perpendicular to the flow direction the boundary layer thickness will change as x to the power half. So it is important that we recognize the fact that the boundary layer thickness will look like this. So if your transport of the reactive species is going to have to happen through this boundary layer thickness then clearly the rate at which species are going to arrive at the surface on the leading edge is going to be very different from the rate at which species are going to arrive on the surface of the trailing edge. It is going to be much slower at the trailing edge which again leads to non-uniformity right. Earlier we had discussed this non-uniformity in another context which is that if you have a pure species that is being fed into the system the concentration of the species will be much higher at the leading edge and then it will start trailing towards the trailing edge and we said you fix that by essentially using a diluent. So instead of feeding in SiH4 at 100% concentration you feed in SiH4 at 1% concentration with 99% hydrogen as diluent and that only but that only keeps the free stream concentration of silane constant across the length of the substrate. But it still does not address this issue that towards the trailing edge of the substrate you are going to have a drop off in the diffusional deposition rate compared to the leading edge of the substrate. So how do you address that? I think the suggestion that was given last time was to tilt at an angle. That solution would still work in this case also. So by tilting the plate at an angle you can essentially force the boundary layer thickness to be much more uniform across the length of the boundary layer. Of course there are other solutions such as again rotating the substrate and so on. Other approaches would include not flowing it in a horizontal direction but as I mentioned in the earlier class perhaps flowing it in a vertical condition to achieve stagnation flow. And so the flow dynamics the point is that it does play a very significant role in how material is delivered to the surface. So this is you know important equation to keep in mind and make sure that we design the system in order to achieve the uniformity as well as rate of deposition that we are looking for. So what happens inside the reactor? I mean we are introducing the flow as a plug flow but it is exiting as a parabolic flow. Where does the transition happen and does that have an influence on again the flow dynamics and its impact on the deposition process? Where do you want the transition to happen? Do you want this transition from plug flow to parabolic flow? Do you want it to happen upstream of where the substrate is or at the midpoint of the substrate or downstream of the substrate? So you want it to happen downstream of the substrate. So you want the material essentially to follow plug flow across the substrate and only achieve laminar flow conditions downstream. Now is that the right strategy? I mean what is the you know if you do if the flow happens this way what is the downside? See the thing you have to remember is material is distributed in this direction right. It is distributed among all the stream lines that are flowing. If the entire flow occurs in a plug flow then only the bottom most layer of the fluid will ever come in contact with the substrate. The rest of the material will just get, so that is not the right way to do it. In fact the transition to a uniform flow which is still a mixed flow you know that is a key thing. You want to have a mixed flow condition before the fluid encounters the substrate but at the same time you do not want turbulent mixing. You essentially want a laminar mixing to occur. The problem with plug flow again is that there is no interlayer transport through diffusion. The difference between a simple plug flow and a parabolic laminar flow is that now there is mixing of the reacting species across the stream lines instead of simply being conveyed along the stream lines. So you do want the transition to happen upstream. So is there a formula for that? I mean how do you calculate where the transition will happen? Have you come across any equations that talk about transition from plug flow to parabolic flow? Actually Schlichting derived an expression which is continues to be used quite effectively. Schlichting's equation says the transition distance that is the downstream distance at which the transition from plug flow to parabolic flow happens can be written as A by 5 squared times the Reynolds number where A again is the radius of the tube. So what it basically means is that the Reynolds number as the Reynolds number increases the transition distance also increases. So that is another argument for keeping the Reynolds number as small as possible so that this transition will happen faster. As you can imagine as the flow velocity is increased and the Reynolds number increases the likelihood of never having the transition take place is higher. I mean the fluid just could just go past the plate in plug flow and then exit before you know what is happening. So by slowing down the flow a lot of good things happen. That is why in a CVD reactor the whole intent is to keep the flow rate as low as possible. It helps us with managing the flow dynamics. It helps ensure that the transition from plug flow to a uniform flow or parabolic flow will happen upstream of the substrate. It helps us ensure that there is no turbulence and all the you know non-uniformities and inconsistencies associated with turbulence. Thirdly it gives a chemically the reacting species sufficient residence time inside the reactor to ensure that contacts happen and chemical reactions happen and equilibrium is achieved. Chemical equilibrium is a time dependent phenomenon. So the longer that a chemical reaction has available to happen the more likely it is that it will push itself towards equilibrium conditions. So by slowing down the flow you can achieve a lot in a CVD reactor right and so again when you are designing the reactor this kind of correlations are very helpful in ensuring that you control the spacing between the inlet and the substrate in such a way that you ensure that the transition happens prior to the fluid encountering the substrate. So we have kind of touched a little bit on the thermodynamic side, touched a little bit on the transport side. So when you are trying to model a CVD reactor in terms of its transport and thermodynamic properties essentially the way you would want to do that is using CFD I mean computational fluid dynamic programs for modeling a CVD reactor are available. In fact most of the commercial CFD packages will have a subroutine for CVD phenomena but the way that they do it sometimes is not accurate because they do not take into account the two way coupling between the chemical reactions and the transport phenomena. Essentially if you look at a conventional program for modeling a CVD reactor it will first deal with the flow and let the flow establish your velocity conditions, your pressure distribution, your temperature distribution and then it will then call upon some chemical kinetic code or equilibrium code to establish the corresponding species concentrations at each location inside the CVD reactor and then based on these conditions it will calculate the rates of transport of the species to the substrate. And then for example if silicon is what you are trying to deposit and silicon is distributed among 10 species it will calculate the rate of deposition of each of these species and from that it will calculate an effective deposition rate of tungsten simply by as a weighted sum of the deposition rates of all species containing the element silicon. That is a conventional approach. The problem with that approach is it may not be self consistent because it is assuming that transport and the thermodynamic processes can be decoupled. For the most part it is true particularly when the depositing species are in dilute amounts. If you are using a carrier gas and for example your silane is only present in trace quantities then one part of the assumption is true that the transport of that species is not going to affect the macroscopic flow characteristics of the system. But if you are not using a diluent and you are feeding in the reacting species at reasonably high concentrations not concentrations that are not much less than 1 then even that assumption is not valid because the diffusional transport of the species can have an influence on the convective transport in the reactor. So there is something called Staphon flow which is we will talk about a little later in this course which is the convective flow that is induced by diffusional flow. Because according to mass conservation if you have a diffusional flow from the bulk of the fluid towards the substrate it must be opposed by something. So the diffusional flow actually induces an equal and opposing convective flow of the fluid which is called Staphon flow which can have a very significant influence on the flow dynamics in the system. That is a very important effect in a CVD reactor which is typically neglected in most of the commercial CFD codes that deal with CVD reactors. The other aspect that again we will deal with in much more detail later on is a self-consistency aspect. Let us say that you are trying to deposit silicon on a substrate right. From a transport view point what is the primary constraint you have to impose on the system to get a meaningful solution. The constraint is the flux of silicon must be positive towards the substrate right. The flux of elemental silicon not every species containing silicon. So if the silicon element is contained in 50 species inside the CVD reactor 40 of them or 25 of them could have a flux towards the substrate, 25 of them could have a flux away from the substrate that is okay as long as the net flux when you add up the contributions of all the species are towards the substrate. That is a self-consistency check. If you have a CVD film that has more than one constituent for example let us say that you are trying to deposit silicon nitride Si3 and 4 then what is the transport constraint that you should impose. There are actually 3 now. The flux of silicon should be positive towards the substrate. The flux of nitrogen should be positive towards the substrate and in order to get a stoichiometric condensate of Si3 and 4 the molar fluxes of silicon and nitrogen must be in the ratio of 3 to 4 right 3 by 4. So all of these constraints must be simultaneously applied in order to have a self-consistent solution. And again commercially available CFD codes won't even try to do that. So these are some of the gaps in the current approach to modeling of CVD processes in commercial codes. So what we will try and emphasize in this class is some of these very interesting interdependent effects and the fact that you obtain the rate of deposition of a CVD film and even its dew point accurately only if you impose not only the thermodynamic and transport constraints that affect the so-called macroscopic environment of the CVD reactor but also the diffusional transport constraints that apply at the interface between the fluid and the substrate and that is where the iterative solution process becomes very important okay. So let us stop at this point and we will continue the discussion in the next class. Any questions on what we have covered so far okay see you at the next class then.