 Good morning and welcome to the next lecture in our course on chemical engineering principles of CVD process. We have been discussing some applications of CVD and some typical reactor designs in the last couple of lectures. So today we will talk about CVD of silicon. Of all the materials that are deposited using CVD, silicon both in crystalline form as well as polycrystalline form is probably the most commonly deposited material. It is essentially a multi-billion dollar industry around the world. So it is important to spend some time talking about this process. One CVD reactors are essentially of three types. One is the horizontal flow, low pressure CVD systems which are quite common and the second most frequently used configuration is the stagnation point flow CVD reactors and the third most popular configuration is a fluidized bed CVD reactor. Now as you know the difference between these systems is in a horizontal flow CVD reactor essentially the wafers are mounted parallel to the flow of the fluid and deposition occurs by diffusion across a boundary layer that develops on top of the substrate or wafer. In the case of the stagnation point flow the flow is vertical to the wafer and deposition occurs in the stagnation region adjacent to the wafer surface. The fluidized bed CVD reactor is particularly useful when you are trying to coat the silicon onto particles. So essentially you can imagine a bed consisting of particles to be coated and then you bubble gas through this bed where you are actually doing in situ thermal pyrolysis to create the precursors of the silicon which then deposit on the outer surfaces of the particles which can be silicon particles in epitaxial silicon film growth and so the film formation happens around the particles that are suspended in this bed. Now if you look at these different configurations as we have discussed earlier in the course each has its advantages and disadvantages in terms of the thermodynamics the kinetics the and also the transport phenomena but one thing that is important to appreciate is the fact that when you talk about a CVD reactor in general there are multiple time scales and also multiple length scales involved. So when you are trying to develop a model to simulate CVD in a reactor you have to be able to deal with these multiple time scales which can be you know very different. For example if you look at the CVD process itself you know we have previously broken it down into you know the first part is essentially the fluid dynamics which on the length scale can be of the order of meters and on a time scale could be of the order of seconds to minutes. It is essentially the time that the carrier gas that is introduced into the CVD reactor has available to traverse the reactor and also establish steady state conditions over the substrate on which the deposition is taking place. Now the second process that happens of course is the boundary layer transport. Now when we talk about transport across the boundary layer that we are obviously talking about very thin dimensions which are typically of the order of micrometers to fractions of a millimeter. How about the times? What is the typical diffusion time scale? How long will a molecule take to traverse you know let us say 10 microns of boundary layer? Will it be like milliseconds or microseconds or picoseconds? It is actually of the order of about 10 to the power minus 8 to 10 to the power minus 10 seconds because diffusion process remember it is not a straight line process. It is a random movement and collision process right. So a molecule is just not going to be able to transport itself convectively. It is going to have to randomly walk around and then if it encounters another molecule it is going to bounce. So it is essentially a sequence of random events that result in a net motion of the vapor or the particles. So the time scales are much slower. Now there are also homogeneous chemical reactions happening and in the case of a homogeneous chemical reaction what is the length scale? Well for a reaction to happen the molecules have to be close on an atomic scale right. So we are really talking about nanometers as a length scale and how about the reaction times? Homogeneous reactions in general have a higher activation energy barrier compared to heterogeneous reactions. So the time scales are roughly comparable to the diffusional time scales. As we have seen before the Damkohler number is an element number that measures the typical time constant for diffusion to the typical time constant for chemical reaction and it can vary from a very large value to a very small value but on an average let us say that these times are comparable. And then you have the heterogeneous chemical reactions. In the case of heterogeneous chemical reactions again the gas phase molecules have to get very close to the substrate or solid surface on an atomic scale. So here again in terms of length scales we are talking about nanometers. However heterogeneous reactions typically produce, proceed much faster than homogeneous reactions because there is a lower activation energy barrier and so the time scales here can actually be of the order of, it can happen very, very quickly compared to the homogeneous reactions or boundary layer transport. So it could be of the order of 10 to the power minus 12 to 10 to the power minus 15 seconds especially as you drive towards equilibrium the reaction is virtually instantaneous which means that the time constant is extremely small. The next step is the adsorption part. You know reaction occurs, you have formed a solid film that is thermodynamically feasible but now it has to be accommodated by the surface. So the adsorption process involves a time constant and here we are really talking about Angstrom's length scale because we are talking about essentially vacant sites on the surface that can present themselves to the molecule that is depositing and allow it to get fixed to the surface. So very, very small time scales, I mean length scales. However the time scales depend on temperature and pressure. The adsorption process can become the rate limiting process if the surface is not sufficiently energized. So the adsorption process itself can have time constants that can range from 10 to the power minus 15 to 10 to the power minus 8. Essentially encompassing the time constants associated with the transport phenomena and the chemical reactions that are taking place. Now sixth process in this is surface diffusion where once a molecule is adsorbed on the surface it migrates on the surface and forms clusters and the clusters eventually grow into a continuous film. So there is a time constant associated with this process as well and again we are talking about length scales that are of the order of Angstrom's and time scales for a molecule to move about on the surface, find adjoining molecules, form clusters then for the clusters to grow, attach themselves and form a continuous film can take much longer compared to the diffusional or the adsorption processes. So we are back to a time scale that can be as much as 10 to the power minus 3 to 10 to the power minus 4 to the order of milliseconds, right. So that I think it is a summary of the complexity of trying to model a CVD reactor because if you look in the literature you know just talk about silicon CVD which as I said is the most researched and most published CVD process in literature. You will see that the models that are presented very few of them try to deal with all of these steps in the process. There are CFD papers that essentially concentrate on the fluid dynamics. They are very good at modeling the macroscopic flow behavior in the system. They are very good at predicting velocity distributions, temperature distributions, pressure distributions and by extension also the mass fraction distributions of the reacting species. However the boundary layer transport part in principle can also be covered by CFD. However they tend to be somewhat weak. They tend to use very very basic definitions of mass transfer coefficients. You know we have in a few lectures we have discussed how your nominal mass transfer coefficients have to be corrected for the various phenomena that are taking place. Virtually none of the CFD codes that are available will provide you these correction factors for thermophoresis or for Stefan flow or for the effect of homogeneous chemical reactions on transport rates or the effect of heterogeneous chemical reactions on transport rates or the combined effect of both. These are enhancements to the theory which are typically not included in the CFD type of papers. However the papers that do deal with these types of phenomena tend to place less emphasis on the CFD part. So unfortunately there is this segmentation of papers in this area. So typically this part, the boundary layer transport part is the province of research groups that are very focused on diffusional phenomena and the various parameters that affect diffusive phenomena. If you look at the homogeneous and heterogeneous chemical reactions that are taking place, again that is an entirely you know different set of research groups that are focused on that area. They essentially specialize in characterizing the various chemical reactions that are taking place in measuring the rate constants, studying the kinetics, developing empirical and semi empirical models for these chemical reactions that are happening both in the gas phase as well as heterogeneously. So that is again an area of literature that has sprung up and grown but it is not necessarily linked very well to these two components. Something like adsorption and surface diffusion is actually more under the province of people that are working in things like Monte Carlo simulations and molecular dynamics because essentially what we are talking about there is what happens at the interface between a surface and a vapor molecule. How does the adsorption process happen? How do the adsorbed molecules move around? What are the driving forces for them to essentially jump from one interstitial site to the next one? What provides the kinetic force? What provides the thermodynamic motivation? You know why does not the molecule just stay where it is? You know what is it that makes it try to find the nearest neighbor and form agglomerates rather than staying as single molecules and then the whole simulation of how it happens because it is you know at that level it is more of a probabilistic study. It is not a deterministic study. You cannot look at a single molecule and try to predict its behavior. Instead what you try to do is propose a multiplicity of possible behaviors and find the ones that gives you the highest probability of it occurring. So that type of simulation is very different from what people are working in these areas are doing. So essentially the point is that if you look at the literature dealing with silicon C V D there is none that really captures the entire phenomenon. They all it is like the blind man and the elephant right. They all look at different aspects of it and study it in a great amount of depth but the linking of these various sub elements is not very strong and the reason again for that is because of the difference in the length scales and the time scales. If all of these phenomena were going on roughly on the same time scale and the same length scale it would not be so difficult for you know one analysis in one aspect to be quickly extended to other aspects as well but because of the huge diversity in length and time scales it is just not possible. For example again let us go back to silicon C V D. You know when you look at fluid dynamics we have talked about it at length. You start with the you know conservation equations, start with mass conservation, momentum conservation, energy conservation, entropy conservation. You apply the appropriate boundary conditions, initial conditions. You apply the appropriate constitutive relationships and you solve and that is essentially how you solve the fluid dynamic part of it particularly the momentum conservation equations both the linear momentum as well as angular momentum enabled you to do that both in the case where force convection is dominant as well as where natural convection is dominant by using the appropriate dimension numbers and so on. When you talk about boundary layer transport here again we have spent quite a bit of time talking about that aspect and we have dealt with not only the baseline process but possible corrections that you have to do to your model to account for it properly. So in terms of what we have covered in this course I would say most of our focus has been actually in this area with a little bit of focus in this area and you know little bit of focus on the other areas but primarily concentrated on this part which you know actually from a chemical engineering view point it is the most unique for our discipline because there are other people that can deal with the other aspects but the 2, 3 and 4 are really the span of control for chemical engineers. That is where we can do the most good that is where we have the most expertise and so primarily our focus in this course has also been on these aspects. So if you look at silicon C V D how would you describe the chemical reactions? We have said before that in a C V D reactor the best way to model the homogeneous rate processes is by using a free energy minimization approach right but that assumes that you are driving every process to equilibrium and it also assumes that you know you have absolutely no insight into the processes that are going on but in the case of specific systems like silicon C V D there has been a lot of work done to diagnose what are the dominant chemical reactions taking place inside the reactor. Particularly in the case of Si H 4 silane going to silicon solid there is quite a bit of literature that tries to describe exactly what the reaction sequence is. For example people would agree that the first step is that it decomposes into Si H 2 plus H 2 and then Si H 4 combines with Si H 2 to make Si 2 H 6 and then Si H 4 plus Si 2 H 6 goes to make Si 3 H 8 I mean Si 3 H 10 Si H 2 plus Si 2 H 6 goes to make Si 3 H 8 and so on. So there are these specific reactions have been identified and people have also try to characterize what is the longest chain length that is achieved before the molecules essentially become too unstable so that they do not stay for very long. And so this type of modeling has been done and the overall kinetics are obviously described by Si H 4 going to Si S plus 2 H 2 but if you look at the homogeneous chemical reactions for example Si H 4 going to Si H 2 plus H 2 that reaction has also been studied in some detail and the reaction rate term corresponding to that has actually been written in the form of K 1 K 2 times C Si H 4 so this is minus R Si H 4 divided by K 1 times C Si H 4 plus K 2 this is called the Lindemann's equation or Lindemann kinetics where K 1 and K 2 are constants and C Si H 4 is the concentration of silane. It has the behavior that when the pressure or concentration actually this is written more in terms of partial pressures rather than concentration so it is K 1 K 2 P Si H 4 squared divided by K 1 times P Si H 4 plus K 2 when the partial pressures are high in other words when the system pressure is close to atmospheric pressure then this behaves like a first order reaction because K 1 P Si H 4 becomes much larger than P 2 and these two cancel so essentially it becomes K 1 times P Si H 4 which is a first order reaction. However as you lower the pressure and you start using LPCVD this term becomes dominant compared to this term so now it becomes a second order reaction. So in a typical Lindemann kinetics depending on the value of the prevailing pressure the reaction can either run as a first order reaction or as a second order reaction. So this is the most conventionally accepted kinetic model. Now the problem is there are models like this for each of these reactions and which have been developed over a long period of time using extensive studies so just describing the chemical kinetics of the system itself is like a lifetime research process right and that is the reason why for someone who is focused on the reactor as a whole it is probably not worth it to try and describe the kinetics of each reaction separately by forming appropriate reaction rate laws and then measuring the constants and so on. Now if you look at for example what is happening inside a fluidized bed CVD reactor. As I said the fluidized bed CVD reactor is one where you have particles in a bed and you are flowing the reactive gases over them and the silicon deposition occurs as a thin film around the particle surface. So if you look at this situation the modeling has been primarily looking at the overall behavior of Si4 H4 going to silicon solid plus hydrogen gas and this has been modeled as or Si H4 equals a kinetic rate constant times A by V times C Si H4 over 1 plus k1 times C Si H4 plus k2 times C H2 where A is the area of the particle the total surface area of the particle and V is the gas volume that is passing by. So it is basically the specific area of the particle area per unit volume and the definition of this parameter A by V is that it obviously depends on the porosity it is actually written as 6 over Dp times 1 minus epsilon by epsilon where Dp is the diameter of the particle comprising the bed and epsilon is the of the pore fraction of the bed and so this is again what I would say more of an empirical relationship which has been observed through many, many measurements. So it is what we call a phenomenologically observed reaction rather than one that is really based upon any fundamental modeling. So if you are trying to model the CVD process as a whole for a fluidized bed CVD reactor you would develop a first principle model for the flow dynamics and for the diffusion process but then for the homogeneous kinetics as well as the heterogeneous kinetics that is presented here you would tend to use these kinetic rate expressions that are experimentally derived. So they have to be essentially fed in as a subroutine to the CFD model which is okay you know it basically becomes a UDF the problem with it is in terms of the rigor there is always a difference between the first principles model and a UDF that is provided strictly based on experimental observations. There is always a question mark when you are providing a subroutine that is experimentally derived because there is no guarantee that if you extend your experimental conditions you will still get the same rate law. All of sorry use a defined function so it is usually a subroutine that you feed to a code to incorporate certain specific behavior which is not covered in the universal code. So the problem is when you try to integrate the two the amount of theoretical soundness is much greater in the first principles model compared to the one that you have provided based on experimental observations. So that leads to essentially a mismatch you have a model that is very good in terms of describing the flow dynamics and the diffusional behavior but is kind of weak in terms of describing the kinetic behavior whereas on the other hand if you are working at it at the other end you may have spent a lot of time in developing the best possible kinetic model but then you are essentially going to be taking in the flow dynamics and the diffusion of the transport phenomena as again a user supplied kind of a subroutine. It is very difficult for a single research group to have sufficient expertise to deal with all the 6 aspects in sufficient depth and so you tend to sacrifice one or the other. Now the third aspect so we kind of talk in the case of silicon C V D you have given some examples of homogeneous and heterogeneous behavior and then there is this adsorption behavior right. Now how do you model adsorption? So adsorption is modeled using significant sigma which is the vacant site on the surface almost like a catalyst. So when you have SiH4 approaching a surface approaching a vacancy what happens is SiH plus or SiH gets absorbed as a separate molecule, H gets absorbed as a separate molecule and then H2 leaves as gas. So the asterisk here essentially indicates an adsorbed molecule. So this is the first step in the adsorption. The second step in the adsorption is SiH2 SiH stars essentially combine through a process of clustering and you get 2 Si adsorbed plus again H2 leaving and similarly 2 of the adsorbed H molecules find each other and leave as H2 which frees up that vacant site for additional adsorption. So this is a sequence that has again been observed specifically for silicon C V D systems and the modeling of adsorption usually follows some outline like this. It is based upon very extensive investigations of the surface on a real time basis essentially you take the surface, expose it to silicon, silane gas and observe what species get absorbed and what is the time dependent behavior, how stable are they, how do they migrate, how do they cluster and so on. So describing this as I said is a separate science all on its own. You really have to have a very good understanding I mean you basically need a surface scientist or an interfacial scientist who understands how an approaching molecule will interact with the surface which has a certain number of vacant sites that need to be filled and how the entire dynamics takes place, how does the initial adsorption happen, how does the later migration and clustering happen and how does the desorption happen because all 3 steps are crucial for the steady state formation of an adsorbed film on the surface. And then you have you know surface diffusion as well, now surface diffusion as I was playing as I was saying plays a major role in how this happens because for this step you know for Si H star going to 2 Si H star requires diffusion to happen. The adsorbed molecules have to migrate, find each other and cause the ejection of the H radical or H iron as a H2 molecule and similarly to adsorbed hydrogen radicals finding each other and becoming gas phase hydrogen. So this is essentially a surface diffusion step, it has again been modeled but surface diffusion cannot be modeled from first principles unlike gas phase diffusion particularly in the case of low density gases and dilute species you can derive the diffusion coefficient from the kinetic theory of gases. But when you are talking about surface diffusion if you remember we wrote down an expression for its you know in one of the earlier lectures it involves things like frequency of jump attempts that cannot be modeled from first principles. It is really a probabilistic phenomenon again, all you can really try and calculate is what is the probability that a molecule that is adsorbed on the surface will move to another location on the surface in a given amount of time and based on your estimated probabilities you can derive an average surface diffusion coefficient. So on the surface you know how do you bring this all together, is there a single model that can at least describe everything that is happening on the surface. The only way to really do that is through population balance modeling. So the latest trend in C V D analysis particularly when it is focused on the surface is to use a population balance approach. A population balance model you know as the name says you take into account birth and death phenomena which causes change in the population but also within the population you know for example if you are looking at a human population and you want to do a model for number of humans between 10 and 20 years of age then your population balance model has to also include essentially terms that indicate growth into that age range and growth out of the age range right. So you need those terms also. Similarly when you talk about a population balance model for adsorbed molecules on a surface you have to decide which molecule are you going to be modeling. So if you want to look at the rate of Si H 4 star you know adsorbed sorry Si H star then you have to clearly look at the rate at which these molecules are adsorbing. So you have to look at D by D T of the Si H star getting on the surface due to an adsorption process you also have to subtract from it the rate of loss of Si H star due to desorption you know the adsorption process itself can be reversible. So an adsorbed Si H molecule can also dissolve so that you have to take into account minus you also have to look at how many of these Si H star molecules are you losing because of agglomeration with an adjacent Si H star molecule and also you have to take into account the diffusional behavior of the molecules separately because the analysis that you are doing this is typically at a particular X because you are treating time as the dependent variable so you are usually looking at how adsorbed molecule concentration is changing at a certain location on the substrate. So you also have to look at simple migration of the molecule away from that location even if it does not result in agglomeration. So that is essentially a rate of change of Si H star due to surface diffusion which would depend on obviously the surface diffusivity of the molecule which again it would be dependent on the prevailing temperature and the pressure conditions. This term the agglomeration term would be dependent again on temperature and pressure and the diffusivity term because all three will play a role in how the adsorbed molecules migrate and find each other but in addition you also have to look at properties like the cohesive behavior of the molecules even if the two of them find each other how likely are they to agglomerate rather than bounce back right. So that is essentially called cohesive behavior and there is a parameter called cohesion coefficient which would depend on parameters like the surface energy of the vapors. It also will depend to some extent on the viscosity of the fluid between them which can also affect how close the particles or the molecules can absorb each other. It really depends on the intermolecular forces as well you know what are the van der Waals forces between the molecules, what are the electrophoretic forces between the molecules, what are the ionic forces between the molecules, what are the surface tension forces, what are the viscous forces. So this cohesion coefficient actually depends on all of these interactions. Now if you look at obviously the adsorption and desorption behavior they are predominantly concerned with the adsorption dynamics. What is for a given temperature and pressure, what is the rate at which molecules can adsorb and what is the rate at which molecules can desorb and this again can be experimentally observed and incorporated into your model. So if you are trying to do a simulation of how a thin film grows on a surface you really have a choice to make. I mean do you try to model all of these phenomena in full detail or do you try to focus on a few of these and essentially take a simplified view of the other processes that are going on. I mean another way to look at it is this is a six dimensional problem essentially but you have a choice to make. You can either try to deal with it as a 6D problem in which case the numerical complexity and the computational effort that is involved can be quite large because of the different time scales. You know trying to imagine building a computer code to deal with this entire process. Certain steps have to be taken in 10 to the power minus 12 or 10 to the power minus 15 second time scales time steps whereas some of the regions of the reactor we have to use much larger time steps of the order of second or meter. So coming up with a code that can really deal with that you know as people that have done a lot of programming will tell you it is very, very hard. It is very difficult to get such codes to converge because of this huge range in the time constants and the length constants. So a preferred approach is essentially either to take a macroscopic view or a microscopic view. So the macroscopic investigators will deal with you know the first 4 aspects in some detail but they will tend to take a very simplistic view of the last 2 whereas a microscopic analysis of the CVD problem would tend to focus a lot more on the last 2 steps and take the others as essentially being given. They will use the simplest possible fluid dynamic model, simplest possible kinetic model, simplest possible transport model and so what they assume you know another way to look at CVD is if you have a substrate there are 2 things that determine how the film grows one is how does the material get to the surface the other is once the material gets to the surface how does it interact with the surface and start to grow as a film. So steps 1 to 4 well 1 to 4 deal with this part how is material delivered to the surface and steps 5 to 6 actually I would say 4 to 6 so 4 is kind of an overlapping step because it involves heterogeneous reaction so it does have some microscopic elements but it also has a macroscopic elements since you are also interacting with the fluid on the other side. So 1 to 4 in this table or in this list deal with the macroscopic view of the CVD system and steps 4 to 6 deal with the microscopic view of the CVD system. The challenge is how do you fit the 2 together right the issue there is not so much that we do not know how to do it I mean it is certainly doable it is just that the computational effort involved does not always justify on the basis of what you are trying to do. For example if you are simply trying to develop a model for how the film thickness changes as a function of time then the model that is most relevant to you is part of the macroscopic model. On the other hand if you really care about not just the rate of growth of the film but also all of its morphological characteristics all of its metallurgical characteristics all of its crystallographic characteristics all the things that material scientist metallurgists would be interested in then you tend to focus a lot more on steps you know 5 and 6. So again a way to think about this is chemical engineers would tend to focus on these steps materials engineers surface scientists would try to concentrate on these steps there are only very few people who are really working in what I would call the interfacial area who try to kind of take both of these into consideration when trying to develop a model for the CVD process and by the way all the discussions we have had have been with one system silicon CVD you can imagine that given the multiplicity of materials that can be deposited using CVD all of these same considerations will apply for each of these systems and essentially as someone who is involved in modeling of CVD systems you have to make a choice you have these 6 different aspects to deal with which ones are you going to give importance to and which ones are you not and your selection should match your application. If you are trying to make high purity silicon with very precisely controlled structural characteristics for a semiconductor or micro electronic application then you cannot afford to neglect the last 2 steps the molecular structure becomes extremely important. Another reason is you know if you look at semiconductor devices the features on a semiconductor device are now of the order of nanometers so unless your model has nanometer resolution you are not going to be able to predict what is going to happen and again if you cannot predict you cannot control you cannot optimize and so depending on certainly the characteristic feature sizes in your product you select the length scale that you want to focus on and based on your selection of length scale that decides your time scale ideally that is approach you should not make the decision of which aspects to focus on based on your capabilities but rather on what the product requirements are what is the functionality that is expected of the CVD film and how am I going to be able to capture that in my model so that should drive your analytical methodologies. So let us stop at this point in the next class we will take up another system of CVD and try to illustrate some of these principles as well. Any questions that you have okay so I will see you at the next class.