 Welcome to the next lecture in our course on chemical engineering principles of CVD process. In the last class we defined something called the mass transfer analogy condition in which the mechanisms of mass transfer and heat transfer are identical therefore you could essentially take an appropriate non-dimensional distribution in that you obtain by energy conservation and from that extract the corresponding mass fraction distribution for example that you will get when you do mass conservation. The ability to apply the mass transfer analogy condition greatly simplifies our analysis of the problem and also reduces the requirements for experimental data collection. But as we also saw there are certain conditions in which the mass transfer analogy can be violated and in such cases you cannot do that I mean you pretty much have to do your experimental measurements separately you have to do measurements for velocity distribution, measurements for temperature distribution and measurements for mass fraction distribution and similarly when you do your analysis, modeling and simulation you have to write the conservation equations and constitutive relationships for mass conservation, energy conservation, momentum conservation separately and solve them completely. And so there are certain conditions which introduce an additional force for mass transfer but which do not violate the mass transfer analogy condition. One example we looked at was the Stefan flux which is induced by the mass diffusion flux towards the surface but it is not considered an analogy breaking condition because the effect of Stefan flow is identical on heat transfer and on mass transfer. So we can still continue to use all the approximations that we normally use when we have mass transfer analogy conditions. In fact the place where you have to pay particular attention to the Stefan flow is when we talk about quiescent conditions within the boundary layer. We have been assuming in all of our flux calculations that there is a laminar sub layer around the substrate in which diffusion dominates and in which convection is absent. Now typically as you know the condition for the importance of convection is set by the Reynolds number. If the Reynolds number is much less than 1 we call that quiescent flow. However when we talk about quiescent flow conditions corresponding to mass transfer it is not sufficient to write it only in terms of Reynolds number. So in terms of simple momentum transfer you could say that Reynolds number much less than 1 implies quiescent flow but again this is on the basis of momentum transfer. If you want to establish quiescent conditions for mass transfer there is an additional condition that applies which says Reynolds number times Schmidt number to the power half must be much less than 1. Another way to think about it is even if you have a flow that is quiescent in nature if the mass transfer conditions are such that the Schmidt number is very high then it is possible that you could violate the quiescent criterion. So you have to really take the multiple of Reynolds number times Schmidt number the square root of that has to be much less than 1. Now this is the case where forced convection dominates in the case where natural convection is dominant the corresponding criterion is in terms of the Rayleigh number that we defined in the last class times Schmidt number and in this case it is to the power 1 by 4. So Rayleigh number times Schmidt number to the power 1 by 4 has to be much smaller than 1 for us to assume quiescence with respect to mass transfer and in fact there is even an additional condition that the Rayleigh number by the way this is the Rayleigh number for mass transfer the additional condition is the Rayleigh number for heat transfer times Prandtl number to the power 1 by 4 must be much smaller than 1. The reason for this is that when you have natural convective heat flow that can again induce mass flow. So finally in the case of natural convection you really have to satisfy both this condition as well as this condition and this condition. So the three conditions that need to be satisfied when you are talking about ensuring quiescence with respect to mass transfer in a natural convective flow situation whereas in the case of forced convective flow there are two criteria which is the Reynolds number must be much smaller than 1 and Reynolds number times Schmidt number to the power half should be less than 1. Now where does the Stefan flow enter into this? Stefan flow essentially induces an additional convective flux which must be taken into account when we do this calculation. So under conditions where Stefan flow is important the Reynolds number will have an additional component to it. So in addition to the mainstream convective velocity U or induced natural convective velocity there will also be the velocity that is associated with Stefan flow. So you now have to define an effective Reynolds number which includes the contribution from Stefan flow as well. So you could again have situations where all the conditions for quiescence are satisfied however the diffusive flow of a non-dilute species induces a sufficiently large opposing flow that it actually contributes to the convection in the system sufficiently that the condition for quiescence is violated. So in other words when you have a CVD process in which the depositing species is not a dilute species that can itself induce a convective flow which can violate your assumption of diffusion domination in the diffusion sublayer. So the Stefan flow effect even though it does not really influence the mass transfer analogy condition it can influence your very fundamental definition of a quiescent layer around the substrate. So it needs to be taken into account in determining whether a quiescent situation really obtains or not. If the condition for quiescence is violated then what you have to do is really apply a correction factor to your calculations to include convective transfer across the boundary layer in addition to diffusive transfer. The other thing that you should also keep in mind is that when we talk about the boundary layer adjacent to the substrate we have so far been assuming in most of our formulation of equations that it is a steady state however the CVD process can take a long time to reach steady state sometimes it may never reach it and so there may be a time dependence of the boundary layer thickness and that is usually when we talk about a transient boundary layer the mass transfer boundary layer has a dependence that looks like this square root of 4 D T where D is the thick diffusivity and T is time so it basically has a square root of time dependence and typically and similarly the heat transfer boundary layer delta H will have a square root of 4 times alpha T dependence. So if you look at the ratio between delta H and delta M it goes as square root of alpha over D. Now in most physical situations thermal diffusivity which is K by rho Cp tends to be much larger than mass diffusivity. So this number is typically larger than 1 and most of the time it is much larger than 1. So the physical implication for that is that the heat transfer boundary layer is much thicker than the mass transfer boundary layer which again you know what that implies for us from a CVD viewpoint is that the effect of gradients in the temperature profile due to the existence of the boundary layer will set in well ahead of the effect of the boundary layer on the mass fraction profiles. So when you are doing your transport calculations you have to make sure that you use the appropriate boundary layer thickness when in doing your calculation. So first you when you calculate your temperature distribution you have to use the appropriate boundary layer thickness and then from that value onwards you have to assume that the temperature profile follows a typical boundary layer profile and of course that is going to then determine the mass fraction distribution and therefore the mass transfer rates associated with it. In a CVD system just like we try to avoid turbulence effects we also try to avoid transient effects. You would like to drive the process towards a steady state as soon as possible because you know you want to run a very stable process where you can control your film thickness to plus or minus 1% of the nominal value. In order to be able to do that you cannot afford to have a time dependent boundary layer thickness because the boundary layer thickness really determines the rate at which the depositing materials are transported to the substrate. So in a sense a square root dependence on time could essentially imply a square root dependence of the film thickness on time as well and that again is not good. So the way to avoid transients in a CVD system typically what is done is you do not even introduce the substrate on which you are trying to deposit the film until you are sure that steady state conditions have been reached. So you first start up your system, do your temperature measurements and watch for a stable temperature distribution. That is usually a good sign that the initial transients have been passed and now you are in the steady or quasi steady state. You can also if you have the ability monitor the plastic distribution, the pressure distribution and provided all these measurements are showing a stable trend you can reasonably assume that your mass transfer also is going to reach steady state around that point. So after you have verified that temperatures and velocities and pressure have stabilized then you introduce the substrate and watch again for the evolution, the time evolution of deposit thickness and hopefully you will see steady state. However some of these parameter values indicate that it will take longer to achieve steady state with respect to mass transfer than it does to achieve steady state with respect to heat transfer and momentum transfer. So you have to be prepared for some surprises, you know just because your temperatures and velocities have stabilized does not automatically mean that your mass transfer rates have stabilized as well. And it also does not mean that you just wait longer and the stabilization will happen because that may not be the case either. The lack of stability is something that can persist over a very long period of time. It all depends on how you have designed your process, how you have set up your process, what are the interdependencies, you know as we have seen from our discussions mass transfer does not exist in isolation. Even though that is the process that we are focused on we have to realize that it depends very much on the momentum and the energy or heat transfer process that are taking place in the system and it also depends on the entropy in the system. You know are we at a stage where we are driving towards equilibrium or are we far away from equilibrium. So mass transfer is a very complex phenomenon in a CVD system which you know as I said has to be solved for iteratively taking into account all the coupling between the mass transfer phenomena and the heat momentum and entropy transfer phenomena. Okay so let us get back to our discussion of analogy breaking mechanisms that can happen. I mentioned in the last class that there are primarily two we should be worried about. One is forces which is an applied force which acts on the species that are diffusing and depositing and it is usually represented in terms of a phoretic velocity C and we said that you can essentially establish a Peclet number which depends on this velocity the phoretic velocity and you can then determine a correction factor for the Nusselt number which will depend on this Peclet factor and we illustrated that for the case of gravitational forces thermophoresis and so on. Today we are going to look at another mechanism that results in breaking of the mass transfer analogy and that is homogeneous chemical reactions that are taking place inside the boundary layer around the substrate. The reason that again that breaks the analogy is because the effect of these homogeneous chemical reactions on the mass fraction profile in the boundary layer will be very very different from their effect on the temperature profile within the boundary layer and so you have to establish a correction factor for the homogeneous chemical reactions to take a take it to account the fact that the mass fraction profiles in the boundary layer are not what you would have predicted if there were no chemical reactions taking place in the gas phase. The condition where we suppress all homogeneous chemical reactions in the gas phase is called the chemically frozen state. That is typically what is assumed when people are doing modeling and simulation of CVD reactors because it simplifies the analysis quite a bit. So when we talk about homogeneous chemical reactions in the boundary layer as an analogy breaking mechanism the extreme case is called the chemically frozen case which implies essentially that all rates of reaction go to 0 inside the boundary layer. This is for all x and t values. So the reason we do that is simply to make it more tractable from an analysis view point. So essentially if you have a substrate on which you are depositing a film and there is a boundary layer that exists on top of the substrate you are assuming that everywhere about this boundary layer chemical reactions are happening you can assume that you are reaching chemical equilibrium a condition known as LTCE local thermochemical equilibrium. You can assume that at the substrate also you have LTCE and you can assume that within this very very short distance over which the boundary layer is present you can assume that chemical reactions are not going to be sufficiently important to affect the mass transfer processes that are going on. So then you make this CF assumption within the boundary layer. But clearly if the homogeneous chemical reactions are happening and they are happening at a sufficiently high rate this assumption is not valid. So you can define an F reaction which is a correction factor that you apply to the Nusselt number or the standard number. So if you remember we can write Num as the F correction factor times Num0 which is the Nusselt number in the absence of that phenomenon and similarly Stm can be written as F times Stm0 and in the case of chemical reactions this F reaction depends on a Damkohler number. Now I am sure you have heard of Damkohler number right. A Damkohler number is a ratio of what to what? What is the basic definition of a Damkohler number? It measures the ratio between two characteristic times which are the characteristic times the other way around T diffusion to T reaction. So in the case of a diffusive deposition process such as CVD the characteristic diffusion time will depend on basically the boundary layer thickness and the diffusivity of the species. So you can write this as delta M0 squared over DA where delta M0 is the thickness of the mass transfer boundary layer in the absence of homogeneous chemical reactions and DA is the diffusivity of species A divided by reaction time which we can write as 1 over K triple prime where K triple prime is a first order rate constant that has units of 1 over time. So basically this has units of time and this has units of time or in other words you can write this as K triple prime times delta M0 squared over DA because this is also K a triple prime and this is the Damkohler number for species A. So the Damkohler number will be specific for each species that you are looking at so it is specific to the species under consideration. So once you have calculated the Damkohler number this way the F factor for chemical reaction can be written in terms of the Damkohler number as Damkohler number squared divided by sine of Damkohler number squared this is for the case where the reaction is a source for the species and you write this as Damkohler number squared divided by the hyperbolic sine sine h of Damkohler number squared in the case where the reaction is a sink for species A and these terms what they essentially imply is as the Damkohler number goes to 0 what happens to F? F will go to 1 so what does that mean? When you say Damkohler number goes to 0 that means the diffusion time is much shorter than the reaction time so what does that imply? The reactions even though they may be occurring they are too slow so by the time that the reaction happens the species has already diffused and gone to the substrate so the homogeneous chemical reactions even if they are happening they are too slow to affect the transport process for species A to the substrate so that is a physical definition. This Damkohler number is also known actually this correction factor F is also known as the Hata factor I do not know if you have heard of that Hata particularly in chemical reaction engineering literature people talk about the Hata number all the time it is the definition of the Hata factor of course is this is equal to Num over Num0 or another way to state this is it is equal to Delta M over Delta M0 and by the way you may recall that previously we had Delta M0 is not easy to measure or to calculate so what you can do is you can write this in terms of the characteristic dimension of the system divided by the Nusselt number for mass transfer in the absence of homogeneous chemical reactions or in terms of Stanton number you can write this as D over U Stanton number so this will be Da and also write this as the diffusivity of species A divided by a characteristic velocity you know just like L is a characteristic dimension you could be your approach velocity for example times the Stanton number for mass transfer in the absence of homogeneous chemical reactions. So in the case where the only reactions that are happening are heterogeneous chemical reactions either at the substrate heterogeneous reactions will happen right here where the species impact the substrate but they can also happen in the gas phase if you have nucleated droplets or particles in the gas phase. Now if heterogeneous chemical reactions are happening they do not necessarily result in a correction. So this the Nusselt number for mass transfer in the presence of heterogeneous chemical reactions can be assumed to be same as the Nusselt number for mass transfer in the absence of heterogeneous chemical reactions. It is only the homogeneous chemical reactions that introduce this particular correction factor that we call the hat-off factor. However heterogeneous chemical reactions will also play a role in altering the rate at which film formation happens. The simplest way to look at it is if you have chemical equilibrium at the substrate then all the corresponding mass fraction values will be at the equilibrium values and the deposition process will happen at its highest rate. However as kinetic constraints come into play the rate at which material is getting transported to the substrate may remain the same but the rate at which it is getting converted to product will slow down. So the presence of a kinetically limited deposition process at the substrate results in a net reduction in the rate at which the film grows on the surface. So it is important to look at the effect of the heterogeneous kinetics on the deposition process. So what will be the effect of heterogeneous chemical reactions? Now again there are 2 pieces to it you know when you form a CVD film on a substrate the first part is the material getting to the substrate and the second part is the material actually getting converted to a film on the substrate. The rate at which material is getting to the substrate is what we have previously calculated using Nusselt number and Stanton number. So our definition of Stanton number says it is minus jAw dot double prime divided by rho times u times omega A infinity minus omega OW. Again all this is under let us say conditions where the heterogeneous chemical kinetic limitations well actually as far as the diffusion process is concerned it does not matter. So this would be essentially the prevailing Stanton number whether or not there is a heterogeneous chemical kinetic constraint at the surface. So we can write the diffusive flux as minus jAw dot double prime equals rho u times Stanton number for mass transfer times omega A infinity minus omega AW where omega A infinity is the mass fraction of species A just outside the boundary layer and omega AW is the mass fraction of species A at the substrate and it is this concentration gradient which is driving the so called reference flux you know the denominator is a reference convective flux which basically assumes a linear profile within the boundary layer with from omega infinity minus omega AW. Now what happens when this material comes to the surface it participates in a chemical reaction and that is a surface reaction. So you can write Ra dot double prime equals some kAw times rho times omega AW where this is a kinetic rate constant has units of velocity rho is the density of the gas and omega AW is the mass fraction of species A close to the substrate. So how do you actually calculate the corresponding fluxes here? How do you calculate the values of omega AW and omega A infinity and therefore the deposition flux? You do that by assuming a jump condition at the substrate. In other words you assume that minus jAw dot double prime equals Ra dot double prime okay it is called a jump condition because it establishes an equality between two values that exist on either side of the boundary. And so when you do that you basically obtain rho times U times STM omega A infinity minus omega AW equals kAw rho omega AW. So from this you can actually simplify this. Let us see this means rho U STM times omega A infinity equals omega AW times kAw rho plus rho U STM right. If I bring it over to the side this means omega A infinity over omega AW equals kAw rho plus rho times U STM divided by rho U STM. So if I take this minus 1 so this becomes omega A infinity minus omega AW over omega AW. So this minus 1 becomes kAw rho divided by rho times U STM the rows cancel out. So you obtain omega A infinity minus omega AW over omega AW equals kAw over U times STM and this is called C which is also known as the surface Damkohler number. It is also known as the catalytic parameter hence that sign the symbol C. And by the way this is we derived this for a situation where convection dominated the delivery of material to the substrate. You can also write the same thing for the case where diffusion is dominating and then you will obtain kAw times delta M over DA. So this should have units of divided by velocity yeah so that is fine. So once you have calculated this value of C as simply I mean actually if you look at it the Damkohler number in this case is simply the difference between the mass fractions of species A in the mainstream minus the mass fraction at the substrate divided by the mass fraction at the substrate. You can actually now go back and write your diffusive flux expression in terms of that C parameter and if you do the calculations it will come out to C over 1 plus C times rho U times stanton number times omega A infinity. So you can calculate the deposition flux in this case simply from a knowledge of the density, the free stream velocity, the prevailing stanton number and the mass fraction of the diffusing species A in the mainstream of your product gases and you have to know what this parameter is, C parameter and again the C parameter can be estimated in several ways. If you know the mass fraction distribution you can estimate the parameter C or if you happen to know the rate constants at the surface you can calculate the parameter C but the correction factor in this case is essentially the C over 1 plus C parameter. Now what does it mean when we say C is a very small value as C tends to let us say 0 what happens to the diffusive flux as C tends to 0 essentially the flux will go to 0 also. As omega A w tends to very small value C tends to a very large value so the smaller the value of omega A w the greater will be the diffusion flux to the surface. So the whole point in a CVD system design should be to keep consuming that species A as it approaches the substrate. As soon as the molecule of species A comes to the substrate it must be immediately heterogeneously reacted and deposited onto the substrate. So that way I mean you try to design the process so that this omega A w goes as close to 0 as possible. I mean obviously if it goes to 0 essentially you get an infinite deposition rate which is not going to happen but you can certainly keep it as a very very small value compared to omega A infinity and that will help enhance the process by which the deposition of the film is happening. Now in reality the value of omega A w at the surface is going to assume its equilibrium value local thermo chemical equilibrium that has been our basic assumption right from the beginning that chemical equilibrium will prevail both in the CVD reaction system as well as on the substrate on which the deposition is happening. So you can always assume that omega A w will tend towards its thermo chemical equilibrium value of course this is again more likely to happen as T w increases. The higher the temperature substrate temperature the more likely that the local mass fraction of species A will tend towards its equilibrium value and typically the equilibrium value of omega A w is also likely to be its lowest value because equilibrium is achieved not homogeneously but heterogeneously at the substrate you have a CVD film that is in contact with the gas. So if you have achieved thermo chemical equilibrium and it results in the formation of a condensate of species A then obviously its mass fraction in the gas phase is going to be very small because most of the species A mass is going to be in the condensed phase. So as long as you drive your process towards equilibrium omega A w is going to tend towards its equilibrium value which is going to be much much smaller than omega A infinity. Now why would that be I mean omega A infinity is also an equilibrium value because we are assuming equilibrium is prevailing everywhere except within the boundary layer. So why would this value be much lower than this? Any idea? There is chemical equilibrium prevailing both in the gas and at the substrate but we are saying that omega A w which is a value at the substrate is much smaller than the value in the gas. The reason is this is a homogeneous equilibrium whereas this is a heterogeneous equilibrium. So in the gas we are assuming equilibrium but everything stays in the gas phase. So the mass fraction of the species still has a very high value but at the substrate we are assuming equilibrium applies but it is a heterogeneous equilibrium which means that most of the mass of species A is in the condensed phase. That is why as long as you drive the entire system towards equilibrium both in the gas phase as well as near the substrate everywhere in fact except the thin boundary layer your diffusion rates are going to be maximum the deposition rates are going to be maximum and so on and by the way within the boundary layer I said that one normal assumption we make it is chemically frozen but that is one extreme assumption. For modeling purposes you can also assume that thermochemical equilibrium prevails everywhere within the boundary layer. Typically that would not affect your deposition rate too much you know whether or not the boundary layer is chemically frozen or chemically reactive. What it does do is add to the complexity of your calculations because now you cannot simply assume that mass transfer across the boundary layer happens only by diffusion. You have to essentially calculate at each location in the boundary layer what is the prevailing chemical equilibrium composition corresponding to the local temperature and pressure conditions and so you essentially have to extend your iterative algorithm to include the boundary layer also. The problem with that is inside the boundary layer things change fast so you have your time steps and your distance steps have to be much shorter. So the assumption of chemical reactions happening inside the boundary layer whether in equilibrium condition or not significantly extends your calculation time because the complexity of characterizing what happens within the boundary layer is much greater than the complexity of characterizing what happens outside the boundary layer because outside the boundary layer you do not have to deal with so many gradients you know it is almost a uniform condition in a well designed CVD reactor but inside a boundary layer by definition things change rapidly and so you know just from a complexity view point we typically prefer to make the assumption that the boundary layer itself is chemically frozen and there are no chemical reactions going on. So this is fundamentally the way you incorporate homogeneous kinetics as well as heterogeneous kinetics into your diffusional deposition model and I think that pretty much brings us to the end of this module where we have been looking at the effect of transport phenomena on CVD rates. By now you should have a reasonably good picture of how to approach the problem of simulating a CVD reactor to obtain the prevailing rates of deposition and growth of the film. It is a very you have to follow a very systematic procedure where you start with the knowns then only known things that you have are the temperature distribution on the outer walls of the CVD reactor, the temperature of the substrate and the pressure that you are applying to the reactor and your input parameters you know what is your feed gas, what is the concentration of your reacting species in the feed gas that is pretty much it. From that you have to be able to apply all the principles that we have talked about the thermodynamic principles as well as the transport principles to estimate an associated rate at which deposition is happening on the surface resulting in the growth of the film. It is not a sequential process it is an iterative process because you know the prevailing velocity distributions and temperature distributions affect your mass fraction distributions but we know that the mass fraction distribution itself can also affect your velocity distribution and your temperature distribution. So it is a highly coupled process which has to be solved iteratively. The other thing that you have to keep in mind is the stoichiometry of the film has to be consistent with the associated transport rates. You cannot violate the elemental flux ratio condition that we have talked about. If you are forming a you know Si3N4 film then the molecular flux of Si2N has to be in the 3 to 4 ratio otherwise you are violating the stoichiometry of the film. So that again introduces an iterative step into your process. You have to calculate diffusion rates assuming a certain thermodynamics at the surface and then once you do that if the fluxes that you calculate are not in the right ratio you have to iterate until you achieve a self consistent solution which satisfies the thermodynamic constraints as well as the transport constraints. And you have to provide when you are calculating the transport rate which directly corresponds to the rate at which the film is growing you have to ensure that you apply all the appropriate correction factors. You have to understand whether or not mass transfer analogy condition holds and if it does not what are the correction factors you have to apply to the Nusselt number and the Stanton number in order to obtain the actual prevailing Nusselt and Stanton numbers and use those to calculate your actual prevailing deposition rates. So there is a lot of work here for chemical engineers to do and that is the reason why you know CVD processes in general employ a lot of chemical engineers in industry. Without a good understanding of the chemical engineering phenomena involved it is very difficult to design an efficient CVD system. So what we will do in subsequent classes is go back to looking at some practical CVD So we will start with CVD reactors that are used for making silicon films and we will also look at some CVD reactors that are used for making metal films and semiconductors and so on. And as we discuss these specific systems we will refer back to this module the transport module because that is going to be you know obviously crucial in our discussions in the following classes as well. So any questions on what we have covered today or about transport process in general in CVD systems? You can, I mean once you know the temperature and you once you know the pressure. Experimentally it is difficult but there are instruments that can do it you know the difficulty is doing it in situ. You can always draw a sample of the material and do offline analysis. For example that you can suck out some vapors and gases, dissolve it in a liquid and then analyze a liquid or you can take the gases directly through a gas chromatograph and analyze. The real problem in CVD reactors is how do you do that in situ without having to suck out the material. That is more difficult to do. So it is also not possible like that. Velocity you can always measure you can have instrumentation inside to measure you know like an animometer or something that can measure the velocity distribution inside. The most difficult thing to measure are the mass fractions particularly when they are at very low concentrations because we are talking about you know 10 to the power minus 6 and smaller values parts per million parts per billion values so it is difficult to measure. See you at the next class.